Soil Mechanics

CalcpadCE worksheet in this section computes stresses and settlements in an elastic half-space under an arbitrary surface pressure by integration of the Boussinesq point-load solution.

The soil stress and deformations sheet integrates the Boussinesq vertical-stress kernel \(3 z^3 / (2 \pi R^5)\) and the two shear-stress kernels over a rectangular footprint with a user-defined pressure distribution \(p(x, y)\). A trapezoidal pressure \(p(x, y) = 100 \text{ kPa} \cdot (1 + x / a)\) on a \(2 \text{ m} \times 3 \text{ m}\) area is taken as the example, and the vertical stress \(\sigma_z\) and shear stresses \(\tau_{xz}\), \(\tau_{yz}\) are mapped on a vertical section through the loaded area. The vertical settlement at the surface follows from \(d(x, y) = (1 - \nu^2) / E_0 \cdot \int \sigma_z \mathrm d z\) over the active depth, so the elastic modulus \(E_0\) and the Poisson ratio \(\nu\) are the only soil parameters required. Both single-variable and double-variable plots ($Plot, $Map) visualise the stress field and the settlement trough in a single document.

Soil Stress and Deformations

Stresses and settlement in an elastic half-space under a trapezoidal surface pressure on a rectangular footprint, obtained by numerical integration of the Boussinesq point-load kernel for \(\sigma_z\), \(\tau_{xz}\) and \(\tau_{yz}\).

Code:

'(by integration of the Boussinesq formula)
'Dimensions of loaded area -'a = 2m', 'b = 3m
'Surface load -'p(x; y) = 100kPa*(1 + (x/a))
'Elastic modulus -'E_0 = 20MPa
'Poisson ratio -'ν = 0.3
'<p><strong>Stress equations</strong></p>
R(x; y; z) = sqrt(x^2 + y^2 + z^2)
'Vertical -'σ_z(x; y; z) = 3/(2*π)*$Integral{$Integral{p(x_p; y_p)*z^3/R(x - x_p; y - y_p; z)^5 @ x_p = -a/2 : a/2} @ y_p = -b/2 : b/2}|kPa
'Shear -'τ_xz(x; y; z) = 3/(2*π)*$Integral{$Integral{p(x_p; y_p)*(x - x_p)*z^2/R(x - x_p; y - y_p; z)^5 @ x_p = -a/2 : a/2} @ y_p = -b/2 : b/2}|kPa
'Shear -'τ_yz(x; y; z) = 3/(2*π)*$Integral{$Integral{p(x_p; y_p)*(y - y_p)*z^2/R(x - x_p; y - y_p; z)^5 @ x_p = -a/2 : a/2} @ y_p = -b/2 : b/2}|kPa
'<p><strong>Deformation equations</strong></p>
d(x; y) = (1 - ν^2)/E_0*$Integral{σ_z(x; y; z) @ z = H_0 : H}|mm
'<h4>Results</h4>
'Maximum depth -'H = 6m
'<!--'Precision = 2*10^-4','PlotStep = 5','H_0 = 0.01*H','k = 1.5'-->
'Plot for plane at'y = 0m
'<!--'PlotWidth = 300','PlotHeight = 60'-->
$Plot{p(x; y)*(abs(x) < a/2) @ x = -k*a : k*a}
'<table><tr><td>
'Vertical stress σ<sub>z</sub>, kPa
'<!--'PlotHeight = PlotWidth*(H - H_0)/(2*k*a)'-->
$Map{σ_z(x; y; z) @ x = -k*a : k*a & z = H : H_0}
'</td><td>
'For'x = 0m
'<!--'PlotWidth = 60'-->
$Plot{σ_z(x; y; z)|-z @ z = H_0 : H}
'</td></tr><tr><td>
'Shear stress τ<sub>xz</sub>, kPa
'<!--'PlotWidth = 300'-->
$Map{τ_xz(x; y; z) @ x = -k*a : k*a & z = H : H_0}
'</td><td>
'For'x = 0.5*a
'<!--'PlotWidth = 60'-->
$Plot{τ_xz(x; y; z)|-z @ z = H_0 : H}
'</td></tr></table>
'Deformations, mm
'<!--'PlotWidth = 300','PlotHeight = 60'-->
$Plot{-d(x; y) & 0mm @ x = -k*a : k*a}

Rendered Output:

(by integration of the Boussinesq formula)

Dimensions of loaded area - a = 2 m , b = 3 m

Surface load - p ( x; y )  = 100 kPa · (1 + xa)

Elastic modulus - E₀ = 20 MPa

Poisson ratio - ν = 0.3

Stress equations

R ( x; y; z )  =    √ x² + y² + z²

Vertical - σ_z ( x; y; z )  = 32 · π ·   b2∫-b2   a2∫-a2 p  ( x_p; y_p )  · z³R  ( x − x_p; y − y_p; z ) ⁵ dx_p dy_p

Shear - τ_xz ( x; y; z )  = 32 · π ·   b2∫-b2   a2∫-a2 p  ( x_p; y_p )  ·  ( x − x_p )  · z²R  ( x − x_p; y − y_p; z ) ⁵ dx_p dy_p

Shear - τ_yz ( x; y; z )  = 32 · π ·   b2∫-b2   a2∫-a2 p  ( x_p; y_p )  ·  ( y − y_p )  · z²R  ( x − x_p; y − y_p; z ) ⁵ dx_p dy_p

Deformation equations

d ( x; y )  = 1 − ν²E₀ ·   H∫H₀ σ_z  ( x; y; z )  dz

Results

Maximum depth - H = 6 m

Plot for plane at y = 0 m

Vertical stress σz, kPa	For x = 0 m
Shear stress τxz, kPa	For x = 0.5 · a = 0.5 · 2 m = 1 m

Deformations, mm

Spotted an error? Edit these examples.