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3.4 Stresses in the soil due to a strip pressure

Stresses applied on the soil surface e.g., from a footing of “infinite” length can be modelled as an infinitely long (strip) pressure acting on the surface of a homogeneous elastic half-space (Figure 3.7a). As plane-strain symmetry conditions apply, additional stresses due to the application of the pressure qext can be calculated as:

Diagram showing a strip pressure applied on the surface of a half-space, and the additional vertical Δσ_z and horizontal stresses Δσ_z that develop at depth z and distance x from the pressure's edge.
Figure 3.7a. Stresses due to strip pressure acting on the surface of a homogeneous elastic half-space.

 

Diagram showing a strip pressure applied on the surface of a half-space, and the additional vertical Δσ_z and horizontal stresses Δσ_z that develop at depth z underneath the axis of the pressure. The formulas providing the angles a and b for this special case are also provided.
Figure 3.7b. Values of angles α and β in the special case where stresses are estimated below the axis of the strip pressure.

(3.10) \Delta {\sigma _z} = \dfrac{{{q_{ext}}}}{\pi }\left[ {\alpha + {\rm{sin}}\alpha \cos \left( {\alpha + 2\beta } \right)} \right]

(3.11) \Delta {\sigma _x} = \dfrac{{{q_{ext}}}}{\pi }\left[ {\alpha - \sin \alpha \cos \left( {\alpha + 2\beta } \right)} \right]

(3.12) \Delta {\tau _{zx}} = \dfrac{{{q_{ext}}}}{\pi }\left[ {\sin \alpha \sin \left( {\alpha + 2\beta } \right)} \right]

Τhe stress increment components and angles α and β are defined in Figure 3.6a. Angles α and β must be input in Eqs. 3.10 to 3.12 in radians, not in degrees, and may attain negative values too, as shown in Figure 3.7b.

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