Explanation Determine the reactions: Entire beam: Show the free body diagram of the entire beam as in Figure 1. Moment about the point A : Determine the vertical reaction at point B by taking moment about point A . ∑ M A = 0 B y ( L ) − 1 2 × w 0 × L 2 × 2 3 × L 2 − 1 2 × w 0 × L 2 × ( 1 3 × L 2 + L 2 ) = 0 (1) Along the vertical direction: Determine the vertical reaction at point A by resolving the vertical component of forces. ∑ F y = 0 A y + B y − 1 2 × w 0 × L 2 − 1 2 × w 0 × L 2 = 0 (2) Along the horizontal direction: Determine the horizontal reaction at point A by resolving the horizontal component of force. ∑ F x = 0 A x = 0 Show the calculation of reaction as follows: Conclusion: Solve Equation (1). B y ( L ) − w 0 L 2 12 − w 0 L 4 × ( 4 L 6 ) = 0 B y = w 0 L 12 + 2 w 0 L 12 B y = w 0 L 4 Substitute w 0 L 4 for B y in Equation (2). A y + w 0 L 4 − w 0 L 4 − w 0 L 4 = 0 A y = w 0 L 4 Strain energy due to Bending: Determine the strain energy in the beam due to bending using the equation. ( U i ) b = ∫ 0 L M 2 d x 2 E I (3) Here, M is the moment in the beam and I is the moment of inertia of the beam. Region 0 ≤ x < L 2 from left end A : Consider a section at a distance x from support A . Due to symmetry of the beam section, determine the strain energy up-to half of the beam and multiply the value by twice. Show the free-body diagram of the section as in Figure 2. Moment about the section: Determine the moment at section by taking moment about the section. ∑ M x = 0 M + [ 1 2 × 2 w 0 L × x × x × 1 3 × x ] − A y ( x ) = 0 (4) Show the calculation of values as follows: Conclusion: Substitute w 0 L 4 for A y in Equation (4). M + w 0 x 3 3 L − w 0 L 4 ( x ) = 0 M = w 0 L x 4 − w 0 x 3 3 L Substitute L / 2 for L and ( w 0 L x 4 − w 0 x 3 3 L ) for M in Equation (3), and integrate

9th Edition

ISBN: 9789810694364

Author: Russell C Hibbeler

Publisher: Pearson Education

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Strain energy

The term strain indicates the variation in an object's length, area, and volume when the object is subjected to an external force. The external force may be of normal, shear, moment, torque, etc. The strain in an object depends on the types of external f…

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Chapter 14.2, Problem 14.17P

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The bending strain energy in the simply supported beam.

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Chapter 14.2, Problem 14.16P

Chapter 14.2, Problem 14.18P

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The bending strain energy in the simply supported beam.

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The bending strain energy in the simply supported beam.

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