Lecture 5 castigliono's theorem

2. Unit 1- Stress and Strain
 Lecture -1 - Introduction, state of plane stress
 Lecture -2 - Principle Stresses and Strains
 Lecture -3 - Mohr's Stress Circle and Theory of
Failure
 Lecture -4- 3-D stress and strain, Equilibrium
equations and impact loading
 Lecture -5 – Castigliono's Theorem
Topics Covered

3. Castigliono’s First
Theorem
 Let P
1
, P
2
,...., P
n
be the forces acting at x
1
, x
2
,......, x
n
from
the left end on a simply supported beam of span L .Let u
1
,
u
2
,..., u
n
be the displacements at the loading P
1
, P
2
,...., P
n

respectively as shown in figure.

4. Castigliono’s First
Theorem
 Now, assume that the material obeys Hooke’s law and
invoking the principle of superposition, the work done by the
external forces is given by
 Work done by external forces is stored in structure as strain
energy.
€
W=
1
2
P
1
u
1
+
1
2
P
2
u
2
+....+
1
2
P
n
u
n
€
U=
1
2
P
1
u
1
+
1
2
P
2
u
2
+....+
1
2
P
n
u
n

5. Castigliono’s First
Theorem
 u
1
(deflection at point of application of P
1
) can be expressed
as
 In general
  = flexibility coeff at i due to unit force applied at j.
 Work done by external forces is stored in structure as strain
energy.
€
u
1
=a
11
P
1
+a
12
P
2
+....+a
1n
P
n
€
U=
1
2
P
1
a
11
P
1
+a
12
P
2
+..
[ ]
+
1
2
P
2
a
21
P
1
+a
22
P
2
+..
[ ]
+....+
1
2
P
n
a
n1
P
1
+a
n2
P
2
+..
[ ]
€
u
1
=a
i1
P
1
+a
i2
P
2
+....+a
in
P
n
€
a
ij

6. Castigliono’s First
Theorem
 In general
 Differentiating the strain energy with force P
1
 This is nothing but displacement at the loading point
€
a
ji
=a
ij
€
U=
1
2
a
11
P
1
2
+a
22
P
2
2
+..+a
nn
P
n
2[ ]
+a
12
P
1
P
2
+a
13
P
1
P
3
+..+a
1n
P
1
P
n
[ ]
€
∂U
∂P
1
=a
11
P
1
+a
12
P
2
+..+a
1n
P
n
[ ]
€
∂U
∂P
n
=u
n

7. Castigliono’s First
Theorem
 Castigliano’s first theorem may be stated as the first
partial derivative of the strain energy of the structure
with respect to any particular force gives the
displacement of the point of application of that force
in the direction of its line of action.
€
∂U
∂P
n
=u
n

8. Castigliono’s Second
Theorem
 Castigliano’s second theorem may be stated as the first
partial derivative of the strain energy of the structure
with respect to any particular displacement gives the
force.
€
∂U
∂u
n
=P
n