Kul Models for beam, plate and shell structures, 08/2016

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1 Kul Models for beam, plate and shell structures, 8/6 Demo problems. Spring geometry is defined by the mapping s s s r ( s) = ( ir cos + jrsin + kε ), R + ε R + ε + ε where R and ε are constants and s is the curve parameter. Use the gradient and the basis vector derivative expressions = e + e + e s s n n b b, es es e n = ε e n s R( ) e + ε b e ε b to derive the expression of curvature κ = ( e n ) c. Answer ε κ = ee s s + ee b s R( + ε ) R( + ε ). Find (some) parametric representations of the following surfaces: (a) Ellipsoid ( r/ a) + ( z/ b) =, (b) Hyperboloid ( r/ a) ( z/ b) = where r = x + y. Answer (a) r( φθ, ) = ia sinθcosφ+ jasinθsinφ+ kb cosθ (b) r( θφ, ) = ia sinhθcosφ+ jasinhθsinφ kb coshθ 3. Compute the expressions of the basis vectors e α, eβ, and e n in terms of i, j,and k, derivatives of the basis vector in terms of e α, eβ and e n, gradient operator, and curvature. Consider the spherical and cylindrical geometries having parametric representations (a) r ( θφ, ) = R(sinθcosφi + sinθsinφ j + cos θk) ( α = θ, β = φ ) (b) r ( φ, z) = R(cosφi + sin φ j) + zk ( α= φβ, = z ) Notice that the order of the coordinates differ from that of the lecture notes, which affect e.g. direction of e n.

2 R R Answer (a) = ( ) eθ + ( ) eφ + en R+ n R θ R+ n Rsinθ φ n κ = ( ee φ φ+ ee θ θ) R+ n (b) R = ( ) eφ + ez + en R+ n R φ z n κ = ee φ φ R + n The demo problems are published in the course homepage on Fridays. The problems are related to the topic of the next weeks lecture (Wed.5-. hall K3 8). Solutions to the problems are explained in the weekly exercise sessions (Thu.5-4. hall K3 8) and will also be available in the home page of the course. Please, notice that the problems of the midterms and the final exam are of this type.

3 Lecture problem Be prepared to derive geometrical quantities of a curved surface. Lecture problems are specified and solved during the lecture (Wed.5-. hall K3 8). The time allocated for this is 3 min.

4 Home problem Consider a conical shell having the mid-surface r ( z, φ ) = zi ( cos φ+ j sin φ+ k ε ), in which ε is constant. Derive the basis vectors, basis vector derivatives, gradient expression, and curvature dyad of the conical mid-surface in the zφn coordinate system. Solution template. Let us start with the relationship between the basis vectors. Definitions give ez r, z/ r, z i i eφ r, φ/ r, φ = j= [ F] j en ez eφ/ ez eφ k k T. Since the basis is orthonormal i.e. [ F] = [ F], the partial derivatives of the basis vectors are given by (the other derivatives vanish) ez e z eφ= ( [ F])[ F] eφ= φ φ en en ez eφ en 4. The gradient expression in concerned with a generic material point so that the mapping between the curvilinear zφn coordinate system and the reference xyz coordinate system is written as r = r + ne n (the mapping needs to define positions of all particles of the body not just those on the mid-surface). Relationship gives

5 [ H ] x y z x y z x y z, φ, φ, φ =, θ, θ, θ =, n, n, n cosφ sinφ ε nε nε ( z)sin φ ( z )cosφ + ε + ε εcosφ εsinφ + ε + ε + ε T F 4. The generic formula for the gradient operator gives (here [ F] = [ ] ) T ez / z T eφ [ F] [ H] / φ + ε = = ez + eφ + en ε z z ε εn φ n + + en / n 5. Finally, curvature of the conical mid-surface can be calculated from κ = ( e n ) c, where the subscript zero denotes mid-surface i.e. n = (derivatives of the basis vectors are needed in the calculation = e n κ = NOTICE. You may use Mathematica notebook Geometry.nb to derive/check the calculations of each step! The compulsory home problems are published in the course homepage on Fridays and the deadline for answers is the next weeks Friday Return your homework answers into the green course mailbox that can be found from the corridor of the K3 building lobby (Puumiehenkuja 5A). Please, use the solution templates given.

6 Kul Models for beam, plate and shell structures INDEX NOTATION (Orthonormal 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 rφ z SYSTEM r = r cosφi + r sinφ j + zk er cφ sφ i er er eφ = sφ cφ j eφ= eφ φ ez k ez ez = er + eφ + ez r r φ z SPHERICAL θφr SYSTEM r( θφ,, r) = r(s θ c φ i + s θ s φ j + c θ k)

7 eθ cθφ c cθφ s sθ i eφ = sφ cφ j er sθφ c sθφ s cθ k eθ cθ eφ eφ= sθer cθeθ φ er sθeφ eθ er eφ =, θ er eθ = eθ + eφ + e r r θ rsinθ φ r THIN BODY snb SYSTEM FOR PLANAR BEAMS r(, s n) = r () s + ne () s es r, s / r, s r, 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 sr rjl 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 ( rφ n) r ( r, φ, n) = [ ir cosφ+ jr sin φ ] + nen Γ ijk = D ir D rjk

8 er cosφ sinφ i eφ = sinφ cosφ j en k er eφ eφ = er φ e n d = r r d r = d = φ φ n n Γ = Γ = φrφ φφr r dv = dndω BEAM GEOMETRY ( snb ) r ( s, n, b) = [ r ( s)] + ne n + be b es r, 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) r ( z, φ, n) = [ ir cosφ+ jrsin φ + kz] + nen ez i ez eφ = sinφ cosφ j eφ = en φ en cos φ sinφ k en eφ d = z z φ = ( ) 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 (minor and major symmetries of the elasticity dyad assumed) ε = [ u + ( u )] c

9 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 stress) 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

10 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) + GIrrφ 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 ) + GIrr( 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 or w w n Nnn Nn or un un = M ns M s or θn θn = N ns Ns or us u s M nn M n or θs θs KIRCHHOFF PLATE ( xyz )

11 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 or un un = N ns Ns or us us REISSNER-MINDLIN PLATE ( rφ z) Q + M Q M or w w M nn M n or w, n + θ s n nss, ss, = [( rn ) + N N ] / r + b [( rnrφ ), r + Nφφ, φ + Nφr] / r + bφ rr, r φr, φ φφ r = Nrr ur, r + ν ( ur + uφφ, )/ r Et Nφφ = u ν rr, + ( ur+ uφ, φ )/ r ν N ( ν )[( u u ) / r+ u ] / rφ r, φ φ φ, r [( rqr), r + Qφφ, ] / r + bz [( rmrr ), r + Mφr, φ Mφφ ] / r Qr + cr = [( rmrφ ), r + Mφφ, φ + Mφr] / r Qφ + cφ Mrr θφ, r + νθ ( φ θr, φ)/ r Mφφ = D νθφ, r + ( θφ θr, φ )/ r M ( ν)[( θ + θ ) / r θ ] / rφ φφ, r rr, Qr w, r + θφ = Gt Qφ w, φ / r θr ROTATION SYMMETRIC KIRCHHOFF PLATE D w+ b z = d d = ( r ) r dr dr 4 r r ( r ) b ( ) z r wr = + a ln + b + cln r+ 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 )

12 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