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AE 321 Homework 10 Due in class on December 2, 2013 Problem 1.
Follow the approach shown in class to find the stresses,
ij ,
strains,
ij ,
and displacements, ui , in a
hollow cylinder with circular cross-section and outer radius R2, inner radius R1 and length L twisted about its axis z by an applied torque M z . The cylinder material is linearly elastic, isotropic and homogeneous and there are no body forces. Solve the problem step by step as in class and in detail to receive full marks.
Hint: Follow the solution for the solid cylinder done in class. The only difference is that the shear stresses due to the applied moment are restricted between R1 and R2.
Problem 2. (a) Explain (i) the concept of superposition in boundary value problems, and (ii) Saint Venant's
principle. (b) A long hollow linearly elastic, isotropic and homogeneous thick walled cylinder is loaded by
uniform axial pressure P and and a torque T as as shown in the figure:
P
P
b
T T
x
a 3
What are the boundary conditions on the end faces, the exterior lateral surface and the interior P , T ? All BCs must be expressed in terms surface of the cylinder subjected cylinder subjected to the combined c ombined load P of stresses on the corresponding surfaces. (c) Use superposition to find the tensor describing the stresses in this cylinder in terms of P , T,
the geometrical parameters of the hollow cylinder and the properties of the material comprising the cylinder. 1