Twelve modules give the stress increase in the ground below loads at the surface, by the theory of elasticity. This page helps you choose the module that matches your load and links to the manual of each.
The Stresses in a Soil Mass family has twelve modules. They answer one question: by how much does a load on the ground raise the stress at a point below it? Pick the module whose load looks like yours.
The load on the ground
Module
One or several concentrated loads, such as a column or a wheel
Each manual follows the same plan: what the module calculates, what you need before you start, a quick start, the screen and its views, 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 twelve 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.
Several loads in one model. The modules add several loads of the same kind to one model and add their effects.
Point A. You choose the point below or beside the loads where you want the stress; it can be placed on the drawing.
Four views. The section with the lines of equal stress, the colour contours, the depth profile under the point, and a 3D view with the stresses on an element.
Loads moved on the drawing. Loads, their edges and point A can be dragged where the module allows it; each manual lists what moves in that module.
Pick the one whose load looks like yours and open its manual for the fields.
Describe the loads
Enter each load with its size and position, or drag it on the drawing before Analyze.
Place point A and the profile depth
Give the point where you want the stress and the depth down to which the profile runs.
Analyze and read the answer
Press Analyze. Read the stress at A, the lines of equal stress and the profile with depth, then save a report with the cover filled in.
Frequently asked questions
How do I calculate the stress increase below a footing with GeoTerva?
Open the module for the shape of the load: Vertical strip load for a strip footing, Rectangle for a rectangular footing, Circle, any point for a circular one. Enter the load and its size, place point A, press Analyze, and read the stress at A with the lines of equal stress and the profile with depth.
Which module gives the stress under a point load?
Point load, by Boussinesq, for one or several loads. Westergaard point load gives the Westergaard value with the Boussinesq value for comparison.
Can I add several loads in one model?
Yes. Each module adds several loads of its kind to one model and gives the total effect at point A.
What is the Influence chart module for?
It handles a loaded area of any polygonal shape on Newmark’s influence chart and compares the chart with the exact integral.
What is a Westergaard material?
A soil with thin rigid seams, such as clay with seams of sand. The Westergaard modules give the stress in it and the Boussinesq value for comparison.
Can I move a load on the drawing?
Yes, in the modules that allow it. Drag the load or its edges on the drawing before Analyze; point A can also be dragged. Each manual lists what moves.
Calculate the vertical and horizontal stress increases at any point below one or several vertical point loads on the ground surface, by Boussinesq’s solution: the lines of equal stress, colour contours, the profile under the point and a 3D view. This manual walks through the screen, the loads you can add and drag, every input, the results and the report, with a worked example you can reproduce.
Point load gives the stress increase in the ground below one or several vertical point loads on the ground surface, by Boussinesq’s solution for a homogeneous, elastic, isotropic half-space. You place the loads on the surface, choose a point A inside the soil, and the module gives the vertical stress increase Δσz and the horizontal stress increases Δσx and Δσy there, the lines of equal stress, the colour contours, the stresses under A against depth and a 3D view.
For one load the stresses depend on the horizontal distance r of the point from the load and on its depth z; L is the distance of the point from the load:
In the horizontal formula x and y are the distances of the point from the load, x − xL and y − yL; Δσy is the same with x and y exchanged. With μ = 0.5 the second terms vanish. Δσz does not depend on Poisson’s ratio μ; the horizontal stresses do. Several loads each act alone on the half-space and their stresses add.
What you can describe
One to eight vertical point loads on the ground surface, each with its force P and its position (xL, yL); no two loads may stand at the same place.
A point A below the surface: its position x, y on the ground and its depth z.
Poisson’s ratio μ of the soil, for the horizontal stresses only.
The depth of the profile zmax: how far down the profile of the stresses under A is drawn.
The module gives the three stress increases at A (and, for one load, the influence factor I1, r and L), the working step by step, the stresses under A at ten depths and at A itself, and a report as a PDF or Word file. The stresses are the increases caused by the loads; they add to the stresses already in the ground.
Each is a vertical force on the ground surface: zero or more. Up to eight.
The position of each load From: your layout
The coordinates xL and yL on the ground surface. The first load may stand at the origin.
The point A From: the place you want the stress
Its position x, y on the ground surface, in the same coordinates as the loads, and its depth z below the surface (at least 0.001 m).
Poisson’s ratio μ From: your soil data
From 0 to 0.5; 0.5 for an undrained clay. It changes the horizontal stresses only.
The depth of the profile zmax From: the depth you want to see
At least 0.1 m.
Quick start in seven steps
Open the module and choose the units
Open Point load from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the load
In Point loads the first load is already in the table. Press its row to change P, xL and yL. To add more, fill the New point load form and press + Add point load.
Place the point A
In Point of interest type x, y and the depth z, or drag the blue dot A on the Section drawing.
Set Poisson’s ratio and the profile depth
In Soil type μ (0.5 for an undrained clay). In Depth profile type zmax, or drag the dashed line at the bottom of the Section.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed — the stresses at the point calculated, the loads lock, and the lines of equal stress appear on the Section (in the Section view).
Read the answer
The Results tab gives Δσz, Δσx and Δσy at A. Drag A on the Section or the Contours to read the stress elsewhere; the answer follows at once.
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, contours, profile and 3D
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing has four views, chosen with the buttons above it:
Section is a vertical cut through the ground. With one load it is the plane through the load and the point A, and the solution is symmetric about the load axis, so the horizontal axis is the distance r from the load. With several loads it is the vertical plane y = y through A, parallel to x, and the loads that stand beside the plane are marked “beside the plane”: they act on it as well. After Analyze it carries the lines of equal vertical stress increase and the value at A.
Contours is the same cut with the vertical stress increase in colour bands on a logarithmic scale, with the legend on the right. Values above the top of the scale take the colour of the top band; values below the bottom, the lowest.
Profile is the stress increase against depth under A: Δσz as a solid line, Δσx and Δσy dashed, with the values at the depth of A.
3D is an oblique view of a block of ground with the loads, the distances x, y, z and an element at A with the three stresses on it.
Section before Analyze: the geometry of the example.
Section after Analyze: lines of equal stress; the blue line is the value at A.
Contours after Analyze.
Profile: the three stresses under A against depth.
3D after Analyze.
The Show chips
Above the Section and the Contours a bar holds two chips (on the Contours it is headed Show). Lines of equal stress draws the lines of equal vertical stress; on the Section it is checked once the model is analysed, on the Contours it starts unchecked. Dimensions draws the dimension lines of the distance r (or x − xL) and of the depth z of A, and is on by default; the label with the value at A stays when it is off. Before Analyze the lines chip is unchecked and cannot be switched on; the Section bar says “The stress and the lines of equal stress appear after Analyze” and the Contours bar says “The colour contours of the stress appear after Analyze”. The Profile and the 3D views have no chips.
Contours with the lines of equal stress on.
Section with Dimensions off.
What you can drag
The point A on the Section and the Contours, before and after Analyze. With one load a horizontal pull moves A along the direction from the load (x and y change together, so r changes) and a vertical pull changes z. With several loads a horizontal pull changes x and y stays. With one load the distance r and the depth z move in steps of 5 cm and x and y follow, rounded to 0.01; with several loads x and z move in steps of 5 cm. A stays inside the frame of the drawing; the boxes of the Point of interest section follow. After Analyze the answer is recalculated at once for the new place.
The dashed line at the bottom of the Section and the Contours sets the depth of the profile zmax (at least 0.1 m), before and after Analyze.
A load, before Analyze only. In the 3D view the arrow of a load is pulled over the ground, so xL and yL change together; when there are several loads, the arrows on the Section and on the Contours are pulled along x as well. All move in steps of 5 cm. A load cannot be pulled onto another one: it stops at the last free place and the module says “Another point load already stands at this position — the value is kept as it was.” With one load the Section has no handle on the load; use the 3D view or the boxes.
Nothing else: the value of Poisson’s ratio and the loads are typed, and after Analyze the loads and μ are locked until you press Edit inputs, which also clears the results.
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, and a zoomed drawing is moved by dragging its background.
Entering the model, section by section
The Model tab has five sections: Model data, Point loads, Point of interest, Soil and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below the last one. In the tables, a unit shown three times, such as kN · tf · kip, gives the units of SI · kN, SI · tf and US in turn; equal ones are merged. Defaults are shown in SI; in US the boxes show the converted values, for example 9.84252 ft for x = 3 m.
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 and the page keeps it as the model of the browser.
Model data.
Point loads
The section is a form over a table. The New point load form takes P, xL and yL of one load; + Add point load puts it into the table, and a counter beside the button shows how many of the eight loads there are. On a fresh page the form opens with P = 5, xL = 4 and yL = 0 (the first load of the model stands at the origin); after you add a load it keeps the values you entered. Press a row of the table to change that load in the form (the form then reads Editing · Point load 1 and offers Done and Delete point load; changes apply as they are entered), or press × to take a load out (it is disabled while only one load remains). With eight loads the add button is disabled.
Point loads: the form over the table.
Load fields (the form, and each row of the table)
Field
Unit
Default
Point load P The vertical force on the ground surface. Zero or more.
kN · tf · kip
5
Position along x xL The place of the load on the ground surface. The form starts with 4.
m · ft
0
Position along y yL The place of the load on the ground surface.
m · ft
0
A row of the table pressed: the load is edited in the form.
Two loads in the table.
Point of interest
The three coordinates of the point A. The note under the fields says that x and y are measured on the ground surface in the same coordinates as the loads, with z downward, and that for one load the solution depends on r and z. Typing the numbers and dragging A on the drawing do the same thing.
Point of interest.
Point fields
Field
Unit
Default
Point A: x on the ground surface x In the same coordinates as the loads.
m · ft
3
Point A: y on the ground surface y In the same coordinates as the loads.
m · ft
4
Depth of the point z Measured down from the ground surface. At least 0.001 m.
m · ft
6
Soil
One number. The note says that Δσz does not depend on μ, that the horizontal stress increases do, and that μ = 0.5 is for an undrained clay.
Soil.
Soil field
Field
Unit
Default
Poisson’s ratio (horizontal stresses only) μ From 0 to 0.5. It does not change Δσz.
—
0.5
Depth profile
One number again: how far down the profile under A is drawn and tabulated. A depth smaller than the depth of A is accepted: the table then ends at the depth of A and the Profile view a little below it, while the dashed line stays at the depth you typed.
Depth profile.
Profile field
Field
Unit
Default
Depth of the profile zmax At least 0.1 m. The Profile view and the table go from the surface down to this depth under A; if it is smaller than the depth of A, the table ends at the depth of A and the Profile view a little below it.
m · ft
20
Choosing the loads, the point and Poisson’s ratio
The module has no method to choose: the solution is Boussinesq’s. What you decide is where the loads and the point stand, and μ.
Choice
What it does
Number of loads
Each load acts alone on the half-space and the stresses add. One load gives the symmetric section about its axis, with I1, r and L on the Results tab; several loads give the stresses of their sum, and the Calculations tab shows each load and then the total.
Position of A
On the axis of a load (x = xL, y = yL) the vertical stress is largest for that depth: at 6 m, I1 = 0.4775 and Δσz = 0.06631 kPa for 5 kN, against 0.01774 kPa at 5 m to the side.
Poisson’s ratio μ
Changes Δσx and Δσy; Δσz stays the same. Use 0.5 for an undrained clay. Under the load axis the two horizontal stresses are equal and, for μ below 0.5, negative: a decrease.
Which should I pick?
Several loads when the loads are distinct and the ground between them is the same: add each, with its own force and position. For a load spread over an area use a module of that shape.
The point where the design needs it: the centre of a layer for a settlement calculation, or the level of a buried structure. Drag A to read the stress elsewhere and compare it with the profile.
μ = 0.5 for an undrained clay; for a drained soil, use the value from your soil data. The vertical stress, which settlement calculations use, does not depend on it.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stresses at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” The page may stay on the Model tab: open the Results tab for the answer.
The Results tab of the example.
Card
Symbol
The example
Vertical stress increase
Δσz
0.01774 kPa
Horizontal stress increase
Δσx
0.004436 kPa
Horizontal stress increase
Δσy
0.007886 kPa
Influence factor (one load only)
I1
0.1278
Horizontal distance from the load (one load only)
r
5.000 m
Distance from the load (one load only)
L
7.810 m
Below the cards, Values, step by step lists r, L, I1 and the three stresses, and a table gives the stresses under A at ten equal depths down to zmax (or down to the depth of A, if that is deeper) and at A itself, whose row is highlighted. For one load the table has the columns z, r/z, I1 and Δσz; for several loads it has z, Δσz, Δσx and Δσy. A line under the cards says: “Increases due to the load(s) only, added to the stresses already in the ground. A negative value is a decrease.”
Using the numbers
The stresses are increases. Add them to the stresses already in the ground (the effective stress from the weight of the soil) before you use them.
The stress is in kPa, t/m² or ksf, by the unit system. The same example gives 0.001809 t/m² and 0.0003706 ksf.
A load of zero is accepted and gives zero stress; the loads add, so a load that stands directly above A adds to Δσz and adds nothing to the horizontal stresses when μ = 0.5.
Changing the unit system after Analyze re-states the cards without a new run. Edit inputs unlocks the loads and clears the results.
Warnings and notes
The bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” A condition that stops the analysis is shown as a message, 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. For one load:
Geometry of the point — r = √(x² + y²) (or with x − xL and y − yL when the load is off the origin), L = √(r² + z²) and r/z.
Vertical stress increase — Δσz = 3Pz³ / (2π L⁵) = (P / z²) I1, the value of I1 and the result.
Horizontal stress increases — the two formulas with μ, the values, and with μ = 0.5 the shorter form.
Stress increase against depth — the same equation down the vertical line through A, as a table.
For several loads, step 1 is Stress of each load (r, L, I1 and Δσz of every load), step 2 is Superposition (Δσz = Σ Δσz of the loads, written as the sum of the numbers) and steps 3 and 4 give the horizontal stresses and the depth table of the sum.
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 (a table of the loads and the point, with the section, the profile and the 3D drawings), 3 Calculation (the steps of the Calculations tab), 4 Results, 5 Summary and 6 References. It states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has six sections: Scope and assumptions, Boussinesq’s solution, The influence factor I1, Lines of equal stress, Worked examples and References. Each opens when you click its title; Open all and Close all open or close them together. The figures and tables 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
Point loads
Point load 1
P = 5 kN at xL = 0, yL = 0
Point of interest
Point A
x = 3 m, y = 4 m, z = 6 m
Soil
Poisson’s ratio
μ = 0.5
Depth profile
Depth
zmax = 20 m
Step 1: the geometry
r = √(3² + 4²) = 5.00 m, L = √(5.00² + 6²) = 7.81 m and r/z = 5/6 = 0.8333.
With μ = 0.5 the formulas reduce to Δσx = 3Px²z / (2π L⁵) = 3 × 5 × 9 × 6 / (2π × 7.810⁵) = 0.004436 kPa and Δσy = 3Py²z / (2π L⁵) = 0.007886 kPa, the ratio of the two being 4²/3². These are the three cards of the Results tab and the steps of the Calculations tab.
On the load axis
Move A to x = 0, y = 0 (still at 6 m): r = 0, I1 = 3/(2π) = 0.4775 and Δσz = 0.4775 × 5/36 = 0.06631 kPa. The module gives 0.06631 kPa, and Δσx = Δσy = 0 for μ = 0.5 (with μ = 0.3 the module gives -0.004421 kPa each, a decrease: −P(1 − 2μ) / (4π z²) = −0.004421).
Two loads
Add a second load of 5 kN at x = 6 m, y = 0. The point A at x = 3, y = 4 is at the distance 5 m from both loads, so each gives 0.01774 kPa and the sum is 0.03549 kPa, with Δσx = 0.008872 kPa and Δσy = 0.01577 kPa. With the form’s own values (5 kN at x = 4 m, y = 0) the second load is at r = √(1² + 4²) = 4.123 m, I1 = 0.1816 and gives 0.02522 kPa; the module gives the total 0.04296 kPa.
Other unit systems
The default example in SI · tf gives Δσz = 0.001809 t/m² and in US 0.0003706 ksf: the same stress in other units (0.01774 kPa is 0.001809 t/m² and 0.0003706 ksf). Three checks by hand:
US. A load of 100 kip at the origin and A at x = 6 ft, y = 8 ft, z = 12 ft: r = 10 ft, r/z = 0.8333, I1 = 0.1278, so Δσz = (100 / 12²) × 0.1278 = 0.08872 ksf. The module gives 0.08872 ksf, with r = 10.00 ft and L = 15.62 ft.
SI · tf. A load of 10 tf and A at x = 2 m, y = 0, z = 3 m: r = 2 m, r/z = 0.6667, I1 = 0.1904, so Δσz = (10 / 9) × 0.1904 = 0.2116 t/m². The module gives 0.2116 t/m² and Δσx = 0.09403 t/m².
SI · kN. The same geometry with 1000 kN: Δσz = (1000 / 9) × 0.1904 = 21.16 kPa. The module gives 21.16 kPa and Δσx = 9.403 kPa.
How each input changes the answer
Each row changes one thing in the example (one load of 5 kN at the origin, A at x = 3 m, y = 4 m, z = 6 m, μ = 0.5) and runs the module again. The numbers are the ones the module gives, so you can reproduce any row. Stresses are in kPa.
The example with one change at a time
Change
Δσz
Δσx · Δσy
The default example: P = 5 kN at the origin, A at 3, 4, 6 m, μ = 0.5 The reference
0.01774
0.004436 0.007886
P = 10 kN All three stresses double: they are proportional to the load
0.03549
0.008872 0.01577
P = 0 All three stresses are zero
0
0 0
z = 3 m The point is shallower: Δσz falls and the horizontal stresses rise
0.009563
0.009563 0.01700
z = 1 m Very shallow: Δσz is small, the horizontal stresses are larger
0.0006926
0.006233 0.01108
z = 12 m Deeper: Δσz is lower than at 6 m, and the horizontal stresses much lower
0.01111
0.0006944 0.001235
A on the load axis, x = 0, y = 0 Δσz is at its largest for this depth; with μ = 0.5 the horizontal stresses are zero
0.06631
0 0
A further out, x = 6, y = 8 (r = 10 m) Everything falls: A is farther from the load
0.002391
0.002391 0.004250
x and y exchanged, x = 4, y = 3 Δσz is unchanged (r is the same); Δσx and Δσy swap
0.01774
0.007886 0.004436
A on the x axis, x = 5, y = 0 Δσz is unchanged; the whole horizontal stress is Δσx
0.01774
0.01232 0
μ = 0.3 Δσz is unchanged; the horizontal stresses fall
0.01774
0.002697 0.005617
μ = 0 Δσz is unchanged; the horizontal stresses fall further
0.01774
0.00008769 0.002212
The load moved to xL = 3, yL = 0 (r = 4 m) A is closer to the load: Δσz rises
0.02645
0 0.01175
A second load, 5 kN at x = 6, y = 0 A is 5 m from both: twice the stresses of the example
0.03549
0.008872 0.01577
A second load, 10 kN at x = 3, y = 4 (directly above A) The load above A adds to Δσz and nothing to the horizontal stresses (μ = 0.5)
0.1504
0.004436 0.007886
Loads 5 kN at the origin, at x = 6, y = 0 and at x = 0, y = 8 The sum of three loads
0.05323
0.01331 0.02366
zmax = 10 m The three stresses at A are the same; only the profile is shorter
0.01774
0.004436 0.007886
zmax = 4 m The three stresses at A are the same
0.01774
0.004436 0.007886
The pattern follows the formulas. The stresses are proportional to the loads; Δσz depends only on r/z and z (and not on μ); the horizontal stresses depend on the direction from the load as well, so that exchanging x and y exchanges Δσx and Δσy; and several loads simply add.
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.
The limits are shown in the units of the unit system chosen.
The window Analyze opens when the depth of the point is zero.
When
Message and what to do
The depth is below 0.001 m
“Depth of the point (z) must be at least 0.001 m.” Enter a depth below the surface.
The depth is empty
“Depth of the point (z) is missing.”
Poisson’s ratio is above 0.5 or below 0
“Poisson’s ratio (horizontal stresses only) (μ) must be at most 0.5.” or “… at least 0.”
A load is negative
“Point load 1 (P1) must be at least 0 kN.” The number of the load is part of the symbol: P2 belongs to load 2.
A load or a coordinate is empty
“Point load 1 (P1) is missing.”, “Point A: x on the ground surface (x) is missing.” or “Position of load 1 along x (xL,1) is missing.”
The depth of the profile is below 0.1 m or empty
“Depth of the profile (zmax) must be at least 0.1 m.” or “… is missing.”
When more than one value is wrong the window lists them together, for example “2 input errors” with the depth and Poisson’s ratio.
Messages on the Point loads section
If you add a load at a position where another already stands, the module says “A point load already stands at this position. Give the new one another position.” and does not add it.
A load added with a negative or an empty force is entered in the table; the error appears when you press Analyze, for example “Point load 2 (P2) must be at least 0 kN.”
A load pulled on a drawing onto another load stops at the last free place: “Another point load already stands at this position — the value is kept as it was.”
Typing a coordinate in the edit form of a load so that it coincides with another load gives the same message and the value reverts.
With usual values the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” A negative horizontal stress, which appears under the load axis for μ below 0.5, is a decrease and not a message.
Good practice and common mistakes
Add the stresses already in the ground. The module gives the increase due to the loads; the effective stress from the weight of the soil is yours to add.
Use the same coordinates for the loads and for A. The positions xL, yL and the point x, y are on the same ground axes; z is downward.
Keep the point below the surface.z is at least 0.001 m; at the surface the stress is zero away from the load.
Pick the vertical stress for settlements and the horizontal ones for the lateral effects you need; Δσz does not depend on μ.
Let each load be a point. For a footing or an area, a module of that shape gives the stress directly; a point load stands for a load small compared with its distance from A.
Drag A to explore, type to fix. Dragging is in steps of 5 cm; type the exact coordinates when the design gives them.
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 stress below a point load calculated?
By Boussinesq’s solution for an elastic half-space: Δσz = 3Pz³ / (2π L⁵) = (P / z²) I1 with L = √(r² + z²) and I1 = (3/2π) / [(r/z)² + 1]5/2. For the module’s example, Δσz = 0.01774 kPa at 6 m below a 5 kN load and 5 m to the side.
Does the result depend on the soil stiffness?
No. The solution does not contain the modulus of elasticity. Poisson’s ratio μ enters the horizontal stresses only; Δσz is the same for any μ.
Is the answer the total stress in the ground?
No, it is the increase caused by the loads. Add it to the stresses already in the ground.
How do I add more loads?
In Point loads fill the New point load form with the force and the position and press Add point load. Up to eight loads; no two at the same place. The stresses of the loads add, and the Calculations tab shows each load and then the sum.
What do the lines of equal stress show?
Lines of equal vertical stress increase in the section: for one load the plane through the load and A, for several loads the plane y = y of A. The blue line is the value at A. The Contours view shows the same field in colour bands on a logarithmic scale.
Can I drag the load and the point on the drawing?
The point A and the dashed profile-depth line can be dragged on the Section and the Contours, before and after Analyze. A load can be dragged before Analyze only: in the 3D view, and, with several loads, along x on the Section and the Contours. After Analyze the loads and Poisson’s ratio are locked until you press Edit inputs.
What happens at the load axis?
The solution is symmetric: r = 0, I1 = 3/(2π) = 0.4775 and Δσz = 0.4775 P/z². The two horizontal stresses are equal there and are zero for μ = 0.5.
What values does the module accept?
The depth of the point at least 0.001 m, Poisson’s ratio from 0 to 0.5, the loads zero or more, the depth of the profile at least 0.1 m, and every box filled. Anything else stops the analysis with a message that names the value.
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 stress is in kPa, t/m² or ksf, the load in kN, tf or kip, and lengths in m or ft. The default example gives 0.01774 kPa, 0.001809 t/m² or 0.0003706 ksf.
What does the report contain?
A cover with your project details and the answer, then six parts: introduction and scope, input data with the section, profile and 3D drawings, the calculation step by step, results, summary and references. It is saved as a PDF or a Word file.
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.
What if the loads are not points?
Use the module of the shape of the load: a line, a strip, a rectangle, a circle, an embankment, or any shape on the Influence chart. For layered ground use the Westergaard modules.
Calculate the increase of the vertical stress in the ground below one or several parallel vertical line loads: the stress at a point, the part of each load, the lines of equal stress and the colour contours, the profile under the point and a 3D view. This manual walks through the screen, what you can drag, every input, the results and the report, with a worked example you can reproduce.
Vertical line load gives the increase of the vertical stress Δσz in the ground below one or several vertical line loads on the ground surface, by the theory of elasticity. A line load is a force q per unit length that acts along a straight line of infinite length, such as a long wall, a rail, or a strip so narrow that its width does not matter. Every cross section across the line is loaded in the same way, so the stresses are those of plane strain and depend only on the horizontal distance x from the line and on the depth z:
The soil is a homogeneous, isotropic, linearly elastic half-space, and the load is flexible. The stresses of several loads add (superposition), each with its own horizontal distance x = xA − xL from the load to the point A. Δσz is the increase caused by the loads; the overburden pressure of the soil above A is not included.
What you can describe
One to eight parallel line loads, each with its intensity q (zero or more) and its position xL along the horizontal axis of the section. No two loads stand at the same position.
A point A at a horizontal position xA and a depth z, where the stress is worked out.
A depth of the profilezmax, down to which the stress under A is tabulated and drawn.
The module gives Δσz at A and the part that each load contributes, the stress under A at ten depths, the lines of equal stress and the colour contours on the section, and a 3D view with the stress on an element at A. It writes out the working and prepares a PDF or Word report.
Which module for which load
A concentrated load at a point on the ground: use Point load.
The pressure that a line load pushes on a wall is a different quantity: see Line load in the Surcharge loads manuals.
Before you start: what you need
Have these ready from your design:
What
Notes
The line loads q Your design loading
A force per unit length of the line, in kN/m (tf/m or kip/ft in the other unit systems). Zero or more. A strip of finite width is a job for the strip modules.
Their positions xL Your layout
Measured along the horizontal axis of the section, from an origin of your choice; to the left of it a position is negative. Only the distances between the loads and the point matter. No two loads at the same position.
The point A The place you want to know
Its horizontal position xA on the same axis, and its depth z below the ground surface. The depth must be at least 0.001 m.
The depth of the profile zmax How deep you want to read
The depth to which the stress under A is tabulated and drawn. At least 0.1 m.
The module needs no soil properties: in the elastic solution the stress increase follows from the loads and the geometry alone.
Quick start in seven steps
Open the module and choose the units
Open Vertical line load from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the line loads
In Line loads fill the New line load form with the intensity q and the position xL and press + Add line load. The load goes to the table below. Press a row to change it; × takes it out.
Place the point A
In Point A type the horizontal position xA and the depth z. You can also drag the point on the drawing.
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line on the drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed, the loads lock, and the lines of equal stress and the stress at A appear on the drawing.
Read the answer
The Results tab gives Δσz at A, the part from each load and the stress under A at ten depths. Contours, Profile and 3D change the view.
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: views, chips and what you can drag
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 is the cross section across the loads, the same in every section along the line. It shows the ground surface, each load as an arrow with its value, the point A with the dimensions x − x1 (the horizontal distance from the first load to A) and z, and the dashed line at the depth of the profile zmax. In the example the frame of the Section is only 7.2 m deep: when zmax is deeper, the dashed line sits at the bottom edge of the frame and its label still reads “Profile depth zmax = 12.00 m”; the Contours view shows the true depth. After Analyze it adds the lines of equal stress, each labelled in kPa, with the line through A drawn heavier in blue, and the value Δσz at A.
Contours is the same section filled with colour bands of Δσz on a logarithmic scale, with a legend on the right. In the example the legend runs from 0.01 to 10 kPa. The part below the depth of the profile is drawn paler.
Profile is the curve of Δσz against depth under A, down to zmax, with the depth of A marked.
3D is an oblique view of the ground with each line load as a row of arrows, the distances x, y and z of A, and, after Analyze, the stress arrow on a small element at A.
Section before Analyze: the loads, the point A and the profile depth.
Section after Analyze: the lines of equal stress.
Contours after Analyze: colour bands on a logarithmic scale.
Profile after Analyze.
3D after Analyze.
The Show chips
A bar of switches above the Section and the Contours views holds two chips, Lines of equal stress and Dimensions; on the Contours view the bar is headed Show. Dimensions is on by default. Lines of equal stress is greyed out before Analyze, with a note: on the Section view “The stress and the lines of equal stress appear after Analyze”, on the Contours view “The colour contours of the stress appear after Analyze”. After Analyze the chip is on in the Section view and off in the Contours view, where you can switch it on to draw the lines over the colours. The Profile and 3D views have no chips.
The Show bar before Analyze.
Zoom
Each view has −, a percentage and + at the lower right, and a button that fits the whole drawing in the pane. Each press of + zooms by 35 %; the range is 100 % (the whole drawing) to 800 %, and − is greyed out at 100 %. The mouse wheel zooms as well.
What you can drag
A load (Section, Contours and 3D, before Analyze, one load or several). Drag the load on the drawing to move it along x; the position follows in steps of 0.05 m and the table of loads follows the drawing. A load cannot be pulled onto another: it stops at the last free place and a message says “Another line load already stands at this position — the value is kept as it was.” The intensity q cannot be dragged; type it.
The point A (Section and Contours). Drag the dot; xA and z follow in steps of 0.05 m. The point stays inside the frame of the drawing, and its depth is at least 0.05 m. The point has no handle in the Profile and 3D views, and the 3D view has no handle at all after Analyze.
The dashed line of the profile depth (Section and Contours). Drag it up or down to set zmax; by dragging it cannot be set shallower than 0.5 m (type a value to go down to 0.1 m).
Entering the model, section by section
The Model tab has four sections: Model data (names, files and quick saves, the same in every module), Line loads, Point A and Depth profile. Each opens when you click its title, and opening one closes the previous one (Model data stays); 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 unrounded converted values, for example 16.4042 ft for 5 m. In the example the loads read 0.513913 and 1.02783 kip/ft at 0 and −16.4042 ft, A stands at 16.4042 ft and 13.1234 ft, zmax reads 39.3701 ft, and the New line load form reads 0.685218 kip/ft and 16.4042 ft.
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.
Line loads
The section is a form over a table. The New line load form takes the intensity q and the position xL of one load. + Add line load puts it into the table, and a counter beside the button shows how many of the eight loads there are (it reads 2 of 8 in the example). The Line loads table lists each load with its number, its intensity and its position. Press a row to change that load in the form: the form then reads Editing · Line load 2 and offers Done and Delete line load, and the note says that changes apply as they are entered. Press × to take a load out.
Line loads: the form over the table.
Line load fields (the form, and each row of the table)
The force per unit length of the line. Zero or more; there is no upper limit. In the table the first load opens with 7.5 and the second with 15.
Position along x xL m · ft · default 5 (form)
The position of the load along the horizontal axis, negative to the left of the origin. No two loads at the same position. In the table the first load opens at 0 and the second at −5.
A row of the table pressed: the load is edited in the form.
The rules of adding loads
No two loads at the same position. Add a load at an occupied position and nothing is added; the message reads “A line load already stands at this position. Give the new one another position.” Type an occupied position into a load you are editing and the old value is kept: “Another line load already stands at this position — the value is kept as it was.”
A blank position is read as 0 when the load is added. If a load already stands at 0, nothing is added and the message above appears. Otherwise the load enters the table with no position, and Analyze then stops with “Position of line load N (xN) is missing.” Enter the position before you add the load.
The intensity is checked at Analyze. A load added with a blank or a negative intensity enters the table, and Analyze then stops and names it; see Messages.
Up to eight loads. At 8 of 8 the Add button is greyed out; remove a load to add another. The form keeps the values you entered last after + Add line load, so change the position before the next one. Loads 4 to 8 take part in the answer like the first three.
Point A
The point where the stress is worked out. The note under the fields says that the stress depends on the horizontal distance xA − xL from each load. The point can also be set by dragging it on the Section and Contours views, and it stays free after Analyze.
Point A.
Point A fields
Field
What to enter
Horizontal position of the point A xA m · ft · default 5
On the same axis as the loads. The stress depends on the distance xA − xL from each load.
Depth of the point A z m · ft · default 4
Below the ground surface. At least 0.001 m.
Depth profile
The depth to which the stress under A is tabulated in the Results tab and drawn in the Profile view. It is set by the dashed line on the Section and Contours views too, and stays free after Analyze.
Depth profile.
Depth profile field
Field
What to enter
Depth of the profile zmax m · ft · default 12
At least 0.1 m. The Profile view and the table of the Results tab go from the surface down to zmax; when zmax is smaller than the depth of A, they go down to A.
Placing the loads and the point
The module has no method to choose: the elastic solution is the one formula. What you decide is where the loads and the point stand, and the stress follows from the distance between them. Three things are worth knowing.
Directly under a line load
When xA = xL the stress is Δσz = 2q/(πz) = 0.637 q/z. Under a line load the stress falls off only as 1/z with depth. With the first load of the example moved to xL = 5 m, directly above A, it gives 1.194 kPa for that load alone: 2 × 7.5 / (π × 4) = 1.194 kPa.
Beside a line load
The stress at a distance x from the load, as a fraction of the stress under it, depends only on x/z. At x/z = 1 it is a quarter of the stress under the load, and beyond x/z = 3 it is below 0.01 of q/z. The Theory tab draws the curve and tabulates it.
Several loads
The stresses add, whatever the order of the loads and however far apart they are. The first load of the example gives 0.1818 kPa and the second 0.04542 kPa at A, together 0.2272 kPa. A third load of 10 kN/m at xL = 5 m, right above A, adds 1.592 kPa and makes the total 1.819 kPa; with one load only the answer is 0.1818 kPa.
The example with a third line load above A: 1.819 kPa at A.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says that the inputs are locked, that Edit inputs clears the results and that the unit system can still be changed. It adds that the point A stays free: drag it, or change its coordinates, and the stress there is calculated at once. The module stays on the Model tab: open the Results tab for the numbers; the drawing stays on the view you were on and already shows the stress.
The Results tab of the example.
Card
What it is
Example
Vertical stress increase at A Δσz
The sum of the stresses of all the loads at the point.
0.2272 kPa
From line load 1 Δσz(1)
The stress that load 1 alone causes at A.
0.1818 kPa
From line load 2 Δσz(2)
The same for load 2.
0.04542 kPa
With one load there is one card. With two or three loads each load has a card of its own. With four to eight loads the big card is followed by a list of the stresses of each load (four loads wrap onto two lines), as in the picture below; the table Values, step by step then lists them. The note under the cards says: “The increase caused by the line loads only; the overburden pressure of the soil above A is not included.”
Four loads: the stresses of the loads in a line below the total.
Two tables follow. Values, step by step lists the stress from each load and the superposition. Vertical stress increase under A gives z and Δσz at ten equal depths down to zmax, with the row of A highlighted (it is inserted when the depth of A is not one of the ten; in the example the depths are 1.2, 2.4 … 12 m and the row of A, 4 m, stands between 3.6 and 4.8 m). When zmax is smaller than the depth of A, the table goes down to A: with zmax = 2 m and A at 4 m, the ten depths are 0.4 to 4 m.
Using the numbers
Δσz is the increase of the vertical stress caused by the loads. Add the overburden pressure of the soil to get the vertical stress itself.
A stress increase of 0.2272 kPa from loads of 7.5 and 15 kN/m is small because A is 5 m beside the nearer load and 4 m down; move A under a load and it grows (see the table below).
Changing the unit system after Analyze re-states the cards, the tables and the report without a new run; the drawings and the table of line loads keep their labels in m, kN/m and kPa, so read the units beside the boxes. Edit inputs unlocks the loads and clears the results.
Warnings
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.”
The Calculations tab, the report and the Theory tab
The Calculations tab opens with Given and what is sought, which lists your loads and the point, and then writes out the working in numbered steps, ending with the answer:
Stress from line load k — one step for each load: the horizontal distance x = xA − xL, the formula, x/z, the ratio Δσz/(q/z), and the stress with your numbers.
Superposition — the sum of the stresses of the loads.
Stress increase against depth — the table of the ten depths with the row of A marked, and then the Answer with the part of each load.
The Calculations tab: the 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 (the table of the loads and the point, and the Section, Profile and 3D drawings), 3 Calculation (one part for each load, the superposition, and the stress against depth), 4 Results, 5 Summary and 6 References. The cover table lists the loads, the point A and the answer with the part of each load, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has five sections: Scope and assumptions, The vertical stress, The nondimensional form, Several line loads and References. It derives the formula by adding the Boussinesq stresses of a row of point loads along the line, draws Δσz/(q/z) against x/z and tabulates it, and works through two line loads by superposition. Each section opens when you click its title, with Open all and Close all above them; 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
Line loads
Line load 1
q1 = 7.5 kN/m at x1 = 0
Line loads
Line load 2
q2 = 15 kN/m at x2 = −5 m
Point A
Position and depth
xA = 5 m, z = 4 m
Depth profile
Depth of the profile
zmax = 12 m
Step 1: the stress of each load
The horizontal distance from load 1 to A is x = 5 − 0 = 5 m, so x/z = 1.250 and
These are the three cards of the Results tab and the steps of the Calculations tab. Load 1, which is 5 m from A, gives about four times the stress of load 2, which is twice as far and twice as heavy.
Two checks by hand
A single load, right above A. One load of 1 kN/m at xL = 0 with A at xA = 0 and z = 1 m: Δσz = 2/π = 0.6366 kPa. The module gives 0.6366 kPa.
The same in the other unit systems. In SI · tf one load of 1 tf/m with A under it at 1 m gives 0.6366 t/m²; in US one load of 1 kip/ft with A under it at z = 10 ft gives 2/(π × 10) = 0.06366 ksf, and the module gives 0.06366 ksf.
Other unit systems
The default example gives 0.02317 t/m² in SI · tf and 0.004745 ksf in US: the same loads and point in other units (0.2272 kPa is 0.02317 t/m² and 0.004745 ksf). In US the loads read 0.5139 and 1.03 kip/ft, at 0 and −16.40 ft, and A stands at 16.40 ft and 13.12 ft deep. The profile table then runs from 3.937 ft to 39.37 ft.
How each input changes the answer
Each row changes one thing in the example (loads of 7.5 kN/m at 0 and 15 kN/m at −5 m, A at xA = 5 m and z = 4 m, zmax = 12 m) and runs the module again. The numbers are the ones the module gives, and you can reproduce any row. The answer is Δσz at A in kPa; the line under the change gives the part of each load.
The example with one change at a time
Change
Δσz (kPa)
The example as it opens The reference Loads: 0.1818 + 0.04542 kPa
0.2272
q1 = 15 kN/m Load 1 doubles and its part doubles; load 2 is unchanged. Loads: 0.3636 + 0.04542 kPa
0.4090
q1 = 30 kN/m The part of load 1 is four times the original. Loads: 0.7271 + 0.04542 kPa
0.7726
q2 = 0 A load of zero is allowed; the answer is that of load 1 alone. Loads: 0.1818 + 0 kPa
q1 = 15 and q2 = 30 kN/m Both loads doubled: the answer exactly doubles, 2 × 0.2272. Loads: 0.3636 + 0.09084 kPa
0.4544
x1 = 2 m Load 1 is 3 m from A instead of 5 m. Loads: 0.4889 + 0.04542 kPa
0.5343
x1 = 5 m Load 1 right above A: its part is 2q/(πz). Loads: 1.194 + 0.04542 kPa
1.239
x1 = 10 m Load 1 is 5 m from A on the other side: the same answer as at 0. Loads: 0.1818 + 0.04542 kPa
0.2272
x2 = 5 m Load 2, the heavier one, right above A. Loads: 0.1818 + 2.387 kPa
2.569
xA = 0 A under load 1. Loads: 1.194 + 0.3636 kPa
1.557
xA = −5 m A under load 2: the same as load 2 at 5 m with A at 5 m. Loads: 0.1818 + 2.387 kPa
2.569
xA = 2.5 m A between the loads. Loads: 0.6173 + 0.1171 kPa
0.7343
xA = 15 m A far from both loads. Loads: 0.005261 + 0.003532 kPa
0.008793
z = 1 m A near the surface. Loads: 0.007063 + 0.0009361 kPa
0.007999
z = 2 m Loads: 0.04542 + 0.007063 kPa
0.05248
z = 8 m Loads: 0.3086 + 0.1818 kPa
0.4904
z = 20 m Smaller than the 0.5660 the example profile gives at 12 m: beside a load the stress rises with depth, reaches a maximum and falls again. Loads: 0.2115 + 0.3056 kPa
0.5170
Load 2 removed Load 1 alone. Loads: kPa
0.1818
A third load, 10 kN/m at xL = 5 m The load of the New line load form, right above A. Loads: 0.1818 + 0.04542 + 1.592 kPa
1.819
zmax = 6 m Only the profile changes: the table has ten depths down to 6 m. Loads: 0.1818 + 0.04542 kPa
0.2272
zmax = 2 m Smaller than the depth of A: the table goes down to 4 m, the depth of A. Loads: 0.1818 + 0.04542 kPa
0.2272
The pattern follows the formula. The stress is proportional to each load and adds over the loads. It falls quickly as the point moves sideways away from a load, and it is largest for a point right under one. With depth, the stress under a load falls as 1/z, but for a point beside a load it first rises and then falls. The depth of the profile changes what is tabulated and drawn, not the answer at A.
Messages: when the module stops, and what it notes
The module checks the numbers it needs when you press Analyze. When one cannot be used it stops, and nothing is calculated: 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. The limits are shown in the units of the unit system chosen.
The window Analyze opens when the depth of A is zero.
Stop when
Message and what to do
A load has a blank intensity
“Line load 1 (q1) is missing.” Enter the intensity, or take the load out.
An intensity is negative
“Line load 1 (q1) must be at least 0 kN/m.” Enter zero or more.
A load has a blank position
“Position of line load 1 (x1) is missing.” Enter the position.
The horizontal position of A is blank
“Horizontal position of the point A (xA) is missing.” Enter xA.
The depth of A is zero or negative
“Depth of the point A (z) must be at least 0.001 m.” Enter a depth in the ground.
The depth of A is blank
“Depth of the point A (z) is missing.” Enter z.
The depth of the profile is below 0.1 m
“Depth of the profile (zmax) must be at least 0.1 m.” (or “… is missing.”) Enter zmax.
When more than one value is wrong the window lists them together, for example “2 input errors” with a blank intensity and a depth of zero. The number of the load is part of the symbol: q1 and x1 belong to line load 1.
Notes at the drawing and the form
A load onto another. Adding at an occupied position gives “A line load already stands at this position. Give the new one another position.” Dragging a load onto another, or typing an occupied position into a load you are editing, gives “Another line load already stands at this position — the value is kept as it was.”
Nothing to warn about. The Warnings (0) button opens “No warnings — the assumptions of the solution hold for these inputs.” The module has no other warnings or notes.
Good practice and common mistakes
Use it for loads that are really long. The solution takes the load as infinite along y. A load that is long compared with its distance from the point acts as a line; a short one is a job for Point load or for the area modules.
Do not turn a wide load into a line. A strip of finite width has its own stress under and beside it: use Vertical strip load. A line is right when the width is small beside the distance to the point.
Mind the sign of the positions. Left of the origin is negative; xA − xL is what counts, so shifting every position by the same amount changes nothing.
Add the overburden pressure yourself.Δσz is the increase only.
Read the point under the load with care. Right under a load the stress is largest at shallow depth and falls with depth; use the Profile view to see how it varies, and keep z above zero.
Keep the profile deep enough to cover the depth you need: zmax sets the range of the table and the Profile view.
Use the drag, then type. Drag the point to explore, then type the exact values in Point A; the loads can be dragged only before Analyze.
Check the units after switching the system: type values in the units shown beside the boxes. The stress answer is in kPa, t/m² or ksf.
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 stress below a vertical line load calculated?
By the elastic solution for a line load of infinite length on a half-space: Δσz = 2qz³ / [π(x² + z²)²], with q the load per unit length, x the horizontal distance from the line to the point and z the depth. For the module’s example, loads of 7.5 kN/m at 0 and 15 kN/m at −5 m, and the point at 5 m and 4 m deep, Δσz = 0.2272 kPa.
How many line loads can I add, and how do they combine?
Up to eight, each with its own intensity and position, and no two at the same position. The soil is elastic, so the stresses of the loads add: the answer at A is the sum of the stresses each load would cause alone. The Results tab gives each part.
What do the positions mean? Where is the origin?
The positions xL and xA lie on one horizontal axis in the plane of the section. The origin is yours; only the distances xA − xL matter. A position to the left of the origin is negative.
Does the answer include the weight of the soil?
No. Δσz is the increase of the vertical stress caused by the loads. The note under the Results cards says that the overburden pressure of the soil above A is not included; add it to get the total vertical stress.
What is the difference between a line load and a strip load?
A line load is a force per unit length with no width; a strip load is a pressure over a width. A strip so narrow that its width does not matter acts as a line load. For a load with a width, use the Vertical strip load module.
Can I put the point A at the ground surface?
No. The depth must be at least 0.001 m; a smaller value stops the analysis with “Depth of the point A (z) must be at least 0.001 m.” On the surface the stress is zero away from the load and infinite under it.
Can I drag the loads and the point on the drawing?
Yes. Before Analyze you can drag each load along x on the Section, Contours and 3D views, in steps of 0.05 m; a load cannot be dragged onto another one. The point A can be dragged on the Section and Contours views, before and after Analyze, and so can the dashed line of the profile depth. After Analyze the loads are locked until you press Edit inputs.
What are the lines of equal stress and the colours?
The lines join the points that have the same Δσz, labelled in kPa; the line through A is drawn heavier. The Contours view fills the section with colour bands of Δσz on a logarithmic scale, with a legend. Use the Lines of equal stress chip to draw the lines over the colours.
What does the profile show, and what is the depth of the profile?
The Profile view and the second table of the Results tab give Δσz against depth under the point A, from the surface down to zmax. The depth of the profile does not change the answer at A. If zmax is smaller than the depth of A, the table goes down to A.
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 loads are in kN/m, tf/m or kip/ft, lengths in m or ft, and the stress in kPa, t/m² or ksf. The default example gives 0.2272 kPa, 0.02317 t/m² or 0.004745 ksf.
What does the report contain?
A cover with your project details and the answer, then six parts: introduction and scope, input data with the section, profile and 3D drawings, the calculation step by step, results, summary and references. 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.
Calculate the vertical stress increase at a point below one or several horizontal or inclined line loads on the ground surface: the stress of the vertical and the horizontal component, the lines of equal stress, colour contours, the profile under the point and a 3D view. This manual walks through the screen, every input, the sign of the inclination, the results and the report, with a worked example you can reproduce.
Horizontal line load gives the vertical stress increaseΔσz at a point in the ground below a long line load that pushes along the ground surface, such as the friction of a long wall base, the braking of a rail track or the thrust of an arch. The module also takes a load that is inclined from the vertical: it splits the load into a vertical and a horizontal component, calculates the stress of each by the theory of elasticity for a half-space (the line-load solutions that follow from Boussinesq’s 1885 point load), and adds them. Several loads can be added; the stresses add.
For a load q per unit length at the angle α from the vertical, and a point at the horizontal distance x from the load and the depth z:
The soil is a homogeneous, isotropic, linearly elastic half-space, and the load is flexible, of infinite length along the line, so every cross section is in plane strain. Δσz is the increase caused by the load alone; it does not include the overburden pressure. A horizontal load increases the vertical stress on the side it pushes toward (x > 0) and decreases it on the other side.
What you can describe
From one to eight line loads, each with its own magnitude q (zero or more, in kN/m), inclination α from the vertical (from −90° to 90°; 90° is a purely horizontal load, 0° a vertical one) and position xL along x. No two loads may stand at the same position.
One point A below the ground surface, by its horizontal position xA and its depth z.
The depth of the profilezmax, down to which the stress under A is drawn and tabulated.
The module gives the stress at A and its two parts, the stress of each load when there are several, and the stress against depth under A. It draws the lines of equal stress, colour contours, the profile and a 3D view, writes out the working, and prepares a PDF or Word report.
Which module for which load
A load that acts straight down on a line: Vertical line load. This module covers it too (α = 0°), but that module is the direct one.
The module gives the vertical stress increase in the ground. For the lateral pressure a vertical line load puts on a wall, see the Surcharge loads manuals.
Before you start: what you need
Have these ready from your design:
What
Notes
The load q Your design loading
A force per unit length of the line, in kN/m (tf/m or kip/ft in the other unit systems). Zero or more. It is the magnitude of the load along its own direction.
The inclination α The direction of the load
The angle from the vertical, from −90° to 90°. It is positive when the horizontal component points toward +x.
The position xL Your layout
Along the x axis, in m (or ft). Any value; the first load of the example stands at 0.
The point A The place you want the stress
The horizontal position xA on the same axis, and the depth z (at least 0.001 m) below the ground surface.
The depth of the profile zmax How deep you want to look
At least 0.1 m. It sets the length of the profile and of its table.
Quick start in seven steps
Open the module and choose the units
Open Horizontal line load from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the first load
In Line loads press the row of Line load 1 and change its magnitude q, inclination α and position xL. Set α = 90 for a purely horizontal load.
Add more loads if you have them
Fill the New line load form and press + Add line load; the load goes to the table. Up to eight loads, no two at the same position.
Place the point A
In Point A type xA and z, or drag A on the Section or Contours drawing.
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line on the drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the loads lock; the point A stays free.
Read the answer and take the report
The Results tab gives Δσz at A and its parts, the Calculations tab writes out the working, and the Report tab saves a PDF or a Word file.
The screen: four views, chips and dragging
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing has four views, chosen with the buttons above it, and every view follows your inputs.
Section is the cross section of the ground with the load or loads as arrows, the point A with its distances x and z, and the dashed line at the depth of the profile. After Analyze it adds the lines of equal stressΔσz and a label with the value at A; the blue line is the one through A.
Contours is the same plane in colour bands on a logarithmic scale, with a legend on the right. Values above the top band take its colour, values below the lowest band take the lowest.
Profile is the stress against depth under A: the total Δσz as a solid line and its vertical and horizontal parts as dashed lines, with the values at A marked. It appears after Analyze.
3D is an oblique view of the ground with the line load, the coordinates of A and a small element at A with the stress on it. After Analyze it shows Δσz; a note says that a negative value is a decrease (tension) in the elastic solution.
Section before Analyze: the geometry only.
Section after Analyze: the lines of equal stress and the value at A.
Contours after Analyze.
Profile after Analyze.
3D after Analyze.
The Show chips
The Section view has two chips: Lines of equal stress and Dimensions. The Contours view has the same two under a Show label. Dimensions is on from the start. Lines of equal stress is dimmed until Analyze; then it comes on in the Section view, while in the Contours view it stays off until you tick it. Before Analyze the Section view says “The stress and the lines of equal stress appear after Analyze” and the Contours view says “The colour contours of the stress appear after Analyze”. The Profile and 3D views have no chips.
Contours with the lines of equal stress put over the colours.
Dragging on the drawing
A load is dragged along x in the Section, Contours and 3D views, before Analyze. Its position xL follows in steps of 0.05 m. A pull that would put the load on another one stops at the last free place, with the message “Another line load already stands at this position — the value is kept as it was.” The drawing has no limit on how far you can pull; the picture rescales.
The point A is dragged in the Section and Contours views, before and after Analyze. xA and z follow in steps of 0.05 m, and a drag does not take A above 0.05 m depth. After Analyze the stress at the new place is calculated at once.
The dashed line at the foot of the Section and Contours views sets the depth of the profile zmax. In the Section view it stands at the foot of the drawing, about 7 m deep for the example, when zmax is deeper; the Contours view shows the true depth. It moves in steps of 0.05 m (a drag does not take it shallower than 0.5 m) and stays free after Analyze.
After Analyze the loads are locked: they have no handles. Edit inputs unlocks them and clears the results.
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, Line loads, Point A and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below. 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 0.685218 kip/ft for the 10 kN/m load and 16.4042 ft for 5 m.
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. These buttons stay available after Analyze.
Model data.
Line loads
The section is a form over a table. The New line load form takes the magnitude, the inclination and the position of one load; + Add line load puts it at the bottom of the table, and a counter beside the button shows how many of the eight loads there are. The Line loads table lists each load with its three values. Press a row to change that load in the form (the form then reads Editing · Line load 1, offers Done and Delete line load, and says “Changes apply as they are entered.”); × takes a load out. With one load left, both delete buttons are dimmed. The form for a new load starts from 10 kN/m, 20° and −5 m; the first load of the example stands at 0 m.
Line loads: the form over the table.
Line load fields (the form, and each row of the table)
Field
Default
What to enter
Line load (magnitude) q kN/m · tf/m · kip/ft
10 (form: 10)
Zero or more. The load along its own direction, per unit length of the line.
Inclination from the vertical (90° = horizontal) α °
20 (form: 20)
From −90° to 90°. Positive when the horizontal component points toward +x; 0° is vertical and 90° purely horizontal.
Position along x xL m · ft
0 (form: −5)
Any value. No two loads may stand at the same position.
A row pressed: the load is edited in the form.
Two loads cannot stand at the same position. Adding a load at the position of another gives the message “A line load already stands at this position. Give the new one another position.”, and changing a position to one that is taken gives “Another line load already stands at this position — the value is kept as it was.”
Point A
The point where you want the stress. Its horizontal position is on the same axis as the loads, positive toward +x; its distance from a load is xA − xL. Place it by typing, or drag it on the Section or Contours drawing.
Point A.
Point A fields
Field
Default
What to enter
Horizontal position of the point A xA m · ft
5
Any value, on the same axis as the loads; its distance from a load is xA − xL.
Depth of the point A z m · ft
4
At least 0.001 m. The stress is calculated inside the ground, not on the surface.
Depth profile
The profile view and its table go from the surface down to zmax under A. The table goes down to zmax (or to z if that is deeper), with the row of A marked; the drawing may run a little deeper when A is close to zmax.
Depth profile.
Depth profile field
Field
Default
What to enter
Depth of the profile zmax m · ft
12
At least 0.1 m. The profile and its table go from the surface down to zmax under A (to z if that is deeper).
Choosing the inclination and the side of the point
Two decisions shape the answer: the direction of each load, and which side of it the point A lies on. Both come down to signs.
Input
Effect
Rule
α = 0°
A vertical load
Only the vertical component acts, and the stress is symmetric about the load. The example load gives Δσz = 0.2424 kPa at A.
α from 0° to 90°
The load leans toward +x
The horizontal part Δσz(H) is positive for x > 0, so it adds to the vertical part on that side and subtracts on the other. At 45°, Δσz = 0.3856 kPa at A.
α = 90°
A purely horizontal load
The vertical part is zero. Δσz = 0.3030 kPa at A, and zero straight below the load (x = 0).
α negative
The load leans toward −x
The horizontal part changes sign. At −20°, Δσz = 0.1241 kPa at A, and at −90° it is -0.3030 kPa.
The side of the point matters in the same way. With the load of the example, moving A from x = 5 m to x = −5 m, at the same depth, takes Δσz(H) from 0.1036 to -0.1036 kPa and Δσz from 0.3314 to 0.1241 kPa; the vertical part, 0.2278 kPa, is the same on both sides.
A horizontal load: an increase on the side it pushes toward, a decrease on the other (shown in the colour of the lowest band).
The profile for a horizontal load.
Which should I pick?
Take the direction of the load from your design. A load that pushes toward +x has a positive α; place your x axis so that this holds.
Check the sign on the result. A negative Δσz is a decrease of the vertical stress in the elastic solution. The 3D view says so under the drawing, and the note under the Results cards says on which side a horizontal load decreases the stress.
One load with α = 0 and one with α = 90 at the same place cannot be entered, since two loads cannot stand at one position. Use one inclined load instead; it gives the same stress.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” The Model tab stays on screen; open the Results tab for the answer, and the drawings show the lines, the contours, the profile and the element.
The Results tab of the example.
Card
Symbol
The example
Vertical stress increase at A
Δσz
0.3314 kPa
From the vertical component (components)
Δσz(V)
0.2278 kPa
From the horizontal component (components)
Δσz(H)
0.1036 kPa
Vertical component (one load only)
qV
9.397 kN/m
Horizontal component (one load only)
qH
3.420 kN/m
With several loads the cards read “From the vertical components” and “From the horizontal components”, and a card From line load k is added for each load (up to three loads; the working of every load is in the Calculations tab). Below the cards, Values, step by step lists each step, and Vertical stress increase under A is a table of z, Δσz(V), Δσz(H) and Δσz at ten equal depths down to zmax, with the row of A marked.
Using the numbers
Δσz is the increase caused by the loads, in kPa (t/m² or ksf in the other unit systems). Add the stress already in the ground to it for the total.
The loads are per unit length: the stress is the same along the whole line, in every cross section.
The depth of the profile does not change the answer at A.
Changing the unit system after Analyze restates the cards without a new run. Edit inputs unlocks the loads and clears the results.
Warnings
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.” This solution has no warnings of its own: the button reads Warnings (0), also when the stress is negative.
The Calculations tab, the report and the Theory tab
The Calculations tab opens with Given and what is sought, then writes out the working in numbered steps. With one load there are five:
Components of the load — qV = q cos α and qH = q sin α with your numbers.
Stress from the vertical component — x/z, the ratio Δσz(V) / (qV/z) and Δσz(V).
Stress from the horizontal component — the same for Δσz(H).
Total vertical stress increase — Δσz = Δσz(V) + Δσz(H).
Stress increase against depth — the table under A, with the row of A marked, and the answer.
With several loads, each load has a step of its own (“Stress from line load k”, with its distance x = xA − xL), followed by a Superposition step that adds them, and the depth step.
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 Components of the load, 3.2 and 3.3 the stress of each component, 3.4 the total and 3.5 the stress against depth), 4 Results, 5 Summary and 6 References. It carries the section, profile and 3D drawings and states the units used.
The Report tab: the cover details, the two buttons and the cover page.
The Theory tab
The Theory tab has four sections: Scope and assumptions, The vertical stress, An inclined line load, and References. Each opens when you click its title, and the figures in it are computed from the equations; the third works the example through.
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
Line loads
Line load 1
q = 10 kN/m; α = 20° from the vertical; xL = 0
Point A
Position and depth
xA = 5 m; z = 4 m
Depth profile
Depth
zmax = 12 m
Step 1: the components
qV = 10 × cos 20° = 9.397 kN/m and qH = 10 × sin 20° = 3.420 kN/m.
Step 2: the stress of each component
The distance of A from the load is x = 5 − 0 = 5 m, so x/z = 1.250. For the vertical part, Δσz(V) / (qV/z) = 2 / {π[(1.25)² + 1]²} = 0.09695, so Δσz(V) = 0.09695 × 9.397 / 4 = 0.2278 kPa. For the horizontal part, Δσz(H) / (qH/z) = 2 × 1.25 / {π[(1.25)² + 1]²} = 0.1212, so Δσz(H) = 0.1212 × 3.420 / 4 = 0.1036 kPa.
Step 3: the total
Δσz = 0.2278 + 0.1036 = 0.3314 kPa
These are the three cards of the Results tab. The same sum at other depths under A gives the table of the profile, for example 0.3314 kPa at 4.000 m, 0.4526 kPa at 6.000 m and 0.4168 kPa at 12.00 m; the stress under A rises to a largest value of about 0.47 kPa near 7 to 8 m and then falls.
Two loads
Add a second load with the form’s defaults (10 kN/m, 20°, at −5 m). Its distance from A is 10 m and it gives 0.05434 kPa; with the first load’s 0.3314 kPa the total is 0.3857 kPa (vertical parts 0.2562 kPa, horizontal parts 0.1295 kPa). The Section view then shows both loads, each in its own colour, with their combined lines of equal stress.
Two loads: the stresses add.
Other unit systems
The default example in SI · tf gives Δσz = 0.03379 t/m² (the load is 1.01972 tf/m), and in US 0.006921 ksf (the load is 0.685218 kip/ft): the same problem in other units. Three checks by hand, all independent of the module:
Vertical load, A below it.q = 10 kN/m, α = 0, A at xA = 0 and z = 4 m: Δσz = 2q/(πz) = 2 × 10 / (π × 4) = 1.592 kPa. The module gives 1.592 kPa.
US.q = 2 kip/ft, α = 0, A below it at 10 ft: 2 × 2 / (π × 10) = 0.1273 ksf. The module gives 0.1273 ksf.
SI · tf.q = 5 tf/m, α = 90°, x = 3 m, z = 3 m: Δσz = 2 × 5 × 3 × 3² / [π (3² + 3²)²] = 270 / (π × 324) = 0.2653 t/m². The module gives 0.2653 t/m².
How each input changes the answer
Each row changes one thing in the example (one load of 10 kN/m at 20° at xL = 0, A at xA = 5 m and z = 4 m) and runs the module again. The numbers are the ones the module gives, so you can reproduce any row. Δσz is in kPa; the parts are under each row.
The example with one change at a time
Change
Δσz (kPa)
The example: 10 kN/m at 20°, xL = 0; A at 5 m and 4 m The reference Vertical part 0.2278 · horizontal part 0.1036 kPa
0.3314
q = 20 kN/m Twice the load: twice the stress Vertical part 0.4555 · horizontal part 0.2072 kPa
0.6628
α = 0° (vertical) Only the vertical part Vertical part 0.2424 · horizontal part 0 kPa
0.2424
α = 45° The horizontal part grows, the vertical part falls Vertical part 0.1714 · horizontal part 0.2142 kPa
0.3856
α = 90° (horizontal) Only the horizontal part Vertical part 0 · horizontal part 0.3030 kPa
0.3030
α = −20° The horizontal part changes sign and subtracts Vertical part 0.2278 · horizontal part -0.1036 kPa
0.1241
α = −90° A horizontal load pushing away from A: a decrease Vertical part 0 · horizontal part -0.3030 kPa
-0.3030
A at xA = 10 m Farther from the load: much smaller Vertical part 0.02845 · horizontal part 0.02589 kPa
0.05434
A at xA = 2 m Nearer the load Vertical part 0.9572 · horizontal part 0.1742 kPa
1.131
A at xA = 0 (under the load) The horizontal part is zero Vertical part 1.496 · horizontal part 0 kPa
1.496
A at xA = −5 m (the other side) The vertical part is the same; the horizontal part is negative Vertical part 0.2278 · horizontal part -0.1036 kPa
0.1241
A at z = 2 m Shallower: smaller at this distance Vertical part 0.05691 · horizontal part 0.05178 kPa
0.1087
A at z = 8 m Deeper: larger at this distance Vertical part 0.3867 · horizontal part 0.08796 kPa
0.4746
Load moved to xL = 2 m A is 3 m from the load: larger Vertical part 0.6126 · horizontal part 0.1672 kPa
0.7798
Load moved to xL = 5 m The load is above A: the horizontal part is zero Vertical part 1.496 · horizontal part 0 kPa
1.496
α = 90° with A at xA = −5 m A decrease of the vertical stress Vertical part 0 · horizontal part -0.3030 kPa
-0.3030
A second load of 10 kN/m, 20°, at xL = −5 m The stresses add Vertical part 0.2562 · horizontal part 0.1295 kPa
0.3857
Two more loads: (10 kN/m, 20°, −5 m) and (5 kN/m, 0°, 8 m) The stresses of the three loads add Vertical part 0.5822 · horizontal part 0.1295 kPa
0.7117
The pattern follows the formulas. The stress is proportional to the load, so twice the load gives twice the stress. At the default 20° the vertical part is the larger one under and near the load, but the balance depends on α and on x/z; the horizontal part changes sign with x. Far from the load both fall. A point on the other side of a horizontal load, or a load that pushes away from the point, gives a decrease. Moving the point deeper does not always raise the stress: at this distance the stress under A rises with depth down to about 7 to 8 m and then falls.
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 numbered.
The limits are shown in the units of the unit system chosen.
The window Analyze opens when a load is negative.
When
Message and what to do
Stop. A magnitude is empty
“Line load 1 (magnitude) (q1) is missing.” Enter q, or take the load out.
Stop. A magnitude is negative
“Line load 1 (magnitude) (q1) must be at least 0 kN/m.” Enter zero or more; give a load that points the other way a negative α instead.
Stop. An inclination is outside −90° to 90°
“Inclination of line load 1 from the vertical (90° = horizontal) (α1) must be at most 90°.” or “… at least -90°.” Enter an angle in that range.
Stop. A position is empty
“Position of line load 1 (x1) is missing.”
Stop. The point has no position or no depth
“Horizontal position of the point A (xA) is missing.” or “Depth of the point A (z) is missing.”
Stop. The depth of the point is below 0.001 m
“Depth of the point A (z) must be at least 0.001 m.” Enter a depth inside the ground.
Stop. The depth of the profile is below 0.1 m
“Depth of the profile (zmax) must be at least 0.1 m.”
When more than one value is wrong the window lists them together, for example “2 input errors” for a negative load and an inclination of 100°. The number after the symbol is the number of the load: q1, α1 and x1 belong to line load 1.
Notes while you build the model
Two loads at one position. A toast reads “A line load already stands at this position. Give the new one another position.” when you add, and “Another line load already stands at this position — the value is kept as it was.” when you change or drag a position onto a taken one.
Results cleared. After Edit inputs a toast reads “Results cleared — the inputs can be edited.”
Good practice and common mistakes
Use one inclined load, not two stacked ones. Two loads cannot stand at the same position, and one load of magnitude q at α gives exactly the stress of its two components.
Mind the sign of α and of x. The horizontal part adds on the side the load pushes toward and subtracts on the other; a negative Δσz is a decrease, not an error.
Read Δσz as an increase. It is caused by the loads alone; add the stress already in the ground.
The loads are lines of infinite length. The stress is the same in every cross section; for a load of limited length, use a point load, strip or area module.
The soil is one elastic half-space. For layered ground use the Westergaard modules.
Choose zmax deep enough to see where the stress under A peaks and falls; it does not change the answer at A.
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. The page remembers your values, but not the analysed state.
Frequently asked questions
How is the stress below a horizontal line load calculated?
By the theory of elasticity for a half-space: Δσz = 2qxz² / [π(x² + z²)²], with x the horizontal distance from the load to the point and z its depth. It is positive on the side the load pushes toward and negative on the other. For the example, a horizontal load of 10 kN/m gives 0.3030 kPa at x = 5 m, z = 4 m.
How does it handle an inclined load?
It splits the load into qV = q cos α and qH = q sin α, calculates the stress of each, and adds them. For 10 kN/m at 20° the parts are 0.2278 and 0.1036 kPa and the total 0.3314 kPa at the example point.
What sign should I give the inclination?
Positive when the horizontal component points toward +x, negative when it points toward −x. The inclination goes from −90° to 90°, where 90° is a purely horizontal load and 0° a vertical one.
Why is the stress negative at my point?
The point is on the side the horizontal component pulls away from, so the vertical stress there decreases in the elastic solution. The 3D view says so under the drawing, and the note under the Results cards says on which side a horizontal load decreases the stress.
How many loads can I add?
Up to eight, each with its own magnitude, inclination and position. No two loads may stand at the same position, and the stresses of all of them add.
Can I drag the loads and the point?
Yes. Loads are dragged along x in the Section, Contours and 3D views before Analyze; the point A in the Section and Contours views before and after Analyze, with the stress at the new place calculated at once; the dashed line in the Section and Contours views sets the depth of the profile. Everything moves in steps of 0.05 m. After Analyze the loads are locked until you press Edit inputs.
Does the depth of the profile change the answer?
No. It sets how deep the profile and its table go; the answer at A stays the same.
What values does the module accept?
A magnitude of zero or more, an inclination from −90° to 90°, any position, a depth of the point of at least 0.001 m and a depth of the profile of at least 0.1 m. Anything else stops the analysis with a message that names the value.
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 load is in kN/m, tf/m or kip/ft, the stress in kPa, t/m² or ksf, and lengths in m or ft. The example gives 0.3314 kPa, 0.03379 t/m² or 0.006921 ksf.
What do the Contours and the lines of equal stress show?
The same field of Δσz in the vertical plane through the point: the Section view as lines at fixed levels, with the blue line through the value at A, and the Contours view as colour bands on a logarithmic scale. Lines can be put over the colours with the Lines of equal stress chip.
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, profile and 3D drawings.
Does the module remember my inputs?
Yes, the loads, the point and the profile depth are kept in the browser, and the page opens again on them. The analysed state is not kept: press Analyze again. Reset to defaults brings back the example; Save to file and Quick Save keep copies.
Calculate the vertical stress increase at a point below one or several flexible strip loads of finite width and infinite length: the stress at A, the lines of equal stress, colour contours, the profile with depth and a 3D view. This manual walks through the screen, dragging, every input, the results and the report, with a worked example you can reproduce.
Vertical strip load gives the increase in vertical stress at a point of the ground below one or several flexible strips of finite width and infinite length, each carrying a uniform pressure q on the ground surface. The ground is taken as a homogeneous, isotropic, elastic half-space, and the problem is plane strain: the stress is the same in every cross section along the strip. You choose the point A by its horizontal position x and its depth z, and the module gives Δσz there, the lines of equal stress, the colour contours, the profile with depth under A and a 3D view.
For one strip of width B the stress at depth z and horizontal distance x from its centre line is found by integrating the line-load solution across the width of the strip:
where α = θ₁ − θ₂ is the angle the strip subtends at A, and θ₁ and θ₂ are the angles the lines from A to the two edges of the strip make with the ground surface. The stress depends on x/B and z/B only, so the result is also given as the influence valueΔσz/q. With several strips the stress of each is found with its own distance from A, and the stresses are added.
What you can describe
One to eight strip loads, each with its own load per unit area q, width B and position xc of its centre line along the x axis.
The point A: its horizontal position x on the same axis and its depth z below the surface.
The depth of the profilezmax, down to which the Profile view and the table under A run.
The result is the increase caused by the strips only; the overburden pressure of the soil above A is not included, so add it to the answer where your design needs the total stress. The module draws the section with the lines of equal stress, colour contours, the profile and a 3D view, writes out the working and prepares a PDF or Word report.
The pressure the strip puts on the ground, zero or more. It is a pressure, not a force per metre of strip.
The width B The footing, road or fill
The width of the loaded band, at least 0.001 m. The strip is taken as infinitely long.
The position xc The layout
The place of the centre line along the x axis. With one strip you can leave it at 0 and measure x from its centre line.
The point A: x and z Where you want the stress
x on the same axis as the centre lines; z at least 0.001 m below the surface.
Quick start in seven steps
Open the module and choose the units
Open Vertical strip load from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the strip
In Strip loads the first strip is already in the table. Press its row to change q, B and xc; or drag the strip on the Section drawing. To add another strip, fill the New strip load form and press + Add strip load.
Place the point A
In Point A type x and z, or drag the blue point on the Section drawing.
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line at the foot of the Section drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the strips lock; the point A and the profile depth stay free.
Read the answer
The Results tab gives Δσz at A, the influence value Δσz/q for one strip, the part from each strip for several, and a table of the stress against depth under A. The lines of equal stress are on the Section drawing, the colour bands in Contours.
Move A and take the report
Drag A or change x and z: the stress there is calculated at once. The Calculations tab writes out the working; the Report tab saves a PDF or a Word file.
The screen: section, contours, profile and 3D
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 is the cross section of the strip or strips on the ground, with the point A, the dimensions x and z and the dashed line of the profile depth. After Analyze it adds the lines of equal stress and a label with Δσz at A.
Contours is the same section filled with colour bands of Δσz on a logarithmic scale, with a legend in kPa and in per cent of q.
Profile is the curve of Δσz against depth under A, with a marker at the depth of A.
3D is an oblique view of the strip on the ground, the axes x, y and z, and a small element at A with the stress on it.
Before Analyze the drawings follow the inputs and say “The stress and the lines of equal stress appear after Analyze” (Section), “The colour contours of the stress appear after Analyze” (Contours) or “The profile appears after Analyze”.
Section before Analyze.
Section after Analyze: the lines of equal stress; the blue line is the value at A.
Contours after Analyze.
Profile after Analyze.
3D after Analyze.
The Show chips
Above the Section and the Contours drawings a bar has two chips: Lines of equal stress and Dimensions. In the Section view the lines chip is off and greyed out until you press Analyze, and then turns on; in Contours it is off by default, so that the lines can be laid over the colours. Dimensions, the second chip, is on by default in both; it shows or hides the dashed axis of the load, the arrow with the distance x and the dashed line with the depth z (A and the profile-depth line stay). The Profile and 3D views have no chips.
Dragging on the drawing
Before Analyze the hint under the Section drawing has two lines: “Drag a load to move it, the rings at its edges to resize it. Drag the dashed line to set the depth of the profile (zmax).” and “Drag the point A to place it. The stress and the lines of equal stress appear after Analyze.” After Analyze only “Drag the dashed line to set the depth of the profile (zmax).” remains. What each pull does, as the module runs it:
The strip. Drag its body to move it: xc follows the pointer in steps of 5 cm. Pulling the strip 2 m to the right changes xc from 0 to 2.
An edge ring. Drag the ring at an edge to change the width: the edge you pull follows the pointer and the other edge stays where it is, so B and xc change together. Pulling the right ring 1 m to the right gives B = 5 and xc = 0.5; pulling the left ring 1 m to the left gives B = 5 and xc = −0.5. The width stops at 0.1 m.
Two strips never overlap. A pull that would put a strip on another stops at the last free place and a message says “That would put strip load 1 and strip load 2 on top of one another — the value is kept as it was.” Edges that meet are allowed.
The point A. Drag the blue point: x and z follow in steps of 5 cm. The depth stops at 0.05 m at the top and at the foot of the drawn frame at the bottom, and x at the sides of the frame. With the example, pulling A far to the upper left gives x = −7.2 m and z = 0.05 m, and far to the lower right x = 7.2 m and z = 8 m.
The dashed line. Drag it up or down to set zmax; it stops at 0.5 m.
After Analyze the strips are locked: pulling them changes nothing. The point A and the dashed line stay free, and so do the boxes x, z and zmax.
The 3D view has its own hint, “Drag a load over the ground to move it, the rings at its edges to resize it.” These pulls, like those of the Section view, work before Analyze only.
Zoom
Each view has −, a percentage and + at the lower right, and a button that fits the whole drawing in the pane. Zoom is a property of the drawing only: it is not saved with the model and does not change the report.
Entering the model, section by section
The Model tab has four sections: Model data, Strip loads, Point A and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below the last one. 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 13.1234 ft for the 4 m strip.
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.
Model data.
Strip loads
The section is a form over a table. The New strip load form takes q, B and xc; + Add strip load puts the strip into the Strip loads table, and a counter beside the button shows how many of the eight strips there are. The table lists each strip with its load, width and position. Press a row to change that strip in the form (the form then reads Editing · Strip load 1 and offers Done and Delete strip load; the note says that changes apply as they are entered), or press × to take a strip out. The first strip is already in the table when the module opens, and Delete strip load and × are greyed out while it is the only one.
Strip loads: the form over the table.
Strip load fields (the form, and each row of the table)
Field
What to enter
Load per unit area q kPa · t/m² · ksf · default 100
The uniform pressure of the strip on the ground. Zero or more.
Width of the strip B m · ft · default 4
At least 0.001 m. The strip is infinitely long.
Position of the centre line xc m · ft · default 0 (strip 1) · 6 (form)
Along the x axis. The New strip load form opens with 6 and keeps the last value after each Add; after deleting strip 1 of 2 the remaining strip is renumbered Strip load 1 and keeps its xc.
A row of the table pressed: the strip is edited in the form.
The strips may not lie on top of one another: two strips may meet at an edge but not overlap. Adding or editing a strip that would overlap another is refused (see Messages). The strips are listed in the order you add them, and their numbers name the results “From strip load 1”, “From strip load 2” and so on.
Two strips in the table.
Point A
The two coordinates of the point where you want the stress. A is under the ground surface at the horizontal position x and the depth z; the drawing shows the same distances as dimensions. The note in the section says that the stress of a strip is the same at equal distances either side of its centre line.
Point A.
Point A fields
Field
What to enter
Horizontal position of the point A x m · ft · default 1
On the same axis as the centre lines. For one strip centred at 0 it is the distance from the centre line, and the stress is the same at equal distances either side.
Depth of the point A z m · ft · default 1
At least 0.001 m below the ground surface.
Depth profile
The depth down to which the Profile view and the table of the stress under A run.
Depth profile.
Depth profile field
Field
What to enter
Depth of the profile zmax m · ft · default 12
At least 0.1 m. The table under A goes down to this depth, or to the depth of A if that is deeper; the Profile view goes to this depth, or to 1.2 times the depth of A if that is deeper.
Several strips and how they combine
The one choice in the module is how many strips there are, from 1 to 8; you make it by adding or deleting strips. The stress of each strip is found as if it were alone, with the distance x − xc of A from its centre line, and the stresses are added. This works because the ground is elastic, so the stresses of separate loads add.
Position is measured along one axis.x and every xc are on the same axis. A strip centred at xc = 3 with A at x = 1 gives the same stress as a strip centred at 0 with A at x = 2 (49.69 kPa in both runs), and a point at x = −1 gives the same as x = 1 (90.22 kPa).
Loads can differ. Each strip has its own q, so a road and a wider fill beside it can be put in one model.
The cards show the parts. The Results tab shows the total and the part from each strip, and the Calculations tab writes out each strip’s working.
With the example strip (100 kPa, 4 m, at 0) and A at x = 1 m, z = 1 m, a second strip of 100 kPa and 4 m centred at 6 m adds only 0.6320 kPa to the 90.22 kPa of the first: 90.86 kPa in all. If the second strip is 50 kPa, it adds 0.3160 kPa and the total is 90.54 kPa. A third strip of 100 kPa centred at 12 m brings the total to 90.87 kPa.
Two strips, 100 kPa and 50 kPa: the lines of equal stress of the sum.
The same two strips in the Contours view.
Strips that meet
Two strips of 100 kPa and 4 m, one centred at 0 and one at 4 m, meet at the edge x = 2 m and together make one strip 8 m wide. With A at x = 2 m, z = 1 m, the edge, each strip gives 49.69 kPa and the total is 99.38 kPa. If the strips are 2 m apart (the second centred at 6 m) and A is at x = 3 m, the gap between them, each gives 8.923 kPa and the total is 17.85 kPa.
Which should I use?
One wide strip or several? One strip when the load is uniform across the whole width; several strips when the pressure or the width changes along the x axis, or when there are gaps.
A load that grows across its width is not several equal strips: use Triangular strip load.
A load of limited length is not a strip: use Rectangle. A strip is the right model when the length is several times the width.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” A message appears briefly, “Analysed — see the Results tab and the Calculations.” The module stays on the Model tab: open Results for the answer. The Section view now shows the lines of equal stress.
The bar after Analyze.
The cards of the example.
Card
Symbol
The example
Vertical stress increase at A
Δσz
90.22 kPa
Influence value (one strip)
Δσz/q
0.9022
From strip load k (two or three strips)
Δσz(k)
part of the total from strip k
Below the cards the Vertical stress increase under A table gives z, Δσz and, for one strip, Δσz/q at ten equal steps down to zmax (the first row is zmax/10), with the row of A inserted and highlighted. With several strips the ratio column is left out. A Values, step by step table above it lists the steps of the Calculations tab with their values.
The depth table of the example.
Two strips: the total and the part from each.
Using the numbers
The result is an increase, not the total stress: add the overburden pressure of the soil above A for the total vertical stress.
The influence value Δσz/q is the part of the surface pressure that reaches A. It is 1 on the surface under the strip and falls with depth and with distance from the strip.
Values are rounded where they are drawn, so the last digit of a figure on the drawing can differ by one from the card.
Change x or z after Analyze and everything follows at once; to change a strip, press Edit inputs, which unlocks the strips and clears the results.
Changing the unit system after Analyze re-states the cards and the report without a new run. The drawings keep their labels in m and kPa.
Warnings and notes
The bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” The runs of this manual, from the default to eight strips, gave no warning.
The Warnings window.
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. For one strip:
The angles the strip subtends at A — B/2, 2x/B and 2z/B, the angles θ₁ and θ₂ with your numbers, and α = θ₁ − θ₂ in radians and degrees.
Stress at A — the equation above, its arctangents and third term with your numbers, then Δσz/q and Δσz = Δσz/q × q.
Stress increase against depth — the same equation down the vertical line through A, as the table of the Results tab, with the row of A marked.
With several strips there is one step for each strip, with its own distance x − xc, then a Superposition step that adds the parts, then the depth table.
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, 4 Results, 5 Summary and 6 References. It carries three drawings, the Section with the lines of equal stress, the Profile and the 3D view, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has six sections: Scope and assumptions, Derivation, The angle form, Pressure bulb and table, Worked example and References. Each opens when you click its title (Open all and Close all open or close them together), 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
Strip loads
Strip load 1
q = 100 kPa; B = 4 m; xc = 0
Point A
Position
x = 1 m; z = 1 m
Depth profile
Depth
zmax = 12 m
Step 1: the angles
The edges of the strip are at ±2 m. Seen from A, the edge lines make the angles θ₁ = atan2(1, 1 − 2) = atan2(1, −1) = 2.3562 rad and θ₂ = atan2(1, 1 + 2) = 0.3218 rad, so α = θ₁ − θ₂ = 2.0344 rad (116.57°). Also 2x/B = 0.5 and 2z/B = 0.5.
Step 2: the stress
The third term is Bz[x² − z² − B²/4] / ([x² + z² − B²/4]² + B²z²) = 4 × 1 × (1 − 1 − 4) / ((1 + 1 − 4)² + 16) = −16/20 = −0.8000.
These are the two cards of the Results tab and the steps of the Calculations tab. The same equation at the depths of the table gives the profile: the module’s table gives 90.22 kPa at 1 m, 86.63 kPa at 1.2 m and 20.57 kPa at 12 m, and the same equation worked independently gives the same three values.
Other unit systems
The same strip in SI · tf, with q = 10 t/m² (the other values unchanged), gives 9.022 t/m² and the influence value 0.9022. In US the default example, which is 2.089 ksf on a strip 13.12 ft wide, gives 1.884 ksf: 90.22 kPa is 1.884 ksf. A check by hand with round US numbers: q = 2 ksf, B = 10 ft, A at x = 2.5 ft, z = 2.5 ft gives 2x/B = 2z/B = 0.5, the same ratio 0.9022, so Δσz = 0.9022 × 2 = 1.804 ksf; the module gives 1.804 ksf.
How each input changes the answer
Each row changes one thing in the example (one strip of q = 100 kPa, B = 4 m, xc = 0, A at x = 1 m, z = 1 m) and runs the module again, in SI · kN. The numbers are the ones the module gives, so you can reproduce any row; an independent calculation with the same equation agrees with every row.
The example with one change at a time
Change
Δσz (kPa)
Δσz/q
The example
90.22
0.9022
q = 200 kPa The stress is proportional to the load; the ratio is unchanged
180.4
0.9022
q = 50 kPa Half the load, half the stress
45.11
0.9022
B = 2 m A narrower strip puts less on the same point
47.97
0.4797
B = 8 m A wider strip: A is deeper inside the loaded width and the stress rises
99.15
0.9915
B = 20 m Nearly an infinite loaded width: the ratio approaches 1
99.96
0.9996
x = 0 (under the centre line) The highest stress of the section at this depth
95.95
0.9595
x = −1 m The same as x = +1 m: the stress is symmetric about the centre line
90.22
0.9022
x = 2 m (under the edge) About half of q (0.497 q) at 1 m depth
49.69
0.4969
x = 4 m (2 m beside the edge) The stress falls to about 2 per cent of the load
1.931
0.01931
x = 10 m Far from the strip the stress is a tiny part of the load
0.02850
0.0002850
z = 0.5 m Near the surface the stress approaches q
97.88
0.9788
z = 0.01 m At the surface under the strip the stress equals q
100.0
1.000
z = 2 m The stress falls with depth
73.47
0.7347
z = 4 m About half of q at one strip width of depth
51.05
0.5105
z = 8 m At twice the strip width, under 30 per cent of q
29.76
0.2976
xc = 3 m (A at x = 1 m) A is 2 m from the centre line: the same as x = 2 with the strip at 0
49.69
0.4969
x = 2 m and z = 2 m The same 2x/B and 2z/B as B = 2 m with A at (1 m, 1 m): the same stress
47.97
0.4797
Several strips, SI · kN, one change at a time from the example (second strip added with the values shown):
Adding strips
Added
Total (kPa)
Parts (kPa)
Strip 2: 100 kPa, 4 m, centre at 6 m A at (1 m, 1 m); the gap between the strips is 2 m (edges at x = 2 m and x = 4 m)
90.86
90.22 + 0.6320
Strip 2: 50 kPa, 4 m, centre at 6 m A at (1 m, 1 m)
90.54
90.22 + 0.3160
Strip 2: 100 kPa, 4 m, centre at 4 m The strips meet at x = 2 m; A at (2 m, 1 m)
99.38
49.69 + 49.69
Strip 2: 100 kPa, 4 m, centre at 6 m A at (3 m, 1 m), in the gap
17.85
8.923 + 8.923
The pattern follows the equation. The stress is proportional to q; it depends on x/B and z/B only, so doubling every length leaves the ratio unchanged; it is highest under the centre line and falls with depth and with distance from the strip; and with z near zero it equals q under the strip and zero beside it. The depth of the profile zmax changes the table and the Profile view but not the stress at A: 90.22 kPa with zmax = 6 m and 90.22 kPa with zmax = 0.5 m, a depth to which the table is extended, down to the depth of A.
Messages: when the module stops, and what it notes
The module checks the numbers it needs before it calculates. When an input cannot be used, press Analyze and 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; nothing is calculated.
The window Analyze opens when the width is zero.
When
Message and what to do
B is 0
“Width of strip 1 (B1) must be at least 0.001 m.” Enter a width of at least 0.001 m.
q is negative
“Load per unit area of strip 1 (q1) must be at least 0 kPa.” Enter zero or a positive load.
z is 0
“Depth of the point A (z) must be at least 0.001 m.” Enter a depth of at least 0.001 m.
zmax is 0
“Depth of the profile (zmax) must be at least 0.1 m.” Enter a depth of at least 0.1 m.
A box is empty
“Width of strip 1 (B1) is missing.” Fill the box or take the strip out.
A new strip would overlap another
A message appears briefly: “The new strip load would lie on strip load 1 — they cannot overlap. Give it another position or width.” The strip is not added.
The messages use the numbered names of the strips, such as “strip 1” and “B1”, even when there is only one strip, and the limits in the units of the unit system chosen. When you type a position or width into a strip already in the table and it would put two strips on one another, a message appears briefly: “That would put strip load 1 and strip load 2 on top of one another — the value is kept as it was.” and the value stays as it was. At eight strips + Add strip load is greyed out and the counter reads 8 of 8.
Notes and warnings
With every input inside its limits the module calculates, and the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” A load of zero is accepted and gives 0 kPa, with a dash for the influence value. Large values are accepted too: a very wide strip such as B = 1000 m is a load that is nearly infinite in width, and with q = 1,000,000 kPa the module gives 1,000,000 kPa at A, an influence value of 1.000.
Good practice and common mistakes
Use the module for a long, uniformly loaded band. The strip is infinitely long and flexible, so the pressure stays q across the width. A footing that is stiff will distribute its pressure differently; the module gives the stress for the pressure you enter.
Enter a pressure, not a force.q is a load per unit area. A line load per metre of strip is q × B; if the width is small compared with the depth, Vertical line load is the closer model.
Mind the axis.x and every xc are on the same axis; the stress of a strip depends on the distance x − xc, so a point left of a strip gives the same as one the same distance right of it.
Do not overlap strips. Strips may meet at an edge but not lie on one another; to describe a stepped load, split it into strips that meet.
Add the overburden. The result is the stress increase; for the total vertical stress add the soil weight above A.
Read the profile with the depth of the strip’s width in mind. Under the centre line of a 4 m strip the stress is 55 per cent of q at 4 m depth, one strip width, and 31 per cent at 8 m, twice the width.
Place A with the drawing, then type exact values. Dragging snaps to 5 cm; type the coordinates when you need a given place.
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 stress below a strip load calculated?
The module integrates the line-load solution of the theory of elasticity across the width of the strip. For a strip of width B and pressure q, the vertical stress increase at the horizontal distance x from its centre line and the depth z is Δσz = (q/π){α − Bz[x² − z² − B²/4]/([x² + z² − B²/4]² + B²z²)}, with α the angle the strip subtends at A. For the module’s example it gives 90.22 kPa.
What is the influence value Δσz/q?
It is the stress increase divided by the strip pressure. It depends only on x/B and z/B, is 1 at the surface under the strip, and is the number to use if you have a table of influence values. The module gives it for one strip; with several strips it gives the stress of each.
Does the result include the weight of the soil?
No. It is the increase caused by the strips only. Add the overburden pressure of the soil above A for the total vertical stress.
How many strips can I add, and can they overlap?
Up to eight, each with its own q, B and xc. They may meet at an edge but not overlap; the module refuses a strip that would lie on another. The stresses of the strips add.
Can I drag the strip and the point on the drawing?
Yes. Drag the strip body to move it, the rings at its edges to change its width, the blue point to place A and the dashed line to set the depth of the profile; they move in 5 cm steps. After Analyze the strips are locked, but A and the dashed line stay free.
What does the point A depend on?
On x, along the same axis as the centre lines, and on z, its depth. The stress of a strip is the same at equal distances either side of its centre line.
What do the lines of equal stress show?
Each line joins the points of the section with the same Δσz, labelled in kPa; the blue line is the value at A. Switch them on and off with the Lines of equal stress chip. The Contours view shows the same stress as colour bands on a logarithmic scale.
What values does the module accept?
q zero or more, B at least 0.001 m, z at least 0.001 m, zmax at least 0.1 m, and x and xc any value. Anything else stops the analysis with a message that names the value.
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. Stresses are in kPa, t/m² or ksf and lengths in m or ft. The default example gives 90.22 kPa, 9.022 t/m² with q = 10 t/m², or 1.884 ksf.
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, profile 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.
When is a strip the wrong model?
When the load is short compared with the depth you ask about, so that its ends matter: use Rectangle. When the pressure is not uniform across the width: use Triangular strip load, or split the load into strips of different q. When the load is horizontal: use Horizontal strip load.
Calculate the vertical stress increase below one or several flexible strips that carry a horizontal or inclined load: the stress at a point, its vertical and horizontal parts, the lines of equal stress, colour contours, the profile with depth and a 3D view. This manual walks through the screen, dragging the strips and the point, every input, the results and the report, with a worked example you can reproduce.
Horizontal strip load gives the vertical stress increase at a point in the ground below one or several flexible strips of finite width and infinite length that carry a uniform load pushing along the ground surface (a horizontal load) or at an angle to the vertical (an inclined load). The ground is a homogeneous, isotropic, linearly elastic half-space and the problem is plane strain: every cross-section along the strip is the same. Such loads are the friction under a sliding footing, the thrust at the base of a retaining wall or the braking of vehicles on a road.
An inclined load of pressure q at the angle α from the vertical is split into a vertical part qv and a horizontal part qh, each is solved apart, and the two stresses add. With b = B/2 and x the horizontal distance of the point A from the centre line of the strip:
ω is the angle that the strip subtends at A and δ is the angle between the vertical through A and the line from A to the edge of the strip on the +x side (signed; negative when A is under the strip). The horizontal part is odd in x: it raises the stress ahead of the load (+x for a load that pushes toward +x) and lowers it behind, and it is zero under the centre line. For several strips each strip is taken with its own x, the distance of A from its centre line, and the stresses are summed.
What you can describe
One to eight strips. Each has its own load per unit area q, inclination α from the vertical (from −90° to 90°; 0° is a vertical strip, 90° a purely horizontal one), width B and the position xc of its centre line along the ground. Strips may meet at an edge but may not overlap.
A point A at the horizontal position x and the depth z.
The depth of the profilezmax: how deep the Profile view and the table under A go.
The module gives the vertical stress increase at A, its vertical and horizontal parts, and, for two or three strips, the part of each strip. It draws the lines of equal stress, the colour contours, the stress against depth under A and a 3D view, writes out the working, and prepares a PDF or Word report. The increase is the one caused by the strip loads only; the overburden pressure is not included, and a negative value is a decrease.
Which module for what
A strip loaded vertically only: Vertical strip load. Here, α = 0° gives the same loading and the horizontal part is zero.
A load of no width: Horizontal line load takes horizontal or inclined line loads.
A strip whose load rises linearly from zero at one edge: Triangular strip load; a long embankment with side slopes: Embankment.
The pressure q per unit area, and its inclination α from the vertical: how much of it is vertical and how much pushes along the surface.
The width B of each strip
Measured across the strip, in the plane of the section. The strip is infinitely long in the other direction.
The position xc of each centre line
Along the same axis as the point A. The first strip is at 0 unless you move it.
The point A: x and z
The horizontal position along the same axis, and the depth below the surface.
The depth of the profile zmax
The table runs from the surface to zmax (to the depth of A instead, if A is deeper), and the Profile view shows at least zmax.
Quick start in seven steps
Open the module and choose the units
Open Horizontal strip load from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US at the top right; you can change it at any time.
Describe the strips
In Strip loads the first strip is already in the table. Press its row to change its load q, inclination α, width B and centre line xc in the form, or fill the New strip load form and press + Add strip load for more.
Place the point A
In Point A type the horizontal position x and the depth z. You can also drag A on the drawing.
Set the depth of the profile
In Depth profile type zmax.
Press Analyze
Press Analyze in the bar at the foot of the Model tab. The bar reads Analysed — the stress at the point calculated, the strips lock, and the drawing shows the lines of equal stress and the value at A. The module stays on the Model tab.
Read the drawings
Switch between Section, Contours, Profile and 3D above the drawing. Drag A to read the stress elsewhere; it is recalculated at once.
Take the numbers, the working and the report
The Results tab gives the stress and its two parts, the Calculations tab the working, and the Report tab saves a PDF or a Word file.
The screen: section, contours, profile and 3D
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 is the cross-section of the strips on the ground surface, the point A with its dimension lines, and the profile depth. After Analyze it adds the lines of equal vertical stress increase and the value at A.
Contours is the same cross-section coloured by the stress, on a logarithmic scale, with a legend in kPa and in per cent of q.
Profile is the stress increase against depth under A: the total, the part from qv and the part from qh. Before Analyze it says “The profile appears after Analyze.”
3D is an oblique block with the loaded strip, the axes x, y, z, the point A and the stress on a small element at A. Before Analyze it says “The stresses on the element appear after Analyze.”
Section before Analyze: the strip, the point A and the dimension lines.
Section after Analyze: lines of equal stress; the blue line is the value at A.
Contours after Analyze.
Profile after Analyze.
3D after Analyze.
The Show chips
Above the Section and the Contours drawings a bar of two chips switches parts of the drawing on and off: Lines of equal stress and Dimensions. In the Section view both are on after Analyze; before it the lines are off and the bar says “The stress and the lines of equal stress appear after Analyze”. In the Contours view the lines are off by default (switch them on to draw them over the colours) and the bar says “The colour contours of the stress appear after Analyze”. The Profile and 3D views have no chips.
Dragging on the drawing
The drawing is a way of entering the model, and the Model tab follows every pull:
A strip (Section, Contours and 3D, before Analyze). Drag the body of the strip along the ground to move its centre line xc. The same pulls work in the 3D view.
The rings at the edges (before Analyze). Drag a ring to move that edge: the other edge stays, so B and xc change together. A pull cannot make a strip narrower than 0.1 m.
The point A (Section and Contours, before and after Analyze). Drag A to move x and z. It stays inside the frame of the drawing, and its depth is at least 0.05 m. After Analyze the stress at A is recalculated at once and the Results tab follows.
The dashed profile line (Section and Contours, before and after Analyze). Drag it up or down to set zmax; a pull cannot go above 0.5 m. When zmax is deeper than the drawing, the line sits at the lower edge with an arrow.
Pulls move in steps of 5 cm. A pull of a strip stops at the last free place, and the module says so (see Messages); two strips may meet at an edge but not overlap. After Analyze the strips and their rings are locked, together with every input of the model: Edit inputs unlocks them and clears the results. Only A and the profile line stay free.
Zoom and pan
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, and dragging the background moves the view.
Entering the model, section by section
The Model tab has four sections: Model data (names, files and quick saves, the same in every module), Strip loads, Point A and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through Strip loads, Point A and Depth profile, and Analyze waits in the bar at the foot of the tab. 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; after a switch to US the boxes show the converted values, for example 3.13282 ksf for the 150 kPa strip, 9.84252 ft for the 3 m width and 7.38189 ft for x = 2.25 m.
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 and keeps it as the model of the browser.
Model data.
Strip loads
The section is a form over a table. The New strip load form takes the load q, the inclination α, the width B and the position xc of one strip. + Add strip load puts it at the bottom of the table, and a counter beside the button shows how many of the eight strips there are (1 of 8). The Strip loads table lists every strip as “q = 150 kPa”, “α = 30°” and “B = 3 m, xc = 0 m”. Press a row to change that strip in the form (the form then reads Editing · Strip load 1 and offers Done and Delete strip load; the note says that changes apply as they are entered), or press × to take a strip out. The stresses of the strips add.
Strip loads: the form over the table.
Strip load fields (the form, and each row of the table)
Field, unit and what to enter
Default
Load per unit areaq Unit: kPa · t/m² · ksf Zero or more. This is the whole pressure, acting at the angle α; the module splits it into qv = q cos α and qh = q sin α.
150
Inclination from the vertical (+ toward +x; 90° = horizontal)α Unit: ° From −90° to 90°. 0° is a vertical load, 90° a horizontal load pushing toward +x, and a negative angle pushes toward −x.
30
Width of the stripB Unit: m · ft At least 0.001 m.
3
Position of the centre linexc Unit: m · ft Any value, along the same axis as the point A. Strip 1 is at 0; the New strip load form offers 5 m, so that a new strip does not lie on the first.
0 · 5 (form)
Two strips may meet at an edge but not overlap. The module checks this when you add a strip, when you change a value, and when you pull a strip on the drawing, and keeps the previous value if the change would put one strip on another (see Messages). After the eighth strip the button is off.
Two strips in the table.
Point A
Two numbers place the point where you want the stress. With one strip at xc = 0, x is the distance from its centre line; with several strips it is the position on the common axis, and the dimension line on the drawing is the distance from the centre line of strip 1 (x − x₁). The note under the fields says that the horizontal part of a load raises the stress ahead of it and lowers it behind. A can also be dragged on the drawing.
Point A.
Point A fields
Field, unit and what to enter
Default
Horizontal position of the point A (+ in the direction of the load)x Unit: m · ft Any value. The axis does not turn with α: it is the axis of the centre lines, and + is the side toward which a load with a positive α pushes.
2.25
Depth of the point Az Unit: m · ft At least 0.001 m. The solution gives the stress inside the ground, so A cannot be on the surface.
3
Depth profile
One number: how deep the table and the Profile view go under A. The table runs from the surface to zmax in ten equal steps (to the depth of A instead, if A is deeper than zmax) and the row of A is added; the Profile view shows at least zmax, and extends to 1.2 times the depth of A if A is deeper. The profile line on the Section and Contours drawings can be dragged to set it.
Depth profile.
Depth profile field
Field, unit and what to enter
Default
Depth of the profilezmax Unit: m · ft At least 0.1 m. The table runs from the surface to zmax in ten equal steps (to the depth of A instead, if A is deeper than zmax) and the row of A is added; the Profile view shows at least zmax, and extends to 1.2 times the depth of A if A is deeper.
12
Describing the load: inclination, direction and several strips
Three decisions shape the answer: the inclination of each strip, which side of the strip the point lies on, and how many strips there are.
Inclination and what it is
kPa at x = 2.25 m
α = 0° A vertical strip: qh = 0.
43.14, all vertical
α = 30° (default) Inclined; the horizontal part pushes toward +x.
The vertical part Δσz(v) is even in x: it is the same on both sides of the centre line. The horizontal part is odd in x: ahead of the load (the side toward which it pushes) the soil is pressed down, behind it the soil is lifted. In the example, A at +2.25 m gives 50.52 kPa (37.36 + 13.15) and A at −2.25 m gives 24.21 kPa (37.36 + (−13.15)); on the centre line, x = 0, the horizontal part is 0 and the total is 71.42 kPa. A negative angle α at x gives the mirror image of a positive angle at −x: the −30° strip at +2.25 m and the 30° strip at −2.25 m both give 24.21 kPa.
Several strips
Add a strip with its own q, α, B and xc. The module finds each strip’s stress with the distance of A from that strip’s centre line and adds them. In the Results tab the stress is “from the vertical parts” and “from the horizontal parts” of all strips, and, for two or three strips, a card gives the part of each strip.
With strip 2 added (100 kPa at 90°, 3 m wide, centre line at 5 m) the example gives 33.98 kPa at A: strip 1 gives 50.52 kPa and strip 2 −16.54 kPa, because A lies behind a strip that pushes away from it. With the point A at x = 2.5 m the total is 28.18 kPa. A strip 2 that meets strip 1 at an edge (centre line at 3 m, so that the edges touch at x = 1.5 m) gives 40.93 kPa. Four strips of the default kind at 0, 5, 10 and 15 m give 65.13 kPa at A, and eight strips at 0, 5 … 35 m give 64.82 kPa.
Two strips: the section with the lines of equal stress. Strip 2 here is 2 m wide.
How to choose
α follows the load. Use 90° for a pure shear along the surface (friction, braking, thrust) and the angle of the resultant for a load that has both parts; a load pushing toward −x takes a negative angle, or place A on the other side.
Mirror to check. Changing the sign of α changes only the sign of the horizontal part; changing the sign of x does the same.
One axis for everything. Strips and A are placed on one axis; type xc of each strip and x of A on it.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” The module stays on the Model tab: the value at A is written beside it on the drawing (Δσz = 50.52 kPa in the example), and the cards are in the Results tab.
The Results tab of the example: the three cards.
Card
Symbol
The example
Vertical stress increase at A
Δσz
50.52 kPa
From the vertical part (“parts” for several strips)
Δσz(v)
37.36 kPa
From the horizontal part (“parts” for several strips)
Δσz(h)
13.15 kPa
From strip load 1, 2 … (two or three strips only)
Δσz(1), Δσz(2)
strip 1 50.52, strip 2 −16.54 kPa in the two-strip case
A note under the cards reads “The increase caused by the strip loads only; a negative value is a decrease. The overburden pressure is not included.” Below it, Values, step by step lists the vertical part, the horizontal part and their sum, and Vertical stress increase under A is a table that runs from the surface to zmax in ten equal steps (to the depth of A instead, if A is deeper than zmax) with the row of A added and marked, each with Δσz(v), Δσz(h) and Δσz.
The depth table of the example; the row of A is marked.
Using the numbers
It is an increase. Add the overburden stress yourself where you need the total vertical stress; the module gives the change caused by the strips.
Negative values are decreases (the elastic solution gives a lift behind a load that pushes away). The 3D view says “A negative Δσ is a decrease (tension) in the elastic solution.”
The value at A and the profile follow the point as you drag it; the Results tab shows the stress at the new position.
Changing the unit system after Analyze re-states the cards and the report without a new run; the drawings keep m and kPa when you change the unit system, while the boxes, cards, tables and report follow it. Edit inputs unlocks the model and clears the results.
Warnings
The bar carries a Warnings (0) button; it opens a window that says “No warnings — the assumptions of the solution hold for these inputs.”
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. For one strip the steps are:
Components of the load — qv = q cos α and qh = q sin α with your numbers: 129.90 kPa and 75.00 kPa in the example.
Stress from the vertical part — the ratios 2x/B = 1.5000 and 2z/B = 2.0000, the strip factor Δσz/qv = 0.2876 and Δσz(v) = 0.2876 × 129.90 = 37.36 kPa.
Stress from the horizontal part — the formula with b, x/b, z/b and x² + z² − b² = 11.8125, the factor 0.1754 and Δσz(h) = 0.1754 × 75.00 = 13.15 kPa.
Stress increase against depth — the same sum down the vertical line through A, as a table with the row of A marked.
With several strips the working has one step for each strip (“Stress from strip load 1”, “… 2”, …), each with its own x, qv, qh and ratios, then the superposition of the strips and the depth table. The numbered steps are followed by an unnumbered Answer line.
The Calculations tab: the first steps with the example’s numbers.
The superposition step.
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. Before Analyze the tab says “Press Analyze on the Model tab to calculate the stress at the point.”
The report has six parts: 1 Introduction and scope, 2 Input data, 3 Calculation (one part for each step of the Calculations tab), 4 Results, 5 Summary and 6 References. It carries the section with the lines of equal stress, the profile and the 3D view, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has four sections: Scope and assumptions, The vertical stress, Worked examples and References. Each opens when you click its title, with Open all and Close all above them. It gives the formula for the horizontal strip, the chart of Δσz/q against x/b at several depths, a table of Δσz/q against z/b and x/b, and two worked examples whose numbers come from the module’s own code.
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
Strip loads
Strip load 1
q = 150 kPa; α = 30°; B = 3 m; xc = 0
Point A
Position and depth
x = 2.25 m; z = 3 m
Depth profile
Depth of the profile
zmax = 12 m
Step 1: the two parts of the load
qv = 150 cos 30° = 129.90 kPa and qh = 150 sin 30° = 75.00 kPa. With B = 3 m, b = 1.5 m.
Step 2: the vertical part
The strip subtends ω = arctan(3.75/3) − arctan(0.75/3) = 51.34° − 14.04° = 37.30°, and δ = 14.04°. In radians ω = 0.6511; sin 37.30° = 0.6061 and cos(ω + 2δ) = cos 65.37° = 0.4168, so the factor Δσz/qv is (0.6511 + 0.6061 × 0.4168)/π = (0.6511 + 0.2526)/π = 0.2876, the value the Calculations tab gives, and Δσz(v) = 0.2876 × 129.90 = 37.36 kPa.
Step 3: the horizontal part
x² + z² − b² = 5.0625 + 9 − 2.25 = 11.8125, so the numerator is 4 × 1.5 × 2.25 × 9 = 121.5 and the denominator π(11.8125² + 4 × 2.25 × 9) = π × 220.54 = 692.8. The factor is 121.5 / 692.8 = 0.1754, and Δσz(h) = 0.1754 × 75.00 = 13.15 kPa.
Step 4: the sum
Δσz = 37.36 + 13.15 = 50.52 kPa
These are the three cards of the Results tab and the steps of the Calculations tab. The Profile view and the depth table give the same stress at the other depths: 50.52 kPa at 3 m, and 50.70 kPa at 2.4 m, 37.79 kPa at 6 m and 21.17 kPa at 12 m.
The point on the other side
Type x = −2.25 m: the vertical part is the same and the horizontal part changes sign, so Δσz = 37.36 − 13.15 = 24.21 kPa.
A purely horizontal strip
For q = 100 kPa, α = 90°, B = 4 m, x = 1 m and z = 1 m, b = 2 and Δσz = 4 × 2 × 100 × 1 × 1² / {π[(1 + 1 − 4)² + 4 × 4 × 1]} = 800 / (π × 20) = 12.73 kPa. The module gives 12.73 kPa, all from the horizontal part.
Other unit systems
The default example in SI · tf gives Δσz = 5.151 t/m² (3.810 + 1.341), and in US 1.055 ksf (0.7803 + 0.2747): the same strip in other units (50.52 kPa is 5.151 t/m² and 1.055 ksf). Two checks by hand:
US. A horizontal strip of 2 ksf, B = 10 ft, x = 5 ft, z = 10 ft: b = 5 ft and Δσz = 4 × 5 × 2 × 5 × 10² / {π[(25 + 100 − 25)² + 4 × 25 × 100]} = 20 000 / (π × 20 000) = 0.3183 ksf. The module gives 0.3183 ksf.
SI · tf. A strip of 10 t/m² at 60° from the vertical, B = 4 m, centre line at 1 m, A at x = 3 m and z = 2 m, so that the distance from the centre line is 2 m: qh = 8.660 and the horizontal part is 4 × 2 × 8.660 × 2 × 2² / {π[(4 + 4 − 4)² + 4 × 4 × 4]} = 554.3 / (π × 80) = 2.205 t/m². The module gives 2.205 t/m² for the horizontal part, 2.399 t/m² for the vertical part and 4.604 t/m² in all.
How each input changes the answer
Each row changes one thing in the example (strip 1: q = 150 kPa, α = 30°, B = 3 m, xc = 0; A at x = 2.25 m, z = 3 m) and runs the module again. The numbers are the ones the module gives, so you can reproduce any row. All values are in kPa; each was also found with an independent calculation. The last column gives the part from the vertical load qv plus the part from the horizontal load qh.
The example with one change at a time
Change
Δσz
qv + qh parts
The default example: q = 150 kPa, α = 30°, B = 3 m, xc = 0; A at x = 2.25 m, z = 3 m The reference
50.52
37.36 + 13.15
α = 0° A vertical strip: the horizontal part is zero
43.14
43.14 + 0
α = 60° A larger horizontal part and a smaller vertical part
44.35
21.57 + 22.78
α = 90° A horizontal strip: the vertical part is zero
26.31
0 + 26.31
α = −30° The horizontal part changes sign: A is now behind the load
24.21
37.36 + (−13.15)
α = −90° A horizontal load pushing away from A: a decrease
−26.31
0 + (−26.31)
q = 300 kPa Twice the load, twice the stress
101.0
74.73 + 26.31
q = 0 No load, no stress
0
0 + 0
B = 6 m A wider strip: A is now under it (edge at x = 3 m)
97.75
81.16 + 16.59
B = 1.5 m A narrower strip
24.60
17.44 + 7.162
xc = 1.5 m The strip moved toward A (A is 0.75 m from the centre line)
73.51
66.32 + 7.190
x = 0 On the centre line the horizontal part is zero
71.42
71.42 + 0
x = −2.25 m The mirror point: the same vertical part, the opposite horizontal part
24.21
37.36 + (−13.15)
x = 6 m Beyond the strip, far from the load
7.807
3.754 + 4.053
z = 0.5 m Near the surface
12.09
5.165 + 6.929
z = 1.5 m Shallower
43.57
27.77 + 15.81
z = 6 m Deeper
37.79
31.45 + 6.339
z = 12 m Deep: both parts fall off
21.17
19.14 + 2.031
Strip 2 added: q = 100 kPa, α = 90°, B = 3 m, xc = 5 m A lies behind strip 2, which lifts it (strip 1 gives 50.52, strip 2 −16.54)
33.98
37.36 + (−3.384)
The same two strips, A at x = 2.5 m A moves 0.25 m nearer strip 2
28.18
32.48 + (−4.297)
Strip 2 with xc = 3 m (its edge meets that of strip 1) Strips that meet at an edge are allowed
40.93
37.36 + 3.566
The pattern follows the formulas. The stress is proportional to q; the vertical part follows how much of the strip is seen from A; the horizontal part changes sign with α and with x and is zero on the centre line; and the stress of several strips is the sum of the single-strip stresses. The total under A is not largest at the surface: in the example it is 50.70 kPa at 2.4 m and 50.52 kPa at 3 m, and it falls to 21.17 kPa at 12 m.
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 its section of the Model tab (the section does not open by itself; open it to find the box).
The limits are shown in the units of the unit system chosen.
The window Analyze opens when the width is zero.
Stop when …
Message and what to do
The width is below 0.001 m or empty
“Width of strip 1 (B1) must be at least 0.001 m.” or “… is missing.” Enter a width.
The load is negative or empty
“Load per unit area of strip 1 (q1) must be at least 0 kPa.” or “… is missing.” Enter zero or a positive load; for a load that pushes the other way, use a negative inclination.
The inclination is outside −90° to 90°
“Inclination of the load of strip 1 from the vertical (+ toward +x; 90° = horizontal) (α1) must be at most 90°.” or “… at least -90°.” Enter an angle in that range.
The centre line is empty
“Position of the centre line of strip 1 (x1) is missing.” Enter the position.
The depth of A is below 0.001 m or empty
“Depth of the point A (z) must be at least 0.001 m.” or “… is missing.” Enter a depth below the surface.
The horizontal position of A is empty
“Horizontal position of the point A (+ in the direction of the load) (x) is missing.”
The depth of the profile is below 0.1 m
“Depth of the profile (zmax) must be at least 0.1 m.”
When more than one value is wrong the window lists them together, for example “2 input errors”: “Width of strip 1 (B1) must be at least 0.001 m.” and “Depth of the point A (z) must be at least 0.001 m.” In the messages the strip number is part of the symbol: B1, q1, α1 and x1 belong to strip 1, B2 to strip 2.
Messages about overlapping strips
Adding a strip on top of another: “The new strip load would lie on strip load 1 — they cannot overlap. Give it another position or width.”
Changing a value that would put two strips on one another: “That would put strip load 1 and strip load 2 on top of one another — the value is kept as it was.”
Pulling a strip on the drawing into another: the same message appears and the strip stays at the last free place, with its edge against the edge of the other (in the example, strip 1 pulled toward a strip at 5 m stops with its centre line at 2 m).
Notes and warnings
With usual values the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” A negative result is not a warning: it is a decrease in the vertical stress, and the Results tab says so under the cards.
Good practice and common mistakes
Choose α from the direction of the pressure. Zero is vertical and 90° horizontal; the sign sets which way the horizontal part pushes along x.
Keep one axis. Type the centre lines xc and the point x on the same axis. The distance that matters is x − xc.
Mind which side A is on. Ahead of a load that pushes toward +x the horizontal part adds to the stress; behind it, it subtracts. Run A on both sides when the point is near a loaded strip.
Enter the pressure, not its vertical part.q is the whole pressure at the angle α; the module takes the vertical and horizontal parts itself.
Strips may meet but not overlap. Split a strip of unequal loading into strips side by side.
Add the overburden yourself. The module gives the increase caused by the strips.
Read the profile as well as the point. The stress under A does not peak at the surface; the profile shows the depth at which it does (in the example near 2.4 to 3 m).
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 vertical stress below a horizontal strip load calculated?
The load q at the angle α from the vertical is split into qv = q cos α and qh = q sin α. The vertical part gives the stress of a vertical strip; the horizontal part gives Δσz(h) = 4bqhxz² / {π[(x² + z² − b²)² + 4b²z²]} with b = B/2; the two add. For the module’s example, Δσz = 50.52 kPa at A, of which 37.36 kPa is from the vertical part and 13.15 kPa from the horizontal part.
What is the difference between this module and Vertical strip load?
Vertical strip load takes loads that are purely vertical. This module takes a horizontal or inclined load, and with α = 0° it is a vertical strip too (the horizontal part is then zero). The vertical part of an inclined strip is found as for a vertical strip.
Which way are α and x positive?
α is measured from the vertical and is positive toward +x, so 90° is a horizontal load pushing toward +x and a negative angle pushes toward −x. x is measured along the axis of the centre lines, and does not turn with α. A strip at −30° read at +2.25 m gives the same stress as a strip at +30° read at −2.25 m (24.21 kPa in the example).
Why is the stress negative at some points?
The horizontal part is odd in x: the soil ahead of a load that pushes along the surface is pressed down, and the soil behind it is lifted, so Δσz is a decrease there. A horizontal strip of 150 kPa at α = −90° gives −26.31 kPa at the example point. A negative value is a decrease, not an error; the module does not call it a warning.
Why is the stress zero under the centre line for a horizontal strip?
The horizontal part is zero at x = 0 (0 kPa at 3 m in the example) because it is odd in x. A purely horizontal strip therefore adds no vertical stress under its centre line; an inclined strip still has the vertical part there (71.42 kPa at x = 0 in the example).
How many strips can I enter, and can they overlap?
Up to eight, each with its own q, α, B and centre line. They may meet at an edge but not overlap; the module keeps the previous value (or stops the pull) when a change would put one strip on another. The stresses of the strips add.
Can I drag the strips and the point A on the drawing?
Yes, before Analyze: drag a strip to move it, the rings at its edges to resize it, A to place the point, and the dashed line to set the profile depth. After Analyze the strips are locked until you press Edit inputs; A and the profile line stay free, and the stress at A is recalculated as you move it.
Does the stress include the weight of the soil?
No. The module gives the vertical stress increase caused by the strip loads. Add the overburden stress where you need the total.
What values does the module accept?
The load zero or more, the inclination from −90° to 90°, the width at least 0.001 m, any centre line, the depth of A at least 0.001 m, any position of A, and the profile depth at least 0.1 m. Anything else stops the analysis with a message that names the value.
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 stress is in kPa, t/m² or ksf, and lengths in m or ft. The default example gives 50.52 kPa, 5.151 t/m² or 1.055 ksf.
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, the profile and the 3D view. 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; an analysed model opens analysed only when nothing was changed after Analyze; if you moved or retyped A, or changed the profile depth, it opens with the new values and waits for Analyze. Reset to defaults brings back the example; Save to file and Quick Save keep copies.
Calculate the vertical stress increase below one or several strip loads that rise linearly from zero at one edge to q at the other: the stress at a point, the lines of equal stress, colour contours, the profile under the point and a 3D view. This manual walks through the screen, every input, the results and the report, with a worked example you can reproduce.
Triangular strip load gives the vertical stress increase Δσz in the ground below a flexible strip load whose intensity rises in a straight line across its width: zero at one edge, q at the other. The strip is infinitely long, so the problem is the same in every cross section (plane strain), and the ground is a homogeneous, isotropic, linearly elastic half-space. It is the load of a fill that tapers out, of a stockpile that is heaped against a wall, or of the toe of an embankment.
Δσz = (q / 2π) [ (2x / B) α − sin 2δ ]
B is the width of the strip, x is measured from the zero edge toward the high edge, α is the angle the strip subtends at the point and δ is the angle from the vertical through the point to the line to the high edge, taken with its sign. You give the strip, the point A and the depth of the profile; the module gives Δσz at A, Δσz/q, the lines of equal stress, colour contours, the profile under A and a 3D view.
What you can describe
Up to eight strips. Each has its own q, width B, left edge xL and direction (zero at the left edge or zero at the right edge). The stresses of the strips add.
A point A at a horizontal position x and a depth z; the stress at A is given and the profile under A runs from the surface to zmax.
Strips that meet at an edge are allowed; strips that overlap are not.
The lateral pressure that a triangular load puts on a wall is a different calculation: Surcharge, triangular load.
This module gives the stress increase caused by the strips. The module states that the overburden pressure of the soil above A is not included; add it, and take the settlement or the bearing check, in your own calculations.
Before you start: what you need
Have these ready:
What
Notes
The load of each strip
q, the load per unit area at the high edge, and the direction: which edge carries zero. The load is zero at the other edge and falls in a straight line between them.
The width B and the place xL
The width of each strip, and the position of its left edge along the horizontal axis x, which you choose. For one strip, put the left edge at 0.
The point A
Its horizontal position x on the same axis (negative to the left of 0) and its depth z below the surface.
The depth of the profile zmax
How deep the profile table and the Profile view go under A.
Quick start in seven steps
Open the module and choose the units
Open Triangular strip load from the Stresses in a Soil Mass family. At the top right choose SI · kN, SI · tf or US; you can change it at any time.
Describe the strip
In Triangular strip loads choose Zero at left or Zero at right in the New strip load form, type q, B and xL, and press + Add strip load. The first strip already exists: edit its row, or press + Add strip load for another.
Add more strips if you have them
Repeat for each strip, up to eight. Strips may touch but not overlap.
Place the point A
Press Next and type x and z in Point A, or drag A on the drawing.
Set the depth of the profile
Press Next and type zmax in Depth profile, or drag the dashed line on the drawing.
Press Analyze
Press Analyze in the bar at the foot of the Model tab. The bar reads Analysed — the stress at the point calculated and the model locks, except the point A and the depth of the profile.
Read the answer, the working and the report
The Results tab gives Δσz at A; Calculations gives the working; Report saves a PDF or a Word file.
The screen: four views, and what you can drag
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: Section, Contours, Profile and 3D. On the right are the tabs Model, Results, Calculations, Report and Theory.
The window as the module opens: the example, not yet analysed.
Section draws the strips on the ground with their loads, the point A and, after Analyze, the lines of equal vertical stress in the vertical plane through A, with the stress at A in a box. Its foot notes say which lines are drawn (4.5, 7, 10, 15, 20, 30 and 50 kPa for the example) and that the blue line is the value at A.
Contours shows the same section as colour bands on a logarithmic scale, with a legend in kPa and as a percentage of q; values below the lowest band take its colour.
Profile plots Δσz against depth under A, with the depth of A marked.
3D shows the strips on the ground, the axes, the coordinates of A and, after Analyze, the stress on an element at A.
Before Analyze the Section view shows the model. The rings at the ends of a strip are its handles. The dimensions mark A.
Section before Analyze.
After Analyze the lines of equal stress appear. The blue line passes through A and the box gives Δσz = 4.470 kPa.
Section after Analyze.
The Contours view paints the field of Δσz. The strip stands on top of it; the section is drawn down to a little below the profile depth.
Contours.
The Profile view is the curve under A. For the example it rises to a maximum at about 4 m and then falls.
Profile.
The 3D view, with the element at A and the stress on it.
3D after Analyze.
The Show chips
Lines of equal stress and Dimensions sit above the Section and the Contours view. Before Analyze the lines chip is unchecked and greyed, with the note “The stress and the lines of equal stress appear after Analyze” (in the Contours view: “The colour contours of the stress appear after Analyze”). After Analyze the Section view has the lines on; the Contours view starts with them off, so you can put the lines on the colours. Dimensions switches the dimension marks of A on or off. Profile and 3D have no chips.
Contours with the lines of equal stress on.
Section with the lines switched off.
Dragging on the drawing
A strip, before Analyze: drag its body along the ground to move it (xL changes, the width stays); drag the ring at its left edge to change the width and the left edge together (the right edge stays); drag the ring at its right edge to change the width. A pull moves in steps of 5 cm; a width cannot go below 0.1 m. The same handles are on the strips of the 3D view.
Strips never overlap. If a pull would put a strip on another, it stops where the strips touch and the notice “That would put strip load 1 and strip load 2 on top of one another — the value is kept as it was.” appears.
The point A: drag the blue point in the Section or the Contours view, before and after Analyze. It moves in steps of 5 cm inside the frame of the drawing, never above 0.05 m depth. After Analyze the stress is recalculated at once for the new place.
The profile depth: drag the dashed line; it stops at 0.5 m at the least. It stays free after Analyze, like the point A.
After Analyze the strips are locked: their handles disappear and their boxes are greyed. Edit inputs unlocks them and clears the results.
Zoomed in.
The − and + buttons (or the mouse wheel over the drawing) zoom; one press of + gives 135%. Once zoomed, drag the background to move the view; the square button fits the drawing again (100%).
Entering the model, section by section
The Model tab is a stack of four sections. Model data and Triangular strip loads are open when the module starts; Next in a section closes it and opens the next, Back goes the other way (Depth profile is the last section and has only Back). Units are shown in the order SI · kN, SI · tf, US, equal ones merged: for example kPa · t/m² · ksf. Defaults are in SI.
Model data
The first section keeps the model. Every change is kept in this browser: after a reload the page opens on the model you last worked on, with the same values (and, if you had pressed Analyze, still analysed). Save to file… and Open file… move a model to a file; Quick Save… asks for a name (it proposes the model name, or “Stress in soil” and the date) and keeps that version in the list, where Load brings it back and Delete (after a confirmation) removes it; Reset to defaults returns to the example at once, without a question, and says so below the list. Model name is a free label.
Model data.
Triangular strip loads
The section has a New strip load form on top and a table of Strip loads below it. In the form choose the direction with the two buttons, type the three values and press + Add strip load; the strip goes into the table, the form returns to its starting values and a counter reads “2 of 8”. Press a row of the table to edit that strip in the form (the title reads “Editing · Strip load 2” and “Changes apply as they are entered”); Done returns to the new-strip form; Delete strip load or the × at the end of a row removes a strip. The table keeps its columns Strip load, Direction, Load, and Width and left edge.
Triangular strip loads: the form and the table.
Field
Unit, default, range
Direction of the load
choice · default Zero at left · Zero at left or Zero at right. The load is zero at that edge and rises to q at the other
Load per unit area at the high edge (q)
kPa · t/m² · ksf · default 100 (first strip and new form) · At least 0
Width of the strip (B)
m · ft · default 2 (first strip and new form) · At least 0.001 m
Position of the left edge (xL)
m · ft · default 0 for the first strip, 3 in the new-strip form · Any value; negative to the left of 0
The new-strip form always starts at q = 100, B = 2 and xL = 3, whichever strips are already in the table. Strips are numbered in the order you add them; the numbers appear on the drawing as q1, q2 and in the working.
Two strips; the second one is being edited.
Overlap
Two strips may meet at an edge (one ends at x = 2, the next begins at 2) but not overlap. The module refuses an overlapping strip at the moment you add it, when you type a value that would make two strips overlap (the value stays as it was) and when you drag one strip onto another. Each refusal is a short notice at the foot of the window.
The notice when a new strip would overlap strip 2.
Point A
The point A where the stress is wanted. x is on the same axis as the strips; for one strip with its left edge at 0 and zero load there, x is measured from the zero edge, negative on the far side of it. The depth z is measured down from the surface.
Point A.
Field
Unit, default, range
Horizontal position of the point A (x)
m · ft · default −1 · Any value
Depth of the point A (z)
m · ft · default 1.5 · At least 0.001 m
Depth profile
How deep the Profile view and the profile table go under A. The table has ten rows at equal steps down to zmax and the row of A itself. If zmax is smaller than z, the profile goes down to z instead.
Depth profile.
Field
Unit, default, range
Depth of the profile (zmax)
m · ft · default 8 · At least 0.1 m
Choosing the direction and the axis
There is one real choice, the direction of each strip, and one convention, the axis along which xL and x are measured.
Direction
The load
x of the equation
Zero at left
Zero at the left edge, q at the right edge
x = xA − xL: the distance of A from the left edge, positive to the right
Zero at right
Zero at the right edge, q at the left edge
x = xL + B − xA: the distance of A from the right edge, positive to the left
The module works this out for you and writes it in the Calculations tab. A strip that is zero at its right edge is the mirror image of the other: the example strip with A at x = −1 m gives 4.470 kPa, and the same strip turned about (zero at right, left edge at 0, B = 2 m) gives 4.470 kPa for A at x = 3 m, which stands 1 m beyond the zero edge on the other side.
Building a load from strips
Two triangles side by side make a roof: strip 1 zero at left over 0 to 2 m and strip 2 zero at right over 2 to 4 m (both q = 100 kPa), a load that peaks at 100 kPa over x = 2 m. A at x = 2 m, z = 1.5 m gives 59.03 kPa, each strip contributing 29.52 kPa. Strips that carry the same load in the same direction in a row make a sawtooth. The module adds whatever you describe; the strips do not have to be the same size or touch.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” Edit inputs unlocks the model. The Warnings (0) button beside it reports no warnings for the runs described here. The module stays on the Model tab; the answer is on Results.
The bar after Analyze.
The cards of the Results tab.
Card
Symbol
The example
Vertical stress increase at A
Δσz
4.470 kPa
Influence value (one strip only)
Δσz/q
0.04470
The cards are followed by a table of the values step by step and by Vertical stress increase under A: a table of z, Δσz and (for one strip) Δσz/q, with the row of A marked. In the example the profile runs 0.8 m: 1.334 kPa, 1.5 m (A): 4.470 kPa, 1.6 m: 4.905 kPa, 4.0 m: 8.936 kPa and 8.0 m: 6.734 kPa.
Several strips
With two or three strips the Results tab has a card for each strip, From strip load 1, From strip load 2, next to the total; with more strips only the total card is shown, and the working lists every strip. The influence value Δσz/q is given for one strip only, and the profile table has no such column for several.
Two strips: the total and each strip’s share.
The two-strip roof: A under the peak.
Moving A after Analyze
Change x or z in the Point A section, or drag A, and the results, the lines and the profile follow without pressing Analyze again; the strips stay locked. The unit system can be changed at any time. The boxes, the Results tab, the Calculations tab and the profile table are stated in the new units; the drawings and the strip table keep SI symbols and values, so read the converted values in the boxes and on the Results tab.
Using the numbers
Δσz is the increase of vertical stress caused by the strips, in the units of q. The module notes that the overburden pressure of the soil above A is not included.
Δσz/q is the influence value for one strip: multiply any q by it for the same geometry. In the example it is 0.04470 for x/B and z/B of −0.5 and 0.75.
The load is zero at one edge, so the strip’s total load per metre of length is qB/2.
Δσz is vertical stress only; combine it with the settlement or bearing check you need.
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. For one strip there are three:
Angles at A — α = tan⁻¹(x/z) − tan⁻¹[(x − B)/z] and δ = tan⁻¹[(x − B)/z], in degrees, with 2δ. For a strip with its left edge away from 0, or zero at right, the step first says how x is found.
Stress at A — the equation, 2x/B and 2z/B, then Δσz/q with the numbers put in and Δσz.
Stress increase against depth — zmax and the table under A, the row of A marked.
The start of the Calculations tab.
With several strips each strip has a step of its own (Stress from strip load 1, …) with its angles and its share, followed by a step Superposition that adds them, and then the profile.
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 the buttons 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 three drawings (the Section with the lines of equal stress, the Profile and the 3D view), the working of every step and the warnings, if any. Before Analyze the tab says “Press Analyze on the Model tab to calculate the stress at the point. The cover entered above is kept.”
The Report tab: the cover and the two download buttons.
The Theory tab
The Theory tab has five sections, each opening on a click (with Open all and Close all): Scope and assumptions; Derivation; Chart and table (a chart of Δσz/q against 2x/B for several depths and a table of its values); 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
Triangular strip loads
Direction
Zero at the left edge
q, B, xL
100 kPa, 2 m, 0
Point A
x, z
−1 m, 1.5 m
Depth profile
zmax
8 m
Step 1: the angles
x = −1 m is measured from the zero edge (the left edge), so x/z = −0.667 and (x − B)/z = −2. α = tan⁻¹(−0.667) − tan⁻¹(−2) = −33.69° + 63.43° = 29.74° = 0.5191 rad, and δ = tan⁻¹(−2) = −63.43°, so 2δ = −126.87°.
This is the number in the Results card, in step 2 of the Calculations tab and in the box beside A in the Section view. The largest value in the profile table is 8.936 kPa at 4 m; the answer at A (1.5 m) is the marked row.
The same strip in SI · tf
Switch to SI · tf. The module converts the model: q = 10.1972 t/m². The answer is Δσz = 0.4558 t/m², the same stress (Δσz/q is unchanged). By hand 4.470 / 9.80665 = 0.4558.
The same strip in US units
Switch to US: q = 2.08854 ksf, B = 6.56168 ft, x = −3.28084 ft, z = 4.92126 ft. The answer is Δσz = 0.09336 ksf; by hand 4.470 kPa / 47.8803 = 0.09336 ksf. The profile rows are in feet: 4.921 ft (A) gives 0.09336, 13.12 ft gives 0.1866.
A second strip
Press + Add strip load with the new-strip form as it stands (zero at left, q = 100, B = 2, xL = 3). The second strip stands 3 to 5 m and A is 4 m from its zero edge on the far side (x = −1 − 3 = −4): its share is 0.2454 kPa by hand, and the module gives a total of 4.715 kPa = 4.470 + 0.245.
How each input changes the answer
Each row changes one thing in the example (one strip, zero at left, q = 100 kPa, B = 2 m, left edge 0, A at x = −1 m, z = 1.5 m) and runs the module again. The numbers are the ones the module gives, and every row has been checked by hand with the equation above.
The example with one change at a time
Change
Δσz (kPa)
Δσz/q
The example The reference: q = 100, B = 2, left edge 0, zero at left, A at (x = −1, z = 1.5)
4.470
0.04470
q = 50 kPa
2.235
0.04470
q = 200 kPa The stress is proportional to q; Δσz/q is unchanged
8.940
0.04470
B = 1 m
4.479
0.04479
B = 4 m
3.259
0.03259
B = 8 m A is 1 m outside the zero edge of a wide strip
1.908
0.01908
Left edge at xL = 1 m A is now 2 m from the zero edge
1.402
0.01402
Left edge at xL = −3 m A stands at the high edge
29.52
0.2952
Zero at right A is now 1 m beyond the high edge
10.10
0.1010
Zero at right, A at x = 3 m The mirror of the example
4.470
0.04470
A at x = 0 (the zero edge)
15.28
0.1528
A at x = 1 m (under the middle)
33.41
0.3341
A at x = 2 m (the high edge)
29.52
0.2952
A at x = 3 m
10.10
0.1010
A at x = 5 m
0.9701
0.009701
A at x = −3 m
0.5393
0.005393
A at x = −5 m
0.1263
0.001263
A at z = 0.5 m
0.4110
0.004110
A at z = 3 m
8.536
0.08536
A at z = 6 m
7.982
0.07982
A at z = 20 m
3.095
0.03095
zmax = 4 m Only the table and the profile change
4.470
0.04470
A second strip (zero at left, 100 kPa, 2 m, left edge 3 m) No influence value for several strips
4.715
—
Second strip zero at right, left edge 2 m A at (−1, 1.5)
5.440
—
What the table shows:
The load acts in proportion: 50, 100 and 200 kPa give 2.235, 4.470 and 8.940 kPa.
The position of A changes the answer by a large factor: from 0.1263 to 33.41 kPa across the strip at 1.5 m depth. Across the strip at 1.5 m depth the stress is 0.1263 kPa at x = −5 m, 4.470 at −1 m, 15.28 at the zero edge, 33.41 under the middle of the strip (x = 1 m, the largest of these runs), 29.52 at the high edge and 10.10 and 0.9701 kPa at 1 and 3 m beyond it.
Depth first raises the stress under a point beside the strip and then lowers it: 0.4110, 4.470, 8.536, 7.982 and 3.095 kPa at 0.5, 1.5, 3, 6 and 20 m.
The width changes the answer moderately for a point beside the strip: 4.479, 4.470, 3.259 and 1.908 kPa for B = 1, 2, 4 and 8 m with the same q.
Direction and the left edge only change where the zero and the high edge are: the strip turned about, with A at the mirror place, gives the same stress.
Further strips add their share; zmax changes only what is shown of the profile.
Messages: when the module stops, and cautions
The module checks its inputs before it calculates. A problem stops the analysis and nothing is calculated: a window titled Cannot analyse — 1 input error (or “n input errors”) lists each one, and a value below its limit is also marked in red before you press Analyze.
A stop: B = 0.
When
The message
q empty
“Load per unit area of strip 1 at its high edge (q1) is missing.”
q below 0
“Load per unit area of strip 1 at its high edge (q1) must be at least 0 kPa.”
B 0 or negative
“Width of strip 1 (B1) must be at least 0.001 m.”
B empty
“Width of strip 1 (B1) is missing.”
z 0 or negative
“Depth of the point A (z) must be at least 0.001 m.”
z empty
“Depth of the point A (z) is missing.”
x empty
“Horizontal position of the point A (x) is missing.”
zmax 0
“Depth of the profile (zmax) must be at least 0.1 m.”
zmax empty
“Depth of the profile (zmax) is missing.”
Several errors are listed together; for strip 2 and on the names read q2, B2 and so on. After the stop the Results tab says “Press Analyze on the Model tab to calculate the stress at the point.”
Notices about the strips
“The new strip load would lie on strip load 2 — they cannot overlap. Give it another position or width.” when you add a strip that overlaps another.
“That would put strip load 1 and strip load 2 on top of one another — the value is kept as it was.” when a typed value or a drag would do so.
Two strips that only touch at an edge are accepted: with strip 2 at xL = 2 m after strip 1 (0 to 2 m) the module gives 5.009 kPa for the example point, 47.97 kPa for A at x = 2 m, z = 1 m.
Cautions
With q = 0 the answer is 0 kPa, but the influence value cannot be formed (the card shows a dash and the Calculations tab prints NaN for it); use q > 0 for a working. The Warnings (0) button stays at 0 in the runs described here; if a warning appears the button reopens it and the report prints it.
Good practice and common mistakes
q is the value at the high edge, not the average: the average over the strip is q/2.
Check the direction. Zero at left and zero at right differ only by a mirror image; a wrong choice puts the heavy edge on the wrong side of A. Look at the drawing before Analyze.
Choose one axis for all strips and for A.xL and x are on the same axis; the drawing labels x − x1 when the first strip does not sit at 0.
Strips may touch, not overlap. For a load that is a trapezoid, a roof or a sawtooth, use several strips side by side.
Read the profile as well as the number. Beside a strip the stress grows with depth first and then falls; a single point can hide that.
Add the overburden yourself. The module gives the increase only.
Use the lines or the colours to find where the stress matters. Drag A over the Section or Contours view and read the box.
After a unit change read the answer from the Results tab; the boxes show the converted values with more decimals.
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 the analysis.
Frequently asked questions
What does the Triangular strip load module calculate?
The vertical stress increase Δσz below one or several flexible strip loads of infinite length whose load rises linearly from zero at one edge to q at the other, for an elastic half-space in plane strain. For the example it gives Δσz = 4.470 kPa (Δσz/q = 0.04470).
What is q?
The load per unit area at the high edge of the strip, the edge that is not zero. The average load over the width is half of it.
How do I put the zero at the right edge?
In the New strip load form (or when editing a strip) choose Zero at right. The strip is then the mirror image: the module measures x from the right edge toward the left.
Can the point A be beside the strip, or under it?
Both. x can be any value on the axis; the equation holds for every position, beside, under or beyond the strip. Depth z must be at least 0.001 m.
How many strips can I use?
Up to eight. Each has its own q, B, xL and direction. They may meet at an edge but not overlap, and their stresses add.
What happens if two strips overlap?
The module refuses: when you add the strip, type the value or drag it, a notice appears and the value is kept as it was. A pull stops where the strips touch.
Can I drag things on the drawing?
Yes. Before Analyze you can drag a strip, and the rings at its edges to change its width, in the Section and 3D views (5 cm steps, width at least 0.1 m). The point A and the dashed profile-depth line can be dragged before and after Analyze; after Analyze the stress at A is recalculated at once.
Does the result include the weight of the soil?
No. The module gives the increase caused by the strips only; the overburden pressure of the soil above A is not included.
What do the lines of equal stress and the colours show?
The same field of Δσz in the vertical plane through A: lines for 4.5, 7, 10, 15, 20, 30 and 50 kPa in the example in the Section view, colour bands on a logarithmic scale in the Contours view. The blue line is the value at A.
What is the depth of the profile for?
It sets how far down the Profile view and the table under A go. It does not change the stress at A. If it is smaller than z, the profile goes down to z.
Does the module remember my inputs?
Yes. Every change is kept in this browser, and the module reopens on the model you last worked on (analysed again if it was analysed). Reset to defaults returns to the example; Save to file and Quick Save keep versions.
What units does it use, and what does the report contain?
SI · kN, SI · tf or US, switchable at any time, even after Analyze (stress in kPa, t/m² or ksf; lengths in m or ft). The boxes and the tabs follow; the drawings and the strip table keep SI symbols and values. The report has a cover and six parts, with the Section, Profile and 3D drawings and the working of every step, as a PDF or a Word file after Analyze.
Calculate the vertical stress increase at a point below one or several long embankments of trapezoidal cross section, under the fill or beside it: the stress at A, the lines of equal stress, colour contours, the profile with depth and a 3D view. This manual walks through the screen, dragging, every input, the results and the report, with a worked example you can reproduce.
Embankment gives the increase in vertical stress at a point of the ground below one or several long embankments of trapezoidal cross section: a road or railway fill, a levee, a long stockpile. The fill is treated as a flexible load on the ground surface, equal to its weight per unit area q0 = γH under the crest and falling in a straight line to zero at the two toes. The ground is a homogeneous, isotropic, elastic half-space and the problem is plane strain: the stress is the same in every cross section along the embankment. You choose the point A by its horizontal position x and its depth z, under the fill or beside it, and the module gives Δσz there, the lines of equal stress, the colour contours, the profile with depth under A and a 3D view.
The module splits each trapezoid into a uniform strip under the crest and two strips that rise or fall linearly under the slopes, finds the stress of each with the theory of elasticity (Boussinesq, integrated across the strip) and adds the three. Under the crest the sum equals Osterberg’s (1957) influence-value form q0(I2 of the left half + I2 of the right half), and the module works that form out as a check. With several embankments the stress of each is found with its own distance from A and the stresses are added.
What you can describe
One to eight embankments, each with its own unit weight γ, height H, crest width Bc, horizontal widths L1 and L2 of the left and right slopes, and position xc of the centre line of its crest along the x axis.
The point A: its horizontal position x and its depth z below the ground surface.
The depth of the profilezmax, down to which the Profile view and the table under A run.
The result is the increase caused by the embankments only; the overburden pressure of the foundation soil is not included, so add it where your design needs the total stress. The module draws the section with the lines of equal stress, the colour contours, the profile and a 3D view, writes out the working and prepares a PDF or Word report.
Which module for which load
A fill with a flat top and sloping sides, or a triangular one (Bc = 0): this module. Slopes may differ, and one side may be vertical.
The unit weight γ and the height H The fill material and the section
They give the load q0 = γH under the crest. Zero or more.
The crest width Bc and the slope widths L1, L2 The section of the fill
Slope widths are horizontal distances from the crest edge to the toe, not slope lengths. Zero or more; at least one of the three must be above zero.
The position xc The layout
The place of the centre line of the crest along the x axis. With one embankment you can leave it at 0 and measure x from the centre line.
The point A: x and z Where you want the stress
x on the same axis as the crests (negative toward the left toe); z at least 0.001 m below the surface.
Quick start in seven steps
Open the module and choose the units
Open Embankment from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the embankment
In Embankments the first embankment is already in the table. Press its row to change γ, H, Bc, L1, L2 and xc; or drag the fill and its four rings on the Section drawing. For another embankment fill the New embankment form and press + Add embankment.
Place the point A
In Point A type x and z, or drag the blue point on the Section drawing.
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line at the foot of the Section drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the embankments lock; the point A and the profile depth stay free.
Read the answer
The Results tab gives Δσz at A, the influence value Δσz/q0 for one embankment, the part from each embankment for two or three, and a table of the stress against depth under A. The lines of equal stress are on the Section drawing, the colour bands in Contours.
Move A and take the report
Drag A or change x and z: the stress there is calculated at once. The Calculations tab writes out the working; the Report tab saves a PDF or a Word file.
The screen: section, contours, profile and 3D
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 is the cross section of the embankment or embankments on the ground, with the load q0 = γH written above the fill, the point A, the dimensions x and z and the dashed line of the profile depth (the dashed line sits at the bottom edge of the frame when zmax is deeper than the frame). After Analyze it adds the lines of equal stress and a label with Δσz at A.
Contours is the same section filled with colour bands of Δσz on a logarithmic scale, with a legend in kPa and in per cent of q0 (of the largest q0 when there are several embankments).
Profile is the curve of Δσz against depth under A, with a marker at the depth of A.
3D is an oblique view of the fill on the ground, the axes x, y and z, and a small element at A with the stress on it.
Before Analyze the drawings follow the inputs and say “The stress and the lines of equal stress appear after Analyze” (Section), “The colour contours of the stress appear after Analyze” (Contours) or “The profile appears after Analyze”. The four rings on the fill are the handles you can pull.
Section before Analyze: the rings are the handles.
Section after Analyze: the lines of equal stress; the line through A is blue.
Contours after Analyze.
Profile after Analyze.
3D after Analyze.
The Show chips
Above the Section and the Contours drawings a bar has two chips: Lines of equal stress and Dimensions. In the Section view the lines chip is off and greyed out until you press Analyze, and then turns on; in Contours it is off by default, so that the lines can be laid over the colours. Dimensions is on by default in both. The Profile and 3D views have no chips.
Dragging on the drawing
Before Analyze two hints stand under the Section drawing: “Drag a load to move it, the rings at its edges to resize it. Drag the dashed line to set the depth of the profile (zmax).” and “Drag the point A to place it. The stress and the lines of equal stress appear after Analyze.” After Analyze they become “Drag the dashed line to set the depth of the profile (zmax).” and a line saying that the blue line is the value at the point A and that you drag A to read the stress elsewhere. What each pull does, as the module runs it:
The fill. Drag its body sideways to move it: xc follows the pointer in steps of 5 cm; a vertical pull changes nothing. Pulling the example 3 m to the right gives xc = 3.
A toe ring. Drag the ring at a toe to change that slope: the crest stays where it is and L1 or L2 changes. Pulling the right toe 4 m to the right changes L2 from 14 to 18; pulling it far to the left stops at L2 = 0, a vertical side. A toe can be pulled well beyond the visible part of the section (the drawing rescales); the only stop is L = 0.
A crest ring. Drag a crest corner to change the crest width with the toe on that side held where it is: the slope takes up what the crest gives. Pulling the left crest corner 1 m to the left gives Bc = 6, L1 = 13 and xc = −0.5; pulling the right one 2 m to the right then gives Bc = 8, L2 = 12 and xc = 0.5. The crest width stops at 0, a triangular fill; there the two crest rings nearly coincide and the right one takes the pull (+1 m gave Bc = 1 with L2 reduced by 1; a pull to the left does nothing).
Two embankments never overlap. A pull that would put the ground one covers, toe to toe, on another’s stops at the last free place and a message says “That would put embankment 1 and embankment 2 on top of one another — the value is kept as it was.” Toes that meet are allowed.
The point A. Drag the blue point: x and z follow in steps of 5 cm. The depth stops at 0.05 m at the top and at the foot of the drawn frame at the bottom (about 26 m in the example; the frame grows when A is deep), and x at the sides of the frame, which grows with the embankments. With the example, pulling A far to the upper left gave x = −18.95 m and z = 0.05 m.
The dashed line. Drag it up or down to set zmax; it stops at 0.5 m.
After Analyze the embankments are locked and their rings disappear: pulling the fill changes nothing. The point A and the dashed line stay free, and so do the boxes x, z and zmax.
The 3D view has its own hint, “Drag a load over the ground to move it, the rings at its edges to resize it.” These pulls, like those of the Section view, work before Analyze only.
Zoom
Each view has −, a percentage and + at the lower right, and a button that fits the whole drawing in the pane. Zoom is a property of the drawing only: it is not saved with the model and does not change the report.
Entering the model, section by section
The Model tab has four sections: Model data, Embankments, Point A and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below the last one. 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. Defaults are shown in SI; in US the boxes show the converted values, for example 111.403 pcf for 17.5 kN/m³.
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.
Model data.
Embankments
The section is a form over a table. The New embankment form takes γ, H, Bc, L1, L2 and xc; + Add embankment puts the embankment into the Embankments table, and a counter beside the button shows how many of the eight there are. The table lists each embankment with its soil and height, its crest and slopes, and its position. Press a row to change that embankment in the form (the form then reads Editing · Embankment 1 and offers Done and Delete embankment; the note says that changes apply as they are entered), or press × to take one out. The first embankment is already in the table when the module opens, and Delete embankment and × are greyed out while it is the only one. Below the table a note describes the load and the rule that embankments may not stand on one another.
Embankments: the form over the table.
Embankment fields (the form, and each row of the table)
Field
What to enter
Unit weight of the embankment soil γ kN/m³ · t/m³ · pcf · default 17.5 (first) · 17.5 (form)
Zero or more. With the height it gives q0.
Height of the embankment H m · ft · default 7 (first) · 3 (form)
Zero or more. A height of 0 is accepted and gives 0 kPa.
Crest width Bc m · ft · default 5 (first) · 4 (form)
Zero or more. 0 gives a triangular embankment.
Horizontal width of the left slope L1 m · ft · default 14 (first) · 6 (form)
Zero or more. The horizontal distance from the left edge of the crest to the left toe; 0 is a vertical side.
Horizontal width of the right slope L2 m · ft · default 14 (first) · 6 (form)
Zero or more. The same on the right.
Position of the centre line of the crest xc m · ft · default 0 (first) · 30 (form)
Along the x axis; any value. The form opens with 30.
A row of the table pressed: the embankment is edited in the form.
The embankments may not lie on top of one another: the ground each covers, from toe to toe, may not be shared, though two toes may meet. Adding or editing an embankment that would overlap another is refused (see Messages). The embankments are numbered in the order you add them, and the numbers name the results “From embankment 1”, “From embankment 2” and so on.
Two embankments in the table.
Point A
The two coordinates of the point where you want the stress. A is under the ground surface at the horizontal position x and the depth z; it may be under an embankment or beside it. The drawing shows the same distances as dimensions.
Point A.
Point A fields
Field
What to enter
Horizontal position of the point A x m · ft · default 0
On the same axis as the crests; for one embankment centred at 0 it is the distance from the centre line of the crest, negative toward the left toe. Any value.
Depth of the point A z m · ft · default 5
At least 0.001 m below the ground surface.
Depth profile
The depth down to which the Profile view and the table of the stress under A run.
Depth profile.
Depth profile field
Field
What to enter
Depth of the profile zmax m · ft · default 30
At least 0.1 m. The Profile view and the table under A go from the surface down to this depth, or to the depth of A if that is deeper.
The shape of the fill and several embankments
There is no method to choose: the module has one solution. What you decide is the shape of each fill and how many fills there are, from 1 to 8.
The shape of one embankment
Crest and slopes are independent.L1 and L2 can differ; a side with a width of 0 is vertical. With L1 = 0 the example gives 89.11 kPa at A, the same as with L2 = 0 (89.11 kPa), because A is on the centre line.
Bc = 0 gives a triangular embankment. The example with Bc = 0 gives 95.75 kPa (influence value 0.7816) against 110.9 kPa with the 5 m crest.
Slope widths are horizontal. For a slope of 1 vertical to 2 horizontal on a 7 m fill, L1 = 14 m.
The load is q0 = γH at the crest. The module treats the fill as a load on the ground; it does not model the stiffness of the fill.
Several embankments
The stress of each embankment is found as if it were alone, with the distance x − xc of A from the centre line of its crest, and the stresses are added. This works because the ground is elastic. Each embankment has its own γ, H, crest and slopes, so a main fill and a lower berm or a second road can be put in one model.
With the example and a second embankment from the form (17.5 kN/m³, 3 m high, crest 4 m, slopes 6 m, centred at 30 m), A at x = 0, z = 5 m receives 110.9 kPa from the first and only 0.05520 kPa from the second: 110.9 kPa in all. A is 22 m from the near toe of the second embankment, so it barely feels it. Moving A to x = 19 m, between the two, gives 9.440 kPa; at x = 30 m, under the second crest, 41.25 kPa. A second embankment equal to the first, centred at 33 m so that the toes meet at 16.5 m, adds 0.2375 kPa and gives 111.1 kPa.
Two embankments: the lines of equal stress of the sum.
The same two embankments in the Contours view.
Which should I use?
One embankment or several? One when the fill is one trapezoid; several when two fills stand apart, or when a berm or a stepped fill is better described by two or three trapezoids that meet at their toes.
A band of uniform load with vertical sides is a strip, not an embankment: use Vertical strip load.
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” A message appears briefly, “Analysed — see the Results tab and the Calculations.” The module stays on the Model tab: open Results for the answer. The Section view now shows the lines of equal stress.
The bar after Analyze.
The cards of the example.
Card
Symbol
The example
Vertical stress increase at A
Δσz
110.9 kPa
Influence value (one embankment)
Δσz/q0
0.9051
Load under the crest (one embankment)
q0
122.5 kPa
From embankment k (two or three)
Δσz(k)
part of the total from embankment k
Below the cards a Values, step by step table lists the steps of the Calculations tab with their values, and the Vertical stress increase under A table gives z, Δσz and, for one embankment, Δσz/q0 at ten equal steps of zmax/10 down to zmax (the first row is at zmax/10, 3.000 m in the example), with the row of A inserted and highlighted: 11 rows unless A falls on a step. With several embankments the ratio column is left out. With two or three embankments each part has its own card; with more, the parts are listed one per line under the total card.
The depth table of the example.
Two embankments: the total and the part from each.
Using the numbers
The result is an increase, not the total stress: add the overburden pressure of the soil above A for the total vertical stress.
The influence value Δσz/q0 is the part of the pressure under the crest that reaches A. It is 1 at the surface under the crest and falls with depth and with distance from the fill.
Values are rounded where they are drawn, so the last digit of a figure on the drawing can differ by one from the card.
Change x or z after Analyze and everything follows at once; to change an embankment, press Edit inputs, which unlocks the embankments and clears the results.
Changing the unit system after Analyze re-states the cards and the report without a new run. The drawings keep their labels in m and kPa.
Warnings and notes
The bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” Every run of this manual that the module accepted, from the default to eight embankments, gave no warning.
The Warnings window.
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. For one embankment:
Load under the crest — q0 = γH with your numbers.
Split into three strip loads — the trapezoid as a uniform strip under the crest plus a rising and a falling strip under the slopes, with the toes and crest edges measured from the centre line of the crest.
Crest strip, Left slope and Right slope — the stress of each part with its ratio to q0 (the slope strips only when their width is above zero).
Sum — the three parts added, and, when A is under the crest, the check with Osterberg’s form: the I2 of the left and right halves and q0(I2 left + I2 right).
Stress increase against depth — the same sum down the vertical line through A, as the table of the Results tab, with the row of A marked.
With several embankments there is one step for each, “Stress from embankment k”, with its own distance x − xc, its toes and the three parts, then a Superposition step that adds them, then the depth table.
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, 4 Results, 5 Summary and 6 References. It carries three drawings, the Section with the lines of equal stress, the Profile and the 3D view, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has five sections: Scope and assumptions, Osterberg’s solution, Superposition of strips, Worked example and References. Each opens when you click its title (Open all and Close all open or close them together), 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
Embankments
Embankment 1
γ = 17.5 kN/m³; H = 7 m; Bc = 5 m; L1 = L2 = 14 m; xc = 0
Point A
Position
x = 0; z = 5 m
Depth profile
Depth
zmax = 30 m
Step 1: the load
q0 = γH = 17.5 × 7 = 122.5 kPa. The crest edges are at ±2.5 m and the toes at ±16.5 m from the centre line.
Step 2: the three parts
A is on the centre line, 5 m deep. The crest strip (width 5 m, 2x/B = 0, 2z/B = 2) gives Δσz/q0 = 0.5498, so 67.35 kPa. Each slope strip gives Δσz/q0 = 0.1776, so 21.76 kPa, the same for both sides because A is on the centre line.
A is under the crest, so the Calculations tab also works out Osterberg’s form for the two halves: for each half the width of crest on that side is 2.5 m and the slope is 14 m, giving I2 = 0.4525 for each, and q0(I2 + I2) = 122.5 × 0.9051 = 110.87 kPa, the same value. A separate calculation that adds up Boussinesq line loads across the whole trapezoid gives 110.87 kPa as well. The depth table gives 118.2 kPa at 3 m, 110.9 kPa at 5 m and 45.06 kPa at 30 m.
Other unit systems
The same fill in SI · tf, with γ = 1.75 t/m³ (the other values unchanged), has q0 = 12.25 t/m² and gives 11.09 t/m² and the influence value 0.9051. In US the default example, a 2.558 ksf load, gives 2.316 ksf: 110.87 kPa is 2.316 ksf. A hand check with round US numbers: γ = 110 pcf, H = 20 ft, Bc = 15 ft, L1 = L2 = 40 ft, A at x = 0, z = 15 ft gives q0 = 2.2 ksf; an independent sum of line loads gives 1.982 ksf, and the module gives 1.982 ksf (influence value 0.9009).
How each input changes the answer
Each row changes one thing in the example (one embankment of γ = 17.5 kN/m³, H = 7 m, Bc = 5 m, L1 = L2 = 14 m, xc = 0, A at x = 0, z = 5 m, q0 = 122.5 kPa) and runs the module again, in SI · kN. The numbers are the ones the module gives, and a separate calculation that adds up Boussinesq line loads across the trapezoid agrees with every row.
The example with one change at a time
Change
Δσz (kPa)
Δσz/q0
The example
110.9
0.9051
γ = 20 kN/m³ More weight, more stress; the ratio is unchanged
126.7
0.9051
γ = 10 kN/m³ Proportional to γ
63.36
0.9051
H = 3.5 m Half the height, half the stress, with the slopes still 14 m wide
55.44
0.9051
H = 14 m Twice the height, twice the stress
221.7
0.9051
Bc = 0 (triangular) Less load near the centre line
95.75
0.7816
Bc = 10 m A wider crest puts more load over A
117.1
0.9562
Bc = 20 m The ratio approaches 1
120.9
0.9867
L1 = 0 (vertical left side) A loses the load of the left slope
89.11
0.7275
L1 = 28 m A longer slope adds a little
113.6
0.9277
L2 = 7 m A steeper right slope lowers the stress
106.5
0.8691
xc = 5 m A is 5 m from the centre line of the crest
93.80
0.7657
xc = −5 m The same as xc = 5: the section is symmetric
93.80
0.7657
x = 2.5 m (under the crest edge) Little lower than on the centre line
106.4
0.8689
x = 7 m (under a slope) The stress falls under the slope
79.91
0.6523
x = 16.5 m (under the toe) About a tenth of q0
13.17
0.1075
x = 25 m (8.5 m beyond the toe) Close to nothing beside the fill
1.032
0.008420
z = 0.01 m At the surface under the crest the stress equals q0
122.5
1.000
z = 1 m Near the surface the stress approaches q0
122.2
0.9978
z = 2.5 m The stress falls slowly under a wide fill
119.6
0.9765
z = 10 m It falls with depth
90.88
0.7419
z = 20 m Still half of q0 at 20 m
61.67
0.5034
Two other changes together: slopes of 7 m on both sides give 102.1 kPa; a very wide flat fill (Bc = 40 m, slopes 1 m) gives 121.8 kPa, an influence value of 0.9942.
The pattern follows the solution. The stress is proportional to γ and H together (q0) when the widths are fixed; it is highest under the crest and falls with depth and with distance from the fill; and with z near zero it equals q0 under the crest and tends to zero beyond the toes. Depth is not a simple fraction of q0 because the slopes keep adding load as the point goes deeper. The depth of the profile zmax changes the table and the Profile view but not the stress at A: 110.9 kPa with zmax = 15 m, and also 110.9 kPa with zmax = 2 m, a depth the module extends to the depth of A.
Messages: when the module stops, and what it notes
The module checks the numbers it needs before it calculates. When an input cannot be used, press Analyze and a window titled Cannot analyse — 1 input error (or 2 input errors …) lists each message and the box of an out-of-range value is marked in red (an empty box is only named in the message); nothing is calculated.
The window Analyze opens when the depth is zero.
When
Message and what to do
γ is negative
“Unit weight of the embankment soil of embankment 1 (γ1) must be at least 0 kN/m³.” Enter zero or more.
H is negative
“Height of embankment 1 (H1) must be at least 0 m.” Enter zero or more.
Bc is negative
“Crest width of embankment 1 (Bc,1) must be at least 0 m.” Enter zero or more; L1 and L2 give the same kind of message.
A box is empty
“Height of embankment 1 (H1) is missing.” Fill the box (the message names the box). An empty x gives “Horizontal position of the point A (x) is missing.”.
z is 0
“Depth of the point A (z) must be at least 0.001 m.” Enter a depth of at least 0.001 m.
zmax is 0
“Depth of the profile (zmax) must be at least 0.1 m.” Enter a depth of at least 0.1 m.
Bc, L1 and L2 are all 0
A second kind of window, “Cannot analyse — a condition of the solution is not met”: “The embankment has no width. The crest and both slopes are zero. Enter the crest width or the slope widths.” Nothing is calculated.
A new embankment would overlap another
A message appears briefly: “The new embankment would lie on embankment 1 — they cannot overlap. Give it another position or width.” The embankment is not added.
The messages use the numbered names of the embankments, such as “embankment 1” and “Bc,1”, even when there is only one, and the limits in the units of the unit system chosen. When you type a position or width into an embankment already in the table and it would put two on one another, a message appears briefly: “That would put embankment 1 and embankment 2 on top of one another — the value is kept as it was.” and the value stays as it was. An embankment whose toe just meets the toe of another (a second one centred at 24.5 m in the example) is accepted. At eight embankments + Add embankment is greyed out and the counter reads 8 of 8.
Notes and warnings
With every input inside its limits the module calculates, and the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” A height of 0 is accepted and gives 0 kPa. A crest of 0.001 m gives 95.75 kPa, the same as a triangular fill. Large values are accepted: γ = 1,000,000 kN/m³ with H = 1000 m gives 905,098,248 kPa at A.
Good practice and common mistakes
Use the module for a long fill. Plane strain means the fill is long compared with the depths you ask about; for a short fill use Rectangle or Influence chart.
Enter horizontal slope widths. A 1:2 slope on a 7 m fill is L = 14 m, not the 15.65 m length of the slope.
Take H from the ground surface to the crest. The load q0 = γH is the weight of the whole fill; if part of the fill sits in an excavation, enter the net height that loads the ground.
Mind the axis.x and every xc are on the same axis; a point to the left of a symmetric fill gives the same as one the same distance to the right.
Do not overlap embankments. They may meet at toes but not share ground; describe a stepped fill as embankments that meet.
Add the overburden. The result is the stress increase; for the total vertical stress add the soil weight above A.
Read the profile with the width of the fill in mind. Under the centre line of the example the stress is still 50 per cent of q0 at 20 m, because the fill is 33 m wide at its base.
Place A with the drawing, then type exact values. Dragging snaps to 5 cm; type the coordinates when you need a given place.
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 stress below an embankment calculated?
The module splits the trapezoid into a uniform strip under the crest and two strips that rise linearly from zero at the toes to q0 at the crest edges, finds the vertical stress of each with the theory of elasticity and adds them. For the module’s example it gives 110.9 kPa at A, and under the crest the same value follows from Osterberg’s influence-value form.
What is the load under the crest?
q0 = γH, the unit weight times the height of the fill: 17.5 × 7 = 122.5 kPa for the example. It acts over the crest and falls linearly to zero at each toe.
What is the influence value Δσz/q0?
It is the stress increase divided by q0, the part of the load under the crest that reaches A. It is 1 at the surface under the crest. The module gives it for one embankment; with several it gives the stress of each.
Does the result include the weight of the soil?
No. It is the increase caused by the embankments only. Add the overburden pressure of the soil above A for the total vertical stress.
Can I describe a triangular embankment or one with a vertical side?
Yes. A crest width of 0 gives a triangular fill, and a slope width of 0 gives a vertical side. At least one of the crest and the two slopes must have a width above zero, or the module stops with “The embankment has no width.”
How many embankments can I add, and can they overlap?
Up to eight, each with its own γ, H, Bc, L1, L2 and xc. They may meet at their toes but not share ground; the module refuses an embankment that would lie on another. The stresses add.
Can I drag the embankment and the point on the drawing?
Yes. Drag the body to move it, a toe ring to change a slope, a crest ring to change the crest width with the toe held, the blue point to place A and the dashed line to set the depth of the profile; they move in 5 cm steps. After Analyze the embankments are locked, but A and the dashed line stay free.
What do the lines of equal stress show?
Each line joins the points of the section with the same Δσz, labelled in kPa; the line through A is blue. Switch them on and off with the Lines of equal stress chip. The Contours view shows the same stress as colour bands on a logarithmic scale.
What values does the module accept?
γ, H, Bc, L1 and L2 zero or more, z at least 0.001 m, zmax at least 0.1 m, and x and xc any value. Anything else stops the analysis with a message that names the value.
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. Stresses are in kPa, t/m² or ksf, unit weight in kN/m³, t/m³ or pcf, and lengths in m or ft. The default example gives 110.9 kPa, or 2.316 ksf; the same fill typed as γ = 1.75 t/m³ gives 11.09 t/m².
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, profile 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. Your values are always kept in the browser, and the page opens again on the model you last worked on; the analysed state is kept as long as you have not changed x, z or zmax after Analyze. Reset to defaults brings back the example; Save to file and Quick Save keep copies.
Calculate the vertical stress increase at any point below or beside one or several flexible circular loads: the stress at A, the lines of equal stress, colour contours, the profile with depth and a 3D view. This manual walks through the screen, dragging, every input, the results and the report, with a worked example you can reproduce.
Circle, any point gives the increase in vertical stress at any point of the ground below or beside one or several flexible circular areas, each carrying a uniform pressure q on the ground surface: a tank, a silo or a round footing. The ground is taken as a homogeneous, isotropic, elastic half-space. You choose the point A by its position x, y on the ground surface and its depth z, and the module gives Δσz there, the lines of equal stress, the colour contours, the profile with depth under A and a 3D view.
For one circle of radius R the stress at depth z and horizontal distance r from its centre is the integral of Boussinesq’s point-load solution over the loaded area. Ahlvin and Ulery (1962) tabulated it in two parts, A′ and B′, which depend on z/R and r/R only; the module evaluates the integral itself instead of reading a table:
Seen from the point below A, the ray in the direction φ crosses the load between ρ1 and ρ2 (ρ1 = 0 when A is under the load); along each ray the integral is closed, and the sum over φ is done numerically. Directly under the centre (r = 0) it reduces to a closed form:
Δσz = q {1 − 1/[(R/z)² + 1]3/2}
The module is not limited to the axis: under the circle off its centre, under its edge and beside it it gives the stress at the point you choose, and the result is also given as the influence valueA′ + B′ = Δσz/q. With several circles the stress of each is found with its own distance from A, and the stresses are added.
What you can describe
One to eight circular loads, each with its own load per unit area q, radius R and the position (xL, yL) of its centre on the ground.
The point A: its position x, y on the ground surface (the same coordinates as the centres) and its depth z.
The depth of the profilezmax, down to which the Profile view and the table under A run.
The result is the increase caused by the circles only; the overburden pressure of the soil above A is not included, so add it to the answer where your design needs the total stress. The module draws the section with the lines of equal stress, colour contours, the profile and a 3D view, writes out the working and prepares a PDF or Word report.
Which module for which load
A load of rectangular shape in plan: Rectangle. A load of any shape in plan, read from a chart: Influence chart.
A layered ground in which stiff layers spread the load: Westergaard areas.
Before you start: what you need
Have these ready from your design:
What
Notes
The load per unit area q Your design loading
The pressure the circle puts on the ground, zero or more. It is a pressure, not the total force of the tank or silo.
The radius R The tank, silo or footing
The radius of the loaded circle, at least 0.001 m; the diameter is 2R.
The centre (xL, yL) The layout
The place of the centre on the ground. With one circle you can leave it at 0, 0 and measure A from the centre.
The point A: x, y and z Where you want the stress
x and y in the same coordinates as the centres; z at least 0.001 m below the surface.
Quick start in seven steps
Open the module and choose the units
Open Circle, any point from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the circle
In Circular loads the first circle is already in the table. Press its row to change q, R, xL and yL; or drag the rings at its edges on the Section drawing. To add another circle, fill the New circular load form and press + Add circular load.
Place the point A
In Point A type x, y and z, or drag the blue point on the Section drawing.
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line at the foot of the Section drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the circles lock; the point A and the profile depth stay free.
Read the answer
The Results tab gives Δσz at A, the influence value A′ + B′ for one circle, the part from each circle for several, and a table of the stress against depth under A. The lines of equal stress are on the Section drawing, the colour bands in Contours.
Move A and take the report
Drag A or change x, y and z: the stress there is calculated at once. The Calculations tab writes out the working; the Report tab saves a PDF or a Word file.
The screen: section, contours, profile and 3D
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 is a vertical section with the loaded circle, the point A, the dimensions r (or x − x1 with several circles) and z and the dashed line of the profile depth. With one circle it is the plane through the centre of the circle and A, and because the problem is axisymmetric the circle is drawn centred on the section. With several circles it is the vertical plane y = yA through A, parallel to the x axis: a circle that this plane cuts shows its chord, and the others act on it from beside. After Analyze it adds the lines of equal stress and a label with Δσz at A.
Contours is the same section filled with colour bands of Δσz on a logarithmic scale, with a legend in kPa and in per cent of q; values below 1 kPa take the colour of the lowest band, and the part below zmax is paler.
Profile is the curve of Δσz against depth under A, with a marker at the depth of A.
3D is an oblique view of the circular load on the ground, the axes x, y and z, and a small element at A with the stress on it.
Before Analyze the drawings follow the inputs and say “The stress and the lines of equal stress appear after Analyze” (Section), “The colour contours of the stress appear after Analyze” (Contours), “The profile appears after Analyze.” (Profile) or “The stresses on the element appear after Analyze.” (3D).
Section before Analyze.
Section after Analyze: the lines of equal stress; the line through A is blue.
Contours after Analyze.
Profile after Analyze.
3D after Analyze.
The Show chips
Above the Section and the Contours drawings a bar has two chips: Lines of equal stress and Dimensions. In the Section view the lines chip is off and greyed out until you press Analyze, and then turns on; in Contours it is off by default, so that the lines can be laid over the colours. Dimensions is on by default in both. The Profile and 3D views have no chips. The levels of the lines are listed under the drawing (10, 13, 15, 20, 30, 50, 70 and 100 kPa for the example); the blue line takes the rounded value at A, here 13, and the levels change with the model.
Dragging on the drawing
The hint under the Section drawing reads: “Drag a load to move it, the rings at its edges to resize it. Drag the dashed line to set the depth of the profile (zmax).” What each pull does, as the module runs it, before Analyze (steps of 5 cm):
One circle, Section view. Drag a ring at the left or right edge of the circle to change the radius: both rings move together, and the centre stays where it is. Pulling the right ring 1 m outward changes R from 3 to 4. The radius stops at 0.1 m. The circle cannot be moved in this view; use the boxes or the 3D view.
Several circles, Section view. A bar over each circle, as wide as its chord, moves that circle along x: xL follows the pointer. The rings are not shown; change R in the table or in the 3D view.
3D view. Drag the disc of a circle over the ground to move it: xL and yL follow the pointer (a short pull, for example 30 px right and 12 px down, moved the example circle to xL = 0.8, yL = 0.15). Drag one of the four rings on its edge to change R; the centre stays.
Circles never overlap. A pull that would put a circle on another stops at the last free place, where the circles touch, and a message says “That would put circular load 1 and circular load 2 on top of one another — the value is kept as it was.” Circles that touch are allowed.
The point A. Drag the blue point: z follows the pointer, and so does the position. With one circle the pull changes the distance r from the centre and keeps the direction of A from it: with the centre at (xL, yL) = (3, 4) and A at (x, y) = (6.6, 6.7), a pull of 1 m to the right and 1 m down gave (x, y) = (7.4, 7.3) and z = 4, which is r = 5.5 on the same ray. Pulled to the other side of the axis it gives the same distance as a positive number. With several circles the pull changes x and z and keeps y. The depth stops at 0.05 m at the top and the position at the sides of the drawn frame.
The dashed line. Drag it up or down to set zmax; it stops at 0.5 m.
After Analyze the circles are locked, and their rings, bars and discs are gone. The point A and the dashed line stay free, and so do the boxes x, y, z and zmax.
The 3D view has its own hint, “Drag a load over the ground to move it, the rings at its edges to resize it.”
Zoom
Each view has −, a percentage and + at the lower right, and a button that fits the whole drawing in the pane. Zoom changes the drawing on the screen only.
Entering the model, section by section
The Model tab has four sections: Model data, Circular loads, Point A and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below the last one. 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 other unit systems the boxes show the converted values with their unrounded digits, for example 9.84252 ft for the 3 m radius and 10.1972 t/m² for 100 kPa.
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.
Model data.
Circular loads
The section is a form over a table. The New circular load form takes q, R, xL and yL; + Add circular load puts the circle into the Circular loads table, and a counter beside the button shows how many of the eight circles there are. The table lists each circle with its load, radius and centre. Press a row to change that circle in the form (the form then reads Editing · Circular load 1 and offers Done and Delete circular load; the note says that changes apply as they are entered), or press × to take a circle out. The first circle is already in the table when the module opens, and × is greyed out while it is the only one.
Circular loads: the form over the table.
Circular load fields (the form, and each row of the table)
Field
Default
What to enter
Load per unit area q kPa · t/m² · ksf
100
The uniform pressure of the circle on the ground. Zero or more.
Radius of the loaded circle R m · ft
3
At least 0.001 m.
Position of the centre along x xL m · ft
0 (circle 1) · 8 (form)
On the ground, in the coordinates of the point A. Circle k of a model made by the module starts at 8 × (k − 1); the New circular load form opens with 8.
Position of the centre along y yL m · ft
0
On the ground, in the coordinates of the point A.
A row of the table pressed: the circle is edited in the form.
The circles may not lie on top of one another: two circles may touch but not overlap, that is, the distance between their centres is at least the sum of their radii. Adding or editing a circle that would overlap another is refused (see Messages). The circles are numbered in the order you add them, and the numbers name the results “From circular load 1”, “From circular load 2” and so on.
Two circles in the table.
Point A
The three coordinates of the point where you want the stress. A is under the ground surface at the position x, y and the depth z. The note in the section says that x and y are measured on the ground surface in the same coordinates as the centres and z downward, and that for one circle the problem is axisymmetric: only the distance r of A from its centre matters, under the circle for r < R and beside it for r > R, not the direction.
Point A.
Point A fields
Field
Default
What to enter
Point A: x on the ground surface x m · ft
4.5
In the coordinates of the centres. For one circle only the distance from its centre counts.
Point A: y on the ground surface y m · ft
0
In the coordinates of the centres.
Depth of the point A z m · ft
3
At least 0.001 m below the ground surface.
Depth profile
The depth down to which the Profile view and the table of the stress under A run.
Depth profile.
Depth profile field
Field
Default
What to enter
Depth of the profile zmax m · ft
12
At least 0.1 m. The Profile view and the table under A go from the surface down to this depth, or to the depth of A if that is deeper.
One circle, several circles and where A is
The one choice in the module is how many circles there are, from 1 to 8; you make it by adding or deleting circles. The stress of each circle is found as if it were alone, with the distance r of A from its centre, and the stresses are added. This works because the ground is elastic, so the stresses of separate loads add.
Only the distance counts for one circle. The stress depends on r, z and R. Moving the centre to (xL, yL) = (3, 4) and A to (6.6, 6.7) keeps r = 4.5 m and gives 12.66 kPa, the same as the example (12.66 kPa); so does A at (x, y) = (0, 4.5) with the circle at the origin (12.66 kPa), and A at x = −4.5 on the other side (12.66 kPa).
Under, at the edge, beside. With z = 3 m and R = 3 m the module gives 64.64 kPa at r = 0, 60.92 kPa at 1 m, 33.22 kPa under the edge (r = R), 12.66 kPa at 4.5 m, 4.181 kPa at 6 m and 0.04884 kPa at 15 m.
Loads and radii can differ. Each circle has its own q and R, so a tank and a smaller silo beside it can be put in one model.
The cards show the parts. The Results tab shows the total and the part from each circle, and the Calculations tab writes out each circle’s working.
With the example circle (100 kPa, radius 3 m, at the origin) and A at (x, y, z) = (4.5, 0, 3), a second circle of 100 kPa and radius 3 m centred at xL = 8 m gives the total 37.71 kPa: 12.66 kPa from the first and 25.05 kPa from the second, whose centre is 3.5 m from A. If the second circle carries 50 kPa, its part is 12.52 kPa and the total is 25.19 kPa. A point midway between two such circles (xL = 0 and 8 m, A at x = 4 m) gets 36.15 kPa, and with the circles touching (xL = 0 and 6 m, A at x = 3 m, the point of contact) 66.45 kPa.
Two circles, 100 kPa and 50 kPa: the lines of equal stress of the sum.
The same two circles in the Contours view.
The same two circles in the 3D view.
Which should I use?
One circle or several? One circle for a tank, silo or round footing on its own; several circles for a group of tanks, or for round loads of different pressures side by side.
A load that is not round is not several circles: use Rectangle for a rectangular footing, or Influence chart for any shape in plan.
A load that is long compared with its width is closer to a strip: use Vertical strip load.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” A message appears briefly, “Analysed — see the Results tab and the Calculations.” The module stays on the Model tab: open Results for the answer. The Section view now shows the lines of equal stress.
The bar after Analyze.
The cards of the example.
Card
Symbol
The example
Vertical stress increase at A
Δσz
12.66 kPa
Influence value (one circle)
A′ + B′
0.1266
From circular load k (several circles)
Δσz(k)
part of the total from circle k
Below the cards the Vertical stress increase under A table gives z, Δσz and, for one circle, z/R and A′ + B′ at ten equal steps down to zmax, with the row of A inserted and highlighted. The Profile view runs a little deeper than A if zmax is smaller than the depth of A. With several circles the table has z and Δσz only, and the card of the influence value is replaced by the part from each circle. A Values, step by step table above it lists the steps of the Calculations tab with their values.
The depth table of the example.
Two circles: the total and the part from each.
Using the numbers
The result is an increase, not the total stress: add the overburden pressure of the soil above A for the total vertical stress.
The influence value A′ + B′ is the part of the surface pressure that reaches A. It is 1 on the surface under the circle and falls with depth on the axis; beside the circle it first rises with depth and then falls (for the example, 4.059 kPa at 1.2 m, 12.66 kPa at 3 m, 13.67 kPa at 4.8 m and 6.487 kPa at 12 m).
The curve in the Profile view is summed over fewer directions than the card, so its last digit can differ by one: the marker at A reads 12.67 where the card says 12.66 (an independent integration gives 12.665).
Change x, y or z after Analyze and everything follows at once; to change a circle, press Edit inputs, which unlocks the circles and clears the results.
Changing the unit system after Analyze re-states the cards and the report without a new run. The drawings keep their labels in m and kPa.
Warnings and notes
The bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” The runs of this manual, from the example to eight circles, gave no warning.
The Warnings window.
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. For one circle:
Stress at A — Δσz = q (A′ + B′), z/R and r/R with your numbers and a note on where A is (“A is outside the loaded circle”, “A is under the loaded circle” or “A is under the edge”), the integral about the point, A′ + B′ and Δσz = A′ + B′ × q, and for comparison the stress on the axis at the same depth. When A is on the axis (r = 0) the closed form is shown first, and the general integral gives the same.
Stress increase against depth — the same integral down the vertical line through A, as the table of the Results tab, with the row of A marked.
With several circles there is one step for each circle, with its own distance r = √[(x − xL)² + (y − yL)²], then a Superposition step that adds the parts, then the depth table.
The Calculations tab: the depth step 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, 4 Results, 5 Summary and 6 References. It carries three drawings, the Section with the lines of equal stress, the Profile and the 3D view, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has five sections: Scope and assumptions, The solution, How the program computes it, Worked example and References. Each opens when you click its title (Open all and Close all open or close them together). It sets out the closed form on the axis, why there is none elsewhere, the integral that the module evaluates and a table of A′ + B′ at several values of z/R and r/R; 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
Circular loads
Circular load 1
q = 100 kPa; R = 3 m; centre at 0, 0
Point A
Position and depth
x = 4.5 m; y = 0; z = 3 m
Depth profile
Depth
zmax = 12 m
Step 1: where A is
The distance from the centre is r = 4.5 m, so r/R = 1.5 (A is outside the loaded circle) and z/R = 1.
Step 2: the stress
The integral about the point has no closed form here. Evaluated by the module, A′ + B′ = 0.12664, so Δσz = 0.12664 × 100 = 12.66 kPa. An independent numerical integration of Boussinesq’s point-load solution over the circle, done outside the module, gives 0.12665. For comparison, directly under the centre at the same depth the closed form gives Δσz = 100 × {1 − 1/[(3/3)² + 1]3/2} = 100 × (1 − 0.35355) = 64.64 kPa, about five times the stress at A; the module prints this comparison in step 1 of the Calculations tab.
Δσz/q = 0.1266, Δσz = 12.66 kPa
These are the two cards of the Results tab and the step of the Calculations tab. The same integral at the depths of the table gives the profile: the module’s table gives 4.059 kPa at 1.2 m, 12.66 kPa at 3 m, 13.67 kPa at 4.8 m and 6.487 kPa at 12 m, and the same independent integration gives 4.059, 12.67, 13.67 and 6.487 kPa.
Directly under the centre
Set x = 0 and the point is on the axis. At z = 1.5 m, R/z = 2 and the closed form gives 100 × (1 − 1/53/2) = 100 × (1 − 0.08944) = 91.06 kPa; the module gives 91.06 kPa. At z = 3 m the module gives 64.64 kPa, the 64.64 kPa above, and at z = 12 m 8.692 kPa, where R/z = 0.25 and 100 × (1 − 1/1.06253/2) = 8.69 kPa by hand.
Other unit systems
The same circle in SI · tf, with q = 10 t/m² (the other values unchanged), gives 1.266 t/m² and the influence value 0.1266. In US the default example, which is 2.089 ksf on a circle of 9.843 ft radius, gives 0.2645 ksf: the same circle and point as 12.66 kPa. A check by hand with round US numbers: q = 2 ksf, R = 10 ft, A at x = 15 ft, z = 10 ft has the same z/R = 1 and r/R = 1.5, so the influence value is 0.12664 and Δσz = 0.12664 × 2 = 0.2533 ksf; the module gives 0.2533 ksf. In SI · tf, q = 10 t/m², R = 2 m, x = 3 m, z = 2 m again has the ratios 1 and 1.5 and gives 1.266 t/m²; the module gives 1.266 t/m².
How each input changes the answer
Each row changes one thing in the example (one circle of q = 100 kPa, R = 3 m at the origin, A at x = 4.5 m, y = 0, z = 3 m) and runs the module again, in SI · kN. The numbers are the ones the module gives, so you can reproduce any row; an independent numerical integration of the point-load solution agrees with every row.
The example with one change at a time
Change
Δσz (kPa)
The example A′ + B′ = 0.1266
12.66
q = 200 kPa The stress is proportional to the load; the influence value is unchanged A′ + B′ = 0.1266
25.33
q = 50 kPa Half the load, half the stress A′ + B′ = 0.1266
6.332
R = 1.5 m A smaller circle: A is now 3 m outside its edge, and the stress falls A′ + B′ = 0.02250
2.250
R = 2 m A smaller circle puts less on the same point A′ + B′ = 0.04411
4.411
R = 4 m A larger circle: A is now 0.5 m outside its edge A′ + B′ = 0.2842
28.42
R = 6 m A is now under the circle, 1.5 m inside its edge A′ + B′ = 0.6923
69.23
R = 20 m A is deep inside a very large circle: the stress approaches q A′ + B′ = 0.9961
99.61
x = 0 (on the axis) The closed form applies; the highest stress at this depth A′ + B′ = 0.6464
64.64
x = 1 m Under the circle, 2 m inside its edge A′ + B′ = 0.6092
60.92
x = 2 m Under the circle, 1 m inside its edge A′ + B′ = 0.4971
49.71
x = 3 m (under the edge) At the edge: about one third of q at this depth A′ + B′ = 0.3322
33.22
x = 4 m 1 m outside the edge A′ + B′ = 0.1808
18.08
x = 6 m Beside the circle, 3 m from the edge A′ + B′ = 0.04181
4.181
x = 9 m Farther out A′ + B′ = 0.006246
0.6246
x = 15 m Far from the circle the stress is a tiny part of the load A′ + B′ = 0.0004884
0.04884
z = 0.01 m At the surface beside the circle the stress is nearly zero A′ + B′ = 0.00000005
0.00000451
z = 0.5 m Close to the surface, beside the circle A′ + B′ = 0.004918
0.4918
z = 1.5 m Beside the circle the stress rises with depth at first A′ + B′ = 0.06044
6.044
z = 6 m Past the largest value, which is at about 4.2 m for this point: the stress now falls A′ + B′ = 0.1265
12.65
z = 9 m The stress keeps falling A′ + B′ = 0.09194
9.194
z = 12 m Four radii deep A′ + B′ = 0.06487
6.487
x = 3 m, z = 3 m Under the edge at a depth equal to the radius A′ + B′ = 0.3322
33.22
xL = 3 m (A at x = 4.5 m) A is 1.5 m from the centre: under the circle A′ + B′ = 0.5622
56.22
R = 6 m, z = 6 m, x = 9 m Every length doubled: the same z/R and r/R, the same stress as the example A′ + B′ = 0.1266
12.66
Several circles, SI · kN, added to the example (A at x = 4.5 m, y = 0, z = 3 m unless the row says otherwise):
Adding circles
Added
Total (kPa)
Parts (kPa)
Circle 2: 100 kPa, R = 3 m, centre at xL = 8 m The centre of circle 2 is 3.5 m from A
37.71
12.66 + 25.05
Circle 2: 50 kPa, R = 3 m, centre at xL = 8 m
25.19
12.66 + 12.52
Circle 2: 100 kPa, R = 3 m, centre at xL = 6 m The circles touch at x = 3 m
68.89
12.66 + 56.22
Circle 2: 200 kPa, R = 1.5 m, centre at xL = 6 m A smaller, more heavily loaded circle
51.86
12.66 + 39.20
Circle 2: 100 kPa, R = 3 m, centre at yL = 6 m Beside the first along y; A stays at y = 0
14.17
12.66 + 1.510
Circles 2 and 3: 100 kPa, R = 3 m, centres at xL = 8 m and 16 m
37.90
12.66 + 25.05 + 0.1850
The pattern follows the integral. The stress is proportional to q; it depends on z/R and r/R only, so doubling every length leaves it unchanged; on the axis it falls steadily with depth, while beside the circle it rises to a largest value and then falls; and with z near zero it equals q under the circle and zero beside it. The depth of the profile zmax changes the table and the Profile view but not the stress at A: 12.66 kPa with zmax = 6 m and 12.66 kPa with zmax = 0.5 m, a depth the module extends to the depth of A.
Messages: when the module stops, and what it notes
The module checks the numbers it needs before it calculates. When an input cannot be used, press Analyze and a window titled Cannot analyse — 1 input error (or 2 input errors …) lists each message; a box whose value is outside its limit is marked in red. Nothing is calculated.
The window Analyze opens when the radius is zero.
When
Message and what to do
R is 0
“Radius of circle 1 (R1) must be at least 0.001 m.” Enter a radius of at least 0.001 m.
q is negative
“Load per unit area of circle 1 (q1) must be at least 0 kPa.” Enter zero or a positive load.
z is 0
“Depth of the point A (z) must be at least 0.001 m.” Enter a depth of at least 0.001 m.
zmax is 0
“Depth of the profile (zmax) must be at least 0.1 m.” Enter a depth of at least 0.1 m.
A box is empty
“Radius of circle 1 (R1) is missing.” or “Point A: x on the ground surface (x) is missing.” Fill the box or take the circle out.
Two values are wrong
The window reads “Cannot analyse — 2 input errors” and lists both: “Radius of circle 1 (R1) must be at least 0.001 m. Depth of the point A (z) must be at least 0.001 m.”
A new circle would overlap another
A message appears briefly: “The new circular load would lie on circular load 1 — they cannot overlap. Give it another position or width.” The circle is not added.
The messages use the numbered names of the circles, such as “circle 1” and “R1”, even when there is only one circle, and the limits in the units of the unit system chosen. When you type a position or radius into a circle already in the table and it would put two circles on one another, a message appears briefly: “That would put circular load 1 and circular load 2 on top of one another — the value is kept as it was.” and the value stays as it was. A radius that is too big does the same. At eight circles + Add circular load is greyed out and the counter reads 8 of 8.
Notes and warnings
With every input inside its limits the module calculates, and the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” A load of zero is accepted and gives 0 kPa, with a dash for the influence value. Large values are accepted too: with q = 1,000,000 kPa and R = 1000 m, a circle that is nearly infinite in extent around A, the module gives 1,000,000 kPa, an influence value of 1.000.
Good practice and common mistakes
Use the module for a loaded circle on the ground. The circle is flexible, so the pressure stays q across it. A stiff footing distributes its pressure differently; the module gives the stress for the pressure you enter.
Enter a pressure, not a force.q is a load per unit area. For a tank, divide the total weight, content included, by the area of the circle, πR².
Use the radius, not the diameter. The drawing labels the diameter 2R so that you can check it.
Mind the coordinates.x, y and the centres are in the same coordinates on the ground; with one circle only the distance between A and the centre matters, with several the position of A counts.
Do not overlap circles. Circles may touch but not lie on one another; the module does not take a circle inside another.
Add the overburden. The result is the stress increase; for the total vertical stress add the soil weight above A.
Read the profile with the radius in mind. Under the centre the stress is 64.64 kPa of 100 at one radius of depth and falls steadily below; beside the circle it first rises: for A at 4.5 m from the centre the largest value, about 13.8 kPa, is at a depth of about 4.2 m.
Place A with the drawing, then type exact values. Dragging snaps to 5 cm; type the coordinates when you need a given place.
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 stress below a circular load calculated?
The module integrates Boussinesq’s point-load solution over the loaded circle, in polar coordinates about the point below A. The result is Δσz = q (A′ + B′), with A′ + B′ depending on z/R and r/R only, where r is the horizontal distance of A from the centre. For the module’s example it gives 12.66 kPa.
Does it work only under the centre of the circle?
No. Directly under the centre there is a closed form, which the module shows first when A is on the axis; at any other point, under the circle off its centre, under its edge or beside it, the module evaluates the integral itself. The example point is 1.5 m beside the edge of the circle.
What is the influence value A′ + B′?
It is the stress increase divided by the pressure of the circle. It depends only on z/R and r/R, and it is the number to use if you have a table of influence values. The module gives it for one circle; with several circles it gives the stress of each.
Does the result include the weight of the soil?
No. It is the increase caused by the circles only. Add the overburden pressure of the soil above A for the total vertical stress.
How many circles can I add, and can they overlap?
Up to eight, each with its own q, R and centre. They may touch but not overlap; the module refuses a circle that would lie on another. The stresses of the circles add.
Can I drag the circle and the point on the drawing?
Yes, before Analyze. In the Section view of one circle, the rings at its edges change the radius; with several circles a bar over each circle moves it along x; in the 3D view you drag the disc to move it and the rings to change the radius. The blue point places A and the dashed line sets the depth of the profile; they move in 5 cm steps. After Analyze the circles are locked, but A and the dashed line stay free.
What does the point A depend on?
On x, y and z. With one circle, only on the distance r of A from the centre and on z: the stress is the same on every point of the same circle around the axis. With several circles the position of A relative to each centre counts.
What do the lines of equal stress show?
Each line joins the points of the section with the same Δσz, labelled in kPa; the line through A is blue. Switch them on and off with the Lines of equal stress chip. The Contours view shows the same stress as colour bands on a logarithmic scale.
What values does the module accept?
q zero or more, R at least 0.001 m, z at least 0.001 m, zmax at least 0.1 m, and x, y and the centres any value. Anything else stops the analysis with a message that names the value.
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. Stresses are in kPa, t/m² or ksf and lengths in m or ft. The default example gives 12.66 kPa; in US, 0.2645 ksf. A circle of 10 t/m² gives 1.266 t/m² in SI · tf.
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, profile 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.
When is a circle the wrong model?
When the loaded area is not round: use Rectangle for a rectangular footing or Influence chart for any other shape. When the load is a long band: use Vertical strip load.
Calculate the vertical stress increase at any point below or beside one or several uniformly loaded flexible rectangles: the stress at A, the lines of equal stress, colour contours, the profile with depth and a 3D view. This manual walks through the screen, dragging, every input, the results and the report, with a worked example you can reproduce.
Rectangle gives the increase in vertical stress at any point of the ground below or beside one or several flexible rectangular areas, each carrying a uniform pressure q on the ground surface. The ground is a homogeneous, isotropic, elastic half-space, and the solution is the one found by integrating Boussinesq’s point-load solution over the rectangle. You place the point A by its three coordinates, x and y on the ground and the depth z, and the module gives Δσz there, the lines of equal stress, the colour contours, the profile with depth under A and a 3D view.
Below a corner of a rectangle with sides L and B the stress is Δσz = qI3, where the influence factor I3 depends on m and n, the two sides divided by the depth:
For any other point the module writes the loaded rectangle as the sum of four rectangles that all have a corner directly above A. Each is added or subtracted, so the point may lie under the area, on its edge or outside it. With several rectangles the stress of each is found in the same way and the stresses are added.
What you can describe
One to eight rectangular loads, each with its own load per unit area q, its side L along x, its side B along y and the position xc, yc of its centre on the ground.
The point A: x and y on the ground surface and the depth z below it.
The depth of the profilezmax, down to which the Profile view and the table under A run.
The result is the increase caused by the rectangles only; the overburden pressure of the soil above A is not included, so add it where your design needs the total stress. The module draws the section, the contours, the profile and a 3D view, writes out the working and prepares a PDF or Word report.
Which module for which load
A footing, raft or slab that is rectangular in plan: this module. A load that is long compared with the depth you ask about is close to a strip: Vertical strip load.
A load of any other shape in plan, such as an L-shaped slab with an irregular edge: Influence chart. A shape made of rectangles can also be built here, as several rectangles that meet at their sides.
Layered ground, where a stiff layer spreads the load: Westergaard areas.
Before you start: what you need
Have these ready from your design:
What
Notes
The load per unit area q Your design loading
The pressure the rectangle puts on the ground, zero or more. It is a pressure, not a total force.
The sides L and B The footing, raft or slab
L is the side along x and B the side along y, each at least 0.001 m. Either may be the longer one.
The centre xc, yc The layout
The place of the centre on the ground. With one rectangle you can leave it at the origin and measure A from the centre.
The point A: x, y, z Where you want the stress
x and y in the same coordinates as the centres; z at least 0.001 m below the surface.
Quick start in seven steps
Open the module and choose the units
Open Rectangle from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the rectangle
In Rectangular loads the first rectangle is already in the table. Press its row to change q, L, B, xc and yc, or drag the rectangle on the drawing. To add another, fill the New rectangular load form and press + Add rectangular load.
Place the point A
In Point A type x, y and z, or drag the blue point on the Section drawing (it sets x and z).
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line at the foot of the Section drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the rectangles lock; the point A and the profile depth stay free.
Read the answer
The Results tab gives Δσz at A, the influence value Δσz/q for one rectangle, the part from each rectangle for several, and a table of the stress against depth under A. The lines of equal stress are on the Section drawing, the colour bands in Contours.
Move A and take the report
Drag A or change x, y and z: the stress there is calculated at once. The Calculations tab writes out the working; the Report tab saves a PDF or a Word file.
The screen: section, contours, profile and 3D
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 is a vertical cut through the ground in the plane y = yA, the plane through A parallel to the x axis, with the rectangles that the plane cuts drawn on the ground. It shows the point A, the dimensions z and x (with several rectangles, or one that is not centred at the origin, x − x₁: the distance of A from the centre of rectangle 1) and the dashed line of the profile depth. After Analyze it adds the lines of equal stress and a label with Δσz at A. Under the load a red note says whether the plane cuts it (“the section cuts the loaded area”) or not (“the loaded area lies beside this section (y outside the rectangle)”). With several rectangles, one the plane misses is drawn dashed with its distance, for example “6.00 m beside the plane”, and still acts on the section from beside; the subtitle says so.
Contours is the same section filled with colour bands of Δσz on a logarithmic scale, with a legend in kPa and in per cent of q (with several rectangles the percentage header reads “% of q max”).
Profile is the curve of Δσz against depth under A, with a marker at the depth of A.
3D is an oblique view of the rectangles on the ground, the axes x, y and z, and a small element at A with the stress on it.
Before Analyze the drawings follow the inputs and say “The stress and the lines of equal stress appear after Analyze” (Section), “The colour contours of the stress appear after Analyze” (Contours), “The profile appears after Analyze” (Profile) or “The stresses on the element appear after Analyze” (3D). The small rings at the edges of the rectangle are its grips.
Section before Analyze.
Section after Analyze: the lines of equal stress; the line through A is blue.
Contours after Analyze.
Profile after Analyze.
3D after Analyze.
The Show chips
Above the Section and the Contours drawings a bar has two chips: Lines of equal stress and Dimensions. In the Section view the lines chip is off and greyed out until you press Analyze, and then turns on; in Contours it is greyed out before Analyze too, and off by default afterwards, so that the lines can be laid over the colours. Dimensions is on by default in both. The Profile and 3D views have no chips.
Dragging on the drawing
Before Analyze two lines under the Section drawing read: “Drag a load to move it, the rings at its edges to resize it. Drag the dashed line to set the depth of the profile (zmax).” and “Drag the point A to place it. The stress and the lines of equal stress appear after Analyze.” After Analyze the first line is reduced to “Drag the dashed line to set the depth of the profile (zmax).” What each pull does, as the module runs it:
The rectangle in the Section and Contours views. Drag its body to move it along x: xc follows the pointer. Pulling the example rectangle 2 m to the right changes xc from 0 to 2. The section shows one plane, so the pull does not change yc.
An edge ring in the Section and Contours views. The rings at the two ends change L: the edge you pull follows the pointer and the other edge stays where it is, so L and xc change together. Pulling the right ring 1 m to the right gives L = 4 and xc = 0.5; pulling the left ring 1 m to the left gives L = 4 and xc = −0.5. The side stops at 0.1 m: pulling the left ring far to the right leaves L = 0.1 with the right edge where it was.
The rectangle in the 3D view. Drag it over the ground to move it in both directions: a pull of the example changed xc to 1.15 and yc to 0.3. Four rings sit on the edges: the two along x change L and xc as above, the two along y change B and yc in the same way. B is changed by dragging in the 3D view, not in the Section view.
Rectangles never overlap. A pull that would put a rectangle on another stops at the last free place, and a message says “That would put rectangular load 1 and rectangular load 2 on top of one another — the value is kept as it was.” Pulling a second rectangle at xc = 5 onto the first stopped at xc = 3, where the two meet at a side.
The point A. Drag the blue point in the Section or Contours view: x and z follow the pointer in steps of 5 cm, and y is not changed. The depth stops at 0.05 m at the top and at the foot of the drawn frame at the bottom, and x at the sides of the frame. With the example, pulling A far to the upper left gave x = −4.8 m and z = 0.05 m, and far to the lower right x = 4.8 m and z = 9 m. The 3D view has no handle for A; type its coordinates.
The dashed line (Section and Contours). Drag it up or down to set zmax; it stops at 0.5 m.
After Analyze the rectangles are locked: their body and rings are gone from the drawing. The point A and the dashed line stay free, and so do the boxes x, y, z and zmax; dragging A after Analyze updates the stress at once (from the example, dragging A to x = 1.5 m, z = 5 m gave 12.01 kPa).
The 3D view has its own hint, “Drag a load over the ground to move it, the rings at its edges to resize it.” These pulls work before Analyze only.
Zoom
Each view has −, a percentage and + at the lower right, and a button that fits the whole drawing in the pane. Zoom is a property of the drawing only: it is not saved with the model and does not change the report.
Entering the model, section by section
The Model tab has four sections: Model data, Rectangular loads, Point A and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below the last one. 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 9.84252 ft for the 3 m side and 3.13282 ksf for the 150 kPa load.
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 to defaults puts the example back at once, whatever you had typed.
Model data.
Rectangular loads
The section is a form over a table. The New rectangular load form takes q, L, B, xc and yc; + Add rectangular load puts the rectangle into the Rectangular loads table, and a counter beside the button shows how many of the eight there are. The table lists each rectangle with its load, sides and centre. Press a row to change that rectangle in the form (the form then reads Editing · Rectangular load 1 and offers Done and Delete rectangular load; the note says that changes apply as they are entered), or press × to take a rectangle out. The first rectangle is already in the table when the module opens, and Delete rectangular load and × are greyed out while it is the only one.
Rectangular loads: the form over the table.
Rectangular load fields (the form, and each row of the table)
Field
Unit
Default
Load per unit area q The uniform pressure of the rectangle on the ground. Zero or more.
kPa · t/m² · ksf
150
Side of the rectangle along x L At least 0.001 m.
m · ft
3
Side of the rectangle along y B At least 0.001 m.
m · ft
2
Position of the centre along x xc Any value. The first rectangle has 0; the New rectangular load form opens with 5.
m · ft
0
Position of the centre along y yc Any value.
m · ft
0
A row of the table pressed: the rectangle is edited in the form.
The rectangles may not lie on top of one another: two rectangles may meet at a side but not overlap. Adding or editing a rectangle that would overlap another is refused (see Messages). The rectangles are listed in the order you add them, and their numbers name the results “From rectangular load 1”, “From rectangular load 2” and so on. The sides stay along x and y: the module does not turn a rectangle.
Two rectangles in the table.
Point A
The three coordinates of the point where you want the stress. A is under the ground surface at the position (x, y) in plan and the depth z. It may be under a rectangle, on its edge or outside it; the corner method handles all of these.
Point A.
Point A fields
Field
Unit
Default
Point A: x on the ground surface x In the same coordinates as the centres. For one rectangle centred at the origin it is measured from the centre.
m · ft
2.5
Point A: y on the ground surface y In the same coordinates. The Section view is the plane through this value of y.
m · ft
−1
Depth of the point A z At least 0.001 m below the ground surface.
m · ft
4
Depth profile
The depth down to which the table of the stress under A and the Profile view run. The table goes down to zmax, or to the depth of A if that is deeper; the Profile view goes to zmax, or to 1.2 times the depth of A if that is deeper.
Depth profile.
Depth profile field
Field
Unit
Default
Depth of the profile zmax At least 0.1 m. The table goes down to this depth, or to the depth of A if that is deeper; the Profile view goes to this depth, or to 1.2 times the depth of A if that is deeper.
m · ft
12
Several rectangles and how they combine
The one choice in the module is how many rectangles there are, from 1 to 8; you make it by adding or deleting rectangles. The stress of each rectangle is found as if it were alone, from the position of A relative to its four corners, and the stresses are added. This works because the ground is elastic, so the stresses of separate loads add.
Positions are on one set of axes.x, y and every xc, yc are in the same coordinates. Moving a rectangle and A together by the same distance changes nothing: the example with the centre at (4, 3) and A at (6.5, 2) gives 10.90 kPa, the same as the example.
Loads can differ. Each rectangle has its own q, L and B, so a heavy core and a lighter surround can be put in one model.
The result lists the parts. The Results tab shows the total and the part from each rectangle, and the Calculations tab writes out each rectangle’s working.
The sides are fixed to the axes. To model a rectangle that is turned, or an irregular shape, use Influence chart.
With the example rectangle (150 kPa, 3 m × 2 m, at the origin) and A at x = 2.5 m, y = −1 m, z = 4 m, a second rectangle of the same size centred at xc = 5 m is the mirror image of the first about A, so it adds the same 10.90 kPa: 21.79 kPa in all. If the second is 75 kPa, it adds 5.448 kPa and the total is 16.34 kPa. A third rectangle of 150 kPa centred at 10 m brings the total of the three to 22.45 kPa.
Two rectangles, 150 kPa and 75 kPa: the lines of equal stress of the sum.
The same two rectangles in the Contours view.
Rectangles that meet
Two rectangles that meet at a side give the same as one rectangle made of both. The example rectangle with a second one of 150 kPa and 3 m × 2 m centred at xc = 3 m meets it at the side x = 1.5 m; the parts are 10.90 kPa and 19.73 kPa and the total 30.63 kPa. One rectangle L = 6 m, B = 2 m centred at xc = 1.5 m gives 30.63 kPa, the same. Likewise a second rectangle meeting along y (centred at yc = 2 m) gives 16.18 kPa, and one rectangle L = 3 m, B = 4 m centred at yc = 1 m gives 16.18 kPa. An L-shaped or stepped load can therefore be built from rectangles that meet at their sides.
Which should I use?
One rectangle or several? One when the pressure is uniform over a rectangle; several when the pressure differs from part to part, or when the shape is made of rectangles.
A load much longer than wide is close to a strip: with L = 30 m the example gives 41.14 kPa; Vertical strip load is the plane-strain model.
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” A message appears briefly, “Analysed — see the Results tab and the Calculations.” The module stays on the Model tab: open Results for the answer. The Section view now shows the lines of equal stress.
The bar after Analyze.
The cards of the example.
Card
Symbol
The example
Vertical stress increase at A
Δσz
10.90 kPa
Influence value (one rectangle)
Δσz/q
0.07264
From rectangular load k (several rectangles)
Δσz(k)
the part of the total from rectangle k
Below the cards the Vertical stress increase under A table gives z, Δσz and, for one rectangle, Δσz/q at ten equal steps from the surface to zmax, with the row of A inserted and highlighted. With several rectangles the ratio column is left out. A Values, step by step table above it lists the steps of the Calculations tab with their values.
The depth table of the example.
Two rectangles: the total and the part from each.
Reading the table
A point beside the loaded area does not have its largest stress at the surface. In the example A lies 1 m outside the short side of the rectangle, and the table rises from 7.668 kPa at 1.2 m to 12.59 kPa at 2.4 m before falling to 10.90 kPa at 4 m and 2.603 kPa at 12 m; the Profile view shows the same bulge. Under the rectangle the stress is largest at the surface and falls with depth: under the centre of the example, 116.2 kPa at 1 m, 22.98 kPa at 4 m and 1.067 kPa at 20 m.
Using the numbers
The result is an increase, not the total stress: add the overburden pressure of the soil above A for the total vertical stress.
The influence value Δσz/q is the part of the surface pressure that reaches A. It is 1 at the surface under the rectangle, 0 beside it, and falls with depth and with distance.
Values are rounded where they are drawn, so the last digit of a figure on the drawing can differ by one from the card.
Change x, y or z after Analyze and everything follows at once; to change a rectangle, press Edit inputs, which unlocks the rectangles and clears the results.
Changing the unit system after Analyze re-states the cards and the report without a new run. The drawings keep their labels in m and kPa.
Warnings and notes
The bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” The runs of this manual, from the example to eight rectangles, gave no warning.
The Warnings window.
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. For one rectangle:
Corner rectangles — the equation for Δσz as a signed sum of I3, with m = |a|/z and n = |b|/z, and the equation of I3.
I3 of each corner rectangle — one line for each of the four corners of the loaded area, with the distances a and b from A, m, n, I3 and the sign. A corner level with A in one direction gives zero.
Sum — the signed sum, Δσz = sum × q. When A is below the centre, a second result for the centre of a rectangle is given as a check.
Stress increase against depth — the same sum down the vertical line through A, as the table of the Results tab, with the row of A marked.
With several rectangles there is one step for each rectangle, with its four corners, then a Superposition step that adds the parts, then the depth table.
The Calculations tab: the given data.
The Calculations tab: the corner rectangles and the sum.
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, 4 Results, 5 Summary and 6 References. It carries three drawings, the Section with the lines of equal stress, the Profile and the 3D view, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has five sections: Scope and assumptions, Below a corner, Any point and the centre, Worked example and References. Each opens when you click its title (Open all and Close all open or close them together), 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
Rectangular loads
Rectangular load 1
q = 150 kPa; L = 3 m; B = 2 m; centre (0, 0)
Point A
Position and depth
x = 2.5 m; y = −1 m; z = 4 m
Depth profile
Depth
zmax = 12 m
Step 1: the corners seen from A
The rectangle runs from x = −1.5 to 1.5 and y = −1 to 1. A is at (2.5, −1), so A is level with the lower long side. Taking each corner of the rectangle in turn, the distances a (along x) and b (along y) from A to the corner are: corner (1.5, 1), a = −1 and b = 2; corner (−1.5, 1), a = −4 and b = 2; the two corners at y = −1 have b = 0, so their rectangles have no area and give zero.
Step 2: the influence factors
With z = 4 m, m = |a|/z and n = |b|/z. Corner (1.5, 1): m = 0.25, n = 0.5, I3 = 0.04753, subtracted. Corner (−1.5, 1): m = 1, n = 0.5, I3 = 0.12018, added. For this second corner the equation gives 2mn√(m² + n² + 1) = 1.5, the fraction 1.5/2.5 × 3.25/2.25 = 0.8667 and the arctangent of 1.5/2.0 = 0.6435, so I3 = (0.8667 + 0.6435)/4π = 0.12018.
The difference of the rounded factors is 0.07265; the module carries the unrounded values and shows 0.07264. These are the two cards of the Results tab and the steps of the Calculations tab. An independent numerical integration of the point-load solution over the rectangle gives the same 10.90 kPa, and the same integration at 1.2 m, 2.4 m and 12 m reproduces the table (7.668, 12.59 and 2.603 kPa).
Other points under the same rectangle
The same rectangle at z = 4 m: A below the centre gives 22.98 kPa (influence value 0.1532, four equal corner factors of 0.03830); A under a corner, at (1.5, 1), gives 16.06 kPa; A under the middle of a short side, at (1.5, 0), gives 18.08 kPa; A far away, at (10, 0), gives 0.2043 kPa. A 2 m square loaded with 150 kPa gives, below its centre at 1 m, 105.1 kPa, an influence value of 0.7009, the value for a point at a depth of half the side below the centre of a square (4 × I3(1, 1)).
Other unit systems
The same rectangle in SI · tf, with q = 15 t/m² (the other values unchanged), gives 1.090 t/m² and the influence value 0.07264. In US the default example, which is 3.13282 ksf on a rectangle 9.84252 ft by 6.56168 ft, gives 0.2276 ksf: 10.90 kPa is 0.2276 ksf. A check with round US numbers, q = 3 ksf, L = 10 ft, B = 6 ft, centred at the origin, A at x = 8 ft, y = −3 ft, z = 12 ft: the same independent integration gives 0.2266 ksf and an influence value of 0.07554, and the module gives 0.2266 ksf and 0.07554.
How each input changes the answer
Each row changes one thing in the example (one rectangle of q = 150 kPa, L = 3 m, B = 2 m centred at the origin, A at x = 2.5 m, y = −1 m, z = 4 m) and runs the module again, in SI · kN; where a row says two things, both change. The numbers are the ones the module gives, so you can reproduce any row; an independent numerical integration of the point-load solution over the rectangle agrees with every row to the digits shown.
The example with one change at a time
Change
Δσz (kPa)
Δσz/q
The example
10.90
0.07264
q = 300 kPa The stress is proportional to the load; the ratio is unchanged
21.79
0.07264
q = 75 kPa Half the load, half the stress
5.448
0.07264
L = 6 m A longer rectangle puts more on A
23.14
0.1543
L = 1.5 m A shorter one puts less
5.232
0.03488
L = 30 m A very long rectangle, close to a strip; A is level with its long side
41.14
0.2742
B = 4 m A wider rectangle: more of the load is near A
20.22
0.1348
B = 1 m A narrower one
5.554
0.03703
B = 30 m A very wide rectangle
38.32
0.2555
L = B = 30 m A is deep inside a large area: the ratio approaches 1
147.9
0.9857
L = 2, B = 3, A at (−1, 2.5) Sides and point turned together: the same as the example
10.90
0.07264
x = 0, y = 0 Larger than below the middle of a short side (next row) or a corner
22.98
0.1532
x = 1.5 m, y = 0 Below the middle of a short side: between the centre and the corner
18.08
0.1205
x = 1.5 m, y = 1 m Below a corner of the rectangle
16.06
0.1071
y = 0 A level with the middle of the long sides; A is closer to the middle
12.10
0.08068
y = +1 m The same as y = −1 m: the rectangle is symmetric about y = 0
10.90
0.07264
x = −2.5 m The same as x = +2.5 m
10.90
0.07264
x = 10 m, y = 0 Far from the rectangle the stress is a tiny part of the load
0.2043
0.001362
z = 1 m Beside the rectangle the stress is small near the surface
5.877
0.03918
z = 2 m It rises with depth first
11.96
0.07977
z = 8 m Then falls
5.028
0.03352
z = 20 m The stress keeps falling with depth
1.021
0.006806
z = 0.01 m At the surface beside the rectangle the stress is almost zero
0.00001479
0.0000001
x = y = 0, z = 1 m Below the centre near the surface the stress approaches q
116.2
0.7746
x = y = 0, z = 0.01 m At the surface under the rectangle the stress equals q
150.0
1.000
xc = 4 m A is then below a corner: the same as x = 1.5 m, y = 1 m
16.06
0.1071
Centre (4, 3), A at (6.5, 2) Both moved by (4, 3): the same as the example
10.90
0.07264
Several rectangles, SI · kN, from the example (second rectangle added with the values shown):
Adding rectangles
Added
Total (kPa)
Parts (kPa)
Rectangle 2: 150 kPa, 3 m × 2 m, centre at (5, 0) The mirror image of rectangle 1 about A
21.79
10.90 + 10.90
Rectangle 2: 75 kPa, 3 m × 2 m, centre at (5, 0)
16.34
10.90 + 5.448
Rectangle 2: 150 kPa, 3 m × 2 m, centre at (3, 0) Meets rectangle 1 at x = 1.5 m
30.63
10.90 + 19.73
Rectangle 2: 150 kPa, 3 m × 2 m, centre at (0, 2) Meets rectangle 1 at y = 1 m
16.18
10.90 + 5.283
Rectangles 2 and 3: 150 kPa, 3 m × 2 m, centres at (5, 0) and (10, 0)
22.45
10.90 + 10.90 + 0.6619
The pattern follows the equations. The stress is proportional to q; it depends on the sides and the position of A only through their ratios to z, so doubling every length leaves the ratio unchanged; at the same depth it is larger below the centre than below an edge or a corner; and with z near zero it equals q under the rectangle and zero beside it. Because it is built from the four corner terms, the result does not change when the rectangle and A are moved together or turned together, as the rows “L = 2 m, B = 3 m …” and “Centre at (4, 3) m” show. The depth of the profile zmax changes the table and the Profile view but not the stress at A: 10.90 kPa with zmax = 6 m and 10.90 kPa with zmax = 0.5 m, a depth the module extends to the depth of A in the table.
Messages: when the module stops, and what it notes
The module checks the numbers it needs before it calculates. When an input cannot be used, press Analyze and 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; nothing is calculated.
The window Analyze opens when a side is zero.
When
Message and what to do
L is 0
“Side of rectangle 1 along x (L1) must be at least 0.001 m.” Enter a side of at least 0.001 m.
B is 0
“Side of rectangle 1 along y (B1) must be at least 0.001 m.” Enter a side of at least 0.001 m.
L and B are 0
The window reads 2 input errors and lists both messages.
q is negative
“Load per unit area of rectangle 1 (q1) must be at least 0 kPa.” Enter zero or a positive load.
z is 0
“Depth of the point A (z) must be at least 0.001 m.” Enter a depth of at least 0.001 m.
zmax is 0
“Depth of the profile (zmax) must be at least 0.1 m.” Enter a depth of at least 0.1 m.
A box is empty
“Side of rectangle 1 along x (L1) is missing.” (and the same for q, x, y, z and the centre). Fill the box; for a rectangle you may instead take it out.
A new rectangle would overlap another
A message appears briefly: “The new rectangular load would lie on rectangular load 1 — they cannot overlap. Give it another position or width.” The rectangle is not added.
The messages use the numbered names of the rectangles, such as “rectangle 1” and “L1”, even when there is only one, and the limits in the units of the unit system chosen. When you type a position or a side into a rectangle already in the table, or drag it, and it would put two rectangles on one another, a message appears briefly: “That would put rectangular load 1 and rectangular load 2 on top of one another — the value is kept as it was.” and the value stays as it was. At eight rectangles + Add rectangular load is greyed out and the counter reads 8 of 8.
Notes and warnings
With every input inside its limits the module calculates, and the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” A load of zero is accepted and gives 0 kPa, with the influence value unchanged at 0.07264 in the cards; in the depth table the ratio column shows a dash except in the row of A, the Section and Contours views draw no lines or legend values, and the Profile view is a zero line. Very large values are accepted too: with q = 1,000,000 kPa and a rectangle of 1000 m × 1000 m the module gives 1,000,000 kPa below its centre, an influence value of 1.000, and a rectangle with a side of 0.001 m gives 0.003431 kPa at the point of the example.
Good practice and common mistakes
Use the module for a flexible, uniformly loaded rectangle. The pressure is q over the whole area. A stiff footing distributes its pressure differently; the module gives the stress for the pressure you enter, and a total load divided by the area is q.
Enter a pressure, not a force.q is a load per unit area: a column load of 600 kN on a 3 m × 2 m footing is q = 100 kPa.
Mind the axes.L is always along x and B along y, and the Section view cuts the plane y = yA. To look at another cut, change y.
Do not overlap rectangles. They may meet at a side but not lie on one another; to describe a stepped load, split it into rectangles that meet.
Add the overburden. The result is the stress increase; for the total vertical stress add the soil weight above A.
Look at more than one depth. Beside a loaded area the stress first rises with depth and then falls, so a single depth can mislead; read the table or the Profile view.
Place A with the drawing, then type exact values. Dragging snaps to 5 cm; type the coordinates when you need a given place.
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 stress below a rectangular load calculated?
The module uses the closed-form integral of Boussinesq’s point-load solution over a rectangle, which gives the stress below one corner as Δσz = qI3. For any other point it adds and subtracts four rectangles that share a corner above A. For the module’s example it gives 10.90 kPa.
What is the influence value Δσz/q?
It is the stress increase divided by the pressure of the rectangle. It depends only on the sides and the position of A divided by the depth, is 1 at the surface under the rectangle, and is the number to use if you have a table of influence values. The module gives it for one rectangle; with several rectangles it gives the stress of each.
Does the result include the weight of the soil?
No. It is the increase caused by the rectangles only. Add the overburden pressure of the soil above A for the total vertical stress.
How many rectangles can I add, and can they overlap?
Up to eight, each with its own q, L, B and centre. They may meet at a side but not overlap; the module refuses a rectangle that would lie on another. The stresses add.
Can I turn a rectangle?
The sides of every rectangle are along x and y. Turn the whole model instead: choose the axes along the sides, and place A in the same axes. For a rectangle at another angle, or an irregular shape, use Influence chart.
Can I drag the rectangle and the point on the drawing?
Yes, before Analyze. In the Section or Contours view drag the rectangle along x and the rings at its ends to change L; in the 3D view drag it over the ground and use all four rings to change L and B. Drag the blue point to place A and the dashed line to set the depth of the profile. After Analyze the rectangles are locked, but A and the dashed line stay free.
What does the Section view show when A is not on the rectangle?
The plane through A parallel to the x axis. A red note under the ground line says whether the plane cuts the load (“the section cuts the loaded area”) or not (“the loaded area lies beside this section (y outside the rectangle)”). With several rectangles, one the plane misses is drawn dashed with its distance, such as “6.00 m beside the plane”, and still acts on the section from beside. The point A stays on that plane, so moving A in y moves the plane.
What values does the module accept?
q zero or more, L and B at least 0.001 m, z at least 0.001 m, zmax at least 0.1 m, and x, y, xc and yc any value. Anything else stops the analysis with a message that names the value.
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. Stresses are in kPa, t/m² or ksf and lengths in m or ft. The default example gives 10.90 kPa, 1.090 t/m² with q = 15 t/m², or 0.2276 ksf.
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, profile 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. It also reopens in the analysed state if nothing was changed after Analyze; after moving A or the profile depth, press Analyze again. Reset to defaults brings back the example at once; Save to file and Quick Save keep copies.
When is a rectangle the wrong model?
When the load is round (use Circle, any point), long and narrow compared with the depth you ask about (use Vertical strip load), of an irregular shape (use Influence chart), or on layered ground with a stiff layer that spreads the load (use Westergaard areas).
Calculate the vertical stress increase at a point below or beside a flexible area of any polygonal shape, drawn on Newmark’s influence chart: the elements the load covers, the stress at A, the lines of equal stress, the profile with depth and a 3D view. This manual walks through the screen, every input, the results and the report, with a worked example you can reproduce.
Influence chart gives the increase in vertical stress at a point of the ground below, or beside, a flexible area of any polygonal shape that carries a uniform pressure q on the ground surface. The ground is taken as a homogeneous, isotropic, elastic half-space (Boussinesq). You enter the corners of the area in plan and the point A with its depth z; the module draws the area on Newmark’s chart at the right scale, shades the elements of the chart that the load covers, and gives Δσz at A, the lines of equal stress, the profile with depth under A and a 3D view.
Newmark’s chart (1942) is built from the stress below the centre of a uniformly loaded circle. Turned around, that solution gives the radius R of the circle that produces a chosen ratio Δσz/q:
R/z = √[ (1 − Δσz/q)−2/3 − 1 ]
Circles with these radii for Δσz/q = 0.1, 0.2 … 0.9, cut by 20 radial lines, make 200 elements, each of which, loaded alone, gives the same share IV = 1/200 = 0.005 of q at the depth z below the centre. The plan of the loaded area is drawn at the scale in which the depth z is the unit length AB of the chart, with the point A over the centre. If the area covers M elements:
Δσz = IVqM = 0.005 qM
The module takes M from the exact integral of Boussinesq’s solution over the polygon, which is what a perfect count of the elements would give, and shows beside it the count read from the shaded chart.
What you can describe
One uniform pressure q on one loaded area of 3 to 6 corners, given in plan by their x and y in order around the area (clockwise or anticlockwise). The area may be convex or concave.
The point A in plan, xA and yA, under the area or beside it, and its depth z below the loaded level.
The depth of the profilezmax, down to which the Profile view and the table under A run.
The result is the increase caused by the load only; the overburden pressure of the soil above A is not included, so add it where your design needs the total stress. The module draws the chart, the section with the lines of equal stress, the profile and a 3D view, writes out the working and prepares a PDF or Word report.
Which module for which load
A load of any shape in plan, one pressure: this module. A rectangle with the corner method and several loads added together: Rectangle; a circle: Circle, any point.
The module takes one area with one pressure. For several areas, add the stresses of separate runs, since the stresses of separate loads add on an elastic ground.
Before you start: what you need
Have these ready from your design:
What
Notes
The load per unit area q
From your design loading. A pressure on a flexible area, zero or more; for a column footing, the column load divided by the footing area.
The corners of the area
From the plan of the footing, slab or fill. Three to six corners as x and y in one co-ordinate system, in order around the area.
The point A: xA, yA, z
Where you want the stress, in the same system; z is the depth below the loaded level, at least 0.001 m.
Quick start in eight steps
Open the module and choose the units
Open Influence chart from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the load
In Load type the pressure q.
Enter the loaded area
In Loaded area (plan) type the x and y of corners 1 to 3, and of corners 4 to 6 if the shape needs them. The Influence chart view shows the plan on the chart as you type.
Place the point A
In Point A type xA, yA and z; or drag the blue point on the Section drawing (before or after Analyze).
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line at the foot of the Section drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the load and the corners lock; the point A and the profile depth stay free.
Read the answer
The Results tab gives Δσz at A, the number of elements M and a table of the stress against depth under A. The Influence chart view shows the shaded elements; the lines of equal stress are on the Section drawing.
Take the working and the report
The Calculations tab writes out the working; the Report tab saves a PDF or a Word file.
The screen: chart, section, profile and 3D
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:
Influence chart is Newmark’s chart, 200 elements, with the plan of the loaded area drawn at the scale AB = z and the point A at the centre. Before Analyze it shows the plan only; after it, the part of each element covered by the load is shaded, and a block at the right gives Elements covered, the two values of M and the working of Δσz.
Section is the vertical plane y = yA through A, parallel to x: the extent of the loaded area along x on the ground, the point A with the dimensions xA and z, and the dashed line of the profile depth. After Analyze it adds the lines of equal stress and a label with Δσz at A.
Profile is the curve of Δσz against depth under A, with a marker and a dashed line at the depth of A.
3D is an oblique view of the loaded area on the ground, the point A with its co-ordinates and an element at the depth z with the stress on it.
The Influence chart view follows the inputs: change a corner, z or the position of A and the plan moves on the chart at once. Before Analyze the Section view says “The stress and the lines of equal stress appear after Analyze”, the Profile view “The profile appears after Analyze” and the 3D view “The stresses on the element appear after Analyze”.
The chart before Analyze: the plan at the scale AB = z.
The chart after Analyze: the shaded elements and the working.
Section after Analyze: the lines of equal stress; the line through A is blue.
Profile after Analyze.
3D after Analyze.
The Show chips
Above the Section drawing a bar has two chips: Lines of equal stress and Dimensions. The lines chip is off and greyed out until you press Analyze, and then turns on; Dimensions is on by default. The Influence chart, Profile and 3D views have no chips.
Section before Analyze.
Dragging on the drawing
The loaded area is entered in the Model tab: the corners cannot be dragged. What can be dragged is on the Section drawing, whose hints read “Drag the point A to place it.” and “Drag the dashed line to set the depth of the profile (zmax).” What each pull does, as the module runs it:
The point A. Drag the blue point: xA and z follow in steps of 5 cm (the boxes update as you pull) and yA stays as typed, because the section is the plane y = yA. The depth stops at 0.05 m at the top, and xA and z stop at the edges of the drawn frame. With the example, pulling A far to the upper left gives xA = −3.9 m and z = 0.05 m, and far to the lower right xA = 3.9 m and z = 7.5 m.
The dashed line. Drag it up or down to set zmax; it stops at 0.5 m and has no upper limit.
After Analyze the load and the corners are locked, but the point A and the dashed line stay free, and so do the boxes xA, yA, z and zmax: pull A and the stress there is calculated at once, and the lines of equal stress follow.
Nothing else is dragged. The Influence chart, Profile and 3D views have no drag handles.
Zoom
Each view has −, a percentage and + at the lower right, and a button that fits the whole drawing in the pane. Zoom is a property of the drawing only: it is not saved with the model and does not change the report.
Entering the model, section by section
The Model tab has five sections: Model data, Load, Loaded area (plan), Point A and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below the last one. 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 3.13282 ksf for a pressure of 150 kPa, or −4.92126 ft for a corner at −1.5 m.
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.
Load
One number, the pressure q of the loaded area. The note in the section reads that it is a uniform pressure on a flexible area at the ground surface, and for a column footing the column load divided by the footing area. The example is 660 kN on 9 m², which the box shows as 73.3333.
Load.
Load field
Field
What to enter
Load per unit area q kPa · t/m² · ksf · default 73.3333
A uniform pressure on a flexible area at the ground surface; zero or more.
Loaded area (plan)
Six pairs of boxes, one pair for each possible corner. List the corners in order around the area, clockwise or anticlockwise; the result is the same either way. Corners 1 to 3 are required; corners 4 to 6 are optional, so leave them empty for a triangle, a quadrilateral or a pentagon. The area may be convex or concave. A corner counts only when both its x and its y are filled in: with one of the two empty the corner is left out, with no message, so look at the plan in the Influence chart view to see which corners the module has taken.
Loaded area: six corners, three of them required.
Loaded area fields
Field
What to enter
Corner 1 of the loaded area: x x1 m · ft · default −1.5
Corners 1, 2 and 3 are required.
Corner 1: y y1 m · ft · default 0
Corner 2: x and y x2, y2 m · ft · default −1.5 and 3
Corner 3: x and y x3, y3 m · ft · default 1.5 and 3
Corner 4: x and y x4, y4 m · ft · default 1.5 and 0
Optional: the x labels of corners 4 to 6 say “(optional)” and empty boxes show “optional”.
Corner 5: x and y x5, y5 m · ft · default empty
Optional.
Corner 6: x and y x6, y6 m · ft · default empty
Optional.
The point A is placed in the same co-ordinate system, so you may put the origin where you like: at a corner of the footing, at its centre, or at a point on the site plan.
Point A
The three co-ordinates of the point where you want the stress: its position in plan and its depth below the loaded level. A can be under the area or beside it. The Section drawing shows xA and z as dimensions.
Point A.
Point A fields
Field
What to enter
x of the point A xA m · ft · default −1.5
In plan, in the co-ordinate system of the corners.
y of the point A yA m · ft · default 1.5
In plan. The Section view is the vertical plane through this value of y.
Depth of the point A below the loaded area z m · ft · default 3
At least 0.001 m below the loaded level.
Depth profile
The depth down to which the Profile view and the table of the stress under A run.
Depth profile.
Depth profile field
Field
What to enter
Depth of the profile zmax m · ft · default 12
At least 0.1 m. The Profile view and the table under A go from the loaded level down to this depth, or to the depth of A if that is deeper; the Profile view runs a little past A.
Choosing the shape and the point
The module has no options to pick: the choices are the shape of the area and the place of A. The stress depends on the plan of the area in units of z, so only the shape, the size relative to the depth and the position of A matter.
The shape of the area
Three corners give a triangle, four a quadrilateral, five and six a polygon with more sides. With the example point and depth, the triangle with corners (−1.5, 0), (−1.5, 3) and (1.5, 0) gives 11.27 kPa, the house shape of five corners (−1.5, 0), (−1.5, 3), (0, 4.5), (1.5, 3), (1.5, 0) gives 20.01 kPa, and an L shape of six corners, the square with the corner (0 to 1.5, 0 to 1.5) cut away, gives 14.97 kPa; the square gives 17.63 kPa.
Concave shapes are accepted. The L shape above has a re-entrant corner and the chart shades only the elements that the load really covers.
The order of the corners can run either way round: the square entered anticlockwise gives 17.63 kPa, as clockwise.
An L shape of six corners on the chart.
A beside the area: the plan lies off the centre of the chart.
The place of A
Under the area, the stress is high and falls with depth. Beside the area it is smaller and falls with distance: A at 1.5 m outside the edge of the example square gives 6.942 kPa, 4.5 m outside 0.7226 kPa.
The Influence chart view shows where A is: it is the centre of the chart, and the plan sits off the centre when A is beside the area.
Each depth has its own chart. The plan is drawn at AB = z, so changing z redraws the plan at another scale. The depth table of the Results tab gives the stress at ten depths, each with its own M.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” A message appears briefly, “Analysed — see the Results tab and the Calculations.” The module stays on the Model tab: open Results for the answer.
The bar after Analyze.
The cards of the example.
Card
Symbol
The example
Vertical stress increase at A
Δσz
17.63 kPa
Elements of the chart covered
M
48.07
Below the cards a Values, step by step table gives the result of the step that ends in a stress, and the Vertical stress increase under A table gives z, M and Δσz at ten equal steps from the loaded level down to zmax, with the row of A inserted and highlighted. In the example the depths are 1.2, 2.4, 3.0 (A), 3.6 … 12 m, and the stress falls from 31.91 kPa at 1.2 m to 2.055 kPa at 12 m.
The depth table of the example.
Reading the chart view
After Analyze the Influence chart view shows Elements covered with two values of M: “M = 48.0 (read from the chart)” and “M = 48.07 (exact integral)”, then the working “Δσz = IV·q·M”, “= 0.005 × 73.33 × 48.07” and “= 17.63 kPa”. The stress is calculated from the exact value; the first is the count taken from the shading. When the load covers the whole chart, M = 200 and Δσz = q: a very large area (x from −1000 to 1000 m, y from 0 to 2000 m, A at its middle) gives M = 200.0 and 73.33 kPa.
Using the numbers
The result is an increase, not the total stress: add the overburden pressure of the soil above A for the total vertical stress.
The ratio Δσz/q = 0.005 M is the part of the surface pressure that reaches A. With a load of zero the stress is 0 kPa and M is shown as a dash, because the ratio has no meaning without a load.
Values are rounded where they are drawn, so the last digit of a figure on the drawing can differ by one from the card.
Change xA, yA, z or zmax after Analyze and everything follows at once; to change the load or a corner, press Edit inputs, which unlocks them and clears the results.
Changing the unit system after Analyze re-states the cards and the report without a new run. The drawings keep their labels in m and kPa.
Warnings and notes
The bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” None of the runs of this manual gave a warning.
The Warnings window.
The Calculations tab, the report and the Theory tab
The Calculations tab opens with Given and what is sought (the pressure, the corners in order, the point A and the depth) and then writes out the working in four numbered steps, ending with the answer:
Plan at the scale of the chart — the 200 elements and their influence IV = 0.005, the radius relation, the area of the plan, AB = z and whether A is under or outside the loaded area.
Elements covered — M from the exact integral over the polygon, with your numbers.
Stress at A — Δσz = IVqM with your numbers.
Stress increase against depth — the same at ten depths down to zmax, the table of the Results tab with the row of A marked.
The Calculations tab: the given values and the first step.
Step 2.
The stress at A and the depth table.
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, 4 Results, 5 Summary and 6 References. It carries four drawings, the chart, the section with the lines of equal stress, the profile and the 3D view, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has five sections: The idea, Newmark’s chart, Using the chart, Worked example and References. Each opens when you click its title (Open all and Close all open or close them together), and the figures in it are computed from the equations; it includes a table of R/z for the pressure ratios 0.05 to 0.95.
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
Load
Pressure
q = 660/9 = 73.3333 kPa
Loaded area (plan)
Corners 1 to 4
(−1.5, 0), (−1.5, 3), (1.5, 3), (1.5, 0) m: a 3 m × 3 m square
Point A
Position and depth
xA = −1.5 m, yA = 1.5 m, z = 3 m: the middle of one side
Depth profile
Depth
zmax = 12 m
Step 1: the plan on the chart
The area is 9.000 m², the scale is AB = z = 3.00 m, and A lies on the edge of the area, so the plan sits against the centre of the chart (the Influence chart view).
Step 2: the number of elements
Through A, the line x = −1.5 m splits the square into two rectangles of 1.5 m × 3 m, each with a corner at A. For one of them m = 1.5/3 = 0.5 and n = 3/3 = 1, and the influence factor of a corner is
With m² + n² = 1.25 the root is √2.25 = 1.5, the first term is 1 × 1.5/2.5 × 3.25/2.25 = 0.8667 and the arctangent is tan⁻¹(0.75) = 0.6435 rad, so I3 = (0.8667 + 0.6435)/(4π) = 0.12018. Two such rectangles give Δσz/q = 0.24035, and
These are the two cards of the Results tab and the step of the Calculations tab. The same hand calculation at other depths gives the table of the Results tab: at z = 6 m the rectangles are 1.5 m × 3 m seen at m = 0.25 and n = 0.5, and the module gives 6.971 kPa; at 12 m, 2.055 kPa. An independent integration of Boussinesq’s solution over the square, done outside the module, gives 17.6257, 6.9715 and 2.0553 kPa at 3, 6 and 12 m.
The same shape at another size
Only the plan in units of z matters. A 6 m × 6 m square with A at the middle of one side (xA = −3 m, yA = 3 m) at z = 6 m gives 17.63 kPa and M = 48.07: the same as the example. At z = 3 m it gives 29.32 kPa, and a 1.5 m × 1.5 m square at z = 3 m gives 6.971 kPa, the value of the 3 m square at 6 m.
A long strip, as a check
A rectangle of 3 m × 30 m, with A at the middle of a long side (xA = −1.5 m, yA = 15 m, z = 3 m), is nearly the strip of the Vertical strip load module with B = 3 m. The module gives 29.98 kPa; the strip equation gives 30.00 kPa under the edge of an infinitely long strip, and with A under the middle line (the module’s 40.29 kPa) 40.32 kPa.
Other unit systems
The default example in SI · tf gives 1.797 t/m² (17.63 kPa is 1.797 t/m²), and in US 0.3681 ksf, with M = 48.07 in both. Two checks by hand with round numbers:
SI · tf.q = 10 t/m² on the example square, A at the middle of one side, z = 3 m: Δσz/q = 0.24035, so Δσz = 2.4035 t/m². The module gives 2.404 t/m².
US.q = 2 ksf on a square of 10 ft × 10 ft (corners at ±5 ft and 0 to 10 ft), A at the middle of one side (xA = −5 ft, yA = 5 ft) and z = 10 ft: the same ratios, Δσz = 0.24035 × 2 = 0.4807 ksf. The module gives 0.4807 ksf.
How each input changes the answer
Each row changes one thing in the example (the 3 m × 3 m square, q = 73.33 kPa, A at (xA, yA) = (−1.5, 1.5) m, z = 3 m) and runs the module again, in SI · kN. The numbers are the ones the module gives, so you can reproduce any row; an independent integration of Boussinesq’s solution over the same polygon, done outside the module, agrees with every row.
The load and the depth
Change
Δσz (kPa)
M
The example The reference
17.63
48.07
q = 150 kPa Proportional to the load; M unchanged
36.05
48.07
q = 50 kPa Proportional to the load; M unchanged
12.02
48.07
q = 0 No load, no stress; M is shown as a dash
0
—
z = 0.01 m At the surface, on the edge of the area: half of q
36.67
100.0
z = 1 m Shallower: the plan covers more of the chart
33.47
91.29
z = 1.5 m
29.32
79.98
z = 6 m Deeper: the plan covers fewer elements
6.971
19.01
z = 12 m
2.055
5.605
z = 30 m Under 1 per cent of q
0.3465
0.9451
The place of A
Change
Δσz (kPa)
M
xA = 0 Under the middle of the square: the highest stress at this depth
24.65
67.22
xA = −1.5 m, yA = 0 Under a corner of the square
12.85
35.04
xA = −3 m 1.5 m outside the edge
6.942
18.93
xA = −6 m 4.5 m outside the edge
0.7226
1.971
xA = −31.5 m 30 m outside the edge: a negligible part of q
0.0002708
0.0007386
yA = 6 m 3 m beyond the end of the square, in line with the edge
1.813
4.944
The size and the shape of the area
Change
Δσz (kPa)
M
6 m × 6 m square, A at the middle of a side, z = 3 m Twice the width at the same depth
29.32
79.98
6 m × 6 m, z = 6 m Everything doubled: the same as the example
17.63
48.07
1.5 m × 1.5 m, z = 3 m Half the size: the same as the example at z = 6 m
6.971
19.01
3 m × 30 m, A at the middle of a long side A long strip: the strip value is 30.00 kPa
29.98
81.76
Triangle (−1.5, 0), (−1.5, 3), (1.5, 0) Half the area of the square, 3 corners
11.27
30.74
House shape of five corners The square with a roof of 1.5 m added
20.01
54.56
L shape of six corners The square with a 1.5 m corner cut away
14.97
40.84
The square, corners anticlockwise The order of the corners makes no difference
17.63
48.07
The depth of the profile
Change
Δσz (kPa)
M
zmax = 6 m Changes the table and the profile only
17.63
48.07
zmax = 2 m Less than z: the table runs to the depth of A
17.63
48.07
zmax = 0.5 m The stress at A is unchanged
17.63
48.07
The pattern follows the theory. The stress is proportional to q; it depends on the plan in units of z only, so doubling every length together leaves M unchanged; it is highest under the middle of the area; under the area it falls with depth, and beside the area it first rises with depth and then falls; and at the surface it equals q under the area and half of q on its edge. The depth of the profile zmax changes the table and the Profile view but not the stress at A.
Messages: when the module stops, and what it notes
The module checks the numbers it needs before it calculates. When an input cannot be used, press Analyze and a window titled Cannot analyse — 1 input error (or 2 input errors …) lists each message; a box whose value is out of range is marked in red, and a box left empty is named in the message. Nothing is calculated.
The window Analyze opens when z is 0.
When
Message and what to do
q is negative
“Load per unit area (q) must be at least 0 kPa.” Enter zero or a positive load.
q is empty
“Load per unit area (q) is missing.” Fill the box.
z is 0 or below 0.001 m
“Depth of the point A below the loaded area (z) must be at least 0.001 m.” Enter a depth of at least 0.001 m.
zmax is below 0.1 m
“Depth of the profile (zmax) must be at least 0.1 m.” Enter a depth of at least 0.1 m.
A co-ordinate of A is empty
“x of the point A (xA) is missing.” and likewise “y of the point A (yA) is missing.”
A required corner is empty
“Corner 1 of the loaded area: x (x1) is missing.” Corners 1 to 3 are required; with corner 3 emptied the window reads “Corner 3 of the loaded area: x (x3) is missing. Corner 3: y (y3) is missing.”
Where a value must reach a limit the message gives it in the units of the unit system chosen. A second kind of stop covers the geometry:
The window opens when the corners lie on one line.
The loaded area is zero. With corners 1 to 3 on one line (for example (0, 0), (1, 0) and (2, 0)) or all four on one line, the window is titled “Cannot analyse — a condition of the solution is not met” and reads “The loaded area is zero. The corners lie on one line. Enter the corners in order around the area.” followed by “Nothing was calculated. Change the inputs marked in red, then press Analyze again.” An outline that crosses itself, such as the square entered as (−1.5, 0), (1.5, 3), (1.5, 0), (−1.5, 3), whose parts cancel to a net area of zero, gets the same message.
A corner with one coordinate empty is not an error: it is left out. With corner 4 as x = 1.5 and no y, the module takes the three corners left, a triangle, and gives 11.27 kPa.
Notes and warnings
With every input inside its limits the module calculates, and the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” Large values are accepted: q = 1,000,000 kPa on the example gives 240,351 kPa, and a point very far from the area, at (1000, 1000) m, gives a value of the order of 1e-12 kPa, shown in scientific notation.
Good practice and common mistakes
Use the module for a flexible area with one uniform pressure. The solution is for a flexible load; a stiff footing distributes its pressure differently, and the module gives the stress for the pressure you enter.
Enter a pressure, not a force.q is a load per unit area: divide a column load by the area of the footing.
List the corners in order around the area. Either direction is accepted. Corners listed in a criss-cross order draw a bow-tie, not the shape you meant; look at the plan in the Influence chart view before you press Analyze.
Fill both co-ordinates of every corner you want. A corner with one of the two empty is left out without a message.
Give A a position clearly under or beside the area rather than exactly on its outline; a point a millimetre inside or outside gives the expected value (for the example square 17.70 kPa at (0, 2.99) and 17.55 kPa at (0, 3.01)).
Mind the depth and the scale. The plan is drawn at AB = z, so a change of z redraws the plan on the chart; the depth table gives ten depths, each with its own count.
Add the overburden. The result is the stress increase; for the total vertical stress add the soil weight above A.
Several areas, several runs. The module takes one area with one pressure; for a second area, run it again and add the two stresses at the same point.
Place A with the drawing, then type exact values. Dragging snaps to 5 cm; type the co-ordinates when you need a given place.
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 does Newmark’s influence chart work?
The chart has circles for the stress ratios 0.1, 0.2 … 0.9 below the centre of a loaded circle and 20 radial lines, which make 200 elements of equal influence 0.005. The plan of the loaded area is drawn at the scale in which the depth z equals the unit length AB of the chart, with the point A over the centre. The stress is Δσz = 0.005 qM, M being the number of elements the plan covers. For the module’s example M = 48.07 and Δσz = 17.63 kPa.
Does the module count the elements, or calculate the stress exactly?
It takes M from the exact integral of Boussinesq’s solution over the polygon, which is what a perfect count would give, and shows beside it the count read from the shaded chart (“M = 48.0 (read from the chart)” and “M = 48.07 (exact integral)” for the example). The stress uses the exact value.
How many corners can the loaded area have, and what shapes?
Three to six, entered as x and y in order around the area, clockwise or anticlockwise. Corners 1 to 3 are required, 4 to 6 optional. The area may be convex or concave.
Where do I put the origin of the co-ordinates?
Anywhere, as long as the corners and the point A use the same system. The default example has the origin at the middle of one side of the footing and A at the middle of the adjacent side.
Can I put A outside the loaded area?
Yes. A may be under the area or beside it; the plan then sits off the centre of the chart. For the example square, A 1.5 m outside an edge gives 6.942 kPa and 4.5 m outside 0.7226 kPa.
Does the result include the weight of the soil?
No. It is the increase caused by the load only. Add the overburden pressure of the soil above A for the total vertical stress.
Can I drag things on the drawing?
On the Section drawing you can drag the blue point A to place it and the dashed line to set the depth of the profile; they move in 5 cm steps and stay free after Analyze. The corners of the area cannot be dragged: you enter them in the Model tab. The other views have no handles.
What do the lines of equal stress show?
Each line joins the points of the vertical plane through A with the same Δσz, labelled in kPa; the line through A is blue and carries the value at A. Switch them on and off with the Lines of equal stress chip.
What values does the module accept?
q zero or more, z at least 0.001 m, zmax at least 0.1 m, the co-ordinates any numbers, three corners at least, and an area that is not zero. Anything else stops the analysis with a message that names the value.
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. Stresses are in kPa, t/m² or ksf and lengths in m or ft. The default example gives 17.63 kPa, 1.797 t/m² or 0.3681 ksf.
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 chart, the section, the profile and the 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.
Calculate the vertical stress increase at a point below one or several vertical point loads on layered ground, in the Westergaard model of soil with thin rigid horizontal layers: the stress at A, the lines of equal stress, colour contours, the profile with depth and a 3D view, the Results, the Profile view and the report giving Boussinesq’s value beside it. This manual walks through the screen, dragging, every input, the results and the report, with a worked example you can reproduce.
Westergaard point load gives the increase in vertical stress at a point of the ground below one or several vertical point loads on the ground surface, in the ground model of Westergaard (1938): an elastic soil reinforced by numerous thin, rigid horizontal layers that allow no lateral strain. It stands for a clay with thin seams of sand, whose horizontal deformation the stiffer layers hold back. You choose the point A by its position x, y on the ground surface and its depth z, and the module gives Δσz there, the lines of equal stress, the colour contours, the profile with depth under A and a 3D view; the Results, the Profile view and the report also give the value that Boussinesq’s solution (a uniform, isotropic half-space) gives for the same load and point, so you see what the layering changes.
For one load P at the horizontal distance r from A and the depth z, in soil whose Poisson’s ratio between the rigid layers is μs:
I5 is the influence value. It depends on r/z and on μs only. With several loads each acts alone with its own distance r from A, and the stresses are added (the ground is elastic, so they superpose).
What you can describe
One to eight vertical point loads, each with its own force P and its own position xL, yL on the ground surface; no two loads may stand at the same place.
The point A: its position x, y on the ground surface and its depth z.
The Poisson’s ratio μs of the soil between the rigid layers.
The depth of the profilezmax, down to which the Profile view and the table under A run.
The result is the increase caused by the loads only; the overburden pressure of the soil above A is not included, so add it where your design needs the total stress. The module draws the section with the lines of equal stress, the colour contours, the profile and a 3D view, writes out the working and prepares a PDF or Word report.
Which module for which load
The same loads on an ordinary uniform half-space: Point load (Boussinesq). Use this module when the ground is layered in the way described above; its Boussinesq column lets you compare the two.
From your design loading. Vertical forces on the ground surface, zero or more, up to eight. A load of zero is accepted and adds nothing.
Their positions xL, yL
From the layout. On the ground surface, in one set of coordinates with the point A; any value; no two loads at the same place.
The point A: x, y, z
Where you want the stress. x and y in the coordinates of the loads; z at least 0.001 m below the surface.
Poisson’s ratio μs
Your judgement of the layered soil. From 0 to 0.49; at 0.5 the soil between the layers could not deform at all, so the module stops at 0.49.
Quick start in seven steps
Open the module and choose the units
Open Westergaard point load from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the loads
In Point loads the first load is already in the table. Press its row to change P, xL and yL. To add another, fill the New point load form and press + Add point load.
Place the point A
In Point of interest type x, y and z, or drag the blue point on the Section drawing.
Set Poisson’s ratio
In Westergaard material type μs (0 to 0.49).
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line at the foot of the Section drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the loads lock; the point A and the profile depth stay free.
Read the answer
The Results tab gives Δσz at A with the Boussinesq value beside it, I5 for one load, the part from each load for several, and a table of the stress against depth under A. The lines of equal stress are on the Section drawing, the colour bands in Contours. The Calculations tab writes out the working and the Report tab saves a PDF or a Word file.
The screen: section, contours, profile and 3D
The layout is the one every GeoTerva module shares, described in the workspace guide. The drawing has four views, chosen with the buttons above it:
Section is a vertical cross section with the load arrow(s) on the ground, the point A with its dimensions and the dashed line of the profile depth. For one load it is the plane through the load and A, with the horizontal distance r from the load as its axis (the solution is the same all round the load). For several loads it is the vertical plane y = yA through A, parallel to x; loads that stand beside the plane are marked “beside the plane” with their distance and act on it as well. After Analyze it adds the lines of equal stress and a label with Δσz at A.
Contours is the same section filled with colour bands of Δσz on a logarithmic scale, with a legend in kPa.
Profile is the curve of Δσz against depth under A for Westergaard’s solution (solid) and Boussinesq’s (dashed), with a marker at the depth of A.
3D is an oblique view of the ground block with the axes x, y and z, the load arrow(s), the distances from the load to A and a small element at A with the stress on it.
Before Analyze the drawings follow the inputs and say “The stress and the lines of equal stress appear after Analyze” (Section), “The colour contours of the stress appear after Analyze” (Contours), “The profile appears after Analyze” (Profile) or “The stresses on the element appear after Analyze” (3D).
Section before Analyze.
Section after Analyze: the lines of equal stress; the line through A (0.011 kPa) is blue.
Contours after Analyze.
Profile after Analyze: Westergaard solid, Boussinesq dashed.
3D after Analyze.
The 3D view ends with the line “A negative Δσ is a decrease (tension) in the elastic solution.” The solution of this module gives positive values for the loads it describes.
The Show chips
Above the Section drawing a bar has two chips: Lines of equal stress and Dimensions. The lines chip is off and greyed out until you press Analyze, and then turns on; Dimensions is on by default. The Contours view has the same two chips under the heading Show; there the lines chip is off by default, so that the lines can be laid over the colours. The Profile and 3D views have no chips.
Contours with the lines of equal stress laid over the colours; the labels sit on blank boxes, so read the values in the Section view.
Dragging on the drawing
The hint under the Section drawing reads “Drag the point A to place it” and “Drag the dashed line to set the depth of the profile”; with several loads it also tells you to drag a load to move it. What each pull does, as the module runs it:
The point A, one load. Drag the blue point in the Section or Contours view: it moves in the plane through the load and A, so x and y change together (the direction from the load stays, the distance r follows the pointer, and x and y are rounded to 1 cm) and z follows in steps of 5 cm. From the example, pulling A 60 px right and 40 px down gave x = 3.9 m, y = 5.2 m, z = 5 m. The depth stops at 0.05 m at the top and at the foot of the drawn frame at the bottom; the distance stops at the sides of the frame.
The point A, several loads. The Section is the plane y = yA, so a pull changes x and z only; y stays.
The dashed line. Drag it up or down to set zmax; it stops at 0.5 m at the top.
A load, several loads. In the Section and Contours views each load arrow can be dragged sideways: xL follows in steps of 5 cm and yL stays. In the 3D view a load can be dragged over the ground and xL and yL both change.
A single load cannot be dragged in the Section or Contours view (its arrow is the axis of the drawing); in the 3D view it can.
No two loads on one place. A pull that would put a load on another stops at the nearest free place and a message says “Another point load already stands at this position — the value is kept as it was.”
After Analyze the loads are locked: the load handles are gone. The point A and the dashed line stay free, and so do the boxes x, y, z and zmax.
Zoom
Each view has −, a percentage and + at the lower right, and a button that fits the whole drawing in the pane. Zoom is a property of the drawing only: it is not saved with the model and does not change the report.
Entering the model, section by section
The Model tab has five sections: Model data, Point loads, Point of interest, Westergaard material and Depth profile. Each opens when you click its title; Next and Back at the foot walk you through them, and Analyze waits below the last one. Lengths are in m in both SI systems and in ft in US; forces are in kN, tf and kip for SI · kN, SI · tf and US. Defaults are shown in SI; in US the boxes show the converted values, for example 20 m is 65.6168 ft and the form opens with 1.12404 kip at 13.1234 ft. In US the row of the Point loads table keeps the labels kN and m although the numbers are converted; read the unit in the form.
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.
Model data.
Point loads
The section is a form over a table. The New point load form takes P, xL and yL; + Add point load puts the load into the Point loads table, and a counter beside the button shows how many of the eight loads there are. The table lists each load with its force and position. Press a row to change that load in the form (the form then reads Editing · Point load 1 and offers Done and Delete point load; the note says that changes apply as they are entered), or press × to take a load out. The first load is already in the table when the module opens, and Delete point load in the form and × in the table are greyed out while it is the only one.
Point loads: the form over the table.
Point load fields (the form, and each row of the table)
Field
Default
What to enter
Point load P kN · tf · kip
5 (load 1) · 5 (form)
The vertical force on the ground surface. Zero or more.
Position along x xL m · ft
0 (load 1) · 4 (form)
Any value. A load added later starts at 4 m.
Position along y yL m · ft
0
Any value.
A row of the table pressed: the load is edited in the form.
The loads are listed in the order you add them, and their numbers name the results “From point load 1”, “From point load 2” and so on. No two loads may stand at the same place: adding a load at a position that is taken, or typing such a position into a row, is refused (see Messages).
Two loads in the table.
Point of interest
The three coordinates of the point where you want the stress. A is under the ground surface at the position x, y and the depth z; the drawing shows the same distances as dimensions. For one load the solution depends on r = √[(x − xL)² + (y − yL)²] and z only.
Point of interest.
Point of interest fields
Field
Default
What to enter
Point A: x on the ground surface x m · ft
3
In the same coordinates as the loads.
Point A: y on the ground surface y m · ft
4
In the same coordinates as the loads.
Depth of the point z m · ft
4
At least 0.001 m below the ground surface.
Westergaard material
The one soil constant: Poisson’s ratio μs of the soil between the rigid layers. The note in the section explains the medium: elastic soil with closely spaced thin rigid horizontal sheets that allow no lateral strain, an idealisation of a clay with thin sand seams, and μs must stay below 0.5.
Westergaard material.
Westergaard material field
Field
Default
What to enter
Poisson’s ratio of the soil between the rigid layers μs —
0.3
From 0 to 0.49.
Depth profile
The depth down to which the Profile view and the table of the stress under A run.
Depth profile.
Depth profile field
Field
Default
What to enter
Depth of the profile zmax m · ft
20
At least 0.1 m. The Profile view and the table under A go from the surface down to this depth, or to the depth of A if that is deeper.
The Westergaard medium, Boussinesq beside it, and several loads
What Poisson’s ratio does
μs enters through η = √[(1 − 2μs)/(2 − 2μs)]: η = 0.7071 at μs = 0, 0.5345 at 0.3 and 0.1400 at 0.49. Its effect depends on where A lies:
Under the load (r = 0) the influence value is 1/(2πη²), so it rises with μs. At z = 4 m under a 5 kN load the module gives 0.09947 kPa with μs = 0 (I5 = 0.3183), 0.1741 kPa with 0.3 (I5 = 0.5570) and 0.2984 kPa with 0.4 (I5 = 0.9549); Boussinesq gives 0.1492 kPa.
At 5 m from the load in the example (r/z = 1.25) a higher μs gives a lower stress: 0.01187 kPa at 0, 0.01129 at 0.2, 0.01058 at 0.3, 0.008930 at 0.4 and 0.003500 at 0.49.
Westergaard beside Boussinesq
The Results, the depth table, the Calculations tab and the report give Boussinesq’s value for the same load and point, and the Profile view draws both curves. In the example (5 kN, r = 5 m, z = 4 m) Westergaard gives 0.01058 kPa and Boussinesq 0.01419 kPa. The Westergaard curve lies below the Boussinesq curve at moderate depths and the two come together deeper down; at z = 20 m they are 0.005175 and 0.005129 kPa. Close to the surface, away from the load, Westergaard’s value is the larger: 0.006364 against 0.004217 kPa at z = 2 m. The Calculations tab gives the ratio Westergaard/Boussinesq (0.745 for the example).
Several loads
The number of loads, from 1 to 8, is made by adding and deleting loads. The stress of each load is found as if it were alone, with its own distance r from A, and the stresses are added; the Boussinesq values are added in the same way.
All positions are in one set of coordinates.xL, yL and A’s x, y refer to the same origin. Only the distance counts: A at (4, 3), (5, 0) or (−3, −4) with the load at the origin gives the same stress, 0.01058 kPa, and so does the load at (2, 1) with A at (5, 5) (0.01058 kPa).
Loads can differ. Each load has its own P.
The cards show the parts. The Results tab shows the total, Boussinesq’s total and a card for the part from each load for two or three loads (from four loads the parts are listed under the cards); the Calculations tab writes out each load’s working.
With the example load and a second load of 5 kN at xL = 4 m, yL = 0 the second load is 4.12 m from A and adds 0.01698 kPa to the 0.01058 kPa of the first: 0.02756 kPa in all (Boussinesq 0.03862 kPa). Make that second load 10 kN and the total is 0.04455 kPa; put it 20 m away and it adds almost nothing (0.01089 kPa). A third load of 5 kN at (0, 6) gives 0.05066 kPa. Eight loads of 5 kN, 1 m apart along x from 0 to 7, give 0.1090 kPa (Boussinesq 0.1525).
Two loads: the lines of equal stress of the sum in the plane through A.
The same two loads in the Contours view.
The same two loads in 3D.
Which should I use?
Layered ground with horizontal seams that restrain lateral strain: this module. Ground you treat as uniform:Point load.
A load spread over an area is not a point load: use Westergaard areas for layered ground or Rectangle for uniform ground. A point load is the right model when the loaded area is small compared with its distance to A.
Analyze and read the results
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” A message appears briefly, “Analysed — see the Results tab and the Calculations.” The module stays on the Model tab: open Results for the answer. The Section view now shows the lines of equal stress.
The bar after Analyze.
The cards of the example.
Card
Symbol
The example
Vertical stress increase (Westergaard)
Δσz
0.01058 kPa
Influence value (one load)
I5
0.03386
Boussinesq, for comparison
Δσz,B
0.01419 kPa
From point load k (two or three loads)
Δσz(k)
the part of the total from load k
Below the cards a Values, step by step table lists the steps of the Calculations tab with their values, and the Vertical stress increase under the point table gives z, Δσz (Westergaard) and Δσz,B (Boussinesq) at ten equal steps from the surface to zmax, with the row of A inserted and highlighted. For one load it also gives r/ηz and I5; with several loads those two columns are left out. If zmax is less than the depth of A, the table runs to the depth of A.
The depth table of the example.
Two loads: the total, Boussinesq and the part from each.
Using the numbers
The result is an increase, not the total stress: add the overburden pressure of the soil above A for the total vertical stress.
The values are small for the example because the load is 5 kN; they scale with the load. Read the digits as the module gives them (four significant figures).
Change x, y or z after Analyze and everything follows at once; to change a load, press Edit inputs, which unlocks the loads and clears the results.
Changing the unit system after Analyze re-states the cards and the report without a new run. The drawings keep their labels in m and kPa.
Values are rounded where they are drawn, so the last digit of a figure on the drawing can differ by one from the card.
Warnings
The bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” The runs of this manual, from the default to eight loads and from μs = 0 to 0.49, gave no warning.
The Warnings window.
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. For one load:
The constant η from μs, with your numbers.
The influence value I5 — the distance r (from the load to A on the ground), r/(ηz), 1/(2πη²) and I5.
The stress — Δσz = (P/z²) I5 with your numbers, Boussinesq’s value at the same point and the ratio Westergaard/Boussinesq.
The stress against depth — the same down the vertical line through A, as the table of the Results tab, with the row of A marked.
With several loads step 2 and 3 become one step for each load (its position, r, r/(ηz), I5, Δσz(k) and Boussinesq’s value), followed by a Superposition step that adds the parts, then the depth table. The working ends with the line “Answer.”.
The Calculations tab of two loads: the steps for each load and the superposition.
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, 4 Results, 5 Summary and 6 References. It carries three drawings, the Section with the lines of equal stress, the Profile and the 3D view, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has four sections: Westergaard’s medium, The solution, Worked example and References. Each opens when you click its title (Open all and Close all open or close them together). They describe the medium of thin rigid sheets, give the equations, draw I5 against r/z for μs = 0, 0.2 and 0.4 beside Boussinesq’s influence value, tabulate I5, and work the module’s own example; the figures 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.
The horizontal distance of A from the load is r = √(3² + 4²) = 5.00 m. Then r/(ηz) = 5/(0.53452 × 4) = 2.3385, 1/(2πη²) = 0.55709 and [2.3385² + 1]−3/2 = 0.06078, so
I5 = 0.55709 × 0.06078 = 0.03386
Step 3: the stress
Δσz = (P/z²) I5 = (5/4²) × 0.03386 = 0.01058 kPa
Boussinesq’s solution gives 0.01419 kPa at the same point, so Westergaard/Boussinesq = 0.745. These are the three cards of the Results tab and the steps of the Calculations tab. The same equations worked independently give every value of the depth table, for example 0.006364 kPa at 2 m, 0.01218 kPa at 6 m and 0.009489 kPa at 12 m.
Other unit systems
The same load in SI · tf, with P = 1 tf (the other values unchanged), gives 0.002116 t/m²: I5 is unchanged and (1/4²) × 0.03386 = 0.002116. In US the default example (5 kN is 1.124 kip) gives 0.0002210 ksf: 0.01058 kPa is 0.000221 ksf. A check by hand with round US numbers: P = 10 kip at the origin and A at x = 9 ft, y = 12 ft, z = 12 ft gives r = 15 ft, r/z = 1.25 as in the example, so I5 = 0.03386 and Δσz = (10/12²) × 0.03386 = 0.002351 ksf; the module gives 0.002351 ksf. With P = 1 kip and A 10 ft from the load at 10 ft depth (r/z = 1) it gives 0.0005835 ksf and I5 = 0.05835, the same as the equations.
How each input changes the answer
Each row changes one thing in the example (one load of P = 5 kN at the origin, A at x = 3 m, y = 4 m, z = 4 m, μs = 0.3, zmax = 20 m) and runs the module again, in SI · kN. The numbers are the ones the module gives, so you can reproduce any row; an independent calculation with the same equations agrees with every row.
The example with one change at a time
Change
Δσz (kPa)
I5
The example
0.01058
0.03386
P = 10 kN Proportional to the load
0.02116
0.03386
P = 20 kN Four times the stress
0.04232
0.03386
z = 0.5 m Near zero close to the surface
0.001694
0.00008471
z = 2 m Larger than Boussinesq here
0.006364
0.005092
z = 6 m Near the maximum
0.01218
0.08767
z = 8 m Past the maximum
0.01195
0.1529
z = 12 m
0.009489
0.2733
z = 20 m Close to Boussinesq
0.005175
0.4140
x = 0, y = 0 A under the load
0.1741
0.5570
x = 2, y = 0 2 m from the load
0.06780
0.2170
x = 10, y = 0 10 m from the load
0.001591
0.005092
x = 6, y = 8 Also 10 m away: same as above
0.001591
0.005092
x = 4, y = 3 Also 5 m away: same as the example
0.01058
0.03386
x = −3, y = −4 Also 5 m away: same as the example
0.01058
0.03386
μs = 0 Higher stress at this distance
0.01187
0.03799
μs = 0.2
0.01129
0.03614
μs = 0.4
0.008930
0.02858
μs = 0.49 The largest value accepted
0.003500
0.01120
Load (2, 1), A (5, 5) Still 5 m apart
0.01058
0.03386
Several loads, SI · kN, from the example (a second load added with the values shown):
Adding loads
Added
Total (kPa)
Parts (kPa)
Load 2: 5 kN at (4, 0) 4.12 m from A
0.02756
0.01058 + 0.01698
Load 2: 10 kN at (4, 0)
0.04455
0.01058 + 0.03396
Load 2: 5 kN at (20, 0) Far from A: almost no change
0.01089
0.01058 + 0.0003124
Load 2: 5 kN at (3, 4) Directly above A, 0 m away
0.1847
0.01058 + 0.1741
The pattern follows the equations. The stress is proportional to P; I5 depends on r/z and μs only. Halving the depth (z = 2 m, r = 5 m) and doubling the distance (r = 10 m, z = 4 m) both give r/z = 2.5, and both rows show the same I5 = 0.005092; the stress is highest under the load and, at a fixed distance, rises to a maximum at some depth and then falls; and at a given r/z the sign of the μs effect follows the position of A, as the section above shows. The depth of the profile zmax changes the table and the Profile view but not the stress at A: 0.01058 kPa with zmax = 6 m and 0.01058 kPa with zmax = 2 m, a depth the table extends to the depth of A.
Messages: when the module stops, and what it notes
The module checks the numbers it needs before it calculates. When an input cannot be used, press Analyze and a window titled Cannot analyse — 1 input error (or 2 input errors …) lists each message; nothing is calculated. A load row whose force is negative is outlined in red.
The window Analyze opens when the depth is zero.
When
Message and what to do
z is 0 or negative
“Depth of the point (z) must be at least 0.001 m.” Enter a depth of at least 0.001 m.
μs is 0.5 or more
“Poisson's ratio of the soil between the rigid layers (μs) must be at most 0.49.” Enter a value up to 0.49.
μs is negative
“Poisson's ratio of the soil between the rigid layers (μs) must be at least 0.” Enter zero or a positive value.
P is negative
“Point load 1 (P1) must be at least 0 kN.” Enter zero or a positive load.
zmax is 0
“Depth of the profile (zmax) must be at least 0.1 m.” Enter a depth of at least 0.1 m.
A box is empty
“Depth of the point (z) is missing.” Fill the box. The same words appear for x, y, μs, P and a load position.
Two inputs are wrong
One window lists both: “Depth of the point (z) must be at least 0.001 m. Poisson's ratio of the soil between the rigid layers (μs) must be at most 0.49.”
A new load would stand on another
A message appears briefly: “A point load already stands at this position. Give the new one another position.” The load is not added.
A position typed or dragged onto another load
A message appears briefly: “Another point load already stands at this position — the value is kept as it was.”
The messages use the numbered names of the loads, such as “Point load 2 (P2)” and “Position of load 1 along x (xL,1)”, and the limits in the units of the unit system chosen. A negative force typed into the New point load form is added to the table with its row marked in red, and Analyze then stops with “Point load 2 (P2) must be at least 0 kN.” At eight loads + Add point load is greyed out and the counter reads 8 of 8.
Notes and warnings
With every input inside its limits the module calculates, and the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” A load of zero is accepted and gives 0 kPa. A point very close to a load gives a very large stress, as the equation does: with A directly under the load at z = 0.01 m the module gives 27,852 kPa for 5 kN (I5 = 0.5570, the value under the load at μs = 0.3). A very large depth is accepted too: z = 100,000 m gives 2.79e-10 kPa. Very small and very large values are shown in exponent form.
Good practice and common mistakes
Use the module for a soil that fits the model. Westergaard’s medium is an idealisation: the rigid layers allow no lateral strain. Where the ground is better described as a uniform half-space, Point load is the model; the Boussinesq column here lets you see how far apart the two are.
Choose μs for the soil between the layers. It sets η; the values the module accepts are 0 to 0.49. Run the example at a few values (the table above) to see how much it matters at your r/z.
Use one set of coordinates.xL, yL of every load and x, y of A refer to the same origin; only the distance between a load and A counts.
Enter forces, not pressures.P is a force. For a load spread over an area use Westergaard areas.
Keep a point load a point. At a distance from A of the order of its own size or less, a real footing is not a point load.
Add the overburden. The result is the stress increase; for the total vertical stress add the soil weight above A.
Place A with the drawing, then type exact values. Dragging snaps to a few centimetres; type the coordinates when you need a given place. In 3D a pull has no stop at the frame, so check the load row after a long pull.
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 stress below a point load calculated here?
By Westergaard’s solution for soil reinforced by thin rigid horizontal layers: Δσz = (P/z²) I5 with I5 = 1/(2πη²) · [(r/ηz)² + 1]−3/2 and η = √[(1 − 2μs)/(2 − 2μs)]. For the module’s example it gives 0.01058 kPa.
What is the influence value I5?
It is the stress increase multiplied by z² and divided by the load. It depends only on r/z and μs, and under the load it equals 1/(2πη²), for example 0.5570 at μs = 0.3. The module gives it for one load; with several loads it gives the stress of each.
How does Westergaard differ from Boussinesq?
Boussinesq treats the ground as a uniform half-space; Westergaard treats it as an elastic soil with thin rigid layers that stop lateral strain, so it spreads the load differently. The module shows both: in the example 0.01058 kPa against 0.01419 kPa. Under the load Westergaard’s stress is the larger at μs = 0.3; at 5 m from it, at moderate depths, the smaller.
Does the result include the weight of the soil?
No. It is the increase caused by the loads only. Add the overburden pressure of the soil above A for the total vertical stress.
How many loads can I add, and can they stand at the same place?
Up to eight, each with its own P, xL and yL. No two may stand at the same place; the module refuses a load that would. The stresses of the loads add.
Can I drag the loads and the point on the drawing?
Yes. Drag A and the dashed line at any time; before Analyze you can also drag the loads: with several loads a load arrow in the Section and Contours views (along x) and over the ground in the 3D view, and one load in 3D only. After Analyze the loads are locked.
What values does the module accept?
P zero or more, z at least 0.001 m, μs from 0 to 0.49, zmax at least 0.1 m, and x, y, xL and yL any value. Anything else stops the analysis with a message that names the value.
Why can Poisson’s ratio not be 0.5?
At μs = 0.5 the constant η is zero: soil that cannot change volume and cannot strain sideways could not deform at all, and the equation breaks down. The module stops at 0.49 (η = 0.1400).
What do the lines of equal stress show?
Each line joins the points of the section with the same Δσz, labelled in kPa; the line through A is blue. Switch them on and off with the Lines of equal stress chip. The Contours view shows the same stress as colour bands on a logarithmic scale.
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. Stresses are in kPa, t/m² or ksf, forces in kN, tf or kip and lengths in m or ft. The default example gives 0.01058 kPa, 0.002116 t/m² with a load of 1 tf, or 0.0002210 ksf.
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, profile 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.
Calculate the vertical stress increase at a point below one or several uniformly loaded circles and rectangles on Westergaard’s material, ground stiffened by thin rigid layers, with Boussinesq’s value beside it. This manual walks through the screen, dragging, every input, Poisson’s ratio, the results and the report, with worked examples you can reproduce.
Westergaard areas gives the increase in vertical stress at a point below one or several uniformly loaded, flexible circles and rectangles on the ground surface, when the ground is Westergaard’s material (Westergaard 1938): an elastic soil that is stiffened by thin, rigid horizontal layers, which stop it spreading sideways. You choose the point A by its position x, y on the ground and its depth z, and the module gives Δσz there, with Boussinesq’s value for the same loads beside it for comparison. It draws the lines of equal stress, the colour contours, the profile with depth under A and a 3D view.
Westergaard’s point load Q gives, at a horizontal distance r from it and the depth z,
where μs is Poisson’s ratio of the soil between the rigid layers. The module integrates this over each loaded area. Two results are closed forms: below the centre of a circle of radius R, and below a corner of a rectangle:
Δσz = q {1 − η / √[η² + (R/z)²]} (circle, on its axis)
For any other point the module integrates Westergaard’s point load over the circle, or adds and subtracts four rectangles that each have a corner above A. With several areas the stress of each is found with its own geometry and the stresses are added. The result is also given as the influence valueΔσz/q when there is one area.
What you can describe
One to eight loaded areas, each a circle (radius R) or a rectangle (L along x, B along y), each with its own load per unit area q and the position (xc, yc) of its centre on the ground.
The point A: x and y on the ground surface, in the same coordinates as the centres, and the depth z.
Poisson’s ratio μs of the soil between the rigid layers, from 0 to 0.49.
The depth of the profilezmax, down to which the Profile view and the table under A run.
The result is the increase caused by the loads only; the overburden pressure of the soil above A is not included, so add it where your design needs the total stress.
This module gives circles and rectangles only. For a strip, an embankment or a load of another shape, use the modules above and add the stresses.
Before you start: what you need
Have these ready from your design:
What
Notes
What Where it comes from
Notes
The loaded areas Your layout
For each: circle or rectangle; the load per unit area q (zero or more, a pressure, not a force); the radius R or the sides L and B (at least 0.001 m); the position of the centre. Areas may touch but not overlap.
The point A Where you want the stress
x and y on the ground in the coordinates of the centres; the depth z at least 0.001 m.
Poisson’s ratio μs Your assumption for the soil between the rigid layers
From 0 to 0.49. The module opens with 0.
The depth of the profile zmax How deep you want to read
At least 0.1 m.
Quick start in eight steps
Open the module and choose the units
Open Westergaard areas from the Stresses in a Soil Mass family. Pick SI · kN, SI · tf or US in the header; you can change it at any time.
Enter the loaded areas
In Loaded areas the first area, a circle, is already in the table. Press its row to change it, or drag it on the drawing. To add another, choose Circle or Rectangle in the New loaded area form, fill it in and press + Add loaded area.
Place the point A
In Point A type x, y and z, or drag the blue point on the Section drawing.
Set the soil
In Westergaard material type Poisson’s ratio μs.
Set the depth of the profile
In Depth profile type zmax, or drag the dashed line near the bottom of the drawing.
Press Analyze
Press Analyze at the foot of the Model tab. The bar reads Analysed and the areas lock; the point A and the profile depth stay free.
Read the answer
The Results tab gives Δσz at A in Westergaard’s material, Boussinesq’s value for comparison, the influence value for one area and the part from each area for several, then a table against depth. The lines of equal stress are on the Section drawing, the colour bands in Contours.
Move A and take the report
Drag A or change x, y and z: the stress there is calculated at once. The Calculations tab writes out the working; the Report tab saves a PDF or a Word file.
The screen: section, contours, profile and 3D
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 is a vertical cross section with the loads on the ground, the point A, the dimensions and the dashed line of the profile depth. For one circle it is the plane through the centre of the circle and A; for a rectangle or for several areas it is the plane y = yA through A, parallel to x, and an area that the plane does not cut is drawn dashed with its distance from the plane. After Analyze it adds the lines of equal stress of Westergaard’s material and a label with Δσz at A.
Contours is the same section filled with colour bands of Δσz on a logarithmic scale, with a legend in kPa and in per cent of q.
Profile is the curve of Δσz against depth under A: Westergaard as a solid blue line and Boussinesq as a dashed orange line, with a marker and the value of each at the depth of A.
3D is an oblique view of the ground with the loaded areas on it, the axes x, y and z and a small element below A with the stress on it.
Before Analyze the drawings follow the inputs and say “The stress and the lines of equal stress appear after Analyze” (Section), “The colour contours of the stress appear after Analyze” (Contours), “The profile appears after Analyze.” (Profile) or “The stresses on the element appear after Analyze.” (3D).
Section before Analyze.
Section after Analyze: the lines of equal stress; the line through A is blue (its value, 130 kPa, is named in the footer and not on the line).
Contours after Analyze.
Profile after Analyze: Westergaard and Boussinesq.
3D after Analyze.
The Show chips
Above the Section and the Contours drawings a bar has two chips: Lines of equal stress and Dimensions. In the Section view the lines chip is off and greyed out until you press Analyze, and then turns on; in Contours it is off by default, so that the lines can be laid over the colours. Dimensions is on by default in both. The Profile and 3D views have no chips.
Dragging on the drawing
Before Analyze the hints under the Section drawing say: “Drag the point A to place it. The stress and the lines of equal stress appear after Analyze.” and “Drag a load to move it, the rings at its edges to resize it. Drag the dashed line to set the depth of the profile (zmax).” After Analyze only the dashed-line sentence stays, with the note on the lines of equal stress and “drag A to read the stress elsewhere”. What each pull does, as the module runs it (pulls move in steps of 5 cm):
One circle, Section. The drawing shows only the rings at its two edges: drag one to change the radius R; it stops at 0.1 m. The circle itself is moved in the 3D view.
A rectangle, or several areas, Section. Drag the bar over an area to move it: xc follows the pointer. A rectangle also has a ring at each end: the edge you pull follows the pointer, the other edge stays where it is, so L and xc change together; the length stops at 0.1 m. A circle among several areas has no rings in the Section, only in 3D.
3D view. Drag the loaded area on the ground to move it along x and y together. A circle has rings at the ends of two diameters, one along x and one along y, and each sets the radius; a rectangle has rings at the ends of L and of B.
The point A. Drag the blue point in the Section or the Contours view. The depth z stops at 0.05 m at the top and at the foot of the drawn frame at the bottom, and the horizontal position at the sides of the frame. With one circle A is placed by its distance from the centre of the circle, along the line from the centre through A, so x and y change together; otherwise x follows the pointer and y stays.
The dashed line. Drag it up or down to set zmax; it stops at 0.5 m.
Areas never overlap. A pull that would put one area on another stops at the last free place, where the two touch, and a message says “That would put loaded area 1 and loaded area 2 on top of one another — the value is kept as it was.”
After Analyze the areas are locked: their rings and bars disappear and a pull changes nothing. The point A and the dashed line stay free, and so do the boxes x, y, z and zmax. There is no handle for A in the 3D view.
The 3D view has its own hint: “Drag a load over the ground to move it, the rings at its edges to resize it.”
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. Zoom is a property of the drawing only: it is not saved with the model and does not change the report.
Entering the model, section by section
The Model tab has five sections: Model data, Loaded areas, Point A, Westergaard material and Depth profile. The four input sections (Loaded areas, Point A, Westergaard material, Depth profile) work as a set: opening one closes the others. Model data opens and closes on its own, and the module starts with Model data and Loaded areas open. Next and Back at the foot walk you through them, and Analyze waits below the last one. 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 13.1234 ft for the 4 m radius.
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.
Model data.
Loaded areas
The section is a form over a table. The New loaded area form starts with the shape; Circle asks for R, Rectangle for L and B. + Add loaded area puts the area into the Loaded areas table, and a counter beside the button shows how many of the eight areas there are. The table lists each area with its shape and load, its size and its centre. Press a row to change that area in the form (the form then reads Editing · Loaded area 1 and offers Done and Delete loaded area; the note says that changes apply as they are entered), or press × to take an area out. The first area is already in the table when the module opens, and Delete loaded area and × are greyed out while it is the only one. A row that is being edited can change its shape with the buttons in the form: the first area becomes a rectangle of 3 m × 2 m when you press Rectangle.
Loaded areas: the form over the table.
Loaded area fields (the form, and each row of the table)
Field
What to enter
Loaded area (shape) — · default Circle
The buttons Circle and Rectangle, each with a small picture. Pressing the other one changes the fields below.
Load per unit area q kPa · t/m² · ksf · default 300
The uniform pressure of the area on the ground. Zero or more.
Radius of the circle R m · ft · default 4
At least 0.001 m. Shown when the shape is Circle.
Side of the rectangle along x L m · ft · default 3
At least 0.001 m. Shown when the shape is Rectangle.
Side of the rectangle along y B m · ft · default 2
At least 0.001 m. Shown when the shape is Rectangle.
Position of the centre along x xc m · ft · default 0 (area 1) · 10 (form)
Any value. The first area is centred at 0; the New loaded area form opens with 10.
Position of the centre along y yc m · ft · default 0
Any value.
A row of the table pressed: the area is edited in the form.
The form with Rectangle chosen.
The areas may not lie on top of one another: two areas may touch but not overlap, whether they are two circles (the centres at least R1 + R2 apart), two rectangles or a circle and a rectangle. Adding or editing an area that would overlap another is refused (see Messages). The areas are listed in the order you add them, and their numbers name the results “From loaded area 1”, “From loaded area 2” and so on.
A circle and a rectangle in the table.
Point A
The three coordinates of the point where you want the stress. A is under the ground surface at (x, y) in the coordinates of the centres, at the depth z. The note in the section says that for a circle the closed form is for the centre, and other points are found by integrating the point-load solution; for a rectangle the closed form is for the corner, and other points by the corner rectangles. You can place A anywhere; you do not need to look for the centre or the corner.
Point A.
Point A fields
Field
What to enter
Point A: x on the ground surface x m · ft · default 0
Any value, in the same coordinates as the centres.
Point A: y on the ground surface y m · ft · default 0
Any value, in the same coordinates as the centres.
Depth of the point A z m · ft · default 4
At least 0.001 m below the ground surface.
Westergaard material
One number, Poisson’s ratio μs of the soil between the rigid layers. The note says that this is Westergaard’s medium, soil between thin rigid horizontal sheets that prevent lateral strain, and that μs < 0.5.
Westergaard material.
Westergaard material field
Field
What to enter
Poisson’s ratio of the soil between the rigid layers μs — · default 0
From 0 to 0.49. It sets the constant η = √[(1 − 2μs) / (2 − 2μs)]; η = 0.7071 at 0.
Depth profile
The depth down to which the Profile view and the table of the stress under A run.
Depth profile.
Depth profile field
Field
What to enter
Depth of the profile zmax m · ft · default 12
At least 0.1 m. The Profile view and the table under A go from the surface down to this depth, or to the depth of A if that is deeper.
Choosing the shape, Poisson’s ratio and the comparison
Three decisions shape the answer: the shape of each area, how many areas there are, and the soil constant μs.
Choice
Options
What it does
Shape (of each area)
Circle, Rectangle
A circle is given by its radius and a rectangle by its two sides along x and y. The sides of a rectangle always run along the axes. The circle of 4 m radius and the rectangle of 3 m × 2 m of the module’s forms put 126.8 kPa and 29.95 kPa on A at the same depth, with A at the centre of each.
Number of areas
1 to 8
Each is calculated alone and the stresses are added. See the next heading.
Poisson’s ratioμs
0 to 0.49
A larger μs gives a smaller η, so more of the load reaches A.
Poisson’s ratio
For the example (one circle, R = 4 m, q = 300 kPa, A on the axis at z = 4 m) the module gives, with Boussinesq’s 193.9 kPa unchanged:
The example for different Poisson’s ratios
μs
kPa
Δσz/q
0
126.8
0.4226
0.2
143.3
0.4778
0.3
158.6
0.5286
0.45
213.4
0.7113
0.49
258.4
0.8613
At μs = 0 Westergaard’s value is about two thirds of Boussinesq’s here; as μs approaches 0.5 the stress below the load rises towards q and passes Boussinesq’s value, so the choice of μs matters. The module accepts values up to 0.49 only; the Calculations tab shows η for the value you enter.
Several areas and how they combine
The stress of each area is found as if it were alone, and the stresses are added. This works because the ground is elastic, so the stresses of separate loads add. The Results tab shows the total and, for two or three areas, the part from each.
Coordinates are shared. The centres and A are in the same coordinates, so moving A and an area by the same amount changes nothing: the circle of the example moved to (6, 2) with A at (6, 2) gives 126.8 kPa again.
Loads and shapes can differ. A circular tank and a rectangular slab can be put in one model, each with its own q.
Far areas add little. With a second circle of 300 kPa and 4 m radius centred at 10 m, A at the origin gets 126.8 kPa from the first and 7.048 kPa from the second: 133.8 kPa in all. With the second area at 150 kPa the second part is 3.524 kPa.
Between two areas. With A at x = 5 m, between the circles, each gives 47.13 kPa and the total is 94.25 kPa. The same circles touching (centres at 0 and 8 m) with A at x = 4 m, the touching point, give 69.51 kPa from each and 139.0 kPa in all.
A circle and a rectangle: the lines of equal stress of the sum.
The same two areas in the Contours view.
Westergaard and Boussinesq side by side
Every result comes with Boussinesq’s value for the same loads and the same point, which is the answer for a ground without rigid layers. Boussinesq’s value does not depend on μs. Westergaard’s is lower than Boussinesq’s below the load at μs = 0, and it can be higher beside the load: with A at 8 m from the axis of the example the module gives 13.75 kPa against Boussinesq’s 12.54 kPa. Read both, and use the one that matches the ground you are modelling.
Which should I use?
Circle or rectangle? Use the shape of the footprint. A tank, a silo or a circular footing is a circle; a slab or a rectangular footing is a rectangle. A footing of another shape is made of rectangles that touch, or can go to Influence chart for a homogeneous ground.
One area or several? One when the load is uniform over a single footprint; several when pressures differ, or when there are gaps.
Press Analyze. The bar changes to Analysed — the stress at the point calculated and says “The inputs are locked. Edit inputs clears the results; the unit system can still be changed. The point A stays free: drag it on the drawing or change its coordinates, and the stress there is calculated at once.” A message appears briefly, “Analysed — see the Results tab and the Calculations.” The module stays on the Model tab: open Results for the answer. The Section view now shows the lines of equal stress.
The bar after Analyze.
The cards of the example.
Card
Symbol
The example
Vertical stress increase (Westergaard)
Δσz
126.8 kPa
Influence value (one area only)
Δσz/q
0.4226
Boussinesq, for comparison
Δσz,B
193.9 kPa
From loaded area k (two or three areas)
Δσz(k)
the part of the total from area k
Below the cards a Values, step by step table lists the steps of the Calculations tab with their values, and the Vertical stress increase under A table gives z, Δσz for Westergaard and for Boussinesq, and, for one area, Δσz/q, at ten equal steps down to zmax (the first at zmax/10), with the row of A inserted and highlighted. With several areas the ratio column is left out; with four or more areas the cards show the total and Boussinesq’s value, and the part from each area is in the Calculations tab. If zmax is less than the depth of A, the table runs to the depth of A.
The depth table of the example.
Two areas: the total and the part from each.
Using the numbers
The result is an increase, not the total stress: add the overburden pressure of the soil above A for the total vertical stress.
The influence value Δσz/q is the part of the surface pressure that reaches A. It is close to 1 just under the loaded area, and it falls with depth and with distance from the load. Beside a load the stress can first rise with depth: with A at x = 4 m, y = 3 m, which is 5 m from the centre of the example’s circle, the table gives 38.76 kPa at 1.2 m and 49.41 kPa at 2.4 m, then a slow fall.
Values are rounded where they are drawn, so the last digit of a figure on the drawing can differ by one from the card.
Change x, y, z or zmax after Analyze and everything follows at once; to change an area or μs, press Edit inputs, which unlocks them and clears the results.
Changing the unit system after Analyze re-states the cards and the report without a new run. The drawings keep their labels in m and kPa.
Warnings and notes
The bar carries a Warnings (0) button. It opens a window that says “No warnings — the assumptions of the solution hold for these inputs.” The runs of this manual, from the default to eight areas, gave no warning.
The Warnings window.
The Calculations tab, the report and the Theory tab
The Calculations tab opens with Given and what is sought: the loads, the point A and μs. Then it writes out the working in numbered steps, ending with the answer.
The constant η — η = √[(1 − 2μs) / (2 − 2μs)] with your μs.
The area — for one circle on its axis, Δσz = q {1 − η / √[η² + (R/z)²]} with R/z and Δσz/q; for a circle with A off its axis, the integral of the point-load solution with r, r/R, z/R and Δσz/q; for a rectangle, Iw for each of the four corner rectangles seen from A, listed with the corner, m, n, Iw and its sign, and their signed sum.
Stress at A — Δσz = Δσz/q × q, Boussinesq’s value at the same point and the ratio of Westergaard’s to Boussinesq’s.
Stress increase against depth — the same down the vertical line through A, as the table of the Results tab, with the row of A marked.
With several areas there is one step for each area, with its own geometry, then a Superposition step that adds the parts, then the depth table. In the corner rectangles m and n are the distances from A to the corner along x and y divided by z; Iw is the same when they swap.
For the example the first steps read:
The Given block of the example.
The step for the circle, 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, 4 Results, 5 Summary and 6 References. It carries three drawings, the Section with the lines of equal stress, the Profile with both curves and the 3D view, and states the units used.
The Report tab: the cover details, the two buttons and the first page.
The Theory tab
The Theory tab has five sections: Scope, Circular area, Rectangular area, Worked example and References. Each opens when you click its title (Open all and Close all open or close them together). It gives the equations above, with three charts and three tables computed from them: the stress below the centre of a circle against R/z, Iw against m and n, and a worked example of Westergaard against Boussinesq below a circle.
Worked example: the module’s own example
The module opens on this example, so you can follow the numbers on your own screen.
Boussinesq’s value for the same circle is q{1 − 1/(1 + (R/z)²)3/2} = 300 × (1 − 0.3536) = 193.9 kPa, and the ratio of the two is 0.654, as the Calculations tab states. These are the three cards of the Results tab.
Doubling the radius or halving the depth gives the same R/z = 2 and so the same ratio, 0.6667, which is a hand check: R = 8 m with z = 4 m, and R = 4 m with z = 2 m, both give 200.0 kPa in the module (200.0 and 200.0 kPa).
A rectangle, below a corner
Take a square of 2 m × 2 m with q = 100 kPa and A below one corner, at (1, 1) with z = 2 m, μs = 0. Then m = n = 1, η² = 0.5, and
The module gives 11.61 kPa, an influence value of 0.1161, and Boussinesq’s 17.52 kPa. For the default rectangle of 3 m × 2 m, with A at its centre at 4 m depth, it gives 29.95 kPa (Boussinesq 45.96 kPa), and the Calculations tab lists the four corners with m = 0.3750, n = 0.2500 and Iw = 0.02496 each.
A circle and a rectangle together
Add a rectangle of q = 200 kPa, L = 6 m, B = 4 m centred at (10, 0) to the example and move A to x = 5 m. A is then 5 m from the centre of the circle, which is 1.25 radii, so the module integrates over the circle; it gives 47.13 kPa from the circle and 15.01 kPa from the rectangle, from the four corners (13, ±2) with signs + and (7, ±2) with signs −. With a third area, a circle of 100 kPa and 3 m radius at x = 20 m, the total is 62.51 kPa, and the third area adds only 0.3730 kPa. Without the third area the total is 62.13 kPa (Boussinesq 84.51 kPa). An independent numerical integration of Westergaard’s point load over each area gives the same values to the digits shown.
Other unit systems
The default example in SI · tf, with q = 30 t/m² (the other values unchanged), gives 12.68 t/m² and the influence value 0.4226. In US the default example, which is 6.266 ksf on a circle of 13.12 ft radius at 13.12 ft depth, gives 2.648 ksf: 126.8 kPa is 2.648 ksf. A check by hand with round US numbers: q = 6 ksf, R = 12 ft and A on the axis at z = 12 ft give R/z = 1, the same ratio 0.4226, so Δσz = 6 × 0.4226 = 2.536 ksf; the module gives 2.536 ksf (Boussinesq 3.879 ksf). For a rectangle, q = 2 ksf, 10 ft × 10 ft, A below a corner at 10 ft depth: m = n = 1, Iw = 0.1161, so Δσz = 0.2322 ksf; the module gives 0.2323 ksf.
How each input changes the answer
Values are in kPa. Each row changes one thing in the example (one circle of q = 300 kPa, R = 4 m at the origin, A at (0, 0, 4 m), μs = 0) and runs the module again, in SI · kN. The numbers are the ones the module gives, so you can reproduce any row; an independent numerical integration of Westergaard’s point load agrees with every row.
The example with one change at a time
Change
Westergaard
Boussinesq
The example Δσz/q 0.4226
126.8
193.9
q = 600 kPa The stress is proportional to the load; the ratio is unchanged. Δσz/q 0.4226
253.6
387.9
q = 100 kPa A third of the load, a third of the stress. Δσz/q 0.4226
42.26
64.64
R = 2 m A smaller circle puts less on A. Δσz/q 0.1835
55.05
85.34
R = 8 m The same R/z as z = 2 m with R = 4 m. Δσz/q 0.6667
200.0
273.2
R = 20 m A wide circle: the ratio rises towards 1. Δσz/q 0.8600
258.0
297.7
z = 0.5 m Near the surface the stress approaches q. Δσz/q 0.9120
273.6
299.4
z = 2 m The stress falls with depth. Δσz/q 0.6667
200.0
273.2
z = 8 m The same R/z as R = 2 m at 4 m depth. Δσz/q 0.1835
55.05
85.34
z = 16 m Deep below the load the stress is a small part of q. Δσz/q 0.05719
17.16
26.08
μs = 0.3 A larger μs puts more stress on A; Boussinesq’s value does not change. Δσz/q 0.5286
158.6
193.9
μs = 0.49 The largest value the module accepts: Westergaard’s stress passes Boussinesq’s. Δσz/q 0.8613
258.4
193.9
A 2 m from the axis (x = 2 m) Inside the circle but off its axis. Δσz/q 0.3732
112.0
168.7
A below the edge (x = 4 m) About 55 per cent of the stress on the axis. Δσz/q 0.2317
69.51
99.67
A 8 m from the axis (x = 8 m) Beside the load Westergaard’s value is the larger of the two. Δσz/q 0.04583
13.75
12.54
x = 4 m, y = 3 m The distance from the centre is 5 m: the same as x = 5 m, y = 0. Δσz/q 0.1571
47.13
64.13
x = −4 m The same as x = +4 m: the circle is symmetric about its centre. Δσz/q 0.2317
69.51
99.67
Area 1 a rectangle, 3 m × 2 m A smaller footprint than the circle; A at its centre. Δσz/q 0.09983
29.95
45.96
Area 1 a rectangle, 8 m × 8 m A square of side 8 m, 64 m² against the circle’s 50 m². Δσz/q 0.4646
139.4
210.3
Rectangle 3 m × 2 m, A below a corner (x = 1.5, y = 1) The stress under a corner is much less than under the centre. Δσz/q 0.06900
20.70
32.12
Rectangle 3 m × 2 m, A at x = 5 m A is 3.5 m from the edge: a small stress. Δσz/q 0.01540
4.619
5.771
Several areas, SI · kN, A at (0, 0, 4 m) unless stated, one change at a time from the example (area 2 added with the values shown):
Adding areas
Added
Total (kPa)
Parts (kPa)
Circle: 300 kPa, R = 4 m, centre (10, 0) A on the first circle’s axis
133.8
126.8 + 7.048
Circle: 150 kPa, R = 4 m, centre (10, 0) Half the load, half the part
130.3
126.8 + 3.524
Circle: 300 kPa, R = 4 m, centre (0, 8) 8 m away along y, touching; A on the first axis
140.5
126.8 + 13.75
Rectangle: 300 kPa, 3 m × 2 m, centre (10, 0) A on the circle’s axis
127.5
126.8 + 0.7463
Circle: 300 kPa, R = 4 m, centre (10, 0) A at x = 5 m, midway
94.25
47.13 + 47.13
Circle: 300 kPa, R = 4 m, centre (10, 0) A at x = 10 m, on the second axis
133.8
7.048 + 126.8
The pattern follows the equations. The stress is proportional to q; for a circle on its axis it depends on R/z only, so doubling every length leaves the ratio unchanged; it falls with depth and with distance from the load, and it rises with μs. The depth of the profile zmax changes the table and the Profile view but not the stress at A: 126.8 kPa with zmax = 6 m and 126.8 kPa with zmax = 2 m; as that is less than the depth of A, the table runs to the depth of A.
Messages: when the module stops, and what it notes
The module checks the numbers it needs before it calculates. When an input cannot be used, press Analyze and a window titled Cannot analyse — 1 input error (or 2 input errors …) lists each message, and a box with an out-of-range value is marked in red; an empty box is named in the list only. Nothing is calculated.
The window Analyze opens when μs is 0.5.
When
Message and what to do
z is 0
“Depth of the point A (z) must be at least 0.001 m.” Enter a depth of at least 0.001 m.
μs is 0.5 or more
“Poisson's ratio of the soil between the rigid layers (μs) must be at most 0.49.” Enter a value from 0 to 0.49.
μs is negative
“Poisson's ratio of the soil between the rigid layers (μs) must be at least 0.” Enter 0 or more.
q is negative
“Load per unit area of area 1 (q1) must be at least 0 kPa.” Enter zero or a positive load.
R is 0
“Radius of circle 1 (R1) must be at least 0.001 m.” Enter a radius of at least 0.001 m.
A side of a rectangle is 0
“Side of rectangle 1 along x (L1) must be at least 0.001 m.” Enter a side of at least 0.001 m.
zmax is 0
“Depth of the profile (zmax) must be at least 0.1 m.” Enter a depth of at least 0.1 m.
A box is empty
“Radius of circle 1 (R1) is missing.” or “Poisson's ratio of the soil between the rigid layers (μs) is missing.” Fill the box or take the area out.
A new area would overlap another
A message appears briefly: “The new loaded area would lie on loaded area 1 — they cannot overlap. Give it another position or width.” The area is not added.
The messages use the numbered names of the areas, such as “area 1”, “circle 1”, “rectangle 1” and “q1”, even when there is only one area, and the limits in the units of the unit system chosen. Two wrong values are listed together: L = 0 and μs = 0.5 give “Cannot analyse — 2 input errors” with both messages. When you type a position or size into an area already in the table and it would put two areas on one another, a message appears briefly: “That would put loaded area 1 and loaded area 2 on top of one another — the value is kept as it was.” and the value stays as it was. At eight areas + Add loaded area is greyed out and the counter reads 8 of 8.
Notes and warnings
With every input inside its limits the module calculates, and the Warnings window reads “No warnings — the assumptions of the solution hold for these inputs.” A load of zero is accepted and gives 0 kPa, with a dash for the influence value. Small and large values are accepted too: a radius of 0.001 m gives 0.00001875 kPa at A, and with q = 1,000,000 kPa and R = 1000 m the module gives 997,172 kPa, an influence value of 0.9972. A point far from the load is accepted: x = −30 m gives 0.2530 kPa.
Good practice and common mistakes
Decide whether the ground is Westergaard’s material. The module is for soil that is stiffened by thin rigid layers; for a homogeneous ground use Boussinesq’s modules, or read the Boussinesq card and column of this module.
Choose μs with care. It changes the answer a great deal: for the example 126.8 kPa at 0, 158.6 kPa at 0.3 and 258.4 kPa at 0.49. The module opens with 0.
Enter a pressure, not a force.q is a load per unit area. A total load Q on a circle is Q/(πR²).
Mind the coordinates. Centres and A are in the same coordinates. A rectangle’s sides run along x and y; swap L and B to turn it.
Do not overlap areas. Areas may touch but not lie on one another; to describe a stepped load, split it into areas that meet.
Add the overburden. The result is the stress increase; for the total vertical stress add the soil weight above A.
Read the profile as well as the point. Beside a load the stress can rise with depth before it falls; the Profile view and the table show it.
Place A with the drawing, then type exact values. Dragging moves in 5 cm steps; type the coordinates when you need a given place.
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
What is Westergaard’s material?
An elastic soil that is stiffened by thin, rigid horizontal layers, which stop it spreading sideways. The soil between the layers has Poisson’s ratio μs, which the module asks for. The vertical stress below a load is then given by Westergaard’s solution instead of Boussinesq’s.
How is the stress below a loaded area calculated?
The module integrates Westergaard’s point load over the area. Below the centre of a circle it is Δσz = q{1 − η/√[η² + (R/z)²]}, and below a corner of a rectangle Δσz = qIw; any other point is found by integrating over the circle or by four signed corner rectangles. For the module’s example it gives 126.8 kPa, with Boussinesq’s 193.9 kPa beside it.
What does Poisson’s ratio do?
It sets η = √[(1 − 2μs)/(2 − 2μs)]. A larger μs puts more stress on a point below the load: for the example 126.8 kPa at 0, 158.6 kPa at 0.3 and 258.4 kPa at 0.49. Boussinesq’s value does not depend on it. The module accepts 0 to 0.49.
Why is there a Boussinesq value as well?
It is the answer for the same loads on a homogeneous elastic ground, given for comparison on the cards, in the table and as the dashed curve of the Profile view. At μs = 0 Westergaard’s value below a load is lower than Boussinesq’s.
Does the result include the weight of the soil?
No. It is the increase caused by the loaded areas only. Add the overburden pressure of the soil above A for the total vertical stress.
How many areas can I add, and can they overlap?
Up to eight, each a circle or a rectangle with its own q and size. They may touch but not overlap; the module refuses an area that would lie on another. The stresses of the areas add.
Can I put A anywhere?
Yes. A is at (x, y) on the ground and at the depth z, in the coordinates of the centres. You do not have to place it at the centre of a circle or the corner of a rectangle; the module integrates for other points.
Can I drag the areas and the point on the drawing?
Yes. In the Section view drag the rings at the edges of a circle or the ends of a rectangle to resize it, and the bar over an area (with a rectangle or several areas) to move it; in the 3D view drag the area on the ground. Drag the blue point to place A and the dashed line to set the depth of the profile; they move in 5 cm steps. After Analyze the areas are locked, but A and the dashed line stay free.
What do the lines of equal stress show?
Each line joins the points of the section with the same Δσz, labelled in kPa; the line through A is blue. Switch them on and off with the Lines of equal stress chip. The Contours view shows the same stress as colour bands on a logarithmic scale.
What values does the module accept?
q zero or more, R, L and B at least 0.001 m, z at least 0.001 m, μs from 0 to 0.49, zmax at least 0.1 m, and x, y, xc and yc any value. Anything else stops the analysis with a message that names the value.
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. Stresses are in kPa, t/m² or ksf and lengths in m or ft. The default example gives 126.8 kPa, 12.68 t/m² with q = 30 t/m², or 2.648 ksf.
Does the module remember my inputs, and what does the report contain?
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. The report has 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, profile and 3D drawings. It is saved as a PDF or a Word file once the model is analysed.
Point loadBoussinesq stresses below one or several point loads.
Vertical line loadStresses below parallel vertical line loads.
Horizontal line loadA horizontal or inclined line load.
Vertical strip loadA uniform vertical load on a strip.
Horizontal strip loadA horizontal or inclined load on a strip.
Triangular strip loadA strip load that increases linearly.
EmbankmentThe load of an embankment with its side slopes.
Circle, any pointAny point below a uniformly loaded circle.
RectangleBelow a corner or any point of a loaded rectangle.
Influence chartAny loaded shape on Newmark’s chart.
Westergaard point loadA point load on layered ground with thin rigid seams.
Westergaard areasLoaded circles and rectangles on a Westergaard material.
