Civil Engineering Reference
In-Depth Information
The axial stiffness terms can be derived directly from strength of materi-
als. The axial stiffness is the inverse of the flexibility, which can be written
as follows. The torsional stiffness would look the same as the axial stiff-
ness with
GI
x
substituted for
EA
x
:
dx
EA
x
=
∫
f
The coplanar X-Z nonprismatic stiffness matrix is shown in Equation 5.12.
The X-Y system will have the same values with the sign of the moments
due to deflection reversed.
1
1
0
0
−
0
0
f
f
S
D
S
D
S
D
SSL
D
−
0
1
−
2
0
−
1
2
1
S
D
S
D
S
D
SL S
D
−
0
−
2
3
0
2
2
3
[
]
=
K
m
1
1
−
0
0
0
0
f
f
S
D
S
D
S
D
SL S
D
−
0
−
1
2
0
1
1
2
2
SSL
D
−
SL S
D
−
SL S
D
−
SL SL S
−
2
+
0
2
1
2
3
0
1
2
1
2
3
D
(5.12)
FIXED-END MOMENTS
The fixed-end forces and moments must be derived for a non-prismatic
member. The changes in stiffness along the length of the member will
change how the forces and moments are distributed by the member.
The following examples derive two of the most common member
loads.
Example 5.5
Non-prismatic member stiffness
Derive the fixed-end forces and moments due to a uniformly dis-
tributed load in the X-Z system for a non-prismatic cross-section using
Castigliano's theorems.
