Civil Engineering Reference
In-Depth Information
Substituting Equation 5.17 into the following equation repeated from ear-
lier, results in Equation 5.18 for the second stiffness value.
2
3
PL
EI
P
LGA
2
M
EI
iy
iy
iz
=
−
z
z
y
(
)
2
4
EI LAG I
LAG EI
+
3
z
y
z
(5.18)
M
=
q
iz
iz
3
+
12
y
z
Example 5.11
Shear stiffness
Derive the local member shear stiffness for ∆
iy
using the Castigliano's
theorems.
The free-body diagram of the beam with an imposed deflection of ∆
iy
is shown in Figure 5.13. Also shown is a left-hand free-body of the beam
cut at any distance
x
from the
i-
end.
Y
M
jz
M
iz
∆
iy
X
P
jy
P
iy
L
M
iz
M
V
P
iy
x
Figure 5.13.
Example 5.11 Shear stiffness.
The internal shear,
V
x
, and moment,
M
x
, are exactly the same as in
Example 5.10. The partial derivatives are also the same.
VP
V
P
V
M
MPxM
M
=−
x
iy
d
d
d
d
x
=−
1
iy
x
=
0
iz
=−
x
iy
iz
