AE 321 Practice Problems
Chapter 7: Failure and Fatigue
1. (Exam question) A linear, elastic, isotropic and homogeneous solid circular cylinder of length L and radius R (no restriction on L and R), Young's modulus E, Poisson's ratio and uniaxial yield str

AE 321 Solutions to Practice Problems Chapter 5: Problem Formulation 1. The figure below shows both the Cartesian coordinates (x, y, z ) and the cylindrical coordinates (r , ! , z ) used to define the boundary conditions of the hollow circular cylinder.
(

AE 321 Practice Problems Chapter 5: Problem Formulation
1. Find all boundary conditions on all surfaces of the hollow circular cylinder (inner radius a, outer radius b) shown below in both (i) Cartesian coordinates and (ii) cylindrical coordinates, where

AE 321 Solution to Practice Problems Chapter 6: Extension, Bending, and Torsion
1. (a) Tractions on the cylindrical surface. Since a > b , we can use the St. Venant's principle, which states that stresses and strains far from the point of application of a

AE 321 Solution to Practice Problems Chapter 8: Plane Problems 1a. The given Airy stress function, ! (x, y ), would provide the stress field for this problem if it satisfies the compatibility equations (equilibrium is identically satisfied), i.e. " 4 ! (x

AE 321 Solution to Practice Problems Chapter 7: Failure and Fatigue 1a. We start with the following trial solution.
& ' P 0 0# [( ]= $ 0 ' P 0! $ ! $ 0 0 0! % "
Note that the entries in the upper left of the stress tensor, i.e. ! 11 , ! 22 , ! 12 = ! 21 ,

AE 321 Solutions to Practice Problems Chapter 4: Material Behavior 1. For a linearly elastic isotropic material starting from
$ ij = 2 ! ij + "# ij ! kk
show that
(4.1)
E! ij = (1+ " )# ij $ "% ij# kk
(4.2)
To this end, we first determine ! kk in terms of

Extra Problem 1. Hollow cylinder under inner pressure Consider the problem of an hollow cylinder (with inner radius a, outer radius b and length L) made of a linearly elastic homogeneous isotropic material (with Young's modulus E and Poisson's ratio ) sub

AE 321 Practice Problems Chapter 6: Extension, Bending and Torsion
1. A rectangular bar of a linearly elastic homogeneous and isotropic material, and unit thickness, is loaded in extension with a load P per unit length linearly distributed over its width

AE 321 Practice Problems Chapter 4: Material Behavior
1. For a linearly elastic isotropic material starting from
! ij = 2 " ij + #$ ij" kk
,
where , are the Lam moduli, show that
E! ij = (1 + " )# ij $ "% ij# kk
,
where E, are the Young's modulus and Pois