Conjugate beam method (structural analysis): For a two-span real beam that contains an internal hinge, identify the correct conjugate beam using the standard mapping rules—fixed end ↔ free end; end pin/roller ↔ same end pin/roller; internal hinge in the real beam ↔ an internal simple support in the conjugate beam.

Introduction / Context:
The conjugate beam method transforms a bending problem into a statics problem by mapping the real beam to a conjugate beam whose shear and reactions represent slopes and deflections. Correctly creating the conjugate beam support conditions is essential; otherwise, the computed slopes and deflections will be wrong.

Given Data / Assumptions:

• Real member: a continuous (two-span) beam.
• There is an internal hinge between the spans.
• Standard conjugate-beam mapping applies.
• Flexural rigidity is uniform enough to apply the usual E I curvature loading concept.

Concept / Approach:

Mapping rules are: (1) a fixed end in the real beam becomes a free end in the conjugate beam; (2) a pin/roller support at an end remains a pin/roller at the same end in the conjugate beam; (3) an internal hinge in the real beam becomes an internal simple support in the conjugate beam. The loading on the conjugate beam is the M/EI diagram of the real beam; the conjugate-beam reactions correspond to real-beam rotations/deflections.

Step-by-Step Solution:

Verification / Alternative check:

At internal hinges, the real beam allows moment release (zero moment, finite rotation). In the conjugate beam, this state is modeled by a simple support (zero deflection, free slope reaction analogy), validating the mapping.

Why Other Options Are Wrong:

• Rigidifying the internal hinge or removing it contradicts the mapping rule.
• Converting all ends to free ends ignores the pin/roller mapping at ends.
• Leaving fixed ends fixed violates the fixed↔free mapping.

Common Pitfalls:

• Forgetting to change internal hinges to internal supports.
• Mixing up slope/deflection analogies for conjugate beam shear/reactions.

Final Answer:

A two-span conjugate beam of the same length with end pins/rollers kept as pins/rollers, fixed ends (if any) changed to free ends, and an INTERNAL SIMPLE SUPPORT placed exactly at the real beam’s internal hinge