From Structural Analysis to FEM. Dhiman Basu

Transcription

1 From Structural Analyss to FEM Dhman Basu

2 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.

3 Structural Modlng Ln lmnt Rfnd Ln Elmnt Dtald Fnt Elmnt Introducton Ln lmnt

4 Ln lmnt Rfnd ln lmnt FEM: Dscrtzaton ovr ntr volum n gnral Analyss Convntonal Structural Analyss Ln lmnt Rfnd ln lmnt FEM Volum dscrtzaton

5 Organzaton ConvntonalStructural Analyss Rvst to Convntonal Analyss Brf fconcptual Rvw of FEM Structural Analyss Smltud btwn both Analyss

6 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 L L { q } K { a } = EA EA L L EI 6EI EI 6EI X L L L L u Y 6EI 4EI 6EI EI v 0 0 M L L L L θ = X EA EA 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

7 Local and Global Coordnat Systms Local coordnat Non orthogonally algnd lmnt axs Global l coordnat Coordnat transformaton by rotaton

8 Orthogonal Transformaton ' x cosθ snθ 0 x y = sn cos 0 y = ' θ 0 0 θ T [ λ] = [ λ] ' { } [ ]{ } ' θ θ δ λ δ

9 Elmnt Equlbrum n Global l Coordnat Transformaton of dsplacmnt and forc vctors u X cosφ snφ u X v sn cos v Y φ φ Y θ M u X = v sn cos 0 v Y φ φ Y θ M θ M L θ { } M a T G = { a } 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

10 Drct Stffnss Mthod Stp : Elmnt Equlbrum n Local Coordnat L L L { q } K { a } = ngatv of th fxd nd forcs du to span loadng Stp : Elmnt Equlbrum n Global Coordnat G G G { q } K { a } = K T T T = T K T, q = T q, a = T a { } { } { } { } G L G L G L 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

11 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

12 Stp 7: Soluton for Elmnt Rspons L G { a } T { a } = L L L L { F } = K { a } { q } = 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

13 Numrcal Exampl EA=8000 kn/m and EI= 0000 knm

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16 Soluton vctor: { , , , } T.

17 Mmbr nd forcs but n global coordnat

18 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 v v= v and = θ at x= 0 x Boundary condtons v v= v and = θ at x= L x v α 3 α L v θ 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}

19 ( ) ( ) θ ( ) ( ) θ ( ) vx= vf x + f x + v f x + f x Dsplacmnt profl ( ) f x ( ) 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

20 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 '' '' '' '' '' = = = v x v 0 x 0 ( ) θ ( ) ( ) θ ( ) ( ) L L L L '' '' '' '' '' '' '' '' = 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 dx= EI + dx= 3 3 L L L L L L L 0 0

21 Applcaton of Raylgh Rtz Mthod 3 '' ( ) ( ) v x = α + α x+ α x + α x v x = α + 6α x L EI U = + x dx= EI L+ L + L 3 ( α3 6α4 ) ( α3 6α3α4 6α4 ) 0 Stran nrgy α α T U = EIL 6EIL = α EIL 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 [ ] [ ] [ ]

22 3 T 3 L L L 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 EI = L L Sam as bfor L 6 6 L L L L L L [ K ]

23 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

24 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)

25 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

26 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

27 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

28 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

29 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

30 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

31 Rmarks FEM whn appld to bam lmnt ld to xactly sam rsults Ths s not tru n gnral

32 Thank You

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