CASTIGLIANO’S THEOREMS CLASS NOTES FOR MECHANICAL ENGINEERING
CASTIGLIANO’S THEOREMS CLASS
NOTES FOR MECHANICAL ENGINEERING
Castiglioni’s Method
This method deals with displacements with respect to strain energy. Strain energy is also called potential energy. It is also named as Internal energy. Deflection is found using strain energy. Strain energy is recoverable as mechanical energy. Displacement in Castilian’s method means

Extension under an axial tensile force, δL= PL/AE

Contraction under an axial compressive force, δL= — PL/AE

Deflection due to Bending Moment and shear force, y= δ_{M} + δ_{V}

Angle of twist due torque (torsional moment), ϴ = TL/GJ

Extension in a spring due to a force, δ =W/k

Contraction in a spring due to a force, δ =W/k and etc.
CASTIGLIANO’S Method determines the displacement in a linearly elastic system at any point. It uses the partial derivatives of the strain energy. There are two Castiglioni’s theorems.
(ii) Castiglioni’s First Theorem OR Castigliano’s Force Finding Method or Castigliano’s Force Theorem
Determines the force at a certain point of known displacement ‘δ_{i}’ within elastic limits. Express strain energy of an elastic structure as a function of generalized displacement δ_{i}; then find the partial derivative of the strain energy with respect to generalized displacement, δ_{i}. This partial derivative gives the force F_{i}. In the mathematical form, it is given as
F_{i} = ∂U/∂δ_{i}
Where U is the total strain energy due to the forces/moments/torques etc.
_{δi} is the displacement at any point ‘i’
F_{i} is the force at the point ‘i’
(ii) Castigliano’s Second Theorem or Castigliano’s Displacement Finding Method or Castigliano’s Displacement Theorem
It determines Displacement at a point of the loaded beam. It uses the partial derivatives of the strain energy with respect to the force acting at that location. If the Displacement is positive, then it is in the direction of the force and vice versa. In the mathematical form, it is given as
δ_{i} = ∂U/∂F_{i}
Where
δ_{i} is the displacement of the point under the force F_{i} in the direction of F_{i}
U is the total strain energy
COR: Castigliano’s theorem
It also determines the slope at a point of the loaded beam. Here it uses the partial derivatives of the strain energy with respect to the bending moment acting at that location. In the mathematical form, it is given as
θ_{i }= ∂U/∂M_{i}
Where U is the total strain energy,
θ_{i} is the slope of the moment M_{i} in the direction of M (Clockwise or anticlockwise)
NOTE: THIS METHOD IS VERY USEFUL FOR OBTAINING THE DEFLECTION OF A POINT WHERE THERE IS NO FORCE. IN THIS CASE A FORCE/ MOMENT (FICTITIOUS) IS ASSUMED TO ACT. THEN MAKE force ZERO AT THAT POINT IN THE PARTIAL DERIVATIVE OF THE STRAIN ENERGY.
MEANING OF GENERALIZED DISPLACEMENTS

In Simple Tension
Extension in tension δL= PL/AE
Strain energy = Average Force x δL = (F/2) (PL/AE) = =F^{2}L/2EA

In Simple Compression
Extension in compression δL= PL/AE
Strain energy = F_{av} δL = (F/2) (PL/AE) = =F^{2}L/2EA

In Simple Bending
Strain energy = Average Bending moment x deflection
= (M/2) (ML/EI) = M^{2}L/2EI

Angle of twist in simple torsion
Strain energy= Average torque x displacement =average torque x angle of twist
= (T/2) (TL/GJ) = T^{2} L/2GJ
STRAIN ENERGY IN A MEMBER FOR ANY ONE TYPE OF LOADING
Sr. No. 
Type of loading 
Strain energywith variable loading 
Strain energy withconstant loading 
1. 
Axial load(tensile or compressive), F 
U =∫ F^{2}dx/2EA 
U =F^{2}L/2EA 
2. 
Bending Moment, M 
U = ∫ M^{2}dx/2EA 
U =M^{2}L/2EI 
3. 
Torque T 
U = ∫ T^{2}dx/2GJ 
U =T^{2}L/2GJ 
4. 
Direct shear force, F 
U = ∫ F^{2}dx/2GA 
U =F^{2}L/2GA 
5. 
Transverse shear force, V 
U = ∫ V^{2}dx/2GA 
U =K V^{2}L/2GA 
INTEGRATION LIMITS ARE FROM 0 to L in the tabular equations.
Where K is a correction factor depending upon the shape of the area
rectangular section K =1.2
solid circular section K =1.11
thin walled tube K =2.0
PROCEDURE FOR APPLICATION OF CASTIGLIANO’S THEOREM
a) To determine deflection y_{f} in the direction of a real force F_{f }

Prepare an expression for the total strain energy.

Obtain the linear deflection y_{f} from the relationship y_{f} = ∂U /∂F_{f}

If the force is fictitious, set F_{f} = 0 and solve the resulting equation)
(b) To determine the slope (an angular displacement) θ_{f} in the direction of a fictitious moment M_{f}

Write the expression for the total strain energy.

Determine the angular deflection from the relationship θ_{f}= ∂U /∂M_{f}

If the moment is fictitious, then set M_{f} = 0 and solve the resulting equation.
EXAMPLE ON CASTIGLIANO’S THEOREM
For a given rectangular beam under a central load W, find the deflection under the load.