Virtual Work-Finite Element Problems
1. Consider the three springs shown below. There are 4 nodal
displacements at the ends of these springs so the global stiffness matrix,
K, is 4x4 and the force vector, F, corresponding to those nodes is a 4x1
vector. Starting with zero values for K and F ,show (using the given
numbering scheme):
(a) the new values for K and F when we add in the contributions from
spring 1,
(b) th
the new values
l
ffor K and
d F when
h we add
dd iin th
the contributions
t ib ti
ffrom
spring 2,
(c) the new values for K and F when we add in the contributions from
spring
p g 3.
(d) Show the modified values of K and F after we apply the boundary
conditions on force and displacement.
(e) Solve for the nodal displacements
(f) Obtain the unknown reaction force and the forces (T or C) in each
spring from those displacements
2
3
4
1
(2)
(3)
(1)
100 lb
30 lb
k1 =20 lb/in
k2 = 50 lb/in
k3 = 10 lb/in
2. Consider the system of hanging weights
and springs shown. If the weights W = 15 lb,
and the spring constants k = 20 lb/in,
determine:
1
k
(1)
W
(a) the nodal displacements (use the given
numbering scheme)
(b) the unknown reaction force at the
support
( ) the
(c)
th fforces (T or C) in
i each
h spring
i
2
(2)
k
(4)
3
W
k
(3)
k
W
k
4
(5)
5
75 lb
3. Consider the truss shown. The two cross members are not connected to
each other. Determine:
(a) the displacements at each node (use the given numbering scheme) in m
(b) the external reactions on the truss in kN
(c) the force in each bar in kN (T or C)
The Young's modulus for each bar is E = 70 GPa
and the cross-sectional area A = 0.004 m2
10 kN
30 kN
2
3
(5)
(1)
35m
3.5
(6)
(3)
1
(4)
(2)
4m
4