Civil Engineering Reference
In-Depth Information
Apply equilibrium on Figure 5.7 to find the
j-
end forces and moments.
FEP
=−
wL FEP
jz
iz
2
wL
FEM
=− −
FEP L FEM
jy
iz
iy
2
Example 5.6
Non-prismatic member stiffness
Derive the fixed-end forces and moments due to a concentrated load in
the X-Z system for a non-prismatic cross-section using Castigliano's
theorems.
The free-body diagram of the beam is shown in Figure 5.8. The pro-
cedure used in Example 5.5 will be repeated here.
P
FEM
iy
FEM
jy
a
b
FEP
iz
FEP
jz
L
Figure 5.8.
Example 5.6 Non-prismatic member stiffness.
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
i-end. In this case, two moment equations
must be written. The first is
M
x
1
, with
x
from the
i
i-end to the point load
(0 ≤
x
≤
a
) and the second is
M
x
2
, from the point load to the
j
-end (
a
≤
x
≤
L
).
M
=−
FEP x FEM
−
x
1
iz
iy
(
)
−
MP xa FEP x FEM
=−
−
−
x
2
iz
iy
The partial derivatives are the same for either of the two moment equations.
d
d
M
FEP
x
=−
x
iz
M
FEM
d
x
=−
1
d
iy
L
d
d
M
FEP
dx
EI
dx
dx
EI
Px xa
dx
EI
(
)
∫
∫
∫
∫
∆
==
0
M
x
=
FEP x
2
+
FEM x
−
−
iz
x
iz
iy
EI
iz
y
y
y
y
a
L
M
FEM
d
dx
EI
dx
EI
dx
EI
dx
EI
(
)
∫
x
∫
∫
∫
q
==
0
M
=
FEP x
+
FEM
−
Px a
−
iy
x
iz
iy
d
iy
y
y
y
a
y
