Stress Distribution in Soil

Stresses due to applied loads

Settlement and stability calculations need the increase in vertical stress $\Delta {\sigma }_z$ at depth caused by foundation loads, embankments and surcharges. These induced stresses add to the existing geostatic stresses (self-weight of soil). They spread out and decrease with depth and with horizontal distance from the load.

Boussinesq's solution — point load

For a vertical point load $Q$ at the surface of a semi-infinite, homogeneous, isotropic, linearly elastic medium, the vertical stress at depth $z$ and radial distance $r$:

Assumptions: soil is elastic, homogeneous, isotropic and semi-infinite; weightless; initially unstressed; the load is concentrated on a point; Hooke's law applies. Although soils are not truly elastic, the results agree well enough for practice.

Observations: ${\sigma }_z$ is independent of $E$ and $\mu$ (for vertical stress); it varies inversely with $z^2$; it is infinite at the load point (the formula is not valid very close to the surface).

Westergaard's solution — stratified soils

For soils with thin, rigid, non-yielding horizontal layers (e.g. alternate sand and clay seams) that prevent lateral strain. With Poisson's ratio $\mu$ = 0:

Below the load: $I_W=1/\pi =0.3183$ — about two-thirds of Boussinesq's value. Westergaard's stresses are lower near the load axis; at larger $r/z$ (beyond about 1.5) the two converge.

Line and strip loads

Line load $q$ per unit length (e.g. a wall on a narrow footing):

Strip load of intensity $q$ and width $B$: if the strip subtends an angle $\alpha$ (radians) at the point, and $\delta$ is the angle between the vertical and the line to the nearer edge: