Kul Models for beam, plate and shell structures, 02/2016

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Ζηνόβιος Κανακάρης-Ρούφος
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1 Kul Models fo beam, plate and shell stuctues, /16 Demo poblems 1. Given the Catesian stain components ε ij ij, {, xy}, deive the coesponding stain components ε αβ αβ, {, φ } of the pola coodinate system (in tems of the Catesian components). Use the invaiance of dyad quantities and elationship e cosφ sinφ i =. eφ sinφ cosφ j Answe cos φ cosφsinφ cosφsinφ sin φ ε xx ε xy ε εφ cosφsinφ cos φ sin φ cosφsinφ = εφ cosφsinφ sin φ cos φ cosφsinφ ε yx εφφ sin cos sin cos sin cos yy φ φ φ φ φ φ ε. Deive the gadient expession of the spheical coodinate system, when the mapping is given by = xi + yj + zk = (sinθcosφi + sinθsinφ j + cos θ k ). Use the geneic ecipe { e e e }[ F] T [ H] 1 α = α β γ β γ αβγ,, θφ,,. and ( ) ( ), whee [ ] x, y, z α α, α e α i H = x, β y, β z, β, eβ = [ F] j x, γ y, γ z, γ eγ k Answe 1 1 = eθ + eφ + e θ sinθ φ 3. In the intinsic coodinate system and XY-plane case, the displacement assumption of a cuved Timoshenko beam model is u = u + θ ρ, whee u = use () s + vse () n, () θ = ψ se b, and ρ = ne n. Deive the small stain component expessions ε ij αβ, {, sn} by using 1 ε = + [ u ( u) ] c, 1 = es + en, 1 n/ R s n es 1 en es = s e R, and = n es n. en Assume that cuvatue 1/ R is constant. R v 1 R 1 Answe εss = ( u, s ψ, sn ) εsn ns ( u v, s ) R n R = ε = R n R + ψ

2 The demo poblems ae published in the couse homepage on Fidays. The poblems ae elated to the topic of the next weeks lectue (Wed hall K3 118). Solutions to the poblems ae explained in the weekly execise sessions (Thu hall K3 118) and will also be available in the home page of the couse. Please, notice that the poblems of the midtems and the final exam ae of this type.

3 Lectue poblem Be pepaed to deive the small stain components in a specified system, when displacement, gadient expession, and deivatives of the basis vectos ae given. Lectue poblems ae specified and solved duing the lectue (Wed hall K3 118). The time allocated fo this is 3 min.

4 Home poblem Deive the small stain expessions in the cylindical coodinate system, when displacement, gadient opeato and non-zeo deives of the basis vectos ae u (,, z) = ue + u e + uze 1 φ φ φ z, = e + eφ + ez φ z Solution template e eφ = eφ, and = e. φ φ, In manipulation of vecto expession containing vectos and dyads, it is impotant to emembe that tenso ( ), coss ( ), inne ( ) poducts ae non-commutative (ode mattes). The basis vectos of a cuvilinea coodinate system ae not constants which should be taken into account if gadient opeato is pat of expession. Othewise, simplifying an expession o finding a specific fom in a given coodinate system is a staightfowad (sometimes tedious) execise. 1. Conside the tems of displacements sepaately to keep expessions shot enough. Substitute the epesentations in the specified system, expand, use the poduct ule fo deivatives, and substitute the deivatives of the basis vectos. ( u 1 e ) = e ( ue ) + e ( ue ) + e z ( ue φ ) φ z u e 1 u 1 e u e = e e + eu + eφ e + eφ u + ez e + ezu φ φ z z u 1 u 1 u = ee + ee φ + ee φ φ u+ ee z φ z ( ue φ φ ) = = = ( ue ) = z z = =

5 . Combine the tems to get the displacement gadient u. Afte that, swap the basis vectos of each tem of u and e-aange to get the conjugate expession ( u ) (in the same ode of the tems as u ) u = ee + ee + ee + φ z c ee + ee + ee φ φ φ φ z + ee + ee + ee z z φ z z ( u) = ee + ee φ + ee z + c ee + ee + ee φ φ φ φ z + ee + ee + ee z z φ z z 3. Use the small stain definition ε = 1 [ u + ( u )] c (symmetic pat of displacement gadient!) ε = ee + ee + ee + φ z ee φ + ee φ φ + ee + φ z ee + ee + ee z z φ z z 4. Wite down the component expessions ε ε = ε z = = φ ε ε = ε z = φ = φφ φ ε z 1 u ( z u = + ) z 1 1 u u ( z φ φ = + ) φ z ε z zz u ε = z z

6 The compulsoy home poblems ae published in the couse homepage on Fidays and the deadline fo answes is the next weeks Fiday Retun you homewok answes into the geen couse mailbox that can be found fom the coido of the K3 building lobby (Puumiehenkuja 5A). Please, use the solution templates given.

7 Kul Models fo beam, plate and shell stuctues INDEX NOTATION (Othonomal basis) ab i i= ab i i= ab 11+ ab+ + anbn i I ai/ xj aij, δ ij ei ej {,1} ( e i e j = δ ij ) ε e ( e e ) { 1,,1} ( e i e j = ε ijk e k ) ijk i j k εijkεimn = δ jmδkn δ jnδ km ε det( a) = ε a a a ijk lmn il jm kn GENERAL a = ae i i a= a ij ee i j a = aijklee i je ke l... I a = a I = a a ( I = ii + jj + kk ) I : a = a: I = a a ( I = iiii + jjjj + kkkk + ijji + jiij + ikki + kiik + kjjk + jkkj ) a= aijee i j ac = aee ij j i a = a c a b = a b b IDENTITIES a ( b c) = ( a b) c a ( b c) = bac ( ) cab ( ) a:( b) = ( a b) ( a) b c CYLINDRICAL φ z SYSTEM = cosφi + sinφ j + zk e cφ sφ i e 1 e eφ = sφ cφ j eφ= 1 eφ φ ez 1 k ez ez 1 = e + eφ + ez φ z SPHERICAL θφ SYSTEM

8 ( θφ,, ) = (s θ c φ i + s θ s φ j + c θ k) eθ cθφ c cθφ s sθ i eθ cθ eφ eφ = sφ cφ j eφ= sθe cθeθ φ e sθφ c sθφ s cθ k e sθeφ eθ e eφ =, θ e eθ 1 1 = eθ + eφ + e θ sinθ φ THIN BODY snb SYSTEM FOR PLANAR BEAMS (, s n) = () s + ne () s n es, s /, s, s es en / R = = = e n ess, / ess, ess, R s en es / R R = es + en R n s n ORTHONORMAL CURVILINEAR COORDINATES eα eα eα 1 i e = ( i[ F])[ F] e = [ D] () i e e = D e en en en β β β i j ijk k T T eα α eα α = e F H = e D e e T 1 β [ ] [ ] β β [ ] β = ed i ij j = ed i i n n n n Γ = e e e = e = ( e e ) D D ( e e ) ijk i j k k i s s jl l k a= ( dae ) i a= ( da + a Γ ) ee i i j k ikj i j a= da +Γ a i i iji j a= ( da +Γ a +Γ a ) e i ij kik ij ikj ik j Γ ijk = D i D jk a= ( a) = dda i i +Γjijda i PLATE GEOMETRY ( φ n)

9 (, φ, n) = [ i cosφ+ j sin φ ] + nen e cosφ sinφ i eφ = sinφ cosφ j en 1 k e eφ eφ = e φ e n d = d 1 = d = φ φ n n Γ = Γ = φφ φφ 1 dv = dndω BEAM GEOMETRY ( snb ) ( s, n, b) = [ ( s)] + ne n + be b es, s es κb es κben en= ess, / ess, en= κb κs en= κseb κbes s eb es en eb κs eb κsen d 1 s = 1 n b) ( s + sb n sn b ( κ κ κ ) d n = n d b = b 1 ssn sns (1 n b) b Γ = Γ = κ κ dv = (1 nκ ) dads b 1 snb Γ sbn = (1 nκb ) κs Γ = CYLINDRICAL SHELL GEOMETRY ( zφ n) ( z, φ, n) = [ ir cosφ+ jrsin φ + kz] + nen ez 1 i ez eφ = sinφ cosφ j eφ = en φ en cos φ sinφ k en eφ d = z z 1 φ = ( ) φ d n = n d R n Γ φφn = Γ φnφ = ( R n) dv = (1 nr ) dn( Rdφ ) dz = (1 nr ) dndω LINEAR ISOTROPIC ELASTICITY σ = E : ε ε = 1 [ u + ( u )] c

10 E ν σ = [ I( I : ε) + ε] 1+ ν 1 ν E ν E = ( II + I ) 1+ ν 1 ν 3 Et D = 1(1 ν ) 1 ε= [(1 + νσ ) ν I( I : σ )] E PRINCIPLE OF VIRTUAL WORK ext int δw = δw + δw = δ u U (a function set) δw = ( σ : δε ) dv + ( f δu) dv + ( t δ u) da V c V A BEAM EQUATIONS F + b F σ = = da M + i F + c M ρ σ F σ E E ρ u + i θ = da = da M ρ σ ρ E ρ E ρ θ E = Eii + Gjj + Gkk TIMOSHENKO BEAM ( xyz ) N + bx Q y + by= Qz + bz T + cx M y Qz + cy= Mz + Qy + cz N EAu ES yψ + ESzθ Qy= GA( v ψ) GSzφ Q z GA( w + θ) + GS yφ T GSz ( v ψ) + GSy ( w + θ) + G( I yy + Izz ) φ M y = ESzu EIzyψ + EIzzθ M z ES yu + EI yyψ EI yzθ TIMOSHENKO BEAM ( snb ) N Qnκb + Qbκn + bs Qn + Nκb Qbκs + bn= Qb Nκn + Qnκs + bb T Mnκb + Mbκn + cs Mn + Tκb Mbκs Qb + cn= Mb Tκn + Mnκs + Qn + cb

11 N EA( u + wκ n vκ b) + ESb( θ + φκb ψκs) ESn( ψ φκn + θκs) Qn= GA( v + uκb wκ s ψ ) GSb( φ + ψκn θκb) Q b GA( w uκn + vκ s + θ ) + GSn( φ + ψκn θκb) T GSn( w uκn + vκ s + θ ) + GI ( φ + ψκn θκb) GSb( v + uκb wκ s ψ ) M n = ESb( u + wκ n vκ b) + EIbb( θ + φκb ψκs ) EIbn( ψ φκn + θκs ) M b ESn( u + wκ n vκ b) EInb( θ + φκb ψκs ) + EInn( ψ φκn + θκs ) PLATE EQUATIONS F + b = ( M Q+ c) k = F = σ dz = iin + ijn + jin + jjn + ( ki + ik ) Q + ( kj + jk ) Q xx xy yx yy x y M = σ zdz = iim + ijm + jim + jjm + ( ki + ik ) R + ( kj + jk ) R xx xy yx yy x y REISSNER-MINDLIN PLATE ( xyz ) Nxx, x + Nyx, y + b x = Nyy, y + Nxy, x + by Qxx, + Qyy, + bz Mxx, x + Myx, y Qx + cx = Myy, y + Mxy, x Qy + cy Qx w, x + θ = Gtk Q w φ y, y Nxx u, x + ν v, y Et Nyy = v, y + νu, x 1 ν N (1 ν )( u + v ) / xy, y, x M xx θ, x νφ, y Myy = D φ, y + νθ, x M (1 ν)( θ φ ) / xy, y, x Qn Q o w w n Nnn Nn o un un = M ns M s o θn θn = N ns Ns o us u s M nn M n o θs θs KIRCHHOFF PLATE ( xyz ) Nxx, x + Nyx, y + b x = Nyy, y + Nxy, x + by Mxx, xx + Mxy, xy + Myy, yy + bz ( Mxx, x + Myx, y Qx + cx ) = ( Myy, y + Mxy, x Qy + cy ) Nxx u, x + ν v, y Et Nyy = v, y + νu, x 1 ν N (1 ν )( u + v ) / xy, y, x Mxx w, xx + ν w, yy Myy = D w, yy + ν w, xx M (1 ν ) w xy, xy

12 Nnn Nn o un un = N ns Ns o us us Q + M Q M o w w M nn M n o w, n + θ s n nss, ss, = REISSNER-MINDLIN PLATE ( φ z) [( N ) + N N ] / + b [( Nφ ), + Nφφ, φ + Nφ] / + bφ, φ, φ φφ = N u, + ν ( u + uφφ, )/ Et Nφφ = u ν, + ( u+ uφ, φ )/ 1 ν N (1 ν )[( u u ) / + u ] / φ, φ φ φ, [( Q), + Qφφ, ] / + bz [( M ), + Mφ, φ Mφφ ] / Q + c = [( Mφ ), + Mφφ, φ + Mφ] / Qφ + cφ Q w, + θφ = Gt Q w θ φ, φ / M θφ, + νθ ( φ θ, φ)/ Mφφ = D νθφ, + ( θφ θ, φ )/ M (1 ν)[( θ + θ ) / θ ] / φ φφ,, ROTATION SYMMETRIC KIRCHHOFF PLATE D w+ b z = 1 d d = ( ) d d MEMBRANE EQUATIONS IN CYLINDRICAL GEOMETRY ( zφ n) 1 Nφz, φ + Nzz, z R bz 1 Nzφ, z + Nφφ, φ + bφ = R b 1 n Nφφ R te 1 [ u zz, + ν ( u φφ, u n)] 1 R Nzz ν te 1 Nφφ = [ ( u φ, φ un) + νuzz, ] 1 ν R Nzφ 1 tg( uz, φ + uφ, z) R MEMBRANE EQUATIONS IN SPHERICAL GEOMETRY ( φθ n ) cscθnφφ, φ + Nθφ, θ + cot θ( Nθφ + Nφθ ) bφ 1 csc θnφθ, φ + Nθθ, θ + cot θ ( Nθθ Nφφ ) + bθ = R Nφφ + Nθθ b n

13 te [ csc θ(cosθu θ + ν sin θuθθ, + uφφ, ) (1 + ν) un] N 1 φφ ν 1 te Nθθ = [ csc θ ( ν cosθu sin u θ + θ θθ, + νuφφ, ) ( 1 + ν) un] R 1 ν Nφθ tg( cscθuθφ, co tθuφ + uφθ, ) SHELL EQUATIONS IN CYLINDRICAL GEOMETRY ( zφ n) κ Nφz, φ + Nzz, z + bz Nzφ, z + κnφφ, φ κqφ + bφ = κqφ, φ + Qzz, + κnφφ + bn Mzφ, z + κmφφ, φ κmφn Qφ + cφ M + κm Q + c = zz, z φz, φ z z Nzz uz, z + νκ( uφφ, un) Et Nφφ = u ν z, z + κ( uφφ, un) 1 ν Nzφ (1 ν)( uφ, z + κuz, φ) / Mzz ωzz, + κνωφφ, κuzz, Mφφ νω zz, + κωφφ, + κ ( uφφ, un) M zφ D (1 ν )( ωφ, z κωz, φ κuφ, z) / = + Mφz (1 ν)( ωφ, z + κωz, φ + κ uz, φ) / M (1 νκκ ) ( u + κu + ω) / φn n, φ φ φ Qz unz, + ωz = tg Q ω + κ( u + u ) φ φ n, φ φ ωz θ φ = ωφ θz

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