To determine The bending strain energy in the beam and the axial strain energy in each of the two posts. Explanation Given information: All members are made of aluminum and have a square cross section 50 mm by 50 mm. Assume the posts only support an axial load. E a l = 70 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: Beam section: Show the free body diagram of the beam section 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 ( 3 ) − 8 × 3 × 3 2 = 0 B y = 12 kN 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 − 8 × 3 = 0 (1) Substitute 12 kN for B y in Equation (1). A y + 12 − 24 = 0 A y = 12 kN 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 (2) Here, M is the moment in the beam and I is the moment of inertia of the beam. Region 0 ≤ x ≤ 3 m from left end A : Consider a section at a distance x from support A . 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 − A y ( x ) + 8 × x × x 2 = 0 (3) Substitute 12 kN for A y in Equation (3). M − 12 ( x ) + 4 x 2 = 0 M = 12 x − 4 x 2 Moment of inertia of the section: Determine the moment of inertia of the square section using the formula. I = b d 3 12 (4) Here, b is width of section and d is depth of section. Substitute 50 mm for b and 50 mm for d in Equation (4). I = 50 × 50 3 12 = 520 , 833.33 mm 4 × 10 − 12 m 4 1 mm 4 = 520 , 833.33 × 10 − 12 m 4 Substitute ( 12 x − 4 x 2 ) for M , 70 GPa for E , and 520 , 833.33 × 10 − 12 m 4 for I in Equation (2), and integrate. ( U i ) b = ∫ 0 3 m ( 12 x − 4 x 2 ) 2 d x 2 E I = 10 6 N 2 2 × 70 GPa × 10 9 N/m 2 1 GPa × 520 , 833

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ISBN: 2810014920922

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 14.2, Problem 14.22P

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The bending strain energy in the beam and the axial strain energy in each of the two posts.

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Chapter 14.2, Problem 14.21P

Chapter 14.2, Problem 14.23P

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