02-06ChapGere.0001 - the middle bar is slightly shorter...

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Problem 2.7-9 A slightly tapered bar AB of rectangular cross section and length L is acted upon by a force P (see figure). The width of the bar varies uniformly from b 2 at end A to b 1 at end B . The thickness t is constant. (a) Determine the strain energy U of the bar. (b) Determine the elongation of the bar by equating the strain energy to the work done by the force P . Solution 2.7-9 Tapered bar of rectangular cross section 144 CHAPTER 2 Axially Loaded Members b 2 b 1 L A B P (a) S TRAIN ENERGY OF THE BAR From Appendix C: dx a bx 1 b ln ( a bx ) L 0 P 2 dx 2 Et b ( x ) P 2 2 Et L 0 dx b 2 ( b 2 b 1 ) x L (1) U [ N ( x ) ] 2 dx 2 E A ( x ) (Eq. 2-41) t B b 2 ( b 2 b 1 ) x L R A ( x ) tb ( x ) b ( x ) b 2 ( b 2 b 1 ) x L Apply this integration formula to Eq. (1): (b) E LONGATION OF THE BAR (E Q . 2-42) N OTE : This result agrees with the formula derived in Prob. 2.3-11. 2 U P PL Et ( b 2 b 1 ) ln b 2 b 1 U P 2 L 2 Et ( b 2 b 1 ) ln b 2 b 1 P 2 2 Et B L ( b 2 b 1 ) ln b 1 L ( b 2 b 1 ) ln b 2 R ( b 2 b 1 ) x L R R 0 L U P 2 2 Et B 1 ( b 2 b 1 )( 1 L ) ln B b 2 b 2 b 1 L A B P dx b(x) x Problem 2.7-10 A compressive load P is transmitted through a rigid plate to three magnesium-alloy bars that are identical except that initially
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Unformatted text preview: the middle bar is slightly shorter than the other bars (see figure). The dimensions and properties of the assembly are as follows: length L 5 1.0 m, cross-sectional area of each bar A 5 3000 mm 2 , modulus of elasticity E 5 45 GPa, and the gap s 5 1.0 mm. (a) Calculate the load P 1 required to close the gap. (b) Calculate the downward displacement d of the rigid plate when P 5 400 kN. (c) Calculate the total strain energy U of the three bars when P 5 400 kN. (d) Explain why the strain energy U is not equal to P d /2. ( Hint: Draw a load-displacement diagram.) L P s A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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