### What is Castigliano’s Theorem? Castigliano’s First & Second Theorem Applications , Advantages

# What is Castigliano’s Theorem? Castigliano’s First & Second Theorem Applications , Advantages

## What is Castigliano’s Theorem?

Castigliano’s theorem, named after Carlo Alberto Castigliano, is a method for calculating the displacements of a linear-elastic system using partial derivatives of energy. He is well-known for two theorems.

### Castigliano’s First Theorem

The first partial derivative of the total internal energy (strain energy) in a structure with respect to any particular deflection component at a point is equal to the force applied at that point and in the direction corresponding to that deflection component.

This first theorem is applicable to linearly or nonlinearly elastic structures in which the temperature is constant and the supports are unyielding.

### Castigliano’s Second Theorem

The first partial derivative of the total internal energy in a structure with respect to the force applied at any point is equal to the deflection at the point of application of that force in the direction of its line of action.

The second theorem of Castigliano is applicable to linearly elastic (Hookean material) structures with constant temperature and unyielding supports

The fundamental principle may be easily grasped by recalling that a change in energy is equal to the producing force multiplied by the consequent displacement.

As a result, the producing force is equal to the energy change divided by the resulting displacement

Alternatively, the resulting displacement equals the energy change divided by the originating force.

To link the originating forces and consequent displacements to the change in energy, partial derivatives are required

Castigliano’s theorem can be used to calculate the deflection of curved beams, but other theorems, such as Macaulay’s theorem, can only be used to calculate the deflection of vertical or horizontal bars.

## The Advantages of Using Castigliano’s Theorem

These are the benefits of employing Castigliano’s theorem over alternative theories.

1. This theorem can measure the deflection of a bent bar, whereas Macaulay approaches can only measure vertical and horizontal deflection.

2. Calculating the value of beam deflection is easier if the load differences are solved first before we integrate or use another solution.

3. This approach uses the ‘dummy,’ whose purpose is to take the point that has not been indicated, the actual load.

## Castigliano’s Theorem FAQs

### 1. When can you use Castigliano’s theorem?

Castigliano’s Theorem lets us use strain energies at the locations of forces to determine the deflections. The Theorem also allows for the determining of deflections for objects with changing cross-sectional areas.

Castigliano’s theorem is a theorem in fluid dynamics and is typically used in everyday calculus in determining the forces for a stationary fluid system, such as the fluid suspended in a tank. It is also sometimes referred to as the law of the lever.

Mathematically, it states that the constitutive force corresponding to an exponential strain rate is -The constitutive force corresponding to an exponential strain rate is proportional to the second power of the exponential strain rate.

Castigliano’s theorem can be applied to maintain equilibrium in a fluid system that is not close to the ideal system.

### 2. How does it work?

Castigliano’s theorem was developed for use in equilibrium problems involving elastic materials and fluids. It calculates the force-deflection relationship for elastic or fluid mechanical systems.

Castigliano’s theorem states that the partial derivative of strain energy with respect to a certain load factor is equal to the force which produces that load factor times the distance of applying that load factor.

### 3. What are the variables in Castigliano’s theorem?

The variables in Castigliano’s theorem are the force, displacement, and stress and strain over some material.

The term “load factor” is also sometimes used in place of “force”. The load factor is the applied force divided by the original cross-sectional area over which that force was applied.

The strain energy, also called the elastic strain energy, is defined as the change in elastic modulus divided by the strain.

### 4. What are the types of problems that Castigliano’s theorem can be used for?

Castigliano’s theorem can be used for different types of problems including those involving a moving load (the loads may include an applied force, a volumetric load, or an imposed force.

It is also sometimes known as the law of the lever when applied to fluid mechanics).

### 5. What does the “second derivative test” of Castigliano’s theorem refer to?

The second derivative test is used in determining whether or not a structure is at equilibrium. Generally, the second derivative test is used to determine whether or not the strains associated with a load on an elastic body are finite (the finite strains lead to infinite forces and are, therefore, at equilibrium).