CE 560 Advanced Mechanics of Materials
Homework Assignments
  1. Prove symmetry of stress tensor.
  2. Derive Cauchy's equation.
  3. Derive the three primary invariants of the stress tensor.
    Fully explain your derivation.
    Simplify the equations by using only the principal stresses.
  4. Derive the transformation equation:
    [S'] = [T][S][T]^T
  5. Derive the equilibrium equations.
  6. Derive the equation for Syy in a cantilever beam fixed at the right end and with a uniform load on the top.
  7. Derive the Lagrangian strain tensor and then simplify to infinitesimal strains.
  8. Derive the compatibility equations for plane strain.
  9. Problem 2.67 in text.
  10. Problem 2.72 in text.
  11. Using the strain energy, prove that an isotropic, elastic material can have only two independent elastic constants.
  12. Derive the equations for shear stress and total angle of twist when a shaft of circular cross-section is subjected to a torque T.
  13. Derive the governing equation for the warping function of a shaft under torsion using Saint-Venant's approach.
  14. Derive the governing equation for the stress function of a shaft under torsion using Prandtl's approach.
  15. Derive the governing equation for the deflection of a tightly stretched membrane (T=force per unit length along any interior boundary) subject to a pressure p. Assume small deflections.
  16. Derive the governing equation for the torque on a shaft in terms of Prandtl's stress function. Assume a multiply connected cross section with one hole.
  17. Derive the governing equation that the stress field must satisfy if it defines strains that are derivable from a single valued warping function.
  18. Using the solution for stress in a shaft of circular cross section, derive Prandtl's stress function in x,y coordinates.
  19. Using the above stress function, determine the torque by using the volume under the surface.
  20. Show that Saint Venant's equations are all satisfied by the elementary theory for torsion of circular shaft.
  21. Solve Problem 6.43 and determine the torsional constant of the cross section shown.
  22. Solve Problem 6.57.
  23. Derive the equation for J for a rod of slender cross-section width w and thickness t.
  24. Solve Problem 6.23.
  25. Torsion Problem
  26. Derive flexure formula clearly showing all assumptions and requirements.
  27. Solve Problem 7.26. Follow all steps shown on class
  28. Solve Problem 8.8
  29. Solve Problem 8.13
  30. Solve Problem 8.21
  31. Solve Problem 8.22
  32. Solve Problem 8.45
  33. MS Shear Problem
  34. Derive formula for flexural stress in a curved beam following approach used in class.
  35. Curved beam, Problem 1
  36. Curved beam, Problem 2
  37. Using winkler.m verify Example 10.4 in text.
  38. Using winkler.m verify any curve of your own choosing shown in Fig. 10.10
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    The following problems are not to be handed in. You will be accountable for the material, however, on the second exam
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  39. Using (a) the Pi-plane, (b) a 3D perspective of stress space, and (c) a 2D, plane stress, stress space, draw the J2 yield surfaces and Tresca yield surfaces. On each, indicate values of $\sigma_y$.
  40. Given a state of stress in terms of principal stresses, e.g.

    +8 +0 +0
    +0 -6 +0
    +0 +0 +4

    plot its location in the Pi-plane. Show all three components as lines in the Pi-plane.
  41. For the above state of stress, determine another state of stress having the same location in the Pi-plane. Show the new state of stress by its three components.
  42. For the following individuals shown on the CE560 homepage, state what topics we covered in class that were attributed to that individual (the names of several of the individuals will appear).