To solve for the unsaturated shear strengths of the soil at the given matric suctions (40 kPa, 120 kPa, and 240 kPa), we will use the following extended Mohr-Coulomb equation for unsaturated soils:

$\tau = c' + (\sigma - u_a) \tan(\phi') + (u_a - u_w) \tan(\phi_b)$

Where:

• $\tau$ = Unsaturated shear strength
• $c'$ = Effective cohesion (given as 10 kPa)
• $\sigma - u_a$ = Net normal stress (given as 100 kPa)
• $\phi'$ = Effective angle of internal friction (given as 35°)
• $u_a - u_w$ = Matric suction (varies: 40 kPa, 120 kPa, 240 kPa)
• $\phi_b$ = Angle related to the rate of increase in shear strength with matric suction (not given explicitly, but we can estimate it from the soil water characteristic curve)

Step 1: Estimate $\phi_b$

To estimate $\phi_b$, we use the soil-water characteristic curve (SWCC). The SWCC graph shows the relationship between matric suction and the degree of saturation. From this, we can estimate the range where the soil is partially saturated and see how the degree of saturation affects shear strength.

From the curve:

• At 40 kPa, the degree of saturation is around 90%.
• At 120 kPa, the degree of saturation is approximately 60%.
• At 240 kPa, the degree of saturation is around 40%.

Typically, for sands and silts, $\phi_b$ is close to the value of $\phi'$ when the soil is nearly saturated and decreases as the soil dries out. We will use a simplified assumption that $\phi_b$ is about half of $\phi'$, since this is a common practice when estimating $\phi_b$.

Thus, $\phi_b \approx 0.5 \times 35^\circ = 17.5^\circ$.

Step 2: Apply the values to the Mohr-Coulomb equation

Now, let's calculate the unsaturated shear strengths for the given matric suctions.

1. For Matric Suction = 40 kPa:

$\tau = 10 + (100) \tan(35^\circ) + (40) \tan(17.5^\circ)$

2. For Matric Suction = 120 kPa:

$\tau = 10 + (100) \tan(35^\circ) + (120) \tan(17.5^\circ)$

3. For Matric Suction = 240 kPa:

$\tau = 10 + (100) \tan(35^\circ) + (240) \tan(17.5^\circ)$

Let me calculate the results for these equations.The calculated unsaturated shear strengths of the soil at the given matric suctions are:

1. For a matric suction of 40 kPa: $\tau = 92.63 \, \text{kPa}$
2. For a matric suction of 120 kPa: $\tau = 117.86 \, \text{kPa}$
3. For a matric suction of 240 kPa: $\tau = 155.69 \, \text{kPa}$

Let me know if you'd like further details or explanations on these calculations.

Here are 5 related questions for further exploration:

1. How does the soil type influence the value of $\phi_b$?
2. What is the relationship between the soil-water characteristic curve and soil permeability?
3. How would the shear strength change if the net normal stress was varied?
4. Can we experimentally determine the exact value of $\phi_b$ for different soils?
5. How does matric suction impact the deformation characteristics of soil?

Tip: The estimation of $\phi_b$ is often an approximation and can vary significantly depending on the soil's characteristics, so performing experimental tests might give more accurate results.