Civil Engineering Reference
In-Depth Information
W
FEM
iy
FEM
jy
FEP
iz
FEP
jz
L
Figure 5.7.
Example 5.5 Non-prismatic member stiffness.
The free-body diagram of the beam is shown in Figure 5.7. The pro-
cedure for Castigliano's second theorem used in Example 5.4 will be
repeated here.
The internal moment,
M
x
, at any point,
x
, can be found from statics
and the partial derivatives of that moment can be found with respect to the
applied force and moment at the
i
-end.
2
wx
M
=− −
FEP x FEM
x
iz
iy
2
d
d
M
FEP
x
=−
x
iz
M
FEM
d
x
=−
1
d
iy
3
d
d
M
F
dx
EI
dx
EI
dx
EI
wx
dx
EI
==
∫
∫
∫
∫
x
2
∆
0
M
=
FEP x
+
FEM x
−
iz
x
iz
iy
EP
2
iz
y
y
y
y
2
dM
FEM
dx
EI
dx
EI
dx
EI
wx
dx
EI
∫
∫
∫
∫
q
==
0
M
x
=
FEP x
+
FEM
−
iy
x
iz
iy
d
2
iy
y
y
y
y
Observing that there is a new term that varies with
x
,
we will substitute
S
4
as follows:
dx
EI
∫
Sx
=
3
4
y
w
S
0
=
FEP
S
+
FEM
S
−
iz
3
iy
2
4
2
w
S
0
=
FEP
S
+
FEM
S
−
iz
2
iy
1
3
2
FEP
FEM
SS
S
S
=
w
iz
3
2
4
SS
2
iy
2
1
3
FEP
FEM
=
S S
SS
−
S
S
SS SS
SSS
−
−
1
ww
D
iz
1
2
4
14 23
=
D
−
22
2
iy
2
3
3
3
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