APPLIED ELASTICITY 27.42 CHAPTER TWENTY-SEVEN Particular Formula Specified displacement From Eq (27-158) for D À 2v " z "z 4vịzị z0 "ị !"ị ẳ 2GD À w 2ð1 À vÞ ð27-161Þ If body force is absent Eq (27-161) becomes " z "z ð3 À 4vÞðzÞ z0 "ị !"ị ẳ 2Gg1 ỵ ig2 ị on C ð27-162Þ where g1 and g2 are functions of z only FORCE AND COUPLE RESULTANTS AROUND THE BOUNDARY (Fig 27-31) The expression for force with components X and Y at point O The expression for couple at O h iB1 " z "z X ỵ iY ẳ i zị þ z0 ð"Þ þ !ð"Þ ð27-163Þ A1 h iB1 ð B1 @z ỵ U ds N ẳ Rl ẫzị z!ðzÞ À z"0 ðzÞ z A1 @s A1 ð27-164Þ GENERALIZED PLANE STRESS The average stress combinations assuming z ¼ 0, a stress free surface, i.e xz ¼ yz ¼ at the surface " and body force potential Uðz; zÞ is independent of z o ẳ x ỵ y 27-165aị ẩo ẳ x y ỵ 2ixy 27-165bị where Âo ¼ pav ¼ B1 τnb 2h τyn ðh Àh ðh Àh σn α N X O 2h ðh Àh È dz p dz O 2h x y x FIGURE 27-31 Èo ¼ z A1 Y  dz; τxn ds y 2h y x FIGURE 27-32 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 The average complex displacement The body force Eq @ẩ @ @ẫ ỵ ỵ ẳ becomes @z @" @" z z Formula Do ¼ uo ỵ ivo ẳ 2h h  h ẳ Taking into consideration the body force, Eq (27-167) and other expression for F and Èo become 2h ðh Àh 27-166ị D dz  @ẩ @ @ẫ ỵ ỵ dz ẳ @z @" @z z 27-167aị @ẩo @o ỵ ẳ0 @z @" z @ @ẩo h ỵ 2Ui ỵ ¼0 @" o z @z  "  @! v @D @ D ỵ ẳ v @z @" z @" z ẩ ẳ 4G 27-167bị 27-168aị @D @" z ẩo ẳ 4G 27-168bị @Do @" z 27-168cị  " 1v @Do @ D ỵ o ẳ 2G @z 1ỵv @" z The equations for generalized plane stress 27.43 " z "z F ẳ 2fzị ỵ z0 "ị ỵ !"ịg ỵ  2GD ẳ 27-168dị 2K w 1ÀK ð27-169Þ  3Àv À 2K " z "z zị z0 "ị !"ị w 1ỵv 21 Kị 27-170ị  " z ẳ 0 zị ỵ 0 "ị @w K @z  À 2K @w " z " z ẩ ẳ 2fz00 "ị ỵ !0 "ịg À K @" z ð27-171Þ ð27-172Þ CONDITIONS ALONG A STRESS-FREE BOUNDARY, Fig 27-33 Adding Eqs (27-169) and (27-170) and putting F ¼ along free boundary, i.e segment AB, the displacement along AB Dẳ zị E SOLUTION INVOLVING CIRCULAR BOUNDARIES (Figs 27-33 and 27-34) From stress strain transformation rules ẳ ẳ r ỵ   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 ð27-173Þ APPLIED ELASTICITY 27.44 CHAPTER TWENTY-SEVEN Particular Formula y B y A θ = constant x r C FIGURE 27-33 r = constant θ x FIGURE 27-34 " z È z i Ài where r ¼ , z ¼ r e , z ¼ r e " È0 ¼ F e2i ẳ r  ỵ 2i ẳ @w ð27-174Þ À K @z   " " À 2K z @w z " z " z È0 ẳ z00 "ị ỵ !0 "ị K z @z z " z ẳ 2f0 zị ỵ 0 "ịg 2GD0 ẳ ei The boundary conditions are F ¼2 ðs  3Àv " z "z zị z0 "ị !"ị 1ỵv  2K w 1K r ỵ ir ỵ Uị 27-175ị 27-176ị @z ds ỵ constant @s 27-177aị "0 "z z zị ỵ z "ị ỵ !"ị ẳ f1 ỵ if2 on C ð27-177bÞ APPLICATION OF CONFORMAL TRANSFORMATION (Fig 27-35) Eqs (27-178) are related stress combinations in rectangular coordinates x and y as ẳ  ỵ  27-178aị ẩ0 ẳ   ỵ 2i The stress combinations after transformation 27-178bị ẳ 27-179aị ẩ0 ẳ ẩ e2i ð27-179bÞ 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 Formula ξ = constant y η η τξη ϑ = constant P 27.45 σξ η = constant Q r ρ α x O (a) z - plane ρ = constant ξ = constant ϑ O ξ O (b) ϕ - plane (c) ξ FIGURE 27-35 An explanation for eÀ2i where z ¼ zị ẳ f ; ị ỵ ig; ị  ẳ  ỵ i f ; ị and g; ị are real and imaginary parts of zị " " e2i ẳ z0 ð Þ=z0 ðÞ ð27-179cÞ or Using Eqs (27-179a) and (27-179b), and Eqs (27-171) and (27-172), when these are no body forces, letting zị ẳ 1 ị and !zị ẳ !1 ðÞ The transformation of a given boundary in the zplane into the unit circle in the -plane Using polar coordinates (, #), the stress components become  " d " " d  ỵ 1 ;  ị 27-180aị ẳ 01 ị dz d" z  " "  1 ðÞ 01 ; ð Þ ỵ " ẳ 27-180bị z0 ị " z ð Þ o n " " " " "1 " " " zịh 00 01  ị ỵ  00  ịi ỵ !01  ị ẩ0 ¼ À z ðÞ ð27-181aÞ or ( ) " "  01 ; ð Þ 0 " ẩ ẳ 27-181bị ỵ !1 ; ị z ị zị z0  ị " " 00 ẳ  þ # È ¼  þ # À 2i# 00 Using polar coordinates Eqs (27-180a) and (27-181) in terms of complex potentials become ð27-182aÞ 00 ð27-182bÞ 00 À2i# where  ¼  and È ¼ È e " # " " 0 ị 0  ị 00 ỵ ẳ2 z ị z0  ị " " ¼ "  È: ð27-183Þ i " 2 h "00 " " "0 " 00 " "0 " ị zịh 1  ị ỵ  1  ịi ỵ !  ị z 27-184aị " #  "0 "  " 2  ð Þ " " ỵ !0  ị 27-184bị zị " ẩ00 ẳ z ị " z  ị ẩ00 ẳ 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 27.46 CHAPTER TWENTY-SEVEN Particular y Formula y T T T x a T σy y τxy dy y z 2b τxy σx 2b x Mb 2h Mb a a FIGURE 27-36 a FIGURE 27-37 Rectangular plate under all round tension Value of complex potentials ðzÞ and !zị assumed zị ẳ Tz; From stress combination Eqs (27-156c) and (27-157) " z ẳ 2f0 zị ỵ 0 "ịg ỵ !xị ẳ 27-185ị @w v @z ẳ 2ẵ1 T ỵ T ¼ 2T 2 ð27-156cÞ À 2v @w " z " z ẩ ẳ 2fz00 "ị ỵ !0 "ịg ỵ v @" z 27-157ị where ẳ x ỵ y and ẩ ẳ x y ỵ 2ixy The stress x and y after equating real and imaginary parts x ¼ T; The displacement from Eq (27-158) after equating real and imaginary parts " z "z 2GD ¼ ð3 À 4vÞðzÞ À z0 ð"Þ À !ð"Þ y ẳ T; ỵ xy ẳ 2v w 21 vị 27-186ị 27-158ị Dẳ T vịx ỵ iyị ẳ u ỵ i E uẳ T vịx; E ẳ T vịy E 27-187ị Rectangular plate under plane flexure Assume values of complex potentials zị and !zị as zị ẳ Az2 !zị ẳ Bz2 Choose A and B, which may be complex, so that edges y ẳ ặb are stress free 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 27.47 Formula Boundary conditions From stress combinations Eqs (27-156) and (27-157) boundary conditions " z " 0 zị ỵ 0 "ị ỵ z00 zị ỵ !0 zị ẳ y ỵ ixy 27-156ị y ẳ 0, xy ¼ throughout the plate A ¼ iC and B ẳ iC where C is real ẳ 0x ỵ 0y ¼ 0x ¼ À8Cy The bending moment Mb ¼ b x 2hy dy ẳ 8CI b 27-188ị where I ¼ moment of inertia about oz C¼ À Mb 8I The values of complex potentials zị ẳ Az2 and !zị ¼ Bz2 are ðzÞ ¼ À The displacement from Eq (27-158) Dẳ iMb z ; 8I !zị ẳ iMb z 8I ð27-188aÞ i h " z "z 4vịzị z0 "ị !"ị ẳ u þ iv 2G when body forces are zero Substituting the values of ðzÞ and !ðzÞ in the above, u and v can be determined Thick cylinder under internal and external pressure Values of complex potentials ðzÞ and !ðzÞ assumed using boundary conditions at r ¼ a or di =2 and r ¼ b or =2 with no body forces, assuming internal pressure pi , external pressure po , values of A and B in Eq (27-189), which are real, can be found From Eqs (27-174) and (27-175) The expressions for  and r at any radius zị ẳ Az and !zị ẳ B z 27-189aị where A and B are real ẵ " z ỵ ẩ0 ẳ r ỵ ir ẳ 0 zị ỵ 0 ð"Þ z " " z À z00 ðzÞ À !0 ð"Þ " z ð27-189bÞ The equations for  and r are given in Eqs (27-101b) and (27-101a) respectively Rotating solid disk and hollow disk of uniform thickness rotating at ! rad/s Values of complex potentials ðzÞ and !ðzÞ assumed ðzÞ ¼ Cz and !ðzÞ ¼ B z where C and B are real 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 ð27-189cÞ APPLIED ELASTICITY 27.48 CHAPTER TWENTY-SEVEN Particular Formula Using boundary conditions at r ịr ẳ b ¼ and ðr Þr ¼ 6¼ for solid disk r ịr ẳ a ẳ and r Þr ¼ b ¼ for hollow disc taking into consideration body forces, values of C and B in Eq (27-189c) which are real can be found Refer Eqs (27-126), (27-127) and (27-128) to (27-131) The radial displacements at the boundaries ur ịr ẳ a ẳ !2 a f1 vịa2 ỵ ỵ vịb2 g 4E 27-189dị ur ịr ẳ b ẳ !2 b f1 vịb2 ỵ þ vÞa2 g 4E ð27-189eÞ Large plate under uniform uniaxial tension with a centrally located unstressed circular hole Values of complex potentials ðzÞ and !ðzÞ assumed Using Eq (27-189b) and above complex potentials Using boundary condition at r ¼ a Tz A ỵ z B C !zị ẳ Tz ỵ ỵ z z where A, B and C are real zị ẳ 27-190ị 3A z B 3C T ỵ T ỵ ỵ ð27-190aÞ " 2 z z" z z z " z     B A Tỵ ỵ T ỵ e2i r ir Þr ¼ a ¼ 2 a a   3C 3A 2i ỵ e 27-190bị a4 a r À ir ¼ A ¼ Ta2 ; B ¼ À Ta2 ; C ¼ a4 ð27-190cÞ since hole is stress free The new values of zị and !zị zị ẳ Using Eqs (27-174), (27-175) and after equating the real and imaginary parts, the stress components are y A T A FIGURE 27-38 θ a T x Tz Ta2 ỵ z 27-190dị Ta2 a4 27-190eị ỵ !zị ẳ Tz À 2z " #    a2 4a2 3a4 r ¼ T À þ À þ cos 2 r r r  ẳ T " 1ỵ a2 r2  #   3a4 ỵ cos 2 r   2a2 3a4 r ¼ T ỵ sin 2 r r 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 ð27-191Þ ð27-192Þ ð27-193Þ APPLIED ELASTICITY APPLIED ELASTICITY Particular The  , r and r at r ¼ a 27.49 Formula The stress concentration factor ð27-194aÞ ð Þr ¼ a ¼ Tð1 À cos 2Þ The maximum tangential stress r ịr ẳ a ẳ r ịr ẳ a ẳ 27-194bị  max ẳ  ịr ẳ a ẳ 3T 27-194cị K ẳ  ịmax 3T ẳ ¼3 T T ð27-195Þ Large plate containing a circular hole under uniform pressure Values of complex potentials ðzÞ and !ðzÞ assumed From Eqs (27-174) and (27-175) in the absence of body forces zị ẳ 0; !zị ẳ A z 27-196ị " z ẳ 2f0 zị ỵ 0 "ịg ẳ r ỵ  ẳ   " z "" z " z ẩ0 ẳ z00 "ị ỵ !0 "ị z ẳ r  ỵ 2ir ẳ Boundary conditions are r ịr ẳ a ẳ p ẳ aị 2A r2 bị 2A a2 A ẳ pa2 The new complex potentials zị ẳ 0; The stress components are r ¼ À !ðzÞ ¼ À pa2 ; r2  ¼ r ẳ The displacement from Eq (27-176) cị pa2 z r ẳ pa2 r2 2GD0 ẳ 2Gur ỵ iu Þ ¼ eÀi A ; 2Gr pa ¼ 2G ður Þ ¼ À ður Þr ¼ a ðdÞ ð27-197Þ  À A " z  ¼À A r u ¼ ð27-198Þ Large plate containing a circular hole filled by an oversize disk Rigid Disk The radius of disk rd rd ẳ a1 ỵ "ị where a ẳ radius of hole 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 ðaÞ APPLIED ELASTICITY 27.50 CHAPTER TWENTY-SEVEN Particular From first of Eq (27-198), the radial displacement The stress components Formula ur ¼ a" ¼ À A 2Ga or A ẳ 2Ga2 " r ẳ  ẳ 2G" bị a2 r2 cị r ẳ dị Elastic Disk The complex potential for all round pressure on the disk 1 zị ẳ pz; The displacement from Eq (27-176) 2G1 D0 ẳ 2G1 ur1 ỵ iu1 ị ¼ eÀi ¼À ur1 ¼ !1 ðzÞ ¼ ðeÞ  vịpz 1ỵv p1 v1 ịpa þ v1  ðfÞ Àpð1 À v1 Þa E1 ðgÞ where subscript for disk and for plate The radial displacement of plate The pressure between disc and plate ur2 ẳ pẳ pa 2G2 hị E1 E2 E1 þ v2 Þ þ E2 ð1 À v1 Þ ð27-198Þ Elliptical hole in a large plate under tension (Fig 27-39) The expression for transformation   m ; zẳC ỵ  m