Civil Engineering Reference
In-Depth Information
be transformed into this system. The basic process to set up and solve the
structure using this stiffness is summarized as follows:
1.
Find the local member stiffness, [
K
m
], using Equation 4.33 (4.23
or 4.32 for 2-D systems). Rotate the local member stiffness to the
global joint stiffness system, [
K
g
]. This is shown in Equation 4.34.
=
[]
[
]
[]
T
(4.34)
KRKR
g
m
2.
Assemble all of the members into the global joint stiffness matrix.
This is done by using joint labeling to order the matrix.
3.
Determine the global joint loading, [
P
g
], from all direct loads on
joints, [
P
&
M
g
], and loads on members. The member loads are
applied as the opposite of the fixed-end forces and moments,
[
FEPM
m
]. These must be rotated from the local system to the global
system, [
R
]
T
.
−
[
]
[]
T
=
(4.35)
PPM
&
FEPM R
g
g
m
4.
Solve the general stiffness equation for global displacements, [∆
g
].
The rows and columns of the matrices corresponding to the defor-
mations restrained by the supports are removed prior to solving the
system of equations.
K
∆
=
P
(4.36)
g
g
g
5.
Determine the reactions at the support, [
P
g
], from the global defor-
mations. Any fixed-end forces must be subtracted from the results.
Only the forces at the supports due to the free deformations need
to be found.
+
[
]
[]
T
=
PK FEPM R
g
∆
(4.37)
g
g
m
6.
Solve for the local member forces and moments, [
P
&
M
m
], for each
member separately from the global joint deformations. The global
joint deformations must be rotated into the member system, [
R
].
The fixed-end forces and moments must be added back to get the
final member end forces.
[
]
=
[
]
[]
+
[
]
PM KR FEPM
m
&
∆
m
g
m
(4.38)
