ME422
FEM Homework #9
Distributed: February 7, 2013
Due: February 14, 2013
Problem 1:
Consider a steel cantilever beam with a pressure load on it:
P
For this beam, we have ρ = 7860 kg/m3 , L = 1 m, E = 200 × 109 Pa,
h = 10.6 mm = 1.06 × 10−2 m, depth into page b = 5.04 mm = 5.04 × 10−3 m,
P = 2380 sin ωt Pa, and ω = 100 rad/s. Suppose we model the beam with a
single beam finite element:
Although we haven’t derived the equations yet (soon!) the equation of motion
for the finite element system is
0.001
156 −22
−22
4
ẅ2
θ̈2
+ 100
1
12 −6
−6
4
w2
θ2
=
−6 sin ωt
+1 sin ωt
Solve the system in time using
(a) Trapezoidal Rule
(b) Central Difference Method
Solve from t=0 up to t=0.5 s. Obviously, the two results should be equal, so
choose a small enough time step that they agree. Give me a plot of the tip
deflection versus time for both methods. When I make this plot, I get
Deflection of the Beam Tip over Time
0.02
0.015
Displacement (meters)
0.01
0.005
0
−0.005
−0.01
−0.015
−0.02
0
0.1
0.2
0.3
0.4
Time (seconds)
2
0.5
0.6
0.7
Problem 2: Consider the simple truss structure, consisting of three trusses,
shown below.
f
3
L
L
2
1
e2 L
The Young’s modulus of each truss is E, and the area is A. Notice
√ that
the length of two of the trusses is L, but the base truss has a length 2L.
(a) From statics, how much of the applied load will each upright truss carry?
What will the force be along each upright truss? What will the stress be in
the upright trusses?
(b) What is the global stiffness matrix for the problem, using finite elements?
(c) What is the global forcing vector for the problem, using finite elements?
(d) Solve for the deflection vector U.
(e) Find the stresses in each truss. Compare your answer with what you got
from part (a). Fix any discrepancies.
Problem 3: No problem 3. Projects!
3