Chapter 8, Problem 8.7P

To determine

Calculate the consolidation settlement under the corner of mat foundation.

 </TO-DETERMINE>

The consolidation settlement under the corner of mat foundation is $\underset{¯}{174\text{mm}}$.

Given information:

The length of the foundation is $L=12\text{m}$.

The width of the foundation is $B=10\text{m}$.

The depth of the foundation $\left(D_f\right)$ is $2.2\text{m}$

The load value Q is $30\text{MN}$.

The preconsolidation pressure $\left({{\sigma }^{\prime }}_c\right)$ is $105{\text{kN/m}}^{\text{2}}$.

The depth values are $x_1=2\text{m}$, $x_2=2\text{m}$ and $x_3=5.2\text{m}$

Calculation:

Find the consolidation settlement at the center of the foundation using the relation.

$S_{c(p)}=\frac{C_c{H}_c}{1+e_o}\log \left(\frac{{{\sigma }^{\prime }}_o+\Delta {{\sigma }^{\prime }}_{av}}{{{\sigma }^{\prime }}_o}\right)$

Find the stress ${{\sigma }^{\prime }}_o$

$\begin{array}{c}{{\sigma }^{\prime }}_o=\left(4.2\text{m}\times \text{16}{\text{kN/m}}^3\right)+2\text{m}\times \left(18-9.81\right)+\frac{5.2\text{m}}{2}\times \left(17.5-9.81\right)\\ =67.2+16.38+19.99\\ =103.57{\text{kN/m}}^{\text{2}}\end{array}$

Height of clay $H_c=5.2\text{m}$

$\begin{array}{l}{C}_c=0.38\\ e_o=0.88\end{array}$

Find the net load per unit area using the relation.

$\begin{array}{c}q=\frac{Q}{A}-\gamma D_f\\ =\frac{\left(30\text{MN}\times \frac{\text{1000}\text{kN}}{1\text{MN}}\right)}{10\times 12}-\left(16\times 2.2\right)\\ =214.8{\text{kN/m}}^{\text{2}}\end{array}$

At the top of the clay layer,

$\begin{array}{c}m=\frac{B}{z}\\ =\frac{10}{2+2}\\ =2.5\end{array}$

$\begin{array}{c}n=\frac{L}{z}\\ =\frac{12}{2+2}\\ =3\end{array}$

Refer Table 6.4, “Variation of influence value I”

Take I value as 0.242 for $m=2.5$ and $n=3$.

Find $\Delta {{\sigma }^{\prime }}_t$.

$\begin{array}{c}\Delta {{\sigma }^{\prime }}_t=I{q}_o\\ =0.242q_o\end{array}$

At the middle of the clay layer,

$\begin{array}{c}m=\frac{B}{z}\\ =\frac{10}{6.2}\\ =1.61\end{array}$

$\begin{array}{c}n=\frac{L}{z}\\ =\frac{12}{6.2}\\ =1.94\end{array}$

Refer Table 6.4, “Variation of influence value I”

Take I value as 0.226 for $m=1.61$ and $n=1.94$.

Find $\Delta {{\sigma }^{\prime }}_m$.

$\begin{array}{c}\Delta {{\sigma }^{\prime }}_m=I{q}_o\\ =0.226q_o\end{array}$

At the bottom of the clay layer,

$\begin{array}{c}m=\frac{B}{z}\\ =\frac{10}{9.2}\\ =1.086\end{array}$

$\begin{array}{c}n=\frac{L}{z}\\ =\frac{12}{9.2}\\ =1.30\end{array}$

Refer Table 6.4, “Variation of influence value I”

Take I value as 0.201 for $m=1.086$ and $n=1.30$.

Find $\Delta {{\sigma }^{\prime }}_m$.

$\begin{array}{c}\Delta {{\sigma }^{\prime }}_m=I{q}_o\\ =0.201q_o\end{array}$

Find the average stress increase in the clay layer below the corner of each rectangular area.

$\begin{array}{c}\Delta {{\sigma }^{\prime }}_{\text{av}}=\frac{1}{6}\left(\Delta {{\sigma }^{\prime }}_t+4\Delta {{\sigma }^{\prime }}_m+\Delta {{\sigma }^{\prime }}_b\right)\\ =\frac{1}{6}\left(214.8\right)\left(0.242+4\times 0.226+0.201\right)\\ =48.2{\text{kN/m}}^{\text{2}}\end{array}$

Find the consolidation settlement at the center of the foundation using the relation.

$\begin{array}{c}{S}_{c(p)}=\frac{C_c{H}_c}{1+e_o}\log \left(\frac{{{\sigma }^{\prime }}_o+\Delta {{\sigma }^{\prime }}_{av}}{{\sigma }^{\prime }}\right)\\ =\frac{0.38\times 5.2}{1+0.88}\log \left(\frac{103.57+48.2}{103.57}\right)\\ =0.174\text{m}\\ \text{=174}\text{mm}\end{array}$

Therefore, the consolidation settlement under the corner of mat foundation is $\underset{¯}{174\text{mm}}$.