See how a concentrated load applied at the ground surface spreads down and outward through the soil mass. Adjust the load, depth and radial distance and watch the vertical stress increase and the shape of the "pressure bulb" update in real time, building intuition for the zone that affects settlement.
Parameters
Point load Q
kN
Vertical load applied at one point on the surface
Depth z
m
Depth below the surface where stress is evaluated
Radial distance from the vertical axis r
m
Horizontal distance from the axis directly under the load
Existing effective stress at that depth
kPa
Stress already present from the soil's own weight before loading
Results
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Influence factor I_b
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Vertical stress increase Δσz (kPa)
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Max stress increase under the load (kPa)
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Increase over existing stress (%)
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Total stress after loading (kPa)
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Influence assessment
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Pressure bulb — how stress spreads in the soil
From the surface point load Q, vertical-stress contours spread down and outward in an onion shape. The cross marks the evaluation point (depth z, radial distance r). Inner, more intense contours carry the larger stress.
Stress increase vs depth (under the load, r=0)
Stress increase vs radial distance (fixed depth z)
Vertical stress increase Δσz at a point at depth z and radial distance r. Q: surface point load. The stress falls off with the square of depth and forms an onion-shaped "pressure bulb" that spreads outward from the load point.
Maximum stress increase directly under the load (r=0) and the dimensionless influence factor I_b. The stress can be written as Δσz = Q·I_b/z². The depth ratio r/z controls I_b — the larger it is, the smaller the influence factor.
What is the Boussinesq Stress Simulator?
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"Stress in soil" — is this about how the weight of a building travels down through the ground once you build it?
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Exactly. When a foundation, an embankment or a storage tank presses on the ground, that force does not stay at the surface — it spreads down and out through the soil mass. The soil grains hand the load on from one to the next as it travels deeper. If you cannot predict how it spreads, you cannot work out how much the ground will settle. That is why, in geotechnical engineering, getting the stress in the soil right is the very first step.
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But the ground looks complicated — sand, clay, different layers. How do you even calculate it?
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Good point. Real soil is messy, so you start by simplifying boldly. In 1885 the French mathematician Boussinesq solved the idealised case: a semi-infinite, homogeneous, isotropic, linearly elastic half-space with a single concentrated load on its surface. "Semi-infinite" means it is flat at the top — the ground surface — but extends forever downward and sideways. Thanks to that idealisation, you only need the depth z and the radial distance r from the load, and a single formula gives you the vertical stress increase Δσz at that point.
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When I raise the depth z on the left, the stress increase Δσz drops fast. Why is that?
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Because directly under the load Δσz is inversely proportional to the square of depth z. A load concentrated at the surface spreads three-dimensionally into the soil like a cone. The deeper you go, the larger the cross-sectional area carrying the same load — and that area grows with the square of depth. A bigger area means the stress per unit area thins out. So the stress weakens rapidly with depth. A handy rule of thumb is the "significant depth": below roughly twice the footing width, the added stress is too small to matter for settlement.
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In the pressure-bulb chart, even at the same depth the spot directly under the load is the most intense, and it fades as you move sideways. Can the formula explain that too?
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It can. That is the term [1/(1+(r/z)²)]^(5/2) in the formula. r/z is the depth ratio: right under the load r=0 and the term equals 1; move sideways and the denominator grows, so the term shrinks fast. So on a horizontal plane at a given depth the stress is largest on the centre-line and smaller outward. Join points of equal stress and you get the onion shape — the "pressure bulb". In design you use that shape to check, say, how far the bulb of a neighbouring building's footing reaches under your own.
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A real foundation pushes over a wide area, not at one point. You can't use the point-load formula directly there, right?
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Sharp. Correct — the point-load formula is a "building block". Rectangular and circular footings, strip wall footings, distributed loads like embankments — you get their stresses by integrating this point-load solution over the loaded area. In practice engineers use influence-factor charts that tabulate those integrals, or the simple 2:1 method. But the intuition — stress decaying with the square of depth, being largest on the centre-line, the idea of a significant depth — all of it comes straight from this Boussinesq point-load solution.
Frequently Asked Questions
The Boussinesq solution gives the increase in vertical stress at any point inside the ground when a concentrated (point) load is applied at the surface of a soil treated as a semi-infinite, homogeneous, isotropic, linearly elastic half-space. The French mathematician Joseph Boussinesq solved it in 1885. For a depth z and a horizontal distance r from the load, the vertical stress increase is given by the closed-form expression that combines a 1/z-squared term with the depth-ratio term in brackets. It is the starting point for predicting how a foundation, embankment or tank load spreads into the soil and therefore how much the ground will settle.
Directly under the load (r=0) the stress increase is 3Q divided by 2-pi-z-squared, so it falls off with the square of the depth z. Physically, a load concentrated at the surface spreads three-dimensionally into the soil mass like a cone, and the cross-sectional area carrying the same load grows with the square of depth. A larger area means a smaller stress per unit area. As a result the added stress weakens rapidly with depth, and below roughly twice the footing width the extra stress is usually too small to matter for settlement. This depth is called the significant depth.
The pressure bulb is the onion-shaped pattern traced by stress contours, the lines that connect points of equal vertical stress increase. The stress is largest on the centre-line directly under the load and decreases outward at any given depth, so the contours start at the load point and spread down and out. In design, engineers treat the contour where the added stress drops to about 10 percent (or 20 percent) of the initial stress as the practical limit of influence, and use its depth and width to set the zone for ground improvement or foundation checks.
The Boussinesq point-load solution is the building block for handling more complex loads. Stresses beneath rectangular and circular footings, strip loads and embankments are obtained by integrating the point-load solution over the loaded area. In practice engineers use influence-factor charts that tabulate those integrals, or the simple 2:1 approximation. This tool deals with the point load itself, but the intuition it gives - that stress decays with the square of depth, is largest on the centre-line, and has a significant depth - carries over to every stress-in-soil calculation.
Real-World Applications
Building and bridge foundation design: Estimating the settlement of spread footings, pile groups and bridge-pier foundations relies on calculating stress in the soil. The load a foundation delivers is represented as a point or distributed load, the Boussinesq family of formulas gives the stress increase at each depth, and that is combined with the soil's compressibility from oedometer tests to compute settlement. When the pressure bulbs of adjacent foundations overlap, the stresses add up, so the same approach is used to check closely spaced construction.
Embankment, road and levee settlement prediction: An embankment or road built on soft ground keeps settling by consolidation over a long period. By treating the fill as a strip or trapezoidal distributed load, computing the stress in the soil and identifying to what depth the stress reaches — the significant depth — engineers decide the zone that needs ground improvement and the height of a preload embankment.
Storage tanks and silos: For heavy, large-area circular structures such as oil tanks and grain silos, differential settlement causes tilt and pipe damage. Treating the load as a circular uniform load and integrating the Boussinesq solution, the stress difference between the centre and the edge predicts the differential settlement. At the design stage, ground improvement under the foundation ring or pre-consolidation by a test load (hydrotest) is considered.
Verifying soil and foundation FEM analyses: When stresses in the ground are computed by finite elements, the result is first checked against the Boussinesq closed-form solution to confirm the mesh and boundary conditions are reasonable. Simple cases such as a point load or a uniform load have analytical solutions, so if the FEM result deviates strongly from them, it is a sanity check pointing to possible errors in element size, elastic moduli or how the boundaries were placed.
Common Misconceptions and Pitfalls
The biggest pitfall is assuming the Boussinesq stress depends on how stiff the ground is (its elastic modulus). Surprisingly, the formula for the vertical stress increase Δσz from a point load contains neither Young's modulus nor Poisson's ratio. In a semi-infinite elastic body the stress distribution is governed by the load and the geometry alone. Stiffness enters only at the next step — turning stress into displacement (settlement). So for the same load, the stress increase at a given depth is the same in stiff and soft ground; what differs is how much the ground compresses under that stress. Mixing up these two roles leads to the mistake of thinking that ground improvement reduces the stress itself.
Next, applying the point-load formula directly to a wide foundation that pushes over an area. The concentrated-load formula gives an infinite stress directly under the load (r=0, z→0). A real foundation presses over a finite area, so no such singularity occurs. At shallow depths relative to the footing width, the point-load approximation and the actual distributed load give very different stresses. As a general rule, if the evaluation point is deeper than about two to three times the footing width the point-load approximation is good enough in practice, but at shallow depths you must always integrate the load as a rectangular or circular distribution, or use influence-factor charts.
Finally, jumping to "large stress means immediate danger". Whether a vertical stress increase Δσz is large is judged not on its own but against the existing effective stress at that depth. That is why this tool reports a percentage increase. As a general guide, an increase below 10 percent of the initial stress has little effect on settlement; 10 to 25 percent is non-negligible; above 25 percent the consolidation settlement should be examined carefully. In addition, immediate settlement, consolidation settlement and secondary (creep) settlement act on different time scales. Beyond the magnitude of the stress, weigh the soil type — sand or clay — and the drainage conditions, so that time and soil character are evaluated together.
How to Use
Enter the concentrated point load (qNum) in kN applied vertically at ground surface.
Set the depth (zNum) in meters below the point of load application where you want to analyze stress distribution.
Define the radial distance (rNum) in meters from the vertical axis through the load point.
The simulator calculates the Boussinesq influence factor I_b and vertical stress increase Δσz using the classical elastic solution for a point load on a semi-infinite, homogeneous soil mass.
Worked Example
For a 500 kN point load applied on clay soil: at depth z=3 m directly under the load (r=0 m), I_b=0.4775, yielding Δσz=79.6 kPa. At the same depth but r=2 m horizontal distance, I_b drops to 0.2156 and Δσz=35.8 kPa. If existing vertical stress at 3 m depth is 60 kPa, the total stress becomes 140 kPa, representing a 59% increase. Beyond r=4 m at this depth, stress increase falls below 10 kPa and becomes negligible for foundation design.
Practical Notes
The Boussinesq solution assumes linear elasticity and applies best to sandy soils with E=15–30 MPa; clay behavior may deviate significantly due to consolidation and plasticity.
Stress bulbs typically extend to 2–3 times the load width; for a point load, stress becomes negligible beyond r/z≈1.5 at practical thresholds (Δσz<5% of surface load).
Use this for preliminary pile foundation settlement checks, raft pressure distribution verification, and assessing stress influence zones for underground utilities.
Results are time-zero (undrained); in saturated clay, pore pressure dissipation over weeks or months redistributes effective stress differently.