# Strain Energy Method

By Mukul Yadav|Updated : October 6th, 2022

The Strain Energy Method is used to determine the deflection of members. In the civil engineering analysis of structure, deflection is one of the most important parameters required for the analysis. The strain energy method is based on the strain energy of structural members.

Strain Energy Method PDF

Strain energy can be defined as the energy stored within an elastic body when the member is deformed under the action of the load. It is also known as the potential energy, which can be calculated as the work done by external loads for deforming the structural member. Here, we will discuss the Strain energy method, its application, and problems based on it.

## What is Strain Energy Method?

Strain energy is calculated by the work done by the structure's member to deflect the member under the action of external loads. Hence, this strain energy can be used to calculate the deflection at any point in the member caused due to external loads. This method of calculating the deformation is known as the strain energy method. The Strain Energy Method is an important part of the GATE CE syllabus for the upcoming exam.

Strain energy can be recovered in the member up to the elastic limit of stresses in the structure. The area under the stress-strain curve up to the elastic limit is the strain energy per unit volume.

## Strain Energy in Beams

Strain energy in beams can be used to calculate the deformation of the beam due to the external loads. In the case of beams, transverse loads cause a bending moment, which introduces the bending stresses. In beams, this strain energy caused by moment can be used to find the deflection at any point, and this method is known as the strain energy method.

## Strain Energy Method for Indeterminate Structures

In the case of indeterminate structures for calculating the deflection, mainly displacement methods are used because deformation and rotations are unknown in the displacement method. For the GATE exam, the displacement method can be preferred over the force method when the number of unknown forces exceeds the number of unknown displacements.

The strain energy method can also calculate deflections in indeterminate structures. This method is based on the Castigliano theorem for deflection. Castigliano’s first theorem is used as the displacement method, while Castigliano’s second theorem is used as the force method for calculating the deflections of indeterminate structures.

## Strain Energy Method for Finding Deflection

The strain energy method for finding deflection is based on the Castigliano theorems. This theorem relates the strain energy of the structure with the external point load at the displacement location. There are two Castigliano’s theorems in the structural analysis related to the displacement of the structure.

• Castigliano’s first theorem: According to the first theorem, the first partial derivative of the total strain energy of structure concerning any deflection component is equal to the point load acting at that point in the same direction as that of deflection.
• Castigliano’s second theorem: According to the second theorem, the first derivative of total strain energy concerning the point load acting at that point is equal to the deflection at the same point of structure in the same direction of deflection

## Problems on Strain Energy Method

The strain energy method is a method to calculate deflection at any point in the structure. This method uses the concept of total strain energy in the structure. Here, the formula of strain energy for some standard cases of beams is given, which helps solve the problems of the strain energy method.

## Different Methods for Finding Deflection

As we know, the calculation of the deflection of a member is a major part of the analysis of a structure. So, there are various methods available for calculating deflections. These methods are widely used in formulating questions in the GATE exam question paper. Some methods are listed below, which can be used to calculate the deflection of beams. The suitability of these methods depends upon the type of loading and support conditions etc.

• Double Integration method
• Macaulay’s method (It is the extension of the double integration method)
• Moment-area method
• Strain energy method (based on Castigliano’s method)
• Superposition method
• Conjugate beam method
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## Strain Energy Method FAQs

• The strain energy method is a method for calculating the deflections of a structure, which is based on the principle of Castigliano’s theorem. And uses the concept of conservation of strain energy for deflection calculation.

• As strain energy is also a form of energy, the unit is also the N-m or Joules. Strain energy can be defined as the energy stored by an elastic body when it is deformed under the action of external loads within the elastic limit.

• The strain energy method is based on the principle of Castigliano’s theorem. According to that first partial derivative of strain energy with respect to the load is equal to the deflection at the same point in the direction of the applied unit load.

• Strain energy is always positive in nature, as strain energy is directly proportional to the square of the load. It will not depend upon the nature of the load. A load may be either tensile or compressive in nature.

• The strain energy method uses the principle of strain energy. This method is used because it is based on the principle of conservation of strain energy, and it is easy for some structures.

### ESE & GATE CE

Civil Engg.GATEGATE CEESE CEESEBARC CEAFCAT CE