Structural Analysis: Castigliano's Theorem

Castigliano's Second Theorem

• Powerful method for finding displacements in elastic structures
• Based on the concept of strain energy
• Formula: ( \delta = \frac{\partial U}{\partial P} )
  • ( \delta ) is the displacement
  • ( U ) is the strain energy
  • ( P ) is the external load

Principle of Least Work

• Basis for energy methods in structural analysis
• External work done on a system equals the change in its internal energy

Analysis of Redundant Frames (Up to Two Degrees of Redundancy)

• Identifying Redundancies: Determine the number of members exceeding the minimum required for equilibrium
• Calculate Reactions and Internal Forces: Use conventional methods for non-redundant members
• Compute Strain Energy: Calculate total strain energy using formulas
  • Bending: ( U = \frac{1}{2} \int M^2 , \frac{dx}{EI} )
  • Axial Loads: ( U = \frac{1}{2} \int N^2 , \frac{dx}{EA} )
• Apply Castigliano's Theorem: Differentiate strain energy with respect to the redundant force to find displacement at the point of redundancy
• Solve the Equations: Set up equations based on calculated displacements and solve for redundant forces

Analysis of Redundant Trusses (Up to Second Degree of Redundancy)

• Truss Structure: Composed of members joined at the ends, designed to support loads through axial forces
• Redundant Members: More members than required for structural stability
• Identify Redundant Members: Determine redundant members and overall redundancy
• Static Equilibrium: Analyze using equilibrium equations for non-redundant members
• Compute Strain Energy: Similar to frames, using the formula: ( U = \frac{1}{2} \sum \frac{N_i^2 L_i}{A_i E_i} )
• Apply Castigliano's Theorem: Differentiate strain energy with respect to redundant force for displacement relationship
• Solve for Redundant Forces: Set up equations based on equilibrium and displacements to find redundant forces