3.3 Stresses in the soil due to a line load

Loading transferred to the soil from e.g., a rail track of limited width (Figure 3.6) can be modelled as an "infinitely long" line load acting on the surface of a homogeneous elastic half-space. As plane-strain symmetry conditions apply, additional soil stresses due to the application of a line load can be calculated as:

(3.5) [latex]\Delta {\sigma _z} = \dfrac{{2{Q_{ext}}{z^3}}}{{\pi {{\left( {{x^2} + {z^2}} \right)}^2}}}[/latex]

(3.6) [latex]\Delta {\sigma _x} = \dfrac{{2{Q_{ext}}{x^2}z}}{{\pi {{\left( {{x^2} + {z^2}} \right)}^2}}}[/latex]

(3.7) [latex]\Delta {\tau _{zx}} = \dfrac{{2{Q_{ext}}x{z^2}}}{{\pi {{\left( {{x^2} + {z^2}} \right)}^2}}}[/latex]

The stress increment components and the vertical z and horizontal x distance from the line load considered in Eqs. 3.5 to 3.7 are defined in Figure 3.6.

In the special case where the line load is acting near the crest of a retaining wall, the horizontal stress distribution along the wall height can be determined as function of the height of the wall H_o (Figure 3.6):

(3.8) [latex]\Delta {\sigma _x} = \dfrac{{4{Q_{ext}}{a^2}b}}{{\pi {H_o}{{\left( {{a^2} + {b^2}} \right)}^2}}}[/latex]

and the resultant horizontal force per running meter of the wall is equal to (Figure 3.6):

(3.9) [latex]\Delta {Q_x} = \dfrac{{2{Q_{ext}}}}{{\pi \left( {{a^2} + 1} \right)}}[/latex]

Note that Eq. 3.8 yields identical results to Eq. 3.5, the only difference between these expressions is that in Eq. 3.8 the coordinates x and z as expressed as function of H_o.