Civil Engineering Reference
In-Depth Information
∆
iz
M
iy
P
iz
M
iy
θ
iy
P
iz
Figure 5.6.
Example 5.4 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
-end.
MPxM
M
P
=− −
x
iz
iy
d
d
d
d
x
=−
x
iz
M
M
x
=−
1
iy
Castigliano's second theorem states that the first partial derivative of strain
energy with respect to a particular force is equal to the displacement of the
point of application of that force in the direction of its line of action. This
can be applied for both ∆
iz
and
q
iy
.
d
d
M
P
dx
EI
Px
dx
EI
Mx
dx
EI
∫
∫
∫
∆
iz
=
M
x
=
2
+
x
iz
iy
iz
y
y
y
d
d
M
M
dx
Px
dx
EI
dx
EI
∫
∫
∫
q
=
M
x
=
+
M
iy
x
iz
iy
EI
iy
y
y
y
Since the cross-sectional properties vary, the moment of inertia,
I
y
, var-
ies. Let the values
S
1
,
S
2
, and
S
3
be used and substituted into the previous
two equations. These values can be pre-derived for the cross-sectional
variation.
dx
EI
∫
S
=
1
y
dx
EI
∫
S
=
x
2
y
dx
