MANE 7100: Homework 2
Problem 1
The stress function 𝜙(𝑥, 𝑦) is defined by the following relationships:
Use the above equations with the equilibrium equations and the equations of compatibility to
obtain the stress function equation.
Equilibrium Equations (2D):
𝜕𝜎𝑥𝑥
𝜕𝑥
+
𝜕𝜎𝑥𝑦
𝜕𝑦
Equation of Compatibility (2D):
=0
𝜕𝜎𝑦𝑦 𝜕𝜎𝑦𝑥
+
=0
𝜕𝑦
𝜕𝑥
Substitute the constitutive equations into the equation of compatibility to create an equation in
terms of stress vice strain.
Constitutive Equations (2D and V = 0):
Simplify to obtain the stress function equation:
Sub into Equation of Compatibility (V = 0):
Problem 2
A simply supported beam has a downward load P=10^5 N applied to it. Use both finite element
analysis and elementary beam theory to determine the deflection with both point and distributed
loads.
Point Load:
FEA Analysis Point Load:
FEA predicts displacement on the order of -14e-5 m.
Elementary Beam Theory (Point Load):
Elementary beam theory predicts a deflection of -25e-5 m.
Distributed Load:
FEA Analysis (Distributed Load):
FEA shows displacement on the order of -6.43e-5 m.
Elementary Beam Theory (Distributed Load):
Beam theory predicts displacement on the order of -15e-5 m.
Problem 3
Stress at a single point inside an isotropic linear elastic body is given by the following stress
tensor:
Find the corresponding strain tensor.
Problem 4
Consider a long thin walled cylinder subjected to inner pressure P and outer pressure Q. Force
balance yields the equation:
Strain Equations and Hooke’s Law Equations:
a) Combine all of the above equations into a single differ eq for σr:
b) Solve the differential equation found in a) and obtain expressions for radial (σr) and hoop
(σΦ) as functions of the radius through the cylinder wall.
c) Assume linear elastic behavior and obtain expressions for u, εr and εΦ as a function of the
radius r:
d) Assume the ends of the cylinder are stress free (σz = 0) and obtain an expression for axial
strain:
e) Obtain an expression for the radial displacement u
f) Plot the stress, strain and displacements:
g) Solve this problem using FEA: