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.
-
AA 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
-
BA two-span conjugate beam where the internal hinge is replaced by a rigid (fixed) connection and all ends are changed to fixed supports
-
CA two-span conjugate beam with the internal hinge removed (no support provided at that section) while keeping all end supports unchanged
-
DA two-span conjugate beam where every end simple support is converted to a free end and the internal hinge remains a hinge in the conjugate beam
-
EA two-span conjugate beam where fixed ends remain fixed and the internal hinge is replaced by a roller at the far end only
Answer
Correct 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
Explanation
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:
Identify supports in the real beam (fixed/pinned/roller) and apply the mapping to each end.Locate the internal hinge in the real beam and replace it with an internal simple support in the conjugate beam.Keep end pins/rollers unchanged, convert fixed ends to free ends, and insert the internal support at the hinge location.Select the option that follows exactly this construction.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