EM 424: Strain Energy and FEM Problems
Some Answers to Energy Methods Problems:
4.5.3
(u )
=
(ux )A
5QL
=
3EI
5qL4
y
Center
768EI
7qL3
(θ )Center =
5760EI
4.5.4
(a)
3
(uy ) =
A
(uz )A
QL3
EI
=0
(b)
(ux )A = 0
(u )
y A
=0
(uz )A =
4FL3 2FL3
+
3EI
GJ
(ux )A =
π Fx R 3
(uy ) =
π Fy R3
4.6.5
A
(ux )A
(u )
y A
k=
=
2EI
2EI
Fx
, so force and displacement are parallel
Fy
Ftotal 2EI
=
∆ total πR 3
EM 424: Strain Energy and FEM Problems
4.7.7
T=
F
6I
2+
5 AL2
4,7.12
FL 2β EI
−
8
L
3
FL
βL
+
(uy )center = 192
EI 4
Mends =
( down)
4.7.16
(a) axial force and bending moment at F are
N = .2453F
M =,3073FR
(b)
(u )
y at F
= 0.0704
FR3
EI
( down)
EM 424: Strain Energy and FEM Problems
Finite Element Problems
1. The bar shown (cross sectional area A and Young’s modulus E) is rigidly fixed at its
right end and is loaded by a uniform body force f 0 (dimensions are force/unit volume)
over two thirds of its length.
f
L/3
0
2L/3
(a) Using three equal length constant strain finite elements, determine the displacement
and stress throughout the bar
(b) Compare your results of (a) with the exact solution
2. The bar shown has a cross sectional area that varies linearly from A0 at x = 0 to 2 A0 at
x = L.
P
L
(a) Using three equal length finite elements, determine the displacement and stress
throughout the bar
(b) Compare your results of (a) to the exact solution