Civil Engineering Reference
In-Depth Information
This equation represents the force in the local system due to deformations
in the local system. To go from local to global we multiply by the rotation
transpose, [
R
]
T
, on both sides of the equation.
[
][][
=
[][]
T
T
KR RP
m
∆
m
m
The right side of the equation now represents the forces in the global sys-
tem, [
P
g
]. The left side represents the force in the global system due to
local deformations. The equation needs to be written in terms of the global
deformation, [∆
g
]. From Section 4.3, the local deformation is the global
deformation multiplied by the rotation matrix, [
R
].
[]
[
]
[]
T
=
RKR
∆
P
m
g
g
Example 4.19
Global joint stiffness
Determine the global joint deformations, support reactions, and local
member forces for the pin-connected bracing structure loaded as shown
in Figure 4.33.
The area of each member,
A
x
, is 10 in
2
and the modulus of elasticity,
E
, is 10,000 ksi. Note that the structure is in the XZ coordinate system.
Since this is a pin connected structure loaded only at the joint, it will
act as a true truss with only axial forces in the members. The stiffness
model will be simplified to only include the axial stiffness components,
AE
/
L
. Furthermore, rotation at the joints will be excluded since there is no
rotational stiffness imparted by the members. Rotation could be included
Z
8k
8k
9k
12k
15'-0"
X
20'-0"
Figure 4.33.
Example 4.19 Global joint stiffness.
