Conjugate beam method: Identify the correct conjugate beam for a two-span real beam that contains an internal hinge. Use the standard mapping rules: fixed end ↔ free end; pin/roller support at an end → pin/roller at the same end; an internal hinge in the real beam → an internal simple support in the conjugate beam. Choose the correct option.

Given data

• Real (actual) beam: two spans with an internal hinge.
• Task: state the correct conjugate beam support configuration.

Concept / Approach
Conjugate Beam Theorems: shear in the conjugate beam corresponds to slope in the real beam; moment in the conjugate corresponds to deflection in the real.Therefore, conjugate beam supports are chosen so that their shear/moment conditions mirror the real beam’s slope/deflection conditions at the corresponding locations.

Support-mapping rules (standard)

• Real fixed end → Conjugate free end (real: θ = 0 and v = 0 ⇒ conjugate: V̄ = 0 and M̄ = 0).
• Real end pin/roller → Conjugate end pin/roller (real: v = 0, θ free ⇒ conjugate: M̄ = 0, V̄ free).
• Real internal hinge → Conjugate internal simple support. This provides a vertical reaction in the conjugate beam (allowing a jump in shear = slope discontinuity in the real beam) while keeping conjugate bending moment continuous (deflection continuous in the real beam).

Step-by-step reasoning
1) The two-span length is unchanged in the conjugate beam.2) If the real beam has end pins/rollers, these remain pins/rollers in the conjugate beam because real v = 0 ⇒ M̄ = 0 at those ends.3) Any real fixed end would become a free end of the conjugate beam (both V̄ and M̄ must be zero there).4) At the real internal hinge, slope can be discontinuous but deflection is continuous. To mirror this, the conjugate beam must have an internal simple support, which introduces a vertical reaction (causing a shear jump = slope jump in the real beam) while preserving moment continuity across the section (deflection continuity in the real beam).

Why the other options are wrong
Option B: Replacing the internal hinge by a fixed connection in the conjugate beam would force shear continuity (no slope jump) and typically impose incorrect moment conditions.Option C: Removing the internal support at that location gives no shear reaction in the conjugate beam, so you cannot model a slope discontinuity of the real beam.Option D: Converting end simple supports to free ends violates the real→conjugate mapping (end pins/rollers stay pins/rollers).

Common pitfalls

• Confusing real internal moment release (hinge) with a hinge in the conjugate beam. In the conjugate beam, that location must be modeled as an internal simple support, not a rigid joint nor “no support”.
• Forgetting that fixed ↔ free mapping applies only at ends, not at interior releases.

Final Answer
Place an internal simple support in the conjugate beam at the real beam’s internal hinge; keep end pins/rollers as pins/rollers; convert any real fixed end to a free end.