Question 2.5: Rainwater collects behind the concrete retaining wall shown ...
Rainwater collects behind the concrete retaining wall shown in Figure 2.7. If the water-saturated soil (specific gravity = 2.2) acts as a fluid, determine the force and center of pressure on a l-m width of the wall.
The force on a unit width of the wall is obtained by integrating the pressure difference between the right and left sides of the wall. Taking the origin at the top of the wall and measuring y downward, the force due to pressure is
F=\int\left(P-P_{\mathrm{atm}}\right)(1) d yThe pressure difference in the region in contact with the water is
P-P_{\mathrm{atm}}=\rho_{\mathrm{H}_{2} \mathrm{O}} g yand the pressure difference in the region in contact with the soil is
P-P_{\mathrm{atm}}=\rho_{\mathrm{H}_{2} \mathrm{O}} g(1)+2.2 \rho_{\mathrm{H}_{2} \mathrm{O}} g(y-1)The force F is
F=\rho_{\mathrm{H}_{2} \mathrm{O}} g \int_{0}^{1} y d y+\rho_{\mathrm{H}_{2} \mathrm{O}} g \int_{0}^{4}[1+2.2 (y-1)] d yF=\left(1000 \mathrm{~kg} / \mathrm{m}^{3}\right)\left(9.807 \mathrm{~m} / \mathrm{s}^{2}\right)(1 \mathrm{~m})\left(13.4 \mathrm{~m}^{2}\right)=131414 \mathrm{~N}\left(29546 \mathrm{lb}_{\mathrm{f}}\right)
The center of pressure of the force on the wall is obtained by taking moments about the top of the wall.
F y_{\text {c.p. }}=\rho_{\mathrm{H}_{2} \mathrm{O}} g\left\{\int_{0}^{1} y^{2} d y+\int_{1}^{4} y[1+2.2(y-1)] d y\right\}
y_{\text {c.p. }}=\frac{1}{(131414 \mathrm{~N})}\left(1000 \mathrm{~kg} / \mathrm{m}^{3}\right)\left(9.807 \mathrm{~m} / \mathrm{s}^{2}\right)(1 \mathrm{~m})\left(37.53 \mathrm{~m}^{3}\right)=2.80 \mathrm{~m}(9.19 \mathrm{ft})
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