Machine Design Databook 2010 Part 16 ppt

N r , N 0 normal forces per unit length in radial and tangential directions in polar co-ordinates, N (lbf) p pressure, MPa (psi) q load per unit length, kN/m (lbf/in) Q x , Q y shearing forces parallel to z-axis per unit length of sections of a plate perpendicular to x and y axis, N/m (lbf/in) N r , N  radial and tangential shearing forces, N (lbf ) r radius, m (in) r x , r y radii of curvature of the middle surface of a plate in xz and yz planes r,  polar co-ordinates t time, s T temperature, 8C tension of a membrane, kN/m (lbf/in) M txy twist of surface u, v, w components or displacements, m (in) V strains energy W weight, N (lbf) w displacement, m (in) displacement of a plate in the normal direction, m (in) deflection, m (in) x, y, z rectangular co-ordinates, m (in) X, Y, Z body forces in x; y; z directions, N (lbf) Z section modulus in bending, cm 3 (in 3 )  density, kN/m 3 (lbf/in 3 ) ! angular speed, rad/s  stress, MPa (psi)  x ,  y ,  z normal components of stress parallel to x, y, and z axis, MPa (psi)  r ,   radial and tangential stress, MPa (psi)  r ,   ,  z normal stress components in cylindrical co-ordinates, MPa (psi)  shearing stress, MPa (psi)  xy ,  yz ,  zx shearing stress components in rectangular co-ordinates, MPa (psi) " unit elongation, m/m (in/in) " x , " y , " z unit elongation in x, y, and z direction, m/m (in/in) " r , "  radial and tangential unit elongation in polar co-ordinates  shearing strain  xy ,  yz ,  zx shearing strain components in rectangular co-ordinate  r ,  z shearing strain in polar co-ordinate  r ,  z ,  rz shearing stress components in cylindrical co-ordinates, MPa (psi)  Poisson’s ratio  stress function angular deflection, deg e ¼ " x þ " y þ " z ¼ " r þ "  þ " z e ¼ " x þ " y þ " z ¼ 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 27.2 CHAPTER TWENTY-SEVEN 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 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) For stresses acting on the part II of solid body cut out from main body in x, y and z directions, Fig. 27-1b 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 Stress ¼  ¼ lim ÁA !0 ÁF ÁA ð27-1Þ where ÁF ¼ force acting normal to the area ÁA ÁA ¼ an infinitesimal area of the body under the action of F  x ¼ lim ÁA x !0 ÁF x ÁA x ð27-2aÞ  xy ¼ lim ÁA x !0 ÁF y ÁA x ð27-2bÞ  xz ¼ lim ÁA x !0 ÁF z ÁA x ð27-2cÞ  x  xy  xz  yz  y  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. Particular Formula Part I o o Part II Part II a a a y x x y N z z dy dx dz a F 1 F 1 F 2 F 2 F 3 F 3 F 4 F 5 F 6 (a) A solid body subject to action of external forces (b) An infineticimal area ∆A of Part II of a solid body under the action of force ∆F at 0 (c) Stresses acting on the faces of a small cube element cut out from the solid body F 7 F 7 ∆F z ∆F y ∆F x σ x σ z σ y τ xy τ zx τ zy τ xz τ yz τ yx ∆F ∆A F 8 F 8 FIGURE 27-1 APPLIED ELASTICITY 27.3 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 Summing moments about x, y and z axes, it can be proved that the cross shears are equal All nine components of stresses can be expressed by a single equation The F Nx , F Ny , and F Nz unknown components of the resultant stress on the plane KLM of elemental tetra- hedron passing through point O (Fig. 27-2) The unknown components of resultant stress F Nx , F Ny and F Nz in terms of direction cosines l, m and n (Fig. 27-4)  xy ¼  yx ; yz ¼  zy ; zx ¼  xz ð27-4Þ  ij ¼ lim ÁA i !0 ÁF j ÁA i ð27-5Þ where i ¼ 1; 2; 3 and j ¼ 1; 2; 3 F Nx ¼  x cos N; x þ  xy cos N; y þ  xz cos N; z F Ny ¼  yx cos N; x þ  y cos N; y þ  yz cos N; z F Nz ¼  zx cos N; x þ  zy cos N; y þ  z cos N; z ð27-6Þ F Nx ¼  x l þ  xy m þ  zx n F Ny ¼  yz l þ y m þ  yx n F Nz ¼  zx l þ zy m þ  z n ð27-7Þ where the direct cosines are l ¼ cos  ¼ cos N; x; m ¼ cos  ¼ cos N; y, n ¼ cos  ¼ cos N; z, l s þ m 2 þ n 2 ¼ðlÞ 02 þðm 0 Þ 2 þðn 0 Þ 2 ¼ 1 τ xy Surface area KLM = A (normal to KLM) x L F z z y K N M T N F y F Ny F Nx F Nz σ z σ y τ yz τ zy τ zx τ xz τ yx σ x F x h o o’ T N = stress vector in N direction F bx , F by , F bz = Body forces in x, y and z - direction FIGURE 27-2 The state of stress at O of an elemental tet- rahedron. x y y’ M o o’ α T x’ σ x’ σ y σ x σ z τ x’y’ τ xz τ xy τ yz τ zy τ yx τ z’x’ γ τ zx β K h N x’ L z z’ FIGURE 27-4 T x 0 , resolved into  x 0 ,  x 0 y 0 and  x 0 z 0 stress components. x + + + + + + + z dx dz dy dy dz dx dz dy y σ y τ zy τ zx τ zx τ xz τ xz τ xy τ xy τ zy τ yz τ yz τ yx τ yx σ z ∂σ x ∂x σ x σ z σ x σ y o dz ∂σ z ∂z ∂σ y ∂y dx ∂τ xy ∂x ∂τ zy ∂z + ∂τ zy ∂z + ∂τ xz ∂x dx ∂τ yx ∂y dy ∂τ yz ∂y 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. Particular Formula 27.4 CHAPTER TWENTY-SEVEN 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 The resultant stress F N on the plane KLM The normal stress which acts on the plane under consideration The shear stress which acts on the plane under consideration 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 The resultant stress vector components in x, y and z directions The resultant stress vector The normal stress which acts on the plane under consideration The shear stress which acts on the plane under consideration The angle between the resultant stress vector T N and the normal to the plane N cos  ¼ l ¼ angle between x axis and Normal N cos  ¼ m ¼ angle between y axis and Normal N cos  ¼ n ¼ angle between z axis and Normal N F N ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2 Nx þ F 2 Ny þ F 2 Nz q ð27-8Þ  N ¼ F Nx cos  þF Ny cos  þ F Nz cos  ð27-8aÞ  N ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2 N À  2 n q ð27-8bÞ T N ¼ lim ÁA !0 ÁF N ÁA ð27-9aÞ where T N coincides with the line of action of the resultant force ÁF n T Nx ¼  x l þ xy m þ  xz n ð27-9bÞ T Ny ¼  xy l þ  y m þ  zy n ð27-9cÞ T Nz ¼  zx l þ  zy m þ  z n ð27-9dÞ T N ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 Nx þ T 2 Ny þ T 2 Nz q ð27-9eÞ where the direction cosines are cosðT N ; xÞ¼T Nx =jT N j, cosðT N ; yÞ¼T Ny =jT N j, cosðT N ; zÞ¼T Nz =jT N j  N ¼jT N jcosðT N ; NÞð27-9f Þ  N ¼ T Nx cosðN; xÞþT Ny cosðN; yÞþT Nz cosðN; zÞ ð27-9gÞ  N ¼jT N jsinðT N ; NÞð27-10aÞ  N ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 N À  2 N q ð27-10bÞ cosðT N ; NÞ¼cosðT N ; xÞcosðN; xÞ þ cosðT N ; yÞcosðN; yÞ þ cosðT N ; zÞcosðN; zÞð27-10cÞ Particular Formula APPLIED ELASTICITY 27.5 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 EQUATIONS OF EQUILIBRIUM The equations of equilibrium in Cartesian coordi- nates which includes body forces in three dimensions (Fig. 27-3) Stress equations of equilibrium in two dimensions TRANSFORMATION OF STRESS The vector form of equations for resultant-stress vectors T N and T 0 N for two different planes and the outer normals N and N 0 in two different planes The projections of the resultant-stress vector T N onto the outer normals N and N 0 Substituting Eqs. (27-9b), (27-9c), (27-9d) and (27-9e) in Eqs. (27-13), equations for T N , N and T N , N 0 The relation between T N , N 0 and T 0 N , N By coinciding outer normal N with x 0 , N with y 0 , and N with z 0 individually respectively and using Eqs. (27-14a) to (27-14b),  x 0 ,  y 0 and  z 0 can be obtained (Fig. 27-4) @ x @x þ @ xy @y þ @ xz @z þ F bx ¼ 0 ð27-11aÞ @ y @y þ @ yz @z þ @ yx @x þ F by ¼ 0 ð27-11bÞ @ z @z þ @ zx @x þ @ zy @y þ F bz ¼ 0 ð27-11cÞ where F bx , F by and F bz are body forces in x, y and z directions @ x @x þ @ xy @y þ F bx ¼ 0 ð27-11dÞ @ y @y þ @ yx @x þ F by ¼ 0 ð27-11eÞ T N ¼ iT Nx þ jT Ny þ kT Nz ð27-12aÞ T N 0 ¼ iT N 0 x þ jT N 0 y þ kT N 0 z ð27-12bÞ N ¼ il þ jm þ kn ð27-12cÞ N 0 ¼ il 0 þ jm 0 þ kn 0 ð27-12dÞ where i, j and k are unit vectors in x, y and z directions, respectively T N ÁN ¼ T Nx l þ T Ny m þ T Nz n ð27-13aÞ T N ÁN 0 ¼ T Nx l 0 þ T Ny m 0 þ T Nz n 0 ð27-13bÞ T N ÁN ¼  x l 2 þ  y m 2 þ  z n 2 þ 2 xy lm þ 2 yz mn þ 2 zx nl ð27-14aÞ T N ÁN 0 ¼  x ll 0 þ  y mm 0 þ  z nn 0 þ  xy ½lm 0 þ ml 0  þ  yz ½mn 0 þ nm 0 þ zx ½nl 0 þ ln 0 ð27-14bÞ T 0 N ÁN ¼ T N ÁN 0 ð27-15Þ  x 0 ¼ T x 0 Áx 0 ¼  x cos 2 ðx 0 ; xÞþ y cos 2 ðx 0 ; yÞ þ  z cos 2 ðx 0 ; zÞþ2 xy cosðx 0 ; xÞcosðx 0 ; yÞ þ 2 yz cosðx 0 ; yÞcosðx 0 ; zÞ þ 2 zx cosðx 0 ; zÞcosðx 0 ; xÞð27-15aÞ Particular Formula 27.6 CHAPTER TWENTY-SEVEN 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 By selecting a plane having an outer normal N co- incident with the x 0 and a second plane having an outer normal N 0 coincident with the y 0 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  x 0 y 0 . Following this procedure and by selecting N and N 0 coincident with the y 0 and z 0 , and z 0 and x 0 axes, the expression for  y 0 z 0 and  z 0 x 0 can be obtained. The expressions for  x 0 y 0 ,  y 0 z 0 and  z 0 x 0 are  y 0 ¼ T y 0 Áy 0 ¼  y cos 2 ðy 0 ; yÞþ z cos 2 ðy 0 ; zÞ þ  x cos 2 ðy 0 ; xÞþ2 yz cosðy 0 ; yÞcosðy 0 ; zÞ þ 2 zx cosðy 0 ; zÞcosðz 0 ; xÞ þ 2 xy cosðy 0 ; xÞcosðy 0 ; yÞð27-15bÞ  z 0 ¼ T z 0 Áz 0 ¼  z cos 2 ðz 0 ; zÞþ x cos 2 ðz 0 ; xÞ þ  y cos 2 ðz 0 ; yÞþ2 zx cosðz 0 ; zÞcosðz 0 ; xÞ þ 2 xy cosðz 0 ; xÞcosðz 0 ; yÞ þ 2 yz cosðz 0 ; yÞcosðz 0 ; zÞð27-15cÞ  x 0 y 0 ¼ T x 0 Áy 0 ¼  x cosðx 0 ; xÞcosðy 0 ; xÞ þ  y cosðx 0 ; yÞcosðy 0 ; yÞþ z cosðx 0 ; zÞcosðy 0 ; zÞ þ  xy ½cosðx 0 ; xÞcosðy 0 ; yÞþcosðx 0 ; yÞcosðy 0 ; xÞ þ  yz ½cosðx 0 ; yÞcosðy 0 ; zÞþcosðx 0 ; zÞcosðy 0 ; yÞ þ  zx ½cosðx 0 ; zÞcosðy 0 ; xÞþcosðx 0 ; xÞcosðy 0 ; zÞ ð27-16aÞ  y 0 z 0 ¼ T y 0 Áz 0 ¼  y cosðy 0 ; yÞcosðz 0 ; yÞ þ  z cosðy 0 ; zÞcosðz 0 ; zÞþ x cosðy 0 ; xÞcosðz 0 ; xÞ þ  yz ½cosðy 0 ; yÞcosðz 0 ; zÞþcosðy 0 ; zÞcosðz 0 ; yÞ þ  zx ½cosðy 0 ; zÞcosðz 0 ; xÞþcosðy 0 ; xÞcosðz 0 ; zÞ þ  xy ½cosðy 0 ; xÞcosðz 0 ; yÞþcosðy 0 ; yÞcosðz 0 ; xÞ ð27-16bÞ  z 0 x 0 ¼ T z 0 Áx 0 ¼  z cosðz 0 ; zÞcosðx 0 ; zÞ þ  x cosðz 0 ; xÞcosðx 0 ; xÞþ y cosðz 0 ; yÞcosðx 0 ; yÞ þ  zx ½cosðz 0 ; zÞcosðx 0 ; xÞþcosðz 0 ; xÞcosðx 0 ; zÞ þ  xy ½cosðz 0 ; xÞcosðx 0 ; yÞþcosðz 0 ; yÞcosðx 0 ; xÞ þ  yz ½cosðz 0 ; yÞcosðx 0 ; zÞþcosðz 0 ; zÞcosðx 0 ; 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 Ox 0 y 0 z 0 coordinate system Particular Formula APPLIED ELASTICITY 27.7 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 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 0 z 0 coincide with z and  is the angle between x and x 0 , Eqs. (27-15a) to (27-15c) and Eqs. (27-16a) to (27-16c) M O L N z N K T Nx T Nz T N , σ N T Ny x y FIGURE 27-5 The stress vector T N . PRINCIPAL STRESSES By referring to Fig. 27-5, where T N coincides with outer normal N, it can be shown that the resultant stress components of T N in x, y and z directions Substituting Eqs. (27-9b) to (27-9d) into (27-18), the following equations are obtained Eq. (27-19) can be written as 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  x 0 ¼  x cos 2  þ  y sin 2  þ 2 xy sin  cos  ¼  x þ  y 2 þ  x À  y 2 cos 2 þ  xy sin 2 ð27-17aÞ  y 0 ¼  y cos 2  þ  x sin 2  À 2 xy sin  cos  ¼  y þ  x 2 þ  y À  x 2 cos 2 À  xy sin 2 ð27-17bÞ  x 0 y 0 ¼  y cos  sin  À x cos  sin  þ  xy ðcos 2  À sin 2 Þ ¼  y À  x 2 sin 2 þ  xy cos 2 ð27-17cÞ  z 0 ¼  z 0 x 0 ¼  y 0 z 0 ¼ 0 ð27-17dÞ T Nx ¼  N l T Ny ¼  N m ð27-18Þ T Nz ¼  N n  x l þ yx m þ  zx n ¼  N l  xy l þ y m þ  xy n ¼  N m ð27-19Þ  xz l þ yz m þ  z n ¼  N n ð x À  N Þl þ yx m þ  zx ¼ 0  xy l þð y À  N Þm þ  zy ¼ 0 ð27-20Þ  xz l þ yz m þð z À  N Þn ¼ 0 cosðN; xÞ¼ 0  yx  zx 0  y À  n  zy 0  yz  z À  N                  x À  N  yx  zx  xy  y À  N  zy  xz  yz  z À  N                 ð27-21Þ  x À  N  yx  zx  xy  y À  N  zy  xz  yz  z À  N                 ¼ 0 ð27-22Þ Particular Formula 27.8 CHAPTER TWENTY-SEVEN 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 Expanding the determinant after making use of Eqs. (27-4) which gives three roots. They are principal stresses For two-dimensional stress system the coordinating system coinciding with the principal directions, Eq. (27-23) becomes The three principal stresses from Eq. (27-23a) are The directions of the principal stresses can be found from From Eq. (27-15) From definition of principal stress Substituting the values of T N1 and T N2 in Eq. (27-15) and simplifying From Eq. (27-20) The three invariant of stresses from Eq. (27-23)  3 N Àð x þ  y þ  z Þ 2 N þð x  y þ  y  z þ  z  x À  2 xy À  2 yz À  2 zx Þ N Àð x  y  z À  x  2 yz À  y  2 zx À  z  2 xy þ 2 xy  yz  zx Þ¼0 ð27-23Þ  3 i Àð x þ  y Þ 2 i þð x  y À  2 xy Þ i ¼ 0 ð27-23aÞ where i ¼ 1; 2; 3  1;2 ¼ 1 2 ð x þ  y ÞÆ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   x À  y 2  2 þ  2 xy s ð27-23bÞ  3 ¼ 0 sin 2ðN 1 ; xÞ¼ 2 xy ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x À  y Þ 2 þ 4 2 xy q ð27-23cÞ cos 2ðN 1 ; xÞ¼  x À  y ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x À  y Þ 2 þ 4 2 xy q ð27-23dÞ tan 2ðN 1 ; xÞ¼ 2 xy  x À  y ð27-23eÞ T N 1 ÁN 2 ¼ T N 2 ÁN 1 ð27-24Þ T N 1 ¼  1 N 1 ð27-25aÞ T N 2 ¼  2 N 2 ð27-25bÞ ð 1 À  2 ÞN 1 ÁN 2 ¼ 0 ð27-26Þ where  1 and  2 are distinct N 1 ÁN 2 ¼ 0 ð27-27Þ which proves that N 1 and N 2 are orthogonal. I 1 ¼  x þ  y þ  z ¼  x 0 þ  y 0 þ  z 0 ð27-28aÞ I 2 ¼  x  y þ  y  z þ  z  x À  2 xy À  2 yz À  2 zx ¼  x 0  y 0 þ  y 0  z 0 þ  z 0  x 0 À  2 x 0 y 0 À  2 y 0 z 0 À  2 z 0 x 0 ð27-28bÞ I 3 ¼  x  y  z À  x  2 yz À  y  2 zx À  z  2 xy þ 2 xy  yz  zx ¼  x 0  y 0  z 0 À  x 0  2 y 0 z 0 À  y 0  2 z 0 x 0 À  z 0  2 x 0 y 0 þ 2 x 0 y 0  y 0 z 0  z 0 x 0 ð27-28cÞ Particular Formula APPLIED ELASTICITY 27.9 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 For the coordinating system coinciding with the principal direction, the expression for invariants from Eq. (27-28) STRAIN (Fig. 27-6) The normal strain or longitudinal strain by Hooke’s law (Fig. 27-6) in x-direction The lateral strains in y and z-direction The normal strains caused by  y and  z THREE-DIMENSIONAL STRESS-STRAIN SYSTEM The general stress-strain relationships for a linear, homogeneous and isotropic material when an element subject to  x ,  y and  z stresses simultaneously where I 1 ¼ first invariant, I 2 ¼ second invariant and I 3 ¼ third invariant of stress I 1 ¼  1 þ  2 þ  3 ð27-29aÞ I 2 ¼  1  2 þ  2  3 þ  3  1 ð27-29bÞ I 3 ¼  1  2  3 ð27-29cÞ " x ¼  x E ð27-30aÞ " y ¼À v E  x ¼Àv" x ð27-30bÞ " z ¼À v E  x ¼Àv" x ð27-30cÞ " y ¼  y E ; " x ¼ " z ¼À v y E ¼Àv" y ð27-31Þ " z ¼  z E ; " x ¼ " y ¼À v z E À v" z ð27-32Þ " x ¼ 1 E ½ x À vð y þ  z Þ ð27-33aÞ " y ¼ 1 E ½ y À vð z þ  x Þ ð27-33bÞ Particular Formula x y z dx’ dy’ dx dz dz’ dy x σ x σ FIGURE 27-6 Uniaxial elongation of an element in the direction of x. y xy M L K N x M M’ K’ K k n m I L L’ N N’ z τ xy τ xy τ xy τ xy τ xy τ xy τ xy τ FIGURE 27-7 A cubic element subject to shear stress,  xy . 27.10 CHAPTER TWENTY-SEVEN 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 The expressions for  x ,  y and  z stresses in case of three-dimensional stress system from Eqs (27-33) BIAXIAL STRESS-STRAIN SYSTEM The normal strain equations, when  z ¼ 0 from Eq. (27-33) The normal stress equation, when  z ¼ 0 from Eq. (27.34) SHEAR STRAINS For a homogeneous, isotropic material subject to shear force, the shear strain which is related to shear stress as in case of normal strain " z ¼ 1 E ½ z À vð x þ  y Þ ð27-33cÞ  x ¼ E ð1 þ vÞð1 À 2vÞ ½ð1 À vÞ" x þ vð" y þ " z Þ ð27-34aÞ  y ¼ E ð1 þ vÞð1 À 2vÞ ½ð1 À vÞ" y þ vð" z þ " x Þ ð27-34bÞ  z ¼ E ð1 þ vÞð1 À 2vÞ ½ð1 À vÞ" z þ vð" x þ " y Þ ð27-34cÞ " x ¼ 1 E ½ x À v y ð27-35aÞ " y ¼ 1 E ½ y À v x ð27-35bÞ " z ¼À v E ½ x þ  y ð27-35cÞ  x ¼ E 1 À v ½" x þ v" y ¼J 1 þ 2" x ¼ 2  þ 2 ð" x þ " y Þþ2" x ð27-36aÞ  y ¼ E 1 À v ½" y þ v" x ¼J 1 þ 2" y ¼ 2  þ 2 ð" y þ " x Þþ2" y ð27-36bÞ  z ¼ J 1 þ 2" z ¼ 0 ð27-36cÞ  xy ¼  xy ;  yz ¼  yz ¼ 0;  zx ¼  zx ¼ 0 ð27-36dÞ  xy ¼  xy G ð27-37aÞ  yz ¼  yz G ð27-37bÞ  zx ¼  zx G ð27-37cÞ Particular Formula APPLIED ELASTICITY 27.11 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 . 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 27.2 CHAPTER TWENTY-SEVEN Downloaded. stresses acting on the faces of a small cube element cut out from the solid body. Particular Formula Part I o o Part II Part II a a a y x x y N z z dy dx dz a F 1 F 1 F 2 F 2 F 3 F 3 F 4 F 5 F 6 (a). to (27-15c) and (27-16a) to (27-16c) reduce to, since  z ¼  zx ¼  yz ¼ 0 z 0 coincide with z and  is the angle between x and x 0 , Eqs. (27-15a) to (27-15c) and Eqs. (27-16a) to (27-16c) M O L N z N K T Nx T Nz T N , σ N T Ny x y FIGURE