Civil Engineering Reference
In-Depth Information
equilibrium equations for a two-plane bending element with axial stiffness are
written in matrix form as
u
1
u
2
v
1
z
1
v
2
z
2
w
1
y
1
w
2
y
2
f
x
1
f
x
2
f
y
1
M
z
1
f
y
2
M
z
2
f
z
1
M
y
1
f
z
2
M
y
2
[
k
axial
]
[0]
[0]
=
(4.67)
[0]
[
k
bending
]
xy
[0]
[0]
[0]
[
k
bending
]
xz
where the
10
×
10
element stiffness matrix has been written in the shorthand form
[
k
axial
]
[0]
[0]
[
k
e
]
=
(4.68)
[0]
[
k
bending
]
xy
[0]
[0]
[0]
[
k
bending
]
xz
The equivalent nodal loads corresponding to a distributed load are computed on
the basis of work equivalence, as in Section 4.6. For a uniform distributed load
q
z
(
x
)
=
q
z
, the equivalent nodal load vector is found to be
q
z
L
2
q
z
L
2
12
q
z
L
2
q
z
L
2
12
f
qz
1
M
qz
1
f
qz
2
M
qz
2
−
=
(4.69)
The addition of torsion to the general beam element is accomplished with
reference to Figure 4.16a, which depicts a circular cylinder subjected to torsion
via twisting moments applied at its ends. A corresponding torsional finite element
L
1
T
1
M
x
1
2
x
x
G
,
J
T
2
M
x
2
(a)
(b)
Figure 4.16
(a) Circular cylinder subjected to torsion. (b) Torsional finite element notation.
