Explanation Given information: The rod has a circular cross section with a polar moment of inertia J and moment of inertia I . A vertical force P is applied at the point A . Consider only the strain energy due to bending. The material constants are modulus of elasticity is E and modulus of rigidity G . Explanation: Consider a section from free end of the circular rod. Show the free-body diagram of the section as in Figure 1. Geometry of the section: Show the free-body diagram of the geometry of the section as in Figure 2. Torsion about the section: Determine the torsion at the section by taking torsion about the section. ∑ T x = 0 T − P r ( 1 − cos θ ) = 0 (1) Moment about the section: Determine the moment at the section by taking moment about thesection. ∑ M x = 0 M − P r sin θ = 0 (2) Show the calculation of values as follows: Conclusion: Solve Equation (1). T = P r ( 1 − cos θ ) Solve Equation (2). M = P r sin θ Strain energy due to Bending: Determine the strain energy in the circular rod due to bending using the equation. ( U i ) b = ∫ 0 s M 2 d s 2 E I (3) Here, d s = r d θ , M is the moment in the beam, and I is the moment of inertia of the beam. Show the calculation of values as follows: Conclusion: Substitute P r sin θ for M and r d θ for d s in Equation (3), and integrate. ( U i ) b = ∫ 0 θ M 2 r d θ 2 E I = ∫ 0 π ( P r sin θ ) 2 r d θ 2 E I = P 2 r 3 2 E I ∫ 0 π ( sin 2 θ ) d θ = P 2 r 3 2 E I ∫ 0 π ( 1 − cos 2 θ 2 ) d θ ( U i ) b = P 2 r 3 4 E I ∫ 0 π ( 1 − cos 2 θ ) d θ = P 2 r 3 4 E I [ θ − sin 2 θ 2 ] 0 π = P 2 r 3 4 E I [ π − sin 2 ( π ) 2 − 0 + 0 ] = π P 2 r 3 4 E I Strain energy due to Torsion: Determine the strain energy in the circular rod due to torsion using the equation. ( U i ) T = ∫ 0 s T 2 d s 2 G J (4) Here, d s = r d θ , T is the torsion in the circular rod, and J is the polar moment of inertia. Show the calculation of values as follows: Conclusion: Substitute P r ( 1 − cos θ ) for T and r d θ for d s in Equation (4), and integrate

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Author: HIBBELER

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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 14.3, Problem 40P

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The vertical displacement at point C.

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Chapter 14.3, Problem 39P

Chapter 14.3, Problem 41P

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The vertical displacement at point C .

Solution Summary: The author explains how to determine the vertical displacement at point C. The free-body diagram shows the geometry of the section.

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The vertical displacement at point C.

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