Chapter 11, Problem 11.1P

A continuous foundation is required in a soil where $c^{\prime }=10\text{kN}/{\text{m}}^{\text{2}}$, ${\varphi }^{\prime }=26°$, and $\gamma =19.0\text{kN}/{\text{m}}^{\text{3}}$. The depth of the footing will be 1.0 m. The dead load and the live load are 600 kN/m and 400 kN/m, respectively. Determine the required width for the foundation based on allowable stress design with FS = 3, using Eq. (6.10) and Table 6.1.

To determine

Find the required width for the foundation.

Answer to Problem 11.1P

The required width for the foundation is $\underset{¯}{3.5\text{m}}$.

Explanation of Solution

Given information:

The unit weight $\left(\gamma \right)$ of the soil is $19{\text{kN/m}}^3$.

The cohesion $\left(c^{\prime }\right)$ is $10{\text{kN/m}}^2$.

The soil friction angle $\left({\varphi }^{\prime }\right)$is $26°$.

The depth $\left(D_f\right)$ of footing is 1 m.

The dead load is 600 kN/m.

The live load is 400 kN/m.

The factor of safety is 3.

Calculation:

Refer Table 6.1, “Terzaghi’s bearing capacity factors” in the text book.

For $\left({\varphi }^{\prime }=26°\right)$:

Take the value of $N_c$ as 27.09.

Take the value of $N_q$ as 14.21.

Take the value of $N_{\gamma }$ as 9.84.

Determine the ultimate bearing capacity of the foundation $q_u$ using the formula;

$\begin{array}{c}{q}_u=\left[c^{\prime }{N}_c+q{N}_q+\frac{1}{2}\gamma B{N}_{\gamma }\right]\\ =c^{\prime }{N}_c\left[c_u{N}_c{F}_{cs}{F}_{cd}+\left(\gamma \times D_f\right)N_q+\frac{1}{2}\gamma B{N}_{\gamma }\right]\\ =\left[10\times 27.09+\left(19\times 1\right)\times 14.21+\frac{1}{2}\times 19\times B\times 9.84\right]\\ =93.84B+540.89{\text{kN/m}}^2\end{array}$

Determine the allowable bearing capacity of the foundation $q_{\text{all}}$ using the formula;

$\begin{array}{c}{q}_{\text{all}}=\frac{q_u}{\text{FS}}\\ =\frac{93.84B+540.89{\text{kN/m}}^2}{3}\\ =31.2B+180.3{\text{kN/m}}^2\end{array}$

Find the design load.

$\begin{array}{c}\text{Design load}=\text{Dead}\text{load}+\text{Live}\text{load}\\ =600+400\\ =1,000\text{kN/m}\end{array}$

Find the required width for the foundation:

$\begin{array}{l}\frac{\text{Design load}}{B}\le q_{\text{all}}\\ \frac{1,000\text{kN/m}}{B}\le 31.2B+180.3{\text{kN/m}}^2\\ 31.2B^2+180.3B-1,000\ge 0\\ B\ge 3.5\text{m}\end{array}$

Hence, the required width for the foundation is $\underset{¯}{3.5\text{m}}$.