Chapter 13, Problem 13.1P

To determine

Find the magnitude of the lateral earth forces $P_o$ per unit length of the wall and find the location $\overline{z}$ of the resultant measured from the bottom of the wall.

Answer to Problem 13.1P

The magnitude of the lateral force is $\underset{\_}{393.1\text{kN/m}}$.

The location of the resultant measured from the bottom of the wall is $\underset{\_}{2.5\text{m}}$.

Explanation of Solution

Given information:

The height (H) of the retaining wall is 7.5 m.

The soil friction angle ${\varphi }^{\prime }$ is $31°$.

The angle of wall friction ${\delta }^{\prime }$ is 0.

The unit weight $\gamma$ of the sand is $19.2{\text{kN/m}}^{\text{3}}$.

The over-consolidation ratio (OCR) is 2.2.

Calculation:

Calculate the earth pressure coefficient $\left(K_o\right)$ using the relation.

$K_o=\left(1-\sin {\varphi }^{\prime }\right){\left(OCR\right)}^{\sin {\varphi }^{\prime }}$

Substitute $31°$ for ${\varphi }^{\prime }$ and $2.2$ for $OCR$.

$\begin{array}{c}{K}_o=\left(1-\sin 31°\right){\left(2.2\right)}^{\sin 31°}\\ =0.48496\times 1.5009\\ =0.7279\end{array}$

Calculate the total force per unit length $\left(P_o\right)$ using the relation.

$P_o=\frac{1}{2}{K}_o\gamma H^2$

Substitute $0.7279$ for $K_o$, $19.2\text{kN}/{\text{m}}^{\text{3}}$ for $\gamma$, and $7.5\text{m}$ for H.

$\begin{array}{c}{P}_o=\frac{1}{2}\times 0.7279\times 19.2\times {\left(7.5\right)}^2\\ =393.1\text{kN/m}\end{array}$

Thus, the magnitude of the lateral earth force per unit length $\left(P_o\right)$ is $\underset{\_}{393.1\text{kN/m}}$.

Calculate the location of the resultant $\left(\overline{z}\right)$ using the relation.

$\overline{z}=\frac{H}{3}$

Substitute $7.5\text{m}$ for H.

$\begin{array}{c}\overline{z}=\frac{7.5}{3}\\ =2.5\text{m}\end{array}$

Thus, the location of the resultant $\left(\overline{z}\right)$ measured from the bottom of the wall is $\underset{\_}{2.5\text{m}}$.