Problems for Chapter 7

 

Introduction to Finite Element Analysis in Solid Mechanics

 

 

 

 

7.2.  A simple finite element program

 

 

7.2.1.      Modify the simple FEA code in FEM_conststrain.mws to solve problems involving plane stress deformation instead of plane strain (this should require a change to only one line of the code).  Check the modified code by solving the problem shown in the figure.  Assume that the block has unit length in both horizontal and vertical directions, use Young’s modulus 100 and Poisson’s ratio 0.3, and take the magnitude of the distributed load to be 10 (all in arbitrary units). Compare the predictions of the FEA analysis with the exact solution.

 

 

 

7.2.2.       Modify the simple FEA code in FEM_conststrain.mws to solve problems involving axially symmetric solids.  The figure shows a representative problem to be solved.  It represents a slice through an axially symmetric cylinder, which is prevented from stretching vertically, and pressurized on its interior surface.   The solid is meshed using triangular elements, and the displacements are interpolated as

u i ( x 1 , x 2 )= u i (a) N a ( x 1 , x 2 )+ u i (b) N b ( x 1 , x 2 )+ u i (c) N c ( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGcca GGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSba aSqaaiaaikdaaeqaaOGaaiykaiabg2da9iaadwhadaqhaaWcbaGaam yAaaqaaiaacIcacaWGHbGaaiykaaaakiaad6eadaWgaaWcbaGaamyy aaqabaGccaGGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcaca WG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiykaiabgUcaRiaadwhadaqh aaWcbaGaamyAaaqaaiaacIcacaWGIbGaaiykaaaakiaad6eadaWgaa WcbaGaamOyaaqabaGccaGGOaGaamiEamaaBaaaleaacaaIXaaabeaa kiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiykaiabgUcaRi aadwhadaqhaaWcbaGaamyAaaqaaiaacIcacaWGJbGaaiykaaaakiaa d6eadaWgaaWcbaGaam4yaaqabaGccaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiyk aaaa@5FD4@

where

N a ( x 1 , x 2 )= ( x 2 x 2 (b) )( x 1 (c) x 1 (b) )( x 1 x 1 (b) )( x 2 (c) x 2 (b) ) ( x 2 (a) x 2 (b) )( x 1 (c) x 1 (b) )( x 1 (a) x 1 (b) )( x 2 (c) x 2 (b) ) N b ( x 1 , x 2 )= ( x 2 x 2 (c) )( x 1 (a) x 1 (c) )( x 1 x 1 (c) )( x 2 (a) x 2 (c) ) ( x 2 (b) x 2 (c) )( x 1 (a) x 1 (c) )( x 1 (b) x 1 (c) )( x 2 (a) x 2 (c) ) N c ( x 1 , x 2 )= ( x 2 x 2 (a) )( x 1 (b) x 1 (a) )( x 1 x 1 (a) )( x 2 (b) x 2 (a) ) ( x 2 (c) x 2 (a) )( x 1 (b) x 1 (a) )( x 1 (c) x 1 (a) )( x 2 (b) x 2 (a) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamOtamaaBaaaleaacaWGHbaabe aakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIha daWgaaWcbaGaaGOmaaqabaGccaGGPaGaeyypa0ZaaSaaaeaadaqada qaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG4bWaa0ba aSqaaiaaikdaaeaacaGGOaGaamOyaiaacMcaaaaakiaawIcacaGLPa aadaqadaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcacaWGJbGa aiykaaaakiabgkHiTiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcaca WGIbGaaiykaaaaaOGaayjkaiaawMcaaiabgkHiTmaabmaabaGaamiE amaaBaaaleaacaaIXaaabeaakiabgkHiTiaadIhadaqhaaWcbaGaaG ymaaqaaiaacIcacaWGIbGaaiykaaaaaOGaayjkaiaawMcaamaabmaa baGaamiEamaaDaaaleaacaaIYaaabaGaaiikaiaadogacaGGPaaaaO GaeyOeI0IaamiEamaaDaaaleaacaaIYaaabaGaaiikaiaadkgacaGG PaaaaaGccaGLOaGaayzkaaaabaWaaeWaaeaacaWG4bWaa0baaSqaai aaikdaaeaacaGGOaGaamyyaiaacMcaaaGccqGHsislcaWG4bWaa0ba aSqaaiaaikdaaeaacaGGOaGaamOyaiaacMcaaaaakiaawIcacaGLPa aadaqadaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcacaWGJbGa aiykaaaakiabgkHiTiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcaca WGIbGaaiykaaaaaOGaayjkaiaawMcaaiabgkHiTmaabmaabaGaamiE amaaDaaaleaacaaIXaaabaGaaiikaiaadggacaGGPaaaaOGaeyOeI0 IaamiEamaaDaaaleaacaaIXaaabaGaaiikaiaadkgacaGGPaaaaaGc caGLOaGaayzkaaWaaeWaaeaacaWG4bWaa0baaSqaaiaaikdaaeaaca GGOaGaam4yaiaacMcaaaGccqGHsislcaWG4bWaa0baaSqaaiaaikda aeaacaGGOaGaamOyaiaacMcaaaaakiaawIcacaGLPaaaaaaabaGaam OtamaaBaaaleaacaWGIbaabeaakiaacIcacaWG4bWaaSbaaSqaaiaa igdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPa Gaeyypa0ZaaSaaaeaadaqadaqaaiaadIhadaWgaaWcbaGaaGOmaaqa baGccqGHsislcaWG4bWaa0baaSqaaiaaikdaaeaacaGGOaGaam4yai aacMcaaaaakiaawIcacaGLPaaadaqadaqaaiaadIhadaqhaaWcbaGa aGymaaqaaiaacIcacaWGHbGaaiykaaaakiabgkHiTiaadIhadaqhaa WcbaGaaGymaaqaaiaacIcacaWGJbGaaiykaaaaaOGaayjkaiaawMca aiabgkHiTmaabmaabaGaamiEamaaBaaaleaacaaIXaaabeaakiabgk HiTiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcacaWGJbGaaiykaaaa aOGaayjkaiaawMcaamaabmaabaGaamiEamaaDaaaleaacaaIYaaaba GaaiikaiaadggacaGGPaaaaOGaeyOeI0IaamiEamaaDaaaleaacaaI YaaabaGaaiikaiaadogacaGGPaaaaaGccaGLOaGaayzkaaaabaWaae WaaeaacaWG4bWaa0baaSqaaiaaikdaaeaacaGGOaGaamOyaiaacMca aaGccqGHsislcaWG4bWaa0baaSqaaiaaikdaaeaacaGGOaGaam4yai aacMcaaaaakiaawIcacaGLPaaadaqadaqaaiaadIhadaqhaaWcbaGa aGymaaqaaiaacIcacaWGHbGaaiykaaaakiabgkHiTiaadIhadaqhaa WcbaGaaGymaaqaaiaacIcacaWGJbGaaiykaaaaaOGaayjkaiaawMca aiabgkHiTmaabmaabaGaamiEamaaDaaaleaacaaIXaaabaGaaiikai aadkgacaGGPaaaaOGaeyOeI0IaamiEamaaDaaaleaacaaIXaaabaGa aiikaiaadogacaGGPaaaaaGccaGLOaGaayzkaaWaaeWaaeaacaWG4b Waa0baaSqaaiaaikdaaeaacaGGOaGaamyyaiaacMcaaaGccqGHsisl caWG4bWaa0baaSqaaiaaikdaaeaacaGGOaGaam4yaiaacMcaaaaaki aawIcacaGLPaaaaaaabaGaamOtamaaBaaaleaacaWGJbaabeaakiaa cIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaa WcbaGaaGOmaaqabaGccaGGPaGaeyypa0ZaaSaaaeaadaqadaqaaiaa dIhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG4bWaa0baaSqaai aaikdaaeaacaGGOaGaamyyaiaacMcaaaaakiaawIcacaGLPaaadaqa daqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcacaWGIbGaaiykaa aakiabgkHiTiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcacaWGHbGa aiykaaaaaOGaayjkaiaawMcaaiabgkHiTmaabmaabaGaamiEamaaBa aaleaacaaIXaaabeaakiabgkHiTiaadIhadaqhaaWcbaGaaGymaaqa aiaacIcacaWGHbGaaiykaaaaaOGaayjkaiaawMcaamaabmaabaGaam iEamaaDaaaleaacaaIYaaabaGaaiikaiaadkgacaGGPaaaaOGaeyOe I0IaamiEamaaDaaaleaacaaIYaaabaGaaiikaiaadggacaGGPaaaaa GccaGLOaGaayzkaaaabaWaaeWaaeaacaWG4bWaa0baaSqaaiaaikda aeaacaGGOaGaam4yaiaacMcaaaGccqGHsislcaWG4bWaa0baaSqaai aaikdaaeaacaGGOaGaamyyaiaacMcaaaaakiaawIcacaGLPaaadaqa daqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcacaWGIbGaaiykaa aakiabgkHiTiaadIhadaqhaaWcbaGaaGymaaqaaiaacIcacaWGHbGa aiykaaaaaOGaayjkaiaawMcaaiabgkHiTmaabmaabaGaamiEamaaDa aaleaacaaIXaaabaGaaiikaiaadogacaGGPaaaaOGaeyOeI0IaamiE amaaDaaaleaacaaIXaaabaGaaiikaiaadggacaGGPaaaaaGccaGLOa GaayzkaaWaaeWaaeaacaWG4bWaa0baaSqaaiaaikdaaeaacaGGOaGa amOyaiaacMcaaaGccqGHsislcaWG4bWaa0baaSqaaiaaikdaaeaaca GGOaGaamyyaiaacMcaaaaakiaawIcacaGLPaaaaaaaaaa@4733@

 

7.2.2.1.            Show that the nonzero strain components in the element can be expressed as

7.2.2.2.             

[ ε ]=[ ε rr ε zz ε θθ 2 ε rz ][ u ]= [ u r (a) u z (a) u r (b) u z (b) u r (c) u z (c) u r (d) u z (d) ] T [ B ]=[ N a r 0 N b r 0 N c r 0 0 N a z 0 N b z 0 N c z N a r 0 N b r 0 N c r 0 N a z N a r N b z N b r N c z N c r ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaamWaaeaacqaH1oqzaiaawUfaca GLDbaacqGH9aqpdaWadaqcaauaauaabeqaeeaaaaqaaiabew7aLPWa aSbaaSqaaiaadkhacaWGYbaabeaaaKaaafaacqaH1oqzkmaaBaaale aacaWG6bGaamOEaaqabaaajaaqbaGaeqyTduMcdaWgaaWcbaGaeqiU deNaeqiUdehabeaaaKaaafaacaaIYaGaeqyTduMcdaWgaaWcbaGaam OCaiaadQhaaeqaaaaaaKaaajaawUfacaGLDbaakiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaadmaabaGaamyDaaGaay5w aiaaw2faaiabg2da9maadmaabaqbaeqabeacaaaaaeaacaWG1bWaa0 baaSqaaiaadkhaaeaacaGGOaGaamyyaiaacMcaaaaakeaacaWG1bWa a0baaSqaaiaadQhaaeaacaGGOaGaamyyaiaacMcaaaaakeaacaWG1b Waa0baaSqaaiaadkhaaeaacaGGOaGaamOyaiaacMcaaaaakeaacaWG 1bWaa0baaSqaaiaadQhaaeaacaGGOaGaamOyaiaacMcaaaaakeaaca WG1bWaa0baaSqaaiaadkhaaeaacaGGOaGaam4yaiaacMcaaaaakeaa caWG1bWaa0baaSqaaiaadQhaaeaacaGGOaGaam4yaiaacMcaaaaake aacaWG1bWaa0baaSqaaiaadkhaaeaacaGGOaGaamizaiaacMcaaaaa keaacaWG1bWaa0baaSqaaiaadQhaaeaacaGGOaGaamizaiaacMcaaa aaaaGccaGLBbGaayzxaaWaaWbaaSqabeaacaWGubaaaaqcaaEaaOWa amWaaeaacaWGcbaacaGLBbGaayzxaaGaeyypa0ZaamWaaeaafaqabe abgaaaaaqaamaalaaabaGaeyOaIyRaamOtamaaBaaaleaacaWGHbaa beaaaOqaaiabgkGi2kaadkhaaaaabaGaaGimaaqaamaalaaabaGaey OaIyRaamOtamaaBaaaleaacaWGIbaabeaaaOqaaiabgkGi2kaadkha aaaabaGaaGimaaqaamaalaaabaGaeyOaIyRaamOtamaaBaaaleaaca WGJbaabeaaaOqaaiabgkGi2kaadkhaaaaabaGaaGimaaqaaiaaicda aeaadaWcaaqaaiabgkGi2kaad6eadaWgaaWcbaGaamyyaaqabaaake aacqGHciITcaWG6baaaaqaaiaaicdaaeaadaWcaaqaaiabgkGi2kaa d6eadaWgaaWcbaGaamOyaaqabaaakeaacqGHciITcaWG6baaaaqaai aaicdaaeaadaWcaaqaaiabgkGi2kaad6eadaWgaaWcbaGaam4yaaqa baaakeaacqGHciITcaWG6baaaaqaamaalaaabaGaamOtamaaBaaale aacaWGHbaabeaaaOqaaiaadkhaaaaabaGaaGimaaqaamaalaaabaGa amOtamaaBaaaleaacaWGIbaabeaaaOqaaiaadkhaaaaabaGaaGimaa qaamaalaaabaGaamOtamaaBaaaleaacaWGJbaabeaaaOqaaiaadkha aaaabaGaaGimaaqaamaalaaabaGaeyOaIyRaamOtamaaBaaaleaaca WGHbaabeaaaOqaaiabgkGi2kaadQhaaaaabaWaaSaaaeaacqGHciIT caWGobWaaSbaaSqaaiaadggaaeqaaaGcbaGaeyOaIyRaamOCaaaaae aadaWcaaqaaiabgkGi2kaad6eadaWgaaWcbaGaamOyaaqabaaakeaa cqGHciITcaWG6baaaaqaamaalaaabaGaeyOaIyRaamOtamaaBaaale aacaWGIbaabeaaaOqaaiabgkGi2kaadkhaaaaabaWaaSaaaeaacqGH ciITcaWGobWaaSbaaSqaaiaadogaaeqaaaGcbaGaeyOaIyRaamOEaa aaaeaadaWcaaqaaiabgkGi2kaad6eadaWgaaWcbaGaam4yaaqabaaa keaacqGHciITcaWGYbaaaaaaaiaawUfacaGLDbaaaaaa@E722@

 

7.2.2.3.            Let σ=[ σ rr σ zz σ θθ σ rz ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCcqGH9aqpdaWadaqaauaabe qabqaaaaqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqabaaakeaa cqaHdpWCdaWgaaWcbaGaamOEaiaadQhaaeqaaaGcbaGaeq4Wdm3aaS baaSqaaiabeI7aXjabeI7aXbqabaaakeaacqaHdpWCdaWgaaWcbaGa amOCaiaadQhaaeqaaaaaaOGaay5waiaaw2faaaaa@48C4@  denote the stress in the element.  Find a matrix [D] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamiraiaac2faaaa@354F@  that satisfies [ σ ]=[ D ][ ε ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiabeo8aZbGaay5waiaaw2 faaiabg2da9maadmaabaGaamiraaGaay5waiaaw2faamaadmaabaGa eqyTdugacaGLBbGaayzxaaaaaa@3DD5@

 

7.2.2.4.            Write down an expression for the strain energy density U el MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaWbaaSqabeaacaWGLbGaam iBaaaaaaa@35A8@  of the element.

 

7.2.2.5.            The total strain energy of each element must be computed.  Note that each element represents a cylindrical region of material around the axis of symmetry.   The total strain energy in this material follows as

W el = A el 2πr U el drdz MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaam4vamaaCaaaleqabaGaamyzaiaadY gaaaGccqGH9aqpdaWdrbqaaiaaikdacqaHapaCcaWGYbGaamyvamaa CaaaleqabaGaamyzaiaadYgaaaGccaWGKbGaamOCaiaadsgacaWG6b aaleaacaWGbbWaaSbaaWqaaiaadwgacaWGSbaabeaaaSqab0Gaey4k Iipaaaa@453D@

The energy can be computed with sufficient accuracy by evaluating the integrand at the centroid of the element, and multiplying by the area of the element, with the result

W el =2π A el r ¯ U ¯ el MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaam4vamaaCaaaleqabaGaamyzaiaadY gaaaGccqGH9aqpcaaIYaGaeqiWdaNaamyqamaaBaaaleaacaWGLbGa amiBaaqabaGcceWGYbGbaebacaaMc8UabmyvayaaraWaaWbaaSqabe aacaWGLbGaamiBaaaaaaa@40DA@

where r ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmOCayaaraaaaa@3311@  denotes the radial position of the element centroid, and U ¯ el MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmyvayaaraWaaWbaaSqabeaacaWGLb GaamiBaaaaaaa@34FC@  is the strain energy density at the element centroid.  Use this result to deduce an expression for the element stiffness, and modify the procedure elstif() in the MAPLE code to compute the element stiffness.

 

7.2.2.6.            The contribution to the potential energy from the pressure acting on element faces must also be computed.  Following the procedure described in Chapter 7, the potential energy is

P= 0 L 2πr t i u i ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfacqGH9aqpcqGHsisldaWdXbqaai aaikdacqaHapaCcaWGYbGaaGPaVlaadshadaWgaaWcbaGaamyAaaqa baGccaWG1bWaaSbaaSqaaiaadMgaaeqaaOGaamizaiaadohaaSqaai aaicdaaeaacaWGmbaaniabgUIiYdaaaa@423A@

where

u i = u i (a) s L + u i (c) ( 1 s L )r= r (a) s L + r (c) ( 1 s L ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWG1bWaa0baaSqaaiaadMgaaeaacaGGOaGaamyyaiaacMca aaGcdaWcaaqaaiaadohaaeaacaWGmbaaaiabgUcaRiaadwhadaqhaa WcbaGaamyAaaqaaiaacIcacaWGJbGaaiykaaaakmaabmaabaGaaGym aiabgkHiTmaalaaabaGaam4CaaqaaiaadYeaaaaacaGLOaGaayzkaa GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamOCaiabg2da9iaadkhadaahaaWcbeqaaiaacIcacaWGHb GaaiykaaaakmaalaaabaGaam4CaaqaaiaadYeaaaGaey4kaSIaamOC amaaCaaaleqabaGaaiikaiaadogacaGGPaaaaOWaaeWaaeaacaaIXa GaeyOeI0YaaSaaaeaacaWGZbaabaGaamitaaaaaiaawIcacaGLPaaa aaa@628F@

and u i (a) , u i (c) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyDamaaDaaaleaacaWGPbaabaGaai ikaiaadggacaGGPaaaaOGaaiilaiaadwhadaqhaaWcbaGaamyAaaqa aiaacIcacaWGJbGaaiykaaaaaaa@3B66@  denote the displacements at the ends of the element face, and r (a) , r (c) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamOCamaaCaaaleqabaGaaiikaiaadg gacaGGPaaaaOGaaiilaiaadkhadaahaaWcbeqaaiaacIcacaWGJbGa aiykaaaaaaa@3984@  denote the radial position of the ends of the element face.  Calculate an expression for P of the form

P element =[ t 1 A t 2 A t 1 B t 2 B ][ u 1 (a) u 2 (a) u 1 (c) u 2 (c) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfadaahaaWcbeqaaiaabwgacaqGSb Gaaeyzaiaab2gacaqGLbGaaeOBaiaabshaaaGccaaMc8UaaGPaVlab g2da9iabgkHiTmaadmaabaqbaeqabeabaaaabaGaamiDamaaBaaale aacaaIXaaabeaakiaadgeaaeaacaWG0bWaaSbaaSqaaiaaikdaaeqa aOGaamyqaaqaaiaadshadaWgaaWcbaGaaGymaaqabaGccaWGcbaaba GaamiDamaaBaaaleaacaaIYaaabeaakiaadkeaaaaacaGLBbGaayzx aaGaeyyXIC9aamWaaeaafaqabeqaeaaaaeaacaWG1bWaa0baaSqaai aaigdaaeaacaGGOaGaamyyaiaacMcaaaaakeaacaWG1bWaa0baaSqa aiaaikdaaeaacaGGOaGaamyyaiaacMcaaaaakeaacaWG1bWaa0baaS qaaiaaigdaaeaacaGGOaGaam4yaiaacMcaaaaakeaacaWG1bWaa0ba aSqaaiaaikdaaeaacaGGOaGaam4yaiaacMcaaaaaaaGccaGLBbGaay zxaaaaaa@5EC1@

where A and B are constants that you must determine.  Modify the procedure elresid() to implement modified element residual.

 

7.2.2.7.            Test your routine by calculating the stress in a pressurized cylinder, which has inner radius 1, exterior radius 2, and is subjected to pressure p=1 on its internal bore (all in arbitrary units), and deforms under plane strain conditions.   Compare the FEA solution for displacements and stresses with the exact solution.  Run tests with different mesh densities, and compare the results with the analytical solution.

 

 

 

7.2.3.      Modify the simple FEA code in FEM_conststrain.mws to solve problems which involve thermal expansion.  To this end

 

7.2.3.1.            Consider a generic element in the mesh.  Assume that the material inside the element has a uniform thermal expansion coefficient α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqySdegaaa@33A1@ , and its temperature is increased by ΔT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeuiLdqKaamivaaaa@3441@ .  Let [B] and [D] denote the matrices of shape function derivatives and material properties defined in Sections 7.2.4, and let q _ =αΔT [ 1,1,0 ] T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmyCayaaDaGaeyypa0JaeqySdeMaeu iLdqKaamivamaadmaabaGaaGymaiaacYcacaaIXaGaaiilaiaaicda aiaawUfacaGLDbaadaahaaWcbeqaaiaadsfaaaaaaa@3E88@  denote a thermal strain vector.  Write down the strain energy density in the element, in terms of these quantities and the element displacement vector u _ element MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaamaaabaGaamyDaaaadaahaaWcbeqaai aabwgacaqGSbGaaeyzaiaab2gacaqGLbGaaeOBaiaabshaaaaaaa@381D@ .

 

7.2.3.2.            Hence, devise a way to calculate the total potential energy of a finite element mesh, accounting for the effects of thermal expansion.

 

7.2.3.3.            Modify the FEA code to read the thermal expansion coefficient and the change in temperature must be read from the input file, and store them as additional material properties.

 

7.2.3.4.            Modify the FEA code to add the terms associated with thermal expansion to the system of equations.   It is best to do this by writing a procedure that computes the contribution to the equation system from one element, and then add a section to the main analysis procedure to assemble the contributions from all elements into the global system of equations.

 

7.2.3.5.            Test your code using the simple test problem

 

 

 

7.2.4.      Modify the simple FEA code in FEM_conststrain.mws to solve plane stress problems using rectangular elements.  Use the following procedure.  To keep things simple, assume that the sides of each element are parallel to the e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaaaaa@323C@  and e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGOmaaqabaaaaa@323D@  axes, as shown in the picture. Let ( u 1 (a) , u 2 (a) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyDamaaBaaaleaacaaIXa aabeaakmaaCaaaleqabaGaaiikaiaadggacaGGPaaaaOGaaiilaiaa dwhadaWgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaacIcacaWGHb GaaiykaaaakiaacMcaaaa@3D91@ , ( u 1 (b) , u 2 (b) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyDamaaBaaaleaacaaIXa aabeaakmaaCaaaleqabaGaaiikaiaadkgacaGGPaaaaOGaaiilaiaa dwhadaWgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaacIcacaWGIb GaaiykaaaakiaacMcaaaa@3D93@ , ( u 1 (c) , u 2 (c) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyDamaaBaaaleaacaaIXa aabeaakmaaCaaaleqabaGaaiikaiaadogacaGGPaaaaOGaaiilaiaa dwhadaWgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaacIcacaWGJb GaaiykaaaakiaacMcaaaa@3D95@ , ( u 1 (d) , u 2 (d) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyDamaaBaaaleaacaaIXa aabeaakmaaCaaaleqabaGaaiikaiaadsgacaGGPaaaaOGaaiilaiaa dwhadaWgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaacIcacaWGKb GaaiykaaaakiaacMcaaaa@3D97@  denote the components of displacement at nodes a, b, c, d.  The displacement at an arbitrary point within the element can be interpolated between values at the corners, as follows

u 1 =(1ξ)(1η) u 1 (a) +ξ(1η) u 1 (b) +ξη u 1 (c) +(1ξ)η u 1 (d) u 2 =(1ξ)(1η) u 2 (a) +ξ(1η) u 2 (b) +ξη u 2 (c) +(1ξ)η u 2 (d) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadwhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaGGOaGaaGymaiabgkHiTiabe67a4jaacMcacaGG OaGaaGymaiabgkHiTiabeE7aOjaacMcacaWG1bWaaSbaaSqaaiaaig daaeqaaOWaaWbaaSqabeaacaGGOaGaamyyaiaacMcaaaGccqGHRaWk cqaH+oaEcaGGOaGaaGymaiabgkHiTiabeE7aOjaacMcacaWG1bWaaS baaSqaaiaaigdaaeqaaOWaaWbaaSqabeaacaGGOaGaamOyaiaacMca aaGccaaMc8UaaGPaVlabgUcaRiabe67a4jabeE7aOjaadwhadaWgaa WcbaGaaGymaaqabaGcdaahaaWcbeqaaiaacIcacaWGJbGaaiykaaaa kiabgUcaRiaaykW7caGGOaGaaGymaiabgkHiTiabe67a4jaacMcacq aH3oaAcaWG1bWaaSbaaSqaaiaaigdaaeqaaOWaaWbaaSqabeaacaGG OaGaamizaiaacMcaaaGccaaMc8oabaGaamyDamaaBaaaleaacaaIYa aabeaakiabg2da9iaacIcacaaIXaGaeyOeI0IaeqOVdGNaaiykaiaa cIcacaaIXaGaeyOeI0Iaeq4TdGMaaiykaiaadwhadaWgaaWcbaGaaG OmaaqabaGcdaahaaWcbeqaaiaacIcacaWGHbGaaiykaaaakiabgUca Riabe67a4jaacIcacaaIXaGaeyOeI0Iaeq4TdGMaaiykaiaadwhada WgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaacIcacaWGIbGaaiyk aaaakiaaykW7cqGHRaWkcqaH+oaEcqaH3oaAcaWG1bWaaSbaaSqaai aaikdaaeqaaOWaaWbaaSqabeaacaGGOaGaam4yaiaacMcaaaGccqGH RaWkcaaMc8UaaiikaiaaigdacqGHsislcqaH+oaEcaGGPaGaeq4TdG MaamyDamaaBaaaleaacaaIYaaabeaakmaaCaaaleqabaGaaiikaiaa dsgacaGGPaaaaaaaaa@9DCB@

where

ξ= x 1 /B,η= x 2 /H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH+oaEcqGH9aqpcaWG4bWaaSbaaS qaaiaaigdaaeqaaOGaai4laiaadkeacaGGSaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8Uaeq4TdGMaeyypa0JaamiEamaaBaaaleaacaaIYaaabeaa kiaac+cacaWGibGaaGPaVdaa@53D6@

 

7.2.4.1.            Show that the components of nonzero infinitesimal strain at an arbitrary point within the element may be expressed as [ ε ]=[B][ u ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaGaeqyTdugacaGLBbGaayzxaa Gaeyypa0Jaai4waiaadkeacaGGDbWaamWaaeaacaWG1baacaGLBbGa ayzxaaaaaa@3A7A@ , where

[ ε ]=[ ε 11 ε 22 ε 12 ][ u ]= [ u 1 (a) u 2 (a) u 1 (b) u 2 (b) u 1 (c) u 2 (c) u 1 (d) u 2 (d) ] T [ B ]=[ 1 B ( 1 x 2 H ) 0 1 B ( 1 x 2 H ) 0 x 2 BH 0 x 2 BH 0 0 1 H ( 1 x 1 B ) 0 x 1 BH 0 x 1 BH 0 1 H ( 1 x 1 B ) 1 2H ( 1 x 1 B ) 1 2B ( 1 x 2 H ) x 1 2BH 1 2B ( 1 x 2 H ) x 1 2BH x 2 2BH 1 2H ( 1 x 1 B ) x 2 2BH ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaamWaaeaacqaH1oqzaiaawUfaca GLDbaacqGH9aqpdaWadaqcaauaauaabeqadeaaaeaacqaH1oqzkmaa BaaajeaqbaGaaGymaiaaigdaaeqaaaqcaauaaiabew7aLPWaaSbaaK qaafaacaaIYaGaaGOmaaqabaaajaaqbaGaeqyTduMcdaWgaaqcbaua aiaaigdacaaIYaaabeaaaaaajaaqcaGLBbGaayzxaaGccaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWadaqaaiaadwhaai aawUfacaGLDbaacqGH9aqpdaWadaqaauaabeqabGaaaaaabaGaamyD amaaDaaaleaacaaIXaaabaGaaiikaiaadggacaGGPaaaaaGcbaGaam yDamaaDaaaleaacaaIYaaabaGaaiikaiaadggacaGGPaaaaaGcbaGa amyDamaaDaaaleaacaaIXaaabaGaaiikaiaadkgacaGGPaaaaaGcba GaamyDamaaDaaaleaacaaIYaaabaGaaiikaiaadkgacaGGPaaaaaGc baGaamyDamaaDaaaleaacaaIXaaabaGaaiikaiaadogacaGGPaaaaa GcbaGaamyDamaaDaaaleaacaaIYaaabaGaaiikaiaadogacaGGPaaa aaGcbaGaamyDamaaDaaaleaacaaIXaaabaGaaiikaiaadsgacaGGPa aaaaGcbaGaamyDamaaDaaaleaacaaIYaaabaGaaiikaiaadsgacaGG PaaaaaaaaOGaay5waiaaw2faamaaCaaaleqabaGaamivaaaaaKaaGh aakmaadmaabaGaamOqaaGaay5waiaaw2faaiabg2da9maadmaajaaq baqbaeqabmacaaaaaeaacqGHsislkmaalaaajaaqbaGaaGymaaqaai aadkeaaaGcdaqadaqcaauaaiaaigdacqGHsislkmaalaaajaaqbaGa amiEaOWaaSbaaKqaafaacaaIYaaabeaaaKaaafaacaWGibaaaaGaay jkaiaawMcaaaqaaiaaicdaaeaakmaalaaajaaqbaGaaGymaaqaaiaa dkeaaaGcdaqadaqcaauaaiaaigdacqGHsislkmaalaaajaaqbaGaam iEaOWaaSbaaKqaafaacaaIYaaabeaaaKaaafaacaWGibaaaaGaayjk aiaawMcaaaqaaiaaicdaaeaakmaalaaajaaqbaGaamiEaOWaaSbaaK qaafaacaaIYaaabeaaaKaaafaacaWGcbGaamisaaaaaeaacaaIWaaa baGcdaWcaaqcaauaaiabgkHiTiaadIhakmaaBaaajeaqbaGaaGOmaa qabaaajaaqbaGaamOqaiaadIeaaaaabaGaaGimaaqaaiaaicdaaeaa cqGHsislkmaalaaajaaqbaGaaGymaaqaaiaadIeaaaGcdaqadaqcaa uaaiaaigdacqGHsislkmaalaaajaaqbaGaamiEaOWaaSbaaKqaafaa caaIXaaabeaaaKaaafaacaWGcbaaaaGaayjkaiaawMcaaaqaaiaaic daaeaakmaalaaajaaqbaGaeyOeI0IaamiEaOWaaSbaaSqaaiaaigda aeqaaaqcaauaaiaadkeacaWGibaaaaqaaiaaicdaaeaakmaalaaaja aqbaGaamiEaOWaaSbaaKqaafaacaaIXaaabeaaaKaaafaacaWGcbGa amisaaaaaeaacaaIWaaabaGcdaWcaaqcaauaaiaaigdaaeaacaWGib aaaOWaaeWaaKaaafaacaaIXaGaeyOeI0IcdaWcaaqcaauaaiaadIha kmaaBaaajeaqbaGaaGymaaqabaaajaaqbaGaamOqaaaaaiaawIcaca GLPaaaaeaacqGHsislkmaalaaajaaqbaGaaGymaaqaaiaaikdacaWG ibaaaOWaaeWaaKaaafaacaaIXaGaeyOeI0IcdaWcaaqcaauaaiaadI hakmaaBaaajeaqbaGaaGymaaqabaaajaaqbaGaamOqaaaaaiaawIca caGLPaaaaeaacqGHsislkmaalaaajaaqbaGaaGymaaqaaiaaikdaca WGcbaaaOWaaeWaaKaaafaacaaIXaGaeyOeI0IcdaWcaaqcaauaaiaa dIhakmaaBaaajeaqbaGaaGOmaaqabaaajaaqbaGaamisaaaaaiaawI cacaGLPaaaaeaacqGHsislkmaalaaajaaqbaGaamiEaOWaaSbaaSqa aiaaigdaaeqaaaqcaauaaiaaikdacaWGcbGaamisaaaaaeaakmaala aajaaqbaGaaGymaaqaaiaaikdacaWGcbaaaOWaaeWaaKaaafaacaaI XaGaeyOeI0IcdaWcaaqcaauaaiaadIhakmaaBaaajeaqbaGaaGOmaa qabaaajaaqbaGaamisaaaaaiaawIcacaGLPaaaaeaakmaalaaajaaq baGaamiEaOWaaSbaaKqaafaacaaIXaaabeaaaKaaafaacaaIYaGaam OqaiaadIeaaaaabaGcdaWcaaqcaauaaiaadIhakmaaBaaajeaqbaGa aGOmaaqabaaajaaqbaGaaGOmaiaadkeacaWGibaaaaqaaOWaaSaaaK aaafaacaaIXaaabaGaaGOmaiaadIeaaaGcdaqadaqcaauaaiaaigda cqGHsislkmaalaaajaaqbaGaamiEaOWaaSbaaKqaafaacaaIXaaabe aaaKaaafaacaWGcbaaaaGaayjkaiaawMcaaaqaaOWaaSaaaKaaafaa cqGHsislcaWG4bGcdaWgaaqcbauaaiaaikdaaeqaaaqcaauaaiaaik dacaWGcbGaamisaaaaaaaacaGLBbGaayzxaaaaaaa@081F@

 

7.2.4.2.            Modify the section of the code which reads the element connectivity, to read an extra node for each element.  To do this, you will need to increase the size of the array named connect from connect(1..nelem,1..3) to connect(1..nelem,1..4), and read an extra integer node number for each element.

 

7.2.4.3.            In the procedure named elstif, which defines the element stiffness, you will need to make the following changes. (a) You will need to modify the [B] matrix to look like the one in Problem 11.1.  Don’t forget to change the size of the [B] array from bmat:=array(1..3,1..6) to bmat:=array(1..3,1..8).  Also, note that as long as you calculate the lengths of the element sides B and H correctly, you can use the [B] matrix given above even if node a does not coincide with the origin.   This is because the element stiffness only depends on the shape of the element, not on its position.  (b) To evaluate the element stiffness, you cannot assume that [B] T [D][B] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWGcbGaaiyxamaaCaaaleqaba GaamivaaaakiaacUfacaWGebGaaiyxaiaacUfacaWGcbGaaiyxaaaa @390E@  is constant within the element, so instead of multiplying  [B] T [D][B] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWGcbGaaiyxamaaCaaaleqaba GaamivaaaakiaacUfacaWGebGaaiyxaiaacUfacaWGcbGaaiyxaaaa @390E@  by the element area, you will need to integrate over the area of the element

[ K elem ]= 0 H 0 B [ B ] T [ D ][ B ] d x 1 d x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaGaam4samaaCaaaleqabaGaae yzaiaabYgacaqGLbGaaeyBaaaaaOGaay5waiaaw2faaiabg2da9maa pehabaWaa8qCaeaadaWadaqaaiaadkeaaiaawUfacaGLDbaadaahaa WcbeqaaiaadsfaaaGcdaWadaqaaiaadseaaiaawUfacaGLDbaadaWa daqaaiaadkeaaiaawUfacaGLDbaaaSqaaiaaicdaaeaacaWGcbaani abgUIiYdGccaWGKbGaamiEamaaBaaaleaacaaIXaaabeaakiaadsga caWG4bWaaSbaaSqaaiaaikdaaeqaaaqaaiaaicdaaeaacaWGibaani abgUIiYdaaaa@4ED0@

Note that MAPLE will not automatically integrate each term in a matrix.  There are various ways to fix this. One approach is to integrate each term in the matrix separately.  Let

[A]= [ B ] T [D][ B ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWGbbGaaiyxaiabg2da9maadm aabaGaamOqaaGaay5waiaaw2faamaaCaaaleqabaGaamivaaaakiaa cUfacaWGebGaaiyxamaadmaabaGaamOqaaGaay5waiaaw2faaaaa@3CFE@

then for i=1..8, j=1..8 let

k ij elem = 0 H 0 B a ij d x 1 d x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaqhaaWcbaGaamyAaiaadQgaae aacaqGLbGaaeiBaiaabwgacaqGTbaaaOGaeyypa0Zaa8qCaeaadaWd XbqaaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaaicdaae aacaWGcbaaniabgUIiYdGccaWGKbGaamiEamaaBaaaleaacaaIXaaa beaakiaadsgacaWG4bWaaSbaaSqaaiaaikdaaeqaaaqaaiaaicdaae aacaWGibaaniabgUIiYdaaaa@4882@

Use two nested int() statements to do the integrals.  Note also that to correctly return a matrix value for elstif, the last line of the procedure must read elstif=k, where k is the fully assembled stiffness matrix.

 

7.2.4.4.            Just before the call to the elstif procedure, you will need to change the dimensions of the element stiffness matrix from k:=array(1..6,1..6) to k:=array(1..8,1..8).

 

7.2.4.5.            You will need to modify the loop that assembles the global stiffness matrix to include the fourth node in each element.  To do this, you only need to change the lines that read

               for i from 1 to 3 do

to

              for i from 1 to 4  do

and the same for the j loop.

 

7.2.4.6.            You will need to modify the part of the routine that calculates the residual forces.  The only change required is to replace the line reading

>pointer := array(1..3,[2,3,1]):

with

>pointer:= array(1..4,[2,3,4,1]):

 

7.2.4.7.            You will need to modify the procedure that calculates element strains.  Now that the strains vary within the element, you need to decide where to calculate the strains.  The normal procedure would be to calculate strains at each integration point within the element, but we used MAPLE to evaluate the integrals when assembling the stiffness matrix, so we didn’t define any numerical integration points.  So, in this case, just calculate the strains at the center of the element. 

 

7.2.4.8.            To test your routine, solve the problem shown in the figure (dimensions and material properties are in arbitrary units).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.2.5.       In this problem you will develop and apply a finite element method to calculate the shape of a tensioned, inextensible cable subjected to transverse loading (e.g. gravity or wind loading).  The cable is pinned at A, and passes over a frictionless pulley at B.  A tension T is applied to the end of the cable as shown. A (nonuniform) distributed load q(x) causes the cable to deflect by a distance w(x) as shown. For w<<L, the potential energy of the system may be approximated as

V(w)= 0 L T 2 ( dw dx ) 2 dx 0 L qwdx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfacaGGOaGaam4DaiaacMcacqGH9a qpdaWdXbqaamaalaaabaGaamivaaqaaiaaikdaaaWaaeWaaeaadaWc aaqaaiaadsgacaWG3baabaGaamizaiaadIhaaaaacaGLOaGaayzkaa aaleaacaaIWaaabaGaamitaaqdcqGHRiI8aOWaaWbaaSqabeaacaaI YaaaaOGaamizaiaadIhacqGHsisldaWdXbqaaiaadghacaWG3bGaam izaiaadIhaaSqaaiaaicdaaeaacaWGmbaaniabgUIiYdaaaa@4B36@

 

To develop a finite element scheme to calculate w, divide the cable into a series of 1-D finite elements as shown.  Consider a generic element of length l  with nodes a, b at its ends. Assume that the load q is uniform over the element, and assume that w varies linearly between values w a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhadaWgaaWcbaGaamyyaaqabaaaaa@3275@ , w b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhadaWgaaWcbaGaamOyaaqabaaaaa@3276@  at the two nodes.

 

7.2.5.1.            Write down an expression for w at an arbitrary distance s from node a, in terms of w a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEhadaWgaa WcbaGaamyyaaqabaaaaa@3847@ , w b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEhadaWgaa WcbaGaamOyaaqabaaaaa@3848@ , s and l.

7.2.5.2.            Deduce an expression for dw/dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgacaWG3b Gaai4laiaadsgacaWG4baaaa@3AB7@  within the element, in terms of w a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEhadaWgaa WcbaGaamyyaaqabaaaaa@3847@ , w b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEhadaWgaa WcbaGaamOyaaqabaaaaa@3848@  and l

7.2.5.3.            Hence, calculate an expression for the contribution to the potential energy arising from the element shown, and show that element contribution to the potential energy may be  expressed as

V elem = 1 2 [ w a , w b ][ T/l T/l T/l T/l ][ w a w b ][ w a , w b ][ ql/2 ql/2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaabwgacaqGSb Gaaeyzaiaab2gaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaa amaadmaabaGaam4DamaaBaaaleaacaWGHbaabeaakiaacYcacaWG3b WaaSbaaSqaaiaadkgaaeqaaaGccaGLBbGaayzxaaWaamWaaeaafaqa beGacaaabaGaamivaiaac+cacaWGSbaabaGaeyOeI0Iaamivaiaac+ cacaWGSbaabaGaeyOeI0Iaamivaiaac+cacaWGSbaabaGaamivaiaa c+cacaWGSbaaaaGaay5waiaaw2faamaadmaabaqbaeqabiqaaaqaai aadEhadaWgaaWcbaGaamyyaaqabaaakeaacaWG3bWaaSbaaSqaaiaa dkgaaeqaaaaaaOGaay5waiaaw2faaiabgkHiTmaadmaabaGaam4Dam aaBaaaleaacaWGHbaabeaakiaacYcacaWG3bWaaSbaaSqaaiaadkga aeqaaaGccaGLBbGaayzxaaWaamWaaeaafaqabeGabaaabaGaamyCai aadYgacaGGVaGaaGOmaaqaaiaadghacaWGSbGaai4laiaaikdaaaaa caGLBbGaayzxaaaaaa@62F3@

 

7.2.5.4.            Write down expressions for the element stiffness matrix and residual vector.

 

7.2.5.5.            Consider the finite element mesh shown in the figure.  The loading q 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadghadaWgaa WcbaGaaGimaaqabaaaaa@3815@  is uniform, and each element has the same length.  The cable tension is T.  Calculate the global stiffness matrix and residual vectors for the mesh, in terms of T, L, and q 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadghadaWgaa WcbaGaaGimaaqabaaaaa@3815@ .

 

7.2.5.6.            Show how the global stiffness matrix and residual vectors must be modified to enforce the constraints w 1 = w 4 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaWG3bWaaSbaaSqaaiaaisdaaeqa aOGaeyypa0JaaGimaaaa@3CDC@

 

7.2.5.7.            Hence, calculate values of w at the two intermediate nodes.