Chapter 12 Extra Example 1 - Direct shear test and Mohr's circle at failure

Summary

This video demonstrates the process of predicting shear stress at failure and determining principal stresses using Mohr's Circle for a directional soil test. Given the normal force and soil properties, the normal stress and shear stress are calculated. Using the Mohr-Coulomb failure criterion, shear stress is predicted, and Mohr’s Circle is constructed to find principal stresses. The major and minor principal stresses are derived graphically, and an additional method to find the inclination of principal planes using the pole method is explained, offering a comprehensive approach to analyzing soil behavior under stress.

Takeaways

• 😀 The soil specimen in this example has a friction angle of 32 degrees and c prime equals zero, as per Example 4.
• 😀 A directional test is conducted with a normal force of 80 pounds to predict the shear stress at failure for this test.
• 😀 The normal stress (σ') is calculated by dividing the normal force (80 pounds) by the cross-sectional area (0.0278), resulting in 2880 psf.
• 😀 The shear stress at failure is determined using Mohr-Coulomb failure criteria: shear stress equals 2880 times the tangent of 32 degrees.
• 😀 Part B involves calculating the principal stresses at failure using Mohr's Circle method.
• 😀 Mohr's Circle is constructed by identifying the point of normal and shear stresses on the failure plane, which is on the Mohr failure envelope.
• 😀 The Mohr Circle's center and radius allow for constructing the circle, and the intersection with the normal stress axis reveals the principal stresses.
• 😀 The major principal stress (σ1') is approximately 6050 psf, while the minor principal stress (σ3') is approximately 1860 psf.
• 😀 The pole method is introduced for calculating the inclination of principal stress planes, though it is not directly asked in the problem.
• 😀 In the pole method, a line is drawn parallel to the failure plane, and its intersection with the Mohr Circle helps identify the pole and inclination of principal planes.
• 😀 Using the pole method, the inclination of the major principal stress plane can be determined by connecting the pole to the major principal stress point (σ1').

Q & A

What is the normal stress (σ') calculated in Part A of the script?

The normal stress (σ') is calculated using the formula σ' = F_n / A, where F_n is the normal force (80 pounds) and A is the cross-sectional area (0.0278 square feet). The result is 2880 psf.

How is the shear stress at failure (τf) calculated in Part A?

The shear stress at failure (τf) is calculated using the Mohr-Coulomb failure criterion, which is τf = σ' * tan(ϕ'), where σ' is the normal stress and ϕ' is the friction angle. For this example, τf = 2880 * tan(32°) ≈ 1800 psf.

What is the significance of the friction angle (ϕ') in predicting the shear stress at failure?

The friction angle (ϕ') is a key parameter in the Mohr-Coulomb failure criterion. It influences the calculation of shear stress at failure, as it is used to multiply the normal stress to predict the shear stress under failure conditions.

What is the value of the major principal stress (σ₁') derived from Mohr's Circle?

From the Mohr's Circle, the major principal stress (σ₁') is approximately 6050 psf, which can be read from the graph where the Mohr circle intersects the normal stress axis.

How is the minor principal stress (σ₃') determined from Mohr's Circle?

The minor principal stress (σ₃') is read directly from the Mohr's Circle at the point where the circle intersects the normal stress axis on the opposite side of the major principal stress. In this case, σ₃' is approximately 1860 psf.

What is the purpose of using Mohr's Circle in this analysis?

Mohr's Circle is used to visually determine the principal stresses at failure, which are crucial for understanding the behavior of soil under stress. It helps in plotting normal and shear stresses to find principal stress values.

What role does the Mohr-Coulomb failure envelope play in the Mohr's Circle construction?

The Mohr-Coulomb failure envelope represents the relationship between shear stress and normal stress at failure. It is tangent to the Mohr's Circle, helping identify the point of failure and determining the shear stress at failure.

What is the significance of identifying the pole in the pole method for determining the inclination of principal planes?

In the pole method, the pole is used to determine the orientation of the principal planes. By connecting the pole to the principal stress points, the inclination of the major and minor principal planes can be determined.

How is the inclination of the major principal plane found using the pole method?

The inclination of the major principal plane is determined by drawing a line from the pole to the major principal stress point (σ₁'). This line is parallel to the major principal plane, indicating its orientation.

Why is the failure plane considered horizontal in a direct shear test?

In a direct shear test, the failure plane is considered horizontal because the soil is forced to fail along a horizontal plane. This assumption is crucial for constructing Mohr's Circle and understanding the stress states at failure.