Studies in Mechanics of Solids
Homework 6
1.- Consider the partial differential equation satisfied by the stress function ϕ in rectangular
Cartesian coordinates
φ
,xxxx
+ 2 φ
,xxyy
+ φ
,yyyy
= 0 a) Perform a coordinate transformation and obtain the corresponding differential equation in two dimensional polar coordinates r, Θ where the stress function is defined such that
σ rr
= (1/r) φ
,r
+ (1/r
2
) φ
, ΘΘ
; σ
ΘΘ
= φ
,rr
and σ r Θ
= -( (1/r) φ
, Θ
), r b) Perform r and Θ force balances on a differential area element dr d Θ and obtain the mechanical equilibrium equations in two-dimensional polar coordinates.
2.- Consider the case of stress distribution symmetrical about the z-axis. a) Write down the equation of mechanical equilibrium for this case and explain. b) Obtain the ordinary differential equation that must be satisfied by the stress function when the body force is zero and integrate it to obtain a general expression for the stress function for this case.
3.- Consider a long hollow steel cylinder subjected to uniform pressures on its inner and outer surfaces and with the following characteristics: Inner radius a =1 m; outer radius b=1.05 m; E =
2e11 Pa; ν = 0.3; P = 10
4
N, p in
= 10
5
Pa , p out
= 10
6
Pa. a) Assume the cylinder ends are stress-free (plane stress condition) and obtain exact values for the displacement, strain and stress fields. b) Use the finite element method to solve the problem approximately and compare with the exact solution.
4.- Consider a steel disk of uniform thickness h = 0.025 m radius b = 1 m that rotates with angular velocity ω = 6000 rpm.

a) Assume axis-symmetric conditions and that the stress-stres function (F) relationship is of the form
σ rr
= (1/r) F ; σ
ΘΘ
= F
,r
+ ρ ω 2 r 2 and obtain the differential equation that must be satisfied by the stress function to satisfy the condition of mechanical equilibrium. b) Solve the resulting equation for F and obtain exact expressions for the stress, strain and displacement fields in the disk. c) Use the finite element method to solve the problem approximately and compare with the exact solution.