From Structural Analyss to FEM Dhman Basu
Acknowldgmnt Followng txt books wr consultd whl prparng ths lctur nots: Znkwcz, OC O.C. andtaylor Taylor, R.L. (000). Th FntElmnt Mthod, Vol. : Th Bass, Ffth dton, Buttrworth Hnmann. Yang, T.Y. (986). Fnt Elmnt Structural Analyss, Prntc Hall Inc. Jan A K (009) Advancd Structural Analyss Nm Chand Jan, A.K. (009). Advancd Structural Analyss, Nm Chand & Bros.
Structural Modlng Ln lmnt Rfnd Ln Elmnt Dtald Fnt Elmnt Introducton Ln lmnt
Ln lmnt Rfnd ln lmnt FEM: Dscrtzaton ovr ntr volum n gnral Analyss Convntonal Structural Analyss Ln lmnt Rfnd ln lmnt FEM Volum dscrtzaton
Organzaton ConvntonalStructural Analyss Rvst to Convntonal Analyss Brf fconcptual Rvw of FEM Structural Analyss Smltud btwn both Analyss
Elmnt Equlbrum 6 6 L L L L Y 6 6 v 4 M EI L L θ = Y L 6 6 v M L L L L θ 6 6 4 L L { q } K { a } = EA EA 0 0 0 0 L L EI 6EI EI 6EI X 0 0 3 3 L L L L u Y 6EI 4EI 6EI EI v 0 0 M L L L L θ = X EA EA 0 0 0 0 u Y L L v M EI 6EI EI 6EI 0 0 θ 3 3 L L L L 6EI EI 6EI 4EI 0 0 L L L L K (, j ) Forc along j th dof whn unt dsplacmnt s appld th dof 3 3 whl all othrs ar rstrant
Local and Global Coordnat Systms Local coordnat Non orthogonally algnd lmnt axs Global l coordnat Coordnat transformaton by rotaton
Orthogonal Transformaton ' x cosθ snθ 0 x y = sn cos 0 y = ' θ 0 0 θ T [ λ] = [ λ] ' { } [ ]{ } ' θ θ δ λ δ
Elmnt Equlbrum n Global l Coordnat Transformaton of dsplacmnt and forc vctors u X cosφ snφ 0 0 0 0 u X v sn cos 0 0 0 0 v Y φ φ Y θ M u X = v 0 0 0 sn cos 0 v Y φ φ Y θ M 0 0 0 0 0 θ M L 0 0 0 0 0 θ { } M a T G = { a } 0 0 0 cosφ snφ 0 u X L G { q } T { q } Transformaton of Equlbrum Equaton = T { q L } = K L { a L } T { q G } = K L T { a G } { q G } = T K L T { a G } { } { } and G G G G T L q = K a K = T K T Sz of th problm rmans sam
Drct Stffnss Mthod Stp : Elmnt Equlbrum n Local Coordnat L L L { q } K { a } = 6 6 6 6 ngatv of th fxd nd forcs du to span loadng Stp : Elmnt Equlbrum n Global Coordnat G G G { q } K { a } = 6 6 6 6 K T T T = T K T, q = T q, a = T a { } { } { } { } G L G L G L 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 Stp 3: Elmnt Equlbrum n Expandd Global Coordnat Exp { q G } Exp K G Exp G { a } = 3N 3N 3N 3N Assumng a plan fram of N nods
Stp 4: Assmbl n Elmnt Equlbrum n Expandd Global Coordnat M M ( ) Exp G Exp G Exp G { q } = ( K { a } ) = = 3 N 3 N 3 N 3 N { q G} { q * G G } K { a } + = 3N 3N 3N 3N 3N Accountng for drctly appld nodal concntratd forcs Stp 5: Effct of Rstrants { q} = [ K] { a} S S S S Stp 6: Soluton for Dsplacmnt G { a } = [ K ] { q } { a } S S S S 3 N
Stp 7: Soluton for Elmnt Rspons L G { a } T { a } = 6 6 6 6 L L L L { F } = K { a } { q } = 6 6 6 6 6 Dsplacmnt n Local coordnat Mmbr nd forcs n Local coordnat Stp 8: Calculaton of Racton Forcs q r Krr K rs ar = 0 = q = K a q K K a s sr ss s r rs s
Numrcal Exampl EA=8000 kn/m and EI= 0000 knm
Soluton vctor: { 0.00356, 0.0075, 0.0058, 0.0078} T.
Mmbr nd forcs but n global coordnat
Rvst to Stffnss Matrx 4 v 0 4 = x Equlbrum of a bam lmnt (constant EI) n th unloadd rgon ( ) v x = α + α x+ α x + α x Assumd soluton 3 3 4 v v= v and = θ at x= 0 x Boundary condtons v v= v and = θ at x= L x v 0 0 0 α 3 α L 0 0 0 v θ 0 0 0 α = 3 α 0 L 0 0 θ 3 v L L L = 3 α = α 3 α3 L 3L L 3L L v θ 0 L 3L α4 α4 L L θ Soluton for coffcnts { } [ H ]{ a}
( ) ( ) θ ( ) ( ) θ ( ) vx= vf x + f x + v f x + f x Dsplacmnt profl ( ) f x ( ) 3 4 3 x x = 3 + L L x x f( x) = x + L L f x 3 f 4 ( x ) 4 3 x x = 3 L L x x = x + L L Spcfc cas v =.0, θ = 0, v = 0, θ = 0, ( ) ( ) v x = f x Dsplacmnt profl assocatd wth frst column of stffnss matrx
Applcaton of Castglano s Thorm P U = a U L EI v = x 0 dx Assumng only flxural dformaton Y EI dx EI v f x f x v f x f x f x dx L L U v v '' '' '' '' '' = = = + + 3 + 4 v x v 0 x 0 ( ) θ ( ) ( ) θ ( ) ( ) L L L L '' '' '' '' '' '' '' '' 3 4 0 0 0 0 = v EI f ( x ) f ( x ) dx+ θ EI f ( x ) f ( x ) dx+ v EI f ( x ) f ( x ) dx+ θ EI f ( x ) f ( x ) dx = K v + K θ + K v + K θ 3 4 L '' j j 0 '' K = EI f ( x) f ( x) dx For xampl, Frst quaton of qulbrum n local coordnat j th lmnt of stffnss matrx 3 '' 3 '' L L x x x x 6 x EI K = EI 3 3 + + dx= EI + dx= 3 3 L L L L L L L 0 0
Applcaton of Raylgh Rtz Mthod 3 '' ( ) ( ) v x = α + α x+ α x + α x v x = α + 6α x 3 4 3 4 L EI U = + x dx= EI L+ L + L 3 ( α3 6α4 ) ( α3 6α3α4 6α4 ) 0 Stran nrgy 0 0 0 0 α 0 0 0 0 α T U = 3 4 0 0 4EIL 6EIL = α 3 3 0 0 6EIL EIL α 4 U kj = α α T { α α α α } { α } k { α } Quadratc form j Extrnal work don Y T T U = a H k { } ([ ] H a [ ]){ } M T W = { v } θ v θ = { a } [ K ]{ a } Y M T K = H k H [ ] [ ] [ ]
3 T 3 L 0 0 0 0 0 0 0 L 0 0 0 3 3 0 L 0 0 0 0 0 0 0 L 0 0 = 3 3 L 3L L 3L L 0 0 4EIL 6EIL L 3 L L 3 L L 3 L L 0 0 6EIL EIL L L 6 6 L L L L 6 6 4 EI = L L Sam as bfor L 6 6 L L L L 6 6 4 L L [ K ]
FEM: A Prlmnary Rvst Dsplacmnt functon Nodal dsplacmnt { } T a = u u x y Dsplacmnt at any pont { x (, ) y (, )} u= u x y u x y a ˆ u u = Nkak N N j... a = j = Na k.. T Shap functons (, ) N x y j j j = j = δ = 0 j An xampl of a plan strss problm u uˆ = Na In gnral
Stran Dsplacmnt Rlaton { ε} { ˆ ε} = [ S]{ u} { ε} { ˆ ε} = [ S]{ u} = [ S][ N]{ a } = [ B]{ a } [ B ] = [ S ][ N ] For plan strss problm { ε} u x 0 x x ε xx u u y x = εyy = = 0 y y uy ε xy u x u y + y x y x Consttutv Rlaton { σ } = [ D]{ ε ε } + { σ } 0 0 { } [ D ] For plan strss problm σ xx ν 0 E σ = σ and 0 yy = ν ν τ xy 0 0 ( ν)
Extrnal Loadng Dstrbutd body forc Dstrbutd surfac loadng Concntratd load drctly actng on th nods Elmnt Equlbrum (Usng Vrtual Work Prncpl) { δ a } Vrtual dsplacmnt at nodal ponts of an lmnt { δu} [ N]{ δa } and { δε} [ B]{ δa } = = At any pont wthn th lmnt EquatngExtrnal Extrnal and Intrnalworks (wthout th concntratdnodalnodal loads) V T T T { δε } { σ } { δ u} { b} dv { δ u} { t } da = 0 A { q } K { a } = = T K [ B] [ D][ B] dv Elmnt V Equlbrum n T T T T { q } = [ B] [ D]{ ε0} dv [ B] { σ0} dv [ N ] {} b dv [ N ] { t } + + da Local V V V A coordnat
Ovrall Analyss Nodal Dsplacmnt Vctor Concptually, rmanng stps followd n drct stffnss mthod wll lad to th soluton for nodal dsplacmnt vctor of th whol structur Strss at Any Pont { σ } = [ D ][ B ]{ a } [ D ]{ ε } + { σ } 0 0
FEM: Wthout Assmblng Elmnt Equlbrum Vrtual work prncpl could hav bn appld drctly on th whol structur Govrnng quaton of qulbrum could b drvd bypassng xplctly lmnt qulbrum Concptually, smlar to formaton of stffnss matrx of th ntr structur
FEM: From th Mnmzaton of Potntal Enrgy Rplac vrtual quanttsby varaton of ral quantts * δw δ { a} T { q } { u} T { b} dv { u} T { t } = + + da V T { } { } V A Du to xtrnal load δu = δ ε σ dv Du to stran nrgy ( ) ( ) 0 δw = δu δ U + W = δ Π = Statonarty of total potntal nrgy T Π Π Π =.. = 0 Formulaton of qulbrum quatons a a a
Exampl: FEM formulaton of Stffnss of a Bam Elmnt Strss Stran Rlaton Rlt n gnralzd form Momnt Curvatur t Rlaton Rlt σ ε dv ε κ = dx d v σ M = EI dx M κ D EI dv dx { a } = v = { v θ} T T Nodal dsplacmnt vctor at a typcal nod th ( ), ( ), j 3( ), 4( ) Shap functons drvd at two nd nods N = f x f x N = f x f x
Formulaton of Stffnss '' '' '' '' B = f ( x), f( x), B j f3 ( x), f4( x) = '' '' '' '' [ B ] = B ( ) ( ) ( ) ( ) B j = f x f x f3 x f4 x = = = T T '' '' [ ] [ ][ ] [ ] ( )[ ] ( ) ( ) K B D BdV B EI Bdx EI f x f j xdx V L L Sam as drvd whn rvstng drct stffnss mthod
Rmarks FEM whn appld to bam lmnt ld to xactly sam rsults Ths s not tru n gnral
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