Civil Engineering Reference
In-Depth Information
2
(0, 80, 0)
1
(0, 0, 0)
2
(30, 30,
20)
2
(50, 40, 30)
1
(0, 0, 0)
1
(10, 10, 0)
1
(0, 0, 0)
2
(0,
80, 0)
(a)
(b)
(c)
(d)
Figure P3.20
3.21
Verify Equation 3.59 via direct computation of the matrix product.
3.22
Show that the axial stress in a bar element in a 3-D truss is given by
u
(
e
1
u
(
e
2
E
d
N
1
d
x
E
[
R
]
U
(
e
)
d
N
2
d
x
1
L
1
L
=
E
ε
=
=
−
and note that the expression is the same as for the 2-D case.
3.23
Determine the axial stress and nodal forces for each bar element shown in
Figure P3.20, given that node 1 is fixed and node 2 has global displacements
U
4
=
U
5
=
U
6
=
0.06 in.
3.24
Use Equations 3.55 and 3.56 to express strain energy of a bar element in terms of
the global displacements. Apply Castigliano's first theorem and show that the
resulting global stiffness matrix is identical to that given by Equation 3.58.
3.25
Repeat Problem 3.24 using the principle of minimum potential energy.
3.26
Assemble the global stiffness matrix of the 3-D truss shown in Figure P3.26 and
compute the displacement components of node 4. Also, compute the stress in
each element.
Y
1
3
X
2
Node
1
2
3
4
X
0
YZ
0
0
0
20
4
0
30
0
25
Z
0
40
30
F
Y
1500 lb
Figure P3.26
Coordinates given in inches. For each
element
E
= 10 × 10
6
psi,
A
= 1.5 in.
2
.
