(a) Explanation Show the free body diagram of the beam as in Figure 1. Calculation: Calculate the vertical reaction at point A by taking moment about point B . ∑ M B = 0 ( w × L × L 2 ) − R A × L = 0 R A = ( w L 2 2 ) L R A = w L 2 Calculate the vertical reaction at point B by resolving the vertical component of forces. ∑ F y = 0 w L 2 − ( w L ) + R B = 0 R B = w L 2 Calculate the moment due to the reaction at A : M 1 = ( w L 2 ) × L 2 = w L 2 4 Calculate the M 1 E I value: Substitute w L 2 4 for M 1 . M 1 E I = w L 2 4 E I = w L 2 4 E I Show the M 1 diagram in below Figure 2. Calculate the area ( A 1 ) using the relation: A 1 = 1 2 ( L A C ) ( M 1 E I ) Substitute L 2 for L B C and w L 2 4 E I for ( M 1 E I ) . A 1 = 1 2 × L 2 × w L 2 4 E I = w L 3 16 E I Calculate the moment due to UDL in portion AC : M 2 = w × L 2 × ( L 2 ) 2 = w L 2 8 Calculate the M 2 E I value: Substitute w L 2 8 for M 1 . M 2 E I = − w L 2 8 E I = − w L 2 8 E I Show the ( M 2 E I ) diagram in below Figure 3. Calculate the area ( A 2 ) using the relation: A 2 = 1 3 ( L A C ) ( M 2 E I ) Substitute L 2 for L A C and − w L 2 8 E I for ( M 2 E I ) . A 2 = 1 3 × ( L 2 ) × ( − w L 2 8 E I ) = − w L 3 48 E I Calculate the moment due to reaction at point B: M 3 = w L 2 × L 2 = w L 2 4 Calculate the M 3 E I value: Substitute w L 2 4 for M 3 , and 2 E I for EI . ( M 3 E I ) = w L 2 4 2 E I = w L 2 8 E I Show the M 3 diagram in below Figure 4. Calculate the area ( A 3 ) using the relation: A 3 = 1 2 ( L C B ) ( M 3 E I ) Substitute ( L 2 ) for L C B and ( w L 2 8 E I ) for ( M 3 E I ) . A 3 = 1 2 ( L 2 ) ( w L 2 8 E I ) = w L 3 32 E I Calculate the moment ( M 4 ) due to UDL in portion CB : M 4 = w × L 2 × ( L 2 ) 2 = − w L 2 8 Calculate the ( M 4 E I ) value; Substitute w L 2 8 for M 1 and 2EI for EI (b) To determine Determine the slope θ B at point B . (c) To determine Find the deflection y A at point C .

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 139P

(a)

To determine

The slope θA at point A.

(b)

To determine

Determine the slope θB at point B.

(c)

To determine

Find the deflection yA at point C.

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Chapter 9.6, Problem 138P

Chapter 9.6, Problem 140P

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The slope θ A at point A .

Solution Summary: The author shows the free body diagram of the beam as in Figure 1. Calculate the vertical reaction at point A by taking moment about point B.

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The slope θA at point A.

Determine the slope θB at point B.

Find the deflection yA at point C.

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Chapter 9 Solutions