Civil Engineering Reference
In-Depth Information
L
mM
EI
0
dQ
∆
=
dx
C
1
2
856 556
1
2
16
() (
)
+
(
)
(
)
ft kft
-
.
k ft
-
ft
16
k
-
ft
3 556
.
kft
-
1
1
k
∆
=
C
EI
+
(
)
(
)
16
ft kft
40
-
2 667
.
k ft
-
23
2958
kk t
EI
kft
EI
-
1
k
∆
=
C
(
)
2958 1728
2000
in ft
33
/
2958
-
3
∆=
=
)
= 1 278
.
in
(
C
kin
/
2
2000
in
4
4.7
cAStigLiAnO'S tHEOREMS
In 1879 Alberto Castigliano published his two theorems on elastic struc-
tures that are known as
Castigliano's Theorems
(Castigliano 1879). The
first theorem states that the first partial derivative of strain energy with
respect to a particular deflection component is equal to the force applied at
the point and in the direction corresponding to that deflection component.
This may be written in mathematical terms as shown in Equation 4.8. The
second theorem is used more often in statically indeterminate structural
analysis and 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 is shown
in Equation 4.9. The equations are written in terms for flexural energy,
M/EI
, of a particular rotation,
q
A
, and moment,
M
A
, relationship and in
terms of a particular deflection, ∆
A
, and force,
P
A
, relationship. They can
be written for any elastic force and deformation relationship.
L
L
MM
Mdx
∂
∂
EI
PM
Mdx
∂
∂
∫
∫
(4.8)
=
and
=
A
A
q
∆
EI
0
A
0
A
L
L
∂
∂
M
M
dx
EI
∂
∂
M
P
dx
EI
∫
∫
(4.9)
q
A
=
M
and ∆
=
M
A
0
A
0
A
Example 4.10
Castigliano's second theorem
Determine the deflection of the beam in Figure 4.21 at point
B
using
Castigliano's second theorem.
Since the deflection at point
B
is desired, a force
P
will be placed at
B
.
This will be the particular force in the direction of the desired deflection.
