Civil Engineering Reference
In-Depth Information
Substituting Equation 5.13 into the following equation repeated from ear-
lier, results in Equation 5.14 for the second stiffness value.
M
EI
2
3
PL
EI
P
LGA
iy
iz
iz
=−
−
y
y
z
(
)
2
4
EI LAG I
LAG EI
+
3
y
z
y
q
iy
(5.14)
M
=
iy
3
+
12
z
y
Example 5.9
Shear stiffness
Derive the local member shear stiffness for
D
iz
using Castigliano's
theorems.
The free-body diagram of the beam with an imposed deflection of ∆
iz
is shown in Figure 5.11. 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
∆
iz
P
jz
P
iz
L
M
iy
M
V
P
iz
x
Figure 5.11.
Example 5.9 Shear stiffness.
The internal shear,
V
x
, and moment,
M
x
, are exactly the same as in
Example 5.8. The partial derivatives are also the same.
VP
V
P
V
M
MPxM
M
=−
x
iz
d
d
d
d
x
=−
1
iz
x
=
0
iy
=− −
x
iz
iy
