A concrete gravity retaininvg wall is shown in Figure 14.13. Determine:
a. The factor of safety against overturning
b. The factor of safety against sliding
c. The pressure on the soil at the toe and heel
(Note: Unit weight of concrete = $\gamma_c$ = 150 lb/ft³.)

Question

A concrete gravity retaininvg wall is shown in Figure 14.13. Determine:
a. The factor of safety against overturning
b. The factor of safety against sliding
c. The pressure on the soil at the toe and heel
(Note: Unit weight of concrete = [latex]\gamma_c[/latex] = 150 lb/ft³.)

Accepted Answer

[latex]H^\prime =15+25=17.5[/latex] ft
[latex]K_a= an^2\left(45-\frac{\phi_1^\prime}{2} ight)= an^2\left(45-\frac{30}{2} ight)=\frac{1}{3}[/latex]
[latex]P_a = \frac{1}{2} \gamma {(H^\prime)}^ 2 Ka=\frac{1}{2}(121)(17.5)^2\left(\frac{1}{3} ight)=[/latex] 6176 lb/ft
= 6.176 kip/ft
Since α = 0
[latex]P_h = P_a =[/latex] 6.176 kip/ft
[latex]P_v =[/latex] 0
Part a: Factor of Safety Against Overturning
The following table can now be prepared to obtain [latex]\sum M_R[/latex].

Area (from Figure 14.13)	Weight (kip)	Moment arm from C (ft)	Moment about C (kip.ft/ft)
1	[latex]\frac{1}{2}(0.8)(15)(\gamma_c)=0.9[/latex]	[latex]1.25+\frac{2}{3}(0.8)=1.783[/latex]	1.605
2	[latex](1.5)(15)(\gamma_c)=3.375[/latex]	[latex]1.25+0.8+0.75=2.8[/latex]	9.45
3	[latex]\frac{1}{2}(5.25)(15)(\gamma_c)=5.906[/latex]	[latex]1.25+0.8+1.5+\frac{25.5}{3}=5.3[/latex]	31.30
4	[latex](10.3)(2.5)(\gamma_c)=3.863[/latex]	[latex]\frac{10.3}{2}=5.15[/latex]	19.89
5	[latex]\frac{1}{2}(5.25)(15)(0.121)=4.764[/latex]	[latex]1.25+0.8+1.5+\frac{2}{3}(5.25)=7.05[/latex]	33.59
6	[latex](1.5)(15)(0.121)=\frac{2.723}{21.531}[/latex]	[latex]1.25+0.8+1.5+5.25+0.75=9.55[/latex]	[latex]\frac{26.0}{121.84}=M_R[/latex]

The overturning moment
[latex]M_O = \frac{H^\prime}{3}P_a =\left(\frac{17.5}{3} ight)(6.176)=[/latex] 36.03 kip/ft
[latex]FS_{(overturning)} = \frac{121.84}{36.03} =[/latex] 3.38
Part b: Factor of Safety Against Sliding
From Eq. (14.10), with [latex]k_1 = k_2 = 2/3[/latex] and assuming that [latex]P_p =[/latex] 0,
[latex]FS_{(sliding)} =\frac{\sum V an \left(\frac{2}{3} ight)\phi_2^\prime+B\left(\frac{2}{3} ight)c_2^\prime}{P_a}[/latex]
=[latex]\frac{21.531 an \left(\frac{2 imes 20}{3} ight)+10.3\left(\frac{2}{3} ight)(1.0)}{6.176}[/latex]
=[latex]\frac{5.1+6.87}{6.176}=[/latex]1.94
Part c: Pressure on the Soil at the Toe and Heel
[latex]e = \frac{B}{2}-\frac{ \sum M_R - \sum M_O}{\sum V}= \frac{10.3}{2}-\frac{ 121.84 - 36.03}{21.531}=5.15-3.99=[/latex] 1.16 ft
[latex]q_{toe} =\frac{ \sum V}{B}\left[1+\frac{6e}{B} ight]=\frac{21.531}{10.3}\left[1+\frac{(6)(1.16)}{10.3} ight]=[/latex] 3.5 kip/ft²
[latex]q_{heel} =\frac{ \sum V}{B}\left[1-\frac{6e}{B} ight]=\frac{21.531}{10.3}\left[1-\frac{(6)(1.16)}{10.3} ight]=[/latex] 0.678 kip/ft²

Question 14.4: A concrete gravity retaininvg wall is shown in Figure 14.13....

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