d 0 = 6 in. d = 7 in. e = pd - pd 0 Ans. = 7-6 = in./in.

Transcription

1 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher n air-filled rubber ball has a diameter of 6 in. If the air pressure within it is increased until the ball s diameter becomes 7 in., determine the average normal strain in the rubber. d 0 6 in. d 7 in. e pd - pd in./in. pd thin strip of rubber has an unstretched length of 15 in. If it is stretched around a pipe having an outer diameter of 5 in., determine the average normal strain in the strip in. p(5 in.) e - 0 5p in.>in The rigid beam is supported b a pin at and wires and E. If the load P on the beam causes the end to be displaced 10 mm downward, determine the normal strain developed in wires E and. E 4 m P 3 E 7 3 (10) mm 3 m 2 m 2 m e E E mm>mm 4000 e mm>mm 1

2 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher. *2 4. The two wires are connected together at. If the force P causes point to be displaced horizontall, determine the normal strain developed in each wire. œ (300)(2) cos mm e e œ mm>mm P 2 5. The rigid beam is supported b a pin at and wires and E. If the distributed load causes the end to be displaced 10 mm downward, determine the normal strain developed in wires E and. 1.5 m E 2 m 2 m 3 m Since the vertical displacement of end is small compared to the length of member, the vertical displacement d of point, can be approimated b referring to the similar triangle shown in Fig. a d ; d 4 mm The unstretched lengths of wires and E are 1500 mm and E 2000 mm. e avg d mm>mm 1500 w e avg E d mm>mm E

3 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher Nlon strips are fused to glass plates. When moderatel heated the nlon will become soft while the glass stas approimatel rigid. etermine the average shear strain in the nlon due to the load P when the assembl deforms as indicated. P 5 mm 5 mm g tan - 1 a 2 b rad If the unstretched length of the bowstring is 35.5 in., determine the average normal strain in the string when it is stretched to the position shown. 18 in. 6 in. 18 in. Geometr: Referring to Fig. a, the stretched length of the string is in. verage Normal Strain: e avg in.>in

4 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher. *2 8. Part of a control linkage for an airplane consists of a rigid member and a fleible cable. If a force is applied to the end of the member and causes it to rotate b u 0.3, determine the normal strain in the cable. Originall the cable is unstretched. u P mm (400)(300) cos 90.3 e mm mm>mm mm 2 9. Part of a control linkage for an airplane consists of a rigid member and a fleible cable. If a force is applied to the end of the member and causes a normal strain in the cable of mm>mm, determine the displacement of point. Originall the cable is unstretched. u P mm + e (500) mm (300)(400) cos a a u p (0.4185) rad (u) 600( p )(0.4185) 4.38 mm mm 4

5 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher The corners and of the square plate are given the displacements indicated. etermine the shear strains at and. 16 mm 16 mm 16 mm 16 mm ppling trigonometr to Fig. a f tan - 1 a 13 p rad b a b rad a tan - 1 a 16 p rad b a b rad the definition of shear strain, g p 2-2f p 2-2(0.6823) rad g p 2-2a p 2-2(0.8885) rad 5

6 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher The corners and of the square plate are given the displacements indicated. etermine the average normal strains along side and diagonal. 16 mm 16 mm 16 mm 16 mm Referring to Fig. a, mm mm Thus, e avg mm>mm 2512 e avg mm>mm 32 6

7 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher. *2 12. The piece of rubber is originall rectangular. etermine the average shear strain g at if the corners and are subjected to the displacements that cause the rubber to distort as shown b the dashed lines. u 1 tan u rad 300 u 2 tan u rad mm g u 1 + u rad The piece of rubber is originall rectangular and subjected to the deformation shown b the dashed lines. etermine the average normal strain along the diagonal and side. 2(400) 2 + (3) mm f tan - 1 a b (300) 2 + (2) w tan - 1 a b a mm 2( ) 2 + ( ) 2-2( )( ) cos ( ) mm 2(300) 2 + (400) mm e e mm>mm (10-3 ) mm>mm 7

8 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher Two bars are used to support a load. When unloaded, is 5 in. long, is 8 in. long, and the ring at has coordinates (0, 0). If a load P acts on the ring at, the normal strain in becomes P 0.02 in.>in., and the normal strain in becomes P in.>in. etermine the coordinate position of the ring due to the load in. 8 in. P verage Normal Strain: œ + e 5 + (0.02)(5) 5.10 in. œ + e 8 + (0.035)(8) 8.28 in. Geometr: a in (9.2268)(8.28) cos u u cos in sin in. -( -a) -( ) in. -( ) -( ) in. 8

9 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher Two bars are used to support a load P. When unloaded, is 5 in. long, is 8 in. long, and the ring at has coordinates (0, 0). If a load is applied to the ring at,so that it moves it to the coordinate position (0.25 in., in.), determine the normal strain in each bar in. 8 in. P Geometr: a in. 2( ) 2 + ( ) in. 2( ) 2 + ( ) in. verage Normal Strain: e in.>in. e in.>in. 9

10 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher. *2 16. The square deforms into the position shown b the dashed lines. etermine the average normal strain along each diagonal, and. Side remains horizontal. Geometr: mm (53)(58) cos mm sin mm mm (53)( ) cos mm 8 mm verage Normal Strain: e mm>mm e mm>mm The three cords are attached to the ring at.when a force is applied to the ring it moves it to point, such that the normal strain in is P and the normal strain in is P. Provided these strains are small, determine the normal strain in. Note that and remain horizontal and vertical, respectivel, due to the roller guides at and. oordinates of ( cos u, sin u) oordinates of ( cos u + e cos u, sin u + e sin u) 2( cos u + e cos u) 2 + ( sin u + e sin u) 2 u 2cos 2 u(1 + 2e + e 2 ) + sin 2 u(1 + 2e + e 2 ) e e Since and are small, 21 + (2 e cos 2 u + 2e sin 2 u) Use the binomial theorem, ( (2 e cos 2 u + 2e sin 2 u)) ( 1 + e cos 2 u + e sin 2 u) Thus, e ( 1 + e cos 2 u + e sin 2 u) - e e cos 2 u + e sin 2 u 10

11 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher The piece of plastic is originall rectangular. etermine the shear strain g at corners and if the plastic distorts as shown b the dashed lines. 5 mm 4 mm Geometr: For small angles, a c rad mm b u rad 403 Shear Strain: (g ) a + b rad rad (g ) -(u + c) rad rad The piece of plastic is originall rectangular. etermine the shear strain g at corners and if the plastic distorts as shown b the dashed lines. 5 mm 4 mm 400 mm Geometr: For small angles, a c rad 403 b u rad 302 Shear Strain: (g ) -(a + b) rad rad (g ) u + c rad rad 11

12 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher. *2 20. The piece of plastic is originall rectangular. etermine the average normal strain that occurs along the diagonals and. Geometr: 5 mm 4 mm mm mm mm verage Normal Strain: 400 mm e e mm>mm mm>mm mm>mm mm>mm The force applied to the handle of the rigid lever arm causes the arm to rotate clockwise through an angle of 3 about pin. etermine the average normal strain developed in the wire. Originall, the wire is unstretched. 600 mm Geometr: Referring to Fig. a, the stretched length of the consine law, can be determined using 2(0.6 cos 45 ) 2 + (0.6 sin 45 ) 2-2(0.6 cos 45 )(0.6 sin 45 ) cos m verage Normal Strain: The unstretched length of wire is 0.6 m. We obtain e avg m>m

13 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher square piece of material is deformed into the dashed position. etermine the shear strain at. g Shear Strain: mm (g ) p p 15 mm mm rad mm mm square piece of material is deformed into the dashed parallelogram. etermine the average normal strain that occurs along the diagonals and mm 15 mm mm mm mm Geometr: (15.18)(15.24) cos mm (15.18)(15.24) cos mm verage Normal Strain: e e mm>mm mm>mm mm>mm mm>mm 13

14 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher. *2 24. square piece of material is deformed into the dashed position. etermine the shear strain at. g mm 15 mm mm mm mm (g ) p p rad The gu wire of a building frame is originall unstretched. ue to an earthquake, the two columns of the frame tilt u 2. etermine the approimate normal strain in the wire when the frame is in this position. ssume the columns are rigid and rotate about their lower supports. u 2 u 2 Geometr: The vertical displacement is negligible (1) 2 p m 180 (4) 2 p m m 1 m 4 m m m m verage Normal Strain: e m>m 14

15 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher The material distorts into the dashed position shown. etermine (a) the average normal strains along sides and and the shear strain g at F, and (b) the average normal strain along line E. 10 mm 15 mm 25 mm E 90 mm 75 mm 80 mm F Referring to Fig. a, E 2(90-75) mm mm mm f tan p rad rad When the plate deforms, the vertical position of point and E do not change ; 13.5 mm EE ; EE mm E 2(90-75) 2 + ( ) mm Thus, e avg >mm e avg mm>mm 80 e avg E E - E E mm>mm Referring to Fig. a, the angle at corner F becomes larger than 90 after the plate deforms. Thus, the shear strain is negative rad 15

16 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher The material distorts into the dashed position shown. etermine the average normal strain that occurs along the diagonals and F. The undeformed length of diagonals and F are F mm 10 mm 15 mm 25 mm E The deformed length of diagonals and F are 2( ) mm 90 mm 75 mm F 2(80-15) mm Thus, e avg >mm 80 mm F e avg F F - F F mm>mm *2 28. The wire is subjected to a normal strain that is defined b Pe - 2, where is in millimeters. If the wire has an initial length, determine the increase in its length. P e 2 d e d e -2 d e -2 d 0 - c 1 2 e-2 d 0 - c 1 2 e d 1 2 [1 - e-2 ] 16

17 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher The curved pipe has an original radius of 2 ft. If it is heated nonuniforml, so that the normal strain along its length is P0.05 cos u, determine the increase in length of the pipe. e 0.05 cos u e d 90 (0.05 cos u)(2 du) 0 u 2 ft cos u du [0.1[sin u] 0 ] ft Solve Prob if P 0.08 sin u. d 2 due 0.08 sin u e d 90 (0.08 sin u)(2 du) sin u du 0.16[-cos u] ft u 2 ft The rubber band has an unstretched length of 1 ft. If it is fied at and attached to the surface at point, determine the average normal strain in the band. The surface is defined b the function ( 2 ) ft, where is in feet. 2 Geometr: 1 ft a d 2 d b d 1 ft However d 2 then d 2 1 ft d 0 1 ft ln ft ft verage Normal Strain: e avg ft>ft

18 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher. *2 32. The bar is originall long when it is flat. If it is subjected to a shear strain defined b g 0.02, where is in meters, determine the displacement at the end of its bottom edge. It is distorted into the shape shown, where no elongation of the bar occurs in the direction. Shear Strain: d d tan g ; d tan (0.02 ) d 0 d tan (0.02 )d 0-50[ln cos (0.02)] The fiber has a length and orientation u. If its ends and undergo ver small displacements u and v, respectivel, determine the normal strain in the fiber when it is in position. v Geometr: 2( cos u - u ) 2 + ( sin u + ) 2 u u u ( sin u - u cos u) verage Normal Strain: e u ( sin u - u cos u) - 1 Neglecting higher terms u 2 and 2 e 1 + 2( sin u - u cos u) Using the binomial theorem: 1 2 R - 1 e sin u - 2u cos u sin u - u cos u 18

19 02 Solutions /6/10 1:45 PM Page Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl eist. No portion of this material ma be reproduced, in an form or b an means, without permission in writing from the publisher If the normal strain is defined in reference to the final length, that is, P n œ s - s lim a b p:p s instead of in reference to the original length, Eq. 2 2, show that the difference in these strains is represented as a œ second-order term, namel, P n -P n P n P œ n. e S - S S e - e œ S - S S - S - S S S 2 - S S - S S + S 2 S S S 2 + S 2-2 S S S S ( S - S)2 S S S - S S - S S S e e œ (Q.E.) 19