Civil Engineering Reference
In-Depth Information
The general stiffness equation can be set-up and solved as follows. The
units are in inches and radians:
[]
=
[[]
[]
=
[][]
PK
d
i
ii
i
−
1
d
KP
i
ii
i
−
1
666 67
.
0
0
0
4 7727
147 73
0
5 3123
[]
=
d
0
2 1818
.
−
181 81
.
−
.
=−
.
i
0
−
181 81
.
21818
.
−
0 0375
.
5.5
SHEAR StiffnESS, X-Z SYStEM
The shear stiffness of a member should be included when it is significant.
This effect was developed by Timoshenko (1921) in 1921. For normal
frame structures, the stiffness contributions due to shear are minor and
are sometimes ignored. For frames and structures with larger or deeper
members, the shear stiffness contribution is appreciable and should be
included. The following two sections and corresponding examples derive
the combined flexural and shear stiffness of members.
Example 5.8
Shear stiffness
Derive the local member shear stiffness for
q
iy
using Castigliano's
theorems.
The free-body diagram of the beam with an imposed rotation of
q
iy
is
shown in Figure 5.10. Also shown is a left-hand free-body of the beam cut
at any distance
x
from the
i-
end.
Z
M
iy
M
jy
X
θ
iy
P
jz
P
iz
L
M
iy
M
V
P
iz
x
Figure 5.10.
Example 5.8 Shear stiffness.
