CE 563 2020 21 HW 1 - HW1 PDF

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CE – 563 Advanced Soil Mechanics Homework I – Due 20/11/2020

1. Sample of a particular sand was tested in shear box. Under a normal stress of 100 kPa, the shear stress is calculated to be 60 kPa. a) Plot the respective Mohr circle for the stress state at failure clearly indicating the pole. b) Calculate the orientation (i.e., inclination from horizontal axis) of major and minor principal planes. Solve the following problems (2, 3 and 4) from Soil Mechanics by Lambe and Whitman. Important note: Use either SI version (1979 Edition) of the book, or convert the stresses given in psi to kPa, assuming 1 psi = 10 kPa when solving. 2. Problem 8.8. 3. Problem 12.1. 4. Problem 12.7. 5. If K0 for a sand is known to be 0.45, determine the Poisson’s ratio for the sand. 6. A soil element at a depth of 8 m below the surface of a saturated normally consolidated clay layer has a unit weight of 20 kN/m3. For this clay K0 = 0.5, c′ = 0 and φ′ = 30º. Draw the stress path for this soil element if it is subjected to i) active stress state and ii) passive stress state. 7. Poisson’s ratio for the stress states of the following cases is expressed as a) Triaxial test: μ = 0.5 (1 – υ / ε1) b) Plane strain condition: μ = (1 – υ / ε1) (where υ = ∆V / V0 (ie, volumetric strain) in the above expressions) Working from the first principles, derive the above two expressions. 8. The state of stress with reference to axes x1, x2 and x3 is given (in MPa) by the matrix below. Determine the normal and shear components σn and σs , respectively, on the plane whose unit normal is:

𝑛 󰇍 =

1 (4𝑖 + 3𝑗) 5

9 12 0 [𝜎] = [12 −9 0] 0 0 5

9. The stress tensor below is given for x, y, z coordinate system. Find the stress tensor if x, y coordinates are rotated 45° counterclockwise.

4 1 2 [𝜎] = [1 6 0] 2 0 8

10. State of stress at a point is given by the following tensor. Determine the principal stresses and the principal directions.

3 5 8 [𝜎] = [5 1 0] 8 0 2...