Maxwell’s reciprocal theorem – applicability in structural analysis Maxwell’s reciprocal theorem (Betti’s theorem) can be applied to which type of structures?

Introduction / Context:Maxwell’s reciprocal theorem underpins many displacement methods and influence-line concepts. It states that the displacement at point A due to a unit load at B equals the displacement at point B due to a unit load at A—under appropriate conditions.

Given Data / Assumptions:

• Small displacements; linear geometry.
• Material behaves linearly elastically (superposition holds).
• Structures may be statically determinate or indeterminate but within elastic range.

Concept / Approach:The reciprocity result (also Betti’s theorem) derives from symmetry of the stiffness matrix in linear elasticity. Plastic or nonlinear behavior breaks superposition and the theorem generally does not apply.

Step-by-Step Solution:Identify requirement: linear elastic behavior → symmetric stiffness.Conclude applicability: theorem valid for linear elastic structures regardless of geometric symmetry.

Verification / Alternative check:Matrix methods: for linear elastic systems, k = k^T; thus, displacement influence coefficients are reciprocal.

Why Other Options Are Wrong:

• Plastic/nonlinear systems violate linear superposition.
• Symmetry of geometry is not required; material linearity is the key.
• “All the above” cannot be true because (a) and (e) are incorrect contexts.

Common Pitfalls:Applying reciprocity across different boundary conditions or after yielding; ensure all loads remain within the elastic range.

Final Answer:Linear elastic structures