CHAPTER 1 PROBLEM 1 1 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown Knowing that the average normal stress must not exceed 175 MPa in rod AB and 150 MPa in rod BC determine the smallest allowable values of d1 and d2 SOLUTION a Rod AB P 40 30 70 kN 70 103 N b AB P AAB d1 4P P 4P 2 d d12 4 1 AB 4 70 103 22 6 10 3 m 175 106 d1 22 6 mm Rod BC P 30 kN 30 103 N BC d2 P ABC 4P BC P 4P 2 d d 22 4 2 4 30 103 15 96 10 3 m 150 106 d 2 15 96 mm PROPRIETARY MATERIAL 2013 The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation A student using this manual is using it without permission PROBLEM 1 2 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown Knowing that d1 50 mm and d 2 30 mm find the average normal stress at the midsection of a rod AB b rod BC SOLUTION a Rod AB P 40 30 70 kN 70 103 N A AB b 4 d12 4 50 2 1 9635 103 mm 2 1 9635 10 3 m 2 70 103 P 35 7 106 Pa A 1 9635 10 3 AB 35 7 MPa Rod BC P 30 kN 30 103 N A BC 4 d 22 4 30 2 706 86 mm 2 706 86 10 6 m 2 30 103 P 42 4 106 Pa 6 A 706 86 10 BC 42 4 MPa PROPRIETARY MATERIAL 2013 The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation A student using this manual is using it without permission PROBLEM 1 3 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown Determine the magnitude of the force P for which the tensile stress in rod AB is twice the magnitude of the compressive stress in rod BC SOLUTION AAB AB 50 2 1963 5 mm 2 4 P P AAB 1963 5 ABC BC 75 2 4417 9 mm 2 4 2 120 P ABC 240 P 4417 9 Equating AB to 2 BC P 2 240 P 1963 5 4417 9 P 112 9 kN PROPRIETARY MATERIAL 2013 The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation A student using this manual is using it without permission PROBLEM 1 4 In Prob 1 3 knowing that P 160 kN determine the average normal stress at the midsection of a rod AB b rod BC PROBLEM 1 3 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown Determine the magnitude of the force P for which the tensile stress in rod AB is twice the magnitude of the compressive stress in rod BC SOLUTION a Rod AB P 160 kN tension AAB AB b 2 d AB 4 4 50 2 1963 5 mm 2 P 160 103 N AAB 1963 5 10 6 m 2 AB 81 5 MPa Rod BC F 160 2 120 80 kN i e 80 kN compression ABC BC 2 d BC 4 4 75 2 4417 9 mm 2 F 80 103 N ABC 4417 9 10 6 m 2 BC 18 11 MPa PROPRIETARY MATERIAL 2013 The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation A student using this manual is using it without permission PROBLEM 1 5 Two steel plates are to be held together by means of 16 mmdiameter high strength steel bolts fitting snugly inside cylindrical brass spacers Knowing that the average normal stress must not exceed 200 MPa in the bolts and 130 MPa in the spacers determine the outer diameter of the spacers that yields the most economical and safe design SOLUTION At each bolt location the upper plate is pulled down by the tensile force Pb of the bolt At the same time the spacer pushes that plate upward with a compressive force Ps in order to maintain equilibrium Pb Ps For the bolt b Fb 4Pb Ab db2 or Pb For the spacer s Ps 4Ps As d s2 db2 or Ps 4 4 bdb2 s d s2 db2 Equating Pb and Ps 4 bdb2 ds 4 s d s2 db2 b 1 d s b 200 1 130 16 d s 25 2 mm PROPRIETARY MATERIAL 2013 The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation A student using this manual is using it without permission PROBLEM 1 6 Two brass rods AB and BC each of uniform diameter will be brazed together at B to form a nonuniform rod of total length 100 m which will be suspended from a support at A as shown Knowing that the density of brass is 8470 kg m3 determine a the length of rod AB for which the maximum normal stress in ABC is minimum b the corresponding value of the maximum normal stress SOLUTION Areas AAB ABC From geometry Weights 4 4 15 mm 2 176 71 mm 2 176 71 10 6 m 2 10 mm 2 78 54 mm 2 78 54 10 6 m 2 b 100 a WAB g AAB AB 8470 9 81 176 71 10 6 a 14 683 a WBC g ABC BC 8470 9 81 78 54 10 6 100 a 652 59 6 526 a Normal stresses At A PA WAB WBC 652 59 8 157a A At B a PA 3 6930 106 46 160 103 a AAB PB WBC 652 59 6 526a B 1 2 PB 8 3090 106 83 090 103 a ABC Length of rod AB The maximum stress in ABC is minimum when A B or 4 6160 106 129 25 …