Istanbul Technical University

MAE 4700

Course nameMAE 4700/5700 (Fall 2011) Exam 1 Solutions
Finite Element Analysis for Mechanical & Aerospace Design Page 1 of 10
October 18th, 7:30-9:30 pm
Problem 1 (35 points) – Consider a one-dimensional elasticity problem with axial
deformation as shown in the figure below. The bar is constrained at both ends (A and C).
Its cross sectional area is constant (A = 0.1 m2 ) on segment AB and varies linearly A =
0.05(x - 1) on BC. The Young’s Modulus is E = 2⋅107 Pa. A distributed load b = 10 N/m
is applied along the left portion of the bar AB and a point force P = 150 N acts at point B.
Use a three-node element on AB and a two-node element on BC as shown below.
Consider the global node numbering as shown. The dimensions in the figure are in
meters.
a) Provide a precise weak statement for the problem and based on this give the final
expressions for the element stiffness matrix and load vector.
b) For each element, compute the element force vector F e and element stiffness
matrix K e and then provide the assembled F and K .
c) Apply the essential boundary conditions and compute the finite element nodal
displacements.
d) Provide expressions for the finite element stress in each element.
Possibly useful formulas: Using standard notation, the basis functions in x-coordinate
for a two-node linear element are:
2 1
1
e e e
e
N x x x x
L
= - -     .
Similarly for a 3-node quadratic element
2 2 3 1 3 1 2
2
e e e e e e e ( )( ) 2( )( ) ( )( )
e
N x x x x x x x x x x x x
L
= - - - - - - - 