A sample of dry sand is subjected to a triaxial test. The angle of internal friction is 36 degrees. If the

cell pressure is 180 kN/m2, at what value of deviator stress will the soil fail?

To determine the deviator stress at which the dry sand will fail in a triaxial test, we can use the relationship between the angle of internal friction (φ), the cell pressure (σ₃), and the deviator stress (σ_d) at failure.

The Mohr-Coulomb failure criterion for triaxial compression can be expressed as:

\[ \sigma_1 = \sigma_3 \cdot \left( \frac{1 + \sin \phi}{1 - \sin \phi} \right) \]

Where:
- \(\sigma_1\) is the major principal stress at failure.
- \(\sigma_3\) is the minor principal stress (cell pressure).
- \(\phi\) is the angle of internal friction.

The deviator stress (\(\sigma_d\)) is the difference between the major and minor principal stresses:

\[ \sigma_d = \sigma_1 - \sigma_3 \]

Given:
- \(\phi = 36^\circ\)
- \(\sigma_3 = 180 \, \text{kN/m}^2\)

First, we calculate \(\frac{1 + \sin \phi}{1 - \sin \phi}\):

\[ \sin 36^\circ \approx 0.5878 \]

\[ \frac{1 + \sin 36^\circ}{1 - \sin 36^\circ} = \frac{1 + 0.5878}{1 - 0.5878} = \frac{1.5878}{0.4122} \approx 3.852 \]

Then, we calculate \(\sigma_1\):

\[ \sigma_1 = 180 \, \text{kN/m}^2 \times 3.852 \approx 693.36 \, \text{kN/m}^2 \]

Finally, we find the deviator stress (\(\sigma_d\)):

\[ \sigma_d = \sigma_1 - \sigma_3 = 693.36 \, \text{kN/m}^2 - 180 \, \text{kN/m}^2 = 513.36 \, \text{kN/m}^2 \]

Therefore, the deviator stress at which the soil will fail is approximately \(513.36 \, \text{kN/m}^2\).

Would you like more details or have any questions about this solution?

Here are 8 related questions to consider:
1. How does the angle of internal friction affect the strength of soil?
2. What is the significance of the Mohr-Coulomb failure criterion in geotechnical engineering?
3. How do you determine the angle of internal friction for a given soil?
4. What are the typical ranges for the angle of internal friction in different soil types?
5. How does the cell pressure influence the failure of soil in a triaxial test?
6. What other parameters can be measured using a triaxial test?
7. How do cohesion and internal friction angle interplay in soil mechanics?
8. How would the deviator stress change if the cell pressure was increased?

**Tip:** In soil mechanics, always ensure that you are using consistent units when performing calculations to avoid errors and obtain accurate results.

Explore how to calculate the deviator stress at failure in a triaxial test of dry sand with an angle of internal friction of 36 degrees and a cell pressure of 180 kN/m². Understand the application of the Mohr-Coulomb failure criterion and the calculation of deviator stress in soil mechanics.

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