Civil Engineering Reference
In-Depth Information
Figure P2.11 depicts an assembly of two bar elements made of different
materials. Determine the nodal displacements, element stresses, and the
reaction force.
2.11
A
1
,
E
1
,
L
1
A
2
,
E
2
,
L
2
20,0
00 lb.
3
1
2
4 in.
2
E
1
15
10
6
2.25 in.
2
E
2
10
10
6
A
1
A
2
lb./in.
2
lb./in.
2
L
1
20 in.
L
2
20 in.
Figure P2.11
Obtain a four-element solution for the tapered bar of Example 2.4. Plot element
stresses versus the exact solution. Use the following numerical values:
2.12
=
10
×
10
6
lb./in.
2
A
0
=
4
in.
2
E
L
=
20
in.
P
=
4000
lb.
A weight
W
is suspended in a vertical plane by a linear spring having spring
constant
k
. Show that the equilibrium position corresponds to minimum total
potential energy.
2.13
For a bar element, it is proposed to discretize the displacement function as
2.14
u
(
x
)
=
N
1
(
x
)
u
1
+
N
2
(
x
)
u
2
with interpolation functions
N
1
(
x
)
=
cos
x
2
L
sin
x
2
L
Are these valid interpolation functions? (Hint: Consider strain and stress
variations.)
N
2
(
x
)
=
The torsional element shown in Figure P2.15 has a solid circular cross section
and behaves elastically. The nodal displacements are rotations
1
and
2
and the
associated nodal loads are applied torques
T
1
and
T
2
. Use the potential energy
principle to derive the element equations in matrix form.
2.15
1
,
T
1
R
L
2
,
T
2
Figure P2.15
