Question

Which method among the following is based on the strain energy principle?

• A. Conjugate beam method
• B. Castigliano's method
• C. Slope-deflection method
• D. Moment distribution method

Hint:

To classify structural analysis methods:
- {Energy Methods:} Castigliano's Theorem, Principle of Virtual Work, Unit Load Method.
- {Displacement/Stiffness Methods:} Slope-Deflection, Moment Distribution, Direct Stiffness Method (basis of FEA).
- {Force/Flexibility Methods:} Method of Consistent Deformations.
- {Geometric Methods:} Conjugate Beam, Moment-Area Method.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks to identify which structural analysis method is derived from the principles of strain energy. Strain energy is the energy stored in a body due to its deformation. Energy methods in structural analysis relate external work done on a structure to the internal strain energy stored within it.
Step 2: Detailed Analysis of Each Method:
- (A) Conjugate beam method: This is a method for finding deflections and slopes in beams. It is based on an analogy between the relationships in a real beam (M, \(\theta\), y) and a fictitious "conjugate" beam (loading, shear, moment). It is a geometric method, not an energy method.
- (B) Castigliano's method: This is a classic energy method. It is based on Castigliano's theorems, which relate the strain energy in a structure to its displacements.
- Castigliano's First Theorem states that the partial derivative of the total strain energy with respect to any displacement component is equal to the force applied at that point in the direction of that displacement.
- Castigliano's Second Theorem (more commonly used) states that the partial derivative of the total strain energy with respect to an applied force is equal to the displacement at the point of application of that force, in the direction of the force. This is used to calculate deflections.
Therefore, this method is fundamentally based on the strain energy principle.
- (C) Slope-deflection method: This is a classical displacement (or stiffness) method of analysis. It relates the moments at the ends of a member to the rotations (slopes) and displacements of its ends. It is based on equilibrium and kinematic compatibility, not directly on energy principles.
- (D) Moment distribution method: This is an iterative method for analyzing indeterminate structures, developed by Hardy Cross. It is a displacement method that is essentially an iterative relaxation of the slope-deflection equations. It is based on equilibrium concepts, not strain energy.
Step 3: Why This is Correct:
Castigliano's method is the only one on the list that explicitly uses the concept of strain energy and its derivatives to solve for forces and displacements in a structure.