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Unformatted text preview: Sec_2.7.qxd 9/25/08 208 11:42 AM Page 208 CHAPTER 2 Axially Loaded Members Problem 2.7-11 A block B is pushed against three springs by a force P
(see figure). The middle spring has stiffness k1 and the outer springs each
have stiffness k2. 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 U1 of the springs
when x 2s.
(c) Explain why the strain energy U1 is not equal to P /2, where Solution 2.7-11 Block pushed against three springs Force P0 required to close the gap:
P0 (a) FORCE-DISPLACEMENT DIAGRAM k1s (1) FORCE-DISPLACEMENT RELATION BEFORE GAP IS CLOSED
P k1x (0 x s)(0 P P0) (2) FORCE-DISPLACEMENT RELATION AFTER GAP IS CLOSED
All three springs are compressed. Total stiffness equals
k1 2k2. Additional displacement equals x s. Force
P equals P0 plus the force required to compress all three
springs by the amount x s.
P P0 (k1 2k2)(x s) 2k2)x k1s k1s (k1 P (k1 2k2)x P1 force P when x Substitute x
P1 2(k1 2k2s (x (b) STRAIN ENERGY U1 WHEN x 2k2s
s); (P P0) (3) U1 Area below force - displacement curve 2s 2s into Eq. (3):
k2)s 2s 1
P s + P0s + (P1
2 (4) P0)s P0s + 1
21 k1s2 + (k1 + k2)s2
U1 (2k1 k2)s2 ; (5) ...
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This note was uploaded on 12/22/2011 for the course MEEG 310 taught by Professor Staff during the Fall '11 term at University of Delaware.
- Fall '11