--- title: "Boundary conditions real to conjugate beam" subject: "Theory of Structures I" chapter: 4 key: "semester_4/theory_of_structures_i:conjugate-boundary-conditions" source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/conjugate-boundary-conditions site: BCT Engineering --- # Boundary conditions real to conjugate beam Theory of Structures I · Chapter 4: Deflection of Beams - Asked in: 2077 Chaitra, 4 marks - Video: [3 Boundary Conditions of a beam](https://www.youtube.com/watch?v=b-HkdI-sySU) — NITIANS ACADEMY - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/conjugate-boundary-conditions - Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-4 ## Question Explain how the boundary conditions are changed while converting a real beam to a conjugate beam with reference to the conjugate beam theorems. ## Short answer In the conjugate beam, slope equals shear and deflection equals moment. Thus a real fixed end (zero slope and deflection) becomes a conjugate free end (zero shear and moment). A real free end becomes fixed. A real simple support remains simple. An interior support becomes an internal hinge, and an internal hinge becomes an interior roller. ## Answer In the conjugate beam method, a real beam is replaced by a conjugate beam so that **slope = shear** and **deflection = bending moment** in the conjugate beam. For this to hold, the support conditions of the conjugate beam must reproduce the boundary conditions of the real beam. | Condition in real beam | Slope $\theta$ | Deflection $y$ | Needed in conjugate | Conjugate support | |---|---|---|---|---| | Fixed end | 0 | 0 | $V' = 0$, $M' = 0$ | **Free end** | | Free end | $\ne 0$ | $\ne 0$ | $V' \ne 0$, $M' \ne 0$ | **Fixed end** | | Simple (pin/roller) end | $\ne 0$ | 0 | $V' \ne 0$, $M' = 0$ | **Simple (pin/roller) end** | | Interior support | continuous, $\ne 0$ | 0 | $V'$ continuous, $M' = 0$ | **Internal hinge** | | Internal hinge | discontinuous | $\ne 0$ | $V'$ jumps, $M' \ne 0$ | **Interior roller support** | ```text Real: fixed ----------- free simple --- interior --- simple Conjugate: free ----------- fixed simple --- hinge ------ simple ``` The conjugate beam must be statically stable and determinate, which these rules guarantee when the real beam is determinate. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A fixed end in the real beam has theta = 0 and y = 0. What support does the conjugate beam have there, and why? A. Fixed end, because the real beam is fixed B. Simple support, because y = 0 means M' = 0 only C. Internal hinge, because the slope is continuous D. Free end, because V' = 0 and M' = 0 are needed Answer: D. Slope = shear and deflection = moment in the conjugate beam. Zero slope and deflection need zero shear and zero moment, which only a free end gives. (Key point: Fixed end becomes free end) 2. A free end of the real beam has non-zero slope and deflection. What does the conjugate beam have at that end? A. A free end B. A roller support C. A fixed end, which can carry shear and moment D. An internal hinge Answer: C. Non-zero slope and deflection need non-zero V' and M', so the conjugate end is fixed. (Key point: Free end becomes fixed end) 3. At an interior support of a continuous real beam, the slope is continuous and y = 0. What is provided in the conjugate beam? A. Internal hinge B. Interior roller support C. Fixed support D. Free end Answer: A. y = 0 requires M' = 0 and a continuous slope requires continuous V', which is exactly an internal hinge. (Key point: Interior support becomes internal hinge) 4. Why does a simple (pin/roller) end of the real beam stay a simple end in the conjugate beam? A. Because the slope is zero there B. Because y = 0 (M' = 0) while the slope is non-zero (V' is non-zero) C. Because both slope and deflection are non-zero there D. Because the real beam is statically determinate Answer: B. A simple end has zero deflection but non-zero slope, so the conjugate end needs M' = 0 and V' not zero, which is a simple support. 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