Explanation Given information: EI is constant. Method adopted: Castigliano’s theorem. Castigliano’s theorem to determine the slope for the beam as follows: θ = ∫ 0 L M ( ∂ M ∂ M ′ ) d x E I (1) Here, θ is the slope of the tangent θ at a point on the elastic curve is to be determined, M is internal moment in the beam, expressed as a function of x and caused by both the force P and the actual loads on the beam, M ′ is the external moment, E is modulus of elasticity of a member, and I is moment of inertia of the cross-sectional area about the neutral axis. Internal moment in the beam ( M ): Apply external moment M ′ at the point where the slope is to be determined. Determine the reactions: Show the free-body diagram of the entire beam as in Figure 1. Moment about point A : Determine the vertical reaction at point B by taking moment about point A . ∑ M A = 0 w × 2 L × 2 L 2 − B y ( L ) − M ′ = 0 B y ( L ) = w × 2 L × 2 L 2 − M ′ B y = 2 w L − M ′ L Along the horizontal direction: Determine the horizontal reaction at point A by resolving the horizontal component of forces. ∑ F x = 0 A x = 0 Along the vertical direction: Determine the vertical reaction at point A by resolving the vertical component of forces. ∑ F y = 0 A y + B y − w × 2 L = 0 (2) Substitute ( 2 w L − M ′ L ) for B y in Equation (2). A y + 2 w L − 2 w L − M ′ L = 0 A y = M ′ L Region 0 ≤ x < L : Show the free body diagram of the beam for the region 0 ≤ x < L as in Figure 2. Moment about the section: Determine the moment at the section by taking moment about the section. ∑ M x 1 = 0 M 1 + w × x 1 × x 1 2 = 0 M 1 = − w x 1 2 2 Region L < x ≤ 2 L : Show the free body diagram of the beam for the region L < x ≤ 2 L as in Figure 3. Moment about the section: Determine the moment at the section by taking moment about the section. ∑ M x 2 = 0 − M 2 + A y ( x 2 ) − w × x 2 × x 2 2 = 0 (3) Substitute M ′ L for A y in Equation (3). − M 2 + M ′ L ( x 2 ) − w ( x 2 ) 2 2 = 0 M 2 = M ′ x 2 L − w x 2 2 2 Calculation of slope: Tabulate the calculated values as in Table 1

9th Edition

ISBN: 9789810694364

Author: Russell C Hibbeler

Publisher: Pearson Education

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Slope and Deflection

The slope in a deflected beam is described as an angle in radians made by the tangent of the section with the original axis of the beam. The deflection at any point on the axis of the beam is defined as the lateral displacement of the point before and af…

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Chapter 14.10, Problem 14.145P

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The slope at point B.

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Chapter 14.10, Problem 14.144P

Chapter 14, Problem 14.146RP

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