“Force method” association in structural analysis In classical structural analysis, the approach often termed the “force method” (flexibility method) is closely associated with which of the following relations/tools for continuous beams?

Introduction / Context:
Two fundamental families of methods exist in structural analysis: displacement (stiffness) methods and force (flexibility) methods. Continuous beams traditionally were analyzed using the force method via classic relationships that link end moments and spans, the most famous being the three-moment equation.

Given Data / Assumptions:

• Prismatic or piecewise-prismatic continuous beams on multiple supports.
• Small deflection linear behavior.
• Compatibility of rotations and deflections imposed at interior supports.

Concept / Approach:

The force method treats redundant reactions or end moments as unknown “forces” and enforces compatibility to solve them. Clapeyron’s three-moment equation directly relates support moments in two adjacent spans to loads and span properties, serving as a principal instrument of the force method for continuous beams.

Step-by-Step Solution (conceptual):

Verification / Alternative check:

Solutions obtained via the force method (three-moment equation) agree with displacement-method (slope-deflection/ stiffness) solutions for the same geometry and loading, validating the association.

Why Other Options Are Wrong:

• Moment-area theorems are deformation evaluation tools, not the core of the force method for continuous beams.
• Maxwell’s reciprocal theorem describes symmetry of deflections, not a direct solving tool for continuous beams.
• Conjugate-beam analogy is a deformation technique, closer to displacement methods.
• “None” is incorrect since a specific relation is indeed tied to the force method.

Common Pitfalls:

• Confusing three-moment (force method) with slope-deflection (displacement method); both solve continuous beams but via different primary unknowns.

Final Answer:

Three-moment equation (Clapeyron’s theorem for three moments).