To determine The slope at end A of the overhang beam ( θ A ) . The deflection at end D of the overhang beam ( Δ D ) . Explanation Given information: The flexural rigidity of the beam is E I . The uniformly distributed load act at entire span of the bam is w . The length of the beam ( L ) is 3 a . The distance of segment L A B is 2 a . The distance of segment L B C is a . The distance of segment ( x ) is a . Calculation: Show the free body and uniform distributed load on the beam is equivalent to the sum of the separate loadings and elastic curve for each separate loading as in Figure 1. Refer to Figure 1. Calculate the moment at B : M B = w × a × a 2 = w a 2 2 Calculate the slope at A ( θ A ) 1 for uniform distributed load ( w ) using the formula: ( θ A ) 1 = w L A B 3 24 E I (1) Substitute 2 a for L A B in Equation (1). ( θ A ) 1 = w ( 2 a ) 3 24 E I = 8 w a 3 24 E I = w a 3 3 E I Calculate the deflection at D ( Δ D ) 1 for uniform distributed load ( w ) using the formula: ( Δ D ) 1 = 5 w L A B 4 384 E I (2) Substitute 2 a for L A B in Equation (2). ( Δ D ) 1 = 5 w ( 2 a ) 4 384 E I = 80 w a 4 384 E I = 5 w a 4 24 E I ↑ Calculate the slope at A ( θ A ) 2 for moment ( M B ) using the formula: ( θ A ) 2 = M B L A B 6 E I (3) Substitute w a 2 2 for M B and 2 a for L A B in Equation (3). ( θ A ) 2 = w a 2 2 × 2 a 6 E I = w a 3 6 E I Calculate the deflection at D ( Δ D ) 2 for moment ( M B ) using the formula: ( Δ D ) 2 = M B x 6 E I L A B ( L A B 2 − x 2 ) (4) Substitute w a 2 2 for M B , a for x , and 2 a for L A B in Equation (4)

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 12.5, Problem 12.86P

To determine

The slope at end A of the overhang beam (θA) .

The deflection at end D of the overhang beam (ΔD) .

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Chapter 12.5, Problem 12.85P

Chapter 12.5, Problem 12.87P

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Chapter 12 Solutions

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The slope at end A of the overhang beam ( θ A ) . The deflection at end D of the overhang beam ( Δ D ) .

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The slope at end A of the overhang beam (θA) . The deflection at end D of the overhang beam (ΔD) .

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Chapter 12 Solutions

The slope at end A of the overhang beam (θA) .

The deflection at end D of the overhang beam (ΔD) .