# Third Edition MECHANICS OF MATERIALS - site. OF MATERIALS Third Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER

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MECHANICS OF MATERIALSThird EditionFerdinand P. BeerE. Russell Johnston, Jr.John T. DeWolfLecture Notes:J. Walt OlerTexas Tech UniversityCHAPTER 2002 The McGraw-Hill Companies, Inc. All rights reserved. 7 Transformations of Stress and Strain 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 2Transformations of Stress and StrainIntroductionTransformation of Plane StressPrincipal StressesMaximum Shearing StressExample 7.01Sample Problem 7.1Mohrs Circle for Plane StressExample 7.02Sample Problem 7.2General State of StressApplication of Mohrs Circle to the Three- Dimensional Analysis of StressYield Criteria for Ductile Materials Under Plane StressFracture Criteria for Brittle Materials Under Plane StressStresses in Thin-Walled Pressure Vesselscontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 3Introduction The most general state of stress at a point may be represented by 6 components,),, :(Notestresses shearing,,stresses normal,,xzzxzyyzyxxyzxyzxyzyx Same state of stress is represented by a different set of components if axes are rotated. The first part of the chapter is concerned with how the components of stress are transformed under a rotation of the coordinate axes. The second part of the chapter is devoted to a similar analysis of the transformation of the components of strain.contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 4Introduction Plane Stress - state of stress in which two faces of the cubic element are free of stress. For the illustrated example, the state of stress is defined by .0,, and xy zyzxzyx State of plane stress occurs in a thin plate subjected to forces acting in the midplane of the plate. State of plane stress also occurs on the free surface of a structural element or machine component, i.e., at any point of the surface not subjected to an external force.contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 5Transformation of Plane Stress Consider the conditions for equilibrium of a prismatic element with faces perpendicular to the x, y, and x axes.contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 6Transformation of Plane Stress The equations may be rewritten to yieldcontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 7Principal Stressescontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 8Principal Stresses The previous equations are combined to yield parametric equations for a circle, 2222222wherexyyxyxaveyxavexRR Principal stresses occur on the principal planes of stress with zero shearing stresses.o22minmax,90by separated angles twodefines :Note22tan22yxxypxyyxyx contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 9Maximum Shearing StressMaximum shearing stress occurs for avex 245by fromoffset and 90by separated angles twodefines :Note22tan2oo22maxyxavepxyyxsxyyxR contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 10Example 7.01For the state of plane stress shown, determine (a) the principal panes, (b) the principal stresses, (c) the maximum shearing stress and the corresponding normal stress.SOLUTION: Find the element orientation for the principal stresses from yxxyp22tan Determine the principal stresses from22minmax,22xyyxyx Calculate the maximum shearing stress with22max2xyyx 2yx contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 11Example 7.01SOLUTION: Find the element orientation for the principal stresses from 1.233,1.532333.1105040222tanpyxxyp 6.116,6.26p Determine the principal stresses from 2222minmax,40302022 xyyxyxMPa30MPa70minmaxMPa10MPa40MPa50xxyxcontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 12Example 7.01MPa10MPa40MPa50xxyx210502yxave The corresponding normal stress isMPa20 Calculate the maximum shearing stress with 2222max40302 xyyxMPa50max 45 ps 6.71,4.18scontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 13Sample Problem 7.1A single horizontal force P of 150 lb magnitude is applied to end D of lever ABD. Determine (a) the normal and shearing stresses on an element at point H having sides parallel to the x and yaxes, (b) the principal planes and principal stresses at the point H.SOLUTION: Determine an equivalent force-couple system at the center of the transverse section passing through H. Evaluate the normal and shearing stresses at H. Determine the principal planes and calculate the principal stresses.contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 14Sample Problem 7.1SOLUTION: Determine an equivalent force-couple system at the center of the transverse section passing through H. inkip5.1in10lb150inkip7.2in18lb150lb150xMTP Evaluate the normal and shearing stresses at H. 421441in6.0in6.0inkip7.2in6.0in6.0inkip5.1JTcIMcxyyksi96.7ksi84.80 yyx contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 15Sample Problem 7.1 Determine the principal planes and calculate the principal stresses. 119,0.6128.184.8096.7222tanpyxxyp 5.59,5.30p 2222minmax,96.7284.80284.8022 xyyxyxksi68.4ksi52.13minmaxcontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 16Mohrs Circle for Plane Stress With the physical significance of Mohrs circle for plane stress established, it may be applied with simple geometric considerations. Critical values are estimated graphically or calculated. For a known state of plane stressplot the points X and Y and construct the circle centered at C. xyyx ,,2222xyyxyxave R The principal stresses are obtained at A and B.yxxypave R22tanminmax,The direction of rotation of Ox to Oa is the same as CX to CA.contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 17Mohrs Circle for Plane Stress With Mohrs circle uniquely defined, the state of stress at other axes orientations may be depicted. For the state of stress at an angle with respect to the xy axes, construct a new diameter XY at an angle 2 with respect to XY. Normal and shear stresses are obtained from the coordinates XY.contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 18Mohrs Circle for Plane Stress Mohrs circle for torsional loading:JTcxyyx 0 0 xyyxJTc0, xyyxAPAPxyyx2 Mohrs circle for centric axial loading:contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 19Example 7.02For the state of plane stress shown, (a) construct Mohrs circle, determine (b) the principal planes, (c) the principal stresses, (d) the maximum shearing stress and the corresponding normal stress.SOLUTION: Construction of Mohrs circle MPa504030MPa40MPa302050MPa2021050222CXRFXCFyxavecontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 20Example 7.02 Principal planes and stresses5020max CAOCOAMPa70max 5020min BCOCOBMPa30min 1.53230402tanppCPFX 6.26pcontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 21Example 7.02 Maximum shear stress 45ps 6.71sRmaxMPa 50max ave MPa 20 contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 22Sample Problem 7.2For the state of stress shown, determine (a) the principal planes and the principal stresses, (b) the stress components exerted on the element obtained by rotating the given element counterclockwise through 30 degrees.SOLUTION: Construct Mohrs circle MPa524820MPa8026010022222FXCFRyxavecontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 23Sample Problem 7.2 Principal planes and stresses4.6724.220482tanppCFXFclockwise7.33 p5280max CAOCOA5280max BCOCOAMPa132max MPa28min contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 24Sample Problem 7.26.52sin526.52cos52806.52cos52806.524.6760180XKCLOCOLKCOCOKyxyx Stress components after rotation by 30oPoints X and Y on Mohrs circle that correspond to stress components on the rotated element are obtained by rotating XY counterclockwise through 602MPa3.41MPa6.111MPa4.48yxyxcontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 25Stresses in Thin-Walled Pressure Vesselscontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 26Stresses in Thin-Walled Pressure Vesselscontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 27Stresses in Thin-Walled Pressure Vessels Cylindrical vessel with principal stresses1 = hoop stress2 = longitudinal stress tprxrpxtFz11 220 Hoop stress: 212222220tprrprtFx Longitudinal stress:contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf7 - 28Stresses in Thin-Walled Pressure Vessels Spherical pressure vessel:tpr221 Mohrs circle for in-plane transformations reduces to a point0constantplane)-max(in21 Maximum out-of-plane shearing stresstpr4121max contents.pptcontents.pptcontents.ppt

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