Chapter 10

 

Approximate theories for solids with special shapes:

 rods, beams, membranes, plates and shells

 

 

 

10.7 Solutions to simple problems involving membranes, plates and shells

 

In this section, we derive solutions to several initial and boundary value problems for plates and shells to illustrate applications of the general theories derived in the preceding sections.

 

 

10.7.1 Thin circular plate bent by pressure applied to one face

 

A thin circular plate, with radius R and thickness h is made from a linear elastic solid with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3240@ , as shown in the figure.  It is subjected to a pressure p= p 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahchacqGH9aqpcaWGWbWaaSbaaSqaai aaiodaaeqaaOGaaCyzamaaBaaaleaacaaIZaaabeaaaaa@3646@  acting perpendicular to the plate, and is simply supported at its edge.  The solution can be derived using the simplified version of plate theory described in Section 10.6.1.  Although the plate is circular, the problem can be solved by expressing all vector and tensor quantities as components in a Cartesian basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3988@  shown in the figure.

 

The deflection of the plate is given by

u 3 = 3(1 ν 2 )p 16E h 3 ( R 2 r 2 )( 5+ν 1+ν R 2 r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpdaWcaaqaaiaaiodacaGGOaGaaGymaiabgkHiTiabe27aUnaa CaaaleqabaGaaGOmaaaakiaacMcacaWGWbaabaGaaGymaiaaiAdaca WGfbGaamiAamaaCaaaleqabaGaaG4maaaaaaGcdaqadaqaaiaadkfa daahaaWcbeqaaiaaikdaaaGccqGHsislcaWGYbWaaWbaaSqabeaaca aIYaaaaaGccaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiaaiwdacqGH RaWkcqaH9oGBaeaacaaIXaGaey4kaSIaeqyVd4gaaiaadkfadaahaa WcbeqaaiaaikdaaaGccqGHsislcaWGYbWaaWbaaSqabeaacaaIYaaa aaGccaGLOaGaayzkaaaaaa@5221@

where r= x α x α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0ZaaOaaaeaacaWG4b WaaSbaaSqaaiabeg7aHbqabaGccaWG4bWaaSbaaSqaaiabeg7aHbqa baaabeaaaaa@3A6D@ .The internal force and moment in the plate are

M αβ = 3 2 (1ν)p x α x β + 3 4 p( r 2 (1+3ν) R 2 (3+ν) ) δ αβ V β = M αβ x α =6(1ν)p x β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamytamaaBaaaleaacqaHXoqycq aHYoGyaeqaaOGaeyypa0ZaaSaaaeaacaaIZaaabaGaaGOmaaaacaGG OaGaaGymaiabgkHiTiabe27aUjaacMcacaWGWbGaamiEamaaBaaale aacqaHXoqyaeqaaOGaamiEamaaBaaaleaacqaHYoGyaeqaaOGaey4k aSYaaSaaaeaacaaIZaaabaGaaGinaaaacaWGWbWaaeWaaeaacaWGYb WaaWbaaSqabeaacaaIYaaaaOGaaiikaiaaigdacqGHRaWkcaaIZaGa eqyVd4MaaiykaiabgkHiTiaadkfadaahaaWcbeqaaiaaikdaaaGcca GGOaGaaG4maiabgUcaRiabe27aUjaacMcaaiaawIcacaGLPaaacqaH 0oazdaWgaaWcbaGaeqySdeMaeqOSdigabeaaaOqaaiaadAfadaWgaa WcbaGaeqOSdigabeaakiabg2da9maalaaabaGaeyOaIyRaamytamaa BaaaleaacqaHXoqycqaHYoGyaeqaaaGcbaGaeyOaIyRaamiEamaaBa aaleaacqaHXoqyaeqaaaaakiabg2da9iaaiAdacaGGOaGaaGymaiab gkHiTiabe27aUjaacMcacaWGWbGaamiEamaaBaaaleaacqaHYoGyae qaaaaaaa@7368@

 

 

Derivation:

  1. The transverse deflection must satisfy the static equilibrium equation

E h 3 12(1 ν 2 ) 4 u 3 x α x α x β x β =p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamyraiaadIgadaahaaWcbe qaaiaaiodaaaaakeaacaaIXaGaaGOmaiaacIcacaaIXaGaeyOeI0Ia eqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaaaadaWcaaqaaiabgk Gi2oaaCaaaleqabaGaaGinaaaakiaadwhadaWgaaWcbaGaaG4maaqa baaakeaacqGHciITcaWG4bWaaSbaaSqaaiabeg7aHbqabaGccqGHci ITcaWG4bWaaSbaaSqaaiabeg7aHbqabaGccqGHciITcaWG4bWaaSba aSqaaiabek7aIbqabaGccqGHciITcaWG4bWaaSbaaSqaaiabek7aIb qabaaaaOGaeyypa0JaaGPaVlaadchacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVdaa@5ADF@

  1. The solution must be axially symmetric, so that u 3 =w( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWG3bWaaeWaaeaacaWGYbaacaGLOaGaayzkaaaaaa@36F7@  where r= x α x α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacqGH9aqpdaGcaaqaaiaadIhada WgaaWcbaGaeqySdegabeaakiaadIhadaWgaaWcbaGaeqySdegabeaa aeqaaaaa@382F@ .  Substituting this expression into the equilibrium equation, and using r/ x α = x α /r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadkhacaGGVaGaeyOaIyRaam iEamaaBaaaleaacqaHXoqyaeqaaOGaeyypa0JaamiEamaaBaaaleaa cqaHXoqyaeqaaOGaai4laiaadkhaaaa@3D52@ , reduces the governing equation to

E h 3 12(1 ν 2 ) ( d 4 w d r 4 + 2 r d 3 w d r 3 1 r 2 d 2 w d r 2 + 1 r 3 dw dr )= E h 3 12(1 ν 2 ) 1 r d dr ( r d dr ( 1 r d dr ( r dw dr ) ) )=p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamyraiaadIgadaahaaWcbe qaaiaaiodaaaaakeaacaaIXaGaaGOmaiaacIcacaaIXaGaeyOeI0Ia eqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaaaadaqadaqaamaala aabaGaamizamaaCaaaleqabaGaaGinaaaakiaadEhaaeaacaWGKbGa amOCamaaCaaaleqabaGaaGinaaaaaaGccqGHRaWkdaWcaaqaaiaaik daaeaacaWGYbaaamaalaaabaGaamizamaaCaaaleqabaGaaG4maaaa kiaadEhaaeaacaWGKbGaamOCamaaCaaaleqabaGaaG4maaaaaaGccq GHsisldaWcaaqaaiaaigdaaeaacaWGYbWaaWbaaSqabeaacaaIYaaa aaaakmaalaaabaGaamizamaaCaaaleqabaGaaGOmaaaakiaadEhaae aacaWGKbGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWc aaqaaiaaigdaaeaacaWGYbWaaWbaaSqabeaacaaIZaaaaaaakmaala aabaGaamizaiaadEhaaeaacaWGKbGaamOCaaaaaiaawIcacaGLPaaa cqGH9aqpdaWcaaqaaiaadweacaWGObWaaWbaaSqabeaacaaIZaaaaa GcbaGaaGymaiaaikdacaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaa leqabaGaaGOmaaaakiaacMcaaaWaaSaaaeaacaaIXaaabaGaamOCaa aadaWcaaqaaiaadsgaaeaacaWGKbGaamOCaaaadaqadaqaaiaadkha daWcaaqaaiaadsgaaeaacaWGKbGaamOCaaaadaqadaqaamaalaaaba GaaGymaaqaaiaadkhaaaWaaSaaaeaacaWGKbaabaGaamizaiaadkha aaWaaeWaaeaacaWGYbWaaSaaaeaacaWGKbGaam4Daaqaaiaadsgaca WGYbaaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaayjkaiaawMca aiabg2da9iaaykW7caWGWbGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7aaa@8768@

  1. This equation can be integrated repeatedly to give

w= 1 ν 2 E h 3 ( 3 16 p r 4 +A r 2 logr+Blogr+C r 2 +D ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhacqGH9aqpdaWcaaqaaiaaigdacq GHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaaakeaacaWGfbGaamiA amaaCaaaleqabaGaaG4maaaaaaGcdaqadaqaamaalaaabaGaaG4maa qaaiaaigdacaaI2aaaaiaadchacaWGYbWaaWbaaSqabeaacaaI0aaa aOGaey4kaSIaamyqaiaadkhadaahaaWcbeqaaiaaikdaaaGcciGGSb Gaai4BaiaacEgacaWGYbGaey4kaSIaamOqaiGacYgacaGGVbGaai4z aiaadkhacqGHRaWkcaWGdbGaamOCamaaCaaaleqabaGaaGOmaaaaki abgUcaRiaadseaaiaawIcacaGLPaaaaaa@5252@

where A,B,C,D are constants of integration.

  1. The curvature is related to w by

κ αβ = 2 u 3 x α x β =( d 2 w d r 2 1 r dw dr ) x α x β r 2 dw dr δ αβ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacqaHXoqycqaHYo GyaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacqGHciITdaahaaWcbeqa aiaaikdaaaGccaWG1bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaeyOaIy RaamiEamaaBaaaleaacqaHXoqyaeqaaOGaeyOaIyRaamiEamaaBaaa leaacqaHYoGyaeqaaaaakiabg2da9iabgkHiTmaabmaabaWaaSaaae aacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaam4DaaqaaiaadsgacaWG YbWaaWbaaSqabeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaa qaaiaadkhaaaWaaSaaaeaacaWGKbGaam4DaaqaaiaadsgacaWGYbaa aaGaayjkaiaawMcaamaalaaabaGaamiEamaaBaaaleaacqaHXoqyae qaaOGaamiEamaaBaaaleaacqaHYoGyaeqaaaGcbaGaamOCamaaCaaa leqabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaadsgacaWG3baaba GaamizaiaadkhaaaWaaSaaaeaacqaH0oazdaWgaaWcbaGaeqySdeMa eqOSdigabeaaaOqaaiaadkhaaaaaaa@66AA@

  1. Substituting this result into the curvature-moment equations gives the following equation for the internal moment distribution

M αβ = E h 3 12( 1 ν 2 ) ( (1ν)( d 2 w d r 2 1 r dw dr ) x α x β r 2 +( ν d 2 w d r 2 + 1 r dw dr ) δ αβ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaeqySdeMaeqOSdi gabeaakiabg2da9maalaaabaGaeyOeI0IaamyraiaadIgadaahaaWc beqaaiaaiodaaaaakeaacaaIXaGaaGOmamaabmaabaGaaGymaiabgk HiTiabe27aUnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaaa daqadaqaaiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykamaabmaaba WaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaam4Daaqaaiaa dsgacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakiabgkHiTmaalaaaba GaaGymaaqaaiaadkhaaaWaaSaaaeaacaWGKbGaam4Daaqaaiaadsga caWGYbaaaaGaayjkaiaawMcaamaalaaabaGaamiEamaaBaaaleaacq aHXoqyaeqaaOGaamiEamaaBaaaleaacqaHYoGyaeqaaaGcbaGaamOC amaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaqadaqaaiabe27aUn aalaaabaGaamizamaaCaaaleqabaGaaGOmaaaakiaadEhaaeaacaWG KbGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaai aaigdaaeaacaWGYbaaamaalaaabaGaamizaiaadEhaaeaacaWGKbGa amOCaaaaaiaawIcacaGLPaaacqaH0oazdaWgaaWcbaGaeqySdeMaeq OSdigabeaaaOGaayjkaiaawMcaaaaa@71ED@

  1. The displacement and curvature of the plate must be finite at r=0, which is only possible if A=B=0 To see this, note that the Blog(r) term in the formula for w is inifite at r=0; similarly, substituting the expression for w into the curvature formula produces a term involving Alog(r) which is also infinite at r=0.    The remaining constants must be determined from the boundary conditions at the edge of the plate.  For a simply supported plate, the boundary conditions are w=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhacqGH9aqpcaaIWaaaaa@3344@  and M αβ n α n β = M αβ x α x β / r 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaeqySdeMaeqOSdi gabeaakiaad6gadaWgaaWcbaGaeqySdegabeaakiaad6gadaWgaaWc baGaeqOSdigabeaakiabg2da9iaad2eadaWgaaWcbaGaeqySdeMaeq OSdigabeaakiaadIhadaWgaaWcbaGaeqySdegabeaakiaadIhadaWg aaWcbaGaeqOSdigabeaakiaac+cacaWGYbWaaWbaaSqabeaacaaIYa aaaOGaeyypa0JaaGimaaaa@49B3@  on r=R.  This yields two equations that can be solved for C and D.  Substituting the results back into the formulas in (3) and (5) then gives the solution.

 

 

 

 

10.7.2 Vibration modes and natural frequencies for a circular membrane

 

A thin circular membrane with thickness h, radius R and mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@  is subjected to a uniform radial force per unit length T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@3247@   acting on its edge.   Our goal is to calculate the mode shapes and natural frequencies of vibration of the membrane (the physical significance of natural frequencies and mode shapes for a continuous system are described in more detail in Section 10.4.1)

 

The natural frequencies of vibration and mode shapes are identified by two integers (m,n) that characterize the mode shape.  The index m=1,2,3… corresponds to the number of circumferential lines (with r=constant) on the membrane that have zero displacement, while n=0,1,2… corresponds to the number of diametral lines (with θ= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9aaa@3344@  constant) that have zero displacement.

 

Roots of Bessel functions of the first kind

 

J 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaaGimaaqabaaaaa@323D@

J 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaaGymaaqabaaaaa@323E@

J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaaGOmaaqabaaaaa@323F@

m=1

2.4048

3.8317

5.1356

m=2

5.5201

7.0155

8.4172

m=3

8.6537

10.1735

11.620

The natural frequencies of vibration are given by the solutions to the equation J n ( ω (m,n) R ρh/ T 0 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaamOBaaqabaGcca GGOaGaeqyYdC3aaSbaaSqaaiaacIcacaWGTbGaaiilaiaad6gacaGG PaaabeaakiaadkfadaGcaaqaaiabeg8aYjaadIgacaGGVaGaamivam aaBaaaleaacaaIWaaabeaaaeqaaOGaaiykaiabg2da9iaaicdaaaa@419A@ ,  where J n (x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaamOBaaqabaGcca GGOaGaamiEaiaacMcaaaa@34D6@  is a Bessel function of the first kind of order n  (This sounds scary, but symbolic manipulation programs have predefined functions that compute roots of Bessel functions).  A few zeros for Bessel functions of order n=0,1,2 are listed in the table.

The first few natural frequencies are ω (1,0) =2.4048 ρh/ R 2 T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaGGOaGaaGymai aacYcacaaIWaGaaiykaaqabaGccqGH9aqpcaaIYaGaaiOlaiaaisda caaIWaGaaGinaiaaiIdadaGcaaqaaiabeg8aYjaadIgacaGGVaGaam OuamaaCaaaleqabaGaaGOmaaaakiaadsfadaWgaaWcbaGaaGimaaqa baaabeaaaaa@426E@ , ω (1,1) =3.8317 ρh/ R 2 T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaGGOaGaaGymai aacYcacaaIXaGaaiykaaqabaGccqGH9aqpcaaIZaGaaiOlaiaaiIda caaIZaGaaGymaiaaiEdadaGcaaqaaiabeg8aYjaadIgacaGGVaGaam OuamaaCaaaleqabaGaaGOmaaaakiaadsfadaWgaaWcbaGaaGimaaqa baaabeaaaaa@4273@   ω (1,2) =5.1356 ρh/ R 2 T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaGGOaGaaGymai aacYcacaaIYaGaaiykaaqabaGccqGH9aqpcaaI1aGaaiOlaiaaigda caaIZaGaaGynaiaaiAdadaGcaaqaaiabeg8aYjaadIgacaGGVaGaam OuamaaCaaaleqabaGaaGOmaaaakiaadsfadaWgaaWcbaGaaGimaaqa baaabeaaaaa@4272@ , and so on.

 

 

The mode shapes are U (m,n) (r,θ)=A J n (r ω (m,n) ρh/ T 0 )sin(nθ+ θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaaiikaiaad2gaca GGSaGaamOBaiaacMcaaeqaaOGaaiikaiaadkhacaGGSaGaeqiUdeNa aiykaiabg2da9iaadgeacaWGkbWaaSbaaSqaaiaad6gaaeqaaOGaai ikaiaadkhacqaHjpWDdaWgaaWcbaGaaiikaiaad2gacaGGSaGaamOB aiaacMcaaeqaaOWaaOaaaeaacqaHbpGCcaWGObGaai4laiaadsfada WgaaWcbaGaaGimaaqabaaabeaakiaacMcaciGGZbGaaiyAaiaac6ga caGGOaGaamOBaiabeI7aXjabgUcaRiabeI7aXnaaBaaaleaacaaIWa aabeaakiaacMcaaaa@55DB@

 

Contour plots of the displacements for the first few vibration modes are shown in the figure below.

 

 

 

Derivation:

  1. The equation that governs transverse motion of a membrane under equibiaxial tension was derived in Section 10.6.5 as

T 0 ( 2 u 3 r 2 + 1 r u 3 r + 1 r 2 2 u 3 θ 2 )=ρh 2 u 3 t 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaGcda qadaqaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyD amaaBaaaleaacaaIZaaabeaaaOqaaiabgkGi2kaadkhadaahaaWcbe qaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaamOCaaaa daWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaaG4maaqabaaakeaacq GHciITcaWGYbaaaiabgUcaRmaalaaabaGaaGymaaqaaiaadkhadaah aaWcbeqaaiaaikdaaaaaaOWaaSaaaeaacqGHciITdaahaaWcbeqaai aaikdaaaGccaWG1bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaeyOaIyRa eqiUde3aaWbaaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaaiabg2 da9iabeg8aYjaadIgadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOm aaaakiaadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciITcaWG0b WaaWbaaSqabeaacaaIYaaaaaaaaaa@5B30@

  1. The general solution to this equation (which can be found by separation of variables, or if you are lazy, using a symbolic manipulation program) is

u 3 (r,θ)=( A J n ( k (m,n) r)sin(nθ+ θ 0 )+B Y n ( k (m,n) r)sin(nθ+ θ 1 ) )cos( ω (m,n) t+ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGcca GGOaGaamOCaiaacYcacqaH4oqCcaGGPaGaeyypa0ZaaeWaaeaacaWG bbGaamOsamaaBaaaleaacaWGUbaabeaakiaacIcacaWGRbWaaSbaaS qaaiaacIcacaWGTbGaaiilaiaad6gacaGGPaaabeaakiaadkhacaGG PaGaci4CaiaacMgacaGGUbGaaiikaiaad6gacqaH4oqCcqGHRaWkcq aH4oqCdaWgaaWcbaGaaGimaaqabaGccaGGPaGaey4kaSIaamOqaiaa dMfadaWgaaWcbaGaamOBaaqabaGccaGGOaGaam4AamaaBaaaleaaca GGOaGaamyBaiaacYcacaWGUbGaaiykaaqabaGccaWGYbGaaiykaiGa cohacaGGPbGaaiOBaiaacIcacaWGUbGaeqiUdeNaey4kaSIaeqiUde 3aaSbaaSqaaiaaigdaaeqaaOGaaiykaaGaayjkaiaawMcaaiGacoga caGGVbGaai4CaiaacIcacqaHjpWDdaWgaaWcbaGaaiikaiaad2gaca GGSaGaamOBaiaacMcaaeqaaOGaamiDaiabgUcaRiabew9aMjaacMca aaa@7175@

where k (m,n) = ω (m,n) ρh/ T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaaiikaiaad2gaca GGSaGaamOBaiaacMcaaeqaaOGaeyypa0JaeqyYdC3aaSbaaSqaaiaa cIcacaWGTbGaaiilaiaad6gacaGGPaaabeaakmaakaaabaGaeqyWdi NaamiAaiaac+cacaWGubWaaSbaaSqaaiaaicdaaeqaaaqabaaaaa@41C2@ , A, B, n, θ 0 , θ 1 ,ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXnaaBaaaleaacaaIWaaabeaaki aacYcacaaMc8UaaGPaVlabeI7aXnaaBaaaleaacaaIXaaabeaakiaa cYcacaaMc8UaaGPaVlaaykW7cqaHvpGzaaa@40B4@  are arbitrary constants, and J n , Y n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaamOBaaqabaGcca GGSaGaamywamaaBaaaleaacaWGUbaabeaaaaa@352D@  are Bessel functions of the first and second kinds, with order n, respectively. 

  1. The solution must satisfy u 3 (θ)= u 3 (2π+θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGcca GGOaGaeqiUdeNaaiykaiabg2da9iaadwhadaWgaaWcbaGaaG4maaqa baGccaGGOaGaeyOmaiJaeqiWdaNaey4kaSIaeqiUdeNaaiykaaaa@3F15@ , which is only possible if n is an integer.
  2. The Bessel function of the second kind is infinite at r=0, so B=0.
  3. The transverse displacement must satisfy the boundary condition u 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaaIWaaaaa@3435@  on the edge of the membrane, which leads to the condition J n ( ω (m,n) R ρh/ T 0 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaamOBaaqabaGcca GGOaGaeqyYdC3aaSbaaSqaaiaacIcacaWGTbGaaiilaiaad6gacaGG PaaabeaakiaadkfadaGcaaqaaiabeg8aYjaadIgacaGGVaGaamivam aaBaaaleaacaaIWaaabeaaaeqaaOGaaiykaiabg2da9iaaicdaaaa@419A@ .

 

 

10.7.3 Estimate for the fundamental frequency of vibration of a simply supported rectangular flat plate

 

The figure shows an initially flat plate, which lies in the e 1 , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaGcca GGSaGaaCyzamaaBaaaleaacaaIYaaabeaaaaa@34ED@  plane and is free of external force.  The plate has Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@ , mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@ , and thickness h.  Its edges are simply supported.  We wish to calculate the lowest natural frequency of vibration for the plate.

 

The exact natural frequencies and modes of vibration for a rectangular plate are best calculated using a numerical method (e.g. finite elements).   However, it is very straightforward to estimate the lowest natural frequency of vibration using the Rayleigh-Ritz method described in Section 5.9.

 

The Rayleigh-Ritz method proceeds as follows:

  1. Select a suitable estimate for the lowest frequency mode of vibration. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  which must satisfy all displacement boundary conditions.  For present purposes, the following mode shape is reasonable

U ^ 3 ( x 1 , x 2 )=( x 1 a)( x 1 +a)( x 2 b)( x 2 +b)+C ( x 1 a) 2 ( x 1 +a) 2 ( x 2 b) 2 ( x 2 +b) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadwfagaqcamaaBaaaleaacaaIZaaabe aakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIha daWgaaWcbaGaaGOmaaqabaGccaGGPaGaeyypa0JaaiikaiaadIhada WgaaWcbaGaaGymaaqabaGccqGHsislcaWGHbGaaiykaiaacIcacaWG 4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyyaiaacMcacaGGOa GaamiEamaaBaaaleaacaaIYaaabeaakiabgkHiTiaadkgacaGGPaGa aiikaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGIbGaai ykaiabgUcaRiaadoeacaGGOaGaamiEamaaBaaaleaacaaIXaaabeaa kiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaaiikai aadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbGaaiykamaa CaaaleqabaGaaGOmaaaakiaacIcacaWG4bWaaSbaaSqaaiaaikdaae qaaOGaeyOeI0IaamOyaiaacMcadaahaaWcbeqaaiaaikdaaaGccaGG OaGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadkgacaGGPa WaaWbaaSqabeaacaaIYaaaaaaa@6769@

where C is a parameter that can be adjusted to obtain the best estimate for the natural frequency.  More terms could be added to obtain a more accurate solution.

 

  1. Calculate the kinetic energy measures

V ^ = E h 3 24(1 ν 2 ) A ( (1ν) 2 U 3 x α x β 2 U 3 x α x β +ν ( 2 U 3 x α x α ) 2 ) dA T ^ = A ( h 2 ρ U 3 U 3 ) dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmOvayaajaGaeyypa0ZaaSaaaeaaca WGfbGaamiAamaaCaaaleqabaGaaG4maaaaaOqaaiaaikdacaaI0aGa aiikaiaaigdacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaGcca GGPaaaamaapefabaWaaeWaaeaacaGGOaGaaGymaiabgkHiTiabe27a UjaacMcadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadw fadaWgaaWcbaGaaG4maaqabaaakeaacqGHciITcaWG4bWaaSbaaSqa aiabeg7aHbqabaGccqGHciITcaWG4bWaaSbaaSqaaiabek7aIbqaba aaaOWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWGvbWa aSbaaSqaaiaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacq aHXoqyaeqaaOGaeyOaIyRaamiEamaaBaaaleaacqaHYoGyaeqaaaaa kiabgUcaRiabe27aUnaabmaabaWaaSaaaeaacqGHciITdaahaaWcbe qaaiaaikdaaaGccaWGvbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaeyOa IyRaamiEamaaBaaaleaacqaHXoqyaeqaaOGaeyOaIyRaamiEamaaBa aaleaacqaHXoqyaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa aGOmaaaaaOGaayjkaiaawMcaaaWcbaGaamyqaaqab0Gaey4kIipaki aadsgacaWGbbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7ceWGubGbaKaacqGH9aqpcaaMc8UaaGPaVlaaykW7da WdrbqaamaabmaabaWaaSaaaeaacaWGObaabaGaaGOmaaaacqaHbpGC caWGvbWaaSbaaSqaaiaaiodaaeqaaOGaamyvamaaBaaaleaacaaIZa aabeaaaOGaayjkaiaawMcaaaWcbaGaamyqaaqab0Gaey4kIipakiaa dsgacaWGbbaaaa@9438@

 

  1. The frequency is estimated as ω 2 V ^ / T ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyYdC3aaWbaaSqabeaacaaIYaaaaO GaeyizImQabmOvayaajaGaai4laiqadsfagaqcaaaa@38FE@  - we therefore need to choose C to minimize V ^ / T ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmOvayaajaGaai4laiqadsfagaqcaa aa@3489@ .  Although an exact formula can be calculated for the resulting upper bound to the natural frequency, the expression is very long, and is best displayed graphically.  The figure shows the variation of normalized natural frequency as a function of the aspect ratio of the plate b/a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamOyaiaac+cacaWGHbaaaa@3482@ .  As a guide to the accuracy of the solution, an exact solution can be calculated for the natural frequency in the limit b/a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamOyaiaac+cacaWGHbGaeyOKH4Qaey OhIukaaa@37E0@  (in this limit the plate is a beam), following the procedure described in Section 10.4.1.  The result is 12(1 ν 2 ) ω 2 ρ a 4 /E h 2 = (π/2) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaaGymaiaaikdacaGGOaGaaGymaiabgk HiTiabe27aUnaaCaaaleqabaGaaGOmaaaakiaacMcacqaHjpWDdaah aaWcbeqaaiaaikdaaaGccqaHbpGCcaWGHbWaaWbaaSqabeaacaaI0a aaaOGaai4laiaadweacaWGObWaaWbaaSqabeaacaaIYaaaaOGaeyyp a0Jaaiikaiabec8aWjaac+cacaaIYaGaaiykamaaCaaaleqabaGaaG inaaaaaaa@4953@ .  It is clear that the Rayleigh-Ritz method gives an excellent estimate of the natural frequency in this limit.

 

 

 

10.7.4 Bending induced by inelastic strain in a thin film on a substrate 

 

The figure illustrates the problem to be solved.   A thin film, with Young’s modulus E f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamOzaaqabaaaaa@3269@ , Poisson’s ratio ν f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUnaaBaaaleaacaWGMbaabeaaaa a@3357@  and thickness h f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIgadaWgaaWcbaGaamOzaaqabaaaaa@328C@  is deposited onto the surface of an initially flat, circular wafer, which has Young’s modulus E s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaam4Caaqabaaaaa@3276@ , Poisson’s ratio ν s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUnaaBaaaleaacaWGZbaabeaaaa a@3364@ , radius R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbaaaa@339D@ , and thickness h s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIgadaWgaaWcbaGaam4Caaqabaaaaa@3299@ .   An inelastic strain ε 11 p = ε 22 p = ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaaIXaGaaGymaa qaaiaadchaaaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaaGOmaiaaikda aeaacaWGWbaaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaaicdaaeqaaa aa@3DB5@  is introduced into the film by some external process, which generates stresses in the film, and also causes the substrate to bend.  Provided the inelastic strain is not too large, the plate adopts a state of uniform curvature κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRbaa@323A@  (its deformed shape can be visualized as a spherical cap, with large radius of curvature 1/κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaGGVaGaeqOUdSgaaa@33A8@  ).  Our goal is to relate the curvature κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRbaa@323A@  to the inelastic strain ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIWaaabeaaaa a@3315@ , and to calculate the stress in the film.  The results are important because stresses in thin films are often determined by measuring the curvature of the substrate. 

 

The inelastic strain may be caused by a number of different processes, including

  1. A mismatch in thermal expansion between the film and the substrate.  In this case the inelastic strain is related to the thermal expansion coefficients α f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHnaaBaaaleaacaWGMbaabeaaaa a@333E@ , α s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHnaaBaaaleaacaWGZbaabeaaaa a@334B@  of the film and substrate and the temperature T by ε 0 =( α f α s )(T T 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIWaaabeaaki abg2da9iaacIcacqaHXoqydaWgaaWcbaGaamOzaaqabaGccqGHsisl cqaHXoqydaWgaaWcbaGaam4CaaqabaGccaGGPaGaaiikaiaadsfacq GHsislcaWGubWaaSbaaSqaaiaaicdaaeqaaOGaaiykaaaa@40E0@ , where T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@3247@  is the temperature at which the system is stress free (many films are approximately free of stress at deposition temperature)
  2. The film may grow epitaxially on the substrate, so that the spacing between atoms in the film is forced to match that of the substrate.  In this case the inelastic strain can be calculated as follows.  Suppose that, in their stress free states, the film and substrate have lattice spacing  a f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamOzaaqabaaaaa@3285@  and a s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaam4Caaqabaaaaa@3292@ , respectively.  Then ε 0 =( a f a s )/ a s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIWaaabeaaki abg2da9iaacIcacaWGHbWaaSbaaSqaaiaadAgaaeqaaOGaeyOeI0Ia amyyamaaBaaaleaacaWGZbaabeaakiaacMcacaGGVaGaamyyamaaBa aaleaacaWGZbaabeaaaaa@3D43@ .
  3. Mismatch strain may develop in the film as a result of the deposition process. 
  4. Mismatch strain may be developed as a result of interdiffusion and possibly chemical reactions between the film and substrate.

 

Solution: The inelastic strain in the film is related to the curvature of the substrate by

ε 0 = κ h s (1+ ρ 4 η 2 +4ρη+4ρ η 3 +6 ρ 2 η) 6ρη(1+ρ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIWaaabeaaki abg2da9maalaaabaGaeqOUdSMaamiAamaaBaaaleaacaWGZbaabeaa kiaacIcacaaIXaGaey4kaSIaeqyWdi3aaWbaaSqabeaacaaI0aaaaO Gaeq4TdG2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGinaiabeg8a YjabeE7aOjabgUcaRiaaisdacqaHbpGCcqaH3oaAdaahaaWcbeqaai aaiodaaaGccqGHRaWkcaaI2aGaeqyWdi3aaWbaaSqabeaacaaIYaaa aOGaeq4TdGMaaiykaaqaaiaaiAdacqaHbpGCcqaH3oaAcaGGOaGaaG ymaiabgUcaRiabeg8aYjaacMcaaaaaaa@5A3B@

where ρ= h f / h s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYjabg2da9iaadIgadaWgaaWcba GaamOzaaqabaGccaGGVaGaamiAamaaBaaaleaacaWGZbaabeaaaaa@3820@ , and η= E f (1 ν s )/( E s (1 ν f )) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeE7aOjabg2da9iaadweadaWgaaWcba GaamOzaaqabaGccaGGOaGaaGymaiabgkHiTiabe27aUnaaBaaaleaa caWGZbaabeaakiaacMcacaGGVaGaaiikaiaadweadaWgaaWcbaGaam 4CaaqabaGccaGGOaGaaGymaiabgkHiTiabe27aUnaaBaaaleaacaWG MbaabeaakiaacMcacaGGPaaaaa@44EA@ .  Note that with the sign convention adopted here, the substrate has a positive curvature if the film is on the convex side of the bent plate.

 

The stress in the film is related to the curvature by

σ 11 = σ 22 = κ E s (1 ν s ) [(1+ ρ 3 η) h s 6ηρz(1+ρ)] 6ρ(1+ρ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGa eyypa0ZaaSaaaeaacqGHsislcqaH6oWAcaWGfbWaaSbaaSqaaiaado haaeqaaaGcbaGaaiikaiaaigdacqGHsislcqaH9oGBdaWgaaWcbaGa am4CaaqabaGccaGGPaaaamaalaaabaGaai4waiaacIcacaaIXaGaey 4kaSIaeqyWdi3aaWbaaSqabeaacaaIZaaaaOGaeq4TdGMaaiykaiaa dIgadaWgaaWcbaGaam4CaaqabaGccqGHsislcaaI2aGaeq4TdGMaeq yWdiNaamOEaiaacIcacaaIXaGaey4kaSIaeqyWdiNaaiykaiaac2fa aeaacaaI2aGaeqyWdiNaaiikaiaaigdacqGHRaWkcqaHbpGCcaGGPa aaaaaa@6147@

where z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQhaaaa@3187@  is the distance above the mid-plane of the film.

 

In most practical situations the thickness of the substrate greatly exceeds the thickness of the film, in which case these results can be approximated by

ε 0 κ h s 2 E s (1 ν f ) 6 h f E f (1 ν s ) σ 11 = σ 22 κ h s 2 E s 6 h f (1 ν s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIWaaabeaaki abgIKi7oaalaaabaGaeqOUdSMaamiAamaaDaaaleaacaWGZbaabaGa aGOmaaaakiaadweadaWgaaWcbaGaam4CaaqabaGccaGGOaGaaGymai abgkHiTiabe27aUnaaBaaaleaacaWGMbaabeaakiaacMcaaeaacaaI 2aGaamiAamaaBaaaleaacaWGMbaabeaakiaadweadaWgaaWcbaGaam OzaaqabaGccaGGOaGaaGymaiabgkHiTiabe27aUnaaBaaaleaacaWG ZbaabeaakiaacMcaaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm3a aSbaaSqaaiaaigdacaaIXaaabeaakiabg2da9iabeo8aZnaaBaaale aacaaIYaGaaGOmaaqabaGccqGHijYUdaWcaaqaaiabgkHiTiabeQ7a RjaadIgadaqhaaWcbaGaam4CaaqaaiaaikdaaaGccaWGfbWaaSbaaS qaaiaadohaaeqaaaGcbaGaaGOnaiaadIgadaWgaaWcbaGaamOzaaqa baGccaGGOaGaaGymaiabgkHiTiabe27aUnaaBaaaleaacaWGZbaabe aakiaacMcaaaaaaa@782B@

These are known as the Stoney equations.

 

Derivation:  It is simplest to derive these results by using the general equations of shell theory to write down the potential energy of the bent plate, and then calculating the values of mid-plane strain and curvature that minimize the potential energy. To this end:

1.      We consider the plate to consist of the film and substrate together, with combined thickness h= h f + h s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIgacqGH9aqpcaWGObWaaSbaaSqaai aadAgaaeqaaOGaey4kaSIaamiAamaaBaaaleaacaWGZbaabeaaaaa@377C@ .  The mid-plane of the plate is at height h/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIgacaGGVaGaaGOmaaaa@32E4@  above the base of the substrate.

2.      We assume that the deformed plate has a small, uniform curvature κ 11 = κ 22 =κ κ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcaaMc8UaeqOUdS2aaSbaaSqaaiaaikdacaaIYaaa beaakiabg2da9iabeQ7aRjaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlabeQ7aRnaaBaaaleaacaaIXaGa aGOmaaqabaGccqGH9aqpcaaIWaaaaa@4F91@ , and mid-plane strain γ 11 = γ 22 =γ γ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcaaMc8UaaGPaVlabeo7aNnaaBaaaleaacaaIYaGa aGOmaaqabaGccqGH9aqpcqaHZoWzcaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlabeo7aNnaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqp caaIWaaaaa@4AC4@ .  As long as the curvature of the plate is small, the in-plane strain is a function only of the in-plane displacement components of the plate, while the curvature is a function only of the out-of-plane displacement (see Sect 10.6.2).  This means that γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNbaa@322F@  and κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRbaa@323A@  can be taken as independent variables that describe the deformed shape of the plate.

3.      The total strain in the substrate follows as ε 11 = ε 22 =γ+ x 3 κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGa eyypa0Jaeq4SdCMaey4kaSIaamiEamaaBaaaleaacaaIZaaabeaaki abeQ7aRbaa@3F67@ , where x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaaaaa@326E@  is the distance from the mid-plane of the plate. 

4.      The stress in the substrate is proportional to the total strain.  We assume that the plate is in a state of plane stress, so that the stress components in the substrate are

σ 11 = σ 22 = E s ( 1 ν s ) ( γ+ x 3 κ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcaaMc8Uaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaakiabg2da9maalaaabaGaamyramaaBaaaleaacaWGZbaabeaaaO qaamaabmaabaGaaGymaiabgkHiTiabe27aUnaaBaaaleaacaWGZbaa beaaaOGaayjkaiaawMcaaaaadaqadaqaaiabeo7aNjabgUcaRiaadI hadaWgaaWcbaGaaG4maaqabaGccqaH6oWAaiaawIcacaGLPaaaaaa@4AB1@

The strain energy density in the substrate can then be calculated as

U s = 1 2 σ ij ε ij = E s ( 1 ν s ) ( γ+ x 3 κ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaam4CaaqabaGccq GH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo8aZnaaBaaaleaa caWGPbGaamOAaaqabaGccqaH1oqzdaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0ZaaSaaaeaacaWGfbWaaSbaaSqaaiaadohaaeqaaaGc baWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd42aaSbaaSqaaiaadohaae qaaaGccaGLOaGaayzkaaaaamaabmaabaGaeq4SdCMaey4kaSIaamiE amaaBaaaleaacaaIZaaabeaakiabeQ7aRbGaayjkaiaawMcaamaaCa aaleqabaGaaGOmaaaakiaaykW7aaa@4FE3@

5.      In the film, the total strain includes contributions from an elastic distorsion of the lattice, together with the inelastic strain, so ε ij = ε ij e + ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWG LbaaaOGaey4kaSIaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam iCaaaaaaa@3F75@ .  The stress in the film is proportional to the elastic strain ε ij e = ε ij ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGPbGaamOAaa qaaiaadwgaaaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaamyAaiaadQga aeqaaOGaeyOeI0IaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam iCaaaaaaa@3F80@ . The nonzero components of elastic strain follow as ε 11 e = ε 22 e =γ+ x 3 κ ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaaIXaGaaGymaa qaaiaadwgaaaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaaGOmaiaaikda aeaacaWGLbaaaOGaeyypa0Jaeq4SdCMaey4kaSIaamiEamaaBaaale aacaaIZaaabeaakiabeQ7aRjabgkHiTiabew7aLnaaBaaaleaacaaI Waaabeaaaaa@44B7@  and the stress in the film is σ 11 = σ 22 = E f ( 1 ν f ) ( γ+ x 3 κ ε 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcaaMc8Uaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaakiabg2da9maalaaabaGaamyramaaBaaaleaacaWGMbaabeaaaO qaamaabmaabaGaaGymaiabgkHiTiabe27aUnaaBaaaleaacaWGMbaa beaaaOGaayjkaiaawMcaaaaadaqadaqaaiabeo7aNjabgUcaRiaadI hadaWgaaWcbaGaaG4maaqabaGccqaH6oWAcqGHsislcqaH1oqzdaWg aaWcbaGaaGimaaqabaaakiaawIcacaGLPaaaaaa@4E1B@ . The strain energy density in the film is

U f = 1 2 σ ij ε ij e = E f ( 1 ν f ) ( γ+ x 3 κ ε 0 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamOzaaqabaGccq GH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo8aZnaaBaaaleaa caWGPbGaamOAaaqabaGccqaH1oqzdaqhaaWcbaGaamyAaiaadQgaae aacaWGLbaaaOGaeyypa0ZaaSaaaeaacaWGfbWaaSbaaSqaaiaadAga aeqaaaGcbaWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd42aaSbaaSqaai aadAgaaeqaaaGccaGLOaGaayzkaaaaamaabmaabaGaeq4SdCMaey4k aSIaamiEamaaBaaaleaacaaIZaaabeaakiabeQ7aRjabgkHiTiabew 7aLnaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaakiaaykW7aaa@542B@

6.      The total potential energy of the system is the integral of the strain energy density

V= 0 R h/2 h/2 h f 2πr U s d x 3 dr + 0 R h/2 h f h/2 2πr U f d x 3 dr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfacqGH9aqpdaWdXbqaaiaaykW7ca aMc8+aa8qCaeaacaaIYaGaeqiWdaNaamOCaiaadwfadaWgaaWcbaGa am4CaaqabaGccaWGKbGaamiEamaaBaaaleaacaaIZaaabeaakiaads gacaWGYbaaleaacqGHsislcaWGObGaai4laiaaikdaaeaacaWGObGa ai4laiaaikdacqGHsislcaWGObWaaSbaaWqaaiaadAgaaeqaaaqdcq GHRiI8aOGaey4kaScaleaacaaIWaaabaGaamOuaaqdcqGHRiI8aOWa a8qCaeaacaaMc8UaaGPaVpaapehabaGaaGOmaiabec8aWjaadkhaca WGvbWaaSbaaSqaaiaadAgaaeqaaOGaamizaiaadIhadaWgaaWcbaGa aG4maaqabaGccaWGKbGaamOCaaWcbaGaamiAaiaac+cacaaIYaGaey OeI0IaamiAamaaBaaameaacaWGMbaabeaaaSqaaiaadIgacaGGVaGa aGOmaaqdcqGHRiI8aaWcbaGaaGimaaqaaiaadkfaa0Gaey4kIipaaa a@6AF2@

The resulting expression is lengthy and will not be written out here.

7.      Finally, the equilibrium values of γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNbaa@322F@  and κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRbaa@323A@  can be calculated from the condition that the potential energy must be a minimum, which requires that

V κ =0 V γ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamOvaaqaaiabgk Gi2kabeQ7aRbaacqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVpaalaaabaGaeyOaIyRaamOvaaqaaiabgkGi2kabeo7aNbaa cqGH9aqpcaaIWaaaaa@52DE@

Solving the resulting linear equations for γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNbaa@322F@  and ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIWaaabeaaaa a@3315@  in terms of κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRbaa@323A@  gives the formula relating ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIWaaabeaaaa a@3315@  and κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRbaa@323A@ ; substituting the results into the formula for stress in (5) and setting x 3 =(h h f )/2+z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaGGOaGaamiAaiabgkHiTiaadIgadaWgaaWcbaGaamOzaaqa baGccaGGPaGaai4laiaaikdacqGHRaWkcaWG6baaaa@3C0F@  gives the formula for stress. 

 

 

 

10.7.5 Bending of a circular plate caused by a through-thickness temperature gradient

 

The figure illustrates the problem to be solved.  An initially flat, circular plate, which lies in the e 1 , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaGcca GGSaGaaCyzamaaBaaaleaacaaIYaaabeaaaaa@34ED@  plane is free of external force.  The plate has Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@ , thermal expansion coefficient α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqySdegaaa@33A1@ , radius R and thickness h.  Its edges are free.  The plate is heated on one face, and cooled on the other, so as to establish a temperature distribution T= T 0 +ΔT x 3 /h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamivaiabg2da9iaadsfadaWgaaWcba GaaGimaaqabaGccqGHRaWkcqqHuoarcaWGubGaamiEamaaBaaaleaa caaIZaaabeaakiaac+cacaWGObaaaa@3C5B@  through the thickness of the plate.  Here, T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamivamaaBaaaleaacaaIWaaabeaaaa a@33C1@  is the temperature of the mid-plane of the plate, while ΔT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeuiLdqKaamivaaaa@3441@  is the drop in temperature across the plate.  The thermal expansion of the plate causes it to bend: our objective is to estimate the curvature of the plate as a function of the temperature gradient ΔT/h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeuiLdqKaamivaiaac+cacaWGObaaaa@35E1@ .  The solution will account for large out-of-plane deflections, and will predict that the plate buckles when the temperature gradient reaches a critical value.

 

We will derive an approximate solution, by assuming that the curvature of the plate is uniform.  Since we are interested in calculating the plate’s shape after buckling, the solution is obtained by means of the Von-Karman theory described in Section 10.6.3. We denote the two principal curvatures of the deformed plate by κ 1 , κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqOUdS2aaSbaaSqaaiaaigdaaeqaaO GaaiilaiabeQ7aRnaaBaaaleaacaaIYaaabeaaaaa@37EF@ .   There are three possible equilibrium configurations, as follows

1.      For temperature gradients satisfying

αΔT R 2 (1+ν) 3/2 4 h 2 <2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaacqaHXoqycqqHuoarcaWGub GaamOuamaaCaaaleqabaGaaGOmaaaakiaacIcacaaIXaGaey4kaSIa eqyVd4MaaiykamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaGcba GaaGinaiaadIgadaahaaWcbeqaaiaaikdaaaaaaOGaeyipaWJaaGOm aaaa@4329@

the plate bends into a spherical cap shape, with two equal principal curvatures.  The curvatures are related to the temperature gradient by

αΔT R 2 (1+ν) 3/2 4 h 2 = κ ¯ ( κ ¯ 2 (1ν)+(1+ν) ) κ 1 = κ 2 = 4h κ ¯ R 2 1+ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaacqaHXoqycqqHuoarcaWGub GaamOuamaaCaaaleqabaGaaGOmaaaakiaacIcacaaIXaGaey4kaSIa eqyVd4MaaiykamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaGcba GaaGinaiaadIgadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0JafqOU dSMbaebadaqadaqaaiqbeQ7aRzaaraWaaWbaaSqabeaacaaIYaaaaO GaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaey4kaSIaaiikaiaa igdacqGHRaWkcqaH9oGBcaGGPaaacaGLOaGaayzkaaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7cqaH6oWAdaWgaaWcbaGaaGymaaqabaGccqGH9aqp cqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaMc8UaaGPaVl aaykW7caaMc8+aaSaaaeaacaaI0aGaamiAaiqbeQ7aRzaaraaabaGa amOuamaaCaaaleqabaGaaGOmaaaakmaakaaabaGaaGymaiabgUcaRi abe27aUbWcbeaaaaaaaa@87E2@

2.      For temperature gradients

αΔT R 2 (1+ν) 3/2 4 h 2 >2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaacqaHXoqycqqHuoarcaWGub GaamOuamaaCaaaleqabaGaaGOmaaaakiaacIcacaaIXaGaey4kaSIa eqyVd4MaaiykamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaGcba GaaGinaiaadIgadaahaaWcbeqaaiaaikdaaaaaaOGaeyOpa4JaaGOm aaaa@432D@

the solution (1) is still a possible equilibrium configuration, but is unstable.  There are infinitely many additional stable configurations, which have two unequal principal curvatures.  One of these solutions can be related to the temperature gradient by

αΔT R 2 (1+ν) 3/2 4 h 2 = κ ^ 2 +1 κ ^ κ 1 = 4h κ ^ R 2 1+ν κ 2 = 4h κ ^ R 2 1+ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaacqaHXoqycqqHuoarcaWGub GaamOuamaaCaaaleqabaGaaGOmaaaakiaacIcacaaIXaGaey4kaSIa eqyVd4MaaiykamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaGcba GaaGinaiaadIgadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0ZaaSaa aeaacuaH6oWAgaqcamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaig daaeaacuaH6oWAgaqcaaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqOUdS2aaSbaaSqaai aaigdaaeqaaOGaeyypa0JaaGPaVlaaykW7caaMc8+aaSaaaeaacaaI 0aGaamiAaaqaaiqbeQ7aRzaajaGaamOuamaaCaaaleqabaGaaGOmaa aakmaakaaabaGaaGymaiabgUcaRiabe27aUbWcbeaaaaGccaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqOUdS2aaSbaaS qaaiaaikdaaeqaaOGaeyypa0JaaGPaVlaaykW7caaMc8+aaSaaaeaa caaI0aGaamiAaiqbeQ7aRzaajaaabaGaamOuamaaCaaaleqabaGaaG OmaaaakmaakaaabaGaaGymaiabgUcaRiabe27aUbWcbeaaaaaaaa@84DB@

The other solutions have the same principal curvatures, but the principal directions are different.

 

These results are displayed by plotting the normalized curvature κ 1 R 2 1+ν /(4h) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqOUdS2aaSbaaSqaaiaaigdaaeqaaO GaamOuamaaCaaaleqabaGaaGOmaaaakmaakaaabaGaaGymaiabgUca Riabe27aUbWcbeaakiaac+cacaGGOaGaaGinaiaadIgacaGGPaaaaa@3DA0@  as a function of the dimensionless temperature gradient αΔT R 2 (1+ν) 3/2 /(4 h 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqySdeMaeuiLdqKaamivaiaadkfada ahaaWcbeqaaiaaikdaaaGccaGGOaGaaGymaiabgUcaRiabe27aUjaa cMcadaahaaWcbeqaaiaaiodacaGGVaGaaGOmaaaakiaac+cacaGGOa GaaGinaiaadIgadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@4365@  in the figure to the right, for a Poisson’s ratio ν=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUjabg2da9iaaicdacaGGUaGaaG 4maaaa@356F@  (the graph is virtually identical for other values of ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3240@  ). To visualize the significance of the graph, suppose that the temperature drop across the plate is gradually increased from zero.  The plate will first deform with two equal principal curvatures, which are related to the temperature by the formula given in (1).  At the critical temperature, the plate will buckle, and assume one of the two possible equilibrium configurations, with two unequal principal curvatures.

 

 

Derivation: The solution is derived by approximating the shape of the plate, and selecting the deformed shape that minimizes the potential energy. 

1.      The displacement of the mid-plane of the plate will be approximated as

u 3 = κ 1 x 1 2 /2+ κ 2 x 2 2 /2 u 1 = A 1 x 1 + A 2 x 1 3 + A 3 x 1 x 2 2 u 2 = B 1 x 2 + B 2 x 2 3 + B 3 x 2 x 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcqaH6oWAdaWgaaWcbaGaaGymaaqabaGccaWG4bWaa0baaSqa aiaaigdaaeaacaaIYaaaaOGaai4laiaaikdacqGHRaWkcqaH6oWAda WgaaWcbaGaaGOmaaqabaGccaWG4bWaa0baaSqaaiaaikdaaeaacaaI YaaaaOGaai4laiaaikdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaadwhadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaWGbbWaaS baaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaacaaIXaaabeaakiab gUcaRiaadgeadaWgaaWcbaGaaGOmaaqabaGccaWG4bWaa0baaSqaai aaigdaaeaacaaIZaaaaOGaey4kaSIaamyqamaaBaaaleaacaaIZaaa beaakiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG4bWaa0baaSqaai aaikdaaeaacaaIYaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWG1bWaaSbaaSqaaiaaikdaaeqaaOGaey ypa0JaamOqamaaBaaaleaacaaIXaaabeaakiaadIhadaWgaaWcbaGa aGOmaaqabaGccqGHRaWkcaWGcbWaaSbaaSqaaiaaikdaaeqaaOGaam iEamaaDaaaleaacaaIYaaabaGaaG4maaaakiabgUcaRiaadkeadaWg aaWcbaGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaam iEamaaDaaaleaacaaIXaaabaGaaGOmaaaaaaa@89E9@

where κ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIXaaabeaaaa a@3321@  and κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIYaaabeaaaa a@3322@  are the two principal curvatures of the plate, and A i , B i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaamyAaaqabaGcca GGSaGaamOqamaaBaaaleaacaWGPbaabeaaaaa@3503@  are six adjustable parameters that must be selected to minimize the potential energy of the plate.

2.      The total strain in the plate must be calculated using the Von-Karman formulas in Section 10.6.3, which yield

ε αβ = 1 2 ( u α x β + u β x α + u 3 x α u 3 x β )+ x 3 κ αβ { ε 11 = A 1 +3 A 2 x 1 2 + A 3 x 2 2 + κ 1 2 x 1 2 /2+ x 3 κ 1 ε 22 = B 1 +3 B 2 x 2 2 + B 3 x 1 2 + κ 2 2 x 2 2 /2+ x 3 κ 2 ε 12 =( A 3 + B 3 ) x 1 x 2 + κ 1 κ 2 x 1 x 2 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacqaHXoqycqaHYo GyaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqa amaalaaabaGaeyOaIyRaamyDamaaBaaaleaacqaHXoqyaeqaaaGcba GaeyOaIyRaamiEamaaBaaaleaacqaHYoGyaeqaaaaakiabgUcaRmaa laaabaGaeyOaIyRaamyDamaaBaaaleaacqaHYoGyaeqaaaGcbaGaey OaIyRaamiEamaaBaaaleaacqaHXoqyaeqaaaaakiabgUcaRmaalaaa baGaeyOaIyRaamyDamaaBaaaleaacaaIZaaabeaaaOqaaiabgkGi2k aadIhadaWgaaWcbaGaeqySdegabeaaaaGcdaWcaaqaaiabgkGi2kaa dwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciITcaWG4bWaaSbaaS qaaiabek7aIbqabaaaaaGccaGLOaGaayzkaaGaey4kaSIaamiEamaa BaaaleaacaaIZaaabeaakiabeQ7aRnaaBaaaleaacqaHXoqycqaHYo GyaeqaaOGaeyO0H49aaiqaaqaabeqaaiabew7aLnaaBaaaleaacaaI XaGaaGymaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaaigdaaeqaaO Gaey4kaSIaaG4maiaadgeadaWgaaWcbaGaaGOmaaqabaGccaWG4bWa a0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSIaamyqamaaBaaale aacaaIZaaabeaakiaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaGc cqGHRaWkcqaH6oWAdaqhaaWcbaGaaGymaaqaaiaaikdaaaGccaWG4b Waa0baaSqaaiaaigdaaeaacaaIYaaaaOGaai4laiaaikdacqGHRaWk caWG4bWaaSbaaSqaaiaaiodaaeqaaOGaeqOUdS2aaSbaaSqaaiaaig daaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaaikdacaaIYaaabeaakiab g2da9iaadkeadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaaIZaGaam OqamaaBaaaleaacaaIYaaabeaakiaadIhadaqhaaWcbaGaaGOmaaqa aiaaikdaaaGccqGHRaWkcaWGcbWaaSbaaSqaaiaaiodaaeqaaOGaam iEamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiabeQ7aRnaa DaaaleaacaaIYaaabaGaaGOmaaaakiaadIhadaqhaaWcbaGaaGOmaa qaaiaaikdaaaGccaGGVaGaaGOmaiabgUcaRiaadIhadaWgaaWcbaGa aG4maaqabaGccqaH6oWAdaWgaaWcbaGaaGOmaaqabaaakeaacqaH1o qzdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Jaaiikaiaadgea daWgaaWcbaGaaG4maaqabaGccqGHRaWkcaWGcbWaaSbaaSqaaiaaio daaeqaaOGaaiykaiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG4bWa aSbaaSqaaiaaikdaaeqaaOGaey4kaSIaeqOUdS2aaSbaaSqaaiaaig daaeqaaOGaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaamiEamaaBaaa leaacaaIXaaabeaakiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGVa GaaGOmaaaacaGL7baaaaa@BEE4@

3.      The plate is assumed to be in a state of plane stress: the stress components can be calculated using the plane stress version of the linear elastic constitutive equations

σ αβ = E 1+ν { ε αβ + ν 1ν ε γγ δ αβ } Eα( T 0 +ΔT x 3 /h) 1ν δ αβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaHXoqycqaHYo GyaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgUcaRiab e27aUbaadaGadaqaaiabew7aLnaaBaaaleaacqaHXoqycqaHYoGyae qaaOGaey4kaSYaaSaaaeaacqaH9oGBaeaacaaIXaGaeyOeI0IaeqyV d4gaaiabew7aLnaaBaaaleaacqaHZoWzcqaHZoWzaeqaaOGaeqiTdq 2aaSbaaSqaaiabeg7aHjabek7aIbqabaaakiaawUhacaGL9baacqGH sisldaWcaaqaaiaadweacqaHXoqycaGGOaGaamivamaaBaaaleaaca aIWaaabeaakiabgUcaRiabfs5aejaadsfacaWG4bWaaSbaaSqaaiaa iodaaeqaaOGaai4laiaadIgacaGGPaaabaGaaGymaiabgkHiTiabe2 7aUbaacqaH0oazdaWgaaWcbaGaeqySdeMaeqOSdigabeaaaaa@681D@

4.      The strain energy density in the plate can be calculated using the formulas given in Section 3.1.7 as U=((1+ν) σ αβ σ αβ ν σ γγ σ αα )/2E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacqGH9aqpcaGGOaGaaiikaiaaig dacqGHRaWkcqaH9oGBcaGGPaGaeq4Wdm3aaSbaaSqaaiabeg7aHjab ek7aIbqabaGccqaHdpWCdaWgaaWcbaGaeqySdeMaeqOSdigabeaaki abgkHiTiabe27aUjabeo8aZnaaBaaaleaacqaHZoWzcqaHZoWzaeqa aOGaeq4Wdm3aaSbaaSqaaiabeg7aHjabeg7aHbqabaGccaGGPaGaai 4laiaaikdacaWGfbaaaa@523D@ .  The result is lengthy and is best calculated using a symbolic manipulation program.

5.      The total strain energy of the plate follows by integrating the strain energy density over the volume of the plate as

Φ= 0 R 0 2π h/2 h/2 U(r,θ, x 3 )d x 3 dθ r dr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfA6agjabg2da9maapehabaWaa8qCae aadaWdXbqaaiaadwfacaGGOaGaamOCaiaacYcacqaH4oqCcaGGSaGa amiEamaaBaaaleaacaaIZaaabeaakiaacMcacaWGKbGaamiEamaaBa aaleaacaaIZaaabeaaaeaacqGHsislcaWGObGaai4laiaaikdaaeaa caWGObGaai4laiaaikdaa0Gaey4kIipakiaadsgacqaH4oqCaSqaai aaicdaaeaacaaIYaGaeqiWdahaniabgUIiYdGccaWGYbaaleaacaaI WaaabaGaamOuaaqdcqGHRiI8aOGaamizaiaadkhaaaa@5534@ .

 To evaluate the integral, the strain energy density can be expressed in polar coordinates by substituting x 1 =rcosθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGYbGaci4yaiaac+gacaGGZbGaeqiUdehaaa@38FC@ , x 2 =rsinθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGYbGaci4CaiaacMgacaGGUbGaeqiUdehaaa@3902@  into the results of (4).  Again, a symbolic manipulation program makes the algebra painless.

6.      The coefficients A i , B i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaamyAaaqabaGcca GGSaGaamOqamaaBaaaleaacaWGPbaabeaaaaa@3503@  and the curvatures must now be determined by minimizing the potential energy Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfA6agbaa@3202@ .  To proceed, we first calculate the coefficients A i , B i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaamyAaaqabaGcca GGSaGaamOqamaaBaaaleaacaWGPbaabeaaaaa@3503@  in terms of the temperature gradient and curvature by solving the six simultaneous equations Φ/ A i =0Φ/ B i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kabfA6agjaac+cacqGHciITca WGbbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGimaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyOaIyRaeu OPdyKaai4laiabgkGi2kaadkeadaWgaaWcbaGaamyAaaqabaGccqGH 9aqpcaaIWaaaaa@4E27@ .  Substituting the resulting formulas for A i , B i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaamyAaaqabaGcca GGSaGaamOqamaaBaaaleaacaWGPbaabeaaaaa@3503@  back into the results of (5), and using the two remaining conditions Φ/ κ 1 =0Φ/ κ 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kabfA6agjaac+cacqGHciITcq aH6oWAdaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIWaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyOaIyRaeuOPdyKaai4lai abgkGi2kabeQ7aRnaaBaaaleaacaaIYaaabeaakiabg2da9iaaicda aaa@4C83@  yields two equations for κ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIXaaabeaaaa a@3321@  and κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIYaaabeaaaa a@3322@

κ 1 κ 2 2 (1 ν 2 ) R 4 +16 h 2 κ 1 +16 h 2 ν κ 2 16(1+ν)αΔTh=0 κ 2 κ 1 2 (1 ν 2 ) R 4 +16 h 2 κ 2 +16 h 2 ν κ 1 16(1+ν)αΔTh=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeqOUdS2aaSbaaSqaaiaaigdaae qaaOGaeqOUdS2aa0baaSqaaiaaikdaaeaacaaIYaaaaOGaaiikaiaa igdacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGGPaGaam OuamaaCaaaleqabaGaaGinaaaakiabgUcaRiaaigdacaaI2aGaamiA amaaCaaaleqabaGaaGOmaaaakiabeQ7aRnaaBaaaleaacaaIXaaabe aakiabgUcaRiaaigdacaaI2aGaamiAamaaCaaaleqabaGaaGOmaaaa kiabe27aUjabeQ7aRnaaBaaaleaacaaIYaaabeaakiabgkHiTiaaig dacaaI2aGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaeqySdeMa euiLdqKaamivaiaadIgacqGH9aqpcaaIWaaabaGaeqOUdS2aaSbaaS qaaiaaikdaaeqaaOGaeqOUdS2aa0baaSqaaiaaigdaaeaacaaIYaaa aOGaaiikaiaaigdacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaa GccaGGPaGaamOuamaaCaaaleqabaGaaGinaaaakiabgUcaRiaaigda caaI2aGaamiAamaaCaaaleqabaGaaGOmaaaakiabeQ7aRnaaBaaale aacaaIYaaabeaakiabgUcaRiaaigdacaaI2aGaamiAamaaCaaaleqa baGaaGOmaaaakiabe27aUjabeQ7aRnaaBaaaleaacaaIXaaabeaaki abgkHiTiaaigdacaaI2aGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGG PaGaeqySdeMaeuiLdqKaamivaiaadIgacqGH9aqpcaaIWaaaaaa@85BB@

7.      Eliminating the temperature from these equations and simplifying the result gives the expression

( κ 1 κ 2 )( κ 1 κ 2 R 4 (1+ν)16 h 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacqaH6oWAdaWgaaWcbaGaaGymaa qabaGccqGHsislcqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccaGGPaGa aiikaiabeQ7aRnaaBaaaleaacaaIXaaabeaakiabeQ7aRnaaBaaale aacaaIYaaabeaakiaadkfadaahaaWcbeqaaiaaisdaaaGccaGGOaGa aGymaiabgUcaRiabe27aUjaacMcacqGHsislcaaIXaGaaGOnaiaadI gadaahaaWcbeqaaiaaikdaaaGccaGGPaGaeyypa0JaaGimaaaa@4B37@

This shows that there are two possible equilibrium configurations: in the first, κ 1 = κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIXaaabeaaki abg2da9iabeQ7aRnaaBaaaleaacaaIYaaabeaaaaa@36CB@ ; in the second, the two curvatures are related by κ 1 R 4 (1+ν)=16 h 2 / κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIXaaabeaaki aadkfadaahaaWcbeqaaiaaisdaaaGccaGGOaGaaGymaiabgUcaRiab e27aUjaacMcacqGH9aqpcaaIXaGaaGOnaiaadIgadaahaaWcbeqaai aaikdaaaGccaGGVaGaeqOUdS2aaSbaaSqaaiaaikdaaeqaaaaa@4153@ .  Finally, these two possible relationships can be substituted back into either of the two equations in (6) to relate the temperature gradient to the curvatures.

 

 

 

10.7.6 Buckling of a cylindrical shell subjected to axial loading

 

The figure illustrates a thin-walled cylinder, with radius a, height L and wall thickness h.  The shell is made from a linear elastic solid with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3240@ .  It is loaded in compression by subjecting its ends to a prescribed axial displacement Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aebaa@31EE@ .   We wish to estimate the critical axial strain Δ/L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaac+cacaWGmbaaaa@3372@  or axial force P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfaaaa@315D@  that will cause the cylinder to buckle.

 

We will derive an approximate solution, by assuming a shape for the buckled cylinder, and calculating the shape that minimizes the potential energy of the system.   Specifically, we assume that the radial displacement of the surface of the cylinder at the instant of buckling has the form u r =C+Asin(λπz/L) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamOCaaqabaGccq GH9aqpcaWGdbGaey4kaSIaamyqaiGacohacaGGPbGaaiOBaiaacIca cqaH7oaBcqaHapaCcaWG6bGaai4laiaadYeacaGGPaaaaa@404A@ .  The solution shows that:

1.      Buckling occurs at a critical axial strain Δ/L=h/( 3 a(1 ν 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaac+cacaWGmbGaeyypa0Jaam iAaiaac+cadaqadaqaamaakaaabaGaaG4maaWcbeaakiaadggacaGG OaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaakiaacM caaiaawIcacaGLPaaaaaa@3F15@

2.      The corresponding axial load is P=2πE h 2 /( (1 ν 2 ) 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfacqGH9aqpcaaIYaGaeqiWdaNaam yraiaadIgadaahaaWcbeqaaiaaikdaaaGccaGGVaWaaeWaaeaacaGG OaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaakiaacM cadaGcaaqaaiaaiodaaSqabaaakiaawIcacaGLPaaaaaa@4050@

3.      The wavelength of the buckling mode is L/λ=π ah / (12) 1/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeacaGGVaGaeq4UdWMaeyypa0Jaeq iWda3aaOaaaeaacaWGHbGaamiAaaWcbeaakiaac+cacaGGOaGaaGym aiaaikdacaGGPaWaaWbaaSqabeaacaaIXaGaai4laiaaisdaaaaaaa@3E57@ .  Note that the buckling mode describes the shape of the cylinder at the instant when buckling begins; it does not correspond to the shape of the cylinder after buckling.

 

The exact buckling strain and load are a factor 1 ν 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaakaaabaGaaGymaiabgkHiTiabe27aUn aaCaaaleqabaGaaGOmaaaaaeqaaaaa@34E1@  smaller than this approximate result.

 

 

Derivation: We must calculate, and then minimize, the potential energy of the cylinder.  It is convenient to work through this problem using curvilinear coordinates: we shall use cylindrical-polar coordinates  ξ 1 z, ξ 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe67a4naaBaaaleaacaaIXaaabeaaki abggMi6kaadQhacaGGSaGaeqOVdG3aaSbaaSqaaiaaikdaaeqaaOGa eyyyIORaeqiUdehaaa@3CE8@  to identify a point on the mid-plane of the shell.  

1.      The position vector of a point in the undeformed shell is r ¯ =acosθ e 1 +asinθ e 2 +z e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahkhagaqeaiabg2da9iaadggaciGGJb Gaai4BaiaacohacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIaamyyaiGacohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaa WcbaGaaGOmaaqabaGccqGHRaWkcaWG6bGaaCyzamaaBaaaleaacaaI Zaaabeaaaaa@45DD@

2.      The natural basis vectors for the undeformed shell are therefore

m ¯ 1 = r ¯ z = e 3 m ¯ 2 = r ¯ θ =asinθ e 1 +acosθ e 2 m ¯ 3 =(cosθ e 1 +sinθ e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaBaaaleaacaaIXaaabe aakiabg2da9maalaaabaGaeyOaIyRabCOCayaaraaabaGaeyOaIyRa amOEaaaacqGH9aqpcaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlqah2gagaqeamaa BaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaeyOaIyRabCOCay aaraaabaGaeyOaIyRaeqiUdehaaiabg2da9iabgkHiTiaadggaciGG ZbGaaiyAaiaac6gacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaO Gaey4kaSIaamyyaiGacogacaGGVbGaai4CaiabeI7aXjaahwgadaWg aaWcbaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7ceWHTbGbaebadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcqGH sislcaGGOaGaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzamaaBaaale aacaaIXaaabeaakiabgUcaRiGacohacaGGPbGaaiOBaiabeI7aXjaa hwgadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@7C98@

and the reciprocal basis vectors are m ¯ 1 = e 3 m ¯ 2 =(sinθ e 1 +cosθ e 2 )/a m ¯ 3 = m ¯ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaCaaaleqabaGaaGymaa aakiabg2da9iaahwgadaWgaaWcbaGaaG4maaqabaGccaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlqah2gagaqeam aaCaaaleqabaGaaGOmaaaakiabg2da9iaacIcacqGHsislciGGZbGa aiyAaiaac6gacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzamaaBaaaleaacaaI YaaabeaakiaacMcacaGGVaGaamyyaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ceWHTbGbaeba daahaaWcbeqaaiaaiodaaaGccqGH9aqpceWHTbGbaebadaWgaaWcba GaaG4maaqabaaaaa@6AD2@

3.      The components of the metric tensor are g ¯ αβ = m ¯ α m ¯ β g ¯ 11 =1, g ¯ 12 =0, g ¯ 22 =1/ a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadEgagaqeamaaCaaaleqabaGaeqySde MaeqOSdigaaOGaeyypa0JabCyBayaaraWaaWbaaSqabeaacqaHXoqy aaGccqGHflY1ceWHTbGbaebadaahaaWcbeqaaiabek7aIbaakiabgk DiElqadEgagaqeamaaCaaaleqabaGaaGymaiaaigdaaaGccqGH9aqp caaIXaGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl qadEgagaqeamaaCaaaleqabaGaaGymaiaaikdaaaGccqGH9aqpcaaI WaGaaiilaiaaykW7caaMc8Uabm4zayaaraWaaWbaaSqabeaacaaIYa GaaGOmaaaakiabg2da9iaaigdacaGGVaGaamyyamaaCaaaleqabaGa aGOmaaaaaaa@5E0C@ , g ¯ αβ = m ¯ α m ¯ β g ¯ 11 =1 g ¯ 12 =0 g ¯ 22 = a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadEgagaqeamaaBaaaleaacqaHXoqycq aHYoGyaeqaaOGaeyypa0JabCyBayaaraWaaSbaaSqaaiabeg7aHbqa baGccqGHflY1ceWHTbGbaebadaWgaaWcbaGaeqOSdigabeaakiabgk DiElqadEgagaqeamaaBaaaleaacaaIXaGaaGymaaqabaGccqGH9aqp caaIXaGaaGPaVlaaykW7caaMc8UaaGPaVlqadEgagaqeamaaBaaale aacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8Uabm4zayaaraWaaSbaaSqaaiaaikdaca aIYaaabeaakiabg2da9iaadggadaahaaWcbeqaaiaaikdaaaaaaa@5E4E@

4.      The covariant components of the curvature tensor for the undeformed shell are

κ ¯ αβ = m ¯ 3 m ¯ α ξ β κ ¯ 11 = κ ¯ 12 =0 κ ¯ 22 =a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeQ7aRzaaraWaaSbaaSqaaiabeg7aHj abek7aIbqabaGccqGH9aqpcqGHsislceWHTbGbaebadaWgaaWcbaGa aG4maaqabaGccqGHflY1daWcaaqaaiabgkGi2kqah2gagaqeamaaBa aaleaacqaHXoqyaeqaaaGcbaGaeyOaIyRaeqOVdG3aaSbaaSqaaiab ek7aIbqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHsh I3caaMc8UaaGPaVlaaykW7caaMc8UafqOUdSMbaebadaWgaaWcbaGa aGymaiaaigdaaeqaaOGaeyypa0JafqOUdSMbaebadaWgaaWcbaGaaG ymaiaaikdaaeqaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cuaH6oWAgaqeamaaBaaaleaacaaIYa GaaGOmaaqabaGccqGH9aqpcqGHsislcaWGHbaaaa@7070@

5.      The position vector of the mid-plane of the deformed shell is approximated as

r=( a+C+Asinλπz/L )( cosθ e 1 +sinθ e 2 )+(1Δ/L)z e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkhacqGH9aqpdaqadaqaaiaadggacq GHRaWkcaWGdbGaey4kaSIaamyqaiGacohacaGGPbGaaiOBaiabeU7a Sjabec8aWjaadQhacaGGVaGaamitaaGaayjkaiaawMcaamaabmaaba Gaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzamaaBaaaleaacaaIXaaa beaakiabgUcaRiGacohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaa WcbaGaaGOmaaqabaaakiaawIcacaGLPaaacqGHRaWkcaGGOaGaaGym aiabgkHiTiabfs5aejaac+cacaWGmbGaaiykaiaadQhacaWHLbWaaS baaSqaaiaaiodaaeqaaaaa@59FA@

It will greatly simplify subsequent calculations to assume a priori that λ>>1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabg6da+iabg6da+iaaigdaaa a@3507@

6.      The natural basis vectors for the deformed shell are therefore

m 1 = r z =( Aλπ/L )sinλπz/L( cosθ e 1 +sinθ e 2 )+(1Δ/L) e 3 m 2 = r θ =( a+C+Asinλπz/L )( sinθ e 1 +cosθ e 2 ) m 3 ( Aλπ/L )cos(λπz/L) e 3 ( cosθ e 1 +sinθ e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaCyBamaaBaaaleaacaaIXaaabe aakiabg2da9maalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2kaadQha aaGaeyypa0ZaaeWaaeaacaWGbbGaeq4UdWMaeqiWdaNaai4laiaadY eaaiaawIcacaGLPaaaciGGZbGaaiyAaiaac6gacqaH7oaBcqaHapaC caWG6bGaai4laiaadYeadaqadaqaaiGacogacaGGVbGaai4CaiabeI 7aXjaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkciGGZbGaaiyA aiaac6gacqaH4oqCcaWHLbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOa GaayzkaaGaey4kaSIaaiikaiaaigdacqGHsislcqqHuoarcaGGVaGa amitaiaacMcacaWHLbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaaCyBam aaBaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaeyOaIyRaaCOC aaqaaiabgkGi2kabeI7aXbaacqGH9aqpdaqadaqaaiaadggacqGHRa WkcaWGdbGaey4kaSIaamyqaiGacohacaGGPbGaaiOBaiabeU7aSjab ec8aWjaadQhacaGGVaGaamitaaGaayjkaiaawMcaamaabmaabaGaey OeI0Iaci4CaiaacMgacaGGUbGaeqiUdeNaaCyzamaaBaaaleaacaaI XaaabeaakiabgUcaRiGacogacaGGVbGaai4CaiabeI7aXjaahwgada WgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaeaacaWHTbWaaSba aSqaaiaaiodaaeqaaOGaeyisIS7aaeWaaeaacaWGbbGaeq4UdWMaeq iWdaNaai4laiaadYeaaiaawIcacaGLPaaaciGGJbGaai4Baiaacoha caGGOaGaeq4UdWMaeqiWdaNaamOEaiaac+cacaWGmbGaaiykaiaahw gadaWgaaWcbaGaaG4maaqabaGccqGHsisldaqadaqaaiGacogacaGG VbGaai4CaiabeI7aXjaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRa WkciGGZbGaaiyAaiaac6gacqaH4oqCcaWHLbWaaSbaaSqaaiaaikda aeqaaaGccaGLOaGaayzkaaaaaaa@B0A1@

7.      The covariant components of the metric tensor and curvature for the deformed shell follow as

g αβ = m α m β g 11 =( λπA/L ) 2 cos 2 (λπz/L)+ ( 1Δ/L ) 2 g 12 =0 g 22 = ( a+C+Asin(λπz/L) ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4zamaaBaaaleaacqaHXoqycq aHYoGyaeqaaOGaeyypa0JaaCyBamaaBaaaleaacqaHXoqyaeqaaOGa eyyXICTaaCyBamaaBaaaleaacqaHYoGyaeqaaOGaeyO0H4Taam4zam aaBaaaleaacaaIXaGaaGymaaqabaGccqGH9aqpdaqadaqaaiabeU7a Sjabec8aWjaadgeacaGGVaGaamitaiaaykW7aiaawIcacaGLPaaaca aMc8+aaWbaaSqabeaacaaIYaaaaOGaci4yaiaac+gacaGGZbWaaWba aSqabeaacaaIYaaaaOGaaiikaiabeU7aSjabec8aWjaadQhacaGGVa GaamitaiaacMcacqGHRaWkdaqadaqaaiaaigdacqGHsislcqqHuoar caGGVaGaamitaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaO qaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWGNbWaaSbaaSqaaiaaigdacaaIYaaabe aakiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWGNbWaaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da9maabm aabaGaamyyaiabgUcaRiaadoeacqGHRaWkcaWGbbGaci4CaiaacMga caGGUbGaaiikaiabeU7aSjabec8aWjaadQhacaGGVaGaamitaiaacM caaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaaa@C686@

κ αβ = m 3 m α ξ β κ 11 (λπ/L) 2 Asin(λπz/L) κ 12 =0 κ 22 (a+C+Asin(λπz/L)) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeqOUdS2aaSbaaSqaaiabeg7aHj abek7aIbqabaGccqGH9aqpcqGHsislcaWHTbWaaSbaaSqaaiaaioda aeqaaOGaeyyXIC9aaSaaaeaacqGHciITcaWHTbWaaSbaaSqaaiabeg 7aHbqabaaakeaacqGHciITcqaH+oaEdaWgaaWcbaGaeqOSdigabeaa aaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiElaaykW7ca aMc8UaaGPaVlaaykW7cqaH6oWAdaWgaaWcbaGaaGymaiaaigdaaeqa aOGaeyisISRaaiikaiabeU7aSjabec8aWjaac+cacaWGmbGaaiykam aaCaaaleqabaGaaGOmaaaakiaadgeaciGGZbGaaiyAaiaac6gacaGG OaGaeq4UdWMaeqiWdaNaamOEaiaac+cacaWGmbGaaiykaiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab eQ7aRnaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaaaba GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeQ7aRnaa BaaaleaacaaIYaGaaGOmaaqabaGccqGHijYUcqGHsislcaGGOaGaam yyaiabgUcaRiaadoeacqGHRaWkcaWGbbGaci4CaiaacMgacaGGUbGa aiikaiabeU7aSjabec8aWjaadQhacaGGVaGaamitaiaacMcacaGGPa aaaaa@EA2C@

8.      The in-plane strain tensor and curvature change tensor may be approximated as

γ αβ =( g αβ g ¯ αβ )/2 γ 11 =Δ/L+ (λπA/L) 2 cos 2 (πλz/L)/2 γ 12 =0 γ 22 aC+Aacos(πλz/L) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4SdC2aaSbaaSqaaiabeg7aHj abek7aIbqabaGccqGH9aqpcaGGOaGaam4zamaaBaaaleaacqaHXoqy cqaHYoGyaeqaaOGaeyOeI0Iabm4zayaaraWaaSbaaSqaaiabeg7aHj abek7aIbqabaGccaGGPaGaai4laiaaikdacaaMc8UaaGPaVlaaykW7 caaMc8UaeyO0H4TaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 Uaeq4SdC2aaSbaaSqaaiaaigdacaaIXaaabeaakiabg2da9iabgkHi Tiabfs5aejaac+cacaWGmbGaey4kaSIaaiikaiabeU7aSjabec8aWj aadgeacaGGVaGaamitaiaacMcadaahaaWcbeqaaiaaikdaaaGcciGG JbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccaGGOaGaeqiWda Naeq4UdWMaamOEaiaac+cacaWGmbGaaiykaiaac+cacaaIYaGaaGPa VdqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 cqaHZoWzdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaaGimai aaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4SdC2aaSbaaSqaaiaa ikdacaaIYaaabeaakiabgIKi7kaadggacaWGdbGaey4kaSIaamyqai aadggaciGGJbGaai4BaiaacohacaGGOaGaeqiWdaNaeq4UdWMaamOE aiaac+cacaWGmbGaaiykaaaaaa@F721@

Δ κ αβ = κ αβ κ ¯ αλ g ¯ λμ g μβ Δ κ 11 (λπ/L) 2 Asin(λπz/L)Δ κ 12 =0Δ κ 22 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeQ7aRnaaBaaaleaacqaHXo qycqaHYoGyaeqaaOGaeyypa0JaaGPaVlaaykW7cqaH6oWAdaWgaaWc baGaeqySdeMaeqOSdigabeaakiabgkHiTiqbeQ7aRzaaraWaaSbaaS qaaiabeg7aHjabeU7aSbqabaGcceWGNbGbaebadaahaaWcbeqaaiab eU7aSjabeY7aTbaakiaadEgadaWgaaWcbaGaeqiVd0MaeqOSdigabe aakiaaykW7caaMc8UaaGPaVlabgkDiElaaykW7caaMc8UaaGPaVlaa ykW7cqqHuoarcqaH6oWAdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaey isISRaaiikaiabeU7aSjabec8aWjaac+cacaWGmbGaaiykamaaCaaa leqabaGaaGOmaaaakiaadgeaciGGZbGaaiyAaiaac6gacaGGOaGaeq 4UdWMaeqiWdaNaamOEaiaac+cacaWGmbGaaiykaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq qHuoarcqaH6oWAdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Ja aGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq qHuoarcqaH6oWAdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyisISRa aGimaaaa@9D74@

9.      The strain energy of the deformed shell can be calculated from

Φ=( A h 2 D αβρμ γ αβ γ ρμ + h 3 24 D αβρμ Δ κ αβ Δ κ ρμ )dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeuOPdyKaeyypa0ZaaeWaaeaadaWdrb qaamaalaaabaGaamiAaaqaaiaaikdaaaGaamiramaaCaaaleqabaGa eqySdeMaeqOSdiMaeqyWdiNaeqiVd0gaaOGaeq4SdC2aaSbaaSqaai abeg7aHjabek7aIbqabaGccqaHZoWzdaWgaaWcbaGaeqyWdiNaeqiV d0gabeaakiabgUcaRaWcbaGaamyqaaqab0Gaey4kIipakmaalaaaba GaamiAamaaCaaaleqabaGaaG4maaaaaOqaaiaaikdacaaI0aaaaiaa dseadaahaaWcbeqaaiabeg7aHjabek7aIjabeg8aYjabeY7aTbaaki abfs5aejabeQ7aRnaaBaaaleaacqaHXoqycqaHYoGyaeqaaOGaeuiL dqKaeqOUdS2aaSbaaSqaaiabeg8aYjabeY7aTbqabaaakiaawIcaca GLPaaacaWGKbGaamyqaaaa@6807@  with D αβρμ = E 2(1 ν 2 ) ( ( g ¯ αρ g ¯ βμ + g ¯ αμ g ¯ βρ )(1ν)+2ν g ¯ αβ g ¯ ρμ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaahaaWcbeqaaiabeg7aHjabek 7aIjabeg8aYjabeY7aTbaakiabg2da9maalaaabaGaamyraaqaaiaa ikdacaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaa aakiaacMcaaaWaaeWaaeaacaGGOaGabm4zayaaraWaaWbaaSqabeaa cqaHXoqycqaHbpGCaaGcceWGNbGbaebadaahaaWcbeqaaiabek7aIj abeY7aTbaakiabgUcaRiqadEgagaqeamaaCaaaleqabaGaeqySdeMa eqiVd0gaaOGabm4zayaaraWaaWbaaSqabeaacqaHYoGycqaHbpGCaa GccaGGPaGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaey4kaSIa aGOmaiabe27aUjqadEgagaqeamaaCaaaleqabaGaeqySdeMaeqOSdi gaaOGabm4zayaaraWaaWbaaSqabeaacqaHbpGCcqaH8oqBaaaakiaa wIcacaGLPaaaaaa@67DD@

Substituting for g ¯ αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadEgagaqeamaaCaaaleqabaGaeqySde MaeqOSdigaaaaa@34F9@  and noting that γ 12 =Δ κ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNnaaBaaaleaacaaIXaGaaGOmaa qabaGccqGH9aqpcqqHuoarcqaH6oWAdaWgaaWcbaGaaGymaiaaikda aeqaaOGaeyypa0JaaGimaaaa@3B67@  reduces this result to

Φ= πahE (1 ν 2 ) 0 L ( γ 11 2 + γ 22 2 / a 4 +2ν γ 11 γ 22 / a 2 )+ h 2 12 ( Δ κ 11 2 +Δ κ 22 2 / a 4 +2νΔ κ 11 Δ κ 22 / a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeuOPdyKaeyypa0ZaaSaaaeaacqaHap aCcaWGHbGaamiAaiaadweaaeaacaGGOaGaaGymaiabgkHiTiabe27a UnaaCaaaleqabaGaaGOmaaaakiaacMcaaaWaa8qCaeaadaqadaqaai abeo7aNnaaDaaaleaacaaIXaGaaGymaaqaaiaaikdaaaGccqGHRaWk cqaHZoWzdaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaOGaai4lai aadggadaahaaWcbeqaaiaaisdaaaGccqGHRaWkcaaIYaGaeqyVd4Ma eq4SdC2aaSbaaSqaaiaaigdacaaIXaaabeaakiabeo7aNnaaBaaale aacaaIYaGaaGOmaaqabaGccaGGVaGaamyyamaaCaaaleqabaGaaGOm aaaaaOGaayjkaiaawMcaaiabgUcaRmaalaaabaGaamiAamaaCaaale qabaGaaGOmaaaaaOqaaiaaigdacaaIYaaaamaabmaabaGaeuiLdqKa eqOUdS2aa0baaSqaaiaaigdacaaIXaaabaGaaGOmaaaakiabgUcaRi abfs5aejabeQ7aRnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaaGc caGGVaGaamyyamaaCaaaleqabaGaaGinaaaakiabgUcaRiaaikdacq aH9oGBcqqHuoarcqaH6oWAdaWgaaWcbaGaaGymaiaaigdaaeqaaOGa euiLdqKaeqOUdS2aaSbaaSqaaiaaikdacaaIYaaabeaakiaac+caca WGHbWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaaleaacaaI WaaabaGaamitaaqdcqGHRiI8aaaa@7FE2@

10.  The potential energy can be evaluated exactly, but the resulting expression is too long to write out in full.  To proceed, we eliminate C by finding the value of C that minimizes the potential energy (set Φ/C=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kabfA6agjaac+cacqGHciITca WGdbGaeyypa0JaaGimaaaa@3809@  and solve for C, then substitute the result back into Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfA6agbaa@3202@  ).  For λ>>1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabg6da+iabg6da+iaaigdaaa a@3507@  the resulting expression for  Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfA6agbaa@3202@  may be simplified to

ΦEπahL ( Δ L ) 2 + π 2 Ehaλ A 2 4(1 ν 2 )L ( ϕ 1 ϕ 2 ( Δ L ) )+O( A 4 ) ϕ 1 = π 3 λ 3 h 2 6 L 2 + 2 L 2 πλ a 2 ϕ 2 =(1 ν 2 )2πλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeuOPdyKaeyisISRaamyraiabec 8aWjaadggacaWGObGaamitamaabmaabaWaaSaaaeaacqqHuoaraeaa caWGmbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgU caRmaalaaabaGaeqiWda3aaWbaaSqabeaacaaIYaaaaOGaamyraiaa dIgacaWGHbGaeq4UdWMaamyqamaaCaaaleqabaGaaGOmaaaaaOqaai aaisdacaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOm aaaakiaacMcacaWGmbaaamaabmaabaGaeqy1dy2aaSbaaSqaaiaaig daaeqaaOGaeyOeI0Iaeqy1dy2aaSbaaSqaaiaaikdaaeqaaOWaaeWa aeaadaWcaaqaaiabfs5aebqaaiaadYeaaaaacaGLOaGaayzkaaaaca GLOaGaayzkaaGaey4kaSIaam4taiaacIcacaWGbbWaaWbaaSqabeaa caaI0aaaaOGaaiykaaqaaiabew9aMnaaBaaaleaacaaIXaaabeaaki abg2da9maalaaabaGaeqiWda3aaWbaaSqabeaacaaIZaaaaOGaeq4U dW2aaWbaaSqabeaacaaIZaaaaOGaamiAamaaCaaaleqabaGaaGOmaa aaaOqaaiaaiAdacaWGmbWaaWbaaSqabeaacaaIYaaaaaaakiabgUca RmaalaaabaGaaGOmaiaadYeadaahaaWcbeqaaiaaikdaaaaakeaacq aHapaCcqaH7oaBcaWGHbWaaWbaaSqabeaacaaIYaaaaaaakiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7cqaHvpGzdaWgaaWcbaGaaGOmaaqa baGccqGH9aqpcaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqaba GaaGOmaaaakiaacMcacaaIYaGaeqiWdaNaeq4UdWMaaGPaVdaaaa@988C@

11.  The buckling load can now be deduced from this result.  Note that both ϕ 1 >0, ϕ 2 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMnaaBaaaleaacaaIXaaabeaaki abg6da+iaaicdacaGGSaGaeqy1dy2aaSbaaSqaaiaaikdaaeqaaOGa eyOpa4JaaGimaaaa@3A2F@ , so the potential energy is minimized with A=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacqGH9aqpcaaIWaaaaa@330E@  (no buckling) if Δ/L< ϕ 1 / ϕ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaac+cacaWGmbGaeyipaWJaeq y1dy2aaSbaaSqaaiaaigdaaeqaaOGaai4laiabew9aMnaaBaaaleaa caaIYaaabeaaaaa@3A92@ , and with A>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacqGH+aGpcaaIWaaaaa@3310@  if Δ/L> ϕ 1 / ϕ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaac+cacaWGmbGaeyOpa4Jaeq y1dy2aaSbaaSqaaiaaigdaaeqaaOGaai4laiabew9aMnaaBaaaleaa caaIYaaabeaaaaa@3A96@ .  The critical axial strain for which buckling is first possible corresponds to the minimum value of ϕ 1 / ϕ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMnaaBaaaleaacaaIXaaabeaaki aac+cacqaHvpGzdaWgaaWcbaGaaGOmaaqabaaaaa@36A4@  with respect to λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@323C@ .  The minimum occurs for λ= (12) 1/4 L/(π ah ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabg2da9iaacIcacaaIXaGaaG OmaiaacMcadaahaaWcbeqaaiaaigdacaGGVaGaaGinaaaakiaadYea caGGVaGaaiikaiabec8aWnaakaaabaGaamyyaiaadIgaaSqabaGcca GGPaaaaa@3F07@ , which gives a critical axial strain of Δ/L=h/( a(1 ν 2 ) 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaac+cacaWGmbGaeyypa0Jaam iAaiaac+cadaqadaqaaiaadggacaGGOaGaaGymaiabgkHiTiabe27a UnaaCaaaleqabaGaaGOmaaaakiaacMcadaGcaaqaaiaaiodaaSqaba aakiaawIcacaGLPaaaaaa@3F15@ .

12.  The axial force at buckling can be computed trivially by noting that the cylinder is in a state of uniaxial axial stress.  The axial force is therefore 2πahEΔ/L=2πE h 2 /( (1 ν 2 ) 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacqaHapaCcaWGHbGaamiAaiaadw eacqqHuoarcaGGVaGaamitaiabg2da9iaaikdacqaHapaCcaWGfbGa amiAamaaCaaaleqabaGaaGOmaaaakiaac+cadaqadaqaaiaacIcaca aIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykamaa kaaabaGaaG4maaWcbeaaaOGaayjkaiaawMcaaaaa@477B@

 

 

10.7.7 Torsion of an open-walled circular cylinder

 

The figure shows a thin-walled tube, with radius a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggaaaa@316E@  and wall-thickness h, which has been slit along a line parallel to its axis.  The tube is made from a linear elastic solid with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3240@ , and is subjected to a twisting moment Λ=Λ e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahU5acqGH9aqpcqqHBoatcaWHLbWaaS baaSqaaiaadQhaaeqaaaaa@3643@  parallel to its axis.  The moment causes the end of the tube at z=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQhacqGH9aqpcaWGmbaaaa@335E@  to twist through an angle ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@3250@  relative to the end at z=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQhacqGH9aqpcaaIWaaaaa@3347@ .

 

The displacement field and the internal forces in the shell can be expressed as components in a cylindrical-polar basis { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiaadQhaaeqaaOGaaiyFaaaa@3B00@  as follows

u= 3Λ(1+ν) π h 3 aE ( aθ e z +z e θ )T= Λ 4π a 2 e z e θ M= Λ 4πa ( e z e θ + e θ e z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpdaWcaaqaaiaaiodacq qHBoatcaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaeaacqaHapaC caWGObWaaWbaaSqabeaacaaIZaaaaOGaamyyaiaadweaaaWaaeWaae aacaWGHbGaeqiUdeNaaCyzamaaBaaaleaacaWG6baabeaakiabgUca RiaadQhacaWHLbWaaSbaaSqaaiabeI7aXbqabaaakiaawIcacaGLPa aacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaahsfacqGH9aqpdaWcaaqaaiabfU5amb qaaiaaisdacqaHapaCcaWGHbWaaWbaaSqabeaacaaIYaaaaaaakiaa ykW7caWHLbWaaSbaaSqaaiaadQhaaeqaaOGaey4LIqSaaCyzamaaBa aaleaacqaH4oqCaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaCytaiabg2da9maalaaabaGaeu 4MdWeabaGaaGinaiabec8aWjaadggaaaWaaeWaaeaacaWHLbWaaSba aSqaaiaadQhaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacqaH4oqCae qaaOGaey4kaSIaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4LIqSa aCyzamaaBaaaleaacaWG6baabeaaaOGaayjkaiaawMcaaaaa@902F@

Note that this is one of the rare shell geometries for which the internal force tensor T is not symmetric.

 

 

Derivation: We choose the cylindrical-polar coordinates  ξ 1 z, ξ 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe67a4naaBaaaleaacaaIXaaabeaaki abggMi6kaadQhacaGGSaGaeqOVdG3aaSbaaSqaaiaaikdaaeqaaOGa eyyyIORaeqiUdehaaa@3CE8@  as our coordinate system.  

1.      The position vector of a point in the undeformed shell is r ¯ =acosθ e 1 +asinθ e 2 +z e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahkhagaqeaiabg2da9iaadggaciGGJb Gaai4BaiaacohacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIaamyyaiGacohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaa WcbaGaaGOmaaqabaGccqGHRaWkcaWG6bGaaCyzamaaBaaaleaacaaI Zaaabeaaaaa@45DD@

2.      The natural basis vectors for the undeformed shell are therefore

m ¯ 1 = r ¯ z = e 3 m ¯ 2 = r ¯ θ =asinθ e 1 +acosθ e 2 m ¯ 3 =(cosθ e 1 +sinθ e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaBaaaleaacaaIXaaabe aakiabg2da9maalaaabaGaeyOaIyRabCOCayaaraaabaGaeyOaIyRa amOEaaaacqGH9aqpcaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlqah2gagaqeamaa BaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaeyOaIyRabCOCay aaraaabaGaeyOaIyRaeqiUdehaaiabg2da9iabgkHiTiaadggaciGG ZbGaaiyAaiaac6gacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaO Gaey4kaSIaamyyaiGacogacaGGVbGaai4CaiabeI7aXjaahwgadaWg aaWcbaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7ceWHTbGbaebadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcqGH sislcaGGOaGaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzamaaBaaale aacaaIXaaabeaakiabgUcaRiGacohacaGGPbGaaiOBaiabeI7aXjaa hwgadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@7C98@

and the reciprocal basis vectors are m ¯ 1 = e 3 m ¯ 2 =(sinθ e 1 +cosθ e 2 )/a m ¯ 3 = m ¯ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaCaaaleqabaGaaGymaa aakiabg2da9iaahwgadaWgaaWcbaGaaG4maaqabaGccaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlqah2gagaqeam aaCaaaleqabaGaaGOmaaaakiabg2da9iaacIcacqGHsislciGGZbGa aiyAaiaac6gacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzamaaBaaaleaacaaI YaaabeaakiaacMcacaGGVaGaamyyaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ceWHTbGbaeba daahaaWcbeqaaiaaiodaaaGccqGH9aqpceWHTbGbaebadaWgaaWcba GaaG4maaqabaaaaa@6AD2@

3.      The components of the metric tensor are g ¯ αβ = m ¯ α m ¯ β g ¯ 11 =1, g ¯ 12 =0, g ¯ 22 =1/ a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadEgagaqeamaaCaaaleqabaGaeqySde MaeqOSdigaaOGaeyypa0JabCyBayaaraWaaWbaaSqabeaacqaHXoqy aaGccqGHflY1ceWHTbGbaebadaahaaWcbeqaaiabek7aIbaakiabgk DiElqadEgagaqeamaaCaaaleqabaGaaGymaiaaigdaaaGccqGH9aqp caaIXaGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl qadEgagaqeamaaCaaaleqabaGaaGymaiaaikdaaaGccqGH9aqpcaaI WaGaaiilaiaaykW7caaMc8Uabm4zayaaraWaaWbaaSqabeaacaaIYa GaaGOmaaaakiabg2da9iaaigdacaGGVaGaamyyamaaCaaaleqabaGa aGOmaaaaaaa@5E0C@ , g ¯ αβ = m ¯ α m ¯ β g ¯ 11 =1 g ¯ 12 =0 g ¯ 22 = a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadEgagaqeamaaBaaaleaacqaHXoqycq aHYoGyaeqaaOGaeyypa0JabCyBayaaraWaaSbaaSqaaiabeg7aHbqa baGccqGHflY1ceWHTbGbaebadaWgaaWcbaGaeqOSdigabeaakiabgk DiElqadEgagaqeamaaBaaaleaacaaIXaGaaGymaaqabaGccqGH9aqp caaIXaGaaGPaVlaaykW7caaMc8UaaGPaVlqadEgagaqeamaaBaaale aacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8Uabm4zayaaraWaaSbaaSqaaiaaikdaca aIYaaabeaakiabg2da9iaadggadaahaaWcbeqaaiaaikdaaaaaaa@5E4E@

4.      The covariant components of the curvature tensor for the undeformed shell are

κ ¯ αβ = m ¯ 3 m ¯ α ξ β κ ¯ 11 = κ ¯ 12 =0 κ ¯ 22 =a κ ¯ β α = κ ¯ βλ g ¯ αλ κ ¯ 1 1 = κ ¯ 2 1 = κ ¯ 1 2 =0 κ ¯ 2 2 =1/a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGafqOUdSMbaebadaWgaaWcbaGaeq ySdeMaeqOSdigabeaakiabg2da9iabgkHiTiqah2gagaqeamaaBaaa leaacaaIZaaabeaakiabgwSixpaalaaabaGaeyOaIyRabCyBayaara WaaSbaaSqaaiabeg7aHbqabaaakeaacqGHciITcqaH+oaEdaWgaaWc baGaeqOSdigabeaaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abgkDiElaaykW7caaMc8UaaGPaVlaaykW7cuaH6oWAgaqeamaaBaaa leaacaaIXaGaaGymaaqabaGccqGH9aqpcuaH6oWAgaqeamaaBaaale aacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlqbeQ7aRzaaraWaaSbaaSqaai aaikdacaaIYaaabeaakiabg2da9iabgkHiTiaadggaaeaacuaH6oWA gaqeamaaDaaaleaacqaHYoGyaeaacqaHXoqyaaGccqGH9aqpcuaH6o WAgaqeamaaBaaaleaacqaHYoGycqaH7oaBaeqaaOGabm4zayaaraWa aWbaaSqabeaacqaHXoqycqaH7oaBaaGccqGHshI3cuaH6oWAgaqeam aaDaaaleaacaaIXaaabaGaaGymaaaakiabg2da9iqbeQ7aRzaaraWa a0baaSqaaiaaikdaaeaacaaIXaaaaOGaeyypa0JafqOUdSMbaebada qhaaWcbaGaaGymaaqaaiaaikdaaaGccqGH9aqpcaaIWaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cuaH6oWAga qeamaaDaaaleaacaaIYaaabaGaaGOmaaaakiabg2da9iabgkHiTiaa igdacaGGVaGaamyyaaaaaa@A549@

5.      The Christoffel symbols for the undeformed shell are Γ ¯ βα λ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbfo5ahzaaraWaa0baaSqaaiabek7aIj abeg7aHbqaaiabeU7aSbaakiabg2da9iaaicdaaaa@38F3@

6.      To proceed, we assume that the internal stresses and moments in the cylinder are uniform; in addition, we assume small strains, so that the geometric terms in the equilibrium equations can be approximated using the geometry of the undeformed shell.  The equilibrium equations therefore reduce to

V 2 =0 T 22 a=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaaikdaaaGccq GH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaadsfadaahaaWcbeqaaiaaikdacaaIYaaaaOGaamyyaiabg2 da9iaaicdaaaa@4411@          V β =0 T 12 T 21 + M 21 /a=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaadAfadaahaaWcbeqaaiabek 7aIbaakiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaadsfadaahaaWcbeqaaiaaigdacaaIYa aaaOGaeyOeI0IaamivamaaCaaaleqabaGaaGOmaiaaigdaaaGccqGH RaWkcaWGnbWaaWbaaSqabeaacaaIYaGaaGymaaaakiaac+cacaWGHb Gaeyypa0JaaGimaaaa@4EF6@

7.      The boundary conditions on θ=0,θ=2π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaaicdacaGGSaGaeq iUdeNaeyypa0JaaGOmaiabec8aWbaa@39E3@  are  T 21 =0 T 22 M 22 /a=0 M 22 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaikdacaaIXa aaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamivamaaCaaale qabaGaaGOmaiaaikdaaaGccqGHsislcaWGnbWaaWbaaSqabeaacaaI YaGaaGOmaaaakiaac+cacaWGHbGaeyypa0JaaGimaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamytamaaCaaaleqabaGaaGOmaiaaikdaaaGccqGH9aqpca aIWaaaaa@6450@

8.      The boundary conditions on z=0,z=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQhacqGH9aqpcaaIWaGaaiilaiaadQ hacqGH9aqpcaWGmbaaaa@36CD@  are T 11 =0 T 12 M 12 /a= P 2 M 11 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaigdacaaIXa aaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGubWaaWbaaSqabeaaca aIXaGaaGOmaaaakiabgkHiTiaad2eadaahaaWcbeqaaiaaigdacaaI YaaaaOGaai4laiaadggacqGH9aqpcaWGqbWaaWbaaSqabeaacaaIYa aaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWGnbWaaWbaaSqabeaacaaIXaGaaGymaaaakiabg2da9iaaic daaaa@5F2D@

9.      The only nonzero components of internal force are the in-plane shear forces T 12 , T 21 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaigdacaaIYa aaaOGaaiilaiaadsfadaahaaWcbeqaaiaaikdacaaIXaaaaaaa@363C@  and the twisting moments M 12 , M 21 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaahaaWcbeqaaiaaigdacaaIYa aaaOGaaiilaiaad2eadaahaaWcbeqaaiaaikdacaaIXaaaaaaa@362E@ .  We therefore assume that the shell deforms in shear, so that the position vector of a point in the deformed shell is

r ¯ =acosθ e 1 +asinθ e 2 +z e 3 +Caθ e 3 +(azϕ/L)(sinθ e 1 +cosθ e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahkhagaqeaiabg2da9iaadggaciGGJb Gaai4BaiaacohacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIaamyyaiGacohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaa WcbaGaaGOmaaqabaGccqGHRaWkcaWG6bGaaCyzamaaBaaaleaacaaI ZaaabeaakiabgUcaRiaadoeacaWGHbGaeqiUdeNaaCyzamaaBaaale aacaaIZaaabeaakiabgUcaRiaacIcacaWGHbGaamOEaiabew9aMjaa c+cacaWGmbGaaiykaiaacIcacqGHsislciGGZbGaaiyAaiaac6gacq aH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaci4yaiaa c+gacaGGZbGaeqiUdeNaaCyzamaaBaaaleaacaaIYaaabeaakiaacM caaaa@6378@

10.  The natural basis vectors for the deformed shell are

m 1 = e 3 +(aϕ/L)(sinθ e 1 +cosθ e 2 ) m 2 =asinθ e 1 +acosθ e 2 +Ca e 3 (zϕa/L)(cosθ e 1 +sinθ e 2 ) m 3 ( cosθ e 1 +sinθ e 2 )+(zϕ/L)( sinθ e 1 cosθ e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaCyBamaaBaaaleaacaaIXaaabe aakiabg2da9iaahwgadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaGG OaGaamyyaiabew9aMjaac+cacaWGmbGaaiykaiaacIcacqGHsislci GGZbGaaiyAaiaac6gacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqa aOGaey4kaSIaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzamaaBaaale aacaaIYaaabeaakiaacMcaaeaacaWHTbWaaSbaaSqaaiaaikdaaeqa aOGaeyypa0JaeyOeI0IaamyyaiGacohacaGGPbGaaiOBaiabeI7aXj aahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbGaci4yaiaa c+gacaGGZbGaeqiUdeNaaCyzamaaBaaaleaacaaIYaaabeaakiabgU caRiaadoeacaWGHbGaaCyzamaaBaaaleaacaaIZaaabeaakiabgkHi TiaacIcacaWG6bGaeqy1dyMaamyyaiaac+cacaWGmbGaaiykaiaacI caciGGJbGaai4BaiaacohacqaH4oqCcaWHLbWaaSbaaSqaaiaaigda aeqaaOGaey4kaSIaci4CaiaacMgacaGGUbGaeqiUdeNaaCyzamaaBa aaleaacaaIYaaabeaakiaacMcaaeaacaWHTbWaaSbaaSqaaiaaioda aeqaaOGaeyisISRaeyOeI0YaaeWaaeaaciGGJbGaai4Baiaacohacq aH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaci4Caiaa cMgacaGGUbGaeqiUdeNaaCyzamaaBaaaleaacaaIYaaabeaaaOGaay jkaiaawMcaaiabgUcaRiaacIcacaWG6bGaeqy1dyMaai4laiaadYea caGGPaWaaeWaaeaaciGGZbGaaiyAaiaac6gacqaH4oqCcaWHLbWaaS baaSqaaiaaigdaaeqaaOGaeyOeI0Iaci4yaiaac+gacaGGZbGaeqiU deNaaCyzamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaaaaaa@A3BB@

11.  The metric tensor for the deformed shell can be approximated by

g αβ = m α m β g 11 1 g 12 =Ca( a 2 ϕ/L) g 22 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgadaWgaaWcbaGaeqySdeMaeqOSdi gabeaakiabg2da9iaah2gadaWgaaWcbaGaeqySdegabeaakiabgwSi xlaah2gadaWgaaWcbaGaeqOSdigabeaakiabgkDiElaadEgadaWgaa WcbaGaaGymaiaaigdaaeqaaOGaeyisISRaaGymaiaaykW7caaMc8Ua aGPaVlaaykW7caWGNbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2 da9iaadoeacaWGHbGaeyOeI0IaaiikaiaadggadaahaaWcbeqaaiaa ikdaaaGccqaHvpGzcaGGVaGaamitaiaacMcacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caWGNbWaaSbaaSqaaiaaikdacaaIYaaa beaakiabgIKi7kaadggadaahaaWcbeqaaiaaikdaaaaaaa@6773@

12.  The strain and curvature components follow as

γ 11 = γ 22 =0, γ 12 =( Ca( a 2 ϕ/L) )/2Δ κ 11 =0,Δ κ 12 =(aϕ/L)Δ κ 22 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHZoWzdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGa eyypa0JaaGimaiaacYcacqaHZoWzdaWgaaWcbaGaaGymaiaaikdaae qaaOGaeyypa0ZaaeWaaeaacaWGdbGaamyyaiabgkHiTiaacIcacaWG HbWaaWbaaSqabeaacaaIYaaaaOGaeqy1dyMaai4laiaadYeacaGGPa aacaGLOaGaayzkaaGaaGPaVlaac+cacaaIYaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeuiLdqKaeq OUdS2aaSbaaSqaaiaaigdacaaIXaaabeaakiabg2da9iaaicdacaGG SaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqqHuoarcqaH6oWAda WgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaaiikaiaadggacqaH vpGzcaGGVaGaamitaiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7cqqHuoarcqaH6oWAdaWgaaWc baGaaGOmaiaaikdaaeqaaOGaeyisISRaaGimaaaa@89D4@

13.  The constitutive equations can be reduced to

T 12 = Eh (1+ν) a 2 γ 12 E h 3 12(1+ν) a 4 γ 12 T 21 = Eh (1+ν) a 2 ( γ 12 h 2 12a Δ κ 12 ) M 12 = E h 3 12(1+ν) a 2 Δ κ 12 M 21 = E h 3 12(1+ν) a 2 ( Δ κ 12 γ 12 /a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamivamaaCaaaleqabaGaaGymai aaikdaaaGccqGH9aqpdaWcaaqaaiaadweacaWGObaabaGaaiikaiaa igdacqGHRaWkcqaH9oGBcqGHPaqkcaWGHbWaaWbaaSqabeaacaaIYa aaaaaakiabeo7aNnaaBaaaleaacaaIXaGaaGOmaaqabaGccqGHsisl caaMc8+aaSaaaeaacaWGfbGaamiAamaaCaaaleqabaGaaG4maaaaaO qaaiaaigdacaaIYaGaaiikaiaaigdacqGHRaWkcqaH9oGBcqGHPaqk caWGHbWaaWbaaSqabeaacaaI0aaaaaaakiaaykW7cqaHZoWzdaWgaa WcbaGaaGymaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWGubWaaWbaaSqabeaacaaIYaGaaGymaaaakiabg2da9maala aabaGaamyraiaadIgaaeaacaGGOaGaaGymaiabgUcaRiabe27aUjab gMcaPiaadggadaahaaWcbeqaaiaaikdaaaaaaOWaaeWaaeaacqaHZo WzdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyOeI0YaaSaaaeaacaWG ObWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGymaiaaikdacaWGHbaaai abfs5aejabeQ7aRnaaBaaaleaacaaIXaGaaGOmaaqabaaakiaawIca caGLPaaacaaMc8oabaGaaGPaVlaad2eadaahaaWcbeqaaiaaigdaca aIYaaaaOGaeyypa0ZaaSaaaeaacaWGfbGaamiAamaaCaaaleqabaGa aG4maaaaaOqaaiaaigdacaaIYaGaaiikaiaaigdacqGHRaWkcqaH9o GBcqGHPaqkcaWGHbWaaWbaaSqabeaacaaIYaaaaaaakiabfs5aejab eQ7aRnaaBaaaleaacaaIXaGaaGOmaaqabaGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caWGnbWaaWbaaSqabeaacaaIYaGaaGymaaaakiabg2da9m aalaaabaGaamyraiaadIgadaahaaWcbeqaaiaaiodaaaaakeaacaaI XaGaaGOmaiaacIcacaaIXaGaey4kaSIaeqyVd4MaeyykaKIaamyyam aaCaaaleqabaGaaGOmaaaaaaGcdaqadaqaaiabfs5aejabeQ7aRnaa BaaaleaacaaIXaGaaGOmaaqabaGccqGHsislcqaHZoWzdaWgaaWcba GaaGymaiaaikdaaeqaaOGaai4laiaadggaaiaawIcacaGLPaaaaaaa @B838@

Note that this is one of the rare shell geometries for which the full coupled constitutive equations must be used.

14.  The equations listed in 6-8 and 12,13 can be solved to show that C=(aϕ/L)(1+ h 2 /12 a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeacqGH9aqpcaGGOaGaamyyaiabew 9aMjaac+cacaWGmbGaaiykaiaacIcacaaIXaGaey4kaSIaamiAamaa CaaaleqabaGaaGOmaaaakiaac+cacaaIXaGaaGOmaiaadggadaahaa WcbeqaaiaaikdaaaGccaGGPaaaaa@40BA@ ,

T 12 = E h 3 12(1+ν) a 3 aϕ L ( 1 h 2 12 a 2 ) M 12 = E h 3 12(1+ν) a 2 aϕ L M 21 = E h 3 12(1+ν) a 2 aϕ L ( 1 h 2 12 a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaigdacaaIYa aaaOGaeyypa0ZaaSaaaeaacaWGfbGaamiAamaaCaaaleqabaGaaG4m aaaaaOqaaiaaigdacaaIYaGaaiikaiaaigdacqGHRaWkcqaH9oGBca GGPaGaamyyamaaCaaaleqabaGaaG4maaaaaaGcdaWcaaqaaiaadgga cqaHvpGzaeaacaWGmbaaamaabmaabaGaaGymaiabgkHiTmaalaaaba GaamiAamaaCaaaleqabaGaaGOmaaaaaOqaaiaaigdacaaIYaGaamyy amaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG nbWaaWbaaSqabeaacaaIXaGaaGOmaaaakiabg2da9maalaaabaGaam yraiaadIgadaahaaWcbeqaaiaaiodaaaaakeaacaaIXaGaaGOmaiaa cIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaadggadaahaaWcbeqaai aaikdaaaaaaOWaaSaaaeaacaWGHbGaeqy1dygabaGaamitaaaacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaad2 eadaahaaWcbeqaaiaaikdacaaIXaaaaOGaeyypa0ZaaSaaaeaacaWG fbGaamiAamaaCaaaleqabaGaaG4maaaaaOqaaiaaigdacaaIYaGaai ikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaamyyamaaCaaaleqabaGa aGOmaaaaaaGcdaWcaaqaaiaadggacqaHvpGzaeaacaWGmbaaamaabm aabaGaaGymaiabgkHiTmaalaaabaGaamiAamaaCaaaleqabaGaaGOm aaaaaOqaaiaaigdacaaIYaGaamyyamaaCaaaleqabaGaaGOmaaaaaa aakiaawIcacaGLPaaaaaa@9117@

15.  The components of internal force and moment may be expressed in terms of cylindrical-polar coordinates by noting that m 1 = e z m 2 =a e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWHLbWaaSbaaSqaaiaadQhaaeqaaOGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaGOmaa qabaGccqGH9aqpcaWGHbGaaCyzamaaBaaaleaacqaH4oqCaeqaaaaa @4709@ , whence

T= T αβ m α m β =a T 12 e z e θ M= M αβ m α m β =a M 12 e z e θ +a M 21 e θ e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaCivaiabg2da9iaadsfadaahaa Wcbeqaaiabeg7aHjabek7aIbaakiaah2gadaWgaaWcbaGaeqySdega beaakiabgEPielaah2gadaWgaaWcbaGaeqOSdigabeaakiabg2da9i aadggacaWGubWaaWbaaSqabeaacaaIXaGaaGOmaaaakiaahwgadaWg aaWcbaGaamOEaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiabeI7aXb qabaaakeaacaWHnbGaeyypa0JaamytamaaCaaaleqabaGaeqySdeMa eqOSdigaaOGaaCyBamaaBaaaleaacqaHXoqyaeqaaOGaey4LIqSaaC yBamaaBaaaleaacqaHYoGyaeqaaOGaeyypa0Jaamyyaiaad2eadaah aaWcbeqaaiaaigdacaaIYaaaaOGaaCyzamaaBaaaleaacaWG6baabe aakiabgEPielaahwgadaWgaaWcbaGaeqiUdehabeaakiabgUcaRiaa dggacaWGnbWaaWbaaSqabeaacaaIYaGaaGymaaaakiaahwgadaWgaa WcbaGaeqiUdehabeaakiabgEPielaahwgadaWgaaWcbaGaamOEaaqa baaaaaa@6E58@

16.  Finally,  external force and couple per unit length acting on the end of the cylinder at z=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQhacqGH9aqpcaWGmbaaaa@335E@  are P= e z TQ= e z M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahcfacqGH9aqpcaWHLbWaaSbaaSqaai aadQhaaeqaaOGaeyyXICTaaCivaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWHrbGaeyypa0JaaCyzamaaBaaaleaaca WG6baabeaakiabgwSixlaah2eaaaa@49A1@ .  The resultant moment about the axis of the cylinder due to these tractions is

Λ=Λ e z = 0 2π ( a e r ×P+Q ) adθ= 2πE h 3 12(1+ν) aϕ L ( 2 h 2 12 a 2 ) πE h 3 3(1+ν) aϕ L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahU5acqGH9aqpcqqHBoatcaWHLbWaaS baaSqaaiaadQhaaeqaaOGaeyypa0Zaa8qCaeaadaqadaqaaiaadgga caWHLbWaaSbaaSqaaiaadkhaaeqaaOGaey41aqRaaCiuaiabgUcaRi aahgfaaiaawIcacaGLPaaaaSqaaiaaicdaaeaacaaIYaGaeqiWdaha niabgUIiYdGccaWGHbGaamizaiabeI7aXjabg2da9maalaaabaGaaG Omaiabec8aWjaadweacaWGObWaaWbaaSqabeaacaaIZaaaaaGcbaGa aGymaiaaikdacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaaWaaS aaaeaacaWGHbGaeqy1dygabaGaamitaaaadaqadaqaaiaaikdacqGH sisldaWcaaqaaiaadIgadaahaaWcbeqaaiaaikdaaaaakeaacaaIXa GaaGOmaiaadggadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzk aaGaaGPaVlabgIKi7oaalaaabaGaeqiWdaNaamyraiaadIgadaahaa WcbeqaaiaaiodaaaaakeaacaaIZaGaaiikaiaaigdacqGHRaWkcqaH 9oGBcaGGPaaaamaalaaabaGaamyyaiabew9aMbqaaiaadYeaaaaaaa@72C2@

It follows that the twist per unit length is related to the twisting moment by ϕ/L=3Λ(1+ν)/( πa h 3 E ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjaac+cacaWGmbGaeyypa0JaaG 4maiabfU5amjaacIcacaaIXaGaey4kaSIaeqyVd4Maaiykaiaac+ca daqadaqaaiabec8aWjaadggacaWGObWaaWbaaSqabeaacaaIZaaaaO GaamyraaGaayjkaiaawMcaaaaa@4344@ .  Substituting this result back into the formulas for T and M and neglecting the terms of order h 2 / a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIgadaahaaWcbeqaaiaaikdaaaGcca GGVaGaamyyamaaCaaaleqabaGaaGOmaaaaaaa@34EA@  gives the result stated.

 

 

 

10.7.8 Membrane shell theory analysis of a spherical dome under gravitational loading

 

The figure shows a thin-walled, spherical dome with radius R, thickness h and mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@ .  We wish to calculate the internal forces induced by gravitational loading of the structure.   Shells that are used for structural applications are usually modeled using a simplified version of general shell theory, known as `Membrane Shell Theory.’   The theory simplifies the governing equations by neglecting internal moments, so that the structure is supported entirely by in-plane forces T αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiabeg7aHjabek 7aIbaaaaa@34CE@ .   The theory is intended to be applied to masonry or concrete structures, which can generally support substantial (compressive) in-plane forces but are weak in bending.  Of course, some bending resistance is critical to ensure stability against buckling; in addition, significant bending moments may develop near the edges of the structure if the boundary conditions constrain the rotation or transverse motion of the shell; so membrane theory must be used with caution.

 

The internal forces are best expressed as components in a spherical-polar basis of unit vectors { e R , e ϕ , e θ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqy1dygabeaakiaacYcacaWH LbWaaSbaaSqaaiabeI7aXbqabaGccaGG9baaaa@3BA9@  shown in the figure.  The solution shows that the in-plane membrane forces are

T=ρgh( 1 (1+cosθ) cosθ ) e ϕ e ϕ ρghR (1+cosθ) e θ e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpcqaHbpGCcaWGNbGaam iAamaabmaabaWaaSaaaeaacaaIXaaabaGaaiikaiaaigdacqGHRaWk ciGGJbGaai4BaiaacohacqaH4oqCcaGGPaaaaiaaykW7caaMc8UaaG PaVlaaykW7cqGHsislciGGJbGaai4BaiaacohacqaH4oqCaiaawIca caGLPaaacaWHLbWaaSbaaSqaaiabew9aMbqabaGccqGHxkcXcaWHLb WaaSbaaSqaaiabew9aMbqabaGccqGHsisldaWcaaqaaiabeg8aYjaa dEgacaWGObGaamOuaaqaaiaacIcacaaIXaGaey4kaSIaci4yaiaac+ gacaGGZbGaeqiUdeNaaiykaaaacaWHLbWaaSbaaSqaaiabeI7aXbqa baGccqGHxkcXcaWHLbWaaSbaaSqaaiabeI7aXbqabaaaaa@67F9@

where g is the gravitational acceleration.

 

A concrete dome should be designed so that the membrane forces are compressive everywhere.  The solution shows that the hoop forces T θθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaeqiUdeNaeqiUde habeaaaaa@34F9@  are always compressive, but the circumferential forces T ϕϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaeqy1dyMaeqy1dy gabeaaaaa@351D@  are compressive only if cosθ>1/(1+cosθ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacogacaGGVbGaai4CaiabeI7aXjabg6 da+iaaigdacaGGVaGaaiikaiaaigdacqGHRaWkciGGJbGaai4Baiaa cohacqaH4oqCcaGGPaaaaa@3F06@ .   The dome should therefore be designed with θ 0 < 51.8 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXnaaBaaaleaacaaIWaaabeaaki abgYda8iaaiwdacaaIXaGaaiOlaiaaiIdadaahaaWcbeqaaiaaicda aaaaaa@3807@

 

 

Derivation: We adopt as curvilinear coordinates the spherical-polar coordinates (ϕ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacqaHvpGzcaGGSaGaeqiUdeNaai ykaaaa@360F@  shown in the figure.  Let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3988@  be the Cartesian basis that is used to provide reference directions for (ϕ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacqaHvpGzcaGGSaGaeqiUdeNaai ykaaaa@360F@  as indicated in the figure.  Then

1.      The position vector of a point in the undeformed shell can be expressed as r ¯ =Rcosϕsinθ e 1 +Rsinϕsinθ e 2 +Rcosϕ e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahkhagaqeaiabg2da9iaadkfaciGGJb Gaai4BaiaacohacqaHvpGzciGGZbGaaiyAaiaac6gacqaH4oqCcaWH LbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamOuaiGacohacaGGPb GaaiOBaiabew9aMjGacohacaGGPbGaaiOBaiabeI7aXjaahwgadaWg aaWcbaGaaGOmaaqabaGccqGHRaWkcaWGsbGaci4yaiaac+gacaGGZb Gaeqy1dyMaaCyzamaaBaaaleaacaaIZaaabeaaaaa@5372@

2.      The natural basis vectors follow as

m ¯ 1 = r ¯ ϕ =Rsinϕsinθ e 1 +Rcosϕsinθ e 2 m ¯ 2 = r ¯ θ =Rcosϕcosθ e 1 +Rsinϕcosθ e 2 Rsinθ e 3 m ¯ 3 =( cosϕsinθ e 1 +sinϕsinθ e 2 +cosθ e 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGabCyBayaaraWaaSbaaSqaaiaaig daaeqaaOGaeyypa0ZaaSaaaeaacqGHciITceWHYbGbaebaaeaacqGH ciITcqaHvpGzaaGaeyypa0JaeyOeI0IaamOuaiGacohacaGGPbGaai OBaiabew9aMjGacohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaaWc baGaaGymaaqabaGccqGHRaWkcaWGsbGaci4yaiaac+gacaGGZbGaeq y1dyMaci4CaiaacMgacaGGUbGaeqiUdeNaaCyzamaaBaaaleaacaaI YaaabeaaaOqaaiqah2gagaqeamaaBaaaleaacaaIYaaabeaakiabg2 da9maalaaabaGaeyOaIyRabCOCayaaraaabaGaeyOaIyRaeqiUdeha aiabg2da9iaadkfaciGGJbGaai4BaiaacohacqaHvpGzciGGJbGaai 4BaiaacohacqaH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4k aSIaamOuaiGacohacaGGPbGaaiOBaiabew9aMjGacogacaGGVbGaai 4CaiabeI7aXjaahwgadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG sbGaci4CaiaacMgacaGGUbGaeqiUdeNaaCyzamaaBaaaleaacaaIZa aabeaaaOqaaiqah2gagaqeamaaBaaaleaacaaIZaaabeaakiabg2da 9iabgkHiTmaabmaabaGaci4yaiaac+gacaGGZbGaeqy1dyMaci4Cai aacMgacaGGUbGaeqiUdeNaaCyzamaaBaaaleaacaaIXaaabeaakiab gUcaRiGacohacaGGPbGaaiOBaiabew9aMjGacohacaGGPbGaaiOBai abeI7aXjaahwgadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkciGGJbGa ai4BaiaacohacqaH4oqCcaWHLbWaaSbaaSqaaiaaiodaaeqaaaGcca GLOaGaayzkaaaaaaa@A226@

3.      The reciprocal base vectors are

m ¯ 1 =( sinϕ e 1 +cosϕ e 2 )/Rsinθ m ¯ 2 =( cosϕcosθ e 1 +sinϕcosθ e 2 sinθ e 3 )/R m ¯ 3 =( cosϕsinθ e 1 +sinϕsinθ e 2 +cosθ e 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGabCyBayaaraWaaWbaaSqabeaaca aIXaaaaOGaeyypa0ZaaeWaaeaacqGHsislciGGZbGaaiyAaiaac6ga cqaHvpGzcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaci4yai aac+gacaGGZbGaeqy1dyMaaCyzamaaBaaaleaacaaIYaaabeaaaOGa ayjkaiaawMcaaiaac+cacaWGsbGaci4CaiaacMgacaGGUbGaeqiUde habaGabCyBayaaraWaaWbaaSqabeaacaaIYaaaaOGaeyypa0ZaaeWa aeaaciGGJbGaai4BaiaacohacqaHvpGzciGGJbGaai4Baiaacohacq aH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaci4Caiaa cMgacaGGUbGaeqy1dyMaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzam aaBaaaleaacaaIYaaabeaakiabgkHiTiGacohacaGGPbGaaiOBaiab eI7aXjaahwgadaWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPaaaca GGVaGaamOuaaqaaiqah2gagaqeamaaCaaaleqabaGaaG4maaaakiab g2da9iabgkHiTmaabmaabaGaci4yaiaac+gacaGGZbGaeqy1dyMaci 4CaiaacMgacaGGUbGaeqiUdeNaaCyzamaaBaaaleaacaaIXaaabeaa kiabgUcaRiGacohacaGGPbGaaiOBaiabew9aMjGacohacaGGPbGaai OBaiabeI7aXjaahwgadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkciGG JbGaai4BaiaacohacqaH4oqCcaWHLbWaaSbaaSqaaiaaiodaaeqaaa GccaGLOaGaayzkaaaaaaa@9226@

4.      The Christoffel symbols for the undeformed shell and its curvature components can be calculated as

Γ βγ α = m ¯ α 2 r ¯ ξ β ξ γ Γ ¯ 11 1 = Γ ¯ 22 1 =0 Γ ¯ 12 1 = Γ ¯ 21 1 = cosθ sinθ Γ ¯ 11 2 =sinθcosθ Γ ¯ 12 2 = Γ ¯ 21 2 = Γ ¯ 22 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo5ahnaaDaaaleaacqaHYoGycqaHZo WzaeaacqaHXoqyaaGccqGH9aqpceWHTbGbaebadaahaaWcbeqaaiab eg7aHbaakiabgwSixpaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYa aaaOGabCOCayaaraaabaGaeyOaIyRaeqOVdG3aaSbaaSqaaiabek7a IbqabaGccqGHciITcqaH+oaEdaWgaaWcbaGaeq4SdCgabeaaaaGccq GHshI3cuqHtoWrgaqeamaaDaaaleaacaaIXaGaaGymaaqaaiaaigda aaGccqGH9aqpcuqHtoWrgaqeamaaDaaaleaacaaIYaGaaGOmaaqaai aaigdaaaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cuqHtoWrgaqeamaaDaaaleaacaaIXa GaaGOmaaqaaiaaigdaaaGccqGH9aqpcuqHtoWrgaqeamaaDaaaleaa caaIYaGaaGymaaqaaiaaigdaaaGccqGH9aqpdaWcaaqaaiGacogaca GGVbGaai4CaiabeI7aXbqaaiGacohacaGGPbGaaiOBaiabeI7aXbaa caaMc8UaaGPaVlaaykW7caaMc8Uafu4KdCKbaebadaqhaaWcbaGaaG ymaiaaigdaaeaacaaIYaaaaOGaeyypa0JaeyOeI0Iaci4CaiaacMga caGGUbGaeqiUdeNaci4yaiaac+gacaGGZbGaeqiUdeNaaGPaVlaayk W7caaMc8UaaGPaVlqbfo5ahzaaraWaa0baaSqaaiaaigdacaaIYaaa baGaaGOmaaaakiabg2da9iqbfo5ahzaaraWaa0baaSqaaiaaikdaca aIXaaabaGaaGOmaaaakiabg2da9iqbfo5ahzaaraWaa0baaSqaaiaa ikdacaaIYaaabaGaaGOmaaaakiabg2da9iaaicdaaaa@A1EC@

κ ¯ αβ = m ¯ 3 m ¯ α ξ β κ ¯ 11 =R sin 2 θ κ ¯ 12 =0 κ ¯ 22 =R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeQ7aRzaaraWaaSbaaSqaaiabeg7aHj abek7aIbqabaGccqGH9aqpcqGHsislceWHTbGbaebadaahaaWcbeqa aiaaiodaaaGccqGHflY1daWcaaqaaiabgkGi2kqah2gagaqeamaaBa aaleaacqaHXoqyaeqaaaGcbaGaeyOaIyRaeqOVdG3aaSbaaSqaaiab ek7aIbqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiElaayk W7caaMc8UaaGPaVlaaykW7cuaH6oWAgaqeamaaBaaaleaacaaIXaGa aGymaaqabaGccqGH9aqpcqGHsislcaWGsbGaci4CaiaacMgacaGGUb WaaWbaaSqabeaacaaIYaaaaOGaeqiUdeNaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlqbeQ7aRzaaraWaaSbaaSqaaiaaig dacaaIYaaabeaakiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UafqOUdSMbae badaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaamOu aaaa@858A@

5.      We now introduce two assumptions (i) the bending resistance of the shell is zero, so that the internal moment components M αβ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaahaaWcbeqaaiabeg7aHjabek 7aIbaakiabg2da9iaaicdaaaa@3691@ ; (ii) the deformations are small enough so that the Christoffel symbols and curvature terms in the equilibrium equations can be approximated using the values for the undeformed shell (this amounts to enforcing equilibrium in the undeformed configuration of the shell). In addition, the shell is in static equilibrium, and the external couples are zero. The equations of motion of Section 10.5.8 can therefore be reduced to

T αβ ξ α + T αβ Γ αγ γ + T αγ Γ γα β + V α κ α β + p β =0 + V α ξ α + V α Γ αβ β T αβ κ αβ + p 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacqGHciITcaWGubWaaW baaSqabeaacqaHXoqycqaHYoGyaaaakeaacqGHciITcqaH+oaEdaWg aaWcbaGaeqySdegabeaaaaGccqGHRaWkcaWGubWaaWbaaSqabeaacq aHXoqycqaHYoGyaaGccqqHtoWrdaqhaaWcbaGaeqySdeMaeq4SdCga baGaeq4SdCgaaOGaey4kaSIaamivamaaCaaaleqabaGaeqySdeMaeq 4SdCgaaOGaeu4KdC0aa0baaSqaaiabeo7aNjabeg7aHbqaaiabek7a IbaakiabgUcaRiaadAfadaahaaWcbeqaaiabeg7aHbaakiabeQ7aRn aaDaaaleaacqaHXoqyaeaacqaHYoGyaaGccqGHRaWkcaWGWbWaaWba aSqabeaacqaHYoGyaaGccqGH9aqpcaaIWaaabaGaey4kaSYaaSaaae aacqGHciITcaWGwbWaaWbaaSqabeaacqaHXoqyaaaakeaacqGHciIT cqaH+oaEdaWgaaWcbaGaeqySdegabeaaaaGccqGHRaWkcaWGwbWaaW baaSqabeaacqaHXoqyaaGccqqHtoWrdaqhaaWcbaGaeqySdeMaeqOS digabaGaeqOSdigaaOGaeyOeI0IaamivamaaCaaaleqabaGaeqySde MaeqOSdigaaOGaeqOUdS2aaSbaaSqaaiabeg7aHjabek7aIbqabaGc cqGHRaWkcaWGWbWaaWbaaSqabeaacaaIZaaaaOGaeyypa0JaaGimaa aaaa@8404@    V β =0 T 12 T 21 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamOvamaaCaaaleqabaGaeqOSdi gaaOGaeyypa0JaaGimaaqaaiaadsfadaahaaWcbeqaaiaaigdacaaI YaaaaOGaeyOeI0IaamivamaaCaaaleqabaGaaGOmaiaaigdaaaGccq GH9aqpcaaIWaaaaaa@3CBD@

6.      We note that and T 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaigdacaaIYa aaaOGaeyypa0JaaGimaaaa@34CF@  by symmetry, and the external force acting on unit area of the shell is p=ρgh e 3 = p i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahchacqGH9aqpcqGHsislcqaHbpGCca WGNbGaamiAaiaahwgadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaWG WbWaaWbaaSqabeaacaWGPbaaaOGaaCyBamaaBaaaleaacaWGPbaabe aaaaa@3E1E@ .  The contravariant components of p can therefore be computed as p i =ρgh m i e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaahaaWcbeqaaiaadMgaaaGccq GH9aqpcqGHsislcqaHbpGCcaWGNbGaamiAaiaah2gadaahaaWcbeqa aiaadMgaaaGccqGHflY1caWHLbWaaSbaaSqaaiaaiodaaeqaaaaa@3E6A@ .   Substituting these, as well as the Christoffel symbols and curvature components reduces the equilibrium equations to

T 11 ϕ =0 T 22 θ + T 22 cosθ sinθ T 11 sinθcosθ+ ρgh R sinθ=0 + T 11 R sin 2 θ+ T 22 R+ρghcosθ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacqGHciITcaWGubWaaW baaSqabeaacaaIXaGaaGymaaaaaOqaaiabgkGi2kabew9aMbaacqGH 9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVpaalaaabaGaeyOaIyRaamivamaaCaaa leqabaGaaGOmaiaaikdaaaaakeaacqGHciITcqaH4oqCaaGaey4kaS IaamivamaaCaaaleqabaGaaGOmaiaaikdaaaGcdaWcaaqaaiGacoga caGGVbGaai4CaiabeI7aXbqaaiGacohacaGGPbGaaiOBaiabeI7aXb aacqGHsislcaWGubWaaWbaaSqabeaacaaIXaGaaGymaaaakiGacoha caGGPbGaaiOBaiabeI7aXjGacogacaGGVbGaai4CaiabeI7aXjabgU caRmaalaaabaGaeqyWdiNaam4zaiaadIgaaeaacaWGsbaaaiGacoha caGGPbGaaiOBaiabeI7aXjabg2da9iaaicdaaeaacqGHRaWkcaWGub WaaWbaaSqabeaacaaIXaGaaGymaaaakiaadkfaciGGZbGaaiyAaiaa c6gadaahaaWcbeqaaiaaikdaaaGccqaH4oqCcqGHRaWkcaWGubWaaW baaSqabeaacaaIYaGaaGOmaaaakiaadkfacqGHRaWkcqaHbpGCcaWG NbGaamiAaiGacogacaGGVbGaai4CaiabeI7aXjabg2da9iaaicdaaa aa@8D90@

7.      Eliminating T 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaigdacaaIXa aaaaaa@3304@  from the second and third equations gives

T 22 θ +2 T 22 cosθ sinθ = ρgh Rsinθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamivamaaCaaale qabaGaaGOmaiaaikdaaaaakeaacqGHciITcqaH4oqCaaGaey4kaSIa aGOmaiaadsfadaahaaWcbeqaaiaaikdacaaIYaaaaOWaaSaaaeaaci GGJbGaai4BaiaacohacqaH4oqCaeaaciGGZbGaaiyAaiaac6gacqaH 4oqCaaGaeyypa0ZaaSaaaeaacqaHbpGCcaWGNbGaamiAaaqaaiaadk faciGGZbGaaiyAaiaac6gacqaH4oqCaaaaaa@4F03@

8.      This equation can be integrated directly by substituting T 22 =Ψ/ sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaikdacaaIYa aaaOGaeyypa0JaeuiQdKLaai4laiGacohacaGGPbGaaiOBamaaCaaa leqabaGaaGOmaaaakiabeI7aXbaa@3BD9@ , with the solution

T 22 = ρgh R sin 2 θ (cosθ1)+C= ρgh R(1+cosθ) +C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaikdacaaIYa aaaOGaeyypa0ZaaSaaaeaacqaHbpGCcaWGNbGaamiAaaqaaiaadkfa ciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaH4oqCaa GaaiikaiGacogacaGGVbGaai4CaiabeI7aXjabgkHiTiaaigdacaGG PaGaey4kaSIaam4qaiabg2da9maalaaabaGaeyOeI0IaeqyWdiNaam 4zaiaadIgaaeaacaWGsbGaaiikaiaaigdacqGHRaWkciGGJbGaai4B aiaacohacqaH4oqCcaGGPaaaaiabgUcaRiaadoeaaaa@56E7@

where C is a constant of integration. 

9.      The constant of integration can be found using the boundary condition at the edge of the shell at θ= θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iabeI7aXnaaBaaale aacaaIWaaabeaaaaa@35E0@ .  The reaction force must act in the plane of the shell, and the vertical component of the force must balance the shell’s weight, so that the force per unit length is P=ρghR(1cos θ 0 ) e θ / sin 2 θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahcfacqGH9aqpcqaHbpGCcaWGNbGaam iAaiaadkfacaGGOaGaaGymaiabgkHiTiGacogacaGGVbGaai4Caiab eI7aXnaaBaaaleaacaaIWaaabeaakiaacMcacaWHLbWaaSbaaSqaai abeI7aXbqabaGccaGGVaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaa caaIYaaaaOGaeqiUde3aaSbaaSqaaiaaicdaaeqaaaaa@4945@ .  The boundary condition requires that e θ T=P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaki abgwSixlaahsfacqGH9aqpcaWHqbaaaa@3868@  at θ= θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iabeI7aXnaaBaaale aacaaIWaaabeaaaaa@35E0@ , which shows that C=0.

10.  T 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaigdacaaIXa aaaaaa@3304@  can be calculated from the third equation in (6), giving

T 11 = ρgh( (1+cosθ) 1 cosθ ) R sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaahaaWcbeqaaiaaigdacaaIXa aaaOGaeyypa0ZaaSaaaeaacqaHbpGCcaWGNbGaamiAamaabmaabaGa aiikaiaaigdacqGHRaWkciGGJbGaai4BaiaacohacqaH4oqCcaGGPa WaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaGPaVlaaykW7caaMc8Ua aGPaVlabgkHiTiGacogacaGGVbGaai4CaiabeI7aXbGaayjkaiaawM caaaqaaiaadkfaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikda aaGccqaH4oqCaaaaaa@549E@

11.  Finally, the components of T in the cylindrical-polar basis can be calculated by noting that m 1 =Rsinθ e ϕ m 2 =R e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGsbGaci4CaiaacMgacaGGUbGaeqiUdeNaaCyzamaaBaaa leaacqaHvpGzaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaaIYaaabeaa kiabg2da9iaadkfacaWHLbWaaSbaaSqaaiabeI7aXbqabaaaaa@503E@ , whence

T= T 11 m 1 m 1 + T 22 m 2 m 2 = T 11 R 2 sin 2 θ e ϕ e ϕ + T 22 R 2 e θ m θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpcaWGubWaaWbaaSqabe aacaaIXaGaaGymaaaakiaah2gadaWgaaWcbaGaaGymaaqabaGccqGH xkcXcaWHTbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamivamaaCa aaleqabaGaaGOmaiaaikdaaaGccaWHTbWaaSbaaSqaaiaaikdaaeqa aOGaey4LIqSaaCyBamaaBaaaleaacaaIYaaabeaakiabg2da9iaads fadaahaaWcbeqaaiaaigdacaaIXaaaaOGaamOuamaaCaaaleqabaGa aGOmaaaakiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaaki abeI7aXjaahwgadaWgaaWcbaGaeqy1dygabeaakiabgEPielaahwga daWgaaWcbaGaeqy1dygabeaakiabgUcaRiaadsfadaahaaWcbeqaai aaikdacaaIYaaaaOGaamOuamaaCaaaleqabaGaaGOmaaaakiaahwga daWgaaWcbaGaeqiUdehabeaakiabgEPielaah2gadaWgaaWcbaGaeq iUdehabeaaaaa@63B2@