Kul Models for beam, plate and shell structures, 10/2016

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1 Kul Models fo beam, plate and shell stuctues, /6 Demo poblems. Conside mapping (, φ, n) = [ cos( φ) i + sin( φ) j] + nen. Compute the expession of the basis vecto deivatives, gadient opeato, and Chistoffel symbols Γ i, jk, {, φ, n}. Is the suface defined by the mapping flat o cuved? ijk Answe e, φ = eφ e φφ, = e = e + eφ + e n n Γ φφ =. Deive the equilibium equations of the plate/shell model in tems of the stess esultants in φn coodinates of poblem. The component foms of the equilibium equation ae given by (indices take values i, jk, {, φ, n} and αβ, {, φ} ) dαfαi +Γ jα jfαi +Γ jkifjk + bi = and dβ Mβα +Γ iβimβα +Γijα Mij Fnα + cα =. Answe ( F, + Fφ, φ + F Fφφ ) + b ( Fφ, + Fφφ, φ + Fφ + Fφ ) + bφ ( Fn, + Fφn, φ + Fn ) + bn = ( M, + Mφ, φ + M Mφφ ) Fn + c ( Mφ, + Mφφ, φ + Mφ + Mφ ) Fnφ + cφ 3. The invaiant fom of the equilibium equations of shell ae given by F ( κ : I)( en F) + b = [ M ( κ : I)( e M) e F + c] e = n n n Deive the component foms in tems of diected deivatives and Chistoffel symbols. Answe If n is excluded fom the index sets of α and β dαfαi +Γ jα jfαi +Γ jkifjk + bi = dβ Mβα +Γ iβimβα +Γijα Mij Fnα + cα =. The demo poblems ae published in the couse homepage on Fidays. The poblems ae elated to the topic of the next weeks lectue (Wed.5-. hall K3 8). Solutions to the poblems ae

2 explained in the weekly execise sessions (Thu.5-4. hall K3 8) 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 Conside mapping (, φ, n) = [ cos( φ) i + sin( φ) j] + nen. Compute the expession of the basis vecto deivatives, gadient opeato, and Chistoffel symbols Γ i, jk, {, φ, n} at n = Is the suface defined by the mapping flat o cuved? Solution In tems of the basis vectos of the Catesian system, expessions of the basis vectos of the φn coodinate system ae ( = cos( φ) i + sin( φ ) j ) eh = = cos( φ) i+ sin( φ ) j h = and e = cos( φ) i + sin( φ) j, eh φ φ= = sin( φ) i+ cos( φ) j ijk hφ = and e sin( ) i φ = φ + cos( φ) j, e = e e = [cos( ) i + sin( ) j] [ sin( ) i + cos( ) j] = k φ φ φ φ φ h =. n In a moe compact fom n e cos( φ) sin( φ) i i eφ = sin( φ) cos( φ ) j= [ F] j in which en k k T [ F] = [ F]. Diect use of the definition gives (just take the deivatives on both sides of the elationship above and use invese of the same elationship to eplace the basis vectos of the Catesian system by the basis vectos of the φn system) e cos( φ) sin( φ) e eφ= sin( φ) cos( φ ) eφ=, en en e sin( φ) cos( φ) cos( φ) sin( φ) e e eφ= cos( φ) sin( φ) sin( φ) cos( φ) eφ= eφ, en en en e cos( φ) sin( φ) e eφ= sin( φ) cos( φ ) eφ=. n en en Gadient of the φn system follows fom the mapping = cos( φ) i + sin( φ) j + nk and the geneic fomula in tems of [ F ] and [ H ]. In an othonomal system

4 x, α y, α z, α hα [ H] = x, β y, β z, β = hβ F = h F x y z h, γ, γ, γ γ [ ] [ ][ ] T T T eα α eα α eα α T T eβ [ F] [ H] β eβ ([ H][ F] ) β eβ [ h] β = = =. eγ γ eγ γ eγ γ The simplified expession fo an othonomal system gives in this case (at n = if the scaling coefficients given by ae used) T e = eφ φ= e + eφ + e n n e n n. Chistoffel symbols ae the components of the basis vecto gadients e e e e = e + e + e = e e n φ n φ φ e e e e = e + e + e = e e n φ φ φ φ φ n φ en en e κ n c = en = e + eφ + e n =. n Γ = e e e φφ φ φ =, Γ φφ = eφ eφ e =, As cuvatue vanishes, mid-suface is flat.

5 Deive the equilibium equations of the plate/shell model in tems of the stess esultants in φn coodinates of poblem. The component foms of the equilibium equation ae given by (indices take values i, jk, {, φ, n} and αβ, {, φ} ) dαfαi +Γ jα jfαi +Γ jkifjk + bi = and dα Mαβ + Mαβ Γ jα j + M jkγjkβ Fn β + cβ =. Solution The diected deivatives and non-zeo Chistoffel symbols ae d =, dφ =, dn =, and n Γ = Γ = φφ φφ. By consideing each foce equilibium equation at a time i = : dαfα +Γ jα jfα +Γ jkfjk + b = df + df φ φ +Γ φφ F +Γ φφfφφ + b = F,, ( ) + Fφ φ + F Fφφ + b =. i = φ : d F +Γ F +Γ F + b = α αφ jα j αφ jkφ jk φ df φ+ df φ φφ+γ φφf φ+γ φφfφ + bφ = F,, ( ) φ + Fφφ φ + F φ + Fφ + bφ =. i = n: dαfαn +Γ jα jfα n +Γ jknfjk + bn = df n + df φ φn +Γ φφ Fn + bn = F,, n + Fφ n φ + F n + b n =. By continuing with the moment equilibium equations (just two) β = : dαmα + MαΓ jα j + M jkγjk Fn + c = M M M M F c, + φ, φ + Γ φφ + φφγφφ n + = M,, ( ) + Mφ φ + Mφ Mφφ F n + c =.

6 β = φ : dα Mαφ + MαφΓ α + M Γ φ F φ + cφ = j j jk jk n M + M + M Γ + M Γ F + c = φ, φφφ, φ φφ φ φφ nφ φ M,, ( ) φ + Mφφ φ + M φ + Mφ F n φ + cφ =.

7 The invaiant fom of the equilibium equations of shell ae given by F ( κ : I)( en F) + b =, [ M ( κ : I)( e M) e F + c] e =. n n n Deive the component foms in tems of diected deivatives and Chistoffel symbols. Solution The diected deivatives, Chistoffel symbols, and cuvatue can be expessed in tems of the gadient opeato and basis vectos. At n =, =, e j = e iγijke k, and κ = e j Γ inj e i. ed i i Let us use notation αβ,, fo the indices not including n and i, j, fo the indices including n. Hence F ( κ : I)( e F) + b = n (summation convention) e d F ee Γ ( e F ee ) + be = k k ij i j knk n ij i j i i (Chistoffel symbols) dfe +Γ Fe + FΓ e Γ F e + be = i ij j kik ij j ij ijk k knk nj j i i (index swapping) dαfαiei+γ jα jfα iei+ FjkΓ jkiei+ be i i= ( dαfαi +Γ jα jfα i + FjkΓ jki + bi ) ei =. The last fom takes into account the fact that the foce esultants do not depend on n. The same steps with the othe equation give [ M ( κ : I)( e M) e F + c] e = n n n (summation convention) [ e d M ee Γ ( e M ee ) e F ee + ce ] e = k k ij i j knk n ij i j n ij i j i i n (Chistoffel symbols) [ djfjiei+γ jα jfα iei+ FjkΓjkiei Fniei+ ce i i] en= (index swapping) [ d jfji +Γ jα jfα i + FjkΓjki Fni + ci ] ei en = dβ Fβα +Γ iβifβα + FijΓijα Fnα + cα =.

8 Kul Models fo beam, plate and shell stuctues INDEX NOTATION (Othonomal basis) ab = ab = ab + a b + + a b i i i I i i n n a / x a i j ij, δ ij ei ej {,} ( e i e j = δ ij ) ε ijk e i ( e j e k ) {,,} ( e i e j = ε ijk e 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= a ee a = aee ij i a = a c j c ij j i 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 e eφ = sφ cφ j eφ= eφ ez k ez ez = e + eφ + ez z SPHERICAL θφ SYSTEM ( θφ,, ) = (s θ c φ i + s θ s φ j + c θ k)

9 eθ cθφ c cθφ s sθ i eφ = sφ cφ j e sθφ c sθφ s cθ k eθ cθ eφ eφ= sθe cθeθ e sθeφ eθ e eφ =, θ e eθ = eθ + eφ + e θ sinθ THIN BODY snb SYSTEM FOR PLANAR BEAMS (, s n) = () s + ne () s es, s /, s, s = = e n ess, / ess, ess, R R = es + en R n s n n es en / R = s en es / R ORTHONORMAL CURVILINEAR COORDINATES eα i α x, α y, α z, α x x eβ = [ F] j β = x, β y, β z, β y= [ H] y en k x, y, z γ γ γ, γ z z eα eα eα i eβ= ( i[ F])[ F] eβ= [ D] () i eβ i e j = D ijk e k en en en T T eα α eα α = e F H = e D e e T β [ ] [ ] β β [ ] β = ed i ij j = ed i i n n n n COMPONENT REPRESENTATIONS 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 a= ( a) = dda i i +Γjijda i PLATE GEOMETRY ( φ n) (, φ, n) = [ i cosφ+ j sin φ ] + nen Γ ijk = D i D jk

10 e cosφ sinφ i eφ = sinφ cosφ j en k e eφ eφ = e e n d = d = d = φ φ n n Γ = Γ = φφ φφ 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 s = n b) ( s + sb n sn b ( κ κ κ ) d n = n d b = b ssn sns ( n b) b Γ = Γ = κ κ dv = ( nκ ) dads b snb Γ sbn = ( nκb ) κs Γ = CYLINDRICAL SHELL GEOMETRY ( zφ n) ( z, φ, n) = [ ir cosφ+ jrsin φ + kz] + nen ez i ez eφ = sinφ cosφ j eφ = en en cos φ sinφ k en eφ d = z z φ = ( ) d n = n d R n Γ φφn = Γ φnφ = ( R n) dv = ( nr ) dn( Rdφ ) dz = ( nr ) dndω LINEAR ISOTROPIC ELASTICITY σ = E: ε = E: u (mino and majo symmeties of the elasticity dyad assumed) ε = [ u + ( u )] c

11 T T ii ν ν ii ij + ji G ij + ji E = jj E ν ν jj + jk + kj G jk + kj kk ν ν kk ki + ik G ki + ik T T ii ν ii ij + ji G ij + ji E E = jj ν jj + jk + kj jk + kj (plane stess) ν kk kk ki + ik ki + ik T T ii E ii ij + ji G ij + ji E = jj jj + jk + kj G jk + kj (beam) kk kk ki + ik G ki + ik T T ii ν ii ij + ji G ij + ji E E = jj ν jj + jk + kj G jk + kj (plate) ν kk kk ki + ik G ki + ik T T ii E ii ij + ji ij + ji E = jj jj + jk + kj jk + kj (uni-axial) kk kk ki + ik ki + ik E Et G = D = ( +ν ) ( ν ) 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 3 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 TIMOSHENKO BEAM ( xyz ) E = Eii + Gjj + Gkk N + bx Q y + by= Qz + bz T + cx M y Qz + cy= Mz + Qy + cz

12 N EAu ESzψ + ES yθ Qy= GA( v ψ) GS yφ Q z GA( w + θ) + GSzφ TIMOSHENKO BEAM ( snb ) T GS y( v ψ) + GSz( w + θ) + GIφ M y = ES yu EIzyψ + EI yyθ M z ESzu + EIzzψ EI yzθ N Qnκ b + bs Qn + Nκb Qbκs + bn= Qb + Qnκ s + bb T Mnκb + cs Mn + Tκb Mbκs Qb + cn= Mb + Mnκ s + Qn + cb N EA( u vκ b) + ESn( θ + φκb ψκ s) ESb( ψ + θκ s) Qn= GA( v + uκ b wκ s ψ ) GSn( φ θκb) Q b GA( w + vκ s + θ ) + GSb( φ θκb) T GSb( w + vκ s + θ ) + GI( φ θκb) GSn( v + uκ b wκ s ψ ) Mn = ESn( u vκ b) + EInn( θ + φκb ψκ s) EIbn( ψ + θκ s) M b ESb( u vκ b) EInb( θ + φκb ψκ s) + EIbb( ψ + θκ s) 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 ν N ( ν )( u + v ) / xy, y, x M xx θ, x νφ, y Myy = D φ, y + νθ, x M ( 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 )

13 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 ν N ( ν )( u + v ) / xy, y, x Mxx w, xx + ν w, yy Myy = D w, yy + ν w, xx M ( ν ) w xy, xy Nnn Nn o un un = N ns Ns o us us REISSNER-MINDLIN PLATE ( φ z) Q + M Q M o w w M nn M n o w, n + θ s n nss, ss, = [( N ) + N N ] / + b [( Nφ ), + Nφφ, φ + Nφ] / + bφ, φ, φ φφ = N u, + ν ( u + uφφ, )/ Et Nφφ = u ν, + ( u+ uφ, φ )/ ν N ( ν )[( u u ) / + u ] / φ, φ φ φ, [( Q), + Qφφ, ] / + bz [( M ), + Mφ, φ Mφφ ] / Q + c = [( Mφ ), + Mφφ, φ + Mφ] / Qφ + cφ M θφ, + νθ ( φ θ, φ)/ Mφφ = D νθφ, + ( θφ θ, φ )/ M ( ν)[( θ + θ ) / θ ] / φ φφ,, Q w, + θφ = Gt Qφ w, φ / θ ROTATION SYMMETRIC KIRCHHOFF PLATE D w+ b z = d d = ( ) d d 4 ( ) b ( ) z w = + a ln + b + cln + d D MEMBRANE EQUATIONS IN CYLINDRICAL GEOMETRY ( zφ n) Nφz, φ + Nzz, z R bz Nzφ, z + Nφφ, φ + bφ = R b n Nφφ R te [ u zz, + ν ( u φφ, u n)] R Nzz ν te Nφφ = [ ( u φ, φ un) + νuzz, ] ν R Nzφ tg( uz, φ + uφ, z) R MEMBRANE EQUATIONS IN SPHERICAL GEOMETRY ( φθ n )

14 cscθnφφ, φ + Nθφ, θ + cot θ( Nθφ + Nφθ ) bφ csc θnφθ, φ + Nθθ, θ + cot θ ( Nθθ Nφφ ) + bθ = R Nφφ + Nθθ b n te [ csc θ(cosθu θ + ν sin θuθθ, + uφφ, ) ( + ν) un] N φφ ν te Nθθ = [ csc θ ( ν cosθu sin u θ + θ θθ, + νuφφ, ) ( + ν) un] R 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) ν Nzφ ( ν)( uφ, z + κuz, φ) / Mzz ωzz, + κνωφφ, κuzz, Mφφ νω zz, + κωφφ, + κ ( uφφ, un) M zφ D ( ν )( ωφ, z κωz, φ κuφ, z) / = + Mφz ( ν)( ωφ, z + κωz, φ + κ uz, φ) / M ( 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