Explanation The following figure shows the load acting on a simply supported beam. Figure (1) Write the expression for the net moment at point A . R 2 l − F ( l + a ) + M A = 0 R 2 = 1 l ( F l + F a − M A ) Here, the reaction at point B is R 2 , the length of the portion AB is l , applied load at point C is F and the moment at point A is M A . Write the expression for the net moment at point B . R 1 l + F a − M A = 0 R 1 = 1 l ( M A − F a ) Here, the reaction at point A is R 1 . Refer to Table 3-1 “Singularity (Macaulay) functions”. Write the expression for load intensity acting on the beam. q = R 1 〈 x 〉 − 1 − M A 〈 x 〉 − 2 + R 2 〈 x − l 〉 − 1 − F 〈 x − ( l + a ) 〉 − 1 (III) Here, the load intensity on the beam is q . Write the expression for the moment. M = ∫ ∫ q d x (IV) Here, the net bending moment is M . From Equation (III) and Equation (IV). M = ∫ ∫ R 1 〈 x 〉 − 1 − M A 〈 x 〉 − 2 + R 2 〈 x − l 〉 − 1 − F 〈 x − ( l + a ) 〉 − 1 = R 1 〈 x 〉 1 − M A 〈 x 〉 0 + R 2 〈 x − l 〉 1 − F 〈 x − ( l + a ) 〉 1 (V) Here, the net bending moment for the portion AB is M A B . Write the equation for the deflection across the beam. E I d 2 y d x 2 = M (VI) Here, the modulus of elasticity is E and the deflection is y . From Equation (V) and Equation (VI). E I d 2 y d x 2 = R 1 〈 x 〉 1 − M A 〈 x 〉 0 + R 2 〈 x − l 〉 1 − F 〈 x − ( l + a ) 〉 1 (VII) Integrate Equation (VIII) with respect to x . E I d y d x = R 1 〈 x 〉 2 2 − M A 〈 x 〉 1 + R 2 〈 x − l 〉 2 2 − F 〈 x − ( l + a ) 〉 1 + C 1 (VIII) Integrate the Equation (IX) with respect to x . E I y = R 1 〈 x 〉 3 6 − M A 〈 x 〉 2 2 + R 2 〈 x − l 〉 3 6 − F 〈 x − ( l + a ) 〉 1 + C 1 x + C 2 (IX) Conclusion: Substitute 0 for y , 0 for x in Equation (IX). E I × 0 = R 1 〈 0 〉 3 6 − M A 〈 0 〉 2 2 + 0 + C 1 × 0 + C 2 0 = 0 + C 2 C 2 = 0 Substitute 0 for y , l for x and 0 for C 2 in Equation (IX)

Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)

10th Edition

ISBN: 9780073398204

Author: Richard G Budynas, Keith J Nisbett

Publisher: McGraw-Hill Education

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Chapter 4, Problem 61P

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The deflection equation at region AB.

The deflection equation for region BC.

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Chapter 4, Problem 60P

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Chapter 4 Solutions

Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)

Ch. 4 - The figure shows a torsion bar OA fixed at O,...Ch. 4 - For Prob. 41, if the simple support at point A...Ch. 4 - A torsion-bar spring consists of a prismatic bar,...Ch. 4 - An engineer is forced by geometric considerations...Ch. 4 - A bar in tension has a circular cross section and...Ch. 4 - Prob. 6P Ch. 4 - Prob. 7P Ch. 4 - Derive the equations given for beam 2 in Table A9...Ch. 4 - Derive the equations given for beam 5 in Table A9...Ch. 4 - The figure shows a cantilever consisting of steel...

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The deflection equation at region AB . The deflection equation for region BC .

Solution Summary: The author compares the deflection equations at region AB and region BC. The following figure shows the load acting on a simply supported beam.

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The deflection equation at region AB.

The deflection equation for region BC.

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Chapter 4 Solutions

The deflection equation at region AB . The deflection equation for region BC .

Solution Summary: The author compares the deflection equations at region AB and region BC. The following figure shows the load acting on a simply supported beam.

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The deflection equation at region AB. The deflection equation for region BC.

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Chapter 4 Solutions

The deflection equation at region AB.

The deflection equation for region BC.