GeoTerva is online geotechnical design software: every module draws the model, does the calculation, writes out the working and gives a report. The manual explains how to use each module and why it gives the answer it does.
Learn the workspace once. Every module opens in the same workspace. The workspace guide covers the drawing, the five tabs, Analyze and the lock, saving, units and the report.
Open the manual of your module. Each one lists what you need, takes you through the inputs, explains the methods, reads the results and ends with a worked example you can reproduce.
Try it on the example. Every module opens on a complete example; press Analyze to see a finished analysis, then change one value at a time.
The manuals by family
The modules are grouped in eleven families. The manuals are being written family by family; the ones that are ready are marked.
What the module calculates, and which module to use for the cases it does not cover.
Before you start: the data you need and where it comes from.
A quick start in a few steps.
Every input, section by section, with its symbol, units, default and what to enter.
The methods the module offers and how to choose between them.
The results, the calculations and the report, read line by line.
A worked example with the numbers the module gives, so you can check your own screen.
How each input changes the answer, run on the module itself.
Messages: the stops, warnings and notes the module gives, and what to do about each.
Answers to common questions.
Frequently asked questions
Where do I start with GeoTerva?
Read the workspace guide once, then open the manual of the module you need. Every module opens on a worked example, so you can press Analyze straight away and then change one value at a time.
Which modules have a manual?
The five Piles modules have manuals now: Pile in sand, Pile in clay, Pile on layered soil, Pile under lateral load and Pile group. The other families follow.
Is the manual the same as the Theory tab?
No. The Theory tab inside a module explains the method, with its equations and references. The manual explains how to use the module: what to enter, what each result means, and what the messages say.
User manual · Getting started
The GeoTerva workspace
Every GeoTerva module opens in the same workspace: a drawing that follows your inputs and a panel with five tabs, Model, Results, Calculations, Report and Theory. Learn it once and every module reads the same way.
Every module of GeoTerva opens in the same workspace. On the left a drawing follows your inputs; on the right a panel holds the model and its answers in five tabs. Learn it once and every module, from a footing to a pile group, reads the same way.
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The workspace, here on Pile in sand after Analyze: the drawing of the model on the left, the five tabs of the panel on the right.
Family menu. Opens the list of the modules of the family, so you can move from one to the next without going back to the catalogue.
Units. SI · kN, SI · tf or US. It can be changed at any time, even after Analyze.
Account menu. Your account, your plan and Sign out.
Views of the drawing. Section, Profile and 3D for the pile modules; every module names its own.
The drawing. Drawn to scale from the inputs; after Analyze it carries the results.
Zoom. Minus, plus and fit; the mouse wheel zooms as well.
The five tabs. Model, Results, Calculations, Report and Theory.
The panel. Here the Results tab: the answers in numbers, step by step, layer by layer.
Layout
Drawing on the left, panel on the right; on a phone the drawing sits above the panel.
Tabs
Model · Results · Calculations · Report · Theory
Units
SI · kN, SI · tf and US, switchable at any time
Your work
Pile and most other modules: kept in the browser as you type, and saved to a model file or a quick save when you want a copy. The Surcharge Loads modules open on their own example each time
Output
A summary report as PDF or Word (.docx)
Works on
The browser of a desktop, a tablet or a phone; nothing to install
The workflow in four moves
Describe the model
Open the Model tab and go through its sections from the top: the ground, the water, the structure, the methods, the design values. The drawing on the left follows every change.
Press Analyze
The Analyze bar at the bottom of the panel checks the model and runs the calculation. When it is done the inputs are locked and the answers are ready.
Read the answers
The Results tab gives the numbers and the verdict, the Calculations tab writes out every step with its working, and the drawing now carries the results.
Take the report
The Report tab fills in the cover page of a summary report and saves it as a PDF or a Word file. The Theory tab explains the method behind every number.
The Model tab
The Model tab is a column of sections: Model data first, then the sections of the module, such as the ground, the water table, the pile, the methods and the design values. A section opens when you click its title; the Next and Back buttons at the foot of a section walk you through them in order.
Model data: names, files and quick saves
The first section is the same in every module.
Model name is a free label for the model, up to 80 characters.
Save to file… saves the whole model to a file on your computer; you choose the folder and the name. Open file… reads such a file back.
Quick Save… keeps a named version of the model in this browser. The list below it holds the quick saves; Load brings one back and Delete removes it.
Reset to defaults returns to the worked example the module opened with.
Every change is also kept in this browser as you work, so the page opens again with the model you last worked on. A file is what you keep with the project, send to a colleague or move to another computer.
The Model data section, the same in every module.
Sections, the (?) buttons and Back / Next
Each section has a short note at its foot that says what it asks for and why. A round (?) next to a title or a choice opens a help window with a drawing and the formula behind that part of the model. Help windows never change anything; close one with Close or the cross.
Choices: buttons that decide what is asked next
A choice is a row of buttons, for example Circular or Square, Driven, Bored or Jacked. The button you press turns the other fields on or off: a method that needs one more number brings its box into view, and a field that no longer applies disappears. The drawing follows at once.
Layers: add, edit, remove
Modules with a ground profile take the layers one at a time from the ground surface down.
Fill in the New layer form at the top: an optional name, the thickness and the soil values.
Press + Add layer. The layer joins the Soil layers table below and the counter ("2 of 8") moves on. Up to eight layers can be entered.
To change a layer, click its row. The form opens with Editing · Layer n in its title; changes apply as you type, and Done closes it. Delete layer removes that layer.
The small cross at the end of a row also removes a layer.
Each layer is drawn in its own colour in the drawing, and the same colour marks it in the table of layers.
The layer entry: the New layer form above, the table of layers below.
Values that fill themselves in
Some boxes carry a value that the module works out from your other inputs: a chart reading, a coefficient that follows the installation of a pile, a default that follows the friction angle. They are filled in for you and kept up to date as you change the model. Type over one to use your own value; empty the box to get the module’s value back.
Analyze, the lock and Edit inputs
The bar at the bottom of the Model tab shows the state of the model. Before the first run it reads Not analysed and tells you what the module still needs; press Analyze.
When the run has finished the bar reports the run (green, or brown when the run has warnings) and reads Analysed. From that moment:
The inputs are locked. The numbers you see are the numbers behind the results, so they cannot be edited by accident.
Edit inputs unlocks the model. It clears the results, and the bar returns to Analyze.
Warnings shows the cautions and notes of the run, with the number of them in brackets. Most of the assumptions and cautions are printed in the report as well.
The unit system can still be changed. Switching units does not need a new run.
If something is missing or impossible, the bar turns brown and a window titled Cannot analyse — a condition of the solution is not met lists what is wrong, in plain words and with what to do about it, for example that the layers must reach the tip of the pile. Nothing is calculated; change the inputs marked in red and press Analyze again.
The drawing: views, zoom and dragging
The drawing is made from your inputs, to scale, as you type. The buttons above it choose the view: for the pile modules Section, Profile and 3D, and more where a module has more to show (the lateral pile and the pile group have views of their own, described in their manuals). After Analyze, the views that show results become available.
Zoom. The + and − buttons and the mouse wheel zoom the drawing; the fit button puts the whole drawing back in the pane.
Dragging. In several modules you can move a feature of the drawing instead of typing it: the footing of the drawing says what can be dragged (for a pile, the water table, a layer base and the pile tip). Dragging changes the input boxes, so the two always agree.
Colours. Layers keep their colour in the drawing, in the table and in the results.
Results and Calculations
The Results tab answers the question the module was built for: the capacity, the pressure, the factor of safety, the settlement. It shows the main numbers on cards, a table of the values step by step, and, where the ground is in layers, what each layer contributes. Where the module has a design check it states the verdict in words, never by a colour alone: PASS or FAIL in the axial pile modules, OK or NOT OK where a module checks a limit.
The Calculations tab is the working. It starts with Given and what is sought and then follows the method step by step; each step has a short explanation, the equation with its symbols, and the equation again with your numbers in it, ending with the result of the step. The chapter marked with a tick at the end is the answer in one sentence. Every number on the Results tab can be traced to a step here.
The Calculations tab: the given data, then every step of the method with its equation and numbers.
Report: PDF and Word
The Report tab produces a calculation report you can hand to a client or a checker.
Fill in the Report cover: project, location, client, report number, date, prepared by, title or qualification, checked by, and optional notes. The details are kept in the browser and saved with the model file; the notes are printed only when they are filled in.
Press Save as PDF or Download Word (.docx). The preview below the buttons shows what the report looks like.
A summary report has the same structure in every module: an introduction and scope, the input data, the calculation step by step, the results, a summary and the references of the methods that were used. The units are stated in the report, with the constants needed to repeat any conversion by hand, and the warnings of the run appear in their own section.
The Report tab: the cover details, the two download buttons and the preview of the report.
Theory and the help windows
The Theory tab is the method behind the module, written as a short article with a contents list, equations, drawings and the references. Chapters open and close; Open all and Close all do it for the whole page. The figures and the worked example in the theory are computed with the same engine as the calculator.
The (?) help windows are the short form: one window per section or choice, with a drawing and the formula that part of the model uses. They answer “what does this box mean?” without leaving the Model tab.
Saving, opening and sharing a model
What
Where it goes
Use it for
Automatic keeping
This browser
Coming back to the model you were working on; nothing to press
Quick Save…
A list in this browser
Keeping several versions of a model to compare, for example two pile lengths
Save to file…
A file on your computer
Keeping the model with the project, sending it to someone, moving to another computer or another browser
Open file…
Reads such a file
Continuing a saved model; open it in the same module it was saved from
Save as PDF, Download Word (.docx)
A report file
The calculation report for the client or the checker
Units
The unit selector in the header offers three systems:
System
Length
Force
Pressure and strength
Unit weight
SI · kN
m
kN
kPa
kN/m³
SI · tf
m
tf
kg/cm² for cohesion, t/m² for pressure
t/m³
US
ft
kip
psf for cohesion, ksf for pressure
pcf
Inside the engine every value is SI; the unit system only changes what you type and what you read. Switching it re-writes the boxes in the new units. After Analyze it re-states the results and the report as well, while the drawing of some modules stays in SI units and says so at its top right (the pile group redraws its views in the unit system you choose). The conversion constants (1 tf = 9.80665 kN, 1 kg/cm² = 98.0665 kPa, unit weight of water 9.81 kN/m³) are printed in every report so that a conversion can be repeated by hand.
Frequently asked questions
Does GeoTerva save my work?
Yes, in two ways. The model is kept in your browser as you type, so the page opens again where you left it. For a copy that travels, use Save to file… to write the model to a file, and Quick Save… to keep named versions in the browser.
Why can I not edit the inputs after Analyze?
After a run the inputs are locked so that the drawing, the results, the calculations and the report all describe the same model. Press Edit inputs to unlock them; this clears the results and the bar returns to Analyze. The unit system can still be changed while the model is locked.
Can I change the units after the analysis?
Yes. The unit selector in the header works at any time. The inputs, the drawing, the results and the report are re-stated in the new system without a new run.
How do I get a report?
Open the Report tab, fill in the cover details if you wish, and press Save as PDF or Download Word (.docx). Run Analyze first: the report is made from the analysed model.
What do the warnings mean?
A warning is a caution or a note from the run: an input outside the range of a chart, a method used outside the case it was fitted to, a check that is close to its limit. The Warnings button on the green bar lists them, and the same text is printed in the report.
How do I move a model to another computer?
Use Save to file… on the first computer, copy the file across, and use Open file… in the same module on the second. The automatic copy and the quick saves stay in the browser they were made in.
Does it work on a phone or a tablet?
Yes. GeoTerva runs in the browser, so it works on a phone, a tablet and a desktop without anything to install. On a narrow screen the drawing sits above the panel.
User manual · Piles
Piles: the user manuals
Five modules take a pile from its axial capacity, through its response to a sideways load, to the group under a cap. This page helps you choose the right one and links to the manual of each.
Each manual follows the same plan: what the module calculates, what you need before you start, a quick start, the screen, every input, the methods, the results, the report, a worked example you can reproduce, a table of how each input changes the answer, the messages the module gives, and answers to common questions.
What the five modules share
The workspace. The drawing, the five tabs, Analyze and the input lock, quick saves and files, units and the report are the same as in every GeoTerva module. See the workspace guide.
Layers. The ground is entered as up to eight layers from the ground surface down, each with its thickness and unit weights and, by soil type, φ′ for sand or cu for clay, with an optional water table.
Installation. Piles are driven, bored or jacked. It decides which methods are open and what the coefficients along the shaft are filled in with. Methods that were fitted to driven piles are refused for a bored pile.
Methods you choose. Where the literature offers more than one way to work out a resistance, the module lets you pick one and run another to compare. Each choice has a (?) help window with a drawing and the formula.
Values that fill themselves in. Coefficients that follow from other inputs, such as the shaft coefficient K or α, are filled in for each layer and can be typed over.
Units. SI · kN, SI · tf and US, switchable at any time.
Reports. A summary report as PDF or Word, with the working, the warnings and the references.
p-multipliers (AASHTO, Rollins et al.) on the p–y curves
A typical pile design in order
The axial capacity of one pile
Run Pile in sand, Pile in clay or Pile on layered soil for the ground you have. Compare two methods for the point and for the shaft and keep each run with a quick save.
The lateral response of one pile
Run Pile under lateral load with the same ground and pile. Read the head deflection and the largest moment, and the load at an allowable deflection.
The group
Run Pile group with the layout of the cap. It reuses the single-pile methods layer by layer and adds the load on every pile, the group effect, the settlement and the lateral response of the rows.
The reports
Save a report from each run. Fill in the cover so that the project, the client and the checker are named on it.
Frequently asked questions
How do I calculate pile capacity with GeoTerva?
Open the module that matches the ground, Pile in sand, Pile in clay or Pile on layered soil. Enter the layers, the pile and the methods, press Analyze, and read the point bearing, the skin friction, the ultimate capacity and the allowable capacity, with every step of the working.
What is the difference between a driven, a bored and a jacked pile in the modules?
Installation decides which methods are open and what the shaft coefficients are filled in with. Methods read from driven piles, such as Coyle & Castello in sand and the λ method in clay, are refused for a bored pile and allowed with a caution for a jacked one. A bored pile gets an at-rest coefficient in sand and a drilled-shaft adhesion factor in clay.
Which module gives the settlement of a pile?
The Pile group module gives the elastic settlement of the pile and the group, and the consolidation settlement of the clay below 2L/3. The axial modules for one pile give the capacity.
Can I analyse a pile in sand and clay together?
Yes. Pile on layered soil takes any stack of sand and clay layers, each typed on its own, and works out the shaft resistance of each layer by the method for its soil and the point bearing by the method for the soil the tip stands in.
Does GeoTerva check lateral loads on piles?
Yes. Pile under lateral load gives the deflection, rotation, moment, shear and soil reaction along a single pile, by p–y curves or an elastic subgrade reaction. The Pile group module adds the p-multipliers of the rows.
Calculate the ultimate and allowable axial capacity of a single driven, bored or jacked pile in layered sand. This manual walks through the screen, every input, the point-bearing and skin-friction methods, the results and the report, with a worked example you can reproduce.
Pile in sand gives the ultimate axial capacityQu and the allowable axial capacityQall of one pile that is pushed, driven or bored into sand. The ultimate capacity is the sum of what the tip carries and what the shaft carries:
Qu = Qp + Qs, Qall = Qu / FS
Qp is the point bearing, the resistance of the sand under the tip; Qs is the skin friction, the resistance along the shaft, worked out layer by layer; FS is the factor of safety. You choose the method for the point and the method for the shaft independently, and can put one factor of safety on the sum or one on each part.
What you can describe
A ground of up to eight sand layers, each with its own thickness, unit weights and friction angle φ′, and an optional water table.
A circular or square pile, driven, bored or jacked, of width or diameter D (at least 0.05 m) and embedded length L (at least 0.5 m).
Three point-bearing methods: Meyerhof (1976), Vesic (1977) and Coyle & Castello (1981).
Two skin-friction methods: the K–δ method and Coyle & Castello (1981), both applied to each layer separately.
What belongs to other modules
This module treats one pile under an axial compression load. Neighbouring piles are covered by the Pile group module, and a sideways load by Pile under lateral load.
The results are the resistance of the ground: the weight of the pile is not deducted, as the module says under its results.
Have these ready from your ground investigation and your design:
What
Where it comes from
Notes
The sand profile: layer thicknesses
Boring logs
The layers must reach at least the pile tip. Describing the sand about two pile widths below the tip is better: the point resistance is governed by the soil at and just below the tip.
Unit weights γ and γsat
Laboratory or correlations
γ applies above the water table, γsat below it.
Friction angle φ′ of each layer
SPT or CPT correlations, tests
An effective (drained) friction angle. The point-bearing chart of Meyerhof is drawn from 20° to 45°.
Water table depth
Piezometers, borings
Optional. Enter it only if it lies within the pile length.
Pile: shape, width or diameter, embedded length, installation
Your design
Installation decides which methods are open and what the module fills in for the shaft.
Factor of safety; applied working load
Your design basis
The module’s hint for the factor is 2.5 to 4. The applied load is optional.
Quick start in eight steps
Open the module and choose the units
Open Pile in sand from the Piles family. Pick SI · kN, SI · tf or US in the header; you can change it at any time. Open Pile in sand.
Describe the sand layers
In Sand layers, fill in the New layer form (thickness, γ, γsat, φ′) and press + Add layer; the module already holds Sand-1 (25 m): click its row to edit it, then add the layers below it, from the ground surface down. Click a row to edit a layer, the cross to remove it. The layers must reach the pile tip.
Set the groundwater
In Groundwater, choose Present and give the depth of the water table, or leave it at None.
Describe the pile
In Pile, choose the shape (Circular or Square) and the installation (Driven, Bored or Jacked), then enter the width or diameter D and the embedded length L.
Choose the method for the point
In Point bearing, pick Meyerhof, Vesic (it asks for m = Es/pa) or Coyle & Castello (it fills in N*q from the chart).
Choose the method for the shaft
In Skin friction, pick the K–δ method or Coyle & Castello. Each layer has its own coefficient and, for the K–δ method, its own δ′; they are filled in and can be typed over.
Set the design values
In Design, enter the factor of safety (one on the total, or one on each part) and, if you want a check, the applied working load.
Press Analyze and read the answer
Press Analyze. The Results tab gives Qu, Qp, Qs and Qall; the Calculations tab shows every step; the Report tab saves a PDF or a Word file.
The screen: three views of the pile
The layout is the one every GeoTerva module shares, described in the workspace guide. For this module the drawing has three views, chosen with the buttons above it:
Section shows the pile in the sand profile, with the water table, the layer boundaries, the critical depth L′ of the K–δ method, the resistances Qp, Qs and Qu (with the method named beside Qp and Qs) and the arrows of the skin friction. You can drag the water table, a layer base, the pile tip or the side of the pile to change them before you analyse.
Profile plots the effective vertical stress σ′ and the unit skin friction f against depth, with the pile tip marked at z = L.
3D shows the pile in a block of the ground with the layers and the water table.
The Section view after Analyze: the pile, the sand, the critical depth L′ = 15D and the resistances.
The Profile view: the effective stress on the left, the unit skin friction on the right, constant below L′.
Entering the model, section by section
The Model tab has six sections after Model data. Each opens when you click its title; Next and Back at the foot walk you through them. A (?) beside a title or a method opens a help window with a drawing and the formula for that part. The tables below list every field with its symbol, the unit in each unit system, and its default. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. A field marked “filled in” carries a value the module works out; you can type over it. Defaults are shown in SI; in another unit system the boxes show the converted values.
Sand layers
Up to eight layers, entered from the ground surface down. The point resistance answers to the layer the tip sits in, and the shaft friction is summed over all the layers the pile passes through. The values in the tables are those of layer 1.
Sand layers: the New layer form above, the table of layers below.
Fields of each sand layer
Field
Symbol
Unit
Default
What to enter
Name (optional)
—
—
empty
A name for the layer, for the drawing and the report. Left empty, the layers are called Sand-1, Sand-2 and so on.
Unit weight (above the water table)
γ
kN/m³ · t/m³ · pcf
18
Unit weight of the sand above the water table, used for the effective stress.
Saturated unit weight
γsat
kN/m³ · t/m³ · pcf
18
Unit weight of the sand below the water table. Below the water table the module uses γsat − γw. Used only when a water table is present.
Friction angle (20–45°)
φ′
°
35
Effective friction angle of the sand in the layer, at least 20°. Meyerhof’s chart is drawn from 20° to 45°; above 45° the end value is held and a warning says so.
Thickness
H
m · ft
25
Thickness of the layer. The layers together must reach at least the pile tip; the smallest value accepted is 0.1 m.
Groundwater
Choose whether there is a water table within the pile length. If there is, enter its depth; below it the effective stress is worked out with the submerged unit weight γsat − γw, with γw = 9.81 kN/m³.
Groundwater: Present or None, then the depth.
Choice
Options
Default
Water table within or below the pile
PresentNone
None
Groundwater field
Field
Symbol
Unit
Default
What to enter
Water table depth
Dw
m · ft
100
Depth of the water table below the ground surface, 0 or more. Shown when the water table is present.
Pile
The shape and the installation are choices; the width and the length are numbers. The section gives the area of the tip and the perimeter of the shaft: a square pile has Ap = D² and a perimeter of 4D; a circular one has Ap = πD²/4 and a perimeter of πD.
Pile: shape, installation, width and length.
Choice
Options
Default
Shape of the cross section
CircularSquare
Square
Installation how the pile is put in the ground
DrivenBoredJacked
Driven
Pile fields
Field
Symbol
Unit
Default
What to enter
Pile width / diameter
D
m · ft
0.407
Width of a square pile or diameter of a circular one. It sets the area of the tip and the perimeter of the shaft.
Embedded length
L
m · ft
20
Embedded length from the ground surface to the tip; the head is taken at the ground surface.
Point bearing
Pick the method for the resistance under the tip. The boxes of the method you choose appear; the others stay out of the way.
Point bearing: one of three methods.
Choice
Options
Default
Method for the point (tip) bearing Qp
MeyerhofVesicCoyle & Castello
Meyerhof
Point-bearing fields
Field
Symbol
Unit
Default
What to enter
Stiffness ratio of the sand
m
—
250
The stiffness ratio Es/pa of the sand: about 100–200 loose, 200–500 medium dense, 500–1000 dense. The smallest value accepted is 50. Shown when point-bearing method is Vesic.
Coyle & Castello N*q (chart value)
N*q(CC)
—
filled in
Read from the Coyle & Castello chart for this pile’s L/D and φ′ of the tip layer, and kept up to date. Type over it to use your own reading; empty the box to get the chart value back. Shown when point-bearing method is Coyle & Castello.
Skin friction
Pick the method for the shaft. Both are worked out layer by layer: every layer of the profile has its own coefficient, named after the layer it belongs to. With the K–δ method each layer also has its own δ′. The values are filled in from the other inputs and can be typed over.
Skin friction: the method, then the coefficients of each layer.
Choice
Options
Default
Method for the skin friction Qs
K–δ methodCoyle & Castello
K–δ method
Shaft coefficients of each layer (layer 1 shown)
Field
Symbol
Unit
Default
What to enter
Earth pressure coefficient K
K
—
filled in
Earth-pressure coefficient on the shaft in this layer. Filled in as K0 = 1 − sin φ′ for a bored pile and 1.4K0 for a driven or jacked one. The smallest value accepted is 0.2. Shown when skin-friction method is K–δ method.
Pile–soil friction angle δ′
δ′
°
filled in
Friction angle between the pile and the sand in this layer. Filled in as 0.75φ′. Shown when skin-friction method is K–δ method.
Coyle & Castello K
K(CC)
—
filled in
Coyle & Castello’s coefficient for this layer, read from their chart at the layer’s φ′ and the pile’s L/D; the interface friction is 0.8φ′. Editable. Shown when skin-friction method is Coyle & Castello.
Design
One factor of safety on the sum, or a separate factor on each part: Qall = Qp/FSp + Qs/FSs. The second form is used when the shaft is trusted more than the point. An applied working load is optional.
Design: factor of safety and the applied load.
Choice
Options
Default
Factor of safety on the total, or on each part
One on the totalOne on each part
One on the total
Design fields
Field
Symbol
Unit
Default
What to enter
Applied working load (optional)
Qapplied
kN · tf · kip
empty
The working load the pile has to carry. Optional: if it is entered it is drawn on the pile and checked against Qall.
Factor of safety (2.5–4 typical)
FS
—
3
One factor of safety on Qu. The module’s hint is 2.5 to 4. Shown when factor of safety is One on the total.
Factor of safety on the point resistance
FSp
—
3
Factor of safety on the point resistance (used with one factor on each part). Shown when factor of safety is One on each part.
Factor of safety on the skin friction
FSs
—
1.5
Factor of safety on the skin friction; lower than the point factor because the shaft is mobilised at smaller settlements. Shown when factor of safety is One on each part.
Choosing the methods
The three point-bearing methods and the two skin-friction methods answer the same question in different ways, so the module lets you run one after the other and compare. What each one needs and how it works:
Point bearing Qp = Apqp
Method
Unit resistance qp
What it asks for
Notes
Meyerhof (1976)
qp = σ′N*q ≤ 0.5 paN*q tan φ′
φ′ of the tip layer only
N*q is read from his own chart for piles, 20°–45°. The limit ql caps the result; the steps say when it governs.
Vesic (1977)
qp = σ̄′oN*σ
φ′ and m = Es/pa
Treats the tip as an expanding cavity: it uses the mean effective stress at the tip and the stiffness of the sand through the reduced rigidity index.
Coyle & Castello (1981)
qp = σ′N*q, no cap
φ′ and the chart value N*q (filled in)
Fitted to field load tests on driven piles in sand; N*q is read against L/D and φ′. Refused for a bored pile. The chart covers φ′ of about 30°–40° (K chart 30°–36°); outside it the end value is used without a message.
Skin friction Qs = perimeter × the sum of f over the layers
Method
Unit skin friction f
Coefficients
Notes
K–δ method
f = Kσ′ tan δ′, with σ′ held constant below L′ = 15D
K and δ′ for each layer
The friction grows with depth like the effective stress until the critical depth of about 15 pile widths, then stays constant. A fixed rule of thumb, not a chart reading.
Coyle & Castello (1981)
f = Kσ̄′o tan δ′, δ′ = 0.8φ′
K read from their chart, for each layer
Uses the mean effective stress of the layer. For a layered profile the module applies it layer by layer, which extends the method, written for a single sand.
Which should I pick?
Start with Meyerhof and the K–δ method. They need the least input and are the module’s defaults.
Use Vesic when you have an estimate of the stiffness of the sand (m). The point resistance then depends on stiffness as well as strength.
Use Coyle & Castello for driven piles when you want the estimates that come from field load tests. The module refuses it for a bored pile and cautions you for a jacked one.
Compare. The methods can differ widely. Run them one after the other and keep the results with Quick Save…; the what-if table shows how far they can differ on the example.
Analyze and read the results
Press Analyze. The bar reports the run, the inputs lock, and the Results tab shows four blocks.
Axial capacity of the pile. Four cards: the ultimate capacity Qu, the point bearing Qp, the skin friction Qs and the allowable capacity Qall.
Values, step by step. One row per step of the calculation, with the numbers: the effective stress at the tip, N*q and qp, L′ and Qs, and the factor of safety.
What each layer gives to the capacity. A table with the skin friction, the point bearing and the total of every layer, and the totals.
Contribution of each mechanism. The share of the point and of the shaft in Qu as a percentage.
The pile in the Section view now carries Qu, Qall, Qs and Qp, with the method named beside Qs and Qp.
The Results tab of the example: capacities, steps, layers and mechanisms.
Checking an applied load
If you entered an applied working loadQapplied, the Results tab adds a check: the ratio Qapplied/Qall and a verdict in words. PASS means the applied load is within the allowable capacity; FAIL means it exceeds it. The report says so in words when the load exceeds even Qu. A failing check also raises a warning with advice: lengthen or widen the pile, use more piles, or revisit the factors of safety.
The weight of the pile
The results do not deduct the weight of the pile. If your design basis takes it into account, add it to the applied load, or deduct it from the capacity, as your standard requires.
The Calculations tab and the report
The Calculations tab writes out the working in chapters: Given and what is sought, then one chapter per step, then the Answer in one sentence. For this module the steps are:
Locate the effective vertical stress at the pile tip — σ′ from the unit weights and the water table.
Compute the point (tip) bearing capacity — the method you chose, with its factor, the limit if it applies, and Qp.
Compute the skin-friction resistance — the method you chose, layer by layer, with L′ for the K–δ method, and Qs.
Apply the factor of safety — Qu and Qall.
Check the applied working load, when one was entered.
The Calculations tab: the given data, then the steps with their equations and numbers.
The Report tab turns the same working into a summary report. Fill in the cover (project, location, client, report number, date, prepared by, title, checked by, notes) and press Save as PDF or Download Word (.docx). The report has six parts: 1 Introduction and scope, 2 Input data, 3 Calculation (one part per step), 4 Results, 5 Summary and 6 References. When a run has notes or warnings about the methods, the report adds an Assumptions and notes section after Results (the parts after it move down by one); the units line states the conversion constants.
The Theory tab explains the method: what the pile carries, the effective stress in the profile, the three point-bearing methods, the skin friction and the critical depth, the allowable capacity, and a worked example.
Worked example: the module’s own example
The module opens on this example. Reset to defaults brings it back at any time, so you can follow the numbers on your own screen.
Inputs of the example
Section
Input
Value
Sand layers
One layer, Sand-1
H = 25 m, γ = γsat = 18 kN/m³, φ′ = 35°
Groundwater
Water table
None
Pile
Shape, installation
Square, driven
Pile
Width, length
D = 0.407 m, L = 20 m
Point bearing
Method
Meyerhof
Skin friction
Method, coefficients
K–δ; K = 0.6 and δ′ = 26.3° (filled in)
Design
Factor of safety
One factor, FS = 3
Step 1: the effective stress at the tip
There is no water table, so σ′ = 18 × 20 = 360.00 kPa at the tip.
Step 2: the point bearing, Meyerhof
For φ′ = 35° the chart gives N*q = 143.00, so σ′N*q = 360 × 143 = 51,480 kPa. That is far above the limiting value ql = 0.5 × 100 × 143 × tan 35° = 5,006.48 kPa, so the limit governs: qp = 5,006.48 kPa. The area of the tip is 0.407² = 0.165649 m², so Qp = 5,006.48 × 0.165649 = 829.32 kN.
Step 3: the skin friction, K–δ
The critical depth is L′ = 15D = 15 × 0.407 = 6.105 m (the module shows 6.11 m). At L′ the effective stress is 18 × 6.105 = 109.89 kPa, so the unit skin friction there is f = 0.6 × 109.89 × tan 26.3° = 32.587 kPa; it grows linearly from zero at the surface to that value, and stays at it down to the tip. With a perimeter of 4 × 0.407 = 1.628 m:
Qu = 829.32 + 899.08 = 1,728.40 kN, and with FS = 3 the allowable capacity is Qall = 1,728.40 / 3 = 576.13 kN. The point carries 48.0 % of the capacity and the shaft 52.0 %.
How each input changes the answer
Each row below changes one thing in the example and runs the module again. The numbers are the results the module gives, in kN, so you can reproduce any row.
The example with one change at a time (kN)
Change
Qp
Qs
Qu
Qall
Why
The default example
829.3
899.1
1,728
576.1
The reference
Point bearing by Vesic (m = 250)
1,721
899.1
2,620
873.3
No limit on qp
Point bearing by Coyle & Castello
2,087
899.1
2,986
995.4
N*q from the chart, no limit
Skin friction by Coyle & Castello
829.3
1,496
2,325
775.0
K from the chart; a note says the chart is projected beyond L/D = 36.1
Both by Coyle & Castello
2,087
1,496
3,583
1,194
Both from the field-test charts
Bored pile
829.3
644.3
1,474
491.2
K = K0 instead of 1.4K0
Jacked pile
829.3
899.1
1,728
576.1
Same as driven for the K–δ method
Circular section
651.3
706.1
1,357
452.5
Area and perimeter are π/4 of the square pile’s
Water table at 5 m
829.3
823.5
1,653
550.9
Lower σ′ below 5 m; Qp stays at its limit
Factor of safety on each part
829.3
899.1
1,728
875.8
FSp = 3 and FSs = 1.5
Friction angle 30
271.1
879.1
1,150
383.4
N*q falls, so Qp drops to a third; the shaft barely changes (K rises as φ′ falls, δ′ falls)
Friction angle 40
2,405
875.2
3,280
1,093
Qp nearly triples; the shaft barely changes (K falls as φ′ rises)
Pile 12 m long
829.3
474.7
1,304
434.7
Only the shaft changes
Pile 24 m long
829.3
1,111
1,941
646.9
Only the shaft changes
Width 0.6 m
1,802
1,787
3,589
1,196
Larger tip and perimeter
A bored pile with Coyle & Castello’s point method is not in the table because the module does not run it: it stops and says why (see Messages).
Messages: stops, warnings and notes
The module tells you in plain words when something prevents the calculation, when a result needs care, and what it assumed. There are three kinds.
A stop: the pile is 30 m long but the layers reach 25 m. Nothing is calculated; the drawing shows where the profile ends.
Kind
When
Message and what to do
Stop (brown bar, “Cannot analyse”)
The layers do not reach the pile tip
“The layers must reach the pile tip. They reach 25.00 m, the pile is 30.00 m long.” (the brown bar reads “Cannot be analysed”, the window “Cannot analyse”) Add a layer, or thicken the last one.
Stop
Coyle & Castello selected for a bored pile
“Coyle & Castello is calibrated on piles driven in sand; a bored pile is outside it.” Use Meyerhof or Vesic for the point and the K–δ method for the shaft.
Warning
Coyle & Castello selected for a jacked pile
“Coyle & Castello is for driven piles; this one is jacked.” Read the result as an estimate, or use the K–δ method.
Warning
An applied load larger than Qall
“The pile does not carry the applied load — 173.6% of Qall.” Lengthen or widen the pile, use more piles, or revisit the factors of safety.
Stop (input error)
A value below the smallest the field accepts, for example φ′ = 19°
“1 input error: Friction angle (20–45°) (φ′1) must be at least 20°.” Correct the field marked in red.
Warning
φ′ above 45°
Meyerhof’s chart is drawn from 20° to 45° only; beyond it the end value is held, not extrapolated.
Warning
K above 1.25K0 of the layer for a bored pile
“K is too high for a bored pile”: boring displaces nothing, so K stays near K0. Use K ≈ K0, or say why this shaft carries more.
Note
K above 1.8K0 for a driven or jacked pile
The table of K runs from K0 to 1.8K0; keep a higher value if a load test or local practice supports it.
Note
The tip is exactly on a layer boundary
The point resistance is worked from the layer below the boundary, the soil beneath the base; the layer above contributes shaft friction only.
Note
The profile ends less than 2D below the tip
The deepest layer is taken to continue below, with the same properties, because the point resistance comes from the soil just under the tip.
Note
Coyle & Castello beyond the chart
The chart is projected beyond L/D = 36.1 (for K); the coefficient boxes stay editable, so a value read off the figure can be used instead.
Note
Bored pile
The friction angle entered at the toe is taken to describe the sand after drilling; no further reduction is applied for the bored construction.
The Warnings (or Notes) button on the green bar lists them after a run. When a run has notes or warnings about the methods, the report adds an Assumptions and notes section after the results; the check on an applied load appears in the calculation and in the summary.
Good practice and common mistakes
Describe the sand below the tip. The point resistance depends on the soil at and just below the tip. Take the profile to at least two pile widths below it, or accept the module’s assumption that the last layer continues.
Match the installation to the pile. It changes K and decides whether Coyle & Castello is open. A driven pile entered as bored gets the lower K and so a lower shaft resistance; a bored pile entered as driven gets the higher K and a higher one.
Do not expect the point bearing to grow with the length when the limit governs. When σ′N*q is above ql, the limit caps qp, and a longer pile adds shaft only.
Run more than one method. The spread of the answers is information. Keep each run with Quick Save….
Keep the layer φ′ an effective friction angle from your investigation, not a total-stress parameter.
Check the units after you switch the system: type values in the units shown beside the boxes.
Read the notes. They list what the module assumed; they are printed in the report.
Frequently asked questions
How do I calculate the capacity of a pile in sand?
The ultimate capacity is the sum of the point bearing under the tip and the skin friction along the shaft, Qu = Qp + Qs. The allowable capacity is Qu divided by a factor of safety. In GeoTerva’s Pile in sand module you describe the sand layers, the pile and the methods, press Analyze, and the module gives Qp, Qs, Qu and Qall with every step of the working.
What is the difference between Meyerhof, Vesic and Coyle & Castello for the point?
Meyerhof reads N*q from a chart against φ′ and caps the unit resistance at a limiting value. Vesic treats the tip as an expanding cavity and also needs the stiffness ratio m = Es/pa. Coyle & Castello read N*q from a chart fitted to field load tests on driven piles, against L/D and φ′, with no cap. On the module’s example they give 829 kN, 1,721 kN and 2,087 kN.
What is the critical depth L′ = 15D?
In the K–δ method the unit skin friction grows with depth like the effective stress until a critical depth of about 15 pile widths, and is held constant below it. The module draws it as a dashed line on the pile and uses L′ = the smaller of 15D and L.
What values of K and δ′ does the module use?
They are filled in for each layer: K = K0 = 1 − sin φ′ for a bored pile and 1.4K0 for a driven or jacked pile, and δ′ = 0.75φ′. You can type over any of them. With the Coyle & Castello method K is read from their chart and δ′ is 0.8φ′.
Why does the point bearing not change when I make the pile longer?
Meyerhof’s method limits the unit point resistance to ql = 0.5 paN*q tan φ′. When the limit governs, as in the module’s example, a longer pile adds shaft friction but no point resistance. The steps say when the limit governs.
Can I use the Coyle & Castello method for a bored pile?
No. Their charts were read from driven piles, so the module stops and asks you to change the method. For a jacked pile it allows the method with a caution.
Does the module deduct the weight of the pile?
No. The capacity is the resistance of the ground; the weight of the pile is not deducted, and the module states this under its results.
How many layers can I enter, and what does the water table do?
Up to eight sand layers. A water table within the pile length makes the module use the submerged unit weight γsat − γw below it for the effective stress, which lowers the shaft friction.
Which units does the module use?
SI · kN, SI · tf or US, switchable in the header at any time, even after Analyze. The report states the units and the conversion constants.
Does it check group effects, settlement or lateral load?
This module treats one pile under axial load. Use Pile group for the group effect and settlement, and Pile under lateral load for the lateral response.
How do I compare two designs?
Run the first, press Quick Save…, press Edit inputs, change a value, run again and quick-save again. Load in the Model tab brings any version back. Save to file… keeps a version with the project.
Calculate the ultimate and allowable axial capacity of a single driven, bored or jacked pile in layered clay, undrained. This manual walks through the screen, every input, the α, λ and β shaft methods, the point-bearing methods, the results and the report, with a worked example you can reproduce.
Pile in clay gives the ultimate axial capacityQu and the allowable axial capacityQall of one pile in clay, loaded undrained. As for every pile in the family, the ultimate capacity is what the tip carries plus what the shaft carries:
Qu = Qp + Qs, Qall = Qu / FS
Qp is the point bearing, worked out on the undrained strength cu of the clay at the tip; Qs is the skin friction, worked out layer by layer. You choose the method for the point and the method for the shaft independently, and can put one factor of safety on the sum or one on each part.
What you can describe
A ground of up to eight clay layers, each with its thickness, undrained shear strength cu, unit weights and, for the β method, its overconsolidation ratio OCR, and a water table.
A circular or square pile, driven, bored or jacked, of width or diameter D (at least 0.05 m) and embedded length L (at least 0.5 m).
Three point-bearing methods: Meyerhof (Qp = 9cuAp), Vesic (the factor N*c from the rigidity index) and, for a bored pile, Reese & O’Neill’s drilled-shaft expression.
Three skin-friction methods: the α method, the λ method and the β method.
What belongs to other modules
This module treats one pile under an axial compression load. Neighbouring piles are covered by Pile group, and a sideways load by Pile under lateral load.
The results are the resistance of the ground: the weight of the pile is not deducted, as the module says under its results.
The layers must reach at least the pile tip; the module stops if they do not.
Undrained shear strength cu of each layer
Unconfined or triaxial tests, vane tests, correlations
Used by the α and λ methods and by the point bearing (the strength of the tip layer governs Qp); the β method does not use it.
Unit weights γ and γsat
Laboratory or correlations
They give the effective stress used by the λ and β methods. The α method works on cu alone.
Water table depth
Piezometers, borings
Below it the submerged unit weight γsat − γw is used for the effective stress (λ and β methods).
For the β method: remoulded friction angle φ′R and OCR of each layer
Drained tests on remoulded clay; the stress history
They describe your clay, so you enter them: φ′R opens at 30° and OCR at 1 (the example’s Clay-3 has 2).
Pile: shape, width or diameter, embedded length, installation
Your design
Installation decides which methods are open and what the filled-in values are.
Factor of safety; applied working load
Your design basis
The module’s hint for the factor is 2.5 to 4. The applied load is optional.
Quick start in eight steps
Open the module and choose the units
Open Pile in clay from the Piles family and pick SI · kN, SI · tf or US in the header. Open Pile in clay.
Describe the clay layers
In Clay layers, fill in the New layer form (thickness, cu, γ, γsat, and OCR for the β method) and press + Add layer; repeat from the ground surface down. The module already holds the example’s three layers: click a row to edit it, the cross to remove it, then add the rest. The layers must reach the pile tip.
Set the groundwater
In Groundwater, choose Present and give the depth, or None.
Describe the pile
In Pile, choose the shape and the installation (Driven, Bored or Jacked), then enter D and L.
Choose the method for the point
In Point bearing, pick Meyerhof, Vesic or, for a bored pile, Reese & O’Neill.
Choose the method for the shaft
In Skin friction, pick the α method, the λ method (driven piles) or the β method. The α of each layer is filled in; φ′R appears in the Skin friction section and OCR in the layer form (click a row of the table to see it); both are yours to enter.
Set the design values
In Design, enter the factor of safety and, if you want a check, the applied working load.
Press Analyze and read the answer
Press Analyze. The Results tab gives Qu, Qp, Qs and Qall and what each layer contributes; Calculations shows every step; Report saves a PDF or Word file.
The screen: three views of the pile
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing has three views:
Section shows the pile in the clay layers with the water table and the resistances Qp, Qs and Qu; the skin-friction arrows follow the unit skin friction of each layer. Before you analyse you can drag the water table, a layer base, the pile tip or the side of the pile.
Profile plots the effective vertical stress σ′ and the unit skin friction f against depth.
3D shows the pile in a block of the ground with the layers and the water table.
The Profile view: the effective stress and the unit skin friction against depth.
Entering the model, section by section
The Model tab has six sections after Model data. Each opens when you click its title; Next and Back walk you through them, and a (?) opens a help window with a drawing and the formula. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. A field marked “filled in” carries a value the module works out; you can type over it.
Clay layers
Up to eight layers from the ground surface down. The shaft is summed over every layer the pile passes through; the point bearing answers to the layer at the tip. The values in the table are those of layer 1.
Clay layers: the New layer form above, the table of layers below.
Fields of each clay layer
Field
Symbol
Unit
Default
What to enter
Name (optional)
—
—
empty
A name for the layer, for the drawing and the report. Left empty, the layers are called Clay-1, Clay-2 and so on.
Undrained shear strength
cu
kPa · kg/cm² · psf
25
Undrained shear strength of the layer, at least 1 kPa. The tip layer’s value gives the point bearing; every layer’s value enters the α and λ shaft methods.
Unit weight (above the water table)
γ
kN/m³ · t/m³ · pcf
16
Unit weight of the clay above the water table.
Saturated unit weight
γsat
kN/m³ · t/m³ · pcf
16
Unit weight below the water table; below it the module uses γsat − γw for the effective stress.
Thickness
H
m · ft
3
Thickness of the layer. The layers together must reach at least the pile tip; the smallest value accepted is 0.1 m.
Overconsolidation ratio
OCR
—
1
Overconsolidation ratio of the layer, at least 1. Used by the β method only. Shown when skin-friction method is β method.
Groundwater
Choose whether there is a water table within the pile length and give its depth. Below it the effective stress uses the submerged unit weight. It matters for the λ and β methods, which work on effective stress; the α method works on cu alone and does not change with the water table.
Groundwater: Present or None, then the depth.
Choice
Options
Default
Water table within or below the pile
PresentNone
Present
Groundwater field
Field
Symbol
Unit
Default
What to enter
Water table depth
Dw
m · ft
3
Depth of the water table below the ground surface, 0 or more. Shown when the water table is present.
Pile
The shape and the installation are choices; the width and the length are numbers. A square pile has Ap = D² and a perimeter of 4D; a circular one Ap = πD²/4 and a perimeter of πD.
Pile: shape, installation, width and length.
Choice
Options
Default
Shape of the cross section
CircularSquare
Circular
Installation how the pile is put in the ground
DrivenBoredJacked
Driven
Pile fields
Field
Symbol
Unit
Default
What to enter
Pile width / diameter
D
m · ft
0.457
Width of a square pile or diameter of a circular one; it sets the area of the tip and the perimeter of the shaft.
Embedded length
L
m · ft
20
Embedded length from the ground surface to the tip; the head is at the ground surface.
Point bearing
Pick the method for the resistance under the tip. Reese & O’Neill is offered for a bored pile only.
Point bearing: the method for the tip.
Choice
Options
Default
Method for the point (tip) bearing Qp
MeyerhofVesicReese & O’Neill
Meyerhof
Skin friction
Pick the method for the shaft. The α and β methods are worked out layer by layer, with an α or a φ′R for each layer in the boxes below, named after the layer they belong to. The λ method is different in kind: it takes one average over the whole embedded length, so it has no value per layer. The shaft is summed over the embedded length only.
Skin friction: the method, then the factor of each layer.
Choice
Options
Default
Method for the skin friction Qs
α methodλ methodβ method
α method
Shaft coefficients of each layer (layer 1 shown)
Field
Symbol
Unit
Default
What to enter
Adhesion factor α
α
—
filled in
Adhesion factor of this layer: the share of cu the shaft develops. Filled in from the correlation that belongs to the installation (a driven-pile curve against cu/pa, or 0.55 falling to 0.45 for a bored pile) and editable; the smallest value accepted is 0.05. Shown when skin-friction method is α method.
Remoulded drained friction angle
φ′R
°
30
Remoulded drained friction angle of this layer, for the β method. Not filled in from the ground; the box opens at 30° and wants the value of your clay. Shown when skin-friction method is β method.
Design
One factor of safety on the sum, or a separate factor on each part: Qall = Qp/FSp + Qs/FSs. An applied working load is optional.
Design: factor of safety and the applied load.
Choice
Options
Default
Factor of safety on the total, or on each part
One on the totalOne on each part
One on the total
Design fields
Field
Symbol
Unit
Default
What to enter
Applied working load (optional)
Qapplied
kN · tf · kip
empty
The working load the pile has to carry. Optional: if it is entered it is drawn on the pile and checked against Qall.
Factor of safety (2.5–4 typical)
FS
—
4
One factor of safety on Qu; the module’s hint is 2.5 to 4. Shown when factor of safety is One on the total.
Factor of safety on the point resistance
FSp
—
4
Factor of safety on the point resistance (used with one factor on each part). Shown when factor of safety is One on each part.
Factor of safety on the skin friction
FSs
—
2
Factor of safety on the skin friction. Shown when factor of safety is One on each part.
Choosing the methods
Point bearing Qp = Apqp
Method
Unit resistance qp
Notes
Meyerhof
qp = 9 cu
cu of the layer at the tip; the strength of the layers above does not enter. The classic deep-foundation value N*c = 9 for saturated clay under undrained loading.
Vesic
qp = N*ccu, with N*c from the rigidity index Irr = 347(cu/pa) − 33, limited to 300
The stiffness of the clay enters through Irr. The correlation starts at about cu = 9.8 kPa; below it the module stops and asks for Meyerhof.
cub is the mean cu over the 2D below the base. It follows the same drilled-shaft procedure as the α the shaft uses. For a base wider than 1.91 m the result is reduced by a factor Fr.
Skin friction Qs
Method
Unit skin friction
Notes
α method
f = αcu, layer by layer
An α for each layer, filled in from cu/pa for the installation and editable. Works on total stress: the water table does not change it.
λ method (driven piles)
Qs = pLfav, fav = λ(σ̄′ + 2c̄u)
One average over the whole embedded length, with λ read from Vijayvergiya & Focht’s correlation at the pile’s L. The total is then shared among the layers by length.
β method
f = βσ′, β = (1 − sin φ′R) tan φ′R √OCR
Effective stress, layer by layer, from the remoulded friction angle and the OCR of each layer. The long-term view: the undrained strength does not enter.
Which should I pick?
Start with Meyerhof and the α method. They need only cu and are the module’s defaults.
Use the λ method for a driven pile when you want the average-condition estimate; it is refused for a bored pile.
Use the β method when you hold effective-stress parameters (φ′R and OCR) and want the long-term shaft resistance.
For a bored pile the module fills in the drilled-shaft α and offers Reese & O’Neill for the base.
Compare. Run the methods one after the other and keep each run with Quick Save…; the what-if table shows how close they come on the example.
Analyze and read the results
Press Analyze. The bar reports the run, the inputs lock, and the Results tab shows four blocks.
Axial capacity of the pile. Cards for Qu, Qp, Qs and Qall.
Values, step by step. The point bearing (N*c, Qp), the shaft (Qs, with λ for that method) and the factor of safety.
What each layer gives to the capacity. The skin friction, point bearing and total of every layer. With Meyerhof or Vesic the tip layer carries the point bearing; with Reese & O’Neill the table has a row for the base bearing zone, and with the λ method the shaft is shown as one total allocated by length.
Contribution of each mechanism. The share of the point and of the shaft in Qu.
The Results tab of the example.
Checking an applied load
If you enter an applied working load, the Results tab adds the ratio Qapplied/Qall and a verdict in words: PASS when the load is within the allowable capacity, FAIL when it exceeds it. A failing check raises a warning with advice: lengthen or widen the pile, use more piles, or revisit the factors of safety.
The Calculations tab and the report
The Calculations tab starts with Given and what is sought and then follows the method:
Compute the point (tip) bearing capacity — the method you chose, with cu at the tip, the factor N*c and Qp.
Compute the skin-friction resistance — the method you chose, with the factors of each layer (α or β) or the averages (λ), and Qs.
Apply the factor of safety — Qu and Qall.
Check the applied working load, when one was entered.
The Report tab fills in a cover and saves the same working as a PDF or a Word file: 1 Introduction and scope, 2 Input data, 3 Calculation, 4 Results, then, when the run has notes or warnings about the methods, 5 Assumptions and notes, and finally the Summary and the References. When a run has notes or warnings about the methods, the report adds an Assumptions and notes section after Results; the units line states the conversion constants. The Theory tab explains the method, from why clay is treated as undrained to the three shaft routes compared, the bored-pile procedure and a worked example.
Worked example: the module’s own example
The module opens on this example; Reset to defaults brings it back.
Inputs of the example
Layer
Thickness H
cu
γ = γsat
OCR
Clay-1
3 m
25 kPa
16 kN/m³
1
Clay-2
7 m
40 kPa
17 kN/m³
1
Clay-3
12 m
90 kPa
18 kN/m³
2
Water table at 3 m; circular driven pile, D = 0.457 m, L = 20 m; Meyerhof for the point, the α method for the shaft, one factor of safety FS = 4.
Point bearing
The tip is in Clay-3, so cu = 90 kPa and N*c = 9. The area of the tip is π × 0.457²/4 = 0.1640 m², so Qp = 9 × 90 × 0.16403 = 132.86 kN.
Skin friction
The module fills in α from cu/pa for a driven pile: 0.87, 0.74 and 0.51 for the three layers. The perimeter is π × 0.457 = 1.4357 m and the pile passes through 3 m of Clay-1, 7 m of Clay-2 and 10 m of Clay-3:
Layer
α
cu (kPa)
Length (m)
αcu perimeter × length (kN)
Clay-1
0.87
25
3
93.68
Clay-2
0.74
40
7
297.48
Clay-3
0.51
90
10
658.99
Total Qs
20
1,050.15
Capacity
Qu = 132.86 + 1,050.15 = 1,183.01 kN, and with FS = 4, Qall = 295.75 kN. The shaft carries 88.8 % of the capacity and the point 11.2 %: in this example, as usually in clay, the shaft is the larger part, and the Results tab shows which layer gives how much.
How each input changes the answer
Each row changes one thing in the example and runs the module again. The numbers are the results the module gives, in kN.
The example with one change at a time (kN)
Change
Qp
Qs
Qu
Qall
Why
The default example
132.9
1,050
1,183
295.8
The reference
Point bearing by Vesic
168.5
1,050
1,219
304.7
Irr = 279.3, N*c = 11.41
Skin friction by the λ method
132.9
1,114
1,247
311.8
λ = 0.1730 at L = 20 m
Skin friction by the β method
132.9
1,058
1,191
297.8
β from φ′R = 30° and the OCR of each layer; a note says the at-rest K is on the safe side
Bored pile
132.9
928.9
1,062
265.4
α = 0.55 in every layer; no friction over the top 1.5 m and the bottom 1D
Jacked pile
132.9
1,050
1,183
295.8
Same as driven for the α method
Jacked pile, λ method
132.9
1,114
1,247
311.8
Allowed with a caution: the λ correlation does not cover jacked piles
Square section
169.2
1,337
1,506
376.6
Larger area and perimeter than the circle of the same D
No water table
132.9
1,050
1,183
295.8
No change: the α method works on cu alone
Water table at 8 m
132.9
1,050
1,183
295.8
No change, for the same reason
Factor of safety on each part
132.9
1,050
1,183
558.3
FSp = 4 and FSs = 2
Pile 14 m long
132.9
654.8
787.6
196.9
Less shaft; the point bearing stays
Width 0.6 m
229.0
1,379
1,608
401.9
Larger tip and perimeter
Strength of the tip layer 60 kPa
88.58
925.2
1,014
253.5
Lower cu at the tip and in Clay-3 lowers both parts
Strength of the first layer 50 kPa
132.9
1,103
1,236
308.9
Only the shaft of Clay-1 changes
Two cases are not in the table because the module does not run them: the λ method for a bored pile, and a pile longer than the layers. It stops and says why (see Messages).
Messages: stops, warnings and notes
The module says in plain words when something prevents the calculation, when a result needs care, and what it assumed.
Kind
When
Message and what to do
Stop (brown bar, “Cannot analyse”)
The layers do not reach the pile tip
“the layers must reach the pile tip. They reach 22.00 m, the pile is 24.00 m long.” Add a layer, or thicken the last one.
Stop
The λ method for a bored pile
“The λ method was read from piles DRIVEN in clay; a bored pile is outside it.” Use the α method or the β method.
Stop
Vesic where cu at the tip is below the start of the correlation
“Vesic’s rigidity index Irr = 347(cu/pa) − 33 has no value at cu = 5.00 kPa; the correlation starts at about 9.80 kPa.” Use Meyerhof for the point.
Stop (input error)
A value below the smallest the field accepts
“1 input error: Undrained shear strength (cu,1) must be at least 1 kPa.” Correct the value; for a layer value the row in the table is outlined in red.
Warning
The λ method for a jacked pile
“The λ method is for driven piles; this one is jacked.” Read the result as an estimate, or use the α method.
Warning
An applied load larger than Qall
“The pile does not carry the applied load — 135.2% of Qall.” Lengthen or widen the pile, use more piles, or revisit the factors of safety.
Warning
A bored pile so short that the drilled-shaft exclusions leave no shaft
“The drilled-shaft exclusions leave no shaft to carry friction” (for a bored pile of about L ≤ 1.5 m + D).
Note
A bored pile with the α method
The adhesion factors come from the drilled-shaft correlation (α = 0.55 up to cu/pa = 1.5, falling to 0.45 at 2.5), and no shaft resistance is counted over the top 1.5 m or the bottom diameter.
Note
A bored pile
The base is taken as clean and the clay beneath it undisturbed; no reduction is applied for the bored construction.
Note
The β method for a driven or jacked pile
Computed with the at-rest K = 1 − sin φ′R; a driven pile pushes the clay sideways, so the true K is higher and the result is on the safe side.
Note
The β method for a bored pile
The correlation is mainly published for driven piles; for a bored pile the installation and the interface are to be assessed separately.
Note
The tip exactly on a layer boundary, or a profile ending just below the tip
The module says which layer carries the base, and that the deepest layer is taken to continue below the described profile.
The Warnings (or Notes) button on the green bar lists them after a run. They are reflected in the report.
Good practice and common mistakes
Check that cu is an undrained strength appropriate to the loading, from tests or a reliable correlation. All the α and λ work rests on it.
Match the installation to the pile. It decides whether the λ method is open and which α is filled in.
Do not expect the water table to move the α method. It acts on the λ and β methods through the effective stress.
Enter φ′R and OCR yourself for the β method. They describe your clay, so you enter them: the boxes open at 30° and 1.
Look at the layer table. In a profile that changes strength with depth the lower layers usually carry most of the shaft, and the table shows which.
Run more than one method and keep each run with Quick Save….
Read the notes. They list what the module assumed.
Frequently asked questions
How do I calculate the capacity of a pile in clay?
The ultimate capacity is the point bearing plus the skin friction, Qu = Qp + Qs, and the allowable capacity is Qu divided by a factor of safety. In GeoTerva’s Pile in clay module you describe the clay layers, the pile and the methods, press Analyze, and read Qp, Qs, Qu and Qall with every step of the working.
What is the α method?
The α method gives the unit skin friction of a clay layer as a share of its undrained shear strength, f = αcu. In the module each layer has its own α, filled in from the correlation for the installation, and the shaft is summed over the layers. For a driven pile the example’s layers get 0.87, 0.74 and 0.51; for a bored pile the module uses 0.55 up to cu/pa = 1.5.
What is the λ method?
The λ method works on the whole embedded length at once: Qs = pLλ(σ̄′ + 2c̄u), with the perimeter p, the mean effective stress and the mean undrained strength over the length and λ read from Vijayvergiya & Focht’s correlation at the pile’s length. It applies to driven piles; the module refuses it for a bored pile.
What is the β method?
The β method works on effective stress: f = βσ′ with β = (1 − sin φ′R) tan φ′R √OCR, layer by layer. It needs the remoulded drained friction angle and the OCR of each layer, which you enter. It is the long-term view of the shaft.
Which point-bearing method should I use in clay?
Meyerhof, qp = 9cu, is the default and needs only cu at the tip. Vesic adds the stiffness of the clay through the rigidity index and gave a 27 % larger Qp on the example (N*c = 11.4 against 9). For a bored pile the module also offers Reese & O’Neill’s drilled-shaft expression.
Why does the water table not change my α-method result?
The α method works on the undrained strength cu alone, so the effective stress and the water table do not enter. They do enter the λ and β methods.
Can I use the λ method for a bored pile?
No. It was read from piles driven in clay, so the module stops and asks you to use the α or β method. For a jacked pile it allows the method with a caution.
What does the module do for a bored pile in clay?
It fills in the drilled-shaft α (0.55 up to cu/pa = 1.5, falling to 0.45 at 2.5), with the α method counts no shaft resistance over the top 1.5 m or the bottom diameter, offers Reese & O’Neill for the base, and notes that the base is taken as clean and the clay beneath it undisturbed.
Does the module deduct the weight of the pile?
No. The capacity is the resistance of the ground; the module states under its results that the weight of the pile is not deducted.
How many clay layers can I enter?
Up to eight. Each has its own thickness, cu, unit weights and OCR; the layers must reach at least the pile tip.
Calculate the ultimate and allowable axial capacity of a single pile through any stack of sand and clay layers. Each layer is typed on its own and worked by the method for its soil; the point follows the soil the tip stands in. This manual walks through the screen, every input, how the layers combine, the results and a worked example you can reproduce.
Pile on layered soil gives the ultimate axial capacityQu and the allowable axial capacityQall of one pile that passes through an arbitrary stack of sand and clay layers. Each layer is typed on its own, and the module works each one by the method for its soil:
Qu = Qp + Qs, Qall = Qu / FS
The point bearingQp follows the soil the tip stands in: a sand tip by Meyerhof, Vesic or Coyle & Castello, a clay tip by Meyerhof (qp = 9cu) or Vesic, and for a bored pile Reese & O’Neill’s expression.
The skin frictionQs is worked out layer by layer: a sand layer by the K–δ method or Coyle & Castello, a clay layer by the α, λ or β method, each layer with its own coefficients.
Up to eight layers, a water table, a circular or square pile, driven, bored or jacked, one factor of safety on the total or one on each part.
Use it whenever the profile is not a single soil: sand over clay, clay over sand, a soft layer in dense sand. It also handles a profile that is all sand or all clay, with the same inputs. The one-soil modules, Pile in sand and Pile in clay, are the shorter forms.
What belongs to other modules
This module treats one pile under an axial compression load, and the results are the resistance of the ground: the weight of the pile is not deducted, as the module says under its results.
The profile: layer thicknesses and the type of each layer, sand or clay
Boring logs
The layers must reach at least the pile tip; the module stops if they do not.
φ′ for each sand layer, cu for each clay layer
SPT or CPT correlations, tests
A sand layer is asked for φ′ and a clay layer for cu, nothing else about strength.
Unit weights γ and γsat
Laboratory or correlations
They give the effective stress through the layers; below the water table γsat − γw is used.
Water table depth
Piezometers, borings
Optional.
Pile: shape, width or diameter, embedded length, installation
Your design
Installation decides which methods are open and what the module fills in.
For a clay layer by the β method: φ′R and OCR
Drained tests on remoulded clay; the stress history
Entered, not correlated: the boxes open at 30° and OCR 1.
Factor of safety; applied working load
Your design basis
The module’s hint for the factor is 2.5 to 4. The applied load is optional.
Quick start in eight steps
Open the module and choose the units
Open Pile on layered soil from the Piles family and pick SI · kN, SI · tf or US. Open Pile on layered soil.
Describe the layers, one at a time
In Soil layers, choose Sand or Clay for the layer, fill in its thickness, γ, γsat and φ′ (sand) or cu (clay), and press + Add layer. The module already holds the example’s three layers: click a row to change it, the cross to remove it, then work from the ground surface down; the layers must reach the pile tip.
Set the groundwater
In Groundwater, choose Present and give the depth, or None.
Describe the pile
In Pile, choose the shape and the installation (Driven, Bored or Jacked), then enter D and L.
Choose the method for the point
In Point bearing, pick the method for the soil the tip stands in. The buttons that do not apply to that soil are not offered.
Choose the method for each layer’s shaft
In Skin friction, every layer has its own method and its own coefficients: K–δ method or Coyle & Castello for sand; α method, λ — extended or β method for clay.
Set the design values
In Design, enter the factor of safety and, if you want a check, the applied working load.
Press Analyze and read the answer
Press Analyze. The Results tab gives Qu, Qp, Qs and Qall and the contribution of every layer; Calculations shows every step; Report saves a PDF or Word file.
The screen: three views of the pile
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing has three views:
Section shows the pile through the sand and clay layers (each type in its own colour) with the water table and the resistances Qp, Qs and Qu; the skin-friction arrows follow the unit skin friction. Before you analyse you can drag the water table, a layer base, the pile tip or the side of the pile.
Profile plots the effective vertical stress σ′ and the unit skin friction f against depth, layer by layer.
3D shows the layers and the pile in a block of the ground.
The Section view after Analyze: sand over clay over sand, the water table and the resistances.
The Profile view: the effective stress and the unit skin friction through the layers.
Entering the model, section by section
The Model tab has six sections after Model data. Each opens when you click its title; Next and Back walk you through them, and a (?) opens a help window with a drawing and the formula. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. A field marked “filled in” carries a value the module works out; you can type over it.
Soil layers
Layers are added one at a time from the ground surface down, each described as a sand or as a clay. A sand layer is asked for φ′ and a clay layer for cu; the box that does not apply is not shown. The values in the table are those of layer 1.
Soil layers: the type of each layer, its values, and the table of layers.
Fields of each layer
Field
Symbol
Unit
Default
What to enter
Name (optional)
—
—
empty
A name for the layer, for the drawing and the report. Left empty, the layers are called Layer-1, Layer-2 and so on.
Thickness
H
m · ft
6
Thickness of the layer. The layers together must reach at least the pile tip; the smallest value accepted is 0.1 m.
Unit weight (above the water table)
γ
kN/m³ · t/m³ · pcf
17
Unit weight above the water table.
Saturated unit weight
γsat
kN/m³ · t/m³ · pcf
18
Unit weight below the water table; below it the module uses γsat − γw.
Friction angle
φ′
°
32
Effective friction angle of a sand layer, at least 20°. Meyerhof’s chart is drawn from 20° to 45°; above 45° the end value is held at the tip layer and a warning says so. Shown when the layer is Sand.
Undrained shear strength
cu
kPa · kg/cm² · psf
40
Undrained shear strength of a clay layer, at least 1 kPa. Shown when the layer is Clay.
Groundwater
Choose whether there is a water table within the pile length and give its depth. Below it the module uses the submerged unit weight γsat − γw for the effective stress.
Groundwater: Present or None, then the depth.
Choice
Options
Default
Water table within or below the pile
PresentNone
Present
Groundwater field
Field
Symbol
Unit
Default
What to enter
Water table depth
Dw
m · ft
3
Depth of the water table below the ground surface, 0 or more. Shown when the water table is present.
Pile
The shape and the installation are choices; the width and the length are numbers.
Pile: shape, installation, width and length.
Choice
Options
Default
Shape of the cross section
CircularSquare
Circular
Installation how the pile is put in the ground
DrivenBoredJacked
Driven
Pile fields
Field
Symbol
Unit
Default
What to enter
Pile width / diameter
D
m · ft
0.4
Width of a square pile or diameter of a circular one.
Embedded length
L
m · ft
14
Embedded length from the ground surface to the tip; the head is at the ground surface.
Point bearing
The tip is worked out by the soil it ends in. Sand tip: Meyerhof, Vesic (it asks for m) or Coyle & Castello (it fills in N*q). Clay tip: Meyerhof gives qp = 9cu and Vesic builds N*c from the rigidity index; Coyle & Castello is a sand correlation, so that button is not offered while the tip is in clay. For a bored pile with a clay tip Reese & O’Neill’s drilled-shaft expression is offered; it averages cu over the 2D below the base, so that depth has to be clay.
Point bearing: the method for the soil at the tip.
Choice
Options
Default
Method for the point (tip) bearing Qp
MeyerhofVesicCoyle & CastelloReese & O’Neill
Meyerhof
Point-bearing fields
Field
Symbol
Unit
Default
What to enter
Stiffness ratio of the sand at the tip
m
—
250
The stiffness ratio Es/pa of the sand at the tip: about 100–200 loose, 200–500 medium dense, 500–1000 dense. Used only when the tip layer is sand and Vesic is selected. Shown when point-bearing method is Vesic.
Coyle & Castello N*q (chart value)
N*q(CC)
—
filled in
Read from the Coyle & Castello chart for the pile’s L/D and φ′ of the tip layer, and kept up to date; type over it or empty the box to get the chart value back. Shown when point-bearing method is Coyle & Castello.
Skin friction
Every layer chooses its own method and carries its own numbers, named after the layer they belong to. A sand layer: the K–δ method (its own K and δ′, with σ′ held constant below L′ = 15D of the whole pile) or Coyle & Castello (its own chart-read K, δ′ = 0.8φ′, the mean σ′ of that layer). A clay layer: the α method (its own adhesion factor on cu), the λ method (λ read at the pile’s embedded length and applied to that layer’s own mean σ′ and cu) or the β method (its own φ′R and OCR).
Skin friction: a method and coefficients for each layer.
The method choice of each layer
Layer is
Methods offered
Default
Sand
K–δ methodCoyle & Castello
K–δ method
Clay
α methodλ — extendedβ method
α method
Shaft coefficients of each layer (layer 1 shown)
Field
Symbol
Unit
Default
What to enter
Earth pressure coefficient K
K
—
filled in
Earth-pressure coefficient on the shaft in this sand layer. Filled in as K0 = 1 − sin φ′ for a bored pile and 1.4K0 for a driven or jacked one; editable. Shown when the layer is Sand and method is K–δ method.
Pile–soil friction angle δ′
δ′
°
filled in
Friction angle between the pile and this sand layer; filled in as 0.75φ′. Shown when the layer is Sand and method is K–δ method.
Coyle & Castello K
K(CC)
—
filled in
Coyle & Castello’s coefficient for this sand layer, read from their chart at the layer’s φ′ and the pile’s L/D; the interface friction is 0.8φ′. Shown when the layer is Sand and method is Coyle & Castello.
Adhesion factor α
α
—
filled in
Adhesion factor of this clay layer. Filled in from cu/pa by the rule that belongs to the installation (a driven-pile curve, or 0.55 falling to 0.45 for a bored pile); editable. Shown when the layer is Clay and method is α method.
Remoulded drained friction angle
φ′R
°
30
Remoulded drained friction angle of this clay layer, for the β method; the box opens at 30°. Shown when the layer is Clay and method is β method.
Overconsolidation ratio
OCR
—
1
Overconsolidation ratio of this clay layer, for the β method; the box opens at 1. Shown when the layer is Clay and method is β method.
Design
One factor of safety on the sum, or a separate factor on each part: Qall = Qp/FSp + Qs/FSs. An applied working load is optional.
Design: factor of safety and the applied load.
Choice
Options
Default
Factor of safety on the total, or on each part
One on the totalOne on each part
One on the total
Design fields
Field
Symbol
Unit
Default
What to enter
Applied working load (optional)
Qapplied
kN · tf · kip
empty
The working load the pile has to carry. Optional: if it is entered it is drawn on the pile and checked against Qall.
Factor of safety (2.5–4 typical)
FS
—
3
One factor of safety on Qu; the module’s hint is 2.5 to 4. Shown when factor of safety is One on the total.
Factor of safety on the point resistance
FSp
—
3
Factor of safety on the point resistance (one factor on each part). Shown when factor of safety is One on each part.
Factor of safety on the skin friction
FSs
—
1.5
Factor of safety on the skin friction. Shown when factor of safety is One on each part.
How the layers combine
The methods themselves are those of the one-soil modules, explained in the manuals of Pile in sand and Pile in clay. What this module adds is the rule for combining them:
Part
Rule
Consequence
Point bearing
Worked by the soil the tip stands in, whatever lies above it.
A strong sand above a soft clay tip gives a clay Qp: the sand does not help the tip. With the example’s tip in sand, Qp = 318 kN; make the tip layer clay and it is 45 kN.
Tip on a boundary
A tip exactly on a layer boundary bears on the layer below it.
The module says which layer it took; the layer above contributes shaft friction only.
Effective stress
Continuous through all the layers, with the unit weights of each and the water table.
σ′ at any depth includes everything above it, so a deep layer of any type has a high σ′.
K–δ critical depth
L′ = 15D of the whole pile, from the ground surface.
In a K–δ sand layer, σ′ is held at its value at L′ at every depth below L′.
Shaft
Summed over the layers the pile passes through, each by its own method.
The Results tab lists each layer’s skin friction and its share.
Which should I pick?
Sand layers: the K–δ method is the default; use Coyle & Castello when you want the load-test charts and the pile is driven.
Clay layers: the α method is the default and works on cu alone. Use β when you hold effective-stress parameters. Treat the λ extension as an estimate.
The point: Meyerhof for either soil is the default; Vesic adds stiffness (m for sand, the rigidity index for clay).
Compare by changing one method at a time and keeping each run with Quick Save…; the what-if table shows the spread on the example.
Analyze and read the results
Press Analyze. The bar reports the run, the inputs lock, and the Results tab shows four blocks.
Axial capacity of the pile. Cards for Qu, Qp, Qs and Qall.
Values, step by step. The point bearing (qp, Qp), the skin friction of the layers (Qs) and the factor of safety.
What each layer gives to the capacity. Every layer with its type and depth range, its skin friction and, for the tip layer, the point bearing.
Contribution of each mechanism. The share of the point and of the shaft in Qu.
The Results tab of the example.
Checking an applied load
If you enter an applied working load, the Results tab adds the ratio Qapplied/Qall and a verdict in words: PASS when the load is within the allowable capacity, FAIL when it exceeds it, with a warning that advises lengthening or widening the pile, using more piles, or revisiting the factors of safety.
The Calculations tab and the report
The Calculations tab starts with Given and what is sought, then:
Compute the point (tip) bearing capacity — governed by the layer at the pile toe. It says which soil the tip is in and the method that follows from it.
Compute the skin-friction resistance of each layer. Every layer by its own method, with its own numbers.
Apply the factor of safety — Qu and Qall.
Check the applied working load, when one was entered.
The Calculations tab: the given data, then the steps.
The Report tab fills in a cover and saves the same working as a PDF or a Word file (1 Introduction and scope, 2 Input data, 3 Calculation, 4 Results, then 5 Assumptions and notes when the run has any, and the Summary and References), with the units line. The Theory tab explains a pile through several layers, the effective stress through them, the rules over a sand and over a clay layer, the bored-pile procedure and a worked example.
Worked example: the module’s own example
The module opens on this example; Reset to defaults brings it back.
Inputs of the example
Layer
Soil
Thickness H
Strength
γ / γsat
Shaft method
Layer-1
Sand
6 m
φ′ = 32°
17 / 18 kN/m³
K–δ
Layer-2
Clay
6 m
cu = 40 kPa
17 / 18 kN/m³
α
Layer-3
Sand
4 m
φ′ = 32°
17 / 18 kN/m³
K–δ
Water table at 3 m; circular driven pile, D = 0.4 m, L = 14 m (the tip is in Layer-3, 2 m below the clay); Meyerhof for the point; one factor of safety FS = 3. The module fills in K = 0.66 and δ′ = 24° for the sand layers and α = 0.74 for the clay.
Point bearing
The tip is in sand, so the point follows Meyerhof’s sand method with φ′ = 32° of Layer-3. N*q = 81.00; the effective stress at the tip is 17 × 3 + (18 − 9.81) × 11 = 141.09 kPa, so σ′N*q = 11,428 kPa, far above the limit ql = 0.5 × 100 × 81 × tan 32° = 2,530.72 kPa, which governs. With Ap = π × 0.4²/4 = 0.1257 m², Qp = 318.02 kN.
Skin friction, layer by layer
The perimeter is π × 0.4 = 1.2566 m, and the critical depth is L′ = 15D = 6 m of the whole pile.
Layer
Depth (m)
How it is worked
Qs (kN)
Layer-1, sand
0–6
f = 0.66 tan 24° σ′ over the whole 6 m (σ′ = 51 kPa at the water table, 75.57 kPa at 6 m): 1.2566 × 0.66 tan 24° × 266.36
98.36
Layer-2, clay
6–12
αcu × perimeter × length = 0.74 × 40 × 1.2566 × 6
223.18
Layer-3, sand
12–14
σ′ is held at its value at L′ = 6 m: f = 0.66 tan 24° × 75.57 = 22.21 kPa; × 1.2566 × 2 m
55.81
Total Qs
377.3
Capacity
Qu = 318.02 + 377.34 = 695.36 kN; with FS = 3, Qall = 231.79 kN. The point carries 45.7 % and the shaft 54.3 %.
How each input changes the answer
Each row changes one thing in the example and runs the module again. The numbers are the results the module gives, in kN.
The example with one change at a time (kN)
Change
Qp
Qs
Qu
Qall
Why
The default example
318.0
377.3
695.4
231.8
The reference
Point bearing by Vesic (sand tip)
632.8
377.3
1,010
336.7
Stiffness enters through m = 250
Point bearing by Coyle & Castello (sand tip)
585.1
377.3
962.4
320.8
N*q from the chart, no limit
Sand layer 1 by Coyle & Castello
318.0
349.6
667.6
222.5
Only Layer-1 changes
Clay layer 2 by the λ method
318.0
438.0
756.1
252.0
A caution: the layer-by-layer λ is an extension
Clay layer 2 by the β method
318.0
372.1
690.1
230.0
At-rest K; a note says it is on the safe side
Bored pile
318.0
275.7
593.7
197.9
K = K0 = 0.47 in the sand, α = 0.55 in the clay; the top 1.5 m and bottom 1D are left out only of clay layers worked by α, and here nothing is cut
Jacked pile
318.0
377.3
695.4
231.8
Same as driven here
Layer-3 made clay (tip in clay)
45.24
395.9
441.2
147.1
qp = 9cu = 360 kPa: the tip is in clay
No water table
318.0
411.5
729.5
243.2
Higher σ′ everywhere
Pile 10 m long (tip in the clay)
45.24
247.1
292.4
97.46
The point is clay: 9cu
Pile 16 m long (tip at the base of the profile)
318.0
433.2
751.2
250.4
A note: the deepest layer is taken to continue below the tip
Diameter 0.6 m
715.5
593.2
1,309
436.3
Larger tip and perimeter, and L′ = 15D = 9 m
Square section
404.9
480.5
885.4
295.1
Larger area and perimeter
Factor of safety on each part
318.0
377.3
695.4
357.6
FSp = 3 and FSs = 1.5
Friction angle of Layer-3 = 36°
766.9
377.7
1,145
381.5
N*q at the tip rises
Clay layer 2: cu = 60 kPa
318.0
434.6
752.7
250.9
Only the clay shaft changes
Cases that the module does not run, such as Coyle & Castello in a layer of a bored pile, or a pile longer than the layers, stop with a message (see Messages).
Messages: stops, warnings and notes
The module says in plain words when something prevents the calculation, when a result needs care, and what it assumed.
Kind
When
Message and what to do
Stop (brown bar, “Cannot analyse”)
The layers do not reach the pile tip
“The layers must reach the pile tip. They reach 16.00 m, the pile is 17.00 m long.” Add a layer, or thicken the last one.
Stop
Coyle & Castello for a bored pile, at the point or in a sand layer
“Change the skin-friction method of Layer-1. Coyle & Castello is calibrated on piles DRIVEN in sand; a bored pile is outside it.” Use the K–δ method in sand.
Stop
The λ method in a clay layer of a bored pile
The λ method was read from driven piles. Use the α or β method.
Stop
Vesic at a clay tip where cu is below the start of the correlation
Use Meyerhof for the point (qp = 9cu).
Stop
Reese & O’Neill (bored pile) when the 2D below the base reaches into a sand layer
“…so that depth has to be clay: here it reaches into Layer-3, which has no undrained strength.” Use Meyerhof or Vesic for the point, or take the pile deeper into the clay.
Stop (input error)
A value below the smallest the field accepts
For example φ′ below 20° in a sand layer. Correct the value; the row of the layer is outlined in red.
Warning
Coyle & Castello, or the λ method, for a jacked pile
Read the result as an estimate, or use the K–δ or the α method.
Warning
The λ method in a clay layer
“The λ method is applied layer by layer here, which is an extension of it.” Read the λ layers as an estimate, or use α or β.
Warning
An applied load larger than Qall
“The pile does not carry the applied load — 129.4% of Qall.” Lengthen or widen the pile, use more piles, or revisit the factors of safety.
Warning
K too high for a bored pile (K above 1.25K0 of the layer)
Boring displaces nothing, so K stays near K0 = 1 − sin φ′. Use K ≈ K0, or say why this shaft carries more.
Warning
φ′ above 45° in a sand layer
Meyerhof’s chart is drawn from 20° to 45° only; beyond it the end value is held.
Note
K well above 1.8K0 for a driven or jacked pile
The range of K for displacement piles is a guide; keep the value if a load test or local practice supports it.
Note
The tip exactly on a layer boundary
The point bearing is worked from the layer below the boundary; the layer above gives shaft friction only.
Note
The profile ends less than 2D below the tip
The deepest layer is taken to continue below, with the same properties.
Note
A bored pile
The drilled-shaft α is used in clay (no friction over the top 1.5 m and the bottom 1D); the friction angle at a sand tip is taken to describe the sand after drilling, and a clay base is taken as clean and undisturbed; no reduction is applied for the bored construction.
Note
The β method for a driven or jacked pile
Computed with the at-rest K = 1 − sin φ′R, which is on the safe side.
Note
Coyle & Castello beyond their charts
The coefficients are projected beyond L/D = 70 (N*q) or 36.1 (K); the boxes stay editable.
The Warnings (or Notes) button on the green bar lists them after a run. Most are printed in the report in an Assumptions and notes section; the K cautions and the applied-load warning stay on screen.
Good practice and common mistakes
Type each layer correctly. Sand takes φ′, clay takes cu; a layer typed wrongly changes both the shaft and, at the tip, the point bearing.
Look at the tip layer. The point bearing answers only to it. The point is worked from the layer the tip stands in (with Reese & O’Neill, from the mean cu over the 2D below the base); a softer layer beneath the tip does not change it.
Match the installation to the pile. It decides which methods are open and what K and α are filled in with.
Read the layer table. It shows which layers carry the shaft; in the example the clay carries most of it.
Remember the whole-pile critical depth. In the K–δ method L′ = 15D counts from the ground surface, so a deep sand layer gets the value at L′.
Run more than one method and keep each run with Quick Save….
Read the notes. They list what the module assumed.
Frequently asked questions
How do I calculate the capacity of a pile in layered soil?
The ultimate capacity is the point bearing plus the skin friction, Qu = Qp + Qs. With layered soil the skin friction is summed over the layers, each by the method for its soil, and the point bearing follows the soil the tip stands in. In GeoTerva’s Pile on layered soil module you type each layer as sand or clay, choose a method for each, press Analyze, and read Qp, Qs, Qu, Qall and the contribution of every layer.
Which soil governs the point bearing?
The soil the tip stands in. A strong layer above a soft tip does not help the tip. If the tip is exactly on a boundary, it bears on the layer below it, and the module says so.
Can I mix methods in one pile?
Yes. Every layer chooses its own: K–δ or Coyle & Castello in a sand layer, α, λ or β in a clay layer, and the point has its own method for the soil at the tip.
What does the K–δ critical depth do in a layered profile?
The critical depth L′ = 15D counts from the ground surface for the whole pile. Sand layers deeper than L′ get the unit friction at L′, held constant. In the example the deep sand layer gets 22.2 kPa, not the larger value its own effective stress would give. A Coyle & Castello layer has no such cut-off.
Why is the λ method called “extended” here?
The λ method was written as one average over the whole embedded length. The module applies it layer by layer, reading λ at the pile’s length and using each clay layer’s own mean stress and strength. It warns you and suggests the α or β method for a layered profile.
Can I use Coyle & Castello for a bored pile in layered soil?
No. It was read from driven piles, so the module stops and asks you to use the K–δ method in sand and Meyerhof or Vesic at the point. A jacked pile gets a caution.
How many layers can I enter?
Up to eight, each typed as sand or clay. They must reach at least the pile tip.
What does the water table change?
Below it the module uses γsat − γw for the effective stress, which lowers the shaft friction in the K–δ, Coyle & Castello, λ and β methods. The α method works on cu and does not change. In the example removing the water table raises Qs from 377 kN to 412 kN.
Is there a free version I can try?
The module opens on a complete example, and a limited version also opens on the Try it page without an account.
Work out how far a single pile deflects under a sideways load, and the bending moment, shear force and soil reaction along it, in ground of up to eight sand and clay layers. This manual walks through the screen, every input, the p–y curves, the results and the report, with a worked example you can reproduce.
Pile under lateral load works out how far one pile deflects when it is pushed sideways at its head, and the bending moment, shear force and soil reaction along its whole length. The ground can be up to eight layers of sand and clay.
The pile is a beam. The ground is a row of independent springs along it, and each spring follows a p–y curve: the reaction p of the ground per unit length of pile, against the deflection y of the pile at that depth. The curve flattens as the soil yields, so the ground gets softer as the load grows. The module solves the beam equation on these springs, at the load you enter:
E Iy″″ + Py″ + p(y, z) = 0
E I is the bending stiffness of the pile, P the axial load, which acts through the deflected shape (the P–δ effect), z the depth below the ground surface, and y the deflection, positive in the direction of the load.
What you can describe
A ground of up to eight layers, each a sand (described by φ′) or a clay (described by cu), with its own thickness and unit weights, and an optional water table.
A pile that is circular, hollow circular (a pipe), square, or of any other section whose second moment of area I you give: its modulus Ep, a factor on E I for a cracked section, its embedded length L, and a free lengthe above the ground where the load acts.
A lateral loadH at the head, a momentM at a free head and an axial loadP; the head free to rotate or held against rotation; static or cyclic loading.
The response of the ground by non-linear p–y curves (the API curve for sand and Matlock’s curve for clay) or by an elastic subgrade reaction.
An optional allowable head deflection and the moment capacity of the section. The module then gives the lateral load at which each is reached.
What you get
The deflection, rotation, bending moment, shear force and soil reaction at every depth, as diagrams and as a table, with the largest values and the depths they occur at.
The p–y curves of the ground at five depths, with the point each depth has reached under the load.
The load–deflection curve of the head, and the load at an allowable deflection or at the moment capacity.
The working step by step, and a summary report as PDF or Word.
What belongs to other modules
This module treats one pile. The load on every pile of a cap, the group effect and the lateral response of its rows belong to Pile group.
The axial capacity of the pile belongs to Pile in sand, Pile in clay and Pile on layered soil. The axial load P is the compression you enter at the head, taken as the same at every depth; it acts through the deflected shape.
The pile is a linear-elastic beam of one section. The strength of that section belongs to the structural design of the pile: you give its moment capacity Mcap and the module compares the largest moment with it, and you represent a cracked concrete section by the factor on E I.
The loads and the limits are yours; the module gives the response to exactly those, and the loads at which your allowable deflection and your moment capacity are reached.
Before you start: what you need
Have these ready from your ground investigation and your design:
What
Where it comes from
Notes
The ground profile: the thickness of each layer and whether it is a sand or a clay
Boring logs
Up to eight layers from the ground surface down. The layers must reach at least the pile tip. A laterally loaded pile moves mainly in the upper part of its length (in the example, the top 8 m or so), so describe that part of the profile with the most care.
Unit weights γ and γsat
Laboratory tests or correlations
γ applies above the water table, γsat below it. They give the effective stress, which sets the ultimate resistance of the ground.
Friction angle φ′ of each sand layer
SPT or CPT correlations, tests
An effective friction angle, from 15° to 50°. It gives the ultimate resistance and, through the API chart, the initial modulus k of the p–y curve.
Undrained shear strength cu of each clay layer
Tests or correlations
At least 1 kPa. It gives the ultimate resistance; the strain ε50 and Matlock’s J are filled in from it, and you can type in measured values.
Water table depth
Piezometers, borings
Optional. Below it the submerged unit weight is used, and the initial modulus of a sand layer is taken for submerged sand.
The pile: section, width or diameter D, modulus Ep, embedded length L
Your design
D is the width the soil reaction acts over. For a section that is not circular, hollow circular or square you also need the second moment of area I from the table of the section.
Cracking of a concrete pile
Structural design
A factor on E I. Often 0.5 to 0.8 for a section that cracks; 1 leaves E I as it is.
The loading at the head
Your design
The lateral load H, the moment M (free head), the axial load P, the height e of the load above the ground, whether a cap holds the head against rotation, and whether the load is sustained or repeated.
Allowable head deflection and moment capacity
Your design criteria
Optional: the limits you want the module to compare the analysis with.
Quick start in nine steps
Open the module and choose the units
Open Pile under lateral load from the Piles family. Pick SI · kN, SI · tf or US in the header; you can change it at any time. Open Pile under lateral load.
Describe the soil layers
In Soil layers, fill in the New layer form: an optional name, Sand or Clay, the thickness, γ and γsat, and φ′ for a sand or cu for a clay. Press + Add layer and repeat from the ground surface down. The layers must reach the pile tip.
Set the groundwater
In Groundwater, choose Present and give the depth Dw of the water table, or choose None.
Describe the pile
In Pile, choose the section (Circular, solid, Circular, hollow (pipe), Square or Other (I given)), then enter the width or diameter D, the modulus Ep, the factor on E I, the embedded length L and the free length e.
Describe the loading
In Loading, choose Free to rotate or Held against rotation and Static (sustained) or Cyclic (repeated), then enter the lateral load H, the moment M (free head only) and the axial load P.
Choose the response of the ground
In Soil response, choose Non-linear p–y curves or Elastic subgrade reaction. For the p–y curves each layer shows constants that fill themselves in: k for a sand, ε50 and J for a clay. Type over them only when you have better values.
Add the limits you design to (optional)
In Design, enter the allowable deflection yall of the head and the moment capacity Mcap of the section. The module then checks the analysis against them and gives the lateral load at which each is reached.
Press Analyze
Press Analyze. The Results tab gives the head deflection, the largest moment and where it is; the drawing now carries the deflected shape of the pile.
Read the diagrams and take the report
Open the Diagrams, p–y curves and Load–deflection views; the Calculations tab shows every step; the Report tab saves a PDF or a Word file.
The screen: four views of the pile
The layout is the one every GeoTerva module shares, described in the workspace guide. For this module the drawing has four views, chosen with the buttons above it. Section follows your inputs; the other three show results, and ask you to press Analyze until there are results.
12345
The module after Analyze: the Section view on the left, the Results tab on the right.
The four views. Section, Diagrams, p–y curves and Load–deflection.
The pile. In its layers, with the deflected shape drawn over it, magnified; the foot of the drawing states the factor.
The results on the drawing. Head deflection, largest moment and the movement of the tip.
The cards. Head deflection, largest moment, its depth, and the rotation at the ground surface.
The steps. The values step by step; below them the table along the pile, depth by depth.
Section
The pile in its layers, with the lateral load H, the free length e, the water table, and the dimensions L and D. A moment and an axial load at the head are drawn too when you enter them. Before Analyze you can drag a layer base (it moves with the layer below it; the base of the last layer has no handle), the water table (Dw), the pile tip (L) or the side of the pile (it changes D by twice the pull, in steps of 1 cm); the input boxes follow, and the handles disappear after Analyze.
After Analyze the deflected shape is drawn over the pile, magnified so that it can be seen (the foot of the drawing says by how much). The largest moment is marked at its depth, the depth of zero deflection is drawn as a line, and a small panel gives the head deflection, the largest moment and the movement of the tip.
Section after Analyze: the deflected shape magnified ×300, with Mmax at 2.13 m and zero deflection at 3.85 m.
Here is the same view before Analyze, with a free length, a moment and an axial load entered. The arrows at the head are the lateral load H (at the top of the free length), the moment M and the axial load P.
Section before Analyze: e = 1.5 m above the ground, M = 30 kN·m and P = 800 kN at the head.
Diagrams
The deflection y, the bending moment M, the shear force V and the soil reaction p along the pile, side by side on a common depth scale, with the coloured strip of the layers beside the depth axis. The largest value of each is marked. Horizontal dashed lines show the layer bases (brown) and the water table (blue).
Two switches above the drawing: Rotation adds the rotation θ (off at first); Limit of the p–y curve (on) draws, in grey on the soil-reaction diagram, the end of the p–y curve at each depth, so you can see where the ground is near its limit.
Diagrams: y, M, V and p along the pile of the example.
p–y curves
The p–y curve of the ground at five depths, spread over the length the pile moves: the soil reaction p per unit length against the deflection y. A dot on each curve marks the point the pile has reached at that depth under the load; the legend gives the depth, the layer, and the deflection and reaction there.
A dot on the straight beginning of a curve is soil still working elastically; a dot near the flat end is soil at its limit. Only the sand curve has a straight beginning; the clay curve rises as the cube root of the deflection. The foot of the drawing says whether the curves are static or cyclic (with the elastic response it says the curves are straight lines, and the choice of static or cyclic has no effect).
The p–y curves: the deeper the curve, the stiffer and the higher it is.
Load–deflection
Two curves against the lateral load H: the head deflection and the largest bending moment. The lateral load and the moment at the head are raised in proportion and the equations are solved again at each load; the applied load is marked on both curves. With Allowables on, a limit you entered in Design is drawn with the load at which the curve reaches it.
The curves bend upwards: the ground gets softer as it is loaded, so doubling the load more than doubles the deflection (on the example, 2.54 times).
Load–deflection: the applied load of 120 kN gives 6.23 mm and 152 kN·m.
Entering the model, section by section
The Model tab has six sections after Model data. Each opens when you click its title; Next and Back at the foot walk you through them. A (?) beside a title or a choice opens a help window with a drawing and the formula for that part. The tables below list every field with its symbol, the unit in each unit system, and its default (defaults are shown in SI · kN). Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. A field marked “filled in” carries a value the module works out; you can type over it, and empty the box to get the module’s value back.
Soil layers
Up to eight layers, entered from the ground surface down, each a sand or a clay. The Soil choice of the New layer form decides which strength is asked for: φ′ for a sand, cu for a clay. The ground is the same at every depth along a layer; the resistance at a depth is worked out from the effective stress there, so the unit weights and the water table matter. The values in the table are those of layer 1.
Soil layers: the New layer form above, the table of layers below.
Choice
Options
Default
Soil of the layer
SandClay
Alternating, starting with a sand: layers 1, 3 … sand, layers 2, 4 … clay
Fields of each layer
Field
Symbol
Unit
Default
What to enter
Name (optional)
—
—
empty
A name for the layer, for the drawing and the report. Left empty, the layers are called Layer-1, Layer-2 and so on.
Thickness
H
m · ft
4
Thickness of the layer, at least 0.1 m. The layers together must reach at least the pile tip. The example has 4 m, 5 m and 8 m.
Unit weight (above the water table)
γ
kN/m³ · t/m³ · pcf
18
Unit weight of the soil above the water table. With the water table it gives the effective vertical stress σ′ at each depth, which sets the ultimate resistance.
Saturated unit weight
γsat
kN/m³ · t/m³ · pcf
19
Unit weight below the water table; there the module uses γsat − γw for the effective stress. Shown when water table is Present.
Friction angle
φ′
°
32
Effective friction angle of a sand layer, from 15° to 50°. It gives the ultimate resistance of the sand and, through the API chart, the filled-in initial modulus k. Shown when the layer is Sand.
Undrained shear strength
cu
kPa · kg/cm² · psf
40
Undrained shear strength of a clay layer, at least 1 kPa. It gives the ultimate resistance of the clay and the filled-in ε50 and J. Shown when the layer is Clay.
Groundwater
Choose whether there is a water table and, if there is, its depth. Below it the effective stress is worked out with the submerged unit weight γsat − γw, with γw = 9.81 kN/m³.
Groundwater: Present or None, then the depth.
Choice
Options
Default
Water table within or below the pile
PresentNone
Present
Groundwater field
Field
Symbol
Unit
Default
What to enter
Water table depth
Dw
m · ft
2
Depth of the water table below the ground surface, 0 or more. Below it the submerged unit weight γsat − γw is used. It also decides whether the filled-in k of a sand layer is that of dry or of submerged sand (by where the middle of the layer lies). Shown when water table is Present.
Pile
The section gives the second moment of area I: πD⁴/64 for the solid circle, π(D⁴ − (D − 2t)⁴)/64 for the pipe, D⁴/12 for the square, or the value you enter for Other (I given). With the modulus Ep and the factor fEI it gives the bending stiffness E I = EpIfEI. The pile is taken to be of one section over its whole length.
Pile: the section, then the width, the stiffness and the lengths.
Choice
Options
Default
Section of the pile
Circular, solidCircular, hollow (pipe)SquareOther (I given)
Circular, solid
Pile fields
Field
Symbol
Unit
Default
What to enter
Pile diameter / width
D
m · ft
0.6
Width of the pile, at least 0.05 m: the diameter of a circular pile. The soil reaction acts over it, so it enters the p–y curves for every section; for the circular, pipe and square sections it also gives I.
Wall thickness of the pipe
t
m · ft
0.02
Wall thickness of the pipe. It must be more than 0 and less than D/2; I = π (D⁴ − (D − 2t)⁴) / 64. Shown when section is Circular, hollow (pipe).
Second moment of area about the axis of bending
I
cm⁴ · in⁴
636170
Second moment of area about the axis the load bends the pile about, from the table of the section. D above is still the width the soil reaction acts over. The value shown is that of the 0.6 m circular pile. Shown when section is Other (I given).
Modulus of elasticity of the pile material
Ep
MPa · kg/cm² · ksi
30000
Modulus of elasticity of the pile material: about 30 000 MPa for a concrete pile, 200 000 MPa for steel. The bending stiffness is E I = EpIfEI.
Factor on E I (cracking of a concrete section; 1 = uncracked)
fEI
—
1
A reinforced concrete pile that cracks under the moment is less stiff than its gross section: often 0.5 to 0.8 for a section that cracks. 1 leaves E I as it is. The field accepts 0.05 to 1.5. Run fEI = 1 as well: the cracked section gives the larger deflection, the gross section the larger moment.
Embedded length
L
m · ft
14
Embedded length from the ground surface to the tip, at least 0.5 m. The pile is of one section over its whole length.
Free length above the ground (height of the load)
e
m · ft
0
The pile stands this far above the ground surface where the load is applied; 0 is a load at the ground surface. The pile above the ground has no soil on it.
Loading
The head is free to rotate or held against rotation, and the loading is static or cyclic. The load acts at the top of the pile, e above the ground. A cap that does not let the head rotate is the held head: the head moves sideways but its slope is zero, and the moment that holds it is worked out. The axial load acts through the deflected shape.
Loading: the head, the type of load and the three loads.
Choice
Options
Default
Pile head free to rotate or held against rotation
Free to rotateHeld against rotation
Free to rotate
Loading static or cyclic
Static (sustained)Cyclic (repeated)
Static (sustained)
Loading fields
Field
Symbol
Unit
Default
What to enter
Lateral load at the pile head
H
kN · tf · kip
120
The lateral load at the head: the load the analysis is run for.
Moment applied at the pile head
M
kN·m · tf·m · kip·ft
0
A moment at the head of a pile free to rotate. It is positive in the sense that adds to the deflection the lateral load causes. With a held head there is no moment to enter: the moment that holds the head is worked out. Shown when pile head is Free to rotate.
Axial load on the pile (compression)
P
kN · tf · kip
0
Axial compression on the pile, taken as constant over the whole length. It acts through the deflected shape (P–δ), so it adds to the deflection and the moments.
Soil response
The constants of the curve of every layer, shown for the soil type of that layer. With Non-linear p–y curves a sand layer shows k, a clay layer ε50 and J; with Elastic subgrade reaction a sand layer shows nh and a clay layer kh. The p–y constants fill themselves in from φ′, cu and the water; the values shown below are layer 1 of the example, a sand.
Soil response: one box for each constant of each layer.
Choice
Options
Default
Response of the ground p–y curves or an elastic subgrade reaction
Non-linear p–y curvesElastic subgrade reaction
Non-linear p–y curves
Constants of each layer (layer 1 shown)
Field
Symbol
Unit
Default
What to enter
Initial modulus k of the p–y curve
k
MN/m³ · t/m³ · pci
filled in
The slope of the sand p–y curve at the origin is kz. It is filled in from φ′ and the water, from the chart of the API recommendation, rounded to 0.1 MN/m³ and never below 5.4 MN/m³ (so φ′ of 28° or less gives 5.4 MN/m³ dry or submerged); a layer whose middle lies exactly at Dw takes the dry value. Type over it to use your own value. Shown when the layer is Sand and response of the ground is Non-linear p–y curves.
Strain ε50 at half the strength
ε50
—
filled in
The strain at half the maximum deviator stress in the undrained test; y50 = 2.5 ε50D. Filled in from cu (0.020 soft, 0.010 medium, 0.005 stiff); type in a measured value if you have one. Shown when the layer is Clay and response of the ground is Non-linear p–y curves.
Matlock J
J
—
filled in
The empirical constant of Matlock’s near-surface wedge: 0.5 for a soft clay, 0.25 for a medium to stiff one. Filled in from cu. Shown when the layer is Clay and response of the ground is Non-linear p–y curves.
Constant nh of the subgrade reaction
nh
MN/m³ · t/m³ · pci
8
Elastic response only. In a sand layer the subgrade reaction is k = nhz, growing with depth. Shown when the layer is Sand and response of the ground is Elastic subgrade reaction.
Coefficient kh of horizontal subgrade reaction
kh
MN/m³ · t/m³ · pci
15
Elastic response only. In a clay layer the subgrade reaction is k = khD, constant over the layer. Shown when the layer is Clay and response of the ground is Elastic subgrade reaction.
Design
Both limits are optional and independent. If you enter one, the module compares the analysis with it, writes the verdict on the Results tab, and gives the lateral load at which the pile reaches it.
Design: the two optional limits.
Design fields
Field
Symbol
Unit
Default
What to enter
Allowable deflection of the pile head (optional)
yall
mm · in
empty
Optional. If it is entered the module checks the head deflection against it and gives the lateral load at which it is reached.
Moment capacity of the pile section (optional)
Mcap
kN·m · tf·m · kip·ft
empty
Optional: the design moment the section can carry, from the structural design of the pile. The module checks the largest moment against it and gives the lateral load at which it is reached.
Choosing the response and the loading
Three choices decide how the pile and the ground answer: the response of the ground, static or cyclic loading, and the pile head. They are independent, so every combination can be run and compared.
The response of the ground
The soil reaction p at a depth z against the deflection y
Ground
Curve
Ultimate resistance
Constants
Sand, non-linear
p = Apu tanh(kzy / (Apu)), with A = max(0.9, 3 − 0.8 z/D) for a static load and 0.9 for a cyclic one
pu = the smaller of (C1z + C2D) σ′ and C3Dσ′, with C1, C2, C3 from φ′
φ′; k (filled in from φ′ and the water)
Clay, non-linear
p = 0.5 pu (y/y50)1/3 up to y = 8 y50, then pu; y50 = 2.5 ε50D
pu = the smaller of (3 + σ′/cu + Jz/D) cuD and 9 cuD
cu; ε50 and J (filled in from cu)
Elastic, either soil
p = ky, a straight line with no limit
None
nh in a sand (k = nhz); kh in a clay (k = khD)
The sand curve is the one of API RP 2A, built by Reese, Cox and Koop (1974) from tests on a pile in sand; the clay curve is Matlock’s (1970) soft-clay curve as API RP 2A states it. The non-linear curves are the default. The elastic option replaces each curve by its straight line: it is a beam on an elastic foundation, good where deflections are small and where you work with a coefficient of subgrade reaction of your own.
Static or cyclic loading, free or held head
Choice
What it does
On the example
Static (sustained)
The static p–y curves.
The reference: yhead = 6.228 mm, Mmax = 151.8 kN·m.
Cyclic (repeated)
The API cyclic curves. In a sand the factor A is 0.9 at every depth. In a clay the curve stops at 0.72 pu from 3 y50 at depth; nearer the surface it then falls to a lower value. The same load gives a larger deflection and moment.
Nothing holds the head from turning: the moment at the head is zero, or the moment you apply. The pile deflects most at the head and its largest moment is a little below the surface.
The reference.
Held against rotation
A cap keeps the slope of the head at zero. The head still moves sideways, but less, and a moment develops at the head to hold it. On the example it is larger than the largest moment of the free pile, but under a larger load or with a free length it can be smaller, so compare the two runs. The module gives it as Moment holding the head.
yhead = 2.241 mm (−64 %), holding moment -174.8 kN·m.
The Diagrams view of the same pile held against rotation. The deflection at the head is 2.241 mm instead of 6.228 mm. The moment is largest at the head, -174.8 kN·m (negative, opposite in sign to the moment of the free pile), and falls to zero and then to a positive peak of 49.6 kN·m about 3.4 m down; the shear still starts at H.
Diagrams with the head held: the largest moment is the one that holds the head.
Which should I pick?
Start with the non-linear p–y curves. They are the default, and the only response that limits the soil reaction by the strength of the ground. The elastic response gives yhead = 10.49 mm on the example, with the module’s own nh and kh; its constants are yours to justify, so it is another ground, not a check of the p–y curves.
Use cyclic loading for loads that repeat many times: wind, waves, traffic, braking. A sustained load, such as earth pressure on the pile, is static.
Choose the head from the connection. (With the elastic response, static and cyclic give identical results.) A rigid pile cap or a stiff superstructure holds the head; take it free to rotate where the connection cannot carry a moment. How near a real cap comes to a held head depends on the stiffness of the cap and the superstructure, and the real condition lies between the two. Free gives the largest deflection and in-ground moment; held gives the largest head moment and the smallest deflection, so each quantity is bounded by the two runs.
Keep the section honest. A concrete pile that cracks is more flexible than its gross section: enter a factor on E I (often 0.5 to 0.8 for a section that cracks). Run fEI = 1 as well: the cracked section gives the larger deflection, the gross section the larger moment.
Analyze and read the results
Press Analyze. The bar turns green (brown when the run raised a warning), the inputs lock, and the Results tab shows three blocks.
Laterally loaded pile. Cards for the deflection of the pile head yhead, the largest bending moment Mmax, the depth zM of the largest moment and the rotation θ0 at the ground surface. Below them, in lines: the depth z0 of zero deflection, the largest shear force Vmax, the largest soil reaction pmax and the deflection of the pile tip ytip.
Values, step by step. One row per step: the bending stiffness E I, the ultimate resistance pu at the middle of each layer, and all the results.
Along the pile. A table of z, y, θ, M, V and p at about 25 depths, from the head to the tip.
With a free length, the deflection y0 at the ground surface is added as a card and the rotation moves to the lines. With a held head, the Moment holding the head is a card. With an allowable value entered, the lateral load at which it is reached is a card (for a moment only, H = 0 with a moment at a free head, the cards read Moment at the allowable deflection and Moment at the moment capacity), and the verdict is written on the card it belongs to.
The Results tab of the example: the cards, the steps and the table along the pile.
The signs
The depth z is measured down from the ground surface; a negative depth is above the ground.
The deflection y is positive in the direction of the load, so the head deflection of a pile pushed forward is positive.
The rotation θ is the slope dy/dz, so it is negative for a pile that leans over in the direction of the load: -0.1499° in the example.
The shear V at the head is H, and M at a free head is the moment you apply (zero if you apply none). The soil reaction p has the sign of the deflection: it is the force per unit length that the ground gives back, against the movement.
Reading the example
For the module’s own example (the full numbers are in the worked example), the Diagrams view and the table tell this story from the top down:
The deflection is largest at the head, 6.228 mm. It passes through zero at 3.851 m: above that depth the pile pushes into the ground, below it the pile moves back by a few tenths of a millimetre at most, and the deflection is practically zero below about 8 m.
The moment is zero at the free head, grows to Mmax = 151.8 kN·m at 2.132 m, where the shear passes through zero, and dies away below. The small reverse moment lower down is in the table.
The shear starts at H = 120 kN, changes sign at the depth of the largest moment, and is zero at the free tip.
The tip hardly moves (ytip = -0.001916 mm, a few micrometres): the pile is long for this load in this ground. When the tip does move, the pile is short and its length governs (see Messages).
The soil reaction peaks at 73.99 kN/m at 1.40 m. The grey limit on the Diagrams view shows the ground is well below its limit there: the reaction is about half of the end of the curve at that depth.
Checking the allowable values
If you entered an allowable head deflectionyall or a moment capacityMcap, the card it belongs to carries a verdict in words: OK, OK, near the limit (the value is within the limit but above 90 % of it) or NOT OK, followed by the limit you entered. (The panel on the Section view and the summary of the report word the same check as “not exceeded” or “EXCEEDED”.) A failing check also raises a warning with advice. The card Lateral load at the allowable deflection or Lateral load at the moment capacity gives the load at which the pile reaches the limit.
The Results tab with yall = 6.5 mm and Mcap = 140 kN·m entered.
The example with one limit entered at a time
Limit entered
Verdict
Lateral load at the limit (kN)
What it means
yall = 10 mm
OK (yall = 10.0 mm)
175.7
The head deflection of 6.228 mm is within the limit; the limit is reached at 1.46 times the load of 120 kN.
yall = 5 mm
NOT OK (yall = 5.00 mm)
98.90
The deflection exceeds the limit. The load that gives 5 mm is 82 % of the load entered.
Mcap = 250 kN·m
OK (Mcap = 250 kN·m)
185.5
The largest moment of 151.8 kN·m is within the capacity; it is reached at 1.55 times the load.
Mcap = 140 kN·m
NOT OK (Mcap = 140 kN·m)
111.4
The largest moment exceeds the capacity; the capacity is reached at 93 % of the load.
Because the ground is non-linear, the load at a limit cannot be read off the working load by proportion. The module raises the load and the moment at the head in proportion (a factor on H and M, with the axial load kept) and solves the equations again at each load until the limit is reached. If the ground cannot carry a load that large, the card reads not reached and the Load–deflection view writes “not reached before equilibrium is lost” beside the limit.
Load–deflection with the limits: the load at which each curve reaches its limit is marked.
The Calculations tab and the report
The Calculations tab writes out the working in chapters: Given and what is sought, then one chapter per step, then the notes and assumptions. For this module the steps are:
The pile — I from the section and E I = EpI (times the factor for cracking, when it is not 1).
The p–y curves of the ground (or The elastic subgrade reaction of the ground) — for each layer, at the middle of the layer within the pile: the effective stress σ′, the coefficients, pu, and the factor A and k of a sand or y50 of a clay.
Analysis of the beam on the springs — the beam equation, the number of beam elements, the iterations to equilibrium, and two checks: the sum of the soil reaction against H, and the moment at the tip by statics.
Results — yhead, θ0, Mmax, zM, z0, Vmax, pmax, zp and ytip, and the table along the pile; the held head adds Mhead and a free length adds y0.
The load at the allowable deflection and at the moment capacity — only when you entered a limit: the load factor at which it is reached and the load.
The Calculations tab: the given data, then each step with its equations and numbers.
The Report tab turns the same working into a summary report. Fill in the cover (project, location, client, report number, date, prepared by, title, checked by, notes) and press Save as PDF or Download Word (.docx). The report has seven parts: 1 Introduction and scope, 2 Input data, 3 Calculation (one part per step), 4 Results, 5 Assumptions and notes, 6 Summary and 7 References. Part 2 holds the table of the inputs followed by four figures, the views of the drawing: the pile with its deflected shape, the diagrams along the pile, the p–y curves, and the load–deflection curve. Part 5 is printed when the run has assumptions or cautions to state, and part 6 is a short summary in words, for example the head deflection and the largest moment with their depths.
The Report tab: the cover details, the two download buttons and the front page of the report.
The Theory tab explains the method in fourteen chapters: a pile under lateral load, the pile as a beam on springs, the elastic solutions, the p–y curve of a sand, the p–y curve of a clay, layers and water, the numerical solution, the axial load, the head and the free length, reading the results, the load–deflection curve and the allowable values, a worked example, what the module assumes, and the references.
Worked example: the module’s own example
The module opens on this example. Reset to defaults brings it back at any time, so you can follow the numbers on your own screen. A beam on non-linear springs is solved numerically, so there is no short hand calculation of the answer; what you can do by hand is the input of the solution, the stiffness of the pile and the p–y curves, and then read the result.
Inputs of the example
Section
Input
Value
Soil layers
Layer-1, sand
H = 4 m, γ = 18 kN/m³, γsat = 19 kN/m³, φ′ = 32°
Soil layers
Layer-2, clay
H = 5 m, γ = 18 kN/m³, γsat = 19 kN/m³, cu = 40 kPa
Soil layers
Layer-3, sand
H = 8 m, γ = 18 kN/m³, γsat = 19 kN/m³, φ′ = 32°
Groundwater
Water table
Present, Dw = 2 m
Pile
Section, width
Circular, solid; D = 0.6 m
Pile
Stiffness, lengths
Ep = 30 000 MPa, fEI = 1; L = 14 m, e = 0
Loading
Head, loading
Free to rotate, static; H = 120 kN, M = 0, P = 0
Soil response
Response
Non-linear p–y curves; k = 20.7 MN/m³ (layer 1) and 11.9 MN/m³ (layer 3), ε50 = 0.02 and J = 0.5 (layer 2), all filled in
Step 1: the pile
I = π × 0.6⁴ / 64 = 0.006362 m⁴, so E I = 30 000 MPa × 0.006362 m⁴ = 190,852 kN·m².
Step 2: the p–y curves at the middle of each layer
The module works out one curve for the middle of each layer, to show you the numbers; the engine uses the same formulas at every depth. For Layer-1, a sand at z = 2.00 m, the effective stress is σ′ = 18 × 2.00 = 36.00 kPa (the water table is at 2 m). With φ′ = 32°, α = 16°, β = 61° and Ka = tan²29° = 0.3073, the coefficients are C1 = 2.2813, C2 = 2.9473 and C3 = 36.814, and
so pu = the smaller = 227.92 kN/m. The factor A = max(0.9, 3 − 0.8 × 2.00/0.6) = 0.9, and the slope of the curve at the origin is kz = 20.7 MN/m³ × 2.00 m = 41,400 kPa.
For Layer-2, a clay at z = 6.50 m, σ′ = 18 × 2 + (19 − 9.81) × 4.5 = 77.36 kPa and y50 = 2.5 × 0.02 × 0.6 = 0.03 m. Then pus = (3 + 77.36/40 + 0.5 × 6.5/0.6) × 40 × 0.6 = 248.41 kN/m and pud = 9 × 40 × 0.6 = 216.00 kN/m, so pu = 216.00 kN/m. For Layer-3, a sand at 11.5 m with σ′ = 123.31 kPa, pu = 2,723.61 kN/m, where the deep value pud governs. These three are the pu of the step-by-step table on the Results tab.
Step 3: the solution
The pile is divided into 140 beam elements with a node at every layer boundary, and the non-linear equations are solved by Newton’s method, which reaches equilibrium in 7 iterations. The two checks the module writes: the sum of the soil reaction is 120.000 kN against H = 120.000 kN, and the moment at the tip, by statics from the load and the soil reaction, is 0.000 kN·m.
Step 4: reading the result
The head deflects yhead = 6.228 mm and the pile rotates θ0 = -0.1499°. The largest bending moment is Mmax = 151.8 kN·m, 2.132 m below the ground, in the sand of Layer-1, a little below the water table. The deflection passes through zero at z0 = 3.851 m, still in Layer-1. The largest shear force is Vmax = 120.0 kN, which is H, at the head. The largest soil reaction is pmax = 73.99 kN/m at 1.40 m, and the tip moves ytip = -0.001916 mm.
How each input changes the answer
Each row below changes one thing in the example and runs the module again. The numbers are the results the module gives, in mm, kN·m and m, so you can reproduce any row.
The example with one change at a time
Change
yhead (mm)
Mmax (kN·m)
at zM (m)
z0 (m)
Why
The example
6.228
151.8
2.132
3.851
The reference
Head held against rotation
2.241
-174.8
0
4.938
The head deflection is 36 % of the free head’s (Mmax shown is the moment that holds the head, at the head, negative)
Moment M = 60 kN·m at the free head
7.755
199.9
1.928
3.619
The moment adds to the deflection the lateral load causes
Load 1.5 m above the ground (e = 1.5 m)
20.23
305.6
1.679
3.373
yhead is at the load; at the ground surface it is 11.12 mm. The moment at the ground surface is He = 180 kN·m; the largest moment is below it
Cyclic loading
7.280
172.0
2.216
3.887
The API cyclic curves; the ground softens
Axial load P = 800 kN
6.324
154.5
2.132
3.829
The P–δ effect adds a little deflection and moment
Cracked section, fEI = 0.6
7.925
140.1
1.951
3.454
A more flexible pile deflects more and takes a smaller largest moment
Ep = 40 000 MPa
5.467
159.3
2.246
4.090
A stiffer pile deflects less and takes more moment
Diameter D = 0.8 m
3.693
184.6
2.637
4.871
D enters I and the width the soil reaction acts over: both stiffen the response
Hollow pipe, D = 0.6 m, t = 20 mm
12.70
124.6
1.691
2.941
The thin wall gives a much smaller I than the solid section
Square section, D = 0.6 m
4.911
166.2
2.350
4.281
I = D⁴/12 is 1.70 times that of the circular pile
Lateral load H = 240 kN (twice)
15.82
347.2
2.330
3.817
The deflection grows 2.54 times, the moment 2.29 times: the ground softens
Lateral load H = 60 kN (half)
2.926
73.36
2.091
3.996
The deflection falls to 0.47 of the reference
Pile 17 m long (to the base of the profile)
6.227
151.8
2.132
3.851
No change: the tip does not move at 14 m
Pile 10 m long
6.229
151.8
2.132
3.849
Practically the same: the tip moves 0.02695 mm
Pile 6 m long
6.562
147.3
2.045
3.603
The tip moves 0.9544 mm: the pile begins to feel its length
No water table
6.224
151.8
2.132
3.863
Almost no change: only the ground below 2 m is affected, and it is working far below its limit
Water table at the surface (Dw = 0)
9.656
182.7
2.497
4.448
The sand is submerged from the top: k falls to the submerged value and so does σ′
Loose sand: φ′ = 28° in layers 1 and 3
13.41
196.9
2.798
5.136
k and the ultimate resistance both fall
Dense sand: φ′ = 38° in layers 1 and 3
3.265
122.0
1.703
3.003
k and the ultimate resistance both rise
Stiffer clay: cu = 100 kPa in layer 2
6.150
152.5
2.142
4.205
The clay is below where the pile works hardest: a small effect
Softer clay: cu = 20 kPa in layer 2
6.339
150.8
2.119
3.595
A small effect, the other way
Elastic subgrade reaction
10.49
172.8
2.488
4.098
With nh = 8 MN/m³ and kh = 15 MN/m³; another ground, not a check of the p–y curves
Three things stand out. The load, the head and the section matter most: holding the head, doubling the load, or a pipe instead of a solid pile change the answer by a factor of two or more. The top of the pile governs: the length beyond about 10 m and the clay at 4 to 9 m change little. The sand at the top counts for much more than the clay below it: a loose sand (φ′ = 28°) gives +115 % on the head deflection, a water table at the surface +55 %.
The same example in US units gives yhead = 0.2452 in, Mmax = 111.9 kip·ft at zM = 6.996 ft: the same answer, in the other units (0.2452 in is 6.228 mm).
Messages: stops, warnings and notes
The module tells you in plain words when something prevents the calculation, when a result needs care, and what it assumed. There are four kinds. The texts below are the module’s own, run on the example with one change.
A stop: the pile is 20 m long but the layers reach 17 m. Nothing is calculated; the drawing shows where the profile ends.
Kind
When
Message and what to do
Stop (brown bar, “Cannot be analysed — …”; the window is titled “Cannot analyse”)
The layers do not reach the pile tip
“The layers must reach the pile tip. They reach 17.00 m, the pile is 20.00 m long. Add a layer, or thicken the last one.”
Stop
No load: H = 0 and no moment (free head)
“No load is given. The pile is loaded by the lateral load or the moment at its head. Enter H or M.” With a held head the message asks for H > 0: “a held head takes no moment of its own: the moment that holds it is worked out”.
Stop (input error)
A value below the smallest the field accepts, for example φ′ = 10° or D = 0.01 m
The title reads “Cannot analyse — 1 input error”: “Friction angle (φ′1) must be at least 15°.” or “Pile diameter / width (D) must be at least 0.05 m.” Correct the field marked in red.
Stop
The wall of a pipe pile is not thinner than its radius
“The wall thickness of the pipe is not between 0 and D/2.” Enter 0 < t < D/2.
Stop
γsat not above the unit weight of water
“The saturated unit weight of Layer-1 is not above that of water.” Enter the saturated unit weight of the soil.
Stop
The load is more than the pile and the ground can resist (here: a 2.5 m pile under 120 kN)
“The load is more than the pile and the ground can resist: no equilibrium.” The message gives the load the pile and the ground do carry (about 90.3 kN here) and the reason. Lower the load, or lengthen or stiffen the pile. For a very large load the reason is that “the ground moves more than five pile widths”; with an axial load it can be that “the axial load acting through the deflection (P–δ) takes the stiffness that is left”.
Stop
The axial load alone buckles the pile in this ground
“The axial load reaches the buckling load of the pile in this ground.” Lower the axial load, or stiffen the pile.
Warning
The tip of the pile moves (here: a 5 m pile)
“The pile tip moves 2.48 mm (31 % of the deflection at the ground).” The pile is short for this load and ground: it rotates as a rigid body and its length governs the result. Lengthen the pile until the tip does not move, or accept the deflection of a short pile. It is raised when the tip moves by more than a tenth of the deflection at the ground and by more than 2 mm.
Warning
The elastic response is chosen
“Elastic analysis: the reaction of the ground is not limited by its strength.” Where the soil would have yielded in reality, the deflection and the moment are too small. Compare the largest reaction with the ultimate resistance of the ground, or use the p–y curves.
Warning
The elastic reaction exceeds the ultimate resistance (here: H = 400 kN)
“The elastic reaction exceeds the ultimate resistance of the ground at z = 0.300 m (218 % of it).” The elastic result is not valid; use the p–y curves.
Warning
The head deflection is larger than the allowable value
“The head deflection 6.23 mm exceeds the allowable 4.00 mm.” Lengthen or stiffen the pile, or lower the load.
Warning
The largest moment is larger than the moment capacity
“The largest moment 152 kN·m exceeds the moment capacity 120 kN·m.” Strengthen the section, or lower the load.
Note
The ground has yielded over more than a fifth of the pile (here: H = 500 kN)
“The ground has reached the end of its p–y curve down to z = 3.00 m.” Over that depth the soil reaction is at its limit: the pile is being pushed through the ground, and the deflection grows quickly with the load. Check the load against the lateral capacity, or lengthen or stiffen the pile.
Note
The head deflection (the deflection at the ground when there is a free length) is more than a tenth of the pile width
“The head deflection is more than a tenth of the pile width.” The p–y curves were derived for deflections up to a few per cent of the width; beyond 60 mm here, the result is an extrapolation of them. Compare the deflection with what the structure can bear.
Assumption (always listed)
Every run
“The pile is a beam on independent springs, with no shear transfer between them.”
Assumption
An axial load is entered
“The axial load acts through the deflected shape, and is taken as constant over the length.”
Assumption
The head is held
“The head is held against rotation but free to move sideways.”
Assumption
Cyclic loading with p–y curves
“Cyclic loading: the API cyclic curves are used.”
The Warnings (or Notes) button on the bar lists the cautions, notes and assumptions after a run; the window is titled Warnings (n) when there are cautions and Notes and assumptions (n) when there are only notes and assumptions. The assumptions, and the cautions about elastic ground and a moving pile tip, are printed in part 5 of the report; the verdicts are printed beside the results; the notes and the limit cautions appear on screen only.
Good practice and common mistakes
Make the pile long enough that its tip does not move. When it moves, the pile turns as a rigid body and its length governs the result; the module cautions you. Lengthen the pile until ytip is practically zero.
Look at the Diagrams view, not only the numbers. The grey limit of the p–y curve on the soil-reaction diagram shows where the ground is near its limit; a reaction that follows the limit over several metres means the pile is being pushed through the soil and the deflection will grow quickly with the load.
Describe the top of the ground with care. It is the upper part of the pile that carries the load, and the water table there changes σ′ and the filled-in k.
Replace a filled-in constant when you have a better one.ε50 from a triaxial test, or a k from your own correlation, beats the standard value.
Run the load you design for. The loads and the limits are yours, and the module gives the response to exactly those. For the load at a limit (a deflection or a moment), enter the limit in Design instead of scaling the working load by hand: the ground is not linear.
Use the moment capacity as a check of the section, not as a design of it. The pile is elastic in the module; a section that yields in bending may fail before the ground does, which is what Mcap is for.
Run the free and the held head when the cap is neither rigid nor flexible: free gives the largest deflection and in-ground moment, held the largest head moment, so each quantity is bounded by the two runs.
Do not read the elastic result as a check of the p–y result. The elastic constants are your own, so it is a different ground.
Keep the layers to the tip. The layers must reach the pile tip; add a layer or thicken the last one.
Read the notes. They list what the module assumed; the assumptions are printed in part 5 of the report, the notes appear on screen only.
Frequently asked questions
How do I calculate the lateral deflection of a pile?
Treat the pile as a beam on non-linear springs, the p–y method: at each depth the ground reacts with a force per unit length p that depends on the deflection y of the pile there. In GeoTerva’s Pile under lateral load module you describe the layers, the pile and the load, press Analyze, and the module solves the beam equation and gives the head deflection, the rotation, the largest bending moment and its depth, and the shear force and soil reaction along the pile.
What is a p–y curve?
A p–y curve is the reaction p of the ground per unit length of pile, against the lateral deflection y of the pile at one depth. It starts steep, flattens as the soil yields and ends on its limit (Apu in sand, pu in clay). GeoTerva uses the API RP 2A curve for sand (Reese, Cox and Koop) and Matlock’s curve for clay, and draws the curves at five depths in the p–y curves view.
What is the difference between a free head and a held head?
With a free head, nothing stops the head turning: the moment at the head is zero (or the one you apply), the pile deflects most at the head and its largest moment is a little below the surface. With a held head a cap keeps the slope at zero: the head moves less and a moment develops at the head to hold it. On the module’s example, holding the head cuts the head deflection from 6.228 mm to 2.241 mm and gives a head moment of -174.8 kN·m, larger than the 151.8 kN·m of the free pile. A real cap lies between the two, so each quantity is bounded by the two runs: free gives the largest deflection and in-ground moment, held the largest head moment.
How does the module treat cyclic loading?
It switches to the cyclic p–y curves of API RP 2A. In sand the factor A is 0.9 at every depth. In clay the curve stops at 0.72 of the ultimate resistance from three times y50 at depth; nearer the surface it then falls to a lower value. On the module’s example, cyclic loading raises the head deflection from 6.228 mm to 7.280 mm.
What does the axial load do to a laterally loaded pile?
The axial load P acts through the deflected shape (the P–δ effect). It is in the beam equation and in the stiffness of every element, so it raises the deflection and the moment. P is the compression you enter at the head, taken as the same at every depth. If it reaches the buckling load of the pile in the ground, the module stops and says so.
What do the signs of the results mean?
The depth z is measured down from the ground surface and the deflection y is positive in the direction of the load. The rotation is the slope dy/dz, so it is negative for a pile that leans over in the direction of the load (-0.1499° on the example). A negative depth is above the ground. The soil reaction p has the sign of the deflection.
How do I find the lateral load for an allowable deflection?
Enter the allowable deflection yall in Design and press Analyze. The module raises the lateral load (and the moment) in proportion, solves the equations again at each load, and gives the lateral load at which the head deflection reaches yall. On the example, 10 mm is reached at 175.7 kN and 5 mm at 98.90 kN. Because the ground is non-linear, the load cannot be scaled from the working load.
When should I use the elastic subgrade reaction instead of p–y curves?
When deflections are small and you work with coefficients of subgrade reaction of your own: nh for a sand (k = nhz) and kh for a clay (k = khD). The elastic soil has no limit, so it overestimates the reaction as the deflection grows; the module cautions you and, where the elastic reaction exceeds the ultimate resistance of the ground, gives the depth. The non-linear p–y curves are the default.
What values do k, ε50 and J take?
They are filled in for each layer, and you can type over them. For a sand, the initial modulus k comes from φ′ and from whether the middle of the layer lies above or below the water table, from the chart of API RP 2A. For a clay, ε50 is 0.020 for cu below 48 kPa, 0.010 from 48 to 96 kPa and 0.005 from 96 kPa; J is 0.5 for cu below 48 kPa and 0.25 above. Replace them with measured values when you have them.
Why does the module say the pile tip moves?
When the tip deflects by more than a tenth of the deflection at the ground and by more than 2 mm, the pile is short for this load in this ground: it turns as a rigid body and its length governs the result instead of the ground near the top. Lengthen the pile until the tip stays put, or accept the deflection of a short pile.
What does “no equilibrium” mean?
The load is more than the pile and the ground can resist: the ground reaches the end of its p–y curves over so much of the pile that it can no longer hold it, or the ground moves more than five pile widths, where the p–y curves no longer describe it. Nothing is calculated. Lower the load, or lengthen or stiffen the pile.
Does the module give the group effect of several piles?
This module analyses one pile. For a cap with several piles, use the Pile group module, which adds the load on every pile, the group effect and the lateral response of the rows with p-multipliers.
Analyse a group of vertical piles under a rigid cap in layered ground: the load on every pile, the capacity of the group as individual piles and as a block, the settlement and the response of the rows to a lateral load. This manual walks through the five views, every input, the methods, the results and the report, with a worked example you can reproduce.
Pile group takes a group of vertical piles under a rigid cap in layered ground and gives you the load on every pile, the capacity of the group, its settlement and, under a sideways load, the deflection of the cap and the load of every row. Four results answer four questions:
Which pile takes how much? Every pile of a rigid cap takes its share of the vertical load and of the moments in proportion to its distance from the centroid of the group: P = V/n + Bx + Cy. The largest and the smallest load, and a pile that is in tension, follow from it.
What can the group carry? The resistance of one pile is worked out from the ground, layer by layer. The group then acts as individual piles, each layer reduced by the efficiency of its soil, or as one block of soil and piles; the lesser is the capacity: Qg(u) = min(Qind, Qblock) and Qall = Qg(u) / FS.
How far does it settle? The elastic settlement, by Vesic’s method, plus the consolidation of every clay layer below the level 2L/3 under the 2:1 spread of the load: S = Sg(e) + Sc. You can leave the settlement out.
How does it move sideways? The cap is rigid, so every pile head has the same deflection Δ. The p–y curves of each row are scaled by the p-multiplier of that row, so the rows carry unequal loads; the module gives Δ, the load and the bending moment of every row.
What you can describe
A rectangular grid of nx rows of ny piles, or a triangular arrangement at the corners of equilateral triangles; up to 100 rows and 100 piles in a row.
A ground of up to eight layers, each sand or clay, and an optional water table.
A circular, hollow circular, square or any other pile section, driven, bored or jacked, with the base of the cap below the ground (zc) or above it (e).
The loads at the base of the cap: the vertical load V, the lateral load H and the moments My and Mx.
For every layer its own skin-friction method (K–δ or Coyle & Castello in sand; α, λ or β in clay) and for the base Meyerhof, Vesic, Coyle & Castello or, for a bored pile in clay, Reese & O’Neill.
Optional limits on the compression and the tension of one pile, the settlement of the group and the deflection of the cap, with a verdict in words.
What belongs to other modules
The resistance of one pile is worked out here by the methods of Pile on layered soil; nothing about it is typed in. For one pile on its own, with the report of that module, use Pile in sand, Pile in clay or Pile on layered soil.
The response of one pile to a sideways load, with its deflection, moment and shear diagrams, is the subject of Pile under lateral load. The rows of this module are solved on the same engine.
The results are loads and movements: the load on every pile and the bending moment along a pile of each row are the loads for the structural design of the pile section, which you make with your own standard.
Before you start: what you need
Have these ready from your ground investigation and your design:
What
Where it comes from
Notes
The layout of the piles: rows, piles in a row, spacings
Your design
The spacings are centre to centre. The module flags a spacing under 2.5 pile widths, and a spacing of the rows under 3 widths for the lateral part.
The pile: section, width or diameter D, length L, modulus Ep, installation
Your design
Installation decides which methods are open and what the module fills in for the shaft and for the efficiency of a sand layer.
The level of the base of the cap
Your design
Below the ground (zc) or above it (e), not both. L is the length of the pile in the ground: measured down from the base of the cap when it is below the ground, from the ground surface when the cap stands above it; the free height e is extra.
The ground in layers, each sand or clay
Boring logs
The layers must reach the pile tips. Sand layers need φ′ (20° to 45°), clay layers cu; both need the unit weights.
e0, Cc and OCR of the clay layers
Oedometer tests or correlations
Needed only when the settlement is calculated. Cs is optional; empty, it is a fifth of Cc.
Water table depth
Piezometers, borings
Optional.
The loads at the base of the cap
Your structural analysis
V is the net vertical load with the weight of the cap included. A lateral load at a height h above the base of the cap gives a moment M = Hh.
Es and νs of the soil below the base
Tests or correlations
Needed for the elastic settlement. Leave Es empty to leave the elastic part out.
Factor of safety; allowable loads and movements
Your design basis
The factor of safety is on the capacity of the group and of one pile. The four limits are optional.
Quick start in ten steps
Open the module and choose the units
Open Pile group from the Piles family. Pick SI · kN, SI · tf or US in the header; you can change it at any time. Open Pile group.
Describe the group
In Group, choose the Rectangular grid or the Triangular (equilateral) arrangement, then give the number of rows nx, the piles in a row ny and the spacings.
Describe the pile and the cap
In Pile, choose the section and the installation (Driven, Bored or Jacked), then enter D, Ep, the embedded length L and the level of the base of the cap.
Describe the soil layers and the water
In Soil layers, fill in the New layer form (sand or clay, thickness, unit weights, φ′ or cu) and press + Add layer; repeat from the ground surface down. The layers must reach the pile tips. In Groundwater choose Present with its depth, or None.
Enter the loads
In Loading, enter V, H, My and Mx at the base of the cap.
Choose the methods
In Point bearing pick the method for the base. In Skin friction every layer shows the methods of its own soil; the coefficients are filled in and can be typed over.
Decide on the settlement
In Settlement choose Calculated and give Es, νs and ξ, or Not calculated to leave the settlement out.
Check the lateral group effect
In Lateral group effect choose whether the pile heads are held against rotation or free, whether the loading is static or cyclic, and whether the load acts in one direction or reverses. The p–y constants and the p-multipliers are filled in from the layers and the spacing.
Set the design values
In Design, enter the factor of safety and, if you want a verdict, the allowable compression and tension of one pile, the allowable settlement and the allowable deflection of the cap.
Press Analyze and read the answer
Press Analyze. The Results tab gives the loads, the capacity, the settlement and the lateral response; the five views of the drawing carry the numbers; the Calculations tab shows every step; the Report tab saves a PDF or a Word file.
The screen: five views of the group
The layout is the one every GeoTerva module shares, described in the workspace guide. For this module the drawing has five views, chosen with the buttons above it. Changing the unit system redraws the views in the new units. Plan and Section follow your inputs as you type; Capacity, Settlement and Lateral show results and are available after Analyze.
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The workspace, here on Pile group after Analyze: the five views above the drawing, the Plan view with the load on every pile, and the Results tab.
Plan. The piles, the cap and the loads; after Analyze the load on every pile.
Section. The piles in the ground, the layers and the spread of the load.
Capacity. The capacity of the group as individuals, as a block, and the efficiency against the spacing.
Settlement. The stress the group puts on each clay layer and what each settles.
Lateral. The rows under the load of the cap: load, deflection and bending moment.
Results box. On the drawing: the main numbers of the run, in the unit system of the page.
Tabs. Model, Results, Calculations, Report and Theory.
Cards. On the Results tab: the capacity, the loads on the piles, the settlement and the lateral response, each with its verdict where a limit applies.
Plan
The piles in plan, with the loads on the cap and the dimensions of the group. The rows are numbered from the front: row 1, the leading row, is the one the lateral load pushes towards, and the rows behind it stand in its shadow. After Analyze the load on every pile is written on it: compression is blue, tension red, and a pile over the load one pile alone is allowed has a heavy red outline. An arrow on each pile gives its share of the lateral load, and a small table lists the p-multipliers of the three kinds of row.
A group of more than eight rows, or of more than eight piles in a row, is drawn in part: the first two rows and the last three rows, and the first two piles and the last two piles of a row, with dots between them, as a matrix is written. The dimensions on the drawing keep the real values.
The Plan view of the example after Analyze: 261, 178 and 94.4 kN on the three rows, and 135, 89.6 and 75.1 kN of lateral load on a pile of each row.
A group of 8 rows of 12 piles, 96 in all, drawn in part: the dots stand for the piles between, and the dimensions are those of the whole group.
Section
The piles in the ground: the layers with their soil, the water table, the cap, the level 2L/3 below the top of the piles and the 2:1 spread of the load below it. After Analyze each layer carries the unit shaft resistance fs, the base the unit resistance qp, and the middle of each clay layer below 2L/3 the stress Δσ′ the group adds there. Two boxes give the resistance of one pile alone (Qs, Qp, Qu) and the start of the spread.
The Section view of the example: the unit shaft resistance of each layer, the level 2L/3 = 9.33 m and the 2:1 spread.
Capacity
Five bars built of the layers they take their resistance from, with the base in a dark colour: the piles alone (n Qu), the piles with their efficiency (Qind), the block (Qblock), the capacityQg(u) (the lesser of the two) and the allowable loadQall. A dashed line marks the vertical load V. Below, the efficiency against the spacings/D: the equation of Converse and Labarre for clay, the chart of Kishida and Meyerhof and the AASHTO rule for sand, and two other equations drawn for comparison, with a vertical line at the spacing of your group.
The Capacity view of the example: the piles with their efficiency (5952 kN) govern; the block carries 25840 kN.
Settlement
The ground on the left with the group and the 2:1 spread; on the right, for every clay layer below 2L/3, a bar at the middle of the layer: grey for the effective stress before the load, orange for the stress Δσ′ the group adds, and the settlement of the layer written beside it. A box gives the elastic part, the consolidation and the total. Nothing is drawn when the settlement is left out.
The Settlement view of the example: 23.4 mm in the upper clay layer and 53.2 mm in the lower.
Lateral
The rows under the load of the cap. Left: the load of a pile against the deflectionΔ for the leading row, the second row, the third and the rows behind it, and an isolated pile, with the deflection of the cap marked. Middle and right: the deflection and the bending moment along a pile of each kind of row, with the largest moment marked.
The Lateral view of the example: the cap deflects 2.92 mm; the leading row takes 135 kN a pile, the second 89.6 kN and the rest 75.1 kN.
Entering the model, section by section
The Model tab has ten sections after Model data. Each opens when you click its title; Next and Back at the foot walk you through them. A (?) beside a title or a method opens a help window with a drawing and the formula for that part. The tables list the fields with their symbol, the unit in each unit system and their default. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. The defaults are those of SI · kN: in US the spacing of 1.8 m shows as 5.90551 ft and V = 1600 kN as 359.694 kip. A field marked “filled in” carries a value the module works out from your other inputs; you can type over it, and emptying the box gives the module’s value back. The fields of a layer are listed once, for layer 1.
Group
The arrangement, the number of rows and of piles in a row, and the spacings. The piles stand in rows across the direction of the lateral load, nx rows of ny piles, under a rigid cap. In the rectangular grid the rows and the piles of a row each have a spacing. In the triangular arrangement one spacing s serves all: the piles of a row are s apart, the rows s √3/2 apart and every second row is shifted by s/2. The module flags a spacing under 2.5 pile widths.
Group: the arrangement, the rows and piles, and the spacings.
Choice
Options
Default
Arrangement a rectangular grid, or the corners of equilateral triangles
Rectangular gridTriangular (equilateral)
Rectangular grid
Group fields
Field
Symbol
Unit
Default
What to enter
Rows in the direction of the lateral load
nx
—
3
Number of rows, one behind the other along the lateral load. A whole number, 1 to 100. Row 1 is the leading row.
Piles in a row (across the load)
ny
—
3
Piles in each row, side by side across the load. A whole number, 1 to 100.
Spacing of the rows (centre to centre)
sx
m · ft
1.8
Spacing of the rows, centre to centre, along the lateral load. It sets the p-multipliers. It must be larger than D. Shown when arrangement of the piles is Rectangular grid.
Spacing of the piles in a row (centre to centre)
sy
m · ft
1.8
Spacing of the piles in a row, centre to centre. It must be larger than D. Shown when arrangement of the piles is Rectangular grid.
Spacing of the piles (centre to centre)
s
m · ft
1.8
One spacing for the triangular arrangement: the piles of a row are s apart, the rows s √3/2 apart, and every second row is shifted by s/2. It must be larger than D. Shown when arrangement of the piles is Triangular (equilateral).
Pile
The section, the installation, the size and stiffness of the pile and the level of the base of the cap. The section gives the area of the base, the perimeter of the shaft and, with Ep and fEI, the bending stiffness EI of the lateral analysis; for Other (I, A and p given) you enter I, A and p yourself. For a hollow pipe the base uses the full outer circle, so a pipe pile gives the same capacity as a solid pile of the same D (5,952 kN for the group of the example); the wall thickness t enters the stiffness and the elastic shortening.
Pile: section, installation, size, stiffness and the level of the cap.
Choice
Options
Default
Section of the pile
Circular, solidCircular, hollow (pipe)SquareOther (I, A and p given)
Circular, solid
Installation driven, bored or jacked
DrivenBoredJacked
Bored
Pile fields
Field
Symbol
Unit
Default
What to enter
Pile diameter / width
D
m · ft
0.6
Diameter of a circular pile or width of a square one. For a section given by I, A and p it is still the width the soil reaction acts over.
Wall thickness of the pipe
t
m · ft
0.02
Wall thickness of the pipe, between 0 and D/2. Shown when section is Circular, hollow (pipe).
Second moment of area about the axis of bending
I
cm⁴ · in⁴
636170
Second moment of area about the axis of bending. Shown when section is Other (I, A and p given).
Cross-section area (also the area of the base)
A
cm² · in²
2827
Area of the section; also the area of the base. Shown when section is Other (I, A and p given).
Perimeter of the pile
p
m · ft
1.885
Perimeter of the section. Shown when section is Other (I, A and p given).
Modulus of elasticity of the pile material
Ep
MPa · kg/cm² · ksi
30000
About 30 000 MPa for a concrete pile, 200 000 MPa for steel.
Factor on E I (cracking of a concrete section; 1 = uncracked)
fEI
—
1
Factor on EI for the lateral analysis: 1 for an uncracked section; 0.5 to 0.8 is common practice for a cracked concrete section. Accepted from 0.05 to 1.5.
Embedded length
L
m · ft
14
Length of the pile in the ground, measured down from the base of the cap when it is below the ground (zc), or from the ground surface when the cap stands above it (e); the free height e is extra.
Depth of the base of the cap below the ground surface
zc
m · ft
0
Depth of the base of the cap below the ground surface. The soil above it counts in the stresses of the axial analysis; the lateral analysis starts at the base of the cap.
Height of the base of the cap above the ground surface
e
m · ft
0
Height of the base of the cap above the ground surface; the piles stand free over this height. Enter zc or e, not both.
Soil layers
Up to eight layers, entered from the ground surface down, each Sand or Clay. A sand layer is asked for φ′, a clay layer for cu and, when the settlement is calculated, for e0, Cc, OCR and optionally Cs. The layers must reach the pile tips; the module takes the deepest layer to continue below the profile when it ends within 2D of the tips, and says so. The New layer form opens at H = 6 m; the table below lists the three layers of the example (4, 6 and 8 m), and the field table shows the defaults of layer 1 (H = 4 m, cu = 40 kPa), while the clays of the example have cu = 50 and 90 kPa. Everything that follows is worked out layer by layer from what you enter here.
Fill in the New layer form and press + Add layer; the layer joins the Soil layers table and the counter (“3 of 8”) moves on.
Click a row to edit a layer; the cross at the end of a row removes it.
Soil layers: the New layer form above, the table of layers below.
Fields of each soil layer (layer 1 shown)
Field
Symbol
Unit
Default
What to enter
Name (optional)
—
—
empty
A name for the layer, for the drawings and the report. Left empty, the layers are called Layer-1, Layer-2 and so on.
Thickness
H
m · ft
4
Thickness of the layer; the smallest value accepted is 0.1 m. The layers together must reach the pile tips.
Unit weight (above the water table)
γ
kN/m³ · t/m³ · pcf
18
Unit weight above the water table.
Saturated unit weight
γsat
kN/m³ · t/m³ · pcf
19
Unit weight below the water table; it must be above the unit weight of water. Used only when a water table is present. Shown when water table is Present.
Friction angle (20–45°)
φ′
°
32
Effective friction angle of a sand layer, 20° to 45°: the base resistance of Meyerhof is tabulated for this range, and outside it the module stops. Shown when the layer is sand.
Undrained shear strength
cu
kPa · kg/cm² · psf
40
Undrained shear strength of a clay layer. Shown when the layer is clay.
Overconsolidation ratio
OCR
—
1
1 is a normally consolidated clay, the safe side for the settlement (it settles most). Used by the consolidation settlement and by the β method. Shown when the layer is clay.
Initial void ratio
e0
—
0.9
Initial void ratio of the clay layer. Shown when the layer is clay and settlement of the group is Calculated.
Compression index
Cc
—
0.28
Compression index of the clay layer. Shown when the layer is clay and settlement of the group is Calculated.
Recompression index (optional)
Cs
—
empty
Recompression index, used where the stress stays below the preconsolidation pressure. Left empty it is a fifth of Cc. Shown when the layer is clay and settlement of the group is Calculated.
Groundwater
Choose whether there is a water table and, if there is, its depth. Below it the effective stresses use the unit weight γsat − γw of the layer, with γw = 9.81 kN/m³.
Groundwater: Present or None, then the depth.
Choice
Options
Default
Water table within or below the piles
PresentNone
Present
Groundwater field
Field
Symbol
Unit
Default
What to enter
Water table depth
Dw
m · ft
2
Depth of the water table below the ground surface, 0 or more. Below it the effective stresses use the submerged unit weights. Shown when water table is Present.
Loading
The loads act at the base of the cap. V is the load the piles carry: the load of the structure with the weight of the cap, less the weight of any soil removed. The lateral load H acts along x, the direction of the rows, and meets the rows one after another. A lateral load at a height h above the base of the cap is entered as H and as the moment My = Hh.
Loading: the vertical and lateral loads and the two moments.
Loading fields
Field
Symbol
Unit
Default
What to enter
Vertical load on the group (net, the weight of the cap included)
V
kN · tf · kip
1600
Net vertical load on the group, in compression, with the weight of the cap included and the weight of the soil removed deducted.
Lateral load on the cap (in the direction of the rows)
H
kN · tf · kip
900
Lateral load on the cap, along x, the direction of the rows, at the base of the cap.
Moment on the group about the y axis (in the plane of the lateral load)
My
kN·m · tf·m · kip·ft
900
Moment on the group about the y axis, in the plane of the lateral load. Positive: more compression on the side the lateral load pushes towards. A lateral load at a height h above the base of the cap gives Hh.
Moment on the group about the x axis (across it)
Mx
kN·m · tf·m · kip·ft
0
Moment about the x axis, across the lateral load. Positive: more compression on the +y side.
Point bearing
Pick the method for the resistance under the tip. The base of a pile is worked out by the soil it ends in: in a sand by Meyerhof, Vesic or Coyle & Castello; in a clay by Meyerhof (qp = 9 cu), by Vesic (which builds N*c from the rigidity index) or, for a bored pile, by the drilled-shaft expression of Reese & O’Neill. Coyle & Castello’s charts are a sand correlation, so that button is not offered while the base is in a clay, and Reese & O’Neill is offered only for a bored pile whose base is in a clay.
Point bearing: the buttons offered for the soil at the base of the example, a clay under a bored pile.
Choice
Options
Default
Method for the point (base) bearing Qp
MeyerhofVesicCoyle & CastelloReese & O’Neill
Meyerhof
Point-bearing fields
Field
Symbol
Unit
Default
What to enter
Es/pa (only used if the base is on a sand and Vesic is selected)
m
—
250
Stiffness ratio Es/pa of the sand at the base, for Vesic’s method. The smallest value accepted is 50. Shown when point-bearing method is Vesic.
Coyle & Castello N*q — chart default for this pile, editable
N*q(CC)
—
filled in
Read from the Coyle & Castello chart for this pile’s L/D and the φ′ of the layer the base stands in, and kept up to date. Type over it to use your own reading. Shown when point-bearing method is Coyle & Castello.
Skin friction
Every layer chooses its own method and carries its own numbers, named after the layer. The methods are those of Pile on layered soil, and each shows only the boxes it needs:
Skin friction: a method and its boxes for every layer.
Soil
Methods
Boxes
Sand layer
K–δ method; Coyle & Castello
K and δ′ for the K–δ method; KCC for Coyle & Castello. All are filled in and can be typed over.
Clay layer
α method; λ — extended; β method
α for the α method (filled in); nothing for the λ method; φ′R for the β method (typed by you, with the OCR of the layer).
Shaft coefficients of each layer (layer 1 shown)
Field
Symbol
Unit
Default
What to enter
Earth pressure coefficient K
K
—
filled in
Earth pressure coefficient on the shaft in this layer. Filled in as K0 = 1 − sin φ′ for a bored pile and 1.4 K0 for a driven or jacked one. The smallest value accepted is 0.2. Shown when the layer is sand and the skin-friction method of the layer is K–δ method.
Pile–soil friction angle δ′
δ′
°
filled in
Friction angle between the pile and the sand in this layer. Filled in as 0.75 φ′. Shown when the layer is sand and the skin-friction method of the layer is K–δ method.
Coyle & Castello K
KCC
—
filled in
Coyle & Castello’s coefficient for this layer, read from their chart at the layer’s φ′ and the pile’s L/D; the interface friction is 0.8 φ′. Editable. Shown when the layer is sand and the skin-friction method of the layer is Coyle & Castello.
Adhesion factor α
α
—
filled in
Adhesion factor on cu. Filled in from the drilled-shaft rule of O’Neill & Reese for a bored pile and from the driven-pile curve otherwise (0.55 in the example). Between 0.05 and 1. Shown when the layer is clay and the skin-friction method of the layer is α method.
Remoulded drained friction angle
φ′R
°
30
Remoulded drained friction angle of the clay, for the β method. It is not filled in: no correlation is given for it, so the box opens at 30° and wants the value of your clay. Shown when the layer is clay and the skin-friction method of the layer is β method.
Settlement
The settlement of the group is not obligatory. Not calculated leaves it out together with its boxes and its allowable value, and the Settlement view draws nothing; the loads, the capacity and the lateral response are worked out either way. With Calculated, the clay parameters entered with the soil layers drive the consolidation, and Es, νs and ξ below the base give the elastic settlement of a pile and, by Vesic’s factor, of the group; leave Es empty to leave the elastic part out.
Settlement: calculated or left out, and the soil below the base.
Choice
Options
Default
Settlement of the group calculated, or left out
CalculatedNot calculated
Calculated
Settlement fields
Field
Symbol
Unit
Default
What to enter
Modulus of the soil at and below the pile base (optional)
Es
MPa · kg/cm² · ksf
30
Modulus of the soil at and below the pile base, for the elastic settlement of a pile and, with it, of the group. Leave it empty to leave the elastic part out. Shown when settlement of the group is Calculated.
Poisson's ratio of the soil below the base
νs
—
0.3
Poisson’s ratio of the soil below the base, 0 to 0.5. Shown when settlement of the group is Calculated.
Distribution of the shaft resistance along the pile
ξ
—
0.6
Distribution of the shaft resistance along the pile, for the shortening of the pile: 0.5 for a uniform one, 0.67 for one increasing with depth. The module accepts 0.5 to 0.67. Shown when settlement of the group is Calculated.
Lateral group effect
How the pile heads are held, how the load acts, and the constants of the p–y curves. Held against rotation is a pile head fixed in the cap; Free to rotate (pinned) a pinned one. Static (sustained) loading uses the p–y curves as they stand; Cyclic (repeated) the cyclic curves of API RP 2A. One direction (a leading row) gives each row its own multiplier; Reversing (the mean of the rows) gives every row the mean of the multipliers. The constants of the p–y curves of each layer and the six p-multipliers are filled in; see Choosing the methods for what they are.
Lateral group effect: the choices, then the p-multipliers filled in from the spacing.
Choice
Options
Default
Pile heads held against rotation by the cap or free to rotate
Held against rotationFree to rotate (pinned)
Held against rotation
Lateral loading static or cyclic
Static (sustained)Cyclic (repeated)
Static (sustained)
Direction of the load one direction or reversing
One direction (a leading row)Reversing (the mean of the rows)
One direction (a leading row)
The p-multipliers (filled in from the spacing of the rows)
Field
Symbol
Unit
Default
What to enter
P-multiplier of the leading row — sand layers
Pm1s
—
filled in
Factor on the p–y curves of the sand layers for a pile of the leading row. Filled in from the spacing of the rows (AASHTO LRFD); type over it to use your own.
P-multiplier of the second row — sand layers
Pm2s
—
filled in
The same for a pile of the second row.
P-multiplier of the third and following row — sand layers
Pm3s
—
filled in
The same for a pile of the third row and of every row behind it.
P-multiplier of the leading row — clay layers
Pm1c
—
filled in
Factor on the p–y curves of the clay layers for a pile of the leading row. Filled in from the spacing of the rows (Rollins et al., 2006); type over it to use your own.
P-multiplier of the second row — clay layers
Pm2c
—
filled in
The same for a pile of the second row.
P-multiplier of the third and following row — clay layers
Pm3c
—
filled in
The same for a pile of the third row and of every row behind it.
The p–y constants of each layer (layer 1 shown)
Field
Symbol
Unit
Default
What to enter
Initial modulus k of the p–y curve
k
MN/m³ · t/m³ · pci
filled in
Initial modulus k of the sand p–y curve: the slope at the origin is kz. Filled in from φ′ and the water; type over it to use your own. Shown when the layer is sand.
Strain ε50 at half the strength
ε50
—
filled in
Strain at half the strength, for the clay p–y curve: y50 = 2.5 ε50D. Filled in from cu (0.020 soft, 0.010 medium, 0.005 stiff). Shown when the layer is clay.
Matlock J
J
—
filled in
Matlock’s constant for the near-surface wedge of a clay: 0.5 for a soft clay, 0.25 for a medium to stiff one. Filled in from cu. Shown when the layer is clay.
Design
The factor of safety turns the capacity of the group, and that of one pile alone, into the allowable load. The four limits are optional: when entered, the module checks the results against them and says OK or NOT OK in words.
Design: the factor of safety and the optional limits.
Design fields
Field
Symbol
Unit
Default
What to enter
Factor of safety on the capacity (the group and one pile)
FS
—
3
Factor of safety on the capacity of the group and on that of one pile. Accepted from 1 to 10.
Allowable compression of one pile (optional)
Pall
kN · tf · kip
empty
Allowable compression of one pile. If entered it is compared with the most loaded pile, and the lesser of this and Qu/FS governs.
Allowable tension of one pile (optional)
Tall
kN · tf · kip
empty
Allowable tension of one pile. Used when a pile is in tension; the lesser of this and Tu/FS governs.
Allowable settlement of the group (optional)
sall
mm · in
empty
Allowable settlement of the group. If entered, the settlement is given a verdict. Shown when settlement of the group is Calculated.
Allowable lateral deflection of the cap (optional)
yall
mm · in
empty
Allowable lateral deflection of the cap. If entered, the deflection is given a verdict.
Choosing the methods and options
A handful of choices decide how the group is analysed. The tables below say what each one does, so that you can pick the one that fits your case and run another to compare.
Rectangular grid or triangular arrangement
Rectangular grid
Triangular (equilateral)
Inputs
nx, ny, sx, sy
nx, ny, one spacing s
Where the piles stand
Rows sx apart; piles of a row sy apart
Piles of a row s apart; rows s √3/2 apart; every second row shifted by s/2, so that every pile has its neighbours at s
Load on the piles
P = V/n + Bx + Cy
The same; the piles are not symmetric about the axes, so the two moments can be coupled through Σxy, and the Calculations tab shows the equation used
Efficiency in clay
Converse–Labarre with the spacing of each direction
The same equation written over the neighbouring pairs of piles, each pair at its own distance; it gives the rectangular form back for a grid
p-multipliers
From sx/D
From the spacing of the rows s √3/2 over D; the shift of the rows is not counted
Los Angeles and Seiler–Keeney equations
Printed beside Converse–Labarre for comparison
Not given
The Plan view of the triangular arrangement of the example, nine piles at s = 1.8 m: the rows are s √3/2 = 1.56 m apart and every second row is shifted by 0.9 m. Dashed lines join each pile to its neighbours. The rows are then closer than 3 pile widths, so the module cautions that the p-multipliers are beyond the range of the tests (see Messages).
The triangular arrangement after Analyze: 274, 178 and 81.6 kN on the three rows.
Group efficiency: how much each pile loses
Piles close together share the soil between them and carry less than the sum of what each would carry alone. The module takes the efficiency layer by layer, by the kind of soil, and applies it to the shaft resistance of that layer; the base takes the efficiency of the layer it bears on:
Estimated for friction piles in clay. The Los Angeles and Seiler–Keeney equations are printed for comparison and do not enter the capacity; the Seiler–Keeney equation, which takes the spacing in feet, is given from 2 ft.
Sand, driven or jacked
The chart of Kishida & Meyerhof at s/D
Never above 1. It is 1 up to φ′ = 40°, and between 40° and 45° it falls towards the curve of 45° of the chart.
Sand, bored
AASHTO LRFD: 0.65 at a spacing of 2.5 D, rising linearly to 1.0 at 4 D
The reduction for a shaft group in a cohesionless soil.
The sand rules take one spacing, the least distance between two piles. The group then acts as Qind = n (Σ ηiQs,i + ηbQp), and the efficiency of the group on the Results tab is η = Qind / (nQu).
Block failure
The piles and the soil between them may fail as one block of the plan size Lg × Bg and the depth of the piles. The block resists by the shear on its sides and by the bearing of its base:
Qblock = 2 (Lg + Bg) Σ τi ΔLi + LgBgcu(p)N*c
In a clay layer the shear is the undrained strength cu; in a sand layer it is the friction K0σ′ tan φ′ on the vertical faces, with K0 = 1 − sin φ′.
Where the base is in a clay, the bearing is cuN*c over the area of the block, with N*c read from the curves for L/Bg and Lg/Bg. Where the base is in a sand, it is the sum of the bases of the piles.
The capacity of the group is the lesser of the piles as individuals and the block: Qg(u) = min(Qind, Qblock). The Capacity view shows both, and the Calculations tab says which one governs.
In the example the piles govern, 5,952 kN against 25,838 kN for the block. The block governs when the soil the group bears on is much weaker than the soil along its sides. As an illustration, take 12 × 12 piles at 2.5 D, 30 m long, in 4 m of sand (φ′ = 32°) over 26 m of stiff clay (cu = 150 kPa) over a soft clay (cu = 8 kPa), with the tips on the boundary of the two clays, V = 20,000 kN and no lateral load or moment. The piles with their efficiency give Qind = 322,484 kN, the block Qblock = 288,857 kN: the block governs, Qg(u) = 288,857 kN, and the Capacity view says “the block governs”.
The Capacity view when the block governs: the capacity is that of the block.
One pile: the methods of each layer
The resistance of one pile alone is worked out in the Calculations tab, step 3, and none of it is typed. Every layer takes the method chosen for it, with its own numbers, and the base takes the method chosen in Point bearing for the soil the pile ends in. Each method is described in full in the manuals of Pile in sand, Pile in clay and Pile on layered soil; in short:
Skin friction Qs = the perimeter × the sum of fΔL over the layers
Soil
Method
Unit skin friction f
Notes
Sand
K–δ method
f = K tan δ′σ′, σ′ held constant below L′ = 15 D
K = K0 for a bored pile, 1.4 K0 for a driven or jacked one; δ′ = 0.75 φ′.
Sand
Coyle & Castello
f = Kσ′ tan(0.8 φ′), on the mean σ′ of the layer
K from their chart at φ′ and L/D. Driven piles; refused for a bored pile, cautioned for a jacked one.
Clay
α method
f = αcu
Bored: the drilled-shaft α of O’Neill & Reese (0.55 up to cu/pa = 1.5, falling to 0.45 at 2.5), and no resistance over the top 1.5 m or the bottom 1 D. Driven or jacked: the driven-pile curve.
Clay
λ — extended
f = λ (σ′ + 2 cu)
λ read at the embedded length of the pile and applied layer by layer, an extension of the correlation. Driven piles; refused for a bored pile.
Clay
β method
f = (1 − sin φ′R) tan φ′R √OCRσ′
Needs φ′R, the remoulded friction angle of your clay.
qp = N*ccu, with N*c from the rigidity index Irr = 347 cu/pa − 33 (at most 300)
Coyle & Castello
Sand only
qp = σ′N*q, N*q from their chart at L/D and φ′
Reese & O’Neill
Bored pile, base in clay
qp = min[6 cub (1 + 0.2 L/D), 9 cub, 40 pa], cub the mean undrained strength over the 2 D below the base
Settlement: calculated or left out
Choice
What is worked out
What goes when it is left out
Calculated
Elastic: a pile at its working load V/n settles by the shortening of the pile, of its base and of its shaft (Vesic’s factors); the group settles Sg(e) = √(Bg/D) se. Needs Es. Consolidation: the load of the group spreads 2:1 from the depth 2L/3 and acts at the middle of every clay layer below it: Δσ′ = V / ((Bg + z) (Lg + z)), z the depth of the middle of the clay layer below 2L/3, ΔSc = CcH / (1 + e0) log[(σ′o + Δσ′) / σ′o]; an over-consolidated clay settles less.
—
Not calculated
Nothing about the settlement.
The settlement step and its answers, the boxes e0, Cc, Cs, Es, νs, ξ and the allowable settlement, what the Settlement view draws and the drawing in the report. The loads, the capacity and the lateral response are unchanged.
With Es left empty only the consolidation is calculated, and the card is labelled “(consolidation)”; with no clay layer below 2L/3 only the elastic part is.
The lateral group effect
A pile in a row behind another stands in the shadow of the pile in front, so its p–y curve is scaled by a p-multiplier that depends on the row. The module uses three: the leading row, the second row, and the third row and every row behind it, by the kind of soil in each layer:
Sand layers: the multipliers of AASHTO LRFD, from the tests of Brown, Morrison & Reese and the centrifuge tests of McVay et al.: 0.8, 0.4 and 0.3 at a spacing of 3 D, 1.0, 0.85 and 0.7 at 5 D, interpolated between and rising to 1.0 at 8 D. Closer than 3 D the values of 3 D are used.
Clay layers: Rollins et al. (2006), from full-scale tests in stiff clay: Pm = 0.26 ln(s/D) + 0.5 for the leading row, 0.52 ln(s/D) for the second and 0.60 ln(s/D) − 0.25 for the rest, none above 1.
With one row in the direction of the load nothing shadows another pile, and every multiplier is 1.
The multipliers the module fills in (leading / second / third and following row)
Spacing of the rows
Sand layers
Clay layers
2 D
0.80 / 0.40 / 0.30
0.68 / 0.36 / 0.17
3 D
0.80 / 0.40 / 0.30
0.79 / 0.57 / 0.41
4 D
0.90 / 0.63 / 0.50
0.86 / 0.72 / 0.58
6 D
1.00 / 0.90 / 0.80
0.97 / 0.93 / 0.83
Choice
Options
What it does
Pile heads
Held against rotation, Free to rotate (pinned)
A head held by the cap does not turn; a pinned head can.
Lateral loading
Static (sustained), Cyclic (repeated)
Static uses the p–y curves as they stand; cyclic uses the cyclic curves of API RP 2A.
Direction of the load
One direction (a leading row), Reversing (the mean of the rows)
One direction gives each kind of row its own multiplier; reversing gives every row the mean of the multipliers of the rows.
Which should I pick?
Set the installation first. It changes K, α, which methods are open and the efficiency of a sand layer.
Start with the defaults for the methods: Meyerhof at the base, the K–δ method in sand and the α method in clay. Change one at a time and compare.
Read the Capacity view. It shows whether the piles with their efficiency or the block govern, and how the efficiency changes with the spacing.
Calculate the settlement when there is clay below the level 2L/3, including clay beside the lower shaft. The consolidation of the clays below the level 2L/3 is usually the larger part (the example: 76.66 mm of 84.60 mm).
Leave the p-multipliers as they are unless you have test evidence for your soil and spacing; they are filled in from the spacing of the rows.
Analyze and read the results
Press Analyze. The bar turns green, the inputs lock, and the Results tab shows its blocks, one below the other.
Pile group. The cards with the main answers (the table below), and under them the lines of the secondary answers: the load one pile alone is allowed, the load of a pile of the leading row, the lateral efficiency, the elastic and the consolidation settlement, the capacity of one pile alone and that of the block.
Values, step by step. One row for each step of the calculation, with its numbers.
Load on every pile. A table of the row, the pile, its position x and y from the centroid of the group, and its load P. It is printed for a group of up to 40 piles.
The rows under the lateral load. For each kind of row the p-multipliers of sand and clay, the lateral load of a pile and of the row, and the largest bending moment and the moment at the head of a pile.
The warnings and notes of the run, if there are any, are listed under the Warnings or Notes button on the green bar and printed in the report. The five views of the drawing now carry the numbers (see the screen).
The Results tab of the example: the cards, the steps, the load on every pile and the rows under the lateral load.
The cards
Card
What it is
Checked against
Capacity of the group Qg(u)
The lesser of the piles as individuals and the block
—
Allowable load of the group Qall
Qg(u) / FS
The vertical load V
Largest pile load Pmax
The load of the most loaded pile of the rigid cap
Qu/FS, the load one pile alone is allowed, or the allowable compression Pall if you entered it and it is smaller
Smallest pile load Pmin
The load of the least loaded pile. Negative: the pile is in tension, and the card is labelled “(tension)”
Only for a pile in tension: its tension capacity over FS, or Tall if smaller
Efficiency of the group η
Qind / (nQu), for a group of more than one pile
—
Settlement of the group S
Elastic plus consolidation, or the one that is worked out
The allowable settlement sall, if entered
Lateral deflection of the cap Δ
Shown when the lateral load H is above zero
The allowable deflection yall, if entered
Largest bending moment in a pile Mmax
The largest moment found in a pile of any row, with its sign
—
Compression and tension
Loads on the piles are positive in compression. A moment on the cap lifts the piles on its far side, and when the vertical load cannot balance it a pile goes into tension. In the Plan view such a pile is drawn in red, the card reads “Smallest pile load (tension)”, and the Notes list says which pile it is. The module then adds two checks, shown as two more lines under the cards:
One pile in tension resists by its shaft alone: a clay layer by the same method as in compression, a sand layer at 0.75 of its resistance in compression. The most tension on a pile Tmax is checked against Tu/FS, or against Tall if you entered it and it is smaller.
The piles in tension together are held by the lesser of their own shafts, each layer with the efficiency of its soil, and the block of soil that encloses them: the shear on its sides and its weight. Their sum ΣT is checked against Tg/FS.
The weight of the pile is not counted in tension. As an example, with My raised to 2700 kN·m the module gives Pmax = 427.8 kN and Pmin = -72.22 kN. The three piles of the back row are in tension (red); the three of the front row are over the load one pile alone is allowed (heavy outline), and the results box on the drawing adds the allowable tension.
A pile in tension: with a moment of 2700 kN·m the back row is lifted (−72.2 kN) and the front row takes 428 kN.
The checks against allowable values
Where a limit applies, the card or the line carries a verdict in words, never by a colour alone, and the figure it was compared with in brackets:
Verdict
Meaning
OK (V = 1,600.00 kN)
The value is within its limit. The example gives this on the allowable load of the group and, beside the largest pile load, “OK (Qu / FS = 302.38 kN)”.
OK, near the limit
The value is within the limit but above 90 % of it.
NOT OK
The value is over the limit. For the load on the group, the most loaded pile and the tension checks the module also raises a warning that says what to change; see Messages.
Two checks are always made: the load on the group against its allowable load, and the most loaded pile against the load one pile alone is allowed. The tension checks are made when a pile is in tension. The settlement and the deflection of the cap are judged only when you enter sall and yall in the Design section; without them the numbers are given and not judged.
The Calculations tab, the report and the theory
The Calculations tab writes out the working in chapters: Given and what is sought, then one chapter per step, then Notes and assumptions. For this module the steps are:
The group of piles — the plan size Lx, Ly of the group, Lg and Bg, and the spacing over the width.
The load on every pile — V/n, Σx², Σy², B and C, the largest and the smallest load, and the table of every pile.
One pile, layer by layer — for each layer the method, its coefficients and f, Qs; the base with qp and Qp; and Qu.
The group effect on the capacity: the piles as individuals — the efficiency of each layer, Qind and η.
The group effect on the capacity: the piles as a block — the shear of each layer, the bearing of the base and Qblock.
The capacity of the group — Qg(u), which one governs, Qall and the share of it that V takes.
The most loaded pile — Pmax against Qu/FS (for a group of more than one pile).
The settlement of the group — the elastic settlement of a pile and of the group, the consolidation of each clay layer, and the total.
The group under the lateral load — the p-multipliers, the deflection of the cap, the load of each row and the moments.
A chapter appears only when it applies: the settlement chapter when the settlement is calculated, the lateral chapter when H is above zero, and The piles in tension, with its own checks, after step 7 when a pile is in tension; it then becomes step 8, the settlement step 9 and the lateral load step 10. Notes and assumptions lists what the run took for granted, such as the drilled-shaft rule for a bored pile or the rigid cap. Every number on the Results tab can be traced to a step here.
The Calculations tab: the given data, then each step with its equation and numbers; a chapter folds with its arrow.
The report
The Report tab turns the same working into a summary report. Fill in the cover (project, location, client, report number, date, prepared by, title, checked by, notes) and press Save as PDF or Download Word (.docx). The report has seven parts: 1 Introduction and scope, 2 Input data, 3 Calculation (one part for each step, 3.1 to 3.9, and 3.10 when a pile is in tension), 4 Results, 5 Assumptions and notes, 6 Summary and 7 References. It carries the drawings of the views that were worked out, and the units line states the conversion constants.
The Report tab: the cover details, the two download buttons and the preview.
The theory
The Theory tab explains the method in fifteen chapters: 1 A group of piles; 2 The load on every pile; 3 The single pile; 4 The efficiency of the group; 5 Piles in sand; 6 Layer by layer; 7 Block failure; 8 Elastic settlement; 9 Consolidation settlement; 10 Lateral load: the p-multipliers; 11 Solving the group; 12 Reading the results; 13 A worked example, whose numbers are computed by the module’s own code; 14 What the module assumes; 15 References.
Worked example: the module’s own example
The module opens on this example. Reset to defaults brings it back at any time, so you can follow the numbers on your own screen.
Inputs of the example
Section
Input
Value
Group
Arrangement, rows, piles in a row, spacings
Rectangular grid, nx = 3 rows of ny = 3 piles, sx = sy = 1.8 m (3 D)
Pile
Section, installation
Circular, solid; bored
Pile
Size, stiffness, length, cap
D = 0.6 m, Ep = 30 000 MPa, fEI = 1, L = 14 m; base of the cap at the ground (zc = e = 0)
Soil layers
Layer-1
Sand, H = 4 m, γ = 18 and γsat = 19 kN/m³, φ′ = 32°
Soil layers
Layer-2
Clay, H = 6 m, γ = 18 and γsat = 19 kN/m³, cu = 50 kPa, e0 = 0.9, Cc = 0.28, OCR = 1
Soil layers
Layer-3
Clay, H = 8 m, γ = 18 and γsat = 19 kN/m³, cu = 90 kPa, e0 = 0.7, Cc = 0.2, OCR = 1
Groundwater
Water table
Present, Dw = 2 m
Loading
Loads at the base of the cap
V = 1600 kN, H = 900 kN, My = 900 kN·m, Mx = 0
Point bearing
Method
Meyerhof
Skin friction
Methods, coefficients
Layer-1 K–δ, K = 0.47 and δ′ = 24° (filled in); Layer-2 and Layer-3 α method, α = 0.55 (filled in)
Settlement
Choice, soil below the base
Calculated; Es = 30 MPa, νs = 0.3, ξ = 0.6
Lateral group effect
Heads, loading, direction
Held against rotation; static; one direction. The multipliers are filled in from sx/D = 3
Design
Factor of safety; limits
FS = 3; no limits
Step 1: the load on every pile
V/n = 1600/9 = 177.78 kN. The piles stand at x = +1.8, 0 and −1.8 m from the centroid, so Σx² = 3 × (1.8² + 0 + 1.8²) = 19.44 m² and B = My/Σx² = 900/19.44 = 46.296 kN/m. The front row takes 177.78 + 46.296 × 1.8 = 261.11 kN a pile, the middle row 177.8 kN and the back row 177.78 − 46.296 × 1.8 = 94.44 kN. The three rows add to 3 × (261.11 + 177.78 + 94.44) = 1600 kN: the moment only moves load from the back to the front.
Step 2: one pile, layer by layer
The perimeter is π × 0.6 m and the area of the base 0.2827 m². The pile is bored, so K stays at K0 and the clay layers take the drilled-shaft α.
Layer-1, sand 0–4 m, K–δ.K = 1 − sin 32° = 0.470, δ′ = 0.75 × 32° = 24°. The critical depth L′ = 15 D = 9 m is not reached in the layer, so the mean effective stress of the layer, 31.60 kPa (the water table is at 2 m), is used: f = 0.470 × tan 24° × 31.60 = 6.61 kPa, which over the shaft of the layer gives Qs = 49.85 kN.
Layer-2, clay 4–10 m, α.α = 0.55: f = 0.55 × 50 = 27.50 kPa and Qs = 27.50 × π × 0.6 × 6 = 311.02 kN.
Layer-3, clay 10–14 m, α.f = 0.55 × 90 = 49.50 kPa, but the drilled-shaft rule leaves out the bottom 1 D = 0.6 m of the shaft, so the friction acts over 3.40 m of the 4.00 m: the mean is 49.50 × 3.40/4.00 = 42.08 kPa and Qs = 317.24 kN.
Base, in Layer-3, Meyerhof.qp = 9 cu = 9 × 90 = 810 kPa and Qp = 810 × 0.2827 = 229.02 kN.
Qu = 49.85 + 311.02 + 317.24 + 229.02 = 907.13 kN. The load one pile alone is allowed is Qu/FS = 907.13/3 = 302.38 kN; the most loaded pile takes 261.11 kN, 86 % of it.
Step 3: the efficiency and the block
Clay layers.θ = tan⁻¹(0.6/1.8) = 18.43° in both directions, so η = 1 − [2 × 3 × 18.43 + 2 × 3 × 18.43] / (90 × 3 × 3) = 0.727.
As individuals.Qind = 9 × (0.767 × 49.85 + 0.727 × (311.02 + 317.24 + 229.02)) = 5,952.28 kN, so the group acts as 5,952.28/8,164.15 = 0.729 of the sum of its piles.
As a block.Lg = Bg = (3 − 1) × 1.8 + 0.6 = 4.20 m, perimeter 16.80 m. The sides give 623.66 + 5,040.00 + 6,048.00 = 11,711.7 kN (τ = K0σ′ tan φ′ = 9.28 kPa in the sand, cu in the clays); the base gives 4.20 × 4.20 × 90 × N*c, with N*c read from the curves as 8.90 (to two decimals), 14,126.5 kN; Qblock = 25,838 kN.
Step 4: the capacity of the group
Qg(u) = min(5,952.28, 25,838.1) = 5,952.28 kN: the piles with their efficiency govern, and the block is far from it. Qall = 5,952.28/3 = 1,984.09 kN, against the V = 1,600 kN the group carries: 81 % of it, OK.
Step 5: the settlement
Elastic. A pile at its working load V/n = 177.78 kN, 75 % of it on the shaft, settles se = 0.2057 + 2.456 + 0.3383 = 3.000 mm (shortening of the pile, base, shaft). The group, wider, settles √(Bg/D) = √(4.20/0.60) times as much: Sg(e) = 7.936 mm.
Consolidation. The load spreads 2:1 from 2L/3 = 9.33 m. Layer-2 has 0.6667 m below that level; its middle is 0.333 m below it, where Δσ′ = 1600/((4.20 + 0.333) × (4.20 + 0.333)) = 77.85 kPa on σ′o = 106.46 kPa, which settles it 23.4 mm. Layer-3, 8 m thick, has its middle 4.67 m below the level, where Δσ′ = 20.35 kPa on σ′o = 146.28 kPa: 53.2 mm. Sc = 76.66 mm.
Total.S = 7.936 + 76.66 = 84.60 mm.
Step 6: the lateral load
The rows are 1.8 m = 3 D apart, so the multipliers are, for the leading, second and third row: sand 0.80, 0.40 and 0.30; clay 0.79, 0.57 and 0.41. The cap deflects Δ = 2.92 mm, at which a pile of the three rows carries 135.3, 89.58 and 75.08 kN, so the rows carry 406.0 + 268.7 + 225.2 = 900 kN = H. An isolated pile under H/n = 100 kN would deflect 1.85 mm; the group deflects 1.58 times as much, and a pile of it takes 65 % of what an isolated pile takes at the same deflection (ηlat = 0.646). The largest moment, −207.4 kN·m, is at the head of a pile of the leading row.
How each input changes the answer
Each row below changes one thing in the example and runs the module again. The numbers are the results the module gives, in kN and mm, so you can reproduce any row; a dash means the module gives no such result in that run. The last column says what changes, and where a check fails or comes within 10 % of its limit (the allowable load of the group against V, Pmax against Qu/FS).
The example with one change at a time
Change
Qu (kN)
η
Qg(u) (kN)
Pmax (kN)
S (mm)
Δ (mm)
What happens
The default example
907.1
0.7291
5,952
261.1
84.60
2.924
The reference
Spacing 2.4 m (4 D)
907.1
0.8035
6,560
240.3
66.96
2.379
η rises; the longer lever arm lowers Pmax; the multipliers rise
Spacing 3.6 m (6 D)
907.1
0.8675
7,083
219.4
47.35
1.981
η 0.87; the multipliers are 1.0 to 0.8 in sand and 0.97 to 0.83 in clay
Spacing 1.2 m (2 D)
907.1
0.6088
4,971
302.8
113.0
2.955
Under 2.5 D: η falls, the lever arm shortens, Pmax is over its limit and Qall is within 10 % of V
4 rows of 3 piles
907.1
0.7129
7,761
183.3
68.65
2.305
Twelve piles: V/n and Pmax fall, η falls a little
2 rows of 3 piles
907.1
0.7613
4,144
433.3
108.7
4.003
Six piles: Pmax is over its limit and Qall is below V
Triangular arrangement, s = 1.8 m
907.1
0.6430
5,250
274.0
80.92
2.934
Rows 1.56 m apart, so the lever arm is shorter and Pmax is within 10 % of its limit; the neighbouring pairs lower η
V = 1200 kN
907.1
0.7291
5,952
216.7
65.18
2.923
Pmin falls to 50.00 kN; the settlement falls with the load
V = 2400 kN
907.1
0.7291
5,952
350.0
121.0
2.925
Pmax is over its limit and Qall is below V
My = 0
907.1
0.7291
5,952
177.8
84.60
2.924
Every pile takes V/n; the capacity and the lateral result do not change
My = 2700 kN·m
907.1
0.7291
5,952
427.8
84.60
2.924
A pile goes into tension, Pmin negative; Pmax is over its limit
H = 450 kN
907.1
0.7291
5,952
261.1
84.60
1.416
Half the load, 0.48 of the deflection
H = 1350 kN
907.1
0.7291
5,952
261.1
84.60
4.577
1.5 times the load, 1.57 times the deflection
Settlement not calculated
907.1
0.7291
5,952
261.1
—
2.924
Nothing else changes
Driven piles
1,030
0.7455
6,908
261.1
83.85
2.924
K = 1.4 K0 and the driven-pile α; the sand has η = 1 below φ′ = 40°. A jacked pile gives the same figures
No water table
914.1
0.7294
6,000
261.1
55.41
2.917
The sand and the clays carry more effective stress; the consolidation falls
L = 10 m
558.8
0.7304
3,673
261.1
129.8
2.924
Base on the boundary of Layer-2 and Layer-3; Qall is below V and Pmax is over its limit; the spread starts higher and the settlement rises
L = 16 m
1,094
0.7287
7,173
261.1
66.68
2.924
More shaft in Layer-3; less settlement
D = 0.8 m
1,286
0.6453
7,471
261.1
79.34
1.836
Spacing now 2.25 D, under 2.5 D; stiffer piles deflect less
Base by Vesic
968.6
0.7289
6,354
261.1
85.78
2.924
Only Qp changes; Reese & O’Neill gives the figures of the default, held at 9 cub
FS = 2.5
907.1
0.7291
5,952
261.1
84.60
2.924
Only Qall and the limit on Pmax move
Pile heads free to rotate
907.1
0.7291
5,952
261.1
84.60
8.494
The head can turn: the cap deflects 2.9 times as much
Cyclic lateral loading
907.1
0.7291
5,952
261.1
84.60
3.045
The API cyclic curves are used; the cap deflects a little more
Reversing lateral load
907.1
0.7291
5,952
261.1
84.60
2.859
Every row takes the mean multiplier; the cap deflects a little less
Clay layers over-consolidated, OCR = 3
907.1
0.7291
5,952
261.1
23.27
2.924
The consolidation falls from 76.66 to 15.33 mm
Layer-3 softer, cu = 60 kPa
725.0
0.7296
4,761
261.1
83.56
2.924
Qu falls; Qall is below V and Pmax is over its limit
Driven, clay layers by the λ method
1,214
0.7426
8,116
261.1
83.01
2.924
λ is read at the length of the pile
Driven, clay layers by the β method
845.0
0.7495
5,700
261.1
85.06
2.924
φ′R = 30°, the box’s default; Pmax is within 10 % of its limit
Read across the table for what each input does. The resistance of one pile (Qu) depends on the length, the diameter, the installation, the methods and the soil, and not on the group. The efficiency follows the spacing, the arrangement and the installation. The load on the piles follows the number of piles, the lever arm of the moment and the loads. The settlement follows the load, the water table, the length and the clay parameters. The lateral result follows the lateral load, the spacing of the rows, the heads, the loading and the stiffness of the piles.
Messages: stops, warnings and notes
The module tells you in plain words when something prevents the calculation, when a result needs care, and what it assumed. There are three kinds, and each message names the fields it concerns and says what to do.
A stop is a window titled “Cannot analyse” (the bar turns brown). Nothing is calculated until you change the inputs marked in red.
A warning is a caution raised by a run: the results are there, and they need care or fail a check. The Warnings button on the green bar lists them.
A note states what the run took for granted. The button on the green bar reads Notes when there are no warnings, and the window is titled “Notes and assumptions”.
A stop: the piles are 20 m long but the layers reach 18 m. Nothing is calculated; the Section view shows where the profile ends.
Kind
When
Message and what to do
Stop
The layers do not reach the pile tips
“The layers must reach the pile tip. They reach 18.00 m, the pile tip is at 20.00 m.” Add a layer, or thicken the last one.
Stop
One row of piles, or one pile in a row, with the moment across it
“The piles stand in a line and cannot take a moment across it.” Add a row or a pile on the other side, or remove that moment.
Stop
The λ method in a bored pile
“Change the skin-friction method of Layer-2. The λ method was read from piles driven in clay; a bored pile is outside it.” Use the α or β method in a clay.
Stop
Coyle & Castello in a bored pile
“Change the skin-friction method of Layer-1. Coyle & Castello is calibrated on piles driven in sand; a bored pile is outside it.” Use the K–δ method in a sand.
Stop
Vesic at the base in a weak clay
“Change the point-bearing method. Vesic’s rigidity index Irr = 347 (cu/pa) − 33 has no value at cu = 5.0 kPa in Layer-3; the correlation starts at about 9.8 kPa.” Use Meyerhof for the base (qp = 9 cu).
Stop
The base of the cap given both below and above the ground
“The base of the cap is either below the ground or above it.” Enter zc or e, not both.
Stop (input error)
A value outside what the field accepts, for example 101 rows or φ′ = 19°
“Rows in the direction of the lateral load (nx) must be at most 100.” and “Friction angle (20–45°) (φ′1) must be at least 20°.” Correct the field marked in red.
Stop
A lateral load more than the group can resist (H = 20,000 kN on the example)
“The lateral load is more than the group can resist: no equilibrium.” The window adds that the rows together carry about 16,092 kN at most, at a deflection of the cap of 2,940 mm (4.9 pile widths). Lower the load, or lengthen or stiffen the piles, or widen the spacing.
Warning
A spacing under 2.5 pile widths
“The spacing of the piles is 2.00 pile widths, less than the 2.5 the source gives as the least.” Widen the spacing.
Warning
Rows closer than 3 pile widths, which includes a triangular group at s = 3 D, whose rows are 2.6 D apart
“The rows are closer than 3 pile widths: the p-multipliers are beyond the range of the tests.” Widen the spacing of the rows, or judge the result accordingly.
Warning
The load on the group over its allowable load
“The load on the group, 6,500.00 kN, exceeds its allowable load 1,984.09 kN.” Lengthen or widen the piles, add piles or widen the spacing.
Warning
The most loaded pile over what one pile alone is allowed
“The most loaded pile takes 427.78 kN, more than the 302.38 kN one pile alone is allowed.” Lengthen or widen the piles, add piles, or place them further from the centre of the group in the direction of the moment.
Warning
A pile in tension over its allowable tension (here Tall = 50 kN)
“The most tension on a pile, 72.22 kN, is more than the 50.00 kN one pile is allowed in tension.” Lengthen the piles, add piles, widen the group in the direction of the moment, or increase the vertical load that holds them down.
Warning
Coyle & Castello in a jacked pile
“Coyle & Castello is for driven piles; this one is jacked.” Read the result as an estimate, or use the K–δ method.
Warning
The λ method in a driven or jacked pile
“The λ method is applied layer by layer here, which is an extension of it.” Read the λ layers as an estimate, or use α or β for a layered profile. A jacked pile raises a second warning, “The λ method is for driven piles; this one is jacked.”
Warning
K too high for a bored pile
“K = 1.20 in Layer-1 is too high for a bored pile.” Use K ≈ K0, or say why this shaft carries more.
Warning
A cap deflection over a fifth of the pile width (H = 9000 kN on the example)
“The cap deflects 329 mm, more than a fifth of the pile width.” The p–y curves are carried beyond the deflections they were derived for, and the result is an estimate. Lower the lateral load, or lengthen or stiffen the piles, or widen the spacing.
Note
A pile in tension
“The pile at x = -1.80 m is in tension, 72.22 kN.” Its shaft resistance in tension, and the piles in tension together against their block, are checked.
Note
A cap deflection over a tenth of the pile width
“The deflection of the cap is more than a tenth of the pile width.” The p–y curves were derived for deflections up to a few per cent of the width; compare the deflection with what the structure can bear.
Note
The base exactly on a layer boundary
“Computed with the base bearing on Layer-3, the layer below the level the base sits on.” The point resistance is worked from the soil beneath the base; the layer above it gives shaft friction only.
Note
The profile ends within 2 D of the tips
“Computed with the deepest layer taken to continue below 18.00 m.” The last layer is taken to continue with the same properties.
Note
A bored pile
“Computed with the drilled-shaft α of O’Neill & Reese in Layer-2, Layer-3.” and “Computed with the base taken as clean and the clay of Layer-3 beneath it undisturbed.” No reduction is applied for the bored construction.
Note
Every run
“The cap is taken as rigid.” and “The group effect is worked out layer by layer, by the kind of soil in each layer.” A cap below the ground adds “The soil above the base of the cap acts in the stresses of the axial analysis.”; cyclic loading adds “Cyclic loading: the API cyclic curves are used.”
The warnings and notes of the run are printed in the report, in its part 5, Assumptions and notes, so the reader of the report sees the same assumptions you saw.
Good practice and common mistakes
Enter V as the net load with the weight of the cap, and the loads at the base of the cap. A lateral load at a height h above the base of the cap is H and the moment My = Hh. Mind the sign: a positive My puts more compression on the side the lateral load pushes towards.
Keep the spacing at 3 pile widths or more where you can. The module flags a spacing under 2.5 D for the capacity, and a spacing of the rows under 3 D for the lateral part, where the multipliers are outside the range of the tests. The what-if table shows what the spacing does to η, Pmax and Δ.
Look at the most loaded pile, not only at the group. The group can be within its capacity while the pile on the side of the moment is over what one pile alone is allowed. The card Pmax and its verdict are there for this.
Describe the ground below the tips. The layers must reach the tips, and the base resistance comes from the soil at and just below them. Take the profile to at least 2 D below the tips, or accept the module’s note that the last layer continues.
Match the installation to the pile. It decides K, α, which methods are open and the efficiency of a sand layer. A driven pile modelled as bored under-predicts the shaft; a bored pile modelled as driven over-predicts it.
Read the Capacity view. It shows whether the piles with their efficiency or the block govern, and where your spacing stands on the efficiency curves.
Take the settlement from clay parameters you trust. The consolidation of the clays below the level 2L/3 is usually the larger part of the settlement. OCR = 1 is the safe side; an over-consolidated clay settles much less (the what-if table: 23.3 mm with OCR = 3 against 84.6 mm).
Treat a large cap deflection with care. The p–y curves were derived for deflections of a few per cent of the width. The module notes a deflection above a tenth of the pile width and cautions above a fifth.
Do not edit the p-multipliers without evidence. They follow the spacing of the rows; type over them only for a soil and a spacing you have test results for.
Compare variants. Keep each run with Quick Save…: two lengths, two spacings, bored and driven. Load brings any of them back.
Check the units after you switch the system: type values in the units shown beside the boxes.
Read the notes. They list what the module assumed and are printed in the report.
Frequently asked questions
How do I calculate the load on each pile of a pile group under a moment?
For a rigid cap each pile takes its share of the vertical load plus a part of the moment in proportion to its distance from the centroid of the group: P = V/n + Myx / Σx² + Mxy / Σy² for a symmetric grid, compression positive. In GeoTerva’s Pile group module you enter V, H, My and Mx at the base of the cap and the module lists the load on every pile, with the largest and the smallest. In the module’s example, 1600 kN and 900 kN·m on 3 × 3 piles at 1.8 m give 261.1, 177.8 and 94.44 kN on the three rows.
What is the group efficiency of a pile group and how does the module estimate it?
The efficiency η is the capacity of the piles acting in the group divided by the sum of what each would carry alone. The module takes it layer by layer, by the kind of soil: Converse–Labarre for a clay layer, the chart of Kishida & Meyerhof (never above 1) for a driven or jacked pile in sand, and the AASHTO LRFD reduction (0.65 at 2.5 diameters rising to 1.0 at 4) for a bored pile in sand. In the example, at a spacing of 3 diameters, the group acts as 0.729 of the sum of its piles.
When does block failure govern the capacity of a pile group?
The module works out the group also as one block of soil and piles, of the plan size of the group and the depth of the piles, resisting by the shear on its sides and the bearing of its base, and takes the lesser of the block and the piles as individuals with their efficiency. The block governs when the soil the group bears on is much weaker than the soil along its sides: 12 × 12 piles at 2.5 diameters, 30 m long, in 4 m of sand over 26 m of stiff clay (cu = 150 kPa) over a soft clay (cu = 8 kPa), with the tips on the boundary of the two clays, a vertical load of 20,000 kN and no lateral load or moment, give 322,484 kN as individuals and 288,857 kN as a block, so the block governs. In the module’s default example the piles govern, 5,952 kN against 25,838 kN for the block. The Capacity view draws both and says which governs.
What spacing should I use between the piles of a group?
The module flags a centre-to-centre spacing under 2.5 pile widths, where the methods are outside their experience, and a spacing of the rows under 3 widths for the lateral part, where the p-multipliers are beyond the range of the tests. Wider spacing raises the efficiency, lengthens the lever arm of the moment and raises the p-multipliers. On the module’s example, 1.2 m (2 D) gives η = 0.6088 and 3.6 m (6 D) gives 0.8675.
How does the module check a pile in tension?
When the moment lifts the piles on the far side of the cap, the smallest load is negative and the card reads “Smallest pile load (tension)”. A pile in tension resists by its shaft alone: a clay layer as in compression, a sand layer at 0.75 of its resistance in compression, with the weight of the pile left out. The most tension on a pile is checked against its tension capacity over the factor of safety, and the piles in tension together against the lesser of their shafts and the block of soil around them.
How is the settlement of a pile group calculated?
In two parts. The elastic settlement of a pile at its working load, by Vesic’s method, is multiplied by the square root of the width of the group over the pile diameter. The consolidation settlement spreads the load of the group 2 vertical to 1 horizontal from a level 2L/3 below the top of the piles and settles every clay layer below it under the stress at its middle, from e0, Cc and OCR. In the example they are 7.936 mm and 76.66 mm, 84.60 mm in all.
Can I leave the settlement out?
Yes. Choose Not calculated in the Settlement section. The settlement step, the clay parameters e0, Cc and Cs, Es, νs, ξ, the allowable settlement and the Settlement view are left out; the loads, the capacity and the lateral response are worked out as before. The Settlement view button stays and says that the settlement of the group is not calculated. Leaving Es empty leaves out only the elastic part. With H = 0 the Lateral view says that no lateral load is given.
How does the module handle a lateral load on a pile group?
The cap is rigid, so every pile head has the same deflection. The p–y curve of every layer of a pile is scaled by the p-multiplier of its row (AASHTO LRFD for a sand layer, Rollins et al. 2006 for a clay layer), so the leading row, which the load pushes towards, takes more than the rows in its shadow. The module finds the deflection of the cap at which the rows carry the load together and gives the load and the moment of every row. In the example 900 kN deflects the cap 2.92 mm, and the leading row takes 135.3 kN a pile against 75.08 kN for the third row.
What do the p-multipliers mean and can I change them?
A p-multiplier is the factor by which the soil reaction of a pile in a group is reduced because it stands behind another pile. The module uses one for the leading row, one for the second row and one for the third row and those behind it, separately for sand and for clay layers, filled in from the spacing of the rows. You can type over any of them; emptying the box returns the module’s value.
Can I use a triangular arrangement of piles?
Yes. Choose Triangular (equilateral) in the Group section and enter one spacing: the piles of a row are that far apart, the rows are 0.866 of it apart (√3/2) and every second row is shifted by half of it. The efficiency of a clay layer is written over the neighbouring pairs of piles, the p-multipliers follow the spacing of the rows, and the block is the rectangle that encloses the piles.
What changes between driven, bored and jacked piles in a group?
Installation decides which methods are open (Coyle & Castello and the λ method are refused for a bored pile), what the boxes are filled in with (K about K0 for a bored pile and 1.4 K0 for a driven or jacked one; α from the drilled-shaft rule for a bored pile) and the efficiency of a sand layer. On the module’s example, driven piles give a single-pile capacity of 1,030 kN against 907.1 kN for bored piles, and a jacked pile gives the same figures as a driven one.
Do I have to enter the capacity of a single pile?
No. The module works the resistance of one pile out from the ground, layer by layer, by the methods of the pile on layered soil, and none of it is typed. Each layer takes the method you choose for it, and the base takes the method you choose in the Point bearing section.
What do OK, NOT OK and OK, near the limit mean on the results?
They are the verdicts of the checks against allowable values, written in words with the figure compared in brackets. OK means the value is within its limit, OK, near the limit means it is within the limit but above 90 % of it, and NOT OK means it is over. The module checks the load on the group against its allowable load, the most loaded pile against the load one pile alone is allowed, a pile in tension against its tension capacity, and, when you enter the limits, the settlement and the deflection of the cap.
Seven modules give the pressure of the ground on a wall: at rest, active and passive, static and in an earthquake. This page helps you choose the right one and links to the manual of each.
The Lateral Earth Pressure family has seven modules. They answer one question, the pressure of the ground on a wall, for different walls, grounds and actions. Pick the one that matches your wall.
Your situation
Module
A wall that does not move, such as a basement wall or a braced wall, with a level backfill
Each manual follows the same plan: what the module calculates, what you need before you start, a quick start, the screen, every input, the results, the report, a worked example you can reproduce, a table of how each input changes the answer, the messages the module gives, and answers to common questions.
What the modules share
The workspace. The drawing, the tabs, Analyze and the input lock, units and the report follow the same layout as the other GeoTerva modules. See the workspace guide.
Layered ground. The backfill is entered in layers, with a water table and a surcharge. See each manual for the soil parameters it takes.
The drawing. The pressure diagram, the resultant and a 3D view follow your inputs.
Cohesive soil. The modules that take c′ and φ′ also show the tension crack in a cohesive layer.
Decide whether the wall is held, free to yield, battered or sloping, and whether the action is static or an earthquake.
Describe the ground
Enter the layers, the water table and the surcharge, and the wall geometry the module asks for.
Analyze and read the diagram
Press Analyze. Read the coefficient of each layer, the pressure diagram, the resultant and the height at which it acts.
Compare and report
Run a second module on the same wall to compare, for example Rankine and Coulomb, then save a report from each run with the cover filled in.
Frequently asked questions
What is the difference between at-rest, active and passive earth pressure?
At-rest pressure acts on a wall that does not move. Active pressure acts when the wall yields away from the soil, and passive pressure when it is pushed into the soil. The At rest, Rankine and Coulomb modules give each of them.
Which module do I use for the earth pressure behind a basement wall?
Use At rest (K₀), which takes a layered level backfill, a water table and a surcharge.
Should I use Rankine or Coulomb?
Use Rankine for a vertical wall with a level backfill. Use Coulomb when the back of the wall has friction, is battered or the backfill is sloping. Run both and compare the resultants.
How do I calculate earth pressure in an earthquake?
Use Mononobe–Okabe for the active force and Passive, earthquake for the passive force. Both take the horizontal and vertical seismic coefficients.
Which module gives the passive pressure on a curved failure surface?
Passive, log spiral, which finds the critical log-spiral trial wedge for a rough wall.
Do the modules include a water table and a surcharge?
Yes. Every module takes layered soil, a water table and a surcharge.
Calculate the earth pressure at rest on a wall that does not move: the coefficient K₀ of each soil layer, the stresses, the resultant force and the height at which it acts, with a water table behind the wall. This manual walks through the screen, every input, the choice of relationship, the results and the report, with worked examples you can reproduce.
At rest (K₀) gives the earth pressure at rest on a vertical wall that does not move: a basement wall braced by its floors, a bridge abutment on piles, a braced wall. The soil behind the wall keeps the state it had in the ground, so the horizontal effective stress is a fixed fraction K0 of the vertical effective stress, and the water adds its own pressure:
σ′h = K0 · σ′v, u = γw (z − zw), σh = σ′h + u
The force on the wall is the area of the pressure diagram, P0 = Ps + Pw, per unit length of wall, and the module also gives the height z̄ at which it acts above the base. For dry, uniform soil this reduces to P0 = ½ K0γH², acting at H/3 above the base.
What you can describe
A wall of height H on level ground, with a water table behind it at any depth, or none.
The backfill in up to six layers, from the top down. Each layer has its own unit weights and its own K0, found by one of four relationships: Jaky / Mayne–Kulhawy, Massarsch (clay), Sherif (compacted sand) or a value you give, Given K₀.
The module draws the section, the pressure on the wall and a 3D view, writes out the working step by step, and prepares a PDF or Word report.
What belongs to other modules
This module is for a wall that does not move. A wall that yields enough for the soil to reach its active state is the case of the other modules of this family, starting with Rankine and Coulomb; a wall pushed into the soil is the passive case, in Passive, log spiral.
The result is the pressure of the soil and the water. A load on the ground behind the wall is added from the Surcharge loads modules, for example Uniform load, and the combination is made in your wall calculation.
Before you start: what you need
Have these ready from your design:
What
Notes
The wall height H From: Your design
From the top of the retained ground to the base of the wall; at least 0.1 m.
The soil layers behind the wall: thickness, γ and γsat From: Boring logs, laboratory tests
From the top down, up to six layers. The last layer runs on to the base of the wall, so it needs no thickness.
For each layer, what its K0 will come from From: Tests or correlations
φ′ and OCR for Jaky / Mayne–Kulhawy; PI and OCR for Massarsch (clay); φ′, γd and γd(min) for Sherif (compacted sand); or K0 itself.
Water table depth zw From: Piezometers, borings
Behind the wall only; there is no water in front of it. Optional: choose None for a dry backfill.
Quick start in seven steps
Open the module and choose the units
Open At rest (K₀) from the Lateral Earth Pressure family. Pick SI · kN, SI · tf or US in the header. Open At rest (K₀).
Enter the wall
In Wall, type the height H.
Describe the soil layers
In Soil layers, choose how K0 is found, fill in the thickness and the soil values, and press + Add layer. The example’s layer is already in the table; add more from the top down.
Set the groundwater
In Groundwater, choose Present and give the depth zw, or None.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the inputs lock. The module stays on the Model tab and the Section view: press Pressure above the drawing to see the pressure.
Read the answer
The Results tab gives P0, its height z̄ above the base and the shares of the soil and the water, with K0 of every layer and the stresses at the breaks of the diagram.
Take the working and the report
The Calculations tab shows the three steps with your numbers. The Report tab saves a PDF or a Word file.
The screen: three views of the wall
The layout is the one every GeoTerva module shares, described in the workspace guide. For this module the drawing has three views, chosen with the buttons above it:
Section shows the wall, the soil layers with their properties and the water table, with the dimensions. It follows your inputs as you type; after Analyze each layer also carries its K0.
Pressure shows the at-rest pressure on the back of the wall after Analyze: σ′h from the soil and u from the water, stacked, the resultants Ps, Pw and P0 with their heights above the base, and σ′v in the backfill beside them on the same scale. Before Analyze it shows the wall, the layers and the water table only.
3D shows a length of the wall with the backfill on its right; after Analyze it adds the pressure prism on the face of the wall, σ′h + u, and the resultant.
Section before Analyze: the geometry alone. The note under the drawing says the drawing follows the inputs.
Section after Analyze: K0 of each layer is added.
Pressure after Analyze: what acts on the wall, and the vertical stress beside it.
3D after Analyze.
The Show chips
Above the drawing a Show bar switches parts of it on and off. In the Section view there is one chip, Dimensions. In the Pressure view there are seven: Soil σ′h, Water u, σ′v in the backfill, Ps soil, Pw water, P₀ total and Dimensions. Before Analyze the six result chips are unchecked and the bar says “The pressures and the resultant appear after Analyze”; after it they are all checked, and you can hide any of them, for example to read the water pressure alone. Dimensions shows or hides the height of the wall and the heights of the resultants. The 3D view has no chips.
The Show bar before Analyze.
The Show bar after Analyze.
Entering the model, section by section
The Model tab has Model data and three sections: Wall, Soil layers and Groundwater. Wall, Soil layers and Groundwater work as an accordion: opening one closes the other two. Model data opens and closes on its own. Next and Back at the foot walk you through them. The tables list each field with its symbol, the unit in each unit system and its default. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. Defaults are those of the example, in SI · kN; after a change of unit the boxes hold the converted values unrounded: in SI · tf the unit weights read 1.60095 and 1.95786 t/m³, and in US the boxes read 14.7638 ft for the wall, 99.9443 pcf, 122.225 pcf and 9.84252 ft (the water depth and the thickness) for the same wall and soil.
Model data
The first section is the one every module has: a Model name, Save to file… and Open file…, Quick Save… (it asks for a name and keeps that version in this browser, to be brought back with Load or removed with Delete) and Reset to defaults, which returns to the example at once. Every change you make is also kept in this browser as you work. If you close the page and open it again the model you last worked on is there, and an analysed model comes back analysed and locked. See the workspace guide.
Wall
One number: the height of the wall. The note in the section says what the module assumes: a vertical wall that does not move, with level ground behind it.
Wall: the height.
Wall field
Field
Unit
Default
Height of the wall H From the top of the retained ground to the base of the wall. At least 0.1 m; an empty box stops the analysis.
m ft
4.5
Soil layers
The layers are entered one at a time from the top down. The New layer form at the top holds the values of the layer you are about to add; the Soil layers table under it lists the layers already in the model, each with its own colour, as on the drawing.
Choose how K0 of the layer is found with the four buttons under K₀ from. The boxes below the buttons follow the choice.
Fill in the thickness and the soil values, and press + Add layer. The layer goes to the bottom of the table and the counter (“1 of 6”) moves on. The form keeps its values, so the next layer starts from them.
To change a layer, press its row in the table: the layer opens in the form and changes apply as you type. The × at the end of a row takes the layer out.
The table shows H, γ, γsat, φ′ and OCR of each layer; a dash stands where a layer has no such value, such as φ′ of a Massarsch layer. The values in the field table are those of the example’s layer; the form for a new layer opens at 2 m, 18 and 20 kN/m³, 30°, OCR 1, PI 20, γd 17, γd(min) 15 and K₀ 0.5.
Soil layers: the New layer form above, the table of layers below.
Fields of each layer
Field
Unit
Default
Name (optional) A name for the drawing and the tables; left empty, the layers are called Layer 1, Layer 2 and so on.
—
automatic
Thickness Hi At least 0.01 m. The thickness of the last layer is not used: it runs on to the base of the wall, and the table shows a dash for it.
m ft
3
Unit weight above the water table γ At least 1 kN/m³.
kN/m³ t/m³ pcf
15.7
Saturated unit weight γsat At least 1 kN/m³. Below the water table the module uses γsat − γw for the vertical effective stress. Shown when the water table is Present.
kN/m³ t/m³ pcf
19.2
Drained friction angle φ′ From 0° to 50°. Shown when K₀ from is Jaky / Mayne–Kulhawy or Sherif (compacted sand).
°
35
Overconsolidation ratio (1 = normally consolidated) OCR At least 1. Shown when K₀ from is Jaky / Mayne–Kulhawy or Massarsch (clay).
—
1.5
Plasticity index (%) PI At least 0. Shown when K₀ from is Massarsch (clay).
—
20
Compacted dry unit weight behind the wall γd At least 1 kN/m³, and not below the next value. Shown when K₀ from is Sherif (compacted sand).
kN/m³ t/m³ pcf
17
Dry unit weight in the loosest state γd(min) At least 1 kN/m³. Shown when K₀ from is Sherif (compacted sand).
kN/m³ t/m³ pcf
15
Coefficient of earth pressure at rest K0 From 0 to 3. Shown when K₀ from is Given K₀.
—
0.5
The boxes of the New layer form change with the relationship chosen. Massarsch (clay) asks for the overconsolidation ratio and the plasticity index and has no friction angle. Sherif (compacted sand) asks for φ′, the compacted dry unit weight γd and the loosest-state value γd(min), and has no OCR. Given K₀ asks for K0 alone.
The form for a Massarsch (clay) layer.
The form for a Sherif (compacted sand) layer.
The form for a given K₀.
Groundwater
Choose Present or None. With a water table the depth zw is asked for, and the layers ask for their saturated unit weight. The water is behind the wall only: the module takes no water in front of it. A water table at the base of the wall or below it gives no water pressure on the wall.
Groundwater: Present or None, then the depth.
Groundwater fields
Field
Unit
Default
Depth of the water table below the top zw Zero or more. Shown when Water table is Present.
m ft
3
Choosing how K₀ is found
K0 is the one decision of this module, and each layer has its own. The four relationships:
Option
Coefficient
Jaky / Mayne–Kulhawy (the default) Use it for sand, gravel and other soils for which you hold φ′; the second form for overconsolidated soil, from clay to gravel.
K0 = 1 − sin φ′ (Jaky 1944) when OCR = 1; K0 = (1 − sin φ′) OCRsin φ′ (Mayne and Kulhawy 1982) when OCR is above 1
Massarsch (clay) Use it for fine-grained soil for which you hold the plasticity index. With OCR = 1 the factor is 1.
K0 = [0.44 + 0.42 PI/100] √OCR (Massarsch 1979 for the normally consolidated value; × √OCR when overconsolidated)
Sherif (compacted sand) Use it for a compacted sand fill behind the wall, where compaction raises the pressure above Jaky’s value.
K0 = (1 − sin φ′) + 5.5 [γd/γd(min) − 1] (Sherif, Fang and Sherif 1984)
Given K₀ Use it for a value from a test or another source.
K0 from 0 to 3, as you enter it
How they compare
The table gives K0 as the module returns it for the example’s layer (φ′ = 35°, γ = 15.7 kN/m³, water table at 3 m) with one relationship changed at a time, and the force P0 on the 4.5 m wall that follows.
One layer, four relationships (P0 in kN/m, z̄ in m)
Relationship
K0
P0
z̄
Jaky, OCR = 1
0.426
75.79
1.394
Mayne–Kulhawy, OCR = 1.5 (the example)
0.538
92.75
1.422
Mayne–Kulhawy, OCR = 3
0.801
132.6
1.460
Massarsch, PI = 20, OCR = 1.5
0.642
108.5
1.440
Sherif, γd = 17, γd(min) = 15 kN/m³
1.160
187.2
1.485
Given K0 = 0.5
0.500
86.97
1.414
Which should I pick?
Start from what you know about the soil. A normally consolidated sand or gravel with a measured φ′: Jaky with OCR = 1. An overconsolidated soil: keep Jaky / Mayne–Kulhawy and enter the OCR.
Use Massarsch for a clay when the plasticity index is what you hold.
Use Sherif only for a compacted fill, with the unit weights it asks for. The module stops if the compacted unit weight is below the loosest-state value.
Use Given K₀ for a value of your own. The module accepts 0 to 3 and, as for any layer, shows a caution above 3 (see the messages).
Different layers, different relationships. A sand fill over a clay can use Jaky / Mayne–Kulhawy in the first layer and Massarsch in the second.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the earth pressure calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed.” The Warnings button beside Edit inputs shows the number of cautions of the run. The Results tab then holds five blocks, and the Pressure view above the drawing carries the pressure. The drawings always read in m, kPa and kN/m; the cards, the tables and the report follow the unit system.
The Results tab of the example.
Card
The example
At-rest force P0 The resultant of the soil and the water pressure per unit length of wall.
92.75 kN/m
Line of action above the base z̄ The height of P0 above the base of the wall.
1.422 m
From the soil Ps The area of the effective-pressure diagram.
81.71 kN/m
From the water Pw The area of the pore-pressure diagram; zero for no water, or a water table at or below the base.
11.04 kN/m
Under the cards the tab lists: Values, step by step (the answer of the last step); Coefficient of earth pressure at rest of each layer (the relationship, its working and K0); Stresses at the breaks of the diagram, with σ′v, σ′h, u and σh = σ′h + u above and below each boundary and at the base; and Areas of the pressure diagrams, each with its force and its arm above the base.
Using the numbers
P0 is a force per unit length of wall, in kN/m (tf/m or kip/ft in the other systems), acting at z̄ above the base. The soil and the water are separate parts: Pw acts at a third of the water’s height above the water’s own base.
Changing the unit system after Analyze re-states the cards, the tables and the report without a new run. Edit inputs unlocks the model and clears the results.
A load on the ground behind the wall is not in these numbers. Add it from the Surcharge loads modules in your wall calculation.
The Calculations tab, the report and the Theory tab
The Calculations tab writes out the working in numbered steps, after a Given and what is sought block:
Coefficient of earth pressure at rest. The relationships used are given first, then the working of every layer with its numbers, and a table of the relationship, the working and K0 of each layer.
Stresses on the wall.σ′v = ΣγiHi (with γ′ = γsat − γw below the water table), σ′h = K0σ′v and u = γw (z − zw), then the table of stresses at the breaks of the diagram.
Resultant and its line of action.P0 is the sum of the areas, z̄ = Σ(area × arm) / P0, with the table of areas and the answer of each part.
The tab ends with an Answer line, for the example: P₀ = 92.75 kN/m, acting 1.42 m above the base. Before Analyze the tab asks you to analyse first.
The Calculations tab: the given data and the first step.
Steps 2 and 3 of the example.
The report
In the Report tab fill in the cover (project, location, client, report number, date, prepared by, title or qualification, checked by, and optional notes). The details are kept in this browser and saved with the model file. After Analyze, Save as PDF and Download Word (.docx) produce the report; the preview below the buttons is the report itself.
The report has six parts (seven when the run has cautions): 1 Introduction and scope, 2 Input data, 3 Calculation (3.1 the coefficient, 3.2 the stresses, 3.3 the resultant), 4 Results, 5 Summary and 6 References. It carries the section, the pressure and the 3D drawings and states the units with the conversion constants. When the run has cautions, a seventh part, Assumptions and notes, is added before the Summary and prints them.
The Report tab: the cover details, the two download buttons and the cover page.
The Theory tab
The Theory tab has six sections that open and close, with Open all and Close all: Earth pressure at rest; The coefficient K₀; Dry backfill; Partially submerged backfill; Worked examples (a wall in sand with a water table, and a wall in sand over clay); and References. The first worked example is the module’s own example.
Worked examples
The module opens on the first example, so you can follow its numbers on your own screen.
At the water table, 3 m down: σ′v = 15.7 × 3 = 47.10 kPa and σ′h = 0.5381 × 47.10 = 25.34 kPa. At the base, with γ′ = 19.2 − 9.81 = 9.39 kN/m³: σ′v = 47.10 + 9.39 × 1.5 = 61.19 kPa (61.185 exactly: the tables show 61.19 and the drawing 61.18), σ′h = 32.92 kPa and u = 9.81 × 1.5 = 14.72 kPa, so σh = 47.64 kPa.
Step 3: the resultant
Force per metre of wall and its arm above the base
Area
kN/m
Arm (m)
Soil, triangle 0–3 m: ½ × 3 × 25.34
38.02
2.500
Soil, rectangle 3–4.5 m: 1.5 × 25.34
38.02
0.7500
Soil, triangle 3–4.5 m: ½ × 1.5 × (32.92 − 25.34)
5.684
0.5000
Water, triangle 3–4.5 m: ½ × 1.5 × 14.72
11.04
0.5000
Ps = 38.02 + 38.02 + 5.684 = 81.71 kN/m, Pw = 11.04 kN/m, so P0 = 92.75 kN/m. The moments about the base give z̄ = (95.04 + 28.51 + 2.84 + 5.52) / 92.75 = 1.422 m. These are the four cards of the Results tab, and the three steps of the Calculations tab.
A second example: sand over clay
A 6 m wall retains 4 m of sand (γ = γsat = 18 kN/m³, φ′ = 34°, OCR = 2) over a clay (γ = γsat = 19 kN/m³, PI = 22, OCR = 3), with the water table at the top of the clay, 4 m down. Set H = 6 m and zw = 4 m, change layer 1 (thickness 4 m) in the table, and add the clay with Massarsch (clay).
K0 of the sand is (1 − sin 34°) × 2sin 34° = 0.4408 × 1.4734 = 0.650, and of the clay [0.44 + 0.42 × 22/100] × √3 = 0.5324 × 1.7321 = 0.922. At 4 m σ′v = 72.00 kPa, so σ′h is 46.76 kPa in the sand and 66.39 kPa in the clay: the diagram steps at the boundary. At 6 m, with γ′ = 19 − 9.81 = 9.19 kN/m³, σ′v = 72.00 + 18.38 = 90.38 kPa, σ′h = 83.34 kPa and u = 19.62 kPa. The areas are 93.53, 132.8 and 16.95 kN/m of soil and 19.62 kN/m of water.
The module gives Ps = 243.3 kN/m, Pw = 19.62 kN/m, P0 = 262.9 kN/m and z̄ = 1.784 m.
The two layers of the second example.
The Pressure view of the second example: the step in σ′h at the boundary of the layers.
A check in US units and in SI · tf
Switch to US, enter a 15 ft wall with γ = 100 pcf, φ′ = 30°, OCR = 1 and no water. K0 = 1 − sin 30° = 0.5 and, by hand, P0 = ½ × 0.5 × 100 × 15² = 5625 lb/ft and z̄ = 15/3 = 5 ft. The module gives 5.625 kip/ft at 5.000 ft.
In SI · tf a 5 m wall with γ = 1.8 t/m³, φ′ = 30°, OCR = 1 and no water gives, by hand, ½ × 0.5 × 1.8 × 5² = 11.25 tf/m at 5/3 = 1.667 m; the module gives 11.25 tf/m at 1.667 m.
How each input changes the answer
Each row changes one thing in the example (4.5 m wall, one layer of φ′ = 35°, OCR = 1.5, γ = 15.7 and γsat = 19.2 kN/m³, water table at 3 m) and runs the module again. The numbers are the ones the module gives, so you can reproduce any row. Forces are in kN/m, heights in m above the base.
The example with one change at a time
Change
K0
P0
z̄
The example The reference: Ps 81.71, Pw 11.04
0.538
92.75
1.422
H = 6 m Both parts grow; the water table is still at 3 m
0.538
180.9
1.840
Water table None Dry soil: P0 = ½ K0γH², at H/3
0.538
85.53
1.500
zw = 1.5 m More soil submerged: Pw 44.15
0.538
114.4
1.374
zw = 0 The whole wall under water: Pw 99.33
0.538
150.5
1.500
zw = 4.5 m No water pressure: the same as dry
0.538
85.53
1.500
φ′ = 30° A smaller φ′ raises K0
0.612
104.0
1.436
φ′ = 40° A larger φ′ lowers K0
0.464
81.44
1.405
OCR = 1 Jaky’s value, the smallest K0 here
0.426
75.79
1.394
OCR = 3 Overconsolidation raises K0
0.801
132.6
1.460
γ = 18, γsat = 20 kN/m³ Heavier soil: Ps 93.34
0.538
104.4
1.440
γsat = 21 kN/m³ Only the soil below the water table changes; Pw does not
0.538
93.84
1.411
Massarsch, PI = 20 A different relationship for the same layer
0.642
108.5
1.440
Sherif, 17 over 15 kN/m³ Compaction raises K0 above 1
1.160
187.2
1.485
Given K0 = 1 Pressure as in a fluid of the soil’s weight
1.000
162.9
1.476
A second layer: φ′ = 30°, OCR = 1, below 3 m The diagram steps where K0 changes
0.538 / 0.500
90.11
1.442
A second layer: Massarsch, PI = 30, OCR = 2, γ = 18, γsat = 19 kN/m³, below 3 m A larger K0 in the lower layer
0.538 / 0.800
113.9
1.292
The pattern follows the formulas. K0 is a property of each layer: it falls as φ′ grows and rises with the OCR. A higher water table adds water pressure, which does not depend on K0, and takes weight off the soil below it, so Ps falls while Pw grows. The height z̄ is H/3 for dry soil. With a water table inside the wall it sits a little below H/3 (1.37 to 1.42 m in these rows) because the water diagram is a triangle on the lower part of the wall; with the water at the surface the whole diagram is triangular again and z̄ returns to H/3 (1.500 m).
Messages: when the module stops, and cautions
The module meets a problem in two ways, and you see each one when you press Analyze. Nothing is calculated until the inputs are corrected.
A value outside its range
A window titled Cannot analyse — 1 input error (or the number of errors) lists each one, numbered, with the name of the field and its symbol. The bar stays at Not analysed. The messages read:
“Height of the wall (H) must be at least 0.1 m.” (also for a negative value), or “… is missing.” for an empty box;
“Thickness (H1) must be at least 0.01 m.”
“Unit weight above the water table (γ1) must be at least 1 kN/m³.”
“Drained friction angle (φ′1) must be at most 50°.”, or at least 0° the other way;
“Overconsolidation ratio (1 = normally consolidated) (OCR1) must be at least 1.”
“Plasticity index (%) (PI1) must be at least 0.”
“Coefficient of earth pressure at rest (K0,1) must be at least 0.” or “… must be at most 3.”
“Depth of the water table below the top (zw) must be at least 0 m.”
The number after a symbol is the layer. Ranges are those of the field tables above. Only the boxes the choices have opened are checked: with Massarsch chosen the friction angle is not asked for.
The window for a wall height of 0.
A condition of the solution
When every value is in range but the model cannot be solved, the window is titled Cannot analyse — a condition of the solution is not met and says “Nothing was calculated. Change the inputs marked in red, then press Analyze again.” The bar turns brown and reads Cannot be analysed followed by the reason. The reasons:
The layers above the last one already reach the base. “The last layer would have no thickness. Reduce the layer thicknesses or the number of layers.”
Sherif: the compacted dry unit weight is below the loosest-state value. The window gives both values, for example “Layer 1: the compacted dry unit weight 14.00 kN/m³ is below the loosest-state value 15.00 kN/m³”, and asks for a compacted value not lower than the minimum.
The saturated unit weight is not above that of water (9.81 kN/m³) in a layer that lies below the water table: the effective unit weight would be zero or negative. Enter the saturated unit weight of the soil. With no water table the saturated weight is not used and this check does not apply.
A condition of the solution: the layers reach the base.
The bar after such a stop.
A caution
A caution does not stop the run. If K0 of a layer comes out above 3, the analysis runs, the bar reads Analysed — the earth pressure calculated — 1 warning, and a Warnings (1) window opens with the text of the caution. The button Warnings on the bar shows it again; the report prints it. With no cautions the window says “No warnings — the assumptions of the solution hold for these inputs.”
A wall of φ′ = 50° and OCR = 50 gives K0 = 4.684, and a Massarsch layer of OCR = 50 gives 3.705; both trigger it. The module still returns the pressure for that K0. For an entered K0 the limit of the box, 3, applies first.
The caution for a high K0.
Good practice and common mistakes
Use this module for a wall that cannot move. A wall that yields is an active-pressure case, and the pressure is smaller; a wall pushed into the soil is passive. See the other modules of the family.
Choose K0 by the soil. Jaky for a normally consolidated sand, Mayne–Kulhawy (the same button, with the OCR) for an overconsolidated one, Massarsch for a clay by its PI, Sherif for a compacted fill. Different layers can use different relationships.
Enter the stress history. A larger OCR raises K0: the example’s OCR of 1.5 gives 0.538 against 0.426 for OCR = 1, and P0 follows.
Mind the water. Water pushes with its full hydrostatic pressure whatever K0 is, and below the table the soil counts at γsat − γw. Give γsat in every layer that lies below the water table.
Describe the layers from the top down and let the last one run to the base. A thickness that brings the layers to the base before the last layer stops the module.
Read P0 per unit length. It is in kN/m; multiply by the length of wall for a total force.
Add the other actions yourself. A surcharge on the ground behind the wall comes from the Surcharge loads modules; the wall calculation combines it with this pressure.
Keep a model file. The browser keeps the model as you work, but a file saved with Save to file… is the copy that travels with the project; use Quick Save… to compare variants.
Check the units after switching the system: type values in the units shown beside the boxes.
Frequently asked questions
What does the at-rest pressure mean?
It is the pressure of the soil on a wall that does not move, so that the soil stays in the state it had in the ground. The horizontal effective stress is K0 times the vertical effective stress. A wall that moves away from the soil gets a smaller pressure (active); one pushed into it gets a larger one (passive).
How is the force on the wall found?
The pressure diagram is built from the stresses at every layer boundary and at the water table, and its area is the force: P0 = Ps + Pw. The height z̄ follows from the moments of the areas about the base. For the module’s example P0 = 92.75 kN/m at 1.422 m.
Which relationship should I choose for K₀?
Jaky / Mayne–Kulhawy for sand and gravel with a measured friction angle, with the OCR for an overconsolidated soil; Massarsch for a clay by its plasticity index; Sherif for a compacted sand fill; Given K₀ for a value of your own, from 0 to 3.
Why is the last layer without a thickness?
The last layer runs on to the base of the wall, so its thickness is the wall height less the layers above. The thickness you type for it is kept and counts as soon as another layer is added below it. If the layers above already reach the base the module stops.
How many layers can I enter?
Up to six, from the top of the wall down. Each has its own unit weights and its own relationship for K0.
How does the water table change the answer?
Below it the soil counts at γsat − γw for the vertical effective stress, and the water adds its hydrostatic pressure u = γw (z − zw). A water table at or below the base gives no water pressure; at the surface the wall is under water over its whole height. Water is taken behind the wall only.
Why did the module stop?
Either a value is outside its range, with a window that names the field, or a condition of the solution is not met: the layers reach the base, a compacted dry unit weight is below the loosest-state value, or a saturated unit weight is not above that of water. Nothing is calculated until you correct it.
What does the caution about K₀ above 3 mean?
It marks a coefficient that is rare even in a heavily overconsolidated clay, so the inputs of that layer deserve a second look. The run is not stopped and the value is used as calculated.
Can I drag the water table or a layer boundary on the drawing?
No. Nothing on the drawing is a handle. Set the wall, the layers and the water table in the Model tab; the drawing follows as you type.
Does the module remember my inputs?
Yes. The model is kept in this browser as you work, so the page opens again where you left it, and an analysed model comes back analysed and locked. For a copy that travels use Save to file…; Quick Save… keeps named versions in the browser; Reset to defaults brings the example back.
Which units does it use?
SI · kN, SI · tf or US, switchable in the header at any time, even after Analyze. The force is in kN/m, tf/m or kip/ft, the pressures in kPa, t/m² or ksf in the tables as the system says, and the heights in m or ft.
Does the result include a surcharge?
No. The result is the pressure of the soil and the water. A load on the ground behind the wall comes from the Surcharge loads modules and is combined with this result in the wall calculation.
Calculate the Rankine active or passive pressure on a vertical wall with a level, layered backfill: the pressure diagram, the force from the soil and from the water, and the height of the resultant above the base. This manual walks through the screen, every input, the state, the tension zone and the strength of the layers, the results and the report, with a worked example you can reproduce.
Rankine gives the earth pressure on a vertical wall from a level backfill, by Rankine’s theory (1857): the pressure diagram on the back of the wall, its resultant per unit length of wall, and the height above the base at which the resultant acts. You choose the active state, when the wall moves away from the soil, or the passive state, when it is pushed into it. The backfill is entered in layers, each with its own unit weights and strength, with a uniform surcharge and a water table.
The pressure at a depth follows from the vertical effective stress σ′v there and the coefficient of the layer:
σ′a = Kaσ′v − 2c′√Ka, σ′p = Kpσ′v + 2c′√Kp
Ka = tan²(45° − φ′/2), Kp = tan²(45° + φ′/2), σ′v = q + ΣγiHi, u = γw (z − zw)
q is the uniform surcharge, γi and Hi are the unit weight and thickness of each layer above the depth z (below the water table the effective unit weight γsat − γw is used), and u is the pore-water pressure below the water table zw. The force is the area of the pressure diagrams, and its line of action is the centroid of those areas about the base.
What you can describe
A vertical wall of height H with a level backfill; the wall face is taken as frictionless.
The state: Active or Passive.
Up to six layers of backfill from the top down, each drained (with c′ and φ′) or undrained (with cu and φu = 0, so that Ka = Kp = 1).
A uniform surcharge q over the whole backfill surface, and a water table behind the wall, or none.
In the active state, how a cohesive backfill is treated near the top: with tension cracks or as the theoretical Rankine pressure.
The module gives the force from the soil, the force from the water, their total, the height of the total above the base, and the depth of the tension zone when there is one. It draws the section, the pressure diagram and a 3D view, writes out the working, and prepares a PDF or Word report.
Which module for which wall
A wall that cannot move (a basement wall, a braced wall) keeps the soil at rest: use At rest.
The module takes one uniform surcharge. For a point, line, strip, triangular, ramp or area load on the backfill, use the Surcharge loads manuals and add the result to the pressure from this module.
Before you start: what you need
Have these ready from your design and your ground investigation:
What
Where it comes from
Notes
The wall height H
Your design
From the top of the backfill to the base of the wall. At least 0.1 m.
The state
How the wall moves
Active when the wall moves away from the soil enough for the soil to reach its limit state; passive when the wall is pushed into the soil.
The layers of backfill
Boring logs and tests
For each layer from the top down: thickness (not for the last one, which runs to the base), unit weight, saturated unit weight if there is water, and the strength: φ′ and c′, or cu for an undrained layer.
The water table
Groundwater readings
The depth zw below the top of the wall, behind the wall; or none.
The surcharge q
Your design loading
A load per unit area on the whole backfill surface. Zero or more.
Quick start in seven steps
Open the module and choose the units
Open Rankine from the Lateral Earth Pressure family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Describe the wall
In Wall, choose Active or Passive, type the height H and the surcharge q. In the active state also choose how a cohesive backfill is treated near the top: Tension cracks — σ′ₐ = 0 above z₀ or No cracks — theoretical Rankine.
Enter the soil layers
In Soil layers fill the New layer form and press + Add layer. The layer goes to the bottom of the table. Press a row of the table to change it; × takes it out.
Enter the groundwater
In Groundwater choose Present and type the depth zw, or choose None.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the inputs lock. Press Pressure above the drawing to see the diagram on the wall.
Read the answer
The Results tab gives the force, its line of action above the base, the parts from the soil and from the water, the coefficient of each layer and the stresses behind the diagram.
Take the working and the report
The Calculations tab writes out the steps with your numbers. The Report tab saves a PDF or a Word file.
The screen: section, pressure and 3D
The layout is the one every GeoTerva module shares, described in the workspace guide. For this module the drawing has three views, chosen with the buttons above it:
Section is the wall, the surcharge, the soil layers with their unit weights and strength, the water table and the dimensions. It follows your inputs; after Analyze it adds the coefficient Ka (or Kp) of each layer.
Pressure is the pressure on the back of the wall. Before Analyze it shows the geometry only; after it, the soil pressure σ′ and the water pressure u stacked on the wall, the tension zone of a cohesive soil on the other side of the wall, the resultants of the soil, of the water and in total at their heights above the base, and the vertical effective stress σ′v in the backfill beside it, on the same scale.
3D is a length of the wall with the backfill on the right. After Analyze it shows the pressure prism on the wall face, the resultant and the pressure at the base.
Section before Analyze: the geometry and the soil of the example.
Section after Analyze: the coefficient of each layer is added.
Pressure after Analyze: the diagram on the wall, the resultants and the vertical stress.
3D after Analyze.
The Show chips
Above the drawing a Show bar switches parts of the drawing on and off. Dimensions is the only chip of the Section view and is on by default. The Pressure view has seven: Soil σ′, Water u, σ′v in the backfill, Ps soil, Pw water, P total and Dimensions. Before Analyze the pressure chips are unchecked and the bar says “The pressures and the resultant appear after Analyze”; after it they are all on. The 3D view has no chips.
The Pressure view before Analyze.
Zoom
Each view has −, a percentage and + at the lower right, and a button that fits the whole drawing in the pane. The mouse wheel zooms as well.
Entering the model, section by section
The Model tab has four sections: Model data (names, files and quick saves, the same in every module), Wall, Soil layers and Groundwater. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below the last one. The tables list every field with its symbol, the unit in each unit system and its default. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. Defaults are shown in SI; in US the boxes show the converted values, for example 19.685 ft for the 6 m wall.
Model data
The first section holds the model name, Save to file…, Open file…, Quick Save… with its list and Load and Delete, and Reset to defaults. It works as described in the workspace guide. Reset returns the example at once.
Model data.
Wall
Two choices and two numbers. State has the buttons Active and Passive. Tension zone has Tension cracks — σ′ₐ = 0 above z₀ and No cracks — theoretical Rankine, each with a (?) that opens a help window; it is shown in the active state only. The note under the fields says that the wall is vertical, the backfill level, the wall face frictionless and that the surcharge acts on the whole backfill surface.
Wall: state, tension zone, height and surcharge.
Wall fields
Field
Unit
Default
What to enter
Height of the wall H
m · ft
6
From the top of the backfill to the base of the wall. At least 0.1 m.
Uniform surcharge on the backfill q
kPa · t/m² · ksf
0
A load per unit area on the whole backfill surface. Zero or more; it adds to σ′v at every depth.
Soil layers
The section is a form over a table. The New layer form takes the name, the strength (Drained — c′, φ′ or Undrained — cᵤ, φᵤ = 0) and the values of one layer. + Add layer puts it at the bottom of the table, and a counter beside the button shows how many of the six layers there are. The Soil layers table lists every layer with its thickness, unit weights and strength, and a swatch of the colour the layer has on the drawing. Press a row to change that layer in the form (the form then reads Editing · Layer 1 and offers Done and Delete layer; the note says that changes apply as they are entered), or press × to take a layer out. The soil is entered from the top down, and the last layer runs on to the base of the wall.
Soil layers: the form over the table.
Layer fields (the form, and each row of the table)
Field
Unit
Default
What to enter
Name
—
automatic
Shown on the drawing and in the tables. Leave it blank for Layer 1, Layer 2 … (optional)
Thickness Hi
m · ft
2 (form)
From the top of the wall down. At least 0.01 m. The last layer has no thickness: it runs on to the base of the wall, and its row of the table shows a dash.
Unit weight above the water table γ
kN/m³ · t/m³ · pcf
18 (form)
At least 1. Used above the water table, or everywhere when there is no water table.
Saturated unit weight γsat
kN/m³ · t/m³ · pcf
20 (form)
At least 1, and above the unit weight of water (9.81 kN/m³) for a layer that reaches below the water table. The effective unit weight below the water table is γsat − γw. Shown when the water table is present.
Effective friction angle φ′
°
30 (form)
From 0° to 50°. It gives Ka or Kp of the layer. Shown when the strength of the layer is Drained.
Effective cohesion c′
kPa · kg/cm² · psf
0 (form)
Zero or more. Shown when the strength of the layer is Drained.
Undrained shear strength cu
kPa · kg/cm² · psf
25 (form)
Zero or more. The layer is taken with φu = 0, so Ka = Kp = 1 and the cohesion term is 2cu. Shown when the strength of the layer is Undrained.
A row of the table pressed: the layer is edited in the form.
Groundwater
Water table has the buttons Present and None. With Present the box for the depth zw appears, and the saturated unit weight is asked for in each layer. With None the water table and the saturated unit weights are left out, the unit weight above the water table is used everywhere and the force from the water is zero.
Groundwater.
Groundwater field
Field
Unit
Default
What to enter
Depth of the water table below the top zw
m · ft
3
Measured down from the top of the wall, behind the wall. Zero or more. A depth at or below the base of the wall has no effect. Shown when the water table is present.
Choosing the state, the tension zone and the strength
Three decisions shape the answer. The state is a decision about the wall; the other two are about the soil.
Choice
Options
What it does
State
Active, Passive
Active uses Ka and subtracts the cohesion term: the wall moves away from the soil. Passive uses Kp and adds it: the wall is pushed into the soil. The example gives 117.2 kN/m active and 927.5 kN/m passive.
Matters when σ′a is negative near the top of a cohesive backfill. See below.
Strength (of each layer)
Drained — c′, φ′, Undrained — cᵤ, φᵤ = 0
Drained uses the effective-stress parameters c′ and φ′. Undrained is a clay in the short term: cu with φu = 0, Ka = Kp = 1.
Tension cracks or theoretical Rankine
In a cohesive backfill the active pressure Kaσ′v − 2c′√Ka is negative near the top, down to the depth z0 where it is zero. Tension cracks — σ′ₐ = 0 above z₀, the default, takes the soil as unable to pull on the wall: cracks open down to z0, the soil loses contact with the wall there, and the negative part of the diagram is left out. No cracks — theoretical Rankine counts the negative area, which subtracts from the force and, being high on the wall, lowers the line of action. The (?) beside each button opens a window with a sketch and with the force for your current inputs.
With a wall of 6 m, the example layers and c′ = 10 kPa in layer 1, the module gives z0 = 2.165 m and, with cracks, Pa = 95.01 kN/m at 1.250 m above the base; without cracks, Pa = 82.51 kN/m at 0.6397 m. The Calculations tab states the force of the other assumption as well.
The (?) window of the tension cracks option.
A cohesive layer with a surcharge: the tension zone is drawn on the other side of the wall and marked “not counted (cracks)”.
Undrained layers
Choose Undrained — cᵤ, φᵤ = 0 for a layer when you work in total stress, for a clay in the short term. The layer takes Ka = Kp = 1, so its pressure is the total vertical stress less 2cu (active) or plus 2cu (passive), with the water pressure inside it. With layer 1 of the example undrained at cu = 25 kPa, the module gives Pa = 93.15 kN/m at 1.209 m above the base and a tension zone down to z0 = 3.000 m, the base of that layer.
Which should I pick?
State follows the wall. Use active for a wall that can move away from the soil, passive for the soil in front of a wall that is pushed into it. A wall that cannot move keeps the soil at rest: use At rest.
Cracks are the usual design assumption for the active pressure on a wall the soil cannot pull; leave the default unless you want the theoretical figure to compare.
Run both tension options on a cohesive backfill to see what the choice changes; the cracks option gives the larger force.
Drained or undrained by the time scale. Drained parameters are for the long term, undrained for a clay in the short term.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the earth pressure calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed.” Press Analyze, then open the Results tab for a block of cards. The Section view now shows the coefficient of each layer; the pressures are in the Pressure view.
The Results tab of the example.
Card
Symbol
What it is
The example
Rankine active force (passive when that state is chosen)
Pa or Pp
The resultant per unit length of wall, soil and water together.
117.2 kN/m
Line of action above the base
z̄
The height above the base at which the resultant acts. A dash when the net force is not a push.
1.781 m
From the soil
Ps
The area of the soil pressure diagram.
73.01 kN/m
From the water
Pw
The area of the water pressure diagram. Zero without a water table.
44.15 kN/m
Depth of the tension zone
z0
Shown only when the active pressure is negative at the top.
not shown
Below the cards four tables give the working in numbers: Values, step by step; Rankine coefficient of each layer (strength, φ′, c′ and 2c′√Ka when a layer has cohesion, and Ka or Kp); Stresses at the breaks of the diagram (z, σ′v, σ′a, u and the total at each break); and Areas of the pressure diagrams (each triangle or rectangle with its force and its arm above the base).
Using the numbers
The force is per unit length of wall, in kN/m (tf/m or kip/ft in the other unit systems); z̄ is a height above the base.
The soil and the water are shown apart so that you can use the water part with your own water level or drainage assumption.
Values are rounded where they are drawn, so the last digit of a figure on the drawing can differ by one from the card.
Changing the unit system after Analyze re-states the cards and the report without a new run; the drawings keep their labels in m, kN/m³, kPa and kN/m. Edit inputs unlocks the model and clears the results.
Warnings and notes
The green bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” When a condition needs your attention the button counts it, and a note about the tension zone reads Notes (1); see Messages.
The Calculations tab, the report and the Theory tab
The Calculations tab opens with Given and what is sought and then writes out the working in numbered steps, ending with the answer:
Rankine coefficients — the relationship (Ka = tan²(45° − φ′/2), or Kp for passive), the working of every layer with your friction angle, and a table of the strength and the coefficient of each layer (with the term 2c′√Ka when a layer has cohesion).
Stresses on the wall — σ′a = Kaσ′v − 2c′√Ka (or the passive form), σ′v = q + ΣγiHi and u = γw (z − zw), then the tables of the stresses at the breaks of the diagram and of the areas of the pressure diagrams.
Tension zone — only when the active pressure is negative at the top: z0, the assumption chosen, and the force that the other assumption would give.
Resultant and its line of action — Pa = Σ areas and z̄ = Σ(area × arm) / Pa, with the soil and the water added.
The Calculations tab: the first steps with the example’s numbers.
The report
In the Report tab fill in the cover (project, location, client, report number, date, prepared by, title or qualification, checked by, and optional notes). The details are kept in this browser and saved with the model file. Save as PDF and Download Word (.docx) produce the summary report; the preview below the buttons is the report itself.
The report has six parts: 1 Introduction and scope, 2 Input data, 3 Calculation (3.1 Rankine coefficients, 3.2 Stresses on the wall, then 3.3 Resultant and its line of action; a Tension zone part is added when there is one), 4 Results, 5 Summary and 6 References. It carries the section, the pressure and the 3D drawings and states the units and the unit weight of water used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has five sections: Active and passive states, Cohesionless backfill, Cohesive backfill, Worked examples and References. Each opens when you click its title, and the figures in it are computed from the equations.
Worked example: the module’s own example
The module opens on this example, so you can follow the numbers on your own screen.
Inputs of the example
Section
Input
Value
Wall
State, tension zone, height, surcharge
Active, cracks; H = 6 m; q = 0
Soil layers
Layer 1
3 m; γ = γsat = 16 kN/m³; φ′ = 30°; c′ = 0
Soil layers
Layer 2
to the base; γ = γsat = 18 kN/m³; φ′ = 35°; c′ = 0
Groundwater
Water table
Present; zw = 3 m
Step 1: the coefficients
Ka = tan²(45° − 30°/2) = 1/3 = 0.3333 for layer 1 and tan²(45° − 35°/2) = 0.2710 for layer 2.
Step 2: the stresses
Layer 1 has no water in it: σ′v = 16 × 3 = 48.00 kPa at 3 m, and σ′a = 0.3333 × 48.00 = 16.00 kPa. At the top of layer 2, σ′a = 0.2710 × 48.00 = 13.01 kPa. Below the water table the effective unit weight is 18 − 9.81 = 8.19 kN/m³, so at 6 m σ′v = 48.00 + 8.19 × 3 = 72.57 kPa and σ′a = 0.2710 × 72.57 = 19.67 kPa. The water pressure at 6 m is u = 9.81 × 3 = 29.43 kPa.
Step 3: the areas and the resultant
The diagrams give four areas: soil, triangle 0–3 m, ½ × 16.00 × 3 = 24.00 kN/m at 4.000 m above the base; soil, rectangle 3–6 m, 13.01 × 3 = 39.02 kN/m at 1.500 m; soil, triangle 3–6 m, ½ × (19.67 − 13.01) × 3 = 9.987 kN/m at 1.000 m; water, triangle 3–6 m, ½ × 29.43 × 3 = 44.15 kN/m at 1.000 m.
z̄ = (24.00 × 4.000 + 39.02 × 1.500 + 9.987 × 1.000 + 44.15 × 1.000) / 117.15 = 1.781 m above the base
These are the four cards of the Results tab, and the steps of the Calculations tab.
The same wall, passive
Choose Passive and press Analyze again (after Edit inputs). The coefficients are Kp = 3.000 and 3.690, the soil gives Ps = 883.4 kN/m, the water the same 44.15 kN/m, and the total is Pp = 927.5 kN/m at 1.985 m above the base.
A single layer of sand
For one layer of 16 kN/m³ sand at φ′ = 36° and no water, 6 m high, the formula P = ½ KγH² applies. Active, Ka = 0.2596, so Pa = ½ × 0.2596 × 16 × 36 = 74.77 kN/m; the module gives 74.77 kN/m at 2.000 m, which is H/3. Passive, Kp = 3.852, so Pp = 1109 kN/m; the module gives 1,109 kN/m at 2.000 m.
Other unit systems
The default example in SI · tf gives Pa = 11.95 tf/m at 1.781 m, and in US 8.028 kip/ft at 5.844 ft: the same wall and soil in other units (117.15 kN/m is 11.95 tf/m and 8.028 kip/ft). Two checks by hand:
US. A single layer of 120 pcf soil at 30°, H = 20 ft, no water and no surcharge: Ka = 1/3, so Pa = ½ × 0.3333 × 0.120 × 20² = 8.000 kip/ft acting at 20/3 = 6.667 ft. The module gives 8.000 kip/ft at 6.667 ft.
SI · tf. A single layer of 1.6 t/m³ soil at 30°, H = 6 m, q = 1 t/m², no water: Pa = ½ × 0.3333 × 1.6 × 36 + 0.3333 × 1 × 6 = 9.60 + 2.00 = 11.60 tf/m, at (9.60 × 2 + 2.00 × 3) / 11.60 = 2.172 m. The module gives 11.60 tf/m at 2.172 m.
How each input changes the answer
Each row changes one thing in the example (active, tension cracks, H = 6 m, q = 0, layers of 16 kN/m³ at 30° and 18 kN/m³ at 35°, water table at 3 m) and runs the module again. The numbers are the ones the module gives, so you can reproduce any row. Forces are Pa, or Pp in the passive rows, in kN/m.
The example with one change at a time
Change
Force
z̄ (m)
The default example: active, cracks, water at 3 m The reference Soil 73.01 · water 44.15 kN/m
117.2
1.781
Passive Kp replaces Ka; the soil force is about twelve times larger, the water force is the same Soil 883.4 · water 44.15 kN/m
927.5
1.985
q = 20 kPa The soil force rises; the water force does not Soil 109.3 · water 44.15 kN/m
153.4
2.106
H = 8 m Layer 2 grows to 5 m; the force rises faster than the height, and the water force more than doubles Soil 116.8 · water 122.6 kN/m
239.4
2.327
H = 4 m The water table now cuts only 1 m of layer 2 Soil 38.12 · water 4.905 kN/m
43.02
1.313
φ′ of layer 1 = 25° A smaller friction angle gives a larger Ka and a larger soil force Soil 78.23 · water 44.15 kN/m
122.4
1.876
φ′ of layer 1 = 35° A larger friction angle gives a smaller force Soil 68.52 · water 44.15 kN/m
112.7
1.693
φ′ of layer 2 = 40° A larger friction angle in layer 2 also lowers the force Soil 63.33 · water 44.15 kN/m
107.5
1.816
Water table: None The water force is zero and the full unit weight acts in layer 2: the total falls Soil 84.97 · water 0 kN/m
84.97
2.077
zw = 1.5 m A higher water table: the water force rises and the soil force falls, but the total rises Soil 57.37 · water 99.33 kN/m
156.7
1.804
zw = 6 m (the base) The same as no water table Soil 84.97 · water 0 kN/m
84.97
2.077
γ of layer 1 = 18 kN/m³ A heavier layer 1 raises σ′v and the soil force Soil 80.89 · water 44.15 kN/m
125.0
1.823
γsat of layer 2 = 20 kN/m³ A heavier saturated layer raises σ′v below the water table Soil 75.45 · water 44.15 kN/m
119.6
1.765
c′ of layer 1 = 10 kPa, tension cracks The tension zone z0 appears; the force falls and the line of action drops Soil 50.87 · water 44.15 kN/m
95.01
1.250
c′ of layer 1 = 10 kPa, no cracks The negative area is counted: a smaller force and a much lower line of action Soil 38.37 · water 44.15 kN/m
82.51
0.6397
c′ of layer 1 = 10 kPa, passive Cohesion adds in the passive state Soil 987.3 · water 44.15 kN/m
1,031
2.238
Layer 1 undrained, cu = 25 kPa Ka = 1 and the cohesion term 2cu acts; tension down to the base of layer 1 Soil 49.01 · water 44.15 kN/m
93.15
1.209
Layer 1 undrained, cu = 25 kPa, passive Passive adds the term Soil 889.4 · water 44.15 kN/m
933.5
2.078
c′ = 40 kPa in both layers, cracks The soil pushes nowhere: only the water acts, and a note says so Soil 0 · water 44.15 kN/m
44.15
1.000
c′ = 40 kPa in both layers, no cracks A negative net force: a warning, and no line of action Soil -190.5 · water 44.15 kN/m
-146.3
—
The pattern follows the formulas. The soil force follows K, the unit weights and the surcharge; a higher water table adds water force faster than it removes soil force; cohesion lowers the active force and raises the passive one; and the line of action depends on where the pressure sits on the wall, so a tension zone at the top moves it down.
Messages: when the module stops, and what it notes
The module checks the numbers it needs before it calculates. When one cannot be used it stops, and nothing is calculated.
If you press Analyze, a window titled Cannot analyse — 1 input error (or 2 input errors …) lists each message, and the box of the value is marked in red in the Model tab.
A condition of the solution that is not met stops it in a window titled Cannot analyse — a condition of the solution is not met, which ends “Nothing was calculated. Change the inputs marked in red, then press Analyze again.”
The limits are shown in the units of the unit system chosen.
The window Analyze opens when the wall height is zero.
Kind
When
Message and what to do
Stop
The wall height is below 0.1 m or empty
“Height of the wall (H) must be at least 0.1 m.” or “… is missing.” Enter a height.
Stop
The surcharge is negative
“Uniform surcharge on the backfill (q) must be at least 0 kPa.” Enter zero or a positive load.
Stop
A thickness is below 0.01 m
“Thickness (H1) must be at least 0.01 m.” Enter a thickness.
Stop
A unit weight is below 1
“Unit weight above the water table (γ1) must be at least 1 kN/m³.” Enter the unit weight of the soil.
Stop
The friction angle is outside 0° to 50°
“Effective friction angle (φ′1) must be at most 50°.” or “… at least 0°.” Enter an angle in that range.
Stop
A cohesion is negative
“Effective cohesion (c′1) must be at least 0 kPa.” Enter zero or more.
Stop
An undrained layer has no strength
“Undrained shear strength (cu,1) is missing.” Enter cu.
Stop
The water depth is negative
“Depth of the water table below the top (zw) must be at least 0 m.” Enter zero or more.
Stop
The layers above the last reach the base
“The layers above the last one already reach the base. The last layer would have no thickness. Reduce the layer thicknesses or the number of layers.”
Stop
A layer below the water table has γsat at or below 9.81 kN/m³
“Layer 2: the saturated unit weight 9.00 kN/m³ is not above that of water. Below the water table the effective unit weight (saturated minus that of water) would be zero or negative, so σ′ᵥ would not grow with depth. Enter the saturated unit weight of the soil (above 9.81 kN/m³).”
When more than one value is wrong the window lists them together, for example “2 input errors” with the wall height and the friction angle. In the table the layer number is part of the symbol: H1, φ′1, c′1 and γ1 belong to layer 1.
Notes and warnings
A note. When the active pressure is negative over the whole height of the wall and cracks are assumed, the soil pushes nowhere and only the water acts. The Warnings button then reads Notes (1) and the window Notes and assumptions (1) says “σ′a is negative over the whole height of the wall.” The card Depth of the tension zone shows the height of the wall.
A warning. With No cracks — theoretical Rankine, the same backfill gives a negative net force. The button reads Warnings (1) and a window opens: “The net active force is not a push: Pₐ = -146.35 kN/m. Counted over the whole height, the tension 2c′√Kₐ outweighs the pressure of the soil, so the soil would have to pull on the wall; the line of action is not given. Tension cracks (the other choice for the tension zone) give the pressure on a wall the soil cannot pull.” The line of action shows a dash.
Both come from c′ = 40 kPa in both layers of the example. With the usual values the window reads “No warnings — the assumptions of the solution hold for these inputs.”
Good practice and common mistakes
Choose the state by what the wall does. Active needs the wall to move away from the soil; if it cannot move, the soil is at rest and At rest gives the pressure. Passive needs a movement into the soil, usually far larger than the active one.
The last layer runs on to the base. Its thickness is not asked for; the layers above it must add up to less than H.
Enter the saturated unit weight above 9.81 kN/m³ for any layer that reaches below the water table.
Use the strength that fits the time scale. Drained c′ and φ′ for the long term; undrained cu with φu = 0 for a clay in the short term. Do not enter cu as c′.
Take cohesion with care. In the active state it lowers the pressure; the cracks option leaves out the tension zone, and the theoretical option counts it. Compare both before you rely on a cohesive backfill.
Read the soil and the water separately.Ps and Pw are on the cards, so you can use your own water level or drainage assumption.
The force is per unit length of wall, in kN/m; z̄ is a height above the base.
This module takes one uniform surcharge, a level backfill and a frictionless wall face. For a sloping backfill use Rankine, sloping backfill; for wall friction and a battered wall use Coulomb.
Check the units after switching the system: type values in the units shown beside the boxes.
Keep a record. The report is the record of an analysis; use Quick Save… or Save to file… to keep the model.
Frequently asked questions
How is the Rankine earth pressure on a wall calculated?
At each depth the pressure is σ′a = Kaσ′v − 2c′√Ka in the active state and σ′p = Kpσ′v + 2c′√Kp in the passive state, with Ka = tan²(45° − φ′/2) and Kp = tan²(45° + φ′/2), σ′v the vertical effective stress including the surcharge, and the water pressure u added below the water table. The force is the area of the diagrams, and the line of action is their centroid above the base. For the module’s example, Pa = 117.2 kN/m at 1.781 m above the base.
When do I use active and when passive?
Use active when the wall moves away from the soil enough for the soil behind it to reach its limit state, and passive for the soil in front of a wall that is pushed into it. For the module’s example the same wall gives 117.2 kN/m active and 927.5 kN/m passive.
What are the tension cracks, and which option should I use?
In a cohesive backfill the active pressure is negative near the top, down to the depth z0. With Tension cracks the soil is taken as unable to pull on the wall, and the negative part is left out. With No cracks — theoretical Rankine it is counted. Cracks is the default and the usual design assumption; the Calculations tab states the force of the other option.
How do I model a clay in the short term?
Choose Undrained — cᵤ, φᵤ = 0 for the layer and enter the undrained shear strength cu. The layer takes Ka = Kp = 1, and its pressure is the total vertical stress less (active) or plus (passive) 2cu, with the water pressure inside it.
How many layers can I enter, and how does the last one work?
Up to six, from the top down. The last layer has no thickness: it runs on to the base of the wall, and the layers above it must add up to less than the wall height.
How do I handle the water table?
In Groundwater choose Present and enter the depth zw below the top of the wall; enter the saturated unit weight of each layer, which must be above 9.81 kN/m³. Below the water table the effective unit weight γsat − γw is used and the water pressure is added. Choose None to leave the water out; a depth at the base of the wall gives the same result.
Does the module include a surcharge?
Yes, one uniform surcharge q over the whole backfill surface; it adds to the vertical stress at every depth. For a point, line, strip, triangular, ramp or area load, use the Surcharge loads manuals and add the pressure to this one.
What values does the module accept?
The wall height at least 0.1 m, the surcharge zero or more, each thickness at least 0.01 m, each unit weight at least 1 kN/m³, the friction angle from 0° to 50°, the cohesion zero or more, and the water depth zero or more. Anything else stops the analysis with a message that names the value.
Can I drag the layers or the water table on the drawing?
No. The drawing has no drag handles; change the values in the Model tab and the drawing follows. After Analyze the inputs are locked until you press Edit inputs.
Which units does it use, and what are the units of the answer?
SI · kN, SI · tf or US, switchable in the header at any time, even after Analyze. The force is in kN/m, tf/m or kip/ft, pressures in kPa, t/m² or ksf, and lengths in m or ft. The default example gives 117.2 kN/m, 11.95 tf/m or 8.028 kip/ft.
What does the report contain?
A cover with your project details and the answer, then six parts: introduction and scope, input data, the calculation step by step, results, summary and references, with the section, pressure and 3D drawings. It is saved as a PDF or a Word file once the model is analysed.
Does the module remember my inputs?
Yes. Every change is kept in the browser, and the page opens again on the model you last worked on, including its analysed state. Reset to defaults brings back the example; Save to file and Quick Save keep copies.
Rankine earth pressure, sloping backfill: user manual
Calculate the Rankine active or passive force on a frictionless wall behind a sloping backfill: the coefficient, the force, its inclination and line of action and the failure wedge, for a vertical or leaning back and for layered soil with water and a surcharge. This manual walks through the screen, every input, the results and the report, with a worked example you can reproduce.
Rankine, sloping backfill gives the active or passive force of the soil on a wall when the ground behind it slopes, and optionally when the back of the wall leans (Rankine 1857, in the form of Chu 1991 for an inclined back). Rankine’s state needs no wall friction: the direction of the force follows from the stress state of the backfill. The module returns the force per unit length of wall, its inclination, its height above the base and the failure wedge.
For one granular layer the force comes from a closed form:
α is the slope of the ground, θ the lean of the wall back from the vertical and φ′ the friction angle. For a passive wall the signs change and Pp uses Kp(R). For θ = 0 and α = 0 the coefficient is tan²(45° − φ′/2) for the active state, the familiar Rankine value.
What you can describe
A frictionless wall of vertical height H, its back inclined at θ from the vertical (+ toward the backfill), active (the wall moves away from the soil) or passive (the wall is pushed into it).
A backfill with a plane surface at α to the horizontal (+ rising away from the wall).
Up to six soil layers parallel to the ground surface, each drained (c′, φ′) or undrained (cᵤ, φᵤ = 0), each with its own unit weight.
A level water table behind the wall with hydrostatic pore pressure, and a uniform surcharge on the backfill surface.
For an active wall in a cohesive layer, the tension zone either as tension cracks or in the theoretical Rankine form.
Beyond the force, the module draws the section, the pressure along the back, the failure wedge, a chart of the coefficient against the slope and a 3D view, writes out the working step by step and prepares a PDF or Word report.
Which module for which wall
Level ground, vertical wall: the Rankine module. A wall that does not move: At rest.
This module: a sloping surface, a leaning back, layers, water and a surcharge, in Rankine’s state with no wall friction.
A rough wall (wall friction δ) or a wedge solution: Coulomb.
A load on the backfill that is not uniform (a strip, a line, a point) belongs to the Surcharge Loads modules, for example Strip load. The module gives the earth pressure; combine it with those and with the wall’s own checks in your wall calculation.
Before you start: what you need
Have these ready from your design and your ground investigation:
What
Where it comes from
Notes
The wall height H and the lean θ of its back
Your design
H is the vertical height. θ is positive when the back leans toward the backfill; θ = 0 is a vertical back. Both can be changed later.
The slope α of the ground
The site
Positive when the ground rises away from the wall, negative when it falls. A level surface is α = 0.
The soil layers
Ground investigation
For each layer: the unit weight γ, and either φ′ and c′ (drained) or cu (undrained clay in the short term). The layers are parallel to the ground surface; give the vertical thickness of each except the last, which runs on below the heel of the wall.
The water table
Ground investigation
The depth zw below the top of the wall and the saturated unit weight γsat of each layer, if water is present.
The surcharge q
Your design loading
A uniform load on the backfill surface, per unit area of the sloping surface. Zero for none.
Quick start in eight steps
Open the module and choose the units
Open Rankine, sloping backfill from the Lateral Earth Pressure family. Pick SI · kN, SI · tf or US in the header; you can change it at any time, even after Analyze.
Describe the wall
In Wall, choose Active or Passive, and for an active wall the tension zone. Enter the vertical height H and the lean θ of the back.
Describe the ground surface
In Backfill, enter the slope α.
Describe the soil
In Soil layers, change Layer 1 to your soil and, for a layered backfill, fill in the New layer form and press Add layer (up to six).
Add water and a surcharge if you need them
In Groundwater choose Present and enter the depth zw. In Surcharge enter q.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed — the earth pressure calculated and the inputs lock.
Read the answer
The Results tab gives the force, its inclination β and, for a layered or loaded model, its height above the base. The Pressure, Chart and 3D views show it.
Take the working and the report
The Calculations tab shows each step with your numbers. The Report tab saves a PDF or a Word file.
The screen: four views of the wall
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing on the left has four views, chosen with the buttons above it, and the model and its results are on the right.
The module after Analyze, on its default example.
Section is the geometry only: the wall with its back at θ, the backfill at α, the layers parallel to the surface, the height H and the soil of each layer. It follows your inputs before Analyze.
Pressure is available after Analyze. It draws the pressure along the back of the wall, the force at its line of action inclined at β to the normal of the back, its horizontal and vertical components, the water pressure and the failure wedge from the heel B. Before Analyze it shows the geometry only.
Chart is available after Analyze: the coefficient against the slope α for the friction angle of each granular layer and the lean θ, from α = 0 to α = φ′, with the model marked. For a cohesive layer the coefficient changes with depth, so there is no single curve.
3D shows a length of wall with the sloping backfill, and the force after Analyze.
Section before Analyze: the drawing follows the inputs.
Pressure after Analyze: the diagram, the force, its components and the failure wedge.
Chart: the coefficient against the slope, the model marked.
3D after Analyze.
The Show chips
Above the drawing a Show bar has chips. The Section view has one, Dimensions. The Pressure view has six: Pressure p, Resultant P, Ph, Pv components, Water u, Failure wedge and Dimensions. All are on by default; each switches its own element on or off, so you can clear the drawing down to what you want to look at.
Pressure with Ph, Pv components and Failure wedge switched off.
Zoom, and nothing to drag
The [−], [+] and fit buttons at the lower right of the drawing zoom it, and the wheel over the drawing does the same; once zoomed in, dragging the background moves the view. The first press of [+] takes the zoom to 135%. The drawing has no handles: the geometry is changed in the Model tab, not by dragging, and the drawing follows.
Entering the model, section by section
The Model tab has six sections. Each opens when you click its title; Next and Back walk you through them and Analyze waits below the last. The unit of a field is shown beside its box; where a unit is listed twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, equal ones merged. The defaults below are in SI; in US the example shows H = 12 ft and γ = 115 pcf.
Model data
The first section is about the model itself, not the soil. Every change is kept in this browser: the page opens again with the model you last worked on, analysed or not, with the unit system. You can also give the model a name, Save to file… and Open file… to keep it with the project or move it to another computer, make named Quick Save… copies and Load or Delete them, and Reset to defaults to return to the example. The details typed on the Report tab are kept with the model.
Model data: the model is kept, and can be saved to a file.
Wall
The state of the wall, the tension zone for an active wall, the vertical height and the lean of the back.
Wall: state, tension zone, height and lean.
Fields of the Wall section
Field
Unit
Default
What to enter
State
—
Active
The wall moves away from the soil (Active) or is pushed into it (Passive).
Tension zone
—
Tension cracks
For a cohesive layer where the active pressure is negative near the top: Tension cracks — σ′ₐ = 0 above z₀ or No cracks — theoretical Rankine. A (?) beside each opens a help window. Shown when the state is Active.
Height of the wall (vertical) H
m · ft
3.6576
The vertical height. The module needs at least 0.1 m.
Inclination of the wall back from the vertical (+ toward the backfill) θ
°
20
From −45° to 45°. Zero is a vertical back.
Backfill
One number: the slope of the ground surface behind the wall. The surface is a plane; the layers of the next section run parallel to it.
Backfill: the slope.
Field of the Backfill section
Field
Unit
Default
What to enter
Slope of the backfill (+ rising away from the wall) α
°
20
From −45° to 45°. Rankine’s state of the slope needs |α| ≤ φ′ in every granular layer.
Soil layers
A New layer form over a table of Soil layers. The module opens with one layer, Layer 1, whose row shows its unit weight, friction angle and cohesion; a — means a value that does not apply (no thickness for the last layer, no saturated unit weight without water, no cᵤ for a drained layer). Press a row of the table to change that layer in the form: the form then reads Editing · Layer 1, Done closes it, and the changes apply as you type. Delete layer and × take a layer out. To add a layer fill in the form and press Add layer; it goes to the bottom of the table. The counter (“1 of 6”) shows how many of the six you have used.
Soil layers: the form and the table, one layer.
Fields of a layer
Field
Unit
Default
What to enter
Name
—
automatic
Optional; shown on the drawing and in the tables. Blank gives Layer 1, Layer 2 …
Strength
—
Drained
Drained — c′, φ′ or Undrained — cᵤ, φᵤ = 0.
Thickness (vertical) Hi
m · ft
2
Measured vertically; the layers lie parallel to the ground surface. The last layer has no thickness: it runs on below the heel of the wall.
Unit weight above the water table γ
kN/m³ · t/m³ · pcf
18 (Layer 1: 18.0651)
Moist or dry unit weight.
Saturated unit weight γsat
kN/m³ · t/m³ · pcf
20
Used below the water table. Shown when a water table is present.
Effective friction angle φ′
°
30
From 0° to 50°. Shown when the layer is drained.
Effective cohesion (vertical back only) c′
kPa · kg/cm² · psf
0
Zero or more. A layer with c′ above zero needs θ = 0. Shown when the layer is drained.
Undrained shear strength (vertical back only) cu
kPa · kg/cm² · psf
25
Total stress, with φᵤ = 0. Needs θ = 0. Shown when the layer is undrained.
The last layer runs on below the heel. The heel of the wall lies at the depth H(1 + tan θ tan α) below the ground surface continued over the wall; the layers above the last one must end above it.
A layer with cohesion or an undrained layer needs a vertical back (θ = 0). With a leaning back the layers are granular.
Only the last layer has no thickness. When you add a layer below Layer 1, Layer 1 shows its thickness in the table (2 m by default) and the new layer, now the last, shows a dash.
The form before Add layer.
Two layers: Layer 1 above, Dense sand below it as the last layer.
Groundwater
A level water table behind the wall, if any, with hydrostatic pore pressure and no seepage. Below it each layer weighs γsat − γw in the effective vertical stress, and the pore pressure u = γw(z − zw) acts normal to the back. Choosing Present adds the depth box here and the Saturated unit weight to each layer.
Groundwater: present, at 3 m.
Fields of the Groundwater section
Field
Unit
Default
What to enter
Water table
—
None
Present or None, a level table behind the wall.
Depth of the water table below the top of the wall zw
m · ft
3
Zero or more. Shown when water is present.
Surcharge
One number: a uniform load on the backfill surface, vertical and per unit area of the sloping surface. Per unit horizontal area it is q/cos α, added to the vertical stress at every depth as a layer of that weight.
Surcharge: the load.
Field of the Surcharge section
Field
Unit
Default
What to enter
Uniform surcharge on the backfill surface q
kPa · t/m² · ksf
0
Vertical, per unit area of the sloping surface. Zero or more; zero for none.
Choosing the state, the strength and the tension zone
Four choices decide which solution the module uses.
Choice
Options
Use it when
State
Active (default), Passive
Active: the wall can move away from the soil. Passive: the wall is pushed into the soil and the soil resists. For the default example the coefficient is 0.7758 active and 2.901 passive, and the force 93.75 and 350.6 kN/m.
Drained for sand and for long-term clay in effective stress. Undrained for a clay in the short term, in total stress. Both are for a vertical back only when the layer has cohesion.
Matters only for a cohesive or undrained layer, where the active pressure is negative near the top. With cracks the soil is taken to pull nothing on the wall and the negative part is left out; with no cracks the negative area is subtracted from the force.
Water table
Present, None (default)
When there is a groundwater level behind the wall.
Which solution the module applies
One granular layer, no water, no surcharge: the closed form of Chu 1991 shown above, with the working of each equation.
A granular layer with a surcharge or water, or several layers: each layer is taken in the Rankine state of the slope under the weight of the soil above it, and the pressure on the back is integrated along the wall. The module states in the Calculations tab that this layered form is an extension made in the program, and that for one layer it reduces to the closed form.
A layer with cohesion (or undrained) behind a vertical back: Mazindrani and Ganjali 1997. The coefficient depends on the cohesion divided by the vertical stress, so it changes with depth and the pressure is not a straight line.
The water table is level, whatever the slope of the ground. A table that meets the back (zw < H) makes the model layered; one at or below the base of the wall has no effect.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the earth pressure calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed.” Edit inputs unlocks the model and clears the results. After a reload an analysed model comes back analysed.
The Results tab has a Rankine force block of cards, then a Values, step by step table that repeats the numbers of each step. The cards depend on the model:
The Results tab of the default example.
Card
Symbol
What it is
Default example
Rankine active (or passive) force
Pa (Pp)
The force per unit length of wall, in kN/m (tf/m or kip/ft).
93.75 kN/m
Coefficient
Ka(R)
Shown for one granular layer without surcharge or water.
0.7758
Inclination to the normal of the back
β
The angle between the force and the normal of the back.
29.95°
Failure plane from the horizontal
η
The side of the failure wedge, from the heel B.
48.42°
Line of action above the base
z̄
Appears, instead of the coefficient and η, when the model is layered, loaded, wet or cohesive. For one granular layer there is no card: the line of action is H/3.
no card; H/3 = 1.219 m is in the Pressure view and the Calculations tab
From the water
Pw
When a water table meets the back: the part of the force from the pore pressure, normal to the back.
—
Depth of the tension zone
z₀
For a cohesive or undrained layer in the active state.
—
A layered, wet or cohesive model adds tables to the Results tab: the layers and their Rankine state (depth, unit weight, strength, coefficient, β, η), the stresses on the back (the vertical stress, the pressure on the back and the pore pressure at the top, the water table, the base and, for a cohesive layer, where the pressure is zero) and the force of each layer with its normal and tangential parts and arm.
Using the numbers
P is the earth force per unit length of wall, inclined at β to the normal of the back; when a water table meets the back it includes the water pressure, and the card From the water shows that part. Resolve it into its horizontal and vertical parts (shown in the Pressure view) for the wall checks you make.
For one granular layer the force acts at H/3 above the base; for any other model read the line of action from the card.
Changing the unit system after Analyze re-states the cards, the Calculations tab and the text of the report without a new run; the drawings stay in SI (m, kN, kPa).
Notes and Warnings. When the module has something to say about a result, the bar shows Notes (n) or Warnings (n); press it to read the text. See the messages below.
The Calculations tab, the report and the Theory tab
The Calculations tab opens with a Given and what is sought card and then numbered steps, each with its equation, the equation with your numbers and the result. For one granular layer there are four:
Angle ψ and the coefficient — ψa and Ka(R), with the numerator and denominator worked out.
Force — Pa = ½ γH² Ka(R), at H/3 above the base.
Direction of the force — β.
Failure wedge — η.
The Given card of the Calculations tab.
Step 1: the coefficient, with the numbers.
A layered, wet or cohesive model has five steps: the Rankine state of each layer, the stresses on the back, the tension zone (when a cohesive layer has one), the force and its direction and the failure wedge. The tables of the Results tab appear in the steps they belong to. Before Analyze the tab asks you to press Analyze.
The report
In the Report tab fill in the cover (project, location, client, report number, date, prepared by, title / qualification, checked by, and optional notes). The details are kept in this browser and saved with the model file. After Analyze, Save as PDF and Download Word (.docx) produce the summary report; the preview below the buttons is the report itself.
The report has six parts: 1 Introduction and scope, 2 Input data, 3 Calculation (one part for each step), 4 Results, 5 Summary and 6 References. It carries the section, the pressure view, the chart and the 3D drawing, and the warnings and notes of the analysis. It states the units with the conversion constants.
The Report tab: the cover details and the two download buttons.
The Theory tab
The Theory tab has six sections, each opening when you click it: Sloping backfill, General case, Cohesive backfill, Layered backfill, Worked example and References. They carry the equations, tables of the coefficient and the figures of the method, computed from the same equations as the module.
Worked example: the module’s own example
The module opens on this example, so you can follow the numbers on your own screen.
Inputs of the example
Section
Input
Value
Wall
State, tension zone
Active, tension cracks
Wall
Height and lean
H = 3.6576 m (12 ft), θ = 20°
Backfill
Slope
α = 20°
Soil layers
One drained layer
γ = 18.0651 kN/m³ (115 pcf), φ′ = 30°, c′ = 0
Groundwater, Surcharge
None, q = 0
Step 1: the angle and the coefficient
sin⁻¹(sin 20° / sin 30°) = sin⁻¹(0.3420 / 0.5000) = 43.160°, so ψa = 43.160° − 20° + 2 × 20° = 63.160°. The numerator is cos(0°) √(1 + 0.5000² − 2 × 0.5000 × cos 63.160°) = 1.0000 × √0.7985 = 0.8936 and the denominator cos²20° × (cos 20° + √(0.5000² − 0.3420²)) = 0.8830 × (0.9397 + 0.3647) = 1.1518, so Ka(R) = 0.8936 / 1.1518 = 0.7758.
Step 2: the force
Pa = ½ × 18.0651 × 3.6576² × 0.7758 = 93.75 kN/m, at H/3 = 1.219 m above the base
Steps 3 and 4: the direction and the failure plane
β = tan⁻¹[sin 30° sin 63.160° / (1 − sin 30° cos 63.160°)] = tan⁻¹(0.4461 / 0.7743) = 29.95° to the normal of the back, and η = 45° + 30°/2 + 20°/2 − ½ × 43.160° = 48.42°. The Pressure view shows the components of the force: Pv = 71.76 and Ph = 60.32 kN/m, which combine to the same 93.75 kN/m.
These are the numbers on the Results tab, the four steps of the Calculations tab and the figures of the Pressure view.
Two simple cases you can check by hand
Vertical wall, level ground: H = 6 m, θ = 0, α = 0, φ′ = 30°, the same soil. The module gives Ka(R) = 0.3333 = tan²30° and Pa = 108.4 kN/m = ½ × 18.0651 × 6² × 0.3333, with β = 0° (horizontal) and η = 60.00°.
Vertical wall, sloping ground: H = 5 m, α = 10°, γ = 18 kN/m³, φ′ = 30°. Ka = cos 10° (cos 10° − √(cos²10° − cos²30°)) / (cos 10° + √(cos²10° − cos²30°)) = 0.3495 and Pa = ½ × 18 × 5² × 0.3495 = 78.64 kN/m, parallel to the ground (β = 10.00°).
Switch to US with the example as it stands (H = 12 ft, γ = 115 pcf). The module gives Pa = 6.424 kip/ft, and by hand ½ × 0.115 × 12² × 0.7758 = 6.424 kip/ft. Enter H = 20 ft and γ = 120 pcf and it gives 18.62 kip/ft: ½ × 0.120 × 20² × 0.7758 = 18.62. In SI · tf the example gives 9.559 tf/m, which is 93.746 / 9.80665, and H = 5 m gives 17.86 tf/m (½ × 18.0651 × 5² × 0.7758 = 175.2 kN/m, divided by 9.80665). The coefficient and the angles do not change with the units.
How each input changes the answer
Each row changes one thing in the default example and runs the module again. The numbers are the ones the module gives, so you can reproduce any row. The first table is one granular layer, where the module returns the coefficient.
The default example with one change at a time (one granular layer)
Change
K
P (kN/m)
β (°)
η (°)
The default example The reference
0.7758
93.75
29.95
48.42
Passive instead of active Kp(R) replaces Ka(R); the force is several times larger and nearly normal to the back
2.901
350.6
7.673
61.58
H = 6 m The force follows H²; the coefficient and the angles do not change
0.7758
252.3
29.95
48.42
H = 2 m The same, downward
0.7758
28.03
29.95
48.42
θ = 0 (vertical back) A smaller coefficient and force; the force is inclined at β = α
0.4142
50.05
20.00
48.42
θ = 10° Between the vertical back and the 20° back
0.5617
67.87
28.30
48.42
θ = −10° (leaning away) The smallest coefficient of the three
0.3433
41.49
3.151
48.42
α = 0 (level ground) Level ground gives a smaller force; η rises to 60°
0.4935
59.64
27.52
60.00
α = 10° Between level and 20°
0.6000
72.50
29.48
54.84
α = −10° (ground falling away) A falling surface lowers the force
0.4165
50.33
23.64
65.16
φ′ = 25° A smaller friction angle: a larger coefficient and force
0.9272
112.0
24.69
40.49
φ′ = 35° A larger friction angle: a smaller coefficient and force
0.6755
81.63
34.98
54.20
γ = 20 kN/m³ The force is proportional to γ; the coefficient does not change
0.7758
103.8
29.95
48.42
Where the back is vertical (θ = 0) the force is inclined at β = α, parallel to the ground. The pattern follows the formulas: P = ½ γH² K is proportional to γ and to H², and K depends on φ′, α and θ only.
Surcharge, water, layers and cohesion
Case
P (kN/m)
β (°)
z̄ (m)
Surcharge q = 15 kPa Adds qHK cos θ / cos(θ − α) = 40.0 kN/m to the 93.75, acting higher
133.7
29.95
1.402
Water table 1.5 m below the top (γsat = 20) The force includes a water part, normal to the back, shown on its own card
103.0
23.18
1.148
Two layers, vertical back, α = 15°, H = 6 m (γ 17 and 19, φ′ 30° and 36°) Each layer in its own Rankine state; β equals α
94.29
15.00
2.054
The same two layers, θ = 15°, α = 10°, H = 5 m An inclined back with granular layers
100.2
32.77
1.706
One layer, γ = 18, φ′ = 20°, c′ = 13.5 kPa, α = 10°, H = 7.5 m, tension cracks The pressure is negative above z₀
133.4
10.00
1.780
The same, no cracks The negative area is subtracted: a smaller force, acting lower
113.0
10.00
0.8786
Undrained, γ = 18, cu = 30 kPa, level, H = 6 m z₀ = 2cu/γ; Pa = ½ × 48 × (6 − 3.333) = 64.0
64.00
0
0.8889
The default example, passive, water table 1.5 m below the top The force with its water part and the line of action
327.7
7.105
1.253
In these cases z̄ is the height of the line of action above the base. The water part of the force, on its own card, is 24.30 kN/m in the water-table row (½ × 9.81 × 2.1576² / cos 20° by hand) and 24.30 kN/m in the passive one. The tension depth in the cohesive row is z₀ = 2.142 m, and in the undrained row 3.333 m.
Messages: when the module stops, notes and warnings
The module checks the model before it calculates. When a condition of the solution is not met it stops and nothing is calculated.
On the Model tab the Analyze bar turns brown and reads Cannot be analysed, followed by the message and a line of explanation.
If you press Analyze, a window titled Cannot analyse — a condition of the solution is not met gives the message and says “Nothing was calculated. Change the inputs marked in red, then press Analyze again.”
A value outside the limit of its box, such as a wall height under 0.1 m, opens Cannot analyse — 1 input error with the name of the field and the limit.
A stop in the Model tab: a granular backfill steeper than its friction angle.
The window that Analyze opens.
Kind
When
Message and what to do
Stop
H is below 0.1 m
“Height of the wall (vertical) (H) must be at least 0.1 m.” Enter a height of at least 0.1 m.
Stop
H is empty
“Height of the wall (vertical) (H) is missing.” Enter a height.
Stop
|α| is above φ′ in a granular layer
“Layer 1: |α| > φ′ — the backfill is steeper than its angle of friction.” Enter |α| ≤ φ′, or give the layer its cohesion.
Stop
A drained layer with φ′ = 0 and c′ = 0
“Layer 1: a granular layer without strength (c′ = 0, φ′ = 0).” Enter φ′ above zero, a cohesion, or choose Undrained for a clay.
Stop
A cohesive or undrained layer with θ not zero
“A cohesive or undrained layer with an inclined back.” Enter θ = 0, or drained granular layers.
Stop
1 + tan θ tan α is close to zero
“The back of the wall is nearly parallel to the ground surface.” Enter a smaller θ or α.
Stop
The layers above the last already reach below the heel
“The layers above the last one already reach below the heel.” Reduce the thicknesses or the number of layers.
Stop
Water is present and a layer’s γsat is not above 9.81 kN/m³
“Layer 1: the saturated unit weight 9 kN/m³ is not above that of water.” Enter the saturated unit weight of the soil.
Stop
A cohesive layer on a slope steeper than its strength allows
“Layer 1: the slope does not stand down to the heel” (“the undrained slope” for an undrained layer), with the stress and the depth at which it fails. Enter a smaller α or a stronger layer.
Notes and warnings after a run
Warning — with No cracks — theoretical Rankine, if the tension of a cohesive layer outweighs the pressure: “The net active force is not a push”, with the value (for example H = 2 m, φ′ = 20°, c′ = 20 kPa, γ = 18 gives Pa = −38.37 kN/m). Choose tension cracks to get the pressure on a wall that the soil cannot pull.
Note — with tension cracks, if the pressure is negative over the whole height: “σ′a is negative over the whole height of the wall”. The force is zero.
Note — when the failure plane from the heel is steeper than the back (for example θ = −35° with level ground): “η = 60.00° is steeper than the wall back (90° + θ = 55°)”. The plane runs behind the back; the coefficient and the force are the Rankine stress state of the backfill acting on the plane of the back.
Good practice and common mistakes
Enter H as the vertical height and θ with its sign: positive when the back leans toward the backfill, negative when it leans away.
Keep |α| ≤ φ′. A granular slope steeper than its friction angle does not stand, and Rankine’s state of the slope has no value for it. Give the layer cohesion, or reduce the slope.
Use a vertical back for cohesive and undrained layers (θ = 0); with a leaning back the layers are granular.
Choose the tension zone deliberately. Tension cracks give the larger active force and are the usual choice for a wall that the soil cannot pull; the no-crack choice is the theoretical Rankine pressure. The Calculations tab states the force with the other choice too.
Enter the surcharge per unit area of the sloping surface, vertical, not per unit horizontal area; the module converts it with 1/cos α.
Do not add the water twice. When a water table meets the back, P already includes the water pressure; the card From the water is the part of it that acts normal to the back.
Read the force as per metre of wall and resolve it into its horizontal and vertical parts for the wall checks; the Pressure view shows both.
The module keeps your model in this browser. Save it to a file to keep it with the project or move it to another computer.
Check the units after switching the system: type values in the units shown beside the boxes.
Frequently asked questions
How is the Rankine force calculated with a sloping backfill and a leaning wall?
For one granular layer the module finds the angle ψa = sin⁻¹(sin α / sin φ′) − α + 2θ, then Ka(R) (Chu 1991) and Pa = ½ γH² Ka(R) at H/3 above the base. For the module’s example Ka(R) = 0.7758 and Pa = 93.75 kN/m, inclined at β = 29.95° to the normal of the back.
Why is the force not horizontal?
Rankine’s state has no wall friction. The pressure on a vertical back acts parallel to the ground surface, at α; on a leaning back it acts at β to the normal of the back, which the module reports. The Pressure view shows its horizontal and vertical components.
What happens when the slope is steeper than the friction angle?
A granular layer cannot stand on a slope steeper than its friction angle and the solution has no real value, so the module stops with “|α| > φ′ — the backfill is steeper than its angle of friction”. Enter |α| ≤ φ′, or give the layer a cohesion.
Can I use a cohesive soil with a leaning wall back?
No. A layer with cohesion, or an undrained layer, needs a vertical back (θ = 0), because the cohesive solution (Mazindrani and Ganjali 1997) is for a vertical back. With a leaning back the layers are drained and granular.
What do the tension-zone choices do?
In an active cohesive layer the pressure is negative near the top, down to a depth z₀. With tension cracks the module takes the soil to pull nothing on the wall and leaves the negative part out, which gives the larger force. With no cracks it subtracts the negative area, which can even make the net force not a push; the module then warns you.
How does the module deal with layers?
The layers are parallel to the ground surface. Each is taken in the Rankine state of the slope with its own unit weight and strength under the weight of the soil above it, and the pressure on the back is integrated along the wall. For one layer this reduces to the closed form. You can have up to six layers; the last runs on below the heel of the wall.
How are water and a surcharge included?
A level water table adds the pore pressure u = γw(z − zw) normal to the back and uses the effective unit weight below it; the water part of the force has its own card. A surcharge is vertical per unit area of the sloping surface and is added to the vertical stress at every depth, as a layer of weight q/cos α.
Active or passive: what is the difference in the results?
For the same soil and wall the passive coefficient is far larger. In the default example Kp(R) = 2.901 against Ka(R) = 0.7758, and the passive force is 350.6 kN/m against 93.75 kN/m. The passive force is nearly normal to the back (β = 7.673°).
What values does the module accept?
The wall height at least 0.1 m, the lean and the slope from −45° to 45°, the friction angle from 0° to 50°, and at most six layers. A condition of the solution that is not met, such as |α| > φ′, stops the analysis with a message that names the layer.
Does the module remember my model?
Yes. Every change is kept in this browser, so the page opens with your last model, analysed or not, and the unit system. Use Save to file to keep a model with the project, Quick Save for named copies and Reset to defaults to return to the example.
Which units does it use, and what are the units of the answer?
SI · kN, SI · tf or US, switchable in the header at any time, even after Analyze. The force per unit length is in kN/m, tf/m or kip/ft, the unit weight in kN/m³, t/m³ or pcf, the cohesion in kPa, kg/cm² or psf and the pressure in kPa, t/m² or ksf.
What does the report contain?
A cover with your project details and a summary, then six parts: introduction and scope, input data, the calculation step by step, results, summary and references, with the section, pressure, chart and 3D drawings and the notes and warnings of the analysis. It is saved as a PDF or a Word file after Analyze.
Calculate Coulomb’s active or passive force on a retaining wall with wall friction, a back that leans and a backfill that slopes, in one soil or in layers, with a water table and a surcharge. This manual walks through the screen, every input, the state and the soil model, the results and the report, with a worked example you can reproduce.
Coulomb gives the active or passive force of the soil on a retaining wall by Coulomb’s wedge theory (1776). It takes into account the friction between the soil and the wall, a wall back that is not vertical and a backfill that slopes. A plane failure surface runs through the heel of the wall; the module finds the wedge that gives the largest force (active) or the smallest (passive) and reports that force, its two components, the angle of the critical plane and where the force acts.
Pa = ½ γH² Ka, Pp = ½ γH² Kp
H is the vertical height of the wall, γ the unit weight, and the coefficient Ka or Kp depends on the friction angle φ′, the wall friction δ′, the inclination θ of the back from the vertical and the slope α of the backfill. The force acts at H/3 above the base for one soil, inclined at δ′ to the normal of the back.
What you can describe
A wall of vertical height H with its back at θ from the vertical (positive toward the backfill, negative away from it), active (the wall moves away from the soil) or passive (the wall is pushed into it).
A backfill that rises or falls at α from the horizontal.
Soil in up to six layers, each with its unit weights, φ′, wall friction δ′ and cohesion c′, with level boundaries.
A level water table behind the wall, and a uniform surcharge on the backfill surface.
How the modules of the family differ
One soil, no water, no surcharge: the closed form, its working, and the Trial wedges curve that shows the largest (or smallest) force being picked out.
Layers, cohesion, a water table or a surcharge: each layer takes its own Coulomb coefficient under the effective vertical stress below the top of the wall, the pressures are summed along the back, and the module reports the force, its inclination, its line of action and, if there is water, the water force. The module states that the layered form is an extension of Coulomb’s solution for one soil.
For a load on the backfill that is not uniform over the whole surface, the Surcharge loads manuals give the pressure it adds to the wall.
Before you start: what you need
Have these ready from your design:
What
Where it comes from
Notes
The wall height H
Your design
The vertical height from the top of the wall to its base, at least 0.1 m.
The inclination θ of the back
Your design
From −40° to 40°, from the vertical. Positive means the back leans toward the backfill; 0 is a vertical back.
The slope α of the backfill
Your design
From −40° to 45°. Positive rises away from the wall.
γ, φ′ and c′ of each soil
Tests or correlations
Unit weight above the water table, effective friction angle (above 0°, up to 50°) and effective cohesion (zero for a granular soil). With a water table, also the saturated unit weight.
The wall friction δ′
Your design judgement
The angle of friction between the wall and each layer, from 0° to 45°. The module’s own note: typically φ′/2 to 2φ′/3, and at most φ′.
Water table and surcharge
Site data and loading
The depth zw of a level water table below the top of the wall, and a uniform surcharge q on the backfill, if any.
Quick start in seven steps
Open the module and choose the units
Open Coulomb from the Lateral Earth Pressure family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Describe the wall
In Wall choose Active or Passive, enter the height H and the inclination θ of the back.
Describe the backfill
In Backfill enter the slope α.
Enter the soil
In Soil layers check the first layer (γ, φ′, δ′, c′); to add another, fill in the New layer form and press + Add layer.
Add water and a surcharge if there are any
In Groundwater choose Present and enter zw; in Surcharge enter q.
Press Analyze
Press Analyze. The bar reads Analysed — the earth pressure calculated, the inputs lock. Open the Pressure view to see the force.
Read the answer and take the report
The Results tab gives the force, its components and the critical plane; Calculations writes out the working; Report saves a PDF or a Word file.
The screen: four views of the wall
The layout is the one every GeoTerva module shares, described in the workspace guide: the drawing on the left, the Model, Results, Calculations, Report and Theory tabs on the right, and the unit selector in the header. This module’s drawing has four views, chosen with the buttons above it:
Section is the geometry alone: the wall, its back at θ, the backfill at α, the layers with their values, the water table and the surcharge. It follows the inputs as you type.
Pressure shows, after Analyze, the pressure along the back, the water pressure, the resultant force with its components, and the critical failure wedge.
Trial wedges plots the force of trial wedges against the angle β of their base plane, with the critical wedge marked. It is drawn for one soil without water or a surcharge.
3D shows a length of the wall with the sloping backfill, and the force after Analyze.
Section before Analyze: the geometry and the soil values.
Pressure after Analyze: the force at H/3, inclined at δ′ to the normal of the back, and the failure wedge.
Trial wedges: Coulomb’s active force is the largest of the trial-wedge forces.
3D after Analyze.
The Show chips
A Show bar above the drawing switches elements on and off. In Section there is one chip, Dimensions. In Pressure there are six: Pressure p, Resultant P, Ph, Pv components, Water u, Failure wedge and Dimensions. Before Analyze the Pressure view shows the geometry only and says “The pressures and the resultant appear after Analyze”.
Zoom, and nothing to change by dragging
Each view has − and + zoom buttons with the zoom shown between them, and a button that fits the whole drawing in the pane. Nothing on the drawing can be dragged to change a value: every number is entered in the Model tab, and the drawing follows. When zoomed in, drag the drawing to move it. After Analyze the inputs are locked.
Pressure with two layers, a water table and a surcharge: the water is drawn as its own triangle, and the failure surface has a piece for each layer.
Entering the model, section by section
The Model tab starts with Model data (the same in every module; see the workspace guide) and then has five sections: Wall, Backfill, Soil layers, Groundwater and Surcharge. Each opens when you click its title; Next and Back walk you through them. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. Defaults are shown in SI; after you switch units the boxes convert the same wall and soil, for example γ = 1.52957 t/m³ in SI · tf and H = 13.1234 ft, γ = 95.4882 pcf in US.
Wall
Two choices and two numbers. State is Active (the wall moves away from the soil, the default) or Passive (the wall is pushed into it). Tension zone appears for the active state and is explained under soil with cohesion.
Wall: the state, the tension zone, the height and the inclination of the back.
Wall fields
Field
Unit
Default
What to enter
Height of the wall (vertical) H
m · ft
4
The vertical height of the wall. At least 0.1 m; a blank box stops the analysis.
Inclination of the wall back from the vertical (+ toward the backfill) θ
°
5
From −40° to 40°. 0 is a vertical back.
Backfill
One number: the slope of the ground surface behind the wall. A plane surface at α to the horizontal, positive when it rises away from the wall.
Backfill: the slope.
Backfill field
Field
Unit
Default
What to enter
Slope of the backfill (+ rising away from the wall) α
°
10
From −40° to 45°.
Soil layers
The soil behind the wall goes in from the top down. The module opens with one layer. To add another, fill in the New layer form and press + Add layer: it goes to the bottom of the Soil layers table below, and the counter (“2 of 6”) moves on. Up to six layers can be entered.
The Default column is the first layer, which the module opens with. The New layer form opens with γ = 18 kN/m³, γsat = 20 kN/m³, φ′ = 30°, δ′ = 15°, c′ = 0 and thickness 2 m.
Soil layers: the New layer form above, the table of layers below.
Soil layer fields
Field
Unit
Default
What to enter
Name
—
empty
Labelled Name (optional); the box shows “automatic” when empty. Shown on the drawing and in the tables; leave blank for the automatic name (Layer 1, Layer 2, …).
Thickness (vertical, at the back) Hi
m · ft
2
Measured vertically at the back of the wall from its top; the boundaries are level. At least 0.01 m. The last layer has no thickness: it runs on below the heel. Shown when there is more than one layer.
Unit weight above the water table γ
kN/m³ · t/m³ · pcf
15
At least 1 kN/m³.
Saturated unit weight γsat
kN/m³ · t/m³ · pcf
19
Used below the water table, where the layer weighs γ′ = γsat − γw. Must be above that of water (9.81 kN/m³). Shown when the water table is present.
Effective friction angle φ′
°
30
Above 0°, up to 50°.
Angle of wall friction against this layer δ′
°
15
From 0° to 45°. Typically φ′/2 to 2φ′/3, at most φ′.
Effective cohesion c′
kPa · kg/cm² · psf
0
Zero for a granular soil. A value above zero adds the cohesion term to the pressure of that layer.
To change a layer, press its row in the table. The form opens with Editing · Layer n in its title; changes apply as you type, Done closes it and Delete layer removes it. The × at the end of a row removes a layer too (not the last one left). The last layer has no thickness to enter: it runs on below the heel of the wall. The layers above it must not already reach the base.
A layer opened for editing from its row.
Groundwater
Water table is None (the default) or Present. With water present the depth zw is asked for, and each layer asks for its saturated unit weight. The table is level, the pore pressure hydrostatic (no seepage): below the water table the layers weigh γ′ = γsat − γw in the effective vertical stress, and the pore pressure u acts normal to the back.
Groundwater: Present, with the depth.
Groundwater field
Field
Unit
Default
What to enter
Depth of the water table below the top of the wall zw
m · ft
2
At least 0. A table at or below the base of the wall has no effect. Shown when Water table is Present.
Surcharge
A uniform surcharge on the backfill surface. In the wedge it acts as the vertical stress q cos θ / cos(θ − α) added at every depth.
Surcharge: one number.
Surcharge field
Field
Unit
Default
What to enter
Uniform surcharge on the backfill surface q
kPa · t/m² · ksf
0
Vertical, per unit area of the sloping surface. Zero or more; 0 for none.
Choosing the state, the angles and the soil model
Active or passive. Choose Active when the wall moves away from the soil, so that the soil pushes it; choose Passive when the wall is pushed into the soil and the soil resists. The example gives Ka = 0.3872 and Pa = 46.46 kN/m active, and Kp = 6.732 and Pp = 807.9 kN/m passive, for the same wall and soil.
The wall friction δ′. It turns the force away from the normal of the back, and it changes the coefficient. The module’s note is that δ′ is typically φ′/2 to 2φ′/3 and never more than φ′. For the active case the coefficient changes little in that range; for the passive case it grows quickly with δ′: the example gives Kp = 5.393 with δ′ = 10° and 6.732 with 15°. With δ′ above φ′/2 in the passive state, the module cautions that the real failure surface is curved and the plane of Coulomb overestimates the force, and points to the Passive, log spiral module.
The back and the slope. A back leaning toward the backfill (θ positive) and a backfill that rises (α positive) both increase the active force; the “How each input changes the answer” table shows by how much. The active case has a solution only for α ≤ φ′, and a back no steeper than φ′ has no active wedge.
One soil, or layers, water and a surcharge
The module chooses the way of working by what you enter:
Model
What you get
Results tab
One layer, no cohesion, no water, no surcharge
Coulomb’s closed form: K, P = ½ γH² K, at H/3, inclined at δ′; the Trial wedges curve
Force, K, the two components, βcr
Layers, cohesion, a water table or a surcharge
Each layer takes its own K under the effective vertical stress; the pressures are summed along the back
Force, inclination β to the normal, line of action z̄, components, the water force Pw and the depth z0 of a tension zone where they apply
The same wall as a one-layer model and as a layered one agree: a single layer with nothing else switched on reduces to the closed form.
Soil with cohesion and the tension zone
A layer with c′ above zero takes the pressure σ′ = Kσ′v − 2c′√K (active) or Kσ′v + 2c′√K (passive), with the Coulomb coefficient of the layer; the module neglects the adhesion between wall and soil. In the active state the pressure is negative near the top of a cohesive layer. Tension zone decides what happens there:
Tension cracks — σ′ₐ = 0 above z₀ (the default): the soil cannot pull on the wall, so no pressure acts above z0, the depth where the pressure first becomes positive.
No cracks — counted in full: the soil is taken to stay in contact with the wall, and the negative area is subtracted from the force.
Example (a 4 m vertical wall, level backfill, φ′ = 26°, c′ = 8 kPa, γ = 15 kN/m³, δ′ = 0): Ka = tan²(32°) = 0.3905, z0 = 2c′/(γ√Ka) = 16/(15 × 0.6249) = 1.707 m, which is the depth the module gives (1.707 m), and the force with cracks is 15.40 kN/m.
Analyze and read the results
Press Analyze. The bar at the foot of the Model tab changes to Analysed — the earth pressure calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed.” Edit inputs unlocks the model and clears the results; Warnings (with the number of them) lists the cautions of the run, and a window with them opens when there are any.
One soil: the cards
The Results tab of the example.
Card
Symbol
The example
Coulomb active force (or passive)
Pa (Pp)
46.46 kN/m
Coefficient
Ka (Kp)
0.3872
Horizontal component
Pa,h
43.66 kN/m
Vertical component, downward (active) or upward (passive) on the wall
Pa,v
15.89 kN/m
Critical failure plane from the horizontal
βcr
55.47 °
The force is per unit length of the wall. The components are Ph = P cos(δ′ + θ) and Pv = P sin(δ′ + θ) in the active state (with δ′ − θ in the passive state): 46.46 × cos 20° = 43.66 kN/m and 46.46 × sin 20° = 15.89 kN/m.
Layers, water or a surcharge: the cards and the tables
The cards become the force P, its inclination β to the normal of the back, the line of action z̄ above the base, the two components, the water force Pw (normal to the back) when there is water, and the depth z0 of the tension zone when a cohesive layer has one. Below them are tables: The layers and their Coulomb coefficient (with K and βcr of each layer), Stresses on the back, and Force of each layer (the normal and tangential parts of each).
The Results tab with two layers, water and a surcharge.
The Trial wedges view
For one soil without water or a surcharge, Trial wedges plots the force of the wedge whose base plane is at β. The active force is the largest of the trial-wedge forces and the passive force the smallest; the critical wedge is marked, here Pa = 46.46 kN/m at β = 55.5°. When layers, water or a surcharge are present the view says that the trial wedges are drawn for one soil without them, and that the report lists the coefficient of each layer.
Trial wedges with layers: the message that replaces the curve.
The Calculations tab, the report and the Theory tab
The Calculations tab writes out the working. It starts with Given and what is sought, then follows the steps:
One soil (four steps): the Coulomb coefficient with its numbers; the force P = ½ γH² K at H/3, inclined at δ′; the components of the force; and the critical failure wedge with βcr. A closing Answer gives K and P.
Layers, cohesion, water or a surcharge: the Coulomb coefficient of each layer with a table of the layers; the stresses on the back (with a table of σ′v, σ′ and u); the tension zone when a cohesive layer has one, with the force the other assumption would give; the force and its direction, with the force of each layer; and the critical failure plane of each layer.
The Calculations tab: the given data and the first steps of the working.
Every number on the Results tab can be traced to a step here.
The report
In the Report tab fill in the cover (project, location, client, report number, date, prepared by, title / qualification, checked by, and optional notes). After Analyze Save as PDF and Download Word (.docx) work; the preview below the buttons is the report itself.
The report has six parts: 1 Introduction and scope, 2 Input data, 3 Calculation (for one soil: 3.1 Coulomb coefficient, 3.2 Force, 3.3 Components of the force, 3.4 Critical failure wedge), 4 Results, 5 Summary and 6 References. It carries the drawings, states the units with their conversion constants and prints the warnings of the run.
The Report tab: the cover details, the two buttons and the preview.
The Theory tab
The Theory tab has six sections: The trial wedge, Coulomb’s active pressure, Coulomb’s passive pressure, Layers, water and surcharge, a worked example, and References.
Worked example: the module’s own example
The module opens on this example (after Reset to defaults), so you can follow the numbers on your own screen.
Inputs of the example
Section
Input
Value
Wall
State, height, back
Active; H = 4 m; θ = 5°
Backfill
Slope
α = 10°
Soil layers
One layer
γ = 15 kN/m³; φ′ = 30°; δ′ = 15°; c′ = 0
Groundwater, Surcharge
None
Water table None; q = 0
Step 1: the coefficient
The square root term is √[sin(15° + 30°) sin(30° − 10°) / (cos(15° + 5°) cos(5° − 10°))] = 0.5083, so
It acts H/3 = 1.33 m above the base, inclined at δ′ = 15° to the normal of the back.
Step 3: the components
Pa,h = 46.46 × cos(15° + 5°) = 43.66 kN/m (horizontal, into the wall); Pa,v = 46.46 × sin 20° = 15.89 kN/m (downward on the wall).
Step 4: the critical plane
The critical wedge has βcr = 55.47° to the horizontal, and its force is the same 46.46 kN/m as from Ka.
The same wall, passive
Choose Passive: Kp = 6.732, Pp = ½ × 15 × 16 × 6.732 = 807.9 kN/m, with Pp,h = 795.6 kN/m and Pp,v = 140.3 kN/m upward on the wall and βcr = 29.56°. Here δ′ = 15° equals φ′/2, so there is no caution; a δ′ of 20° brings one (see Messages).
A check with a vertical back and a level backfill
With θ = 0, α = 0, δ′ = 20°, H = 5 m, γ = 18 kN/m³ and φ′ = 30° the module gives Ka = 0.2973 and Pa = 66.90 kN/m, which is ½ × 18 × 25 × 0.2973 by hand. Passive, with δ′ = 15°: Kp = 4.977 and Pp = 1,120 kN/m (½ × 18 × 25 × 4.977 = 1,120).
The same wall in other units
In SI · tf the module gives Pa = 4.738 tf/m (46.46 kN/m ÷ 9.80665). In US, switching converts the boxes to H = 13.1234 ft and γ = 95.4882 pcf, and the module gives Pa = 3.184 kip/ft, Pa,h = 2.992 kip/ft and Pa,v = 1.089 kip/ft, with Ka = 0.3872 and βcr = 55.47° unchanged. Typing the rounded 13.12 ft and 95.49 pcf instead gives 3.182 kip/ft.
With a surcharge
Enter q = 10 kPa. The module now works in its layered form and gives Pa = 61.95 kN/m inclined at 15.00° (that is δ′) with the line of action z̄ = 1.500 m. By hand: the stress added is q cos 5° / cos(5° − 10°) = 10 kPa, so Pa = Ka (qH + ½ γH²) = 0.3872 × (40 + 120) = 61.95 kN/m, and the moments about the base give (15.49 × 2 + 46.46 × 1.333) / 61.95 = 1.50 m.
How each input changes the answer
Each row changes one thing in the example and runs the module again; the numbers are the ones the module gives, so you can reproduce any row. First, one soil without water or a surcharge:
The example with one change at a time (kN/m; the passive row is passive)
φ′: a larger friction angle gives a smaller coefficient and force: Ka falls from 0.4686 at 25° to 0.2607 at 40°.
δ′: the active force changes little with wall friction (49.79, 46.46 and 47.22 kN/m for 0°, 15° and 30°), but the vertical component grows from 4.339 to 27.08 kN/m.
α and θ: a rising backfill and a back leaning toward the fill raise the active force. At α = φ′ the force is 105.7 kN/m, with the critical plane parallel to the surface (βcr = 30.00°).
H and γ: the force is proportional to γ and to H²: 46.46 × (6/4)² = 104.5 kN/m, and 46.46 × 18/15 = 55.75 kN/m.
Now water and a surcharge (the module works in its layered form, so the cards are the force, its inclination β to the normal, the line of action z̄ and the water force):
Surcharge, layers and water
Change
P (kN/m)
β (°)
z̄ (m)
Pw (kN/m)
q = 10 kPa
61.95
15.00
1.500
—
Two layers (lower: 18 kN/m³, 35°, 17°) and q = 10 kPa
56.26
16.31
1.556
—
The same with a water table at 2.5 m
64.18
13.50
1.414
11.08
Water table at zw = 1 m (one soil)
79.97
6.754
1.188
44.31
Water table at zw = 2 m (one soil)
61.20
10.22
1.163
19.69
Water table at zw = 3 m (one soil)
50.11
13.54
1.255
4.924
A surcharge adds KqeH to the force, where qe = q cos θ / cos(θ − α) (equal to q in this example), and moves its line of action up (from 1.333 m to 1.500 m).
Water adds its own force normal to the back and lowers the effective weight of the soil below it: the shallower the table, the larger the total force (79.97, 61.20 and 50.11 kN/m for 1, 2 and 3 m, against 46.46 without water).
Messages: when the module stops, and cautions
When a value cannot be used the module stops and nothing is calculated. A window titled Cannot analyse lists what is wrong; it says either “1 input error” (a value missing or outside its range) or “a condition of the solution is not met” (angles or layers that admit no solution), and the second kind ends with “Nothing was calculated. Change the inputs marked in red, then press Analyze again.”
The input errors name the field and the limit:
A stop: a friction angle of zero.
“Height of the wall (vertical) (H) must be at least 0.1 m.” (and “… is missing.” for an empty box).
“Slope of the backfill … (α) must be at most 45°.” and, in the same form, the limit of θ (40°), and “Uniform surcharge … (q) must be at least 0 kPa.”
Kind
When
What to do
Stop
φ′ is 0: “Layer 1: a granular layer needs φ′ > 0.”
Enter φ′ above 0; Coulomb’s wedge carries its weight by friction on its base plane.
Stop
Active, α above φ′: “no Coulomb solution for these angles … the backfill is steeper than φ′ (α > φ′).”
Reduce α to φ′ or less.
Stop
Active, the back no steeper than φ′: “no active wedge — the back is no steeper than φ′.”
Increase θ or reduce φ′.
Stop
Passive, α below −φ′: “no Coulomb solution for these angles”; or φ′ + δ′ + α − θ reaches 90°
Use α ≥ −φ′; reduce δ′ or α, or increase θ.
Stop
The layers above the last one reach the base: “The layers above the last one already reach the base.”
Reduce the thicknesses or the number of layers.
Stop
With water, a saturated unit weight not above 9.81 kN/m³: “the saturated unit weight … is not above that of water.”
Enter the saturated unit weight of the soil.
Caution
δ′ above φ′: “Layer 1: δ′ > φ′.”
The wall friction cannot exceed that of the soil; use δ′ ≤ φ′.
Caution
Passive, δ′ above φ′/2: “Coulomb’s passive force with δ′ > φ′/2.”
The real failure surface is curved and the plane overestimates the force; consider the log-spiral module.
Caution
Active, cohesive layer, “No cracks” chosen, and the tension outweighs the pressure: “The net active force has no line of action on the wall.”
Tension cracks give the pressure on a wall the soil cannot pull.
A caution does not stop the analysis: the results appear and a Warnings window opens with the text; the same text is printed in the report.
A caution: δ′ above φ′.
Good practice and common mistakes
Choose δ′ deliberately. Use the typical φ′/2 to 2φ′/3. In the passive state it matters most: the force grows quickly with δ′ (Kp from 5.393 at 10° to 6.732 at 15° in the example), and above φ′/2 the plane of Coulomb overestimates it.
Measure the wall from its top.H is the vertical height; the layers’ thicknesses are measured vertically at the back, from the top of the wall, with level boundaries.
Mind the sign of θ and α.θ is positive when the back leans toward the backfill, α is positive when the ground rises away from the wall.
Remember the force is inclined. It acts at δ′ to the normal of the back; use the horizontal and vertical components Ph and Pv in the stability checks, and the line of action given.
Check the saturated unit weight when you add water: it must be above 9.81 kN/m³, and it is asked for in every layer.
Do not enter a load that does not cover the surface as q.q covers the whole backfill surface; for a strip, a line or a point load use the Surcharge loads modules and add their pressure to this force.
Run active and passive separately. They are different states of the wall; the module analyses one at a time.
Read the Warnings. A caution tells you where the model leaves the range where Coulomb’s plane is reliable.
Frequently asked questions
How is Coulomb’s active force calculated?
A plane failure surface runs through the heel; the force on the wall follows from the equilibrium of the wedge (its weight, the reaction on the failure plane at φ′ to its normal, and the wall force at δ′ to the normal of the back). The critical wedge gives the largest force: Pa = ½ γH² Ka. For the module’s example, Ka = 0.3872 and Pa = 46.46 kN/m at H/3, inclined at 15° to the normal of the back.
What is the difference between Coulomb and Rankine?
Rankine assumes a smooth wall and a force parallel to the backfill. Coulomb takes the friction δ′ between the soil and the wall into account, and the inclination θ of the back. With δ′ = 0, θ = 0 and a level backfill the two coefficients are the same.
What do θ and α mean, and which sign is which?
θ is the inclination of the back from the vertical, positive when it leans toward the backfill; α is the slope of the backfill from the horizontal, positive when it rises away from the wall. θ is limited to ±40°, α from −40° to 45°.
What value of the wall friction should I enter?
The module’s note: typically φ′/2 to 2φ′/3 and at most φ′. A δ′ above φ′ is accepted with a caution. For the passive state, a δ′ above φ′/2 brings a caution that Coulomb’s plane overestimates the force.
Can I use layers, water and a surcharge?
Yes: up to six layers with level boundaries, a level water table and a uniform surcharge on the backfill. Each layer takes its own Coulomb coefficient under the effective vertical stress below the top of the wall, and the pressures are summed along the back. The module states that this layered form is an extension of Coulomb’s solution for one soil.
Does the module handle cohesion?
Yes: a layer with c′ above zero takes the pressure K σ′v − 2c′√K (active) or + 2c′√K (passive), neglecting the adhesion between wall and soil. In the active state the Tension zone choice decides whether the soil above z₀ is taken to lose contact with the wall (the default) or to stay in contact.
Where does the force act, and in what direction?
For one soil it acts H/3 above the base, inclined at δ′ to the normal of the back. With layers, cohesion, water or a surcharge the module gives the line of action z̄ above the base and the inclination β to the normal. The Results tab gives the horizontal and vertical components, and says whether the vertical one is downward (active) or upward (passive) on the wall.
Why does the Trial wedges view show no curve?
The curve is drawn for one soil without water or a surcharge. With layers, a water table or a surcharge the view says so, and the coefficient of each layer is listed on the Results tab and in the report.
What values stop the analysis?
A wall height under 0.1 m or empty, φ′ of 0, a backfill steeper than φ′ in the active state, a back no steeper than φ′, a passive case whose angles leave no wedge, layers that reach the base, or a saturated unit weight not above that of water. The window names the value and what to change.
Can I drag the wall or the backfill on the drawing?
No. Nothing on the drawing can be dragged to change a value; when zoomed in, drag the drawing to move it. Enter every number in the Model tab and the drawing follows. After Analyze the inputs are locked until you press Edit inputs.
Does the module remember my inputs?
Yes. The model, its analysed state and the unit system are kept in the browser, so the page opens again on the model you last worked on; Reset to defaults brings the example back. For a copy that travels use Save to file…, and Quick Save… for named versions in the browser.
Which units does it use, and what does it give?
SI · kN, SI · tf or US, switchable at any time, even after Analyze. The force per unit length is in kN/m, tf/m or kip/ft, the pressures in kPa, t/m² or ksf, the angles in degrees. For the example 46.46 kN/m is 4.738 tf/m, and in US units (the boxes convert to 13.1234 ft and 95.4882 pcf) 3.184 kip/ft; typing the rounded 13.12 ft and 95.49 pcf gives 3.182 kip/ft.
Mononobe–Okabe seismic earth pressure: user manual
Calculate the active force on a wall during an earthquake: the Mononobe–Okabe coefficient, the dynamic force with its static and seismic parts, its line of action, components and critical failure plane, for a layered granular backfill with water and a surcharge. This manual walks through the screen, every input, the results and the report, with a worked example you can reproduce.
Mononobe–Okabe gives the active force on a retaining wall during an earthquake, by the pseudo-static method: the shaking is replaced by two constant forces on the soil wedge behind the wall, khW horizontally toward the wall and kvW vertically, and the force on the wall is that of the wedge that pushes hardest. You enter the two seismic coefficients; the module does not derive them from a site or a code.
ψ is the seismic angle, K′a is the Mononobe–Okabe coefficient (it depends on φ′, the wall friction δ′, the inclination θ of the back, the slope α of the backfill and ψ) and Pae is the dynamic active force per unit length of wall. The module also splits it into the static Coulomb force Pa and the seismic increment ΔPae, and gives the height of the line of action, the force components and the critical failure plane.
What you can describe
A wall of vertical height H whose back is inclined at θ from the vertical (positive toward the backfill).
A backfill surface sloping at α (positive rising away from the wall).
A granular backfill in up to six layers, each with its own unit weights, φ′ and wall friction δ′ (the soil is cohesionless).
The earthquake by kh and kv.
A level water table behind the wall, with pore water that either moves with the soil (restrained) or does not (free).
A uniform surchargeq on the backfill surface.
One soil, no water and no surcharge give the closed form, its working and the curve of trial wedges. Layers, water or a surcharge give the layered form, where each layer takes its own coefficient.
This module gives the earth pressure. Combine it with the weight of the wall, the passive resistance and the other actions in your own stability checks.
Before you start: what you need
Have these ready from your design:
What
Notes
The wall: H, θ
The vertical height from the top of the wall to its base, and the inclination of the back (0 for a vertical back).
The backfill slope α
Rising away from the wall is positive. The backfill must be stable: α cannot exceed φ′ − ψ.
The soil: γ, φ′, δ′ for each layer
Effective friction angle of the granular soil, unit weights, and the wall friction against each layer. The module notes that δ′ is usually taken as φ′/2 to 2φ′/3 and warns above φ′.
The seismic coefficients kh, kv
The horizontal and vertical accelerations as fractions of g. You choose them in your design; the module has no site or code table.
The water table, if any
Its depth below the top of the wall, the saturated unit weight of each layer, and whether the pore water is restrained or free (for free water also the dry unit weight).
The surcharge q
A uniform load per unit area on the backfill surface; 0 for none.
Quick start in seven steps
Open the module and choose the units
Open Mononobe–Okabe from the Lateral Earth Pressure family. Pick SI · kN, SI · tf or US at the top right; you can change it at any time.
Describe the wall and the backfill
In Wall type H and θ; in Backfill type α. Press Next to move on.
Describe the soil
In Soil layers edit the first layer in the table (press its row, change the values, press Done) or add layers with + Add layer. Each layer has γ, φ′ and δ′; every layer but the last also has a thickness.
Enter the earthquake
In Earthquake type kh and kv.
Add water and a surcharge if you have them
In Groundwater choose Present and enter the depth zw and the pore-water case; in Surcharge type q.
Press Analyze
Press Analyze in the bar at the foot of the Model tab. The bar reads Analysed — the earth pressure calculated, the inputs lock, and the drawings carry the pressure and the force.
Read the answer, the working and the report
The Results tab gives Pae, K′a, ΔPae, the line of action and the components; Calculations gives the working; Report saves a PDF or a Word file.
The screen: four views of the wall
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing on the left has four views, chosen with the buttons above it, and a pane on the right with the tabs Model, Results, Calculations, Report and Theory.
The window as the module opens: the example, not yet analysed.
Section is the geometry only: the wall, its back at θ, the backfill at α, the layers with their values, the water table and the surcharge. It follows the inputs as you type.
Pressure adds, after Analyze, the static pressure along the back, the seismic increment drawn with its resultant at 0.6H, the water on the back, the force Pae at its line of action with its components Ph and Pv, and the critical failure plane.
Trial wedges plots the force of the trial wedges against the angle η of their base plane; the dynamic force is the highest point.
3D shows a length of the wall with the backfill and, after Analyze, the force.
Before Analyze the Section view shows the model. This one has a water table at 2 m.
Section: the geometry follows the inputs.
After Analyze the Pressure view shows the example: the static pressure (orange), the seismic increment (red), the failure wedge, and Pae with its components.
Pressure after Analyze.
The Trial wedges view is the curve of P(η); the dashed line marks the critical plane and the force there.
Trial wedges: the largest wedge force is the answer.
The 3D view shows the wall, the backfill and the force.
3D after Analyze.
The Show chips
The Show bar above the Section view has one chip, Dimensions. In the Pressure view it has seven: Static pressure, Seismic increment, Resultant Pae, Ph, Pv components, Water, Failure wedge and Dimensions. Before Analyze the first six are unchecked and greyed, with the note “The pressures and the resultant appear after Analyze”; after it they are all checked and you can switch each one.
Pressure before Analyze: the geometry alone.
The Seismic increment chip switched off.
Zoom, and nothing to drag
No value can be dragged on the drawing; every value is typed in the Model tab. The − and + buttons (or the mouse wheel over the drawing) zoom it (one press of + gives 135%); once zoomed, drag the background to move the view, and the square button fits the drawing again (100%).
Entering the model, section by section
The Model tab is a stack of sections; the one you work in is open, Next and Back walk you through them, and Analyze waits in the bar below. Units are shown in the order SI · kN, SI · tf, US, equal ones merged: for example m · ft. Defaults are in SI; the same example in US units shows 13.12 ft and 98.67 pcf.
Model data
The first section keeps the model. Every change is kept in this browser, so the page opens again with the model you last worked on. Save to file… and Open file… move a model to a file; Quick Save… keeps a named version in the list below it, with Load and Delete; Reset to defaults returns to the example. Model name is a free label.
Model data.
Wall
The wall moves away from the soil during the earthquake (active). H is the vertical height; θ is positive when the back leans toward the backfill, and 0 is a vertical back. The force of each layer acts at its δ′ to the normal of the back.
Wall.
Field
Unit, default, range
Height of the wall (vertical) (H)
m · ft · default 4 · At least 0.1 m
Inclination of the wall back from the vertical (θ)
° · default 0 · −40 to 40. + toward the backfill
Backfill
The ground surface is a plane at α to the horizontal; the failure surface of each soil is a plane through the heel of the wall.
Backfill.
Field
Unit, default, range
Slope of the backfill (α)
° · default 0 · −40 to 45. + rising away from the wall
Soil layers
The soil is granular (c′ = 0). Layers run from the top down; their thicknesses are measured vertically at the back and their boundaries are level. The last layer has no thickness: it runs on below the heel. The section has a New layer form with + Add layer (a counter shows “1 of 6”), and a table of the layers below it. Press a row of the table to edit it in the form; × takes a layer out. With one layer the table shows the default row: γ = 15.5, φ′ = 30, δ′ = 15. The table always has γsat and γd columns; they hold values only when water is present (γd only for free pore water), and the form asks for them only then.
Soil layers: the form and the table.
Field
Unit, default, range
Name (optional)
text · default automatic · Shown on the drawing and in the tables
Thickness (vertical, at the back) (H)
m · ft · default 2 · At least 0.01 m. Not asked for the last layer
Unit weight above the water table (γ)
kN/m³ · t/m³ · pcf · default 15.5 (first layer), 18 (new layer) · At least 1
Saturated unit weight (γsat)
kN/m³ · t/m³ · pcf · default 19 (first layer), 20 (new layer) · At least 1, and above that of water. Asked only when water is present
Dry unit weight (γd)
kN/m³ · t/m³ · pcf · default 15 (first layer), 16 (new layer) · At least 1, not above γsat. Asked only for free pore water
Effective friction angle (φ′)
° · default 30 · 0 to 50, and above 0 to analyse
Angle of wall friction against this layer (δ′)
° · default 15 (first layer), 15 (new layer) · 0 to 45; typically φ′/2 to 2φ′/3, at most φ′
Two layers: the first has a thickness, the last runs on below the heel.
Earthquake
Pseudo-static: the wedge carries khW toward the wall and kvW (upward positive, so the weight becomes (1 − kv)W). A soil stays in equilibrium only while α ≤ φ′ − ψ.
— · default 0 · −0.5 to 0.5. + upward, reduces the weight
Groundwater
Choose Present or None (default). With water present the module asks for the depth and for the pore-water case, and the layer form and table gain γsat (and γd for free water). Below a level water table the layers weigh γ′ = γsat − γw in the effective vertical stress and the pore pressure acts normal to the back.
Field
Unit, default, range
Water table
choice · default None · Present or None
Depth of the water table below the top of the wall (zw)
m · ft · default 2 · At least 0. A table at or below the base has no effect
Pore water in the earthquake
choice · default Restrained · Restrained or Free, each with a (?) window
Restrained — moves with the soil (for soils of low permeability, k < 5×10⁻⁴ m/s): the saturated soil carries the inertia, with a seismic angle ψ of its own below the water table.
Free — highly permeable backfill (k > 5×10⁻⁴ m/s): only the dry mass of the soil carries inertia below the water table, and the free water adds a hydrodynamic pressure on the back.
The (?) beside each case opens a window with a sketch, the formulae and a line headed With the current inputs, which gives Pae for both cases on your model: for the example with a water table at 2 m the window gives 92.9 kN/m restrained and 85.7 kN/m free (the Results tab shows 92.92 and 85.69). Reading it changes nothing.
Surcharge
A uniform vertical surcharge per unit area of the sloping surface. It acts as the vertical stress q cos θ / cos(θ − α) added at every depth, and its mass carries inertia like the soil.
Surcharge.
Field
Unit, default, range
Uniform surcharge on the backfill surface (q)
kPa · t/m² · ksf · default 0 · At least 0; 0 for none
A layered model in the Section view: two layers, water at 2.5 m and q = 10 kPa.
Choosing the soil model and the pore-water case
The module has two forms, and which one runs follows your model:
Model
What runs
You get
One layer, no water on the wall, no surcharge
The closed form: K′a from the Mononobe–Okabe expression and Pae = ½ γH² (1 − kv) K′a
Six cards, the Trial wedges curve, a six-step working. The force is at δ′ to the normal of the back.
Two or more layers, or water above the base, or q > 0
The layered form: each layer takes its own K′a under the effective vertical stress below the top of the wall, and the forces are summed along the back
The same cards without K′a (each layer’s K′a is in the table “The layers and their coefficients” on the Results tab and in the Calculations tab), the inclination β of the force, a five-step working. The Trial wedges view says it is drawn for one soil without water or surcharge.
For a single dry soil the two forms agree. The layered form is an extension of the one-soil solution; for a layered backfill the static and the seismic parts are approximate.
Restrained or free pore water?
Pick Restrained when the backfill is not very permeable, so the water moves with the soil. Pick Free for a highly permeable backfill such as clean gravel or coarse sand; the module then needs γd and adds a hydrodynamic pressure. If you cannot tell, run both: the (?) window gives both results at once.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the earth pressure calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed.” Edit inputs unlocks the model. A Warnings (n) button sits beside it; with no warning it shows 0.
The Results tab of the example.
Card
Symbol
The example
Dynamic active force
Pae
56.05 kN/m
Mononobe–Okabe coefficient
K′a
0.4520
Seismic increment over the static Coulomb force
ΔPae
18.68 kN/m
Line of action above the base
z̄
1.689 m
Horizontal component
Pae,h
54.14 kN/m
Vertical component, downward on the wall (the module labels it upward when it is)
Pae,v
14.51 kN/m
Below the cards are the static Coulomb force Pa = 37.38 kN/m, the seismic angle ψ = 11.31° and the critical failure plane ηc = 45.32° from the horizontal, then a table of the values of each step.
Layers, water and a surcharge
In the layered form the cards change: the force is “on the back”, the card K′a gives way to the inclination β of the force to the normal of the back, and two cards appear for water, Pw (the water on the back) and, for free pore water, Pwd (the hydrodynamic part). The tab also lists each layer with its coefficients.
The Results tab of the layered example (two layers, water at 2.5 m, free pore water, q = 10 kPa).
The Pressure view of the layered example.
Using the numbers
Pae is the force per unit length of wall, inclined to the normal of the back: at δ′ in the one-soil form, at β in the layered form. Pae,h is horizontal into the wall; Pae,v is vertical, downward on the wall.
z̄ is the height of the line of action above the base: the static part is placed at H/3 (where its pressures put it in the layered form) and the seismic increment at 0.6H. The module notes that this split is approximate and does not model the dynamics of the soil.
Changing the unit system after Analyze re-states the cards, the drawings and the report without a new run.
The Calculations tab, the report and the Theory tab
The Calculations tab opens with a card Given and what is sought and then writes out the working in numbered steps. Each step shows its equation, the equation with your numbers and the result. The one-soil form has six steps:
Seismic angle — ψ = tan⁻¹[kh / (1 − kv)], and the limit kh(cr) = (1 − kv) tan(φ′ − α), which for the example is 0.5774.
Mononobe–Okabe coefficient — K′a with every number substituted.
Dynamic active force — Pae = ½ γH² (1 − kv) K′a.
Line of action (Seed & Whitman) — the static Ka and Pa, ΔPae = Pae − Pa and z̄.
Components of the force — Pae,h and Pae,v.
Critical failure wedge — the trial-wedge force Pae(η) and the critical plane ηc, with a check that it gives the same force as K′a.
Steps 2 and 3 of the Calculations tab.
The layered form has five steps: the seismic angle (for each water zone), the coefficient of each layer (with a table of the layers), the stresses on the back, the force with its seismic increment and line of action, and the critical failure plane of each layer.
The report
In the Report tab fill in the cover: project, location, client, report no., date, prepared by, title / qualification, checked by, and optional notes. The cover details are kept in this browser and saved with the model file. Save as PDF and Download Word (.docx) produce the report; the preview below is the report itself.
The report has six parts: 1 Introduction and scope, 2 Input data, 3 Calculation, 4 Results, 5 Summary and 6 References. It carries the Section, Pressure, Trial wedges and 3D drawings, the working of every step and the warnings, if any.
The Report tab: the cover details and the two download buttons.
The Theory tab
The Theory tab has seven sections, each opening on a click (with Open all and Close all): The pseudo-static wedge; Mononobe–Okabe coefficient; The limit of equilibrium; Line of action; Layers, water and surcharge; Worked example; References.
Worked example: the module’s own example
The module opens on this example, so you can follow the numbers on your own screen.
Inputs of the example
Section
Input
Value
Wall
H, θ
4 m, 0°
Backfill
α
0°
Soil layers
γ, φ′, δ′
15.5 kN/m³, 30°, 15°
Earthquake
kh, kv
0.2, 0
Groundwater, Surcharge
None, q = 0
Step 1: the seismic angle
ψ = tan⁻¹(0.2 / 1) = 11.31°. The limit kh(cr) = tan 30° = 0.5774, so kh = 0.2 is well inside it.
Step 2: the coefficient
K′a = 0.4520 (the module writes the substituted numbers in the Calculations tab). With kh = 0 the same module gives the static coefficient Ka = 0.3014.
Pa = ½ × 15.5 × 4² × 0.30141 = 37.38 kN/m, so ΔPae = 56.05 − 37.38 = 18.68 kN/m (from the unrounded values). The line of action: z̄ = (37.38 × 4/3 + 18.68 × 2.4) / 56.05 = 1.689 m above the base.
Step 5: components and critical plane
Pae,h = 56.05 × cos 15° = 54.14 kN/m and Pae,v = 56.05 × sin 15° = 14.51 kN/m, downward on the wall. The critical failure plane is at ηc = 45.32° from the horizontal.
These are the numbers on the Results tab, the six steps of the Calculations tab and the peak of the Trial wedges curve.
The same kind of wall in US units
Switch to US and enter H = 15 ft and γ = 110 pcf with φ′ = 30°, δ′ = 15°, kh = 0.2. The module gives K′a = 0.4520 and Pae = 5.594 kip/ft at z̄ = 6.333 ft; by hand ½ × 110 × 15² × 0.4520 / 1000 = about 5.59 kip/ft.
The same kind of wall in SI · tf
Switch to SI · tf and enter γ = 1.6 t/m³ with the other values of the example: Pae = 5.786 tf/m, ΔPae = 1.928 tf/m, z̄ = 1.689 m. By hand ½ × 1.6 × 4² × 0.4520 = 5.786 tf/m.
A layered model: two layers, water and a surcharge
Add a second layer (γ = 19, γsat = 21, γd = 17 kN/m³, φ′ = 36°, δ′ = 20°), give the first layer a thickness of 2.5 m, put the water table at 2.5 m with Free pore water and enter q = 10 kPa. The module gives Pae = 86.14 kN/m on the back, of which the soil’s seismic increment is ΔPae = 30.80 kN/m, at z̄ = 1.727 m above the base, with Pw = 11.04 kN/m and Pwd = 2.575 kN/m. By hand Pw = ½ × 9.81 × 1.5² = 11.04 kN/m and Pwd = 7/12 × 0.2 × 9.81 × 1.5² = 2.575 kN/m. With Restrained pore water the same model gives Pae = 90.98 kN/m.
How each input changes the answer
Each row changes one thing in the example (H = 4 m, γ = 15.5 kN/m³, φ′ = 30°, δ′ = 15°, kh = 0.2) and runs the module again. The numbers are the ones the module gives, so you can reproduce any row. A dash marks a layered run, which has no single K′a.
The example with one change at a time
Change
Pae (kN/m)
z̄ (m)
The example: kh = 0.2 K′a = 0.4520 · ΔPae = 18.68. The reference
56.05
1.689
kh = 0 K′a = 0.3014 · ΔPae = 0. The static Coulomb force: ΔPae = 0 and the line of action at H/3
37.38
1.333
kh = 0.1 K′a = 0.3679 · ΔPae = 8.244
45.62
1.526
kh = 0.3 K′a = 0.5626 · ΔPae = 32.38
69.76
1.829
kh = 0.4 K′a = 0.7178 · ΔPae = 51.64
89.01
1.952
kh = 0.5 K′a = 0.9715 · ΔPae = 83.10. Close to the limit kh(cr) = 0.5774
120.5
2.069
kv = +0.1 (upward) K′a = 0.4739 · ΔPae = 15.51. The weight is reduced: the force is less
52.89
1.646
kv = −0.1 (downward) K′a = 0.4351 · ΔPae = 21.97. The force is more
59.35
1.728
φ′ = 25° K′a = 0.5387 · ΔPae = 21.77. Weaker soil: more force
66.80
1.681
φ′ = 35° K′a = 0.3793 · ΔPae = 16.31. Stronger soil: less force
47.03
1.703
δ′ = 0° K′a = 0.4733 · ΔPae = 17.35. A smooth wall: the force is horizontal
58.68
1.649
δ′ = 20° K′a = 0.4540 · ΔPae = 19.42
56.29
1.701
γ = 18 kN/m³ K′a = 0.4520 · ΔPae = 21.69. K′a unchanged; the force follows γ
65.09
1.689
H = 6 m K′a = 0.4520 · ΔPae = 42.02. The force goes with H²; the line of action with H
126.1
2.533
H = 8 m K′a = 0.4520 · ΔPae = 74.71
224.2
3.377
θ = +10° (leans toward the backfill) K′a = 0.5416 · ΔPae = 20.24. More force
67.16
1.655
θ = −10° K′a = 0.3799 · ΔPae = 17.71. Less force
47.11
1.734
α = 10° K′a = 0.5619 · ΔPae = 27.12. A rising backfill: more force
69.67
1.749
q = 10 kPa ΔPae = 24.70. Layered form (no K′a card)
74.13
1.797
q = 20 kPa ΔPae = 30.73. The surcharge part doubles
92.21
1.863
Water at 2 m, restrained ΔPae = 40.26. Includes the water on the back
92.92
1.673
Water at 2 m, free ΔPae = 28.52
85.69
1.527
Water at 5 m (below the base) K′a = 0.4520 · ΔPae = 18.68. No effect on the pressure on the wall
56.05
1.689
What the table shows:
The seismic coefficient raises the force steeply: from 37.38 kN/m at kh = 0 to 120.5 kN/m at kh = 0.5, and the line of action rises from H/3 toward the upper part of the wall.
The soil matters twice: through γ (the force is proportional to it) and through φ′ (a weaker soil pushes harder).
The geometry acts through H² and, for the coefficient, through θ and α.
Water and a surcharge add to the force and move the module to the layered form.
Messages: when the module stops, and cautions
The module checks its inputs and the conditions of the solution. A problem stops the analysis and nothing is calculated; a questionable value only warns and the analysis runs.
Stops
A value outside its range or empty opens a window titled Cannot analyse — 1 input error (or more), for example “Height of the wall (vertical) (H) must be at least 0.1 m.”, “Height of the wall (vertical) (H) is missing.”, “Effective friction angle (φ′1) must be at most 50°.” or “Horizontal seismic coefficient (horizontal acceleration / g) (kh) must be at least 0.”
A condition of the solution that is not met opens a window titled Cannot analyse — a condition of the solution is not met, with the cause, what to do and the line “Nothing was calculated. Change the inputs marked in red, then press Analyze again.”
The Results tab then says “Press Analyze on the Model tab to calculate the earth pressure.”
A stop: φ′ = 0.
When
Message and what to do
kh too large for φ′ and α (for example kh = 0.6 with φ′ = 30°, α = 0)
“Layer 1: the backfill slope α exceeds φ′ − ψ. φ′ − α − ψ = −0.96° < 0 with ψ = 30.96°: the inertia force exceeds what the friction of the soil can carry, so no finite active force exists …” Reduce kh or α, or increase φ′.
α steeper than φ′ (for example α = 35°, kh = 0)
“Layer 1: the backfill slope is steeper than φ′. α = 35° > φ′ = 30°: the backfill is not stable even without the earthquake …” Reduce α or increase φ′.
φ′ = 0
“Layer 1: a granular layer needs φ′ > 0. The wedge carries its weight and its inertia by the friction on its base plane.” Enter φ′ > 0.
δ′ + θ + ψ ≥ 90°
“Layer 1: no finite active force — δ′ + θ + ψ ≥ 90°. δ′ + θ + ψ = 111.6° ≥ 90° …” Reduce δ′, θ or kh.
The back no steeper than φ′ − ψ (for example φ′ = 50°, θ = −40°, kh = 0)
“Layer 1: no active wedge — the back is no steeper than φ′ − ψ. The back rises at 90° + θ = 50° from the horizontal, no steeper than φ′ − ψ …” Increase θ or reduce φ′.
Layers above the last already reach the base
“The layers above the last one already reach the base. The last layer would not meet the wall.” Reduce the thicknesses or the number of layers.
γsat not above that of water (water present)
“Layer 1: the saturated unit weight 9 kN/m³ is not above that of water.” Enter γsat above 9.81 kN/m³.
γd zero or above γsat (free pore water)
“Layer 1: the dry unit weight γd must be above zero and not above γsat.” Enter γd of the layer.
Cautions
If δ′ is above φ′ the analysis runs and the Results tab shows an orange banner, “1 warning — Layer 1: δ′ > φ′”, and the bar reads Analysed — the earth pressure calculated — 1 warning; a window Warnings (1) opens with “The wall friction cannot exceed the friction of the soil itself; δ′ is usually taken as φ′/2 to 2φ′/3. Use δ′ ≤ φ′.” The Warnings (n) button reopens it, and the warnings are printed in the report.
The Warnings window for δ′ = 40° with φ′ = 30°.
The banner on the Results tab.
Good practice and common mistakes
Choose kh and kv from your design basis. The module takes them as given and stops only where no solution exists; the force rises steeply as kh approaches the limit kh(cr).
Check the static case first. Set kh = 0: the module returns the static Coulomb force, the part of the answer that does not come from the earthquake.
Keep δ′ at or below φ′ (usually φ′/2 to 2φ′/3); above φ′ the module warns.
Enter thicknesses vertically at the back. The last layer has none: it runs on below the heel.
Match the pore-water case to the soil. Restrained for soils of low permeability, free for clean gravels and coarse sands; compare both with the (?) window.
Read Pae per unit length and in its direction: inclined to the normal of the back, with the horizontal and vertical components given.
Use the module for the pressure only. The wall’s own weight, the passive resistance and the other actions belong to your stability checks.
Your typed values are kept in the browser; save the model to a file to keep it with the project, and use the report as the record of an analysis.
Check the units after switching the system: type values in the units shown beside the boxes.
Frequently asked questions
What does the Mononobe–Okabe module calculate?
The active force on a retaining wall during an earthquake by the pseudo-static method: Pae = ½ γH² (1 − kv) K′a for one soil, with K′a from φ′, δ′, θ, α and the seismic angle ψ. For the module’s example it gives K′a = 0.4520 and Pae = 56.05 kN/m.
Where do I get kh and kv?
From your design basis. The module takes kh (0 to 0.9) and kv (−0.5 to 0.5) as inputs and does not derive them.
What is the difference between Pae and the static Coulomb force?
The static Coulomb force Pa is the active force with kh = kv = 0; ΔPae = Pae − Pa is the seismic increment. In the example Pa = 37.38 kN/m and ΔPae = 18.68 kN/m.
Where does the force act?
The line of action is z̄ above the base, with the static part at H/3 and the seismic increment at 0.6H; in the example z̄ = 1.689 m of 4 m. The module notes that the split is approximate.
Why does the module stop with “the backfill slope α exceeds φ′ − ψ”?
The seismic angle ψ grows with kh, and the soil can carry the inertia only while α ≤ φ′ − ψ. For a level backfill that means kh ≤ (1 − kv) tan φ′, which is 0.5774 for φ′ = 30°. Reduce kh or α or increase φ′.
Can I use layers, water and a surcharge?
Yes: up to six granular layers, a level water table with restrained or free pore water, and a uniform surcharge. The module then uses its layered form, where each layer takes its own coefficient.
What is the difference between restrained and free pore water?
Restrained water moves with the soil, so the saturated mass carries inertia; free water, in a highly permeable backfill, does not, so only the dry mass does, and the water adds a hydrodynamic pressure. The (?) window gives the force for both cases on your model.
Does it include cohesion?
No. The soil is granular, c′ = 0; a layer needs φ′ above 0.
Why are there no trial wedges with layers or water?
The Trial wedges view is drawn for one soil without water or surcharge. With layers, water or a surcharge each layer takes its own coefficient, and the Calculations tab and the report list them.
Does the module remember my inputs?
Yes. Every change is kept in this browser, so the page opens on the model you last worked on; Reset to defaults returns to the example, and Save to file or Quick Save keeps a version of a model.
Can I drag things on the drawing?
No value can be dragged; every value is typed in the Model tab and the drawing follows. Zoom with the buttons or the wheel, drag the background to pan when zoomed, and use the square button to fit it.
What units does it use, and what does the report contain?
SI · kN, SI · tf or US, switchable at any time, even after Analyze (forces in kN/m, tf/m or kip/ft). The report has a cover and six parts: introduction and scope, input data, the calculation step by step, results, summary and references, with the four drawings, as a PDF or a Word file after Analyze.
Passive earth pressure by the log spiral: user manual
Find the passive force that soil in front of a wall can give, from the critical log-spiral failure surface: its size, its direction and the height at which it acts, for one soil or several layers, an inclined back, a sloping surface, water and a surcharge. This manual walks through the screen, every input, the results and the report, with worked examples you can reproduce.
Passive, log spiral gives the passive force that the soil in front of a wall can put up when the wall is pushed into it: the force per metre of wall, its direction, and the height at which it acts. The failure surface is not a plane but a logarithmic spiral that leaves the heel of the wall and runs up to a straight line to the ground surface (Terzaghi and Peck). A rough wall pushes the soil up its back, and a curved surface follows that movement; a plane (Coulomb) overstates the resistance as wall friction grows.
Pp = ½ γH² Kp (one soil, a vertical back, a level surface)
Kp is the passive coefficient the module finds from the smallest trial wedge, H is the height of the wall and γ the unit weight of the soil. For an inclined back, a sloping surface, several layers, water or a surcharge the module works on the pressure along the back instead and adds the parts, as described under Choosing what to enter.
What you can describe
A wall of height H whose back is vertical or inclined at θ (from −30° to 30°).
A ground surface in front of the wall, level or sloping at α (from −30° to 30°).
One to six layers of granular soil (no cohesion), each with its own unit weights, friction angle φ′ and wall friction δ′.
A level water table at a depth you give, and a uniform surcharge on the surface.
The module gives the passive force, its inclination to the normal of the back and its height above the base, draws the pressure on the back and the critical failure surface, shows the force of the trial wedges, writes out the working and prepares a PDF or Word report.
Which module for which job
This module: passive resistance on a rough wall, a curved failure surface, soils in layers.
Rankine and Rankine, sloping backfill give active and passive pressure on a smooth back; Coulomb uses a plane failure surface with wall friction and a battered wall; Mononobe–Okabe gives the active pressure in an earthquake; At rest gives the pressure behind a wall that does not move.
A load on the ground behind a wall is the subject of the Surcharge loads modules.
Before you start: what you need
Have these ready from your design:
What
Where it comes from
Notes
The wall height H
Your design
The vertical height of the wall over which the passive pressure acts. At least 0.1 m.
The inclination θ of the back
Your design
Measured from the vertical, positive toward the backfill (the soil). 0° is a vertical back; the limits are −30° and 30°.
The slope α of the ground surface
Survey
Positive when the ground rises away from the wall; from −30° to 30°. The size of α is limited by φ′: see Messages.
The soil layers
Tests or correlations
For each layer: the unit weight γ, the saturated unit weight γsat when there is water, the effective friction angle φ′ (10° to 45°; granular, c′ = 0) and the thickness, measured vertically at the back.
The wall friction δ′
Judgement or tests
The friction angle between the wall and each layer, from 0° up to φ′ of that layer. The Theory tab gives δ′ ≈ φ′ for loose granular soil and φ′/2 to 2φ′/3 for dense soil.
The water table
Ground investigation
If there is one: its depth below the top of the wall. The module takes it level, with hydrostatic pressure and no seepage.
The surcharge q
Your design loading
A uniform load on the ground surface in front of the wall. Zero for none.
Quick start in eight steps
Open the module and choose the units
Open Passive, log spiral from the Lateral Earth Pressure family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Describe the wall
In Wall, enter the height H and the inclination θ of the back.
Describe the surface
In Backfill, enter the slope α of the ground surface (0 for level ground).
Enter the soil
In Soil layers, press the first layer in the table to change it (the form turns to EDITING · LAYER 1), or fill the New layer form and press + Add layer for each further layer. Each layer has its unit weights, φ′ and δ′.
Add water and a surcharge if you have them
In Groundwater choose Present and enter the depth zw. In Surcharge enter q.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed — the earth pressure calculated and the inputs lock.
Read the answer
The Results tab gives the passive force, its direction and its height above the base. The Pressure view draws the pressure on the back and the critical failure surface.
Take the working and the report
The Calculations tab shows every step with your numbers. The Report tab saves a PDF or a Word file.
The screen: four views of the wall
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing on the left has four views, chosen with the buttons above it; the tabs on the right hold the model and the answers.
The whole window after Analyze. The bar at the foot of the Model tab turns green and the inputs are locked.
Section is a section through the wall and the soil: the wall with its back at θ, the ground surface at α, the layers, the water table and the surcharge, and the height H. It shows the geometry only and follows the inputs as you type.
Pressure shows, after Analyze, the passive pressure along the back, the water pressure on it, the resultant force with its inclination, its horizontal and vertical components, and the critical failure surface.
Trial wedges plots the force of the log-spiral trial wedges against the depth of point C; the smallest is the passive force and is marked. It is drawn for one soil, a vertical back and a level surface without water or surcharge.
3D shows a length of the wall with the soil and, after Analyze, the force as an arrow.
Section before Analyze: the geometry of the default example.
Pressure after Analyze: the pressure on the back, the resultant, its horizontal component and the critical surface.
Trial wedges: the passive force is the minimum of the curve.
3D after Analyze.
Until you press Analyze the Pressure view says “The pressures and the resultant appear after Analyze”, the Trial wedges view says “The chart appears after Analyze” and the 3D view says “The force appears after Analyze”.
The Show chips
Above the drawing a Show bar switches parts of the picture on and off. The Section view has Dimensions. The Pressure view has six: Pressure p, Resultant P, Ph, Pv components, Water u, Failure surface and Dimensions. The Trial wedges and 3D views have no chips.
Zoom
At the lower right of the drawing, + enlarges it (100%, 135%, 182%, 246%, 332%), − reduces it (it stops at 100%), and the third button fits the whole drawing to the pane again. The reading in the middle shows the zoom; the mouse wheel zooms too, and dragging pans a zoomed drawing.
Entering the model, section by section
The Model tab has a Model data section and five input sections: Wall, Backfill, Soil layers, Groundwater and Surcharge. Click a title to open or close a section; Next and Back at the foot of a section walk you through them, and Analyze waits at the foot of the tab. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged. Defaults are given in SI; in SI · tf the unit weight of the example reads 1.60095 t/m³, and in US the wall reads 9.84252 ft and the unit weight 99.9443 pcf for the same wall and soil.
Model data
At the top of the tab you can name the model and keep it. Save to file… and Open file… keep the model in a file with the project. Quick Save… asks for a name and lists the version in the box below; choose it there and press Load to come back to it, or Delete to remove it. Reset to defaults puts the example back. Every change is also kept in this browser, so the page opens again with the model you last worked on, analysed or not.
Model data: names, files and quick saves.
Wall
Two numbers: the height and the inclination of the back.
Wall: the height and the inclination.
Wall fields
Field
Symbol
Unit
Default
Height of the wall (vertical) The vertical height of the wall. At least 0.1 m; a blank or smaller box stops the analysis.
H
m · ft
3
Inclination of the wall back from the vertical (+ toward the backfill) Positive toward the backfill (the soil), negative when the back leans away from it. From −30° to 30°; 0° is a vertical back.
θ
°
0
Backfill
One number: the slope of the ground surface in front of the wall.
Backfill: the slope of the surface.
Backfill field
Field
Symbol
Unit
Default
Slope of the surface (+ rising away from the wall) Positive when the surface rises away from the wall. From −30° to 30°. It must be smaller than φ′ in size, and a falling surface also needs φ′ + |α| ≤ 50°: see Messages.
α
°
0
Soil layers
A New layer form on top, a table of the layers below it. The first layer is already in the table. To add a layer, fill the form and press + Add layer: it goes to the bottom of the table. To change a layer, press its row: the form turns to EDITING · LAYER n, changes apply as you type, and Done closes it; Delete layer and the × at the end of a row remove a layer (a single layer cannot be removed). Up to six layers are allowed. The layers run from the top of the wall down; every layer has the fields below, and the table lists them with their units.
Fields of a layer
Field
Symbol
Unit
Default
Name (optional) Shown on the drawing and in the tables; leave it blank for “Layer 1”, “Layer 2” and so on.
—
—
automatic
Thickness (vertical, at the back) Measured vertically at the back of the wall from its top; the layer boundaries are taken level. At least 0.01 m. The last layer has no thickness: it runs on below the heel, and its thickness shows as — in the table.
H
m · ft
2
Unit weight above the water table The unit weight of the layer. At least 1.
γ
kN/m³ · t/m³ · pcf
15.7
Saturated unit weight The weight of the layer below the water table. At least 1, and above that of water (9.81 kN/m³) when the water table is present. Shown when the water table is present.
γsat
kN/m³ · t/m³ · pcf
19
Effective friction angle (granular, c′ = 0) From 10° to 45°.
φ′
°
30
Angle of wall friction against this layer (0 ≤ δ′ ≤ φ′) From 0° up to φ′ of the same layer. The force of the soil on the wall is inclined at δ′ to the normal, turned upward.
δ′
°
15
Soil layers before any layer is added: the form and Layer 1.
Two layers: the last one has no thickness.
Editing a layer: press its row in the table.
Groundwater
Two buttons choose Present or None. With water present the depth zw appears, and each layer asks for its saturated unit weight. Below the table the layers weigh γsat − γw in the effective vertical stress, and the water pressure u = γw (z − zw) acts normal to the back and is added to the force; γw is 9.81 kN/m³.
Groundwater with the water table present.
Groundwater fields
Field
Symbol
Unit
Default
Water table Choose Present or None. It is on the side of the soil.
—
—
None
Depth of the water table below the top of the wall At least 0. A table deeper than the wall has no effect on the pressure on it. Shown when the water table is present.
zw
m · ft
2
Surcharge
One number: a uniform load on the surface. It is taken as vertical, per unit area of the sloping surface, and its part of the passive force has its own coefficient Kpq from its own critical wedge.
Surcharge: one number.
Surcharge field
Field
Symbol
Unit
Default
Uniform surcharge on the surface Vertical, per unit area of the sloping surface. Zero or more; zero for none.
q
kPa · t/m² · ksf
0
How the passive force is found
There is nothing to choose between methods: the module always uses the critical log-spiral surface. What differs is how much of your model it can use in the simple form, and what it shows for comparison.
One soil, a vertical back, a level surface
With one soil, θ = 0, α = 0, no water and no surcharge, the module runs the trial wedges of Terzaghi and Peck. A trial wedge is bounded by the spiral BC from the heel B and the zone ACD in Rankine’s passive state; moments about the centre of the spiral give the force of the trial wedge, and the passive force is the smallest of them. Then Pp = ½ γH² Kp, acting H/3 above the base and inclined at δ′ to the normal of the wall. With δ′ = 0 the spiral flattens to a plane and the result is Rankine’s, Kp = tan²(45° + φ′/2).
In this form the module also lists Kp by other solutions for the same wall: Lancellotta (a lower bound), Shields and Tolunay (the method of slices), Zhu and Qian (for φ′ from 30° to 45°), Coulomb (plane surface) and Rankine (δ′ = 0). They are for comparison; the passive force of the module is the log spiral.
Anything more: the general form
When there is an inclined back, a sloping surface, more than one layer, water or a surcharge, the module generalises the wedge. It finds two coefficients for each layer, Kpγ for the weight of the soil and Kpq for what lies above the layer, each from its own critical wedge, and takes the smaller of the log spiral and Coulomb’s plane for each. Then it builds the pressure along the back layer by layer from the effective vertical stress, adds the water pressure, and sums the force:
σ′v = qe + Σ γihi, σ′p = [Kpqσ′v,top + Kpγ (σ′v − σ′v,top)] cos θ, qe = q cos θ / cos(θ − α)
The force of each layer is the area of its pressure diagram along the back, Pi; its normal part is Ni = Pi cos δ′ and its part along the back Ti = Pi sin δ′, up the back. The total is Pp = √(N² + T²) at an angle tan⁻¹(T/N) to the normal, and the line of action comes from the moment of N about the base. The Calculations tab writes out each of these steps with your numbers. The wedge shown in the Pressure view is that of the layer at the heel.
What the module says about the generalised form
The layers are taken with level boundaries, their thicknesses measured vertically at the back, and the water table is level with hydrostatic pressure and no seepage.
For one soil it reduces to the classic wedge; where the wall friction equals the inclination of Rankine’s stress on the back it gives Rankine’s passive force of the slope.
A back inclined within ±30° and a rising or falling surface are covered. For a falling surface the limit is φ′ + |α| ≤ 50°.
How to choose δ′
δ′ is a property of the wall face and the soil, not a calculation result. The module accepts any value from 0° to φ′ of the layer.
The Theory tab gives δ′ ≈ φ′ for loose granular soil and φ′/2 to 2φ′/3 for dense soil; the curvature of the failure surface matters when δ′ is above φ′/2, and the results table shows how much the answer changes: see How each input changes the answer.
δ′ = 0 gives Rankine’s value, the lowest passive force for the same soil.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the earth pressure calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed.” It also carries a Warnings (0) or Notes (n) button and Edit inputs. The Results tab shows the cards; the Pressure view now carries the pressure, the force and the failure surface.
One soil, vertical back, level surface
The Results tab of the example.
Card
Symbol
What it is
The example
Passive force, log spiral
Pp
The resultant per unit length of wall, in kN/m (tf/m or kip/ft in the other systems).
326.0 kN/m
Passive coefficient, log spiral
Kp
The coefficient of the critical wedge.
4.615
Horizontal component
Pp,h
Pp cos δ′, horizontal, into the wall.
314.9 kN/m
Vertical component
Pp,v
Pp sin δ′, upward on the wall.
84.38 kN/m
Lancellotta (lower bound)
Kp,L
The coefficient by Lancellotta, for comparison.
4.439
Coulomb (plane)
Kp,C
The coefficient for a plane surface, for comparison; shown when it has a finite value.
4.977
Under the cards a line gives the depth of point C, the end of the spiral, then a table of Values, step by step, and a table of Kp by each solution with the passive force each would give.
Anything more
For an inclined back, a sloping surface, layers, water or a surcharge the cards change: Passive forcePp, Inclination to the normal of the backβ, Line of action above the basez̄, the Horizontal component and the Vertical component; with a single layer also the coefficients Passive coefficient of the soil weightKpγ and Passive coefficient of a surchargeKpq; with water, From the water (normal to the back)Pw. Below them are tables of the passive coefficients of the layers, the stresses on the back and the force of each layer.
The Results tab of the layered example in the worked example below.
Using the numbers
Pp is the force of the soil on the wall per unit length of wall. Its direction is inclined at δ′ (or β in the general form) to the normal of the back, turned upward: the wall pushes the soil up and its friction holds it down, so the force of the soil on the wall is inclined at δ′ to the normal, turned upward.
The force acts at the height shown above the base: H/3 for one soil with a vertical back and a level surface without water or surcharge.
In the general form the water pressure Pw is part of the total.
Changing the unit system after Analyze re-states the cards, the drawing and the report without a new run. Edit inputs unlocks the model and clears the results.
The Calculations tab, the report and the Theory tab
The Calculations tab opens with Given and what is sought, then numbers the steps. For one soil with a vertical back and a level surface they are Trial wedge, Critical wedge (the position of point C and the spiral’s dimensions: d, ω, r0, r1, W, Pd and the lever arms), Passive force, Components of the force, and Other solutions with the table of coefficients; the Answer closes the tab. For the general form the steps are Log-spiral wedge, Passive coefficients of each layer, Stresses on the back, Force and its direction and Critical surface. Before Analyze the tab asks you to press Analyze.
The Calculations tab: the critical wedge and the passive force of the example.
The report
In the Report tab fill in the cover: project, location, client, report number, date, prepared by, title or qualification, checked by, and optional notes. The details are kept in this browser and saved with the model file. Save as PDF and Download Word (.docx) make the report; the preview below the buttons is the report itself.
The report has a summary cover and six parts: 1 Introduction and scope, 2 Input data, 3 Calculation (the steps of the Calculations tab), 4 Results, 5 Summary and 6 References. It carries the Section, Pressure, Trial wedges and 3D drawings, states the units with their conversion constants, and lists the notes of the analysis.
The Report tab: the cover details and the two download buttons.
The Theory tab
The Theory tab has eight sections: Wall friction and the failure surface; The logarithmic spiral; The log-spiral trial wedge; Coefficient of passive pressure; Caquot and Kerisel; Inclined back, sloping surface, layers; Worked example; and References.
Worked examples
The module’s own example
The module opens on this example, so you can follow the numbers on your own screen.
Inputs of the example
Section
Input
Value
Wall
Height, inclination
H = 3 m, θ = 0°
Backfill
Slope
α = 0°
Soil layers
One layer
γ = 15.7 kN/m³, φ′ = 30°, δ′ = 15°
Groundwater, Surcharge
—
None, q = 0
The Calculations tab gives the critical wedge at d = 1.899 m below the surface (d/H = 0.633, ω = 23.97°), and then
Pp = ½ γH² Kp = ½ × 15.70 × 3.00² × 4.615 = 326.0 kN/m
A hand check: ½ × 15.70 × 9 = 70.65 kN/m, and 70.65 × 4.6147 = 326.0. The force acts H/3 = 1.00 m above the base, inclined at 15° to the normal, so
Pp,h = Pp cos 15° = 314.9 kN/m, Pp,v = Pp sin 15° = 84.38 kN/m (upward)
The same wall by each solution (the module’s table)
Solution
Kp
Pp (kN/m)
Log spiral
4.615
326.0
Lancellotta (lower bound)
4.439
313.6
Shields and Tolunay
4.280
302.4
Zhu and Qian
4.575
323.2
Coulomb (plane)
4.977
351.6
Rankine (δ′ = 0)
3.000
212.0
The plane overstates the resistance by about 8 % here, and Rankine’s smooth wall understates it by about a third. Setting δ′ = 0 in the module gives Kp = 3.000 = tan²60° and Pp = 212.0 kN/m, the last row.
The same example in other units
Switch to SI · tf: the module gives Pp = 33.25 tf/m, with the same Kp = 4.615. Switch to US: Pp = 22.34 kip/ft with Kp = 4.615. They are the same wall and soil in other units (326.03 kN/m ÷ 9.80665 = 33.25 tf/m).
A layered example with water and a surcharge
A 4 m wall with a vertical back and a level surface, a surcharge of 10 kPa, two layers and the water table 2 m below the top of the wall:
Inputs of the layered example
Layer
H (m)
γ
γsat
φ′, δ′
Layer 1
1.5
17
19
32°, 16°
Layer 2 (to the heel)
—
19
21
36°, 18°
Press Analyze. The module gives for the layers Kpγ = 5.2732 and Kpq = 5.1628 (Layer 1), 7.0264 and 6.8282 (Layer 2). The pressure on the back is built from these: at the top of Layer 1, σ′v = q = 10 kPa and σ′p = 5.1628 × 10 = 51.63 kPa; at its base σ′v = 10 + 17 × 1.5 = 35.5 kPa and σ′p = 5.1628 × 10 + 5.2732 × 25.5 = 186.1 kPa; at the top of Layer 2, 6.8282 × 35.5 = 242.4 kPa; at the base, 466.4 kPa, with the water pressure 19.62 kPa there. The force of each layer is the area of its pressure diagram along the back: P1 = 178.3 kN/m for Layer 1 and P2 = 913.4 kN/m for Layer 2. Their normal parts are 178.3 cos 16° = 171.4 and 913.4 cos 18° = 868.7 kN/m, and the water adds 19.62, so N = 171.4 + 868.7 + 19.62 = 1059.7 kN/m. Their parts along the back are 178.3 sin 16° = 49.14 and 913.4 sin 18° = 282.3, so T = 49.14 + 282.3 = 331.4 kN/m, up the back. Then
Pp = √(1059.7² + 331.4²) = 1110 kN/m, β = tan⁻¹(331.4 / 1059.7) = 17.37°, z̄ = 1.441 m above the base
The module gives Pp,h = 1060 kN/m and Pp,v = 331.4 kN/m, and the water 19.62 kN/m (= ½ × 9.81 × 2²).
The Pressure view of the layered example: the pressure steps at the layer boundary and the water adds to the force.
How each input changes the answer
Each row changes one thing in the example (H = 3 m, γ = 15.7 kN/m³, φ′ = 30°, δ′ = 15°, level surface, vertical back) and runs the module again; for the first rows δ′ follows φ′/2. The numbers are the ones the module gives, so you can reproduce any row. In the rows from θ down, the module is in its general form and the coefficient is Kpγ.
The example with one change at a time
Change
Kp or Kpγ
Pp (kN/m)
Pp,h · Pp,v
The example: φ′ = 30°, δ′ = 15°
4.615
326.0
314.9 · 84.38
φ′ = 25°, δ′ = 12.5°
3.388
239.3
233.7 · 51.80
φ′ = 35°, δ′ = 17.5°
6.522
460.8
439.4 · 138.6
φ′ = 40°, δ′ = 20°
9.672
683.3
642.1 · 233.7
δ′ = 0°
3.000
212.0
212.0 · 0
δ′ = 10°
4.020
284.0
279.7 · 49.32
δ′ = 20°
5.260
371.6
349.2 · 127.1
δ′ = 30° (equal to φ′)
6.677
471.8
408.6 · 235.9
H = 6 m
4.615
1,304
1,260 · 337.5
γ = 18 kN/m³
4.615
373.8
361.1 · 96.75
θ = +10° (back leans into the soil)
3.728
263.4
262.4 · 22.96
θ = −10° (back leans away)
6.004
424.2
384.4 · 179.3
α = +10° (rising)
6.402
452.3
436.9 · 117.1
α = −10° (falling)
3.135
221.5
213.9 · 57.33
q = 10 kPa
4.615
462.0
446.2 · 119.6
Water table at 1 m
4.615
284.9
276.5 · 68.83
What the rows show:
φ′ is the strongest soil input.Kp rises from 3.388 at 25° to 9.672 at 40°, with δ′ = φ′/2 throughout.
Wall friction raises the force a lot. From δ′ = 0 to δ′ = φ′ the force rises from 212.0 to 471.8 kN/m. At δ′ = 0 the result is Rankine’s, Kp = 3.000.
The force follows H² and γ. Doubling the wall height multiplies the force by four ( 1,304 kN/m, with the same Kp); γ = 18 against 15.7 scales it by 18/15.7.
The back and the surface move Kpγ the way you would expect. A back leaning into the soil or a falling surface lowers it; a back leaning away or a rising surface raises it.
A surcharge adds to the force and lifts its line of action; with water the water pressure adds Pw and the effective weight below the table falls.
Messages: when the module stops, and notes
The module checks the numbers it needs. When one of them cannot be used it stops and nothing is calculated.
A condition of the solution (the first table below): the bar at the foot of the Model tab turns brown and reads Cannot be analysed followed by the message, the field is marked in red, and the drawing shows a red box, “Not analysed — the drawing shows the geometry only. Why:”. Pressing Analyze opens “Cannot analyse — a condition of the solution is not met” with the message and “Nothing was calculated. Change the inputs marked in red, then press Analyze again.”
A value outside its range (an empty or out-of-limits box): the bar stays blue and reads Not analysed, the row of the box is marked, and the message is written in italics in the drawing. Pressing Analyze opens “Cannot analyse — 1 input error” (or n input errors) with the message and a Close button.
The window that Analyze opens for δ′ > φ′.
The bar after a stop.
Conditions of the solution
Kind
When
Message and what to do
Stop
δ′ is above φ′ of a layer
“Layer 1: δ′ > φ′.” The wall friction cannot exceed the friction of the soil. Use δ′ ≤ φ′.
Stop
|α| is not below φ′
“Layer 1: the surface is as steep as φ′.” Below the surface there is no passive Rankine state, on which the wedge is built. Keep the size of α below φ′. With φ′ = 30°, α = 25° is accepted and α = 30° stops.
Stop
A falling surface with φ′ + |α| above 50°
“Layer 1: a falling surface with φ′ + |α| > 50°.” For a surface falling this steeply the wedge overstates the force, so the module does not use it. Use a flatter surface or a smaller φ′. α = −25° with φ′ = 30° stops; −20° gives an answer.
Stop
The layers above the last reach the base
“The layers above the last one already reach the base.” The last layer would not meet the wall. Reduce the thicknesses or the number of layers.
Stop
Water present and γsat not above 9.81 kN/m³
“Layer 1: the saturated unit weight … is not above that of water.” Below the water table the effective weight would be zero or negative. Enter the saturated unit weight of the soil.
Values outside their range
A box outside its limits stops the analysis with a message that names it, for example “Height of the wall (vertical) (H) must be at least 0.1 m.”, “Effective friction angle (granular, c′ = 0) (φ′1) must be at least 10°.”, “Inclination of the wall back from the vertical (+ toward the backfill) (θ) must be at most 30°.” An empty box reads “… is missing.” The limits are in the field tables above.
Notes
Some results carry a note: the bar shows Notes (1) instead of Warnings (0), and pressing it opens “Notes and assumptions”. The notes are also listed in the report.
The note for δ′ above φ′/2.
Coulomb’s Kp above the log spiral for δ′ > φ′/2 (one soil, vertical back, level surface). For δ′ = 20° and φ′ = 30° the plane gives Kp = 6.11 against 5.26 from the log spiral, and the passive force is taken from the log spiral.
A back leaning away from the soil with large wall friction (θ < 0 and δ′ above φ′/2). For θ < 0 and δ′ = φ′ the log spiral gives up to about 14 % more than published charts; keep δ′ moderate there. The note appears when θ < 0 and a layer has δ′ above φ′/2 (and the log spiral, not Coulomb’s plane, governs).
Good practice and common mistakes
Pick δ′ deliberately. It changes the passive force a great deal (from 212.0 kN/m at 0° to 471.8 kN/m at 30° for the example). Run the analysis with δ′ at φ′/2 and at 2φ′/3, and use the lower force if you are unsure.
Match the force to the movement your wall allows. The passive force is the resistance at failure; the movement needed to mobilise it is part of your design judgement, and the force you rely on may be a fraction of it.
Check what the force is used for. Combine the passive force with the other forces on the wall in the wall calculation, with the factors your design method requires.
Take the vertical component with care.Pp,v acts upward on the wall; whether you may count on it is a design decision.
Check the soil near the surface. The soil in front of a wall is often disturbed, softened or removed later; the first layers carry little of the force but they are the first to go.
Use layers for real changes. Add a layer when the soil changes, not for small variations; the thicknesses add up from the top of the wall.
Remember the soil is granular. The module takes c′ = 0 and φ′ from 10° to 45°.
Enter the thicknesses as measured vertically at the back, not along an inclined back.
Save your model. The module keeps the last model in the browser, but the file from Save to file… is the record that goes with the project.
Frequently asked questions
How is the passive force on a wall calculated with a log-spiral surface?
The module takes trial wedges bounded by a log spiral from the heel of the wall and a zone in Rankine’s passive state up to the surface; moments about the centre of the spiral give the force of each, and the smallest is the passive force. For one soil, a vertical back and a level surface it is Pp = ½ γH² Kp. For the module’s example (H = 3 m, γ = 15.7 kN/m³, φ′ = 30°, δ′ = 15°), Kp = 4.615 and Pp = 326.0 kN/m.
Why not use Coulomb’s plane for passive pressure?
A plane failure surface is a fair approximation for active pressure, but for passive pressure on a rough wall it overstates the resistance, and the more so the larger the wall friction. In the example Coulomb gives Kp = 4.977 against 4.615 from the log spiral. The module shows both and takes the log spiral.
What wall friction should I use?
Enter δ′ for the wall face and the soil, from 0° up to φ′ of the layer. The Theory tab gives φ′/2 to 2φ′/3 for dense soil and about φ′ for loose soil. The force is sensitive to it: for φ′ = 30° the example gives 212.0 kN/m at δ′ = 0°, 326.0 at 15° and 471.8 at 30°.
What happens when δ′ is zero?
The spiral flattens to a plane and the result is Rankine’s, Kp = tan²(45° + φ′/2). For φ′ = 30° that is 3.000, and the force is 212.0 kN/m for the example wall.
Can I use it with several layers, water and a surcharge?
Yes. Up to six granular layers with level boundaries, a level water table with hydrostatic pressure, and a uniform surcharge. The module works on the pressure along the back from the coefficients Kpγ and Kpq of each layer and adds the water pressure. It also takes an inclined back (−30° to 30°) and a rising or falling surface.
Why does Analyze say “the surface is as steep as φ′”?
When the surface slope reaches the friction angle, the soil below it has no passive Rankine state, on which the log-spiral wedge is built. Keep the size of α below φ′. A falling surface is also limited to φ′ + |α| ≤ 50°.
Is the force horizontal?
No. It is inclined at δ′ to the normal of the back, turned upward, because the wall pushes the soil up and its friction holds it down. The module gives the horizontal component Pp,h = Pp cos δ′ and the vertical one Pp,v = Pp sin δ′ (314.9 and 84.38 kN/m for the example).
Where does the force act on the wall?
For one soil, a vertical back and a level surface without water or surcharge, at H/3 above the base. In the general form the module gives the height of the line of action above the base: 1.441 m for the layered example of the worked examples (4 m wall, two layers, water at 2 m, surcharge 10 kPa).
Which units does it use, and what are the units of the answer?
SI · kN, SI · tf or US, switchable in the header at any time, even after Analyze. The force is in kN/m, tf/m or kip/ft, the pressure in kPa, t/m² or ksf, and the unit weight in kN/m³, t/m³ or pcf. The example gives 326.0 kN/m, 33.25 tf/m or 22.34 kip/ft.
Does the module remember my inputs?
Yes. Every change is kept in this browser, so the page opens again with the model you last worked on, analysed or not. Reset to defaults brings back the example; Save to file… and Quick Save… keep versions you name.
Can I drag the wall or the water table on the drawing?
No. The drawing follows the Model tab and has no handles. You can zoom with the + and − buttons or the mouse wheel, pan a zoomed drawing by dragging it, and fit it with the third button; each view keeps its own zoom, and the 3D view does not rotate.
What does the report contain?
A cover with your project details and a summary, then six parts: introduction and scope, input data, the calculation step by step, results, summary and references, with the Section, Pressure, Trial wedges and 3D drawings. It is saved as a PDF or a Word file.
Passive earth pressure in an earthquake: user manual
Find the passive force that soil in front of a vertical wall can give while the ground shakes, with the horizontal and vertical seismic coefficients you choose: its size, its direction and the height at which it acts, for one soil or several layers, with a water table and a surcharge. This manual walks through the screen, every input, the results and the report, with worked examples you can reproduce.
Passive, earthquake gives the passive force that the soil in front of a vertical wall can put up when the wall is pushed into it and the ground is shaking: the force per metre of wall, its direction and the height at which it acts. The shaking is taken as a pseudo-static load: a horizontal coefficient kh and a vertical coefficient kv, both entered by you, act on the soil wedge in front of the wall. In an earthquake the inertia of the wedge helps to push it out, so the passive resistance is lower than the static value.
The module follows Subba Rao and Choudhury (2005). For one soil without water the passive force is
Ppe = [ ½ γH² Kpγ(e) + qHKpq(e) + 2 c′HKpc′(e) ] / cos δ′
Kpγ(e), Kpq(e) and Kpc′(e) are the passive coefficients for the weight of the soil, the surcharge and the cohesion, taken normal to the wall; the division by cos δ′ turns the normal force into the force inclined at the wall friction δ′. The module reads the coefficients for your φ′, δ′, kh and kv from the published curves and tables of the method; you do not look anything up. For several layers or a water table it builds the pressure along the wall layer by layer instead and adds the parts, as described under How the passive force is found.
What you can describe
A vertical wall of height H, with its back vertical and the ground surface in front of it level.
One to six layers of soil, each with its unit weights, friction angle φ′ (10° to 40°), wall friction δ′ (from φ′/2 to φ′) and cohesion c′.
The wall adhesion: none, or equal to the cohesion times tan δ′/tan φ′.
The earthquake: kh from 0 to 0.5, kv from 0 to kh (upward).
A level water table with restrained or free pore water, and a uniform surcharge on the surface.
The module gives the passive force, its inclination, its components and the height of its line of action, draws the pressure on the wall, plots the coefficient Kpγ(e) against kh, writes out the working and prepares a PDF or Word report.
Which module for which job
This module: passive resistance of a vertical wall in an earthquake, with kh and kv given.
Passive, log spiral gives the static passive force, with an inclined back and a sloping surface.
Mononobe–Okabe gives the active pressure in an earthquake.
A load on the ground behind a wall is the subject of the Surcharge loads modules.
Before you start: what you need
Have these ready from your design:
What
Source
Notes
The wall height H
Your design
The height of the vertical back over which the passive pressure acts. At least 0.1 m.
The soil layers
Tests or correlations
For each layer: the unit weight γ, with water also γsat (and γd for free pore water), φ′ from 10° to 40°, δ′ and c′, and the thickness, measured from the top of the wall.
The wall friction δ′
Judgement or tests
From φ′/2 to φ′ of each layer. A value outside that range stops the analysis.
The seismic coefficients
Your design
kh from 0 to 0.5 and kv from 0 to kh. You enter them; the module has no map or code table.
The water table
Ground investigation
If there is one: its depth below the top of the wall, and whether the pore water moves with the soil or not.
The surcharge q
Your design loading
A uniform load on the surface in front of the wall. Zero for none.
Quick start in eight steps
Open the module and choose the units
Open Passive, earthquake from the Lateral Earth Pressure family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Describe the wall
In Wall, choose the wall adhesion and enter the height H.
Enter the soil
In Soil layers, check the first layer in the table and change it, or fill the New layer form and press + Add layer for each further layer.
Enter the earthquake
In Earthquake, enter kh and kv.
Add water and a surcharge if you have them
In Groundwater choose Present, enter the depth zw and the pore water. In Surcharge enter q.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed — the earth pressure calculated and the inputs lock. The module stays on the Model tab and the Section view: open Results to read the answer and press Pressure to see the pressure on the wall.
Read the answer
Open the Results tab for the passive force, its line of action and its components; press Pressure above the drawing for the pressure on the wall and the force.
Take the working and the report
The Calculations tab shows every step with your numbers. The Report tab saves a PDF or a Word file.
The screen: four views of the wall
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing on the left has four views, chosen with the buttons above it; the tabs on the right hold the model and the answers.
The whole window after Analyze, with the Results tab open.
Section is a section through the wall and the soil: the wall, the level surface, the layers, the water table and the surcharge, and the height H. It shows the geometry only and follows the inputs as you type. Before Analyze it says “Not analysed yet: the drawing follows the inputs.”
Pressure shows, after Analyze, the passive pressure along the wall, the water pressure on it, the resultant Ppe with its inclination, its horizontal and vertical components and the height of its line of action.
Kpγ(e) against kh plots the coefficient Kpγ(e) for the φ′, δ′/φ′ and kv of the first layer against kh, with your kh marked. It appears after Analyze (“The chart appears after Analyze”).
3D shows a length of the wall with the soil and, after Analyze, the force as an arrow.
Section before Analyze: the geometry of the default example.
Pressure after Analyze.
The coefficient against kh, drawn for the φ′, δ′/φ′ and kv of the first layer.
3D after Analyze.
The Show chips
Above the drawing a Show bar switches parts of the picture on and off. The Section view has Dimensions. The Pressure view has five: Pressure p, Resultant Ppe, Ph, Pv components, Water u and Dimensions. The other two views have no chips.
The Show bar of the Pressure view.
Zoom
At the lower right of the drawing, + enlarges it (100 %, 135 %, 182 %, … up to 800 %), − reduces it and the third button fits the whole drawing again. The mouse wheel over the drawing zooms in and out; once zoomed, drag the drawing with the mouse to move it. − and the fit button are greyed at 100 %.
Entering the model, section by section
The Model tab has a Model data section and five input sections: Wall, Soil layers, Earthquake, Groundwater and Surcharge. Click a title to open or close a section; Next and Back at the foot of a section walk you through them, and Analyze waits at the foot of the tab. Where a unit is shown twice or three times, such as m · ft, the units are those of SI · kN, SI · tf and US in turn, and equal ones are merged.
Model data
At the top of the tab you can name the model and keep it. Save to file… and Open file… keep the model in a file with the project. Quick Save… asks “Quick Save — a name for this version:”; the version is listed in the box below with its date. Choose it there and press Load to come back to it, or Delete to remove it (the module asks you to confirm). Reset to defaults puts the example back at once. Every change is also kept in this browser. If you reload after Analyze the results come back and the inputs are still locked; press Edit inputs to change them. Quick saves and model files also keep the analysed state. The model section says “Every change is kept in this browser, so the page opens again with the model you last worked on.”
Model data: names, files and quick saves.
Wall
A choice and one number: the wall adhesion and the height of the wall. The back is vertical; the module does not take an inclined back. The ? beside each adhesion button opens a window that explains it and shows what it does to your force: with the example, Ppe = 3608.1 kN/m without adhesion and 3610.2 kN/m with adhesion equal to the cohesion times tan δ′/tan φ′.
Wall: the adhesion and the height.
Wall fields
Field
Default
What to enter
Wall adhesion
c′ₐ = 0
Two buttons: no adhesion between the wall and a cohesive soil, or adhesion equal to the cohesion times tan δ′/tan φ′. Only the cohesion part of the force depends on this choice.
Height of the wall (vertical back) H m · ft
4
At least 0.1 m; a blank or smaller box stops the analysis.
Soil layers
A New layer form on top, a table of the layers below it. The first layer is already in the table. To add a layer, fill the form and press + Add layer: it goes to the bottom of the table. To change a layer, press its row: the form turns to the layer and Done closes it; the × at the end of a row removes a layer (a single layer cannot be removed). Up to six layers are allowed; the counter in the form reads “2 of 6” with two layers. The layers run from the top of the wall down, and the table lists the unit weights, φ′, δ′ and c′ of each (the saturated and dry unit weights show — until water is present). Layer 1 opens with γ 17.5, γsat 19.5, γd 16, φ′ 40°, δ′ 40° and c′ 10; the New layer form opens with H 2 m, γ 18, γsat 20, γd 16, φ′ 30°, δ′ 20° and c′ 0.
Fields of a layer
Field
Default
What to enter
Name (optional)
automatic
Shown on the drawing and in the tables; leave it blank for “Layer 1”, “Layer 2” and so on.
Thickness (vertical) H m · ft
2
Measured from the top of the wall; the layer boundaries are taken level. At least 0.01 m. The last layer has no thickness: it runs on below the heel.
Unit weight above the water table γ kN/m³ · t/m³ · pcf
17.5
At least 1.
Saturated unit weight γsat kN/m³ · t/m³ · pcf
19.5
At least 1, and above that of water (9.81 kN/m³) in a layer that is below the water table. Shown when the water table is present.
Dry unit weight γd kN/m³ · t/m³ · pcf
16
Above zero and not above γsat. Only the dry mass carries inertia below the water table when the pore water is free. Shown when the water table is present and the pore water is free.
Friction angle (10–40°, the range of the charts) φ′ °
40
From 10° to 40°; outside it the analysis stops.
Angle of wall friction (φ′/2 ≤ δ′ ≤ φ′, the range of the charts) δ′ °
40
From φ′/2 to φ′ of the same layer. The force of the layer is inclined at δ′ to the normal of the wall.
Cohesion c′ kPa · kg/cm² · psf
10
Zero or more.
Soil layers before any layer is added: the form and Layer 1.
Two layers: the last one has no thickness.
Editing a layer: press its row in the table.
Earthquake
Two numbers, the horizontal and the vertical seismic coefficient. The section says that each chart curve ends where its passive resistance runs out, and that a kv between 0 and kh is interpolated linearly. The module takes kv as upward, the sense that reduces the weight of the soil and so the passive force.
Earthquake: two numbers.
Earthquake fields
Field
Default
What to enter
Horizontal seismic coefficient kh —
0.3
From 0 to 0.5. Each curve of the method ends where its passive resistance runs out, so the usable kh is smaller for a smaller φ′: see Messages.
From 0 to 0.5, and not above kh. Upward, the sense unfavourable to a passive force.
Groundwater
Two buttons choose Present or None. With water present, a second choice describes the pore water in the earthquake and the depth zw appears; each layer then asks for its saturated unit weight, and for its dry unit weight when the pore water is free. Below the table the layers weigh γsat − γw in the effective vertical stress, and the water pressure u = γw (z − zw) acts normal to the wall; γw is 9.81 kN/m³.
Groundwater fields
Field
Default
What to enter
Water table
None
Choose Present or None. It is on the side of the soil.
Pore water in the earthquake
Restrained
Choose Restrained (moves with the soil) or Free (highly permeable soil). The permeability limit between the two is 5×10⁻⁴ m/s. Shown when the water table is present.
Depth of the water table below the top of the wall zw m · ft
2
At least 0. A table at or below the base of the wall has no effect on the result. Shown when the water table is present.
Groundwater with no water table.
With water present: the pore water and the depth.
Soil layers with water present: the saturated unit weights are filled in; the dry one is asked for with free pore water only.
Surcharge
One number: a uniform load on the level surface. It enters the force through its own coefficient Kpq(e).
Surcharge: one number.
Surcharge field
Field
Default
What to enter
Uniform surcharge on the surface q kPa · t/m² · ksf
10
Zero or more; zero for none.
How the passive force is found
There is no method to choose: the module always uses the coefficients of Subba Rao and Choudhury (2005), which come from a curved failure surface analysed with the pseudo-static forces khW and kvW on the wedge. What your inputs decide is which coefficients are read and how they are combined.
The coefficients
For a layer the module reads Kpγ(e) and Kpq(e) from the curves of the method, which are drawn for φ′ = 10°, 20°, 30° and 40°, δ′/φ′ = 0.5 and 1, and kv = 0 and kv = kh. Between them the Calculations tab says the module interpolates linearly in kh and in δ′/φ′ and log-linearly in φ′, and linearly in kv between the curves of kv = 0 and kv = kh. Kpc′(e) comes from a table for the same four angles, in two columns: no adhesion, or adhesion equal to the cohesion times tan δ′/tan φ′.
For the example (φ′ = δ′ = 40°, kh = 0.3, kv = 0) the module gives Kpγ(e) = 13.20, Kpq(e) = 11.00 and Kpc′(e) = 5.950. All three are components normal to the wall.
One soil, no water
The formula of the first section applies. The weight part acts at H/3 above the base, the surcharge and cohesion parts at H/2, so the line of action is
z̄ = [ Pγ (H/3) + (Pq + Pc′) (H/2) ] / Ppe
The force is inclined at δ′ to the normal of the wall, turned upward: the wall pushes the soil up and the friction of the wall holds it down. Its components are Ppe,h = Ppe cos δ′, horizontal, into the wall, and Ppe,v = Ppe sin δ′, upward on the wall.
Layers or water: the pressure along the wall
With more than one layer, or with a water table above the base, the module works on the pressure instead. Each layer takes its own coefficients; the weight of the layers above it and the surcharge act on it as a surcharge, through Kpq(e); its own weight acts through Kpγ(e); its cohesion through Kpc′(e). The pressure normal to the wall, per unit depth, is
σn = Kpq(e)σ′v,top + Kpγ(e) (σ′v − σ′v,top) + 2 c′Kpc′(e), along the wall σn tan δ′ (upward)
The module sums the normal part N and the part along the wall T over the height of the wall. Then Ppe = √(N² + T²), the inclination is β = tan⁻¹(T/N) to the normal, Ppe,h = N and Ppe,v = |T|, and the line of action comes from the moment of N about the base. The Calculations tab writes out each step. For one soil without water this reduces to the formula above.
Water in the earthquake
Effective stress. Below the level water table the layers weigh γ′ = γsat − γw in σ′v, and the water pressure u acts normal to the wall and is added to the force as Pw.
Restrained pore water (soil of low permeability: the water moves with the soil). Below the water table the curves are read at a larger coefficient, kh,w = khγsat / (γsat − γw). With the layered example below, kh = 0.1 is read at 0.196 in the layer under water.
Free pore water (highly permeable soil: the water does not move with the soil). Only the dry mass carries inertia, so kh,w = khγd / (γsat − γw), and the hydrodynamic pressure of the water, ⅞ khγw √(dw (z − zw)), is taken off the resistance. The module shows it as Hydrodynamic, taken off the resistance.
The reading is limited. The curves end at a largest kh; below the water table the larger kh,w must still be inside them (see Messages).
The Theory tab calls the layered form with a water table the module’s own extension of the single-soil solution. For one soil without water it gives the same force as the formula.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the earth pressure calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed.” It also carries Warnings (0) and Edit inputs. After Analyze the screen stays on the Model tab and the Section view. Open Results for the cards; press Pressure above the drawing to see the pressure and the force.
The bar after Analyze.
One soil, no water
The Results tab of the example.
Card
Symbol
The example
Passive force with earthquake
Ppe
3,608 kN/m
Line of action above the base
z̄
1.554 m
Horizontal component
Ppe,h
2,764 kN/m
Vertical component, upward on the wall
Ppe,v
2,319 kN/m
Coefficient, unit weight (normal)
Kpγ(e)
13.20
Coefficient, surcharge (normal)
Kpq(e)
11.00
Coefficient, cohesion (normal)
Kpc′(e)
5.950
Under the cards a table of Values, step by step repeats the coefficients, the force, the line of action and the components. The force is per metre of wall, in kN/m, tf/m or kip/ft.
Layers or water
When there is more than one layer or water above the base, the cards are Passive force with earthquake, Line of action above the base, Inclination to the normal of the wall (β), Horizontal component and Vertical component, upward on the wall. With water a card From the water (normal to the wall) (Pw) is added, and with free pore water Hydrodynamic, taken off the resistance (Pwd). Under the cards the tab lists Values, step by step, The chart coefficients of the layers, Pressure on the wall and Force of each layer, instead of the three coefficient cards.
The Results tab of the layered example in the worked example below.
Using the numbers
Ppe is the force of the soil on the wall per unit length of wall, inclined at δ′ (or β) to the normal of the wall, turned upward.
The module gives the passive force during the earthquake; combine it with the other forces on the wall in your wall calculation.
Changing the unit system after Analyze re-states the cards, the drawing and the report without a new run. Edit inputs unlocks the model and clears the results.
The Calculations tab, the report and the Theory tab
The Calculations tab opens with Given and what is sought, then numbers the steps. For one soil without water they are Coefficients, Passive force, Line of action and Components of the force, then the Answer. For layers or water they are Coefficients of each layer (with a table of the φ′, δ′, kh read and the three coefficients of each layer and zone), Pressure on the wall (the σ′v, σn and u at each level, with the worked arithmetic) and Force, its direction and line of action (the N and T of each layer), then the Answer.
Step 2 for the example: the three parts and their sum.
Step 2 for the layered example with water.
Step 3 for the layered example.
The report
In the Report tab fill in the cover: project, location, client, report number, date, prepared by, title or qualification, checked by, and optional notes. The details are kept in this browser and saved with the model file. Save as PDF and Download Word (.docx) make the report; the preview below the buttons is the report itself.
The report has a summary cover and six parts: 1 Introduction and scope, 2 Input data, 3 Calculation (the steps of the Calculations tab), 4 Results, 5 Summary and 6 References. It carries the Section, Pressure, coefficient and 3D drawings, states the units with their conversion constants, and gives the warnings and notes when there are any.
The Report tab: the cover details and the two download buttons.
The Theory tab
The Theory tab has five sections, with Open all and Close all above them: The pseudo-static passive force; The coefficients; Layers, water and surcharge; Worked example; and References. It writes the formula, the line of action and the layered pressure, and the reading of the coefficients below the water table, as used in the Calculations.
Worked examples
The module’s own example
The module opens on this example, so you can follow the numbers on your own screen.
Inputs of the example
Section
Input
Value
Wall
Adhesion, height
c′a = 0, H = 4 m
Soil layers
One layer
γ = 17.5 kN/m³, φ′ = 40°, δ′ = 40°, c′ = 10 kPa
Earthquake
kh, kv
0.3, 0
Groundwater, Surcharge
—
None, q = 10 kPa
The Calculations tab gives the coefficients Kpγ(e) = 13.20, Kpq(e) = 11.00, Kpc′(e) = 5.950, and then
Ppe = (1847.86 + 440.12 + 476.00) / cos 40° = 3,608 kN/m
A hand check: ½ × 17.5 × 16 = 140 and 140 × 13.199 = 1847.9; 10 × 4 × 11.003 = 440.1; 2 × 10 × 4 × 5.950 = 476; the sum 2764.0 divided by cos 40° = 0.7660 gives 3608.1. The weight part acts at H/3 and the other two at H/2, so
z̄ = (2412.21 × 1.3333 + 1195.91 × 2) / 3608.12 = 1.554 m above the base
The force is inclined at δ′ = 40° to the normal, so Ppe,h = Ppe cos 40° = 2,764 kN/m, which is the sum 2764.0 of the three parts, and Ppe,v = Ppe sin 40° = 2,319 kN/m, upward.
The same example in other units
Switch to SI · tf: the module gives Ppe = 367.9 tf/m at z̄ = 1.554 m (3608.1 kN/m ÷ 9.80665 = 367.9 tf/m). Switch to US: Ppe = 247.2 kip/ft at z̄ = 5.099 ft, with Ppe,h = 189.4 and Ppe,v = 158.9 kip/ft. The coefficients stay 13.20, 11.00 and 5.950: they have no units.
A check in US units, typed in
Switch to US and enter H = 13 ft, one layer with γ = 110 pcf, φ′ = δ′ = 35° and c′ = 200 psf, kh = 0.2 and q = 0.2 ksf. The module gives Kpγ(e) = 8.533, Kpq(e) = 7.435 and Kpc′(e) = 4.511. By hand: ½ × 110 × 13² × 8.533 = 79 312 lb/ft = 79.31 kip/ft; 0.2 × 13 × 7.435 = 19.33; 2 × 0.2 × 13 × 4.511 = 23.46 (cohesion 200 psf = 0.2 ksf). Their sum 122.10 kip/ft divided by cos 35° = 0.8192 gives Ppe = 149.1 kip/ft, and the line of action is 5.093 ft.
Two layers with water, restrained pore water
Keep the example, set kh = 0.1 (the water table needs it: see Messages), choose Present with zw = 2 m and Restrained, and add a second layer with γ = 18 kN/m³, γsat = 20, φ′ = 32°, δ′ = 22° and c′ = 0. Layer 1 (0 to 2 m) is the example’s soil; its γsat of 19.5 kN/m³ is the default.
Inputs of the layered example
Layer
γ, γsat (kN/m³)
φ′, δ′, c′
Layer 1 (H = 2 m)
17.5, 19.5
40°, 40°, 10 kPa
Layer 2 (to the heel)
18, 20
32°, 22°, 0
Press Analyze. Below the water table the curves are read at kh,w = 0.1 × 20 / (20 − 9.81) = 0.196 in Layer 2, and the module gives Kpγ(e) = 14.110 and Kpq(e) = 11.075 for Layer 1 (read at 0.1), and 5.352 and 5.163 for Layer 2. The pressure on the wall is built from them. At the base of Layer 1, σ′v = 10 + 17.5 × 2 = 45 kPa and
At the top of Layer 2 the surcharge of everything above acts through Kpq(e): σn = 5.163 × 45 = 232.3 kPa. At the base, σ′v = 45 + (20 − 9.81) × 2 = 65.38 kPa and σn = 5.163 × 45 + 5.352 × 20.38 = 341.4 kPa, with the water pressure u = 9.81 × 2 = 19.62 kPa. The normal forces of the layers are the areas of the pressure: (229.8 + 723.6)/2 × 2 = 953.4 kN/m and (232.3 + 341.4)/2 × 2 = 573.7 kN/m, and the water Pw = ½ × 19.62 × 2 = 19.62 kN/m, so N = 1546.7 kN/m. Along the wall T = 953.35 tan 40° + 573.71 tan 22° = 800.0 + 231.8 = 1031.8 kN/m, upward. Then
Ppe = √(1546.7² + 1031.8²) = 1,859 kN/m, β = tan⁻¹(1031.8 / 1546.7) = 33.71°, z̄ = 2.099 m
The module gives Ppe,h = 1,547 kN/m, Ppe,v = 1,032 kN/m and Pw = 19.62 kN/m. The same soil with Free pore water (the dry unit weight of Layer 1 is the default 16 kN/m³, that of Layer 2 is 16 kN/m³) gives Ppe = 1,864 kN/m, with the hydrodynamic pressure of 2.289 kN/m taken off (7/12 × 0.1 × 9.81 × 2² = 2.289 by hand). Without water the layers give 1,954 kN/m, and restrained water with kv = 0.05 gives 1,770 kN/m.
The Pressure view of the layered example: the pressure steps at the layer boundary and the water adds to the force.
How each input changes the answer
Each row changes one thing in the example (H = 4 m, γ = 17.5 kN/m³, φ′ = δ′ = 40°, c′ = 10 kPa, q = 10 kPa, kh = 0.3, kv = 0, no water) and runs the module again. The numbers are the ones the module gives, so you can reproduce any row.
One soil without water: one change at a time
Change
Kpγ(e)
Ppe
Ppe,h / Ppe,v
The example
13.20
3,608
2,764 / 2,319
kh = 0 (no earthquake)
14.49
3,850
2,949 / 2,475
kh = 0.1
14.11
3,778
2,894 / 2,429
kh = 0.2
13.66
3,693
2,829 / 2,374
kh = 0.4
12.74
3,523
2,699 / 2,265
kh = 0.46
12.47
3,472
2,660 / 2,232
kv = 0.05
12.28
3,410
2,613 / 2,192
kv = 0.15
10.44
3,015
2,310 / 1,938
kv = kh = 0.3
7.687
2,422
1,855 / 1,557
φ′ = δ′ = 30°
4.945
1,345
1,165 / 672.5
φ′ = 35°, δ′ = 25°
6.607
1,646
1,492 / 695.7
δ′ = 30° (φ′ = 40°)
10.85
2,673
2,315 / 1,336
δ′ = 20° (φ′/2)
8.495
1,985
1,865 / 678.9
c′ = 0
13.20
2,987
2,288 / 1,920
c′ = 20 kPa
13.20
4,229
3,240 / 2,719
Adhesion equal to the cohesion times tan δ′/tan φ′
13.20
3,610
2,766 / 2,321
H = 6 m
13.20
7,221
5,532 / 4,642
H = 2 m
13.20
1,201
920.0 / 772.0
q = 0
13.20
3,034
2,324 / 1,950
q = 20 kPa
13.20
4,183
3,204 / 2,689
γ = 20 kN/m³
13.20
3,953
3,028 / 2,541
Forces in kN/m.
What the rows show:
kh lowers the force slowly, kv lowers it fast. From kh = 0 to 0.4 the force falls from 3,850 to 3,523 kN/m, less than 9 %; kv = 0.05 alone takes 3,410 kN/m, and kv = kh = 0.3 takes 2,422 kN/m.
φ′ and δ′ are the strongest inputs. At φ′ = δ′ = 30° the force is 1,345 kN/m, less than half of the 40° value; at φ′ = 40° halving δ′ from 40° to 20° takes it from 3,608 to 1,985 kN/m.
Cohesion and surcharge add to the force and lift its line of action.c′ = 0 gives z̄ = 1.462 m against 1.554 m; the weight part acts at H/3, the others at H/2.
The adhesion choice changes little here: 3,610 against 3,608 kN/m; with c′ = 0 it changes nothing.
The height works through H².H = 6 m gives 7,221 kN/m against 3,608 kN/m at 4 m; the surcharge and cohesion parts grow with H, the weight part with H².
In the second table the layered cases use the two layers of the worked example (restrained pore water unless stated); the single-soil cases use the example with the water table as named, restrained unless stated.
Layers and water
Case
Ppe
β, z̄
Pw
Two layers, dry, kh = 0.3 (second: 30°, 20°)
1,743
33.63°, 2.129 m
—
Two layers, dry, kh = 0.1
1,954
33.35°, 2.026 m
—
The same with the water table at 2 m, restrained
1,859
33.71°, 2.099 m
19.62
The same, free pore water
1,864
33.70°, 2.095 m
19.62
The same, restrained, kv = 0.05
1,770
33.73°, 2.105 m
19.62
One soil, dry, kh = 0.1
3,778
40°, 1.545 m
—
One soil, water 2 m, kh = 0.1
3,451
39.79°, 1.621 m
19.62
One soil, water 1 m, kh = 0.1
3,106
39.48°, 1.653 m
44.15
One soil, water 2 m, kh = 0.2
3,320
39.78°, 1.637 m
19.62
One soil, water 2 m, kh = 0.2, free
3,356
39.83°, 1.630 m
19.62
Ppe and Pw in kN/m.
A water table lowers the force (about 9 % here: 3,778 to 3,451 kN/m for the single soil at kh = 0.1) and adds Pw; the higher the table, the larger Pw and the lower the force. Restrained and free pore water differ by about 1 % in these runs. For the single soil without water the inclination is δ′ (40° here).
Messages: when the module stops
The module checks the numbers it needs. When one of them cannot be used it stops and nothing is calculated.
For a condition of the solution the bar at the foot of the Model tab turns brown and reads Cannot be analysed followed by the message. For a value outside its range the box is marked in red and the bar stays Not analysed; press Analyze and a window opens: “Cannot analyse — 1 input error” (or “2 input errors”) with the boxes named, or “Cannot analyse — a condition of the solution is not met”. Both add: “Nothing was calculated. Change the inputs marked in red, then press Analyze again.”
The window that Analyze opens for kh = 0.5.
The bar after a stop.
Conditions of the solution
When
Message and what to do
kh is beyond the end of the curves of the layer
“Layer 1: kₕ beyond the charts.” The message gives where the curves end: for φ′ = 40°, δ′/φ′ = 1 and kv = 0 they end at kh = 0.463, so kh = 0.46 runs and 0.47 stops. With φ′ = δ′ = 20° they end at 0.279. Use a smaller kh or a larger φ′.
kh read below the water table is beyond the curves
“Layer 1 (below the water table): kₕ beyond the charts.” With the example and water at 2 m, restrained, the curves are read at kh,w = 0.604 and the module stops; free pore water is read at 0.495 and stops too. Use a smaller kh (the layered example uses 0.1).
kv is above kh
“kᵥ above kₕ.” The curves are for kv = 0 and kv = kh and are not extended beyond. Use 0 ≤ kv ≤ kh.
δ′/φ′ is outside 0.5 to 1
“Layer 1: δ′/φ′ outside 0.5–1.” For φ′ = 40°, δ′ = 20° runs and 19° stops. The message gives the range: here 20.0° to 40°.
The layers above the last reach the base
“The layers above the last one already reach the base.” The last layer would not meet the wall. Reduce the thicknesses or the number of layers.
Water present and γsat not above 9.81 kN/m³
“Layer 1: the saturated unit weight is not above that of water.” Enter the saturated unit weight of the soil.
Free pore water and γd not above zero, or above γsat
“Layer 1: the dry unit weight γd must be above zero and not above γsat.” Enter γd of the layer.
Values outside their range
A box outside its limits stops the analysis with a message that names it, for example “Height of the wall (vertical back) (H) must be at least 0.1 m.”, “Friction angle (10–40°, the range of the charts) (φ′1) must be at most 40°.”, “Horizontal seismic coefficient (kh) must be at most 0.5.”, “Uniform surcharge on the surface (q) must be at least 0 kPa.” An empty box reads “… is missing.” Two errors at once are counted: φ′ = δ′ = 41° gives “2 input errors”. The limits are in the field tables above.
Warnings
The bar carries Warnings (0); pressing it opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” In the examples in this manual the window reads this.
The Warnings window.
Good practice and common mistakes
Choose kh and kv from your design basis. They are inputs; the module does not derive them. The force depends on them (3,850 kN/m at kh = 0, 3,608 at 0.3, 2,422 with kv = 0.3).
Pick δ′ deliberately. It is limited to φ′/2 to φ′ and changes the force a great deal: 1,985 kN/m at φ′/2 against 3,608 kN/m at φ′ for the example.
Use the lower force if you are unsure. Run the analysis at both ends of the range of δ′ and of kv and use the case that governs your design.
Keep c′ in view. Cohesion is a large part of the force in the example (476 of 2764 kN/m before the division by cos δ′); the adhesion choice and a smaller c′ show how much your result relies on it.
Match the force to the movement your wall allows. The passive force is the resistance at failure; the movement needed to mobilise it, and the factors your design method requires, are yours to apply.
Take the vertical component with care.Ppe,v acts upward on the wall; whether you may count on it is a design decision.
Check the soil in front of the wall. It is often disturbed, softened or removed later; the first layers carry little of the force but they are the first to go.
Use layers for real changes. Add a layer when the soil changes; the thicknesses add up from the top of the wall, and the last layer takes the rest.
Keep the angles inside the range.φ′ from 10° to 40°, δ′ from φ′/2 to φ′, and kh inside the end of the curves; with water, check kh,w too.
Save your model. The module keeps the last model in the browser, but the file from Save to file… is the record that goes with the project.
Frequently asked questions
How is the passive force in an earthquake calculated?
For one soil without water, Ppe = [½ γH² Kpγ(e) + qHKpq(e) + 2 c′HKpc′(e)] / cos δ′, with the coefficients of Subba Rao and Choudhury (2005) read for your φ′, δ′, kh and kv. For the module’s example (H = 4 m, φ′ = δ′ = 40°, c′ = 10 kPa, q = 10 kPa, kh = 0.3) it is 3,608 kN/m.
Where do I get kh and kv?
From your design basis. The module takes them as inputs: kh from 0 to 0.5 and kv from 0 to kh, upward. It does not look them up from a map or a code table.
Why does the module stop with “kₕ beyond the charts”?
The curves of the method end at a largest kh where the passive resistance runs out, and the module does not extend them. For φ′ = δ′ = 40° and kv = 0 they end at 0.463; for 20° at 0.279. The message gives the value. Use a smaller kh or a larger φ′. With a water table the curves are read at a larger kh,w below it, so the limit on kh is usually lower.
Why must δ′ be between φ′/2 and φ′?
The coefficients are given for δ′/φ′ = 0.5 and 1 and the module interpolates between them; it does not extend beyond. δ′ below φ′/2 (19° for φ′ = 40°) or above φ′ stops the analysis.
What do restrained and free pore water mean?
Restrained pore water moves with the soil (low permeability, below 5×10⁻⁴ m/s): the curves are read at kh,w = khγsat / (γsat − γw). Free pore water does not (highly permeable soil): they are read at khγd / (γsat − γw) and the hydrodynamic pressure of the water is taken off the resistance. In the layered example the two give 1,859 and 1,864 kN/m.
Can I use several layers?
Yes, up to six, each with its own γ, φ′, δ′ and c′. Each takes its own coefficients, and the weight above it acts on it as a surcharge. The last layer runs to the heel; the layers above it must be shorter than the wall.
Is the force horizontal?
No. It is inclined at δ′ to the normal of the wall (at β for layers or water), turned upward. The module gives the horizontal component Ppe,h (2,764 kN/m for the example) and the vertical one Ppe,v (2,319 kN/m).
Where does the force act?
For one soil without water the weight part acts at H/3 above the base and the surcharge and cohesion parts at H/2, so the line of action is between: 1.554 m for the example. With layers or water the module finds it from the moment of the normal pressure about the base: 2.099 m in the layered example.
Can the wall be inclined or the ground sloping?
No. The module is for a vertical back and a level surface. The Section view says so (“vertical back”, “level backfill”). For an inclined back or a sloping surface without an earthquake use Passive, log spiral (granular soil).
Which units does it use?
SI · kN, SI · tf or US, switchable in the header at any time, even after Analyze. The force is in kN/m, tf/m or kip/ft. The example gives 3,608 kN/m, 367.9 tf/m or 247.2 kip/ft.
Does the module remember my inputs?
Yes. Every change is kept in this browser, so the page opens again with the model you last worked on. Reset to defaults brings back the example; Save to file… and Quick Save… keep versions you name.
Can I drag the wall or the water table on the drawing?
No. The drawing follows the Model tab and has no handles. You can zoom with the + and − buttons or the mouse wheel, pan a zoomed drawing by dragging it, and fit it with the third button.
What does the report contain?
A cover with your project details and a summary, then six parts: introduction and scope, input data, the calculation step by step, results, summary and references, with the Section, Pressure, coefficient and 3D drawings. It is saved as a PDF or a Word file.
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Sheet-Pile WallsEmbedment, anchor forces and the steel section of a sheet-pile wall.3 manuals
Braced CutsStrut loads, wales and sheeting of a strutted excavation.3 manuals
Retaining WallsCantilever, gravity and reinforced-soil (MSE) walls: stability and bearing, static and seismic.3 manuals
Lateral Earth PressureAt-rest, active and passive pressure on a wall, static and seismic.7 manuals
Surcharge LoadsThe lateral pressure on a wall from loads on the ground behind it.7 manuals
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LiquefactionLiquefaction triggering from SPT blow counts or CPT cone resistance for a site with several boreholes or soundings.2 manuals
