Civil Engineering Reference
In-Depth Information
where
V
is total volume of the element. Substituting for the stress and strain per
Equations 4.5 and 4.6,
y
2
d
2
v
d
x
2
2
E
2
U
e
=
d
V
(4.36)
V
which can be written as
2
d
2
v
d
x
2
L
E
2
d
x
y
2
d
A
U
e
=
(4.37)
0
A
Again recognizing the area integral as the moment of inertia
I
z
about the cen-
troidal axis perpendicular to the plane of bending, we have
d
2
v
d
x
2
2
L
EI
z
2
U
e
=
d
x
(4.38)
0
Equation 4.38 represents the strain energy of bending for any constant cross-
section beam that obeys the assumptions of elementary beam theory. For the
strain energy of the finite element being developed, we substitute the discretized
displacement relation of Equation 4.27 to obtain
d
2
N
1
d
x
2
2
2
L
d
2
N
2
d
x
2
d
2
N
3
d
x
2
d
2
N
4
d
x
2
EI
z
2
U
e
=
v
1
+
1
+
v
2
+
d
x
(4.39)
0
as the approximation to the strain energy. We emphasize that Equation 4.39 is an
approximation because the discretized displacement function is not in general an
exact solution for the beam flexure problem.
Applying the first theorem of Castigliano to the strain energy function with
respect to nodal displacement
v
1
gives the transverse force at node 1 as
d
2
N
1
d
x
2
2
d
2
N
1
d
x
2
L
d
2
N
2
d
x
2
d
2
N
3
d
x
2
d
2
N
4
d
x
2
∂
U
e
∂
v
1
=
F
1
=
EI
z
v
1
+
1
+
v
2
+
d
x
0
(4.40)
while application of the theorem with respect to the rotational displacement
gives the moment as
d
2
N
1
d
x
2
2
d
2
N
2
d
x
2
L
d
2
N
2
d
x
2
d
2
N
3
d
x
2
d
2
N
4
d
x
2
∂
U
e
∂
1
=
M
1
=
EI
z
v
1
+
1
+
v
2
+
d
x
0
(4.41)
