APPLIED ELASTICITY 27.2 CHAPTER TWENTY-SEVEN Nr , N0 p q Qx , Qy Nr , N r rx , ry r, t T Mtxy u, v, w V W w x, y, z X, Y, Z Z ! x , y , z r , r , , z xy , yz , zx " "x , "y , "z "r , " xy , yz , zx r , z r , z , rz normal forces per unit length in radial and tangential directions in polar co-ordinates, N (lbf) pressure, MPa (psi) load per unit length, kN/m (lbf/in) shearing forces parallel to z-axis per unit length of sections of a plate perpendicular to x and y axis, N/m (lbf/in) radial and tangential shearing forces, N (lbf ) radius, m (in) radii of curvature of the middle surface of a plate in xz and yz planes polar co-ordinates time, s temperature, 8C tension of a membrane, kN/m (lbf/in) twist of surface components or displacements, m (in) strains energy weight, N (lbf ) displacement, m (in) displacement of a plate in the normal direction, m (in) deflection, m (in) rectangular co-ordinates, m (in) body forces in x; y; z directions, N (lbf ) section modulus in bending, cm3 (in3 ) density, kN/m3 (lbf/in3 ) angular speed, rad/s stress, MPa (psi) normal components of stress parallel to x, y, and z axis, MPa (psi) radial and tangential stress, MPa (psi) normal stress components in cylindrical co-ordinates, MPa (psi) shearing stress, MPa (psi) shearing stress components in rectangular co-ordinates, MPa (psi) unit elongation, m/m (in/in) unit elongation in x, y, and z direction, m/m (in/in) radial and tangential unit elongation in polar co-ordinates shearing strain shearing strain components in rectangular co-ordinate shearing strain in polar co-ordinate shearing stress components in cylindrical co-ordinates, MPa (psi) Poisson’s ratio stress function angular deflection, deg e ¼ "x ỵ "y ỵ "z ẳ "r ỵ " ỵ "z e ẳ "x ỵ "y ỵ "z ẳ volume expansion shearing components in cylindrical co-ordinates Note: and with subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website APPLIED ELASTICITY APPLIED ELASTICITY Particular 27.3 Formula STRESS AT A POINT (Fig 27-1) The stress at a point due to force ÁF acting normal to an area dA (Fig 27-1b) Stress ¼ ¼ lim A ! F A 27-1ị where F ẳ force acting normal to the area ÁA ÁA ¼ an infinitesimal area of the body under the action of F x ẳ lim Fx Ax 27-2aị xy ẳ lim Fy Ax 27-2bị xz ẳ lim For stresses acting on the part II of solid body cut out from main body in x, y and z directions, Fig 27-1b ÁFz ÁAx ð27-2cÞ ÁAx ! ÁAx ! ÁAx ! Similarly the stress components in xy and xz planes can be written and the nine stress components at the point O in case of solid body made of homogeneous and isotropic material x yz xy y xz yz zx zy z ð27-3Þ Fig 27-1c shows the stresses acting on the faces of a small cube element cut out from the solid body F4 F5 a F1 Part I Part II ∆Fy ∆A a N ∆F o F8 a F6 y F2 F3 F7 (a) A solid body subject to action of external forces z ∆Fz a y F2 F3 F1 Part II ∆Fx x F8 F7 (b) An infineticimal area ∆A of Part II of a solid body under the action of force ∆F at σy τyz τzy dy σ z o τzx τyx τxy τxz σx x dz dx z (c) Stresses acting on the faces of a small cube element cut out from the solid body FIGURE 27-1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website APPLIED ELASTICITY 27.4 CHAPTER TWENTY-SEVEN Particular Formula Summing moments about x, y and z axes, it can be proved that the cross shears are equal xy ¼ yx ; yz ¼ zy ; All nine components of stresses can be expressed by a single equation ij ¼ lim Fj Ai Ai ! zx ẳ xz 27-4ị 27-5ị where i ẳ 1; 2; and j ẳ 1; 2; The FNx , FNy , and FNz unknown components of the resultant stress on the plane KLM of elemental tetrahedron passing through point O (Fig 27-2) FNx ẳ x cos N; x ỵ xy cos N; y þ xz cos N; z FNy ¼ yx cos N; x ỵ y cos N; y ỵ yz cos N; z FNz ẳ zx cos N; x ỵ zy cos N; y ỵ z cos N; z The unknown components of resultant stress FNx , FNy and FNz in terms of direction cosines l, m and n (Fig 27-4) y xy Fz FNx ẳ x l ỵ xy m ỵ zx n FNy ẳ yz l ỵ y m ỵ yx n FNz ẳ zx l ỵ zy m ỵ z n Surface area KLM = A TN N (normal to KLM) FNy τzx σz τxz Fx FNz ho’ τzy FNx K x o τyz Fy l ¼ cos ¼ cos N; x; m ¼ cos ¼ cos N; y, n ẳ cos ẳ cos N; z, l s ỵ m2 ỵ n2 ẳ lị 02 ỵ m0 ị2 ỵ n0 ị2 ẳ yx y y L TN = stress vector in N direction Fbx, Fby, Fbz = Body forces in x, y and z - direction z σy+ FIGURE 27-2 The state of stress at O of an elemental tetrahedron + τyz y σx M Tx’ τx’y’ σz τxz τzx L γ τxy o τyx z’ β o’ h α τzy τyz N ∂τyz ∂y dy τ zx τxz ∂τzy ∂σy dy ∂y ∂τyx + τyx dy σz ∂τ τxy+ xy dx ∂y ∂x τzy ∂τxzσx + + τzy ∂z dz + τxz ∂x dx ∂τzy τxy o dy + τzx ∂z dz ∂σz σz+ ∂z dz τ τyz dz yx σy y’ σx ð27-7Þ where the direct cosines are M σx ð27-6Þ x’ σx’ ∂σx ∂x dx x dx K x τz’x’ σ y z FIGURE 27-3 Small cube element removed from a solid body showing stresses acting on all faces of the body in x, y and z directions z FIGURE 27-4 Tx0 , resolved into x0 , x0 y0 and x0 z0 stress components Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website APPLIED ELASTICITY 27.5 APPLIED ELASTICITY Particular Formula cos ¼ l ¼ angle between x axis and Normal N cos ¼ m ¼ angle between y axis and Normal N The resultant stress FN on the plane KLM cos ¼ n ¼ angle between z axis and Normal N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 FN ẳ FNx ỵ FNy ỵ FNz ð27-8Þ The normal stress which acts on the plane under consideration N ẳ FNx cos ỵ FNy cos ỵ FNz cos The shear stress which acts on the plane under consideration N ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F N À 2 n ð27-8aÞ ð27-8bÞ Equations (27-1), (27-2) and (27-7) to (27-8) can be expressed in terms of resultant stress vector as follows (Fig 27-2) The resultant stress vector at a point ÁFN ð27-9aÞ ÁA where TN coincides with the line of action of the resultant force ÁFn TN ¼ lim ÁA ! 27-9bị 27-9cị TNz ẳ zx l ỵ zy m þ z n The resultant stress vector TNx ¼ x l ỵ xy m ỵ xz n TNy ẳ xy l ỵ y m ỵ zy n The resultant stress vector components in x, y and z directions ð27-9dÞ TN ẳ q 2 TNx ỵ TNy ỵ TNz 27-9eị where the direction cosines are cosTN ; xị ẳ TNx =jTN j, cosTN ; yị ẳ TNy =jTN j, cosTN ; zị ẳ TNz =jTN j The normal stress which acts on the plane under consideration N ¼ jTN j cosTN ; Nị 27-9f ị N ẳ TNx cosN; xị ỵ TNy cosN; yị ỵ TNz cosN; zị 27-9gị The shear stress which acts on the plane under consideration N ẳ jTN j sinTN ; Nị 27-10aị q T N 2 N 27-10bị N ẳ The angle between the resultant stress vector TN and the normal to the plane N cosTN ; Nị ẳ cosTN ; xị cosN; xị ỵ cosTN ; yị cosN; yị ỵ cosTN ; zÞ cosðN; zÞ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð27-10cÞ APPLIED ELASTICITY 27.6 CHAPTER TWENTY-SEVEN Particular Formula EQUATIONS OF EQUILIBRIUM @x @ xy @ xz ỵ ỵ ỵ Fbx ẳ @x @y @z 27-11aị @y @ yz @ yx ỵ ỵ ỵ Fby ẳ @y @z @x 27-11bị @z @ zx @ zy ỵ ỵ ỵ Fbz ẳ @z @x @y The equations of equilibrium in Cartesian coordinates which includes body forces in three dimensions (Fig 27-3) ð27-11cÞ where Fbx , Fby and Fbz are body forces in x, y and z directions @x @ xy ỵ ỵ Fbx ẳ @x @y 27-11dị @y @ yx ỵ ỵ Fby ẳ @y @x 27-11eị TN ẳ iTNx ỵ jTNy ỵ kTNz 27-12aị TN ẳ iTN x þ jTN y þ kTN z ð27-12bÞ N ẳ il ỵ jm ỵ kn Stress equations of equilibrium in two dimensions ð27-12cÞ TRANSFORMATION OF STRESS The vector form of equations for resultant-stress vectors TN and TN for two different planes and the outer normals N and N in two different planes 0 0 N ẳ il ỵ jm ỵ kn 27-12dị where i, j and k are unit vectors in x, y and z directions, respectively Substituting Eqs (27-9b), (27-9c), (27-9d) and (27-9e) in Eqs (27-13), equations for TN , N and TN , N TN N ẳ TNx l ỵ TNy m ỵ TNz n 27-13aị TN N ẳ TNx l ỵ TNy m0 ỵ TNz n0 The projections of the resultant-stress vector TN onto the outer normals N and N 27-13bị TN N ẳ x l ỵ y m2 ỵ z n2 ỵ 2 xy lm ỵ 2 yz mn ỵ 2 zx nl 0 27-14aị 0 TN N ẳ x ll ỵ y mm ỵ z nn ỵ xy ẵlm ỵ ml ỵ yz ẵmn0 ỵ nm0 ỵ zx ẵnl ỵ ln0 The relation between TN , N and TN , N TN ÁN ¼ TN ÁN By coinciding outer normal N with x0 , N with y0 , and N with z0 individually respectively and using Eqs (27-14a) to (27-14b), x0 , y0 and z0 can be obtained (Fig 27-4) 27-14bị x0 ẳ Tx0 x0 ẳ x cos2 x0 ; xị ỵ y cos2 x0 ; yị 27-15ị ỵ z cos2 x0 ; zị ỵ 2 xy cosx0 ; xị cosx0 ; yị ỵ 2 yz cosx0 ; yị cosx0 ; zị ỵ 2 zx cosx0 ; zị cosx0 ; xÞ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð27-15aÞ APPLIED ELASTICITY APPLIED ELASTICITY Particular 27.7 Formula y0 ¼ Ty0 Áy ¼ y cos2 ðy0 ; yị ỵ z cos2 y0 ; zị ỵ x cos2 y0 ; xị ỵ 2 yz cosy0 ; yị cosy0 ; zị ỵ 2 zx cosy0 ; zị cosz0 ; xị þ 2 xy cosðy0 ; xÞ cosðy0 ; yÞ ð27-15bÞ z0 ẳ Tz0 z0 ẳ z cos2 z0 ; zị ỵ x cos2 z0 ; xị ỵ y cos2 z0 ; yị ỵ 2 zx cosz0 ; zị cosz0 ; xị ỵ 2 xy cosz0 ; xị cosz0 ; yị ỵ 2 yz cosz0 ; yÞ cosðz0 ; zÞ By selecting a plane having an outer normal N coincident with the x0 and a second plane having an outer normal N coincident with the y0 and utilizing Eq (27-14b) which was developed for determining the magnitude of the projection of a resultant stress vector on to an arbitrary normal can be used to determine x0 y0 Following this procedure and by selecting N and N coincident with the y0 and z0 , and z0 and x0 axes, the expression for y0 z0 and z0 x0 can be obtained The expressions for x0 y0 , y0 z0 and z0 x0 are ð27-15cÞ x0 y0 ẳ Tx0 y0 ẳ x cosx0 ; xị cosy0 ; xị ỵ y cosx0 ; yị cosy0 ; yị ỵ z cosx0 ; zị cosy0 ; zị ỵ xy ẵcosx0 ; xị cosy0 ; yị ỵ cosx0 ; yị cosy0 ; xị ỵ yz ẵcosx0 ; yị cosy0 ; zị ỵ cosx0 ; zị cosy0 ; yị ỵ zx ẵcosx0 ; zị cosy0 ; xị ỵ cosx0 ; xị cosy0 ; zị 27-16aị y0 z0 ẳ Ty0 z0 ẳ y cosy0 ; yị cosz0 ; yị ỵ z cosy0 ; zị cosz0 ; zị ỵ x cosy0 ; xị cosz0 ; xị ỵ yz ẵcosy0 ; yị cosz0 ; zị ỵ cosy0 ; zị cosz0 ; yị ỵ zx ẵcosy0 ; zị cosz0 ; xị ỵ cosy0 ; xị cosz0 ; zị ỵ xy ẵcosy0 ; xị cosz0 ; yị ỵ cosy0 ; yị cosz0 ; xị 27-16bị z0 x0 ẳ Tz0 x0 ẳ z cosz0 ; zị cosx0 ; zị ỵ x cosz0 ; xị cosx0 ; xị ỵ y cosz0 ; yị cosx0 ; yị ỵ zx ẵcosz0 ; zị cosx0 ; xị ỵ cosz0 ; xị cosx0 ; zị ỵ xy ẵcosz0 ; xị cosx0 ; yị ỵ cosz0 ; yị cosx0 ; xị ỵ yz ẵcosz0 ; yị cosx0 ; zị ỵ cosz0 ; zÞ cosðx0 ; yÞ ð27-16cÞ Equations (27-15a) to (27-15c) and Eqs (27-16a) to (27-16c) can be used to determine the six Cartesian components of stress relative to the Oxyz coordinate system to be transformed into a different set of six Cartesian components of stress relative to an Ox0 y0 z0 coordinate system Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website APPLIED ELASTICITY 27.8 CHAPTER TWENTY-SEVEN Particular For two-dimensional stress fields, the Eqs (27-15a) to (27-15c) and (27-16a) to (27-16c) reduce to, since z ¼ zx ¼ yz ¼ z0 coincide with z and is the angle between x and x0 , Eqs (27-15a) to (27-15c) and Eqs (27-16a) to (27-16c) y TNy K TNz M O L x0 ẳ x cos2 ỵ y sin2 ỵ 2 xy sin cos ẳ x þ y x À y þ cos 2 þ xy sin 2 2 27-17aị y0 ẳ y cos2 þ x sin2 À 2 xy sin cos ẳ N y ỵ x y x ỵ cos 2 À xy sin 2 2 ð27-17bÞ x0 y0 ¼ y cos sin À x cos sin TN, N TNx N Formula ỵ xy cos2 À sin2 Þ x z FIGURE 27-5 The stress vector TN ẳ y x sin 2 ỵ xy cos 2 z0 ¼ z0 x0 ¼ y0 z0 ẳ 27-17cị 27-17dị PRINCIPAL STRESSES By referring to Fig 27-5, where TN coincides with outer normal N, it can be shown that the resultant stress components of TN in x, y and z directions TNx ¼ N l Substituting Eqs (27-9b) to (27-9d) into (27-18), the following equations are obtained x l ỵ yx m ỵ zx n ¼ N l TNy ¼ N m ð27-18Þ TNz ¼ N n xy l ỵ y m ỵ xy n ẳ N m 27-19ị xz l ỵ yz m ỵ z n ¼ N n Eq (27-19) can be written as x N ịl ỵ yx m ỵ zx ẳ xy l ỵ y N ịm ỵ zy ẳ 27-20ị xz l ỵ yz m ỵ z N ịn ẳ From Eq (27-20), direction cosine (N, x) is obtained and putting this in determinant form Putting the determinator of determinant into zero, the non-trivial solution for direction cosines of the principal plane is cosðN; xị ẳ x N yx zx y À N zy xy xz yz z À N ð27-21Þ x À N ... zx zy z ð27 -3? ? Fig 27-1c shows the stresses acting on the faces of a small cube element cut out from the solid body F4 F5 a F1 Part I Part II ∆Fy ∆A a N ∆F o F8 a F6 y F2 F3 F7 (a) A solid... in three dimensions (Fig 27 -3) ð27-11cÞ where Fbx , Fby and Fbz are body forces in x, y and z directions @x @ xy ỵ ỵ Fbx ẳ @x @y 27-11dị @y @ yx ỵ ỵ Fby ẳ @y @x 27-11eị TN ẳ iTNx ỵ jTNy ỵ kTNz... ELASTICITY 27.6 CHAPTER TWENTY-SEVEN Particular Formula EQUATIONS OF EQUILIBRIUM @x @ xy @ xz ỵ ỵ ỵ Fbx ẳ @x @y @z 27-11aị @y @ yz @ yx ỵ ỵ ỵ Fby ẳ @y @z @x 27-11bị @z @ zx @ zy ỵ ỵ ỵ Fbz ¼ @z