CHAPTER 7 PR ROBLEM 7 1 7 4 ksi 3 ksi 708 8 ksi Foor the given sttate of stress determine thee normal and shearing stressses exerted onn the oblique face of the shhaded trianguular element shown s Use a method of anaalysis based on o the equilibrrium of that ellement as waas done in the derivations of Sec 7 1A SOLUTION F 0 A 8 A cos 20 2 cos 20 3 A cos 20 sin 20 2 3 A sin 200 cos 20 4 A sin 20 sin 20 2 0 8cos 2 20 3cos 20 sin 20 3 sin 20 cos 20 4sin 2 20 0 9 46 ksi F 0 A 8 A cos 200 sin 20 3A A cos 20 cos 20 3 A sin 200 sin 20 4A A sin 20 cos 200 0 8coos 20 sin 20 3 cos2 20 sin 2 20 4sin 4 20 cos 200 1 013 ksi PRO OPRIETARY MAT TERIAL Copyrigght 2015 McGrraw Hill Educatioon This is proprietary material soleely for authorizedd instructor use Not authorized a for salee or distribution inn any manner Thiis document may not n be copied scannned duplicated forwarded distribbuted or posted on a website w in whole or part 1027 60 MPa PRO OBLEM 7 2 2 For th he given statee of stress dettermine the noormal and sheearing stressess exerted on the oblique face off the shaded triangular t elem ment shown Use U a methodd of analysis based d on the equilibbrium of that element e as waas done in the derivations off Sec 7 1A 608 90 MPa M SO OLUTION F 0 A 90 9 A sin 30 coss 30 90 A cos 30 sin 30 60 A cos 30 ccos 30 0 180sin 30 cos c 30 60 coos 2 30 32 9 3 M Pa F 0 A 900 A sin 30 sin 30 3 90 A cos 30 cos 30 60 A cos 30 sinn 30 0 90 cos 2 30 sin 2 30 60 cos 30 sin 30 71 0 7 M Pa PRO OPRIETARY MATERIAL Copyriight 2015 McG Graw Hill Educattion This is proprrietary material soolely for authorizedd instructor use Not authorized for salle or distribution in i any manner Thhis document may not be copied scaanned duplicated forwarded distributed or posted on a website in wholee or part 1028 PROBLEM M 7 3 10 ksi 6 ksi 758 For the giveen state of sttress determiine the normaal and sheariing stresses exerted on thhe oblique faace of the shaaded triangulaar element shoown Use a method of annalysis based on the equilibbrium of that element as was w done in the derivationns of Sec 7 1A A 4 ksi SOLUTION F 0 A 4 A cos15 sin15 100 A cos15 cos115 6 A sin15 sin15 4 A sin15 cos155 0 4 coos15 sin15 10 cos 2 15 6sin 2 15 4sin15 4 cos155 10 93 1 ksi F 0 A 4 A cos15 cos15 10 A cos15 sin 15 6 A sin15 cos15 4 A sin15 sin155 0 4 ccos2 15 sin 2 15 10 6 cos15 sin15 0 536 0 ksi PRO OPRIETARY MAT TERIAL Copyrigght 2015 McGrraw Hill Educatioon This is proprietary material soleely for authorizedd instructor use Not authorized a for salee or distribution inn any manner Thiis document may not n be copied scannned duplicated forwarded distribbuted or posted on a website w in whole or part 1029 PROBLEM P 7 4 80 MPa M 40 MPa 558 For F the given state s of stress determine thhe normal andd shearing streesses exerted on o the obliquee face of the shaded trianggular element shown Use a method of analysis a based on the equilibbrium of that element e as was w done in thee derivations of o Sec 7 1A SO OLUTION Streesses Areas Forces F 0 0 A 80 A cos 55 cos555 40 A sin 55 sin 55 0 80 cos 2 55 40sin 2 55 0 5521 MPa F 0 0 A 80 A cos 55 sin 55 5 40 A sin 55 cos 55 56 4 5 MPa 120 1 cos 55 sin 55 PRO OPRIETARY MATERIAL Copyriight 2015 McG Graw Hill Educattion This is proprrietary material soolely for authorizedd instructor use Not authorized for salle or distribution in i any manner Thhis document may not be copied scaanned duplicated forwarded distributed or posted on a website in wholee or part 1030 PROBLEM 7 5 40 MPa 35 MPa For the given state of stress determine a the principal planes b the principal stresses 60 MPa SOLUTION x 60 MPa y 40 MPa xy 35 MPa a tan 2 p 2 xy x y 2 35 3 50 60 40 2 p 74 05 b max min x y 2 p 37 0 53 0 2 x y 2 xy 2 2 60 40 60 40 2 35 2 2 50 36 4 MPa max 13 60 MPa min 86 4 MPa PROPRIETARY MATERIAL Copyright 2015 McGraw Hill Education This is proprietary material solely for authorized instructor use Not authorized for sale or distribution in any manner This document may not be copied scanned duplicated forwarded distributed or posted on a website in whole or part 1031 PROBLEM 7 6 10 ksi For the given state of stress determine a the principal planes b the principal stresses 2 ksi 3 ksi SOLUTION x 2 ksi a tan 2 p 2 xy x y y 10 ksi xy 3 ksi 2 3 0 750 2 10 2 p 36 87 b max min x y 2 p 18 4 108 4 2 x y 2 xy 2 2 2 10 2 10 2 3 2 2 6 5 ksi max 11 00 ksi min 1 000 ksi PROPRIETARY MATERIAL Copyright 2015 McGraw Hill Education This is proprietary material solely for authorized instructor use Not authorized for sale or distribution in any manner This document may not be copied scanned duplicated forwarded distributed or posted on a website in whole or part 1032 PROBLEM 7 7 30 MPa For the given state of stress determine a the principal planes b the principal stresses 150 MPa 80 MPa SOLUTION x 150 MPa a tan 2 p 2 xy x y y 30 MPa xy 80 MPa 2 80 MPa 1 33333 MPa 150 MPa 30 MPa 2 p 53 130 and 126 870 p 26 6 and 63 4 b max min x y 2 x y 2 xy 2 2 150 MPa 30 MPa 150 MPa 30 MPa 2 80 MPa 2 2 90 MPa 100 MPa max 190 0 MPa min 10 00 MPa PROPRIETARY MATERIAL Copyright 2015 McGraw Hill Education This is proprietary material solely for authorized instructor use Not authorized for sale or distribution in any manner This document may not be copied scanned duplicated forwarded distributed or posted on a website in whole or part 1033 PROBLEM 7 8 12 ksi 8 ksi For the given state of stress determine a the principal planes b the principal stresses 18 ksi SOLUTION y 12 ksi x 18 ksi a tan 2 p 2 xy x …