Chapter 17, Problem 17.10P

To determine

Find the modulus elasticity of the foundation soil with the various depth.

Answer to Problem 17.10P

The modulus elasticity of the foundation soil at the depth $1\text{m}$ is $\underset{\_}{9,335\text{kN}/{\text{m}}^{\text{2}}}$.

The modulus elasticity of the foundation soil at the depth $2.5\text{m}$ is $\underset{\_}{12,446\text{kN}/{\text{m}}^{\text{2}}}$.

The modulus elasticity of the foundation soil at the depth $4\text{m}$ is $\underset{\_}{17,115\text{kN}/{\text{m}}^{\text{2}}}$.

The modulus elasticity of the foundation soil at the depth $5.5\text{m}$ is $\underset{\_}{20,227\text{kN}/{\text{m}}^{\text{2}}}$.

The modulus elasticity of the foundation soil at the depth $7\text{m}$ is $\underset{\_}{21,784\text{kN}/{\text{m}}^{\text{2}}}$.

The modulus elasticity of the foundation soil at the depth $8.5\text{m}$ is $\underset{\_}{24,896\text{kN}/{\text{m}}^{\text{2}}}$.

Explanation of Solution

Given information:

The atmospheric pressure is $\left(p_a\right)$ $100\text{kN}/{\text{m}}^{\text{2}}$.

Assume the grain size $\left(D_{50}\right)$ $0.46\text{mm}$.

Show the depth and number of blows shown in table:

Depth,(m)	$N_{60}$
1	6
2.5	8
4	11
5.5	13
7	14
8.5	16

Calculation:

Show the expression of correlation between $q_c\text{and}{N}_{60}$ using the Equation:

$\frac{\left(\frac{q_c}{p_a}\right)}{N_{60}}=c{D}_{50}^a$ (1)

Here, $p_a$ is atmospheric pressure, $N_{60}$ is standard penetration resistance, c and a as developed from various studies.

For c and a value given by Kulhawy and Mayne(1990).

Refer Table 17.7 in section 17.11 “Cone penetration test” in the textbook.

c	a
5.44	0.26

Find the variation of cone penetration resistance with various depth $1\text{m}$

Substitute $100\text{kN}/{\text{m}}^{\text{2}}$ for $p_a$, $0.46\text{mm}$ for $D_{50}$, 5.44 for c, 0.26 for a, and 6 for $N_{60}$ in Equation (1).

$\begin{array}{l}\frac{\left(\frac{q_c}{100}\right)}{6}=5.44{\left(0.46\right)}^{0.26}\\ \frac{\left(\frac{q_c}{100}\right)}{6}=4.4454\\ \left(\frac{q_c}{100}\right)=26.67\\ q_c=2,667\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Find the variation of cone penetration resistance with various depth $2.5\text{m}$.

Substitute $100\text{kN}/{\text{m}}^{\text{2}}$ for $p_a$, $0.46\text{mm}$ for $D_{50}$, 5.44 for c, 0.26 for a, and 8 for $N_{60}$ in Equation (1).

$\begin{array}{l}\frac{\left(\frac{q_c}{100}\right)}{6}=5.44{\left(0.46\right)}^{0.26}\\ \frac{\left(\frac{q_c}{100}\right)}{8}=4.4454\\ \left(\frac{q_c}{100}\right)=35.56\\ q_c=3,556\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Find the variation of cone penetration resistance with various depth $4\text{m}$.

Substitute $100\text{kN}/{\text{m}}^{\text{2}}$ for $p_a$, $0.46\text{mm}$ for $D_{50}$, 5.44 for c, 0.26 for a, and 11 for $N_{60}$ in Equation (1).

$\begin{array}{l}\frac{\left(\frac{q_c}{100}\right)}{6}=5.44{\left(0.46\right)}^{0.26}\\ \frac{\left(\frac{q_c}{100}\right)}{11}=4.4454\\ \left(\frac{q_c}{100}\right)=48.89\\ q_c=4,890\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Find the variation of cone penetration resistance with various depth $5.5\text{m}$.

Substitute $100\text{kN}/{\text{m}}^{\text{2}}$ for $p_a$, $0.46\text{mm}$ for $D_{50}$, 5.44 for c, 0.26 for a, and 13 for $N_{60}$ in Equation (1).

$\begin{array}{l}\frac{\left(\frac{q_c}{100}\right)}{6}=5.44{\left(0.46\right)}^{0.26}\\ \frac{\left(\frac{q_c}{100}\right)}{13}=4.4454\\ \left(\frac{q_c}{100}\right)=57.79\\ q_c=5,779\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Find the variation of cone penetration resistance with various depth $7\text{m}$.

Substitute $100\text{kN}/{\text{m}}^{\text{2}}$ for $p_a$, $0.46\text{mm}$ for $D_{50}$, 5.44 for c, 0.26 for a, and 14 for $N_{60}$ in Equation (1).

$\begin{array}{l}\frac{\left(\frac{q_c}{100}\right)}{6}=5.44{\left(0.46\right)}^{0.26}\\ \frac{\left(\frac{q_c}{100}\right)}{14}=4.4454\\ \left(\frac{q_c}{100}\right)=62.236\\ q_c=6,224\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Find the variation of cone penetration resistance with various depth $8.5\text{m}$.

Substitute $100\text{kN}/{\text{m}}^{\text{2}}$ for $p_a$, $0.46\text{mm}$ for $D_{50}$, 5.44 for c, 0.26 for a, and 16 for $N_{60}$ in Equation (1).

$\begin{array}{l}\frac{\left(\frac{q_c}{100}\right)}{6}=5.44{\left(0.46\right)}^{0.26}\\ \frac{\left(\frac{q_c}{100}\right)}{16}=4.4454\\ \left(\frac{q_c}{100}\right)=71.13\\ q_c=7,113\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Show the expression of modulus elasticity $\left(E_s\right)$ as follows:

$E_s=3.5q_c$ (1)

Here, $q_c$ is cone penetration resistance.

Find the modulus elasticity of the foundation soil at the depth $1\text{m}$ using the Equation:

Substitute $2,667\text{kN}/{\text{m}}^{\text{2}}$ for $q_c$ in Equation (1).

$\begin{array}{c}{E}_s=3.5\times 2,667\\ =9,334.5\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Thus, the modulus elasticity of the foundation soil at the depth $1\text{m}$ is $\underset{\_}{9,335\text{kN}/{\text{m}}^{\text{2}}}$.

Find the modulus elasticity of the foundation soil at the depth $2.5\text{m}$ using the Equation:

Substitute $3,556\text{kN}/{\text{m}}^{\text{2}}$ for $q_c$ in Equation (1).

$\begin{array}{c}{E}_s=3.5\times 3,556\\ =12,446\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Thus, the modulus elasticity of the foundation soil at the depth $2.5\text{m}$ is $\underset{\_}{12,446\text{kN}/{\text{m}}^{\text{2}}}$.

Find the modulus elasticity of the foundation soil at the depth $4\text{m}$ using the Equation:

Substitute $4,890\text{kN}/{\text{m}}^{\text{2}}$ for $q_c$ in Equation (1).

$\begin{array}{c}{E}_s=3.5\times 4,890\\ =17,115\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Thus, the modulus elasticity of the foundation soil at the depth $4\text{m}$ is $\underset{\_}{17,115\text{kN}/{\text{m}}^{\text{2}}}$.

Find the modulus elasticity of the foundation soil at the depth $5.5\text{m}$ using the Equation:

Substitute $5,779\text{kN}/{\text{m}}^{\text{2}}$ for $q_c$ in Equation (1).

$\begin{array}{c}{E}_s=3.5\times 5,779\\ =20,227\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Thus, the modulus elasticity of the foundation soil at the depth $5.5\text{m}$ is $\underset{\_}{20,227\text{kN}/{\text{m}}^{\text{2}}}$.

Find the modulus elasticity of the foundation soil at the depth $7\text{m}$ using the Equation:

Substitute $6,224\text{kN}/{\text{m}}^{\text{2}}$ for $q_c$ in Equation (1).

$\begin{array}{c}{E}_s=3.5\times 6224\\ =21,784\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Thus, the modulus elasticity of the foundation soil at the depth $7\text{m}$ is $\underset{\_}{21,784\text{kN}/{\text{m}}^{\text{2}}}$.

Find the modulus elasticity of the foundation soil at the depth $8.5\text{m}$ using the Equation:

Substitute $7,113\text{kN}/{\text{m}}^{\text{2}}$ for $q_c$ in Equation (1).

$\begin{array}{c}{E}_s=3.5\times 7,113\\ =24,896\text{kN}/{\text{m}}^{\text{2}}\end{array}$

Thus, the modulus elasticity of the foundation soil at the depth $8.5\text{m}$ is $\underset{\_}{24,896\text{kN}/{\text{m}}^{\text{2}}}$.