A sample of clay is isotropically consolidated to a mean effective stress of 200kPa and then

subjected to standard undrained shear test. Calculate values of q, p’, v at failure. The soil

constants are:

M=0.87, N=2.16, =2.09 and =0.11

To solve this problem, we'll break it down step by step, using the given soil parameters. The variables \( q \), \( p' \), and \( v \) represent:

- \( q \): Deviatoric stress (also known as shear stress)
- \( p' \): Mean effective stress
- \( v \): Specific volume (volume per unit mass)

The soil parameters given are:

- \( M = 0.87 \) (slope of the critical state line in \( q \)-\( p' \) space)
- \( N = 2.16 \) (intercept of the normal consolidation line at \( p' = 1 \) kPa)
- \( \Gamma = 2.09 \) (specific volume at critical state corresponding to \( p' = 1 \) kPa)
- \( \lambda = 0.11 \) (slope of the normal consolidation line in specific volume versus logarithm of pressure)

### 1. **Calculate Mean Effective Stress \( p' \) at Failure**
   
For an undrained test, the mean effective stress \( p' \) at failure remains the same as the initial mean effective stress because no volume change occurs. Therefore:

\[
p' = 200 \text{ kPa}
\]

### 2. **Calculate Deviatoric Stress \( q \) at Failure**

The critical state line in \( q \)-\( p' \) space is defined by:

\[
q = M \cdot p'
\]

Substituting the given values:

\[
q = 0.87 \times 200 \text{ kPa} = 174 \text{ kPa}
\]

### 3. **Calculate Specific Volume \( v \) at Failure**

The critical state specific volume \( v \) at failure can be calculated using the following relation:

\[
v = \Gamma - \lambda \cdot \log_{10} p'
\]

Substituting the given values:

\[
v = 2.09 - 0.11 \times \log_{10} 200
\]

Calculating the logarithm:

\[
\log_{10} 200 \approx 2.301
\]

So,

\[
v = 2.09 - 0.11 \times 2.301 = 2.09 - 0.2531 = 1.8369
\]

### **Summary of Results:**

- \( p' \) at failure = \( 200 \) kPa
- \( q \) at failure = \( 174 \) kPa
- \( v \) at failure = \( 1.8369 \)

These are the values of \( q \), \( p' \), and \( v \) at failure for the given soil sample under the specified test conditions.

Would you like further details or have any questions? 

Here are some related questions you might be interested in:

1. What is the significance of the critical state line in soil mechanics?
2. How does the value of \( M \) affect the strength of the soil?
3. What would happen if the soil was overconsolidated instead of normally consolidated?
4. How does the value of \( \lambda \) affect the compressibility of the soil?
5. How is the undrained shear strength related to the mean effective stress?

**Tip:** Understanding the critical state framework is essential in predicting the behavior of soils under various stress conditions, especially for designing safe and effective foundations.

Learn how to calculate the values of q, p', and v at failure in an undrained shear test for a clay sample consolidated to a mean effective stress of 200 kPa. Includes step-by-step solution using soil constants M=0.87, N=2.16, \Gamma=2.09, and \lambda=0.11. Useful for soil mechanics studies and engineering analysis.

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