GATE CE 2016 Set 1 — Question 42
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Structural Engineering → Indeterminate Analysis & Influence Lines → Analysis of Beams
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Question
Two beams PQ (fixed at P and with a roller support at Q, as shown in Figure I, which allows vertical movement) and XZ (with a hinge at Y) are shown in the Figures I and II respectively. The spans of PQ and XZ are and respectively. Both the beams are under the action of uniformly distributed load () and have the same flexural stiffness, (where, and respectively denote modulus of elasticity and moment of inertia about axis of bending). Let the maximum deflection and maximum rotation be and , respectively, in the case of beam PQ and the corresponding quantities for the beam XZ be and , respectively.
📷 Figure: Figure I showing beam PQ with span L, fixed at P and guided roller at Q under UDL W
📷 Figure: Figure II showing beam XZ with span 2L, fixed at X and Z, with a hinge at Y under UDL W
Which one of the following relationships is true?Correct answer
(D) δₘₐₓ₁ = δₘₐₓ₂ and θₘₐₓ₁ = θₘₐₓ₂
Solution
Consider Beam PQ (Figure I): It is fixed at P and has a guided roller at Q. A guided roller prevents rotation but allows vertical translation. Thus, the boundary conditions are and .Consider Beam XZ (Figure II): It is fixed at both ends X and Z with a hinge at the center Y. Due to the symmetry of the structure and the loading (UDL over the entire span ), the slope at the center (hinge Y) must be zero (). Because , the segment XY of beam XZ behaves exactly like beam PQ. Both have a span , are fixed at one end, and have a zero-slope condition at the other end while being subjected to the same UDL . Since the governing differential equations and boundary conditions for beam PQ and segment XY are identical, their deflection and rotation profiles will be identical. Therefore, and .
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