Explanation Given information: The L-shaped frame is made from two segments, each of length L and flexural stiffness EI . Method adopted: Castigliano’s theorem. Explanation: Castigliano’s theorem to determine the slope for the beam is as follows: Δ = ∫ 0 L M ( ∂ M ∂ P ) d x E I (1) Here, Δ is displacement of the point caused by the real loads acting on the beam, P is an external force of variable magnitude applied to the beam at the point and in the direction of Δ , 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, 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 an external force P at the point where the displacement is to be determined. Region CB 0 ≤ x < L from top: Show the free body diagram of the beam for the region 0 ≤ x < L as in Figure 1. 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 (2) SolveEquation (2). M 1 = w ( x 1 ) 2 2 Region AB 0 < x ≤ L from point B : 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 2 = 0 M 2 − w × L × L 2 + P ( x 2 ) = 0 (3) Solve Equation (3). M 2 = w L 2 2 − P x 2 Calculation: Tabulate the calculated values as in Table 1. Limit M ∂ M ∂ P M ( P = 0 ) 0 ≤ x < L M 1 = w ( x 1 ) 2 2 ∂ M 1 ∂ P = 0 M 1 = w ( x 1 ) 2 2 L < x ≤ 2 L M 2 = w L 2 2 − P x 2 ∂

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Axial Load

When a bar of the uniform cross-section is acted by a load, such that the load acts along the longitudinal axis of the bar, such kinds of loadings are known as axial loadings. In general terms, a load is an external force acting on a component. An axial …

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Chapter 14.10, Problem 14.143P

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The vertical displacement of point B.

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Chapter 14.10, Problem 14.142P

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The vertical displacement of point B .

Solution Summary: The author explains Castigliano's theorem to determine the slope for the beam.

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The vertical displacement of point B.

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