Castigliano’s
Theorems
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Castigliano’s Theorem
• This method is a powerful new way to determine deflections in many
types of structures – bars, beams, frames, trusses, curved beams, etc.
• We can calculate both horizontal and vertical displacements and
rotations (slopes).
• There are actually two Castigliano’s Theorems.
• The first can be used for structures made of both linear and non-linear elastic
materials.
• The second is restricted to structures made of linear elastic materials only. This
is the one we will use.
Castigliano’s Theorem
• For linearly elastic structure, the Castigliano’s first theorem may be
defined 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
i 
Fi
Where:
Fi = Force at i-th application point
δi = Displacement at i-th point in the direction of Fi
U = Total strain energy
Castigliano’s Theorem
• Consider an elastic beam AB subjected to loads W1 and W2, acting at
points 1 and 2 respectively
Castigliano’s Theorem
 1   11   12
 2   21   22
If 11W1  11
 21W1   21
Castigliano’s First theorem derivation
• Now applying W2 at Point 2 first
and then applying W1 at Point 1,
1  12  11  12W2  11W1
Similarly,
 2   22   21   22W2   21W1
Strain energy, Ui
1
1
 W2 22  W2 21  W111
2
2
1
1
 W2   22W2  W2   21W1  W1  11W1
2
2
1
1
2
  22W2   21W1W2  11W12 - - - - - - - (IV)
2
2
Castigliano’s First theorem derivation
• Considering equation (III) and (IV), and equating them, it can
be shown that
1
1
 11W12   22W22   12W1W2
2
2
1
1
  22W22   21W1W2   11W12
2
2
Ui 
12   21
This is called Betti – Maxwell’s reciprocal
theorem
Deflection at point 2 due to a unit load at point 1 is equal to the
deflection at point 1 due to a unit load at point 2.
Castigliano’s First theorem derivation
• From Eqn. (III), Ui  1 11W12  1  22W22  12W1W2
2
2
U i
 11W1  12W2   1
W1
From Eqn. (IV), Ui 
1
1
 22W22   21W1W2  11W12
2
2
U i
  22W2   21W1   2
W2
U i
 i
Wi
This is Castigliano’s first theorem.
Castigliano’s second theorem
It states,
“If U is total strain energy stored up in a frame work in
equilibrium under an external system of forces, its magnitude is
always minimum.”
Similarly the energy Ui can be express in terms of spring stiffnesses k11,
k12 (or k21), & k22 and deflections δ1 and δ2; then it can be shown that
U i
 W1
 1
U i
 W2
 2
This is Castigliano’s second theorem.
When rotations are to be determined,
U
i 
M i
Strain Energy in Common Members

Beam subject to bending
I,E
M
M
L
2
L
1M L
M 2 ( x)
U
or U  
dx
2 EI
2 E ( x) I ( x)
0
Castigliano’s Theorem - Frame

For the structure and loading shown below, determine the vertical
deflection at point B. Neglect axial force in the column.
P
L2
B
E, I
L1
A
Castigliano’s Theorem - Frame

For the structure and loading shown below, determine the vertical
and horizontal deflection at point B. Neglect axial force in the
column.
L2
w
B
E, I
L1
A
Statically Indeterminate Problems

For the structure and loading shown below, find the
fixed end reactions.
w
A
B
L
Thank
You…