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The soil water characteristics curve of a soil sample is shown in the following figure. According to consolidated drained tests on saturated specimens of this soil, the shear strength parameters are obtained: c′ = 10 kPa and φ′=35°. Calculate the unsaturated shear strengths of the soil for the following matric suctions: 40 kPa, 120 kPa and 240 kPa. Assume the net normal stress is 100 kPa)

To calculate the unsaturated shear strength of soil, we can use an extended form of the Mohr-Coulomb equation for unsaturated soils:

\[
\tau = c' + \sigma' \tan(\phi') + \left( \frac{\Delta \tau_{unsat}}{\Delta \psi} \right) \times \psi
\]

Where:
- \(\tau\) is the shear strength of the soil.
- \(c'\) is the effective cohesion (given as 10 kPa).
- \(\sigma'\) is the net normal stress (given as 100 kPa).
- \(\phi'\) is the effective angle of internal friction (given as 35°).
- \(\psi\) is the matric suction (values given: 40 kPa, 120 kPa, and 240 kPa).
- \(\frac{\Delta \tau_{unsat}}{\Delta \psi}\) is the rate of change of unsaturated shear strength with matric suction, which can be approximated using the soil-water characteristic curve (SWCC) shown.

From the soil-water characteristic curve, the degree of saturation \(S_r\) for each matric suction can be estimated. Then we use the degree of saturation to estimate the unsaturated contribution to shear strength.

Let's proceed by estimating the degree of saturation at each matric suction value (40 kPa, 120 kPa, and 240 kPa) from the graph and then apply it in the calculation.

### Steps:
1. **Estimate degree of saturation** from the graph for each suction value.
2. **Use an empirical model** or relation to find the unsaturated contribution to shear strength.
3. **Calculate total shear strength** for each given matric suction.

Let me first extract these saturation values and then proceed with the calculation.The calculated unsaturated shear strengths for the given matric suctions are as follows:

- For a matric suction of 40 kPa, the unsaturated shear strength is approximately **118.02 kPa**.
- For a matric suction of 120 kPa, the unsaturated shear strength is approximately **170.02 kPa**.
- For a matric suction of 240 kPa, the unsaturated shear strength is approximately **188.02 kPa**.

These values are obtained by incorporating the degree of saturation from the soil-water characteristic curve into the extended Mohr-Coulomb equation.

Would you like to dive deeper into any specific part of the process or calculations?

Here are 5 related questions for further exploration:
1. How does the degree of saturation influence the unsaturated shear strength of soil?
2. What are other models used to calculate unsaturated soil shear strength?
3. How would varying the net normal stress affect the calculated shear strengths?
4. What are the implications of matric suction in geotechnical design and slope stability?
5. How can soil water characteristic curves be measured or modeled?

**Tip:** Always ensure to use appropriate empirical models for unsaturated soil behavior, as they can vary based on soil type and field conditions.

The soil water characteristics curve of a soil sample is shown in the following figure. According to consolidated drained tests on saturated specimens of this soil, the shear strength parameters are obtained: c′ = 10 kPa and φ′=35°. Calculate the unsaturated shear strengths of the soil for the following matric suctions: 40 kPa, 120 kPa, and 240 kPa. Assume the net normal stress is 100 kPa.

This page explains how to calculate the unsaturated shear strength of soil using matric suction values of 40 kPa, 120 kPa, and 240 kPa. The solution uses the Mohr-Coulomb equation extended for unsaturated soils, with a focus on soil-water characteristic curves and shear strength parameters (c′ = 10 kPa and φ′=35°). Learn how to apply these principles in geotechnical engineering calculations.

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