02-06ChapGere.0003

# 02-06ChapGere.0003 - A block B is pushed against three...

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(b) S TRAIN ENERGY U 1 WHEN x 5 2 s (5) (c) S TRAIN ENERGY U 1 IS NOT EQUAL TO For (This quantity is greater than U 1 .) U 1 5 area under line OAB. under a straight line from O to B , which is larger than U 1 . Thus, is not equal to the strain energy because the force-displacement relation is not linear. P d 2 P d 2 5 area d 5 2 s : P d 2 5 1 2 P 1 (2 s ) 5 P 1 s 5 2( k 1 1 k 2 ) s 2 P d 2 U 1 5 (2 k 1 1 k 2 ) s 2 5 k 1 s 2 1 ( k 1 1 k 2 ) s 2 5 1 2 P 0 s 1 P 0 s 1 1 2 ( P 1 2 P 0 ) s 5 P 0 s 1 1 2 P 1 s U 1 5 Area below force-displacement curve 146 CHAPTER 2 Axially Loaded Members Force P 0 required to close the gap: P 0 5 k 1 s (1) F ORCE - DISPLACEMENT RELATION BEFORE GAP IS CLOSED P 5 k 1 x (0 # x # s )(0 # P # P 0 ) (2) F ORCE - DISPLACEMENT RELATION AFTER GAP IS CLOSED All three springs are compressed. Total stiffness equals k 1 1 2 k 2 . Additional displacement equals x 2 s . Force P equals P 0 plus the force required to compress all three springs by the amount x 2 s . ( x \$ s ); ( P \$ P 0 ) (3) P 1 5 force P when x 5 2 s Substitute x 5 2 s into Eq. (3): P 1 5 2( k 1 1 k 2 ) s (4) (a) F ORCE - DISPLACEMENT DIAGRAM P 5 ( k 1 1 2 k 2 ) x 2 2 k 2 s Ê 5 k 1 s 1 ( k 1 1 2 k 2 ) x 2 k 1 s 2 2 k 2 s P 5 P 0 1 ( k 1 1 2 k 2 )( x 2 s ) P B x k 2 k 1 k 2 s Force P P 1 P 0 0 s 2 s Displacement x Eq (2) Eq (3) B Slope = k 1 + 2k 2 Slope = k 1 A Problem 2.7-11
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Unformatted text preview: A block B is pushed against three springs by a force P (see figure). The middle spring has stiffness k 1 and the outer springs each have stiffness k 2 . Initially, the springs are unstressed and the middle spring is longer than the outer springs (the difference in length is denoted s ). (a) Draw a force-displacement diagram with the force P as ordinate and the displacement x of the block as abscissa. (b) From the diagram, determine the strain energy U 1 of the springs when x 5 2 s . (c) Explain why the strain energy U 1 is not equal to P d /2, where d 5 2 s . Solution 2.7-11 Block pushed against three springs P B x k 2 k 1 k 2 s = + + A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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## This note was uploaded on 09/20/2009 for the course COE 3001 taught by Professor Armanios during the Spring '08 term at Georgia Tech.

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