Practice Question: Factor of Safety Against Overturning for a Retaining Wall

---

---

Solution

Answer: B) 3.8

The factor of safety against overturning is calculated by taking the ratio of the resisting moment to the overturning moment. 

The resisting moment is caused by the self-weight of concrete and soil above the wall footing resisting the overturning of the wall.

The overturning moment is caused by the active pressure of soil retained by the wall. 

Both overturning moment and resisting moment are calculated based on rotation about the toe. Moments are calculated in units of pound-foot per foot of the wall.

Calculate the resisting moment based on weight and distance to center of each component from toe:

 

$$ W_{wall}=1.5\ ft\ \ast\ 12\ ft\ \ast\ 150\ pcf=2700\ plf $$
$$ x_{wall}=4.75\ ft $$
$$ W_{ftg}=2.5\ ft\ \ast\ (4.0\ ft+1.5\ ft+8.0\ ft)\ \ast\ 150\ pcf=5062.5\ plf $$
$$ x_{ftg}=6.75\ ft $$
$$ W_{toesoil}=4\ ft\ \ast\ 3\ ft\ast120\ pcf=1440\ plf $$
$$ x_{toesoil}=2\ ft $$
$$ W_{heelsoil\ }=12\ ft\ \ast\ 8\ ft\ast120\ pcf=11520\ plf $$
$$ x_{heelsoil}=9.5\ ft $$
$$ M_{resisting}=W_{wall}\ \ast\ x_{wall}+W_{ftg}\ \ast{\ x}_{ftg}+W_{toesoil}\ \ast\ x_{toesoil}+W_{heelsoil\ }\ \ast\ x_{heelsoil} $$
$$ M_{resisting}=159316.9\frac{lb\ \ast\ ft}{ft} $$

Calculate the overturning moment based on the active earth pressure. 

The equation for active earth pressure:
$$ p_a=p_v\ \ast\ k_a-2\ \ast\ c\ \ast\ \sqrt{k_a}  $$

where:
$$ p_v=vertical\ pressure $$
$$ k_a=Rankine\ active\ earth\ pressure\ coefficient $$
$$ c=cohesion\ (given) $$

Rankine active earth pressure coefficient, since ϕ = 0 (saturated clay):
$$ k_a=1 $$
Vertical pressure:
$$ p_v=\gamma\ \ast\ H=120\ pcf\ \ast\ \left(12\ ft+2.5\ ft\right)=1740\ psf $$
Active earth pressure:
$$ p_a=1740\ psf\ \ast\ 1-2\ast275\ psf\ast\sqrt1=1190\ psf $$
Resultant force due to active earth pressure:
$$ R_a=\frac{1}{2}\ \ast\ p_a\ \ast\ H=\frac{1}{2}\ \ast\ 1190\ psf\ \ast\ \left(12\ ft+2.5\ ft\right)=8627.5\ plf $$

Since the resultant force acts at H/3 above the base (triangular distribution), we can calculate the overturning moment as follows:
$$ M_{overturning}=R_a\ \ast\ \frac{H}{3}=8627.5\ plf\ \ast\ \frac{12\ ft+2.5\ ft}{3}=41699.6\frac{lb\ \ast\ ft}{ft} $$

Determine Factor of Safety:

$$ FS=\frac{M_{resisting}}{M_{overturning}}\approx 3.8 $$

---