Civil Engineering Reference
In-Depth Information
beam length. As shown by the shear force diagram, a jump discontinuity exists at
the point of application of the concentrated force, and the magnitude of the dis-
continuity is the magnitude of the applied force. Similarly, the bending moment
diagram shows a jump discontinuity in the bending moment equal to the magni-
tude of the applied bending moment. Therefore, if the beam were to be divided
into two finite elements with a connecting node at the midpoint, the net force at
the node is the applied external force and the net moment at the node is the ap-
plied external moment.
EXAMPLE 4.1
Figure 4.7a depicts a statically inderminate beam subjected to a transverse load applied at
the midspan. Using two flexure elements, obtain a solution for the midspan deflection.
■
Solution
Since the flexure element requires loading only at nodes, the elements are taken to be of
length
L
/
2
, as shown in Figure 4.7b. The individual element stiffness matrices are then
12
6
L
/
2
−
12
6
L
/
2
k
(1)
=
k
(2)
=
EI
z
(
L
/
6
L
/
2
L
2
/
4
−
12
−
6
L
/
2 2
−
6
L
/
2
6
L
/
2
L
2
/
4
−
6
L
/
2
L
2
2)
3
/
4
−
6
L
/
2
L
2
/
4
12
3
L
−
12
3
L
8
EI
z
L
3
3
L
2
−
3
LL
2
/
2
=
−
12
−
3
L
12
−
3
L
3
L
2
/
2
−
3
L
2
Note particularly that the length of each element is
L
/
2
.
The appropriate boundary con-
ditions are
v
1
=
1
=
v
3
=
0
and the element-to-system displacement correspondence
table is Table 4.1.
P
L
2
L
2
v
1
v
2
v
3
2
3
1
1
1
2
2
3
(b)
(a)
v
2
v
1
1
0
v
3
0
3
2
(c)
Figure 4.7
(a) Loaded beam of Example 4.1. (b) Element and displacement designations.
(c) Displacement solution.
