ME422
FEM Homework #8
Distributed: January 31, 2013
Due: February 7, 2013
Consider a thin slab which is fixed to a wall and has a pressure, P , on its
free edge:
w
L
E ν
pressure,
P
2L
Problem 1: Using standard Mechanics of Materials and elasticity equations
and concepts, and assuming that the wall doesn’t have any effect except to
keep the slab from moving sideways,
(a) approximately what is the τxx stress in the slab?
(b) approximately what is the deflection of the end of the slab?
(c) approximately what is the strain xx in the slab?
(d) approximately what is the strain yy in the slab due to the Poisson’s ratio
effect? Use the approximation ν = 1/3.
(e) approximately what is the change in height of the slab due to the Poissson’s ratio effect? (ν = 1/3.)
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Problem 2: Discretize the top half of the slab with a single 4-node isoparametric finite element. There will be three degrees of freedom in the system.
v1
2
1
3
4
u1
pressure,
P
u4
(a) Find the 3×3 global stiffness matrix for the system. Again, use ν = 1/3,
and assume that the system is in plane stress. (It is.) You may find Maple
useful for some of these calculations, although actually there are only 4 unique
values in the 3×3 matrix.
(b) Find the forcing vector for the system (3 entries, 2 of which are non-zero).
(c) Solve for the displacements of the nodes. Sketch the new shape of the
element. Compare the displacements to the answer you got for 1b and 1e
above. I would expect that the 1b answer would agree quite closely, and the
1e answer is surprisingly close as well.
(d) Solve for the three components of stress in the element at the element
center (s=0, t=0). Compare the appropriate stress to the answer you found
in 1a, and discuss any differences you find.
Problem 3: No problem 3. Projects!
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