Explanation Calculation: Show the free body diagram of the beam as in Figure 1. Calculate the vertical reaction at point A by taking moment about point B . ∑ M B = 0 − R A L + ( 1 2 w 0 L ) × L 3 = 0 − R A L = − w 0 L 2 6 R A = w 0 L 6 Calculate the moment due to the reaction at A as below: M 1 = w 0 L 6 × L = w 0 L 2 6 Calculate the M 1 E I value; Substitute w 0 L 2 6 for M 1 . M 1 E I = w 0 L 2 6 E I = w 0 L 2 6 E I Show the M 1 E I diagram in below Figure 2. Calculate the area ( a ) using the relation: A 1 = 1 2 ( L A B ) ( M 1 E I ) Substitute L for L A B and w 0 L 2 6 E I for ( M 1 E I ) . A 1 = 1 2 ( L ) ( w 0 L 2 6 E I ) = w 0 L 3 12 E I Calculate the moment due to the UVL load: M 2 = − w 0 L 2 ( L 3 ) = − w 0 L 2 6 Calculate the M 2 E I value; Substitute − w 0 L 2 6 for M 1 . M 2 E I = − w 0 L 2 6 E I = − w 0 L 2 6 E I Show the ( M 2 E I ) diagram in below Figure 3. Calculate the area ( A 2 ) using the relation: A 2 = 1 4 ( M 2 E I ) L Substitute − w 0 L 2 6 E I for ( M 2 E I ) . A 2 = 1 4 ( − w 0 L 2 6 E I ) L = − w 0 L 3 24 E I Calculate the tangential deviation of B with respect to A using the relation: t B / A = A 1 × L × ( 1 3 × L ) + A 2 ( L 5 ) Substitute w 0 L 3 12 E I for A 1 and − w 0 L 3 24 E I for A 2 . t B / A = w 0 L 3 12 E I × ( 1 3 × L ) + ( − w 0 L 3 24 E I ) ( L 5 ) = w 0 L 4 36 E I − w 0 L 4 120 E I = 7 w 0 L 4 360 E I Calculate slope ( θ A ) at point A using the relation: θ A = − t B / A L Substitute 7 w 0 L 4 360 E I for t B / A . θ A = − 7 w 0 L 4 360 E I L = − 7 w 0 L 3 360 E I Show deflection of the beam and reference tangent in below Figure 4. Let point k be the location of largest downward deflection. Show the point k the location of largest downward deflection in below Figure 5. Calculate the moment due to the reaction at A : M k 1 = w 0 L 6 × x m = w 0 L x m 6 Calculate the M k 1 E I value: Substitute w 0 L x m 6 for M 1 . M k 1 E I = w 0 L x m 6 E I = w 0 L x m 6 E I Show the ( M k 1 E I ) diagram in below Figure 6. Calculate the area ( A k 1 ) using the relation: A k 1 = 1 2 ( x m ) M k 1 E I Substitute w 0 L x m 6 E I for ( M k 1 E I )

7th Edition

ISBN: 9780100257061

Author: BEER

Publisher: YUZU

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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 9.6, Problem 142P

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The magnitude (ym) and location of the largest downward deflection (xm).

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Chapter 9.6, Problem 141P

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The magnitude ( y m ) and location of the largest downward deflection ( x m ) .

Solution Summary: The author calculates the vertical reaction at point A by taking moment about point B.

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The magnitude (ym) and location of the largest downward deflection (xm).

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