Explanation Given information: A load of 5 kN is applied to the center of the A992 steel beam. The steel beam is supported on two springs, each having a stiffness of k = 8 MN/m . Moment of inertia of the beam is I = 4.5 ( 10 6 ) mm 4 . Consideration: For A992 steel beam, The Young’s modulus of the beam is E = 200 GPa . Assumption: Consider that the right side force as positive and left side as negative in horizontal components of forces. Consider that the upward force as positive and downward as negative in vertical components of forces. Consider that the clockwise moment as negative and anticlockwise moment as positive. Determine the reactions: Entire beam: Show the free body diagram of the beam as in Figure 1. Moment about the point A : Determine the force in the spring at point B by taking moment about point A . ∑ M A = 0 ( F s p ) B ( 6 ) − 5 × 3 = 0 ( F s p ) B = 2.5 kN Along the vertical direction: Determine the force in the spring at point A by resolving the vertical component of forces. ∑ F y = 0 ( F s p ) A + ( F s p ) B − 5 = 0 (1) Substitute 2.5kNfor ( F s p ) B in Equation (1). ( F s p ) A + 2.5 − 5 = 0 ( F s p ) A = 2.5 kN Strain energy in the spring: Determine the strain energy in the spring using the Equation: ( U i ) s p = 1 2 k δ s p 2 (2) Here, k is stiffness and δ s p is spring deformation. Determine the spring deformation using the relation. δ s p = F s p k (3) Show the calculation of strain energy as follows: Conclusion: Substitute 2.5kN for F s p and 8 MN/m for k in Equation (3). δ s p = 2.5 kN × 1,000 N 1 kN 8 MN/m 2 × 10 6 N/m 2 1 MN/m 2 = 0.0003125 m Substitute 8 MN/m for k and 0.0003125 m for δ s p in Equation (2). ( U i ) s p = 1 2 × 8 MN/m 2 × 10 6 N/m 2 1 MN/m 2 × ( 0.0003125 ) 2 = 0

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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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 14.21P

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The strain energy in each of the springs and the bending strain energy in the beam.

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Chapter 14.2, Problem 14.20P

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