Chapter 14, Problem 14.1P

To determine

Find the relative proportions of the load carried by the piles and the raft (X) in terms of $K_r$ and $K_P$.

Find the stiffness of the piled raft $\left(K_{pr}\right)$ in terms of $K_r$ and $K_P$.

Answer to Problem 14.1P

The relative proportions of the load carried by the piles and the raft in terms of $K_r$ and $K_P$ is $\frac{K_p-0.85K_r}{K_p-0.7K_r}$.

The stiffness of the piled raft $\left(K_{pr}\right)$ in terms of $K_r$ and $K_P$ is $\frac{K_p-0.7K_r}{1-0.72\frac{K_r}{K_p}}$

Explanation of Solution

Given information:

The pile raft interaction factor ${\alpha }_{rp}$ as 0.85.

Calculation:

Consider the ratio of load on raft to total load of pile $\left(\frac{P_r}{P_t}\right)$ is equal to relative proportion carried by the raft (X)

Find the relative proportion carried by the raft (X) using the formula.

$\begin{array}{c}\frac{P_r}{P_t}=X\\ =\frac{K_r\left(1-{\alpha }_{rp}\right)}{K_p+K_r\left(1-2{\alpha }_{rp}\right)}\\ =\frac{K_r\left(1-0.85\right)}{K_p+K_r\left(1-2\times 0.85\right)}\\ =\frac{0.15K_r}{K_p-0.7K_r}\end{array}$

Find the ratio of load on pile to total load $\left(\frac{P_p}{P_t}\right)$.

$\begin{array}{c}\left(\frac{P_p}{P_t}\right)=1-X\\ =1-\frac{0.15K_r}{K_p-0.7K_r}\\ =\frac{K_p-0.7K_r-0.15K_r}{K_p-0.7K_r}\\ =\frac{K_p-0.85K_r}{K_p-0.7K_r}\end{array}$

Therefore, relative proportions of the load carried by the piles and the raft in terms of $K_r$ and $K_P$ is $\frac{K_p-0.85K_r}{K_p-0.7K_r}$.

Find the stiffness of the piled raft $\left(K_{pr}\right)$ using the formula:

$\begin{array}{c}{K}_{pr}=\frac{K_p+K_r\left(1-2{\alpha }_{rp}\right)}{1-{\alpha }_{rp}^2\frac{K_r}{K_p}}\\ =\frac{K_p+K_r\left(1-2\times 0.85\right)}{1-{0.85}^2\frac{K_r}{K_p}}\\ =\frac{K_p-0.7K_r}{1-0.72\frac{K_r}{K_p}}\end{array}$

Therefore, the stiffness of the piled raft $\left(K_{pr}\right)$ in terms of $K_r$ and $K_P$ is $\frac{K_p-0.7K_r}{1-0.72\frac{K_r}{K_p}}$