ME 522 Spring 2013-2014
Homework 1
Problem 1: Consider the deformation (displacement δ) of a tapered bar under an applied force:
x=0
x
X=L
F
For this problem, the governing differential equation is
d
dδ
(EA ) = 0
dx
dx
and the boundary conditions are
at x = 0
at x = L
δ=0
dδ
=F
EA
dx
What is the weak form for this specific problem?
1
Problem 2: Consider the free vibration of a tapered bar:
x=0
x
X=L
For this problem, the governing differential equation is
dδ
d
(EA ) = −ρAω 2 δ
dx
dx
and the boundary conditions are
at x = 0
δ=0
dδ
at x = L EA
=0
dx
What is the weak form for this specific problem? (Note: don’t worry that it is “vibration”– ω is just another constant here.)
2
Problem 3: The governing equation for pressure waves in a car exhaust system is
∂p
∂
βA
∂x
∂x
= ρA
∂2p
∂t2
Here β is the bulk modulus of the air in the pipe (a material property), A is the cross-sectional area of the pipe, x is the distance
along the pipe, ρ is the density of the air in the pipe, and p is the pressure. (p is the unknown.) What is the weak form for the
problem?
Problem 4: Consider the three-node one-dimensional conduction element shown below. Unlike ordinary isoparametric elements,
this one has node three at r = − 12 . What would the shape functions (N1 , N2 , N3 ) be for this element?
1
r=-1/2
2
r=-1
3
r=+1
3
Problem 5: We can often choose several different models for the same object. Consider the steel beam shown below:
3000 N
y
clamped
x
0.05 m
1m
0.1 m
z
NOT TO SCALE
(a) Make a hand calculation for the tip deflection and the maximum normal stress half-way along the beam and add it to the
table on the next page. Use the default properties for steel from ANSYS.
(b) Then solve the problem with ANSYS by several methods, and add those results to the table as well. Also, attach a
printout of the τxx plot for each case.
– Use 3D elements that are bricks.
– Use 3D elements that are tetrahedrons. (Mesh – Insert – Method – Method – Tetrahedrons)
– Use plate/shell elements on an x-y plane.
– Use plate/shell elements on an x-z plane.
– Use plane stress elements on the x-y plane that is at z=0.
– Use beam elements on a line (one of the long edges).
You should find that the tip deflections and stresses half-way along the beam are very similar for each case. If you find that
this isn’t the case, you have made an error and you should fix the error before you turn in your homework.
4
Model
hand calculation
# nodes # elements
–
Tip Deflection
(mm)
Max Equivalent Stress
at Wall (MPa)
–
–
bricks
tetrahedrons
plate/shell
x-y plane
plate/shell
x-z plane
plane stress
x-y plane, z=0
beams
5
Max τxx half-way
along beam (MPa)