Civil Engineering Reference
In-Depth Information
The forces at the
j-
i-end of the member due to the motions at the
i-
end can
be found using the transmission matrix. Then, the forces at the
i-
end due
to the motions at the
j-
end can be found by symmetry of the stiffness
matrix. Finally, the forces at the
j-
end due to motions at the
j-
end can be
found using the transmission matrix. This process was illustrated previ-
ously in Chapter 4. The resulting terms are shown in the matrices given as
Equations 5.21 through 5.24.
5.6
SHEAR StiffnESS, X-Y SYStEM
The following contains the combined flexural and shear stiffness of mem-
bers in the X-Y system.
Example 5.10
Shear stiffness
Derive the local member shear stiffness for
q
iz
using Castigliano's
theorems.
The free-body diagram of the beam with an imposed rotation of
q
iz
is
shown in Figure 5.12. Also shown is a left-hand free-body of the beam cut
at any distance
x
from the
i-
end.
Y
M
iz
M
jz
θ
iz
X
P
jy
P
iy
L
M
iz
M
V
P
iy
x
Figure 5.12.
Example 5.10 Shear stiffness.
The internal shear,
V
x
, and moment,
M
x
, at any point,
x
, can be found
from statics and the partial derivatives of that shear and moment can be
found with respect to the applied force and moment at the
i
-end.
