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.