Kul Models for beam, plate and shell structures, 09/2016

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1 Kul Models for beam, plate and shell structures, 9/6 Demo problems. Derive the component forms of the membrane equations in spherical φθ n coordinate system and geometry. Use the component form N + b = ( d N +Γ N +Γ N + b) e = j ji kjk ji jki jk i i and expressions Γ φφn =Γ θθn =, Γ φθφ = cot θ, dφ = φ, sinθ dθ = θ, and d n = n. Answer T [csc θnφφ, φ + Nθφ, θ + cot θ ( Nθφ + Nφθ )] + bφ e φ eθ [csc θnφθ, φ + Nθθ, θ + cot θ ( Nθθ Nφφ )] + bθ = e n ( Nφφ + Nθθ ) + bn. Consider a simply supported (long) circular cylindrical shell of radius, thickness t, and filled with liquid of density ρ in cylindrical φn coordinates. Determine the mid-surface stress resultants N φφ, N φ and N by assuming that there are no axial forces at the ends of the shell and bending deformation is negligible. (J.N.eddy: Example.3.) g L x Answer N p ρg cosφ φφ Nφ = ρg ( L )sin φ+ A N ( ρg L )cos φ (A and p are constants) 3. Consider a truncated cone, as shown in the figure. Determine the mid-surface stress resultants due to its own weight. Acceleration by gravity g, density of the material ρ, and thickness of the cell t wall are constants. Use cylindrical φn coordinates and assume that the cone stands freely on a frictionless foundation. The coordinate value of the free end is. (J.N.eddy: Problem.8 modified somewhat). y α x g

2 Answer N φφ tρgtan α Nφ = N tρg( + tan α) ( ) 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 write component forms of equilibrium and constitutive equations of membrane model using directed derivatives and Christoffel symbols. 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 cylindrical shell of semicircular cross section supporting its own weight, which is assumed to be distributed uniformly over the surface of the shell. Using the membrane theory, determine Nφφ, Nφ and N assuming that there are no axial forces at the ends of the tube. (J.N.eddy: Problem.) x g y L Solution template The membrane equations written in cylindrical φ coordinate system and the relationship between the basis vector of the Cartesian and cylindrical system are, N T φ, φ + N, + b e eφ Nφ, + Nφφ, φ + bφ = en Nφφ + bn i sinφ cosφ e j= cosφ sinφ eφ k en. External distributed force due to gravity expressed in the basis of the cylindrical coordinate system is (here t = ) T e t/ t/ b = f = [ ] dn e t/ = t/ φ e n. Membrane equations of the present case are ( Nφ = Nφ) T e eφ en = 3. Solution to the eqution associated with direction e n is given by

5 N φφ = 4. Knowing the solution above, the equation associated with direction e φ gives (notice that integration constants of partial differential equations are not constants but arbitrary functions of φ. Denote the function A( φ) here) N φ =, N φ = 5. Knowing the solution above, the equation associated with direction e gives (again: notice that integration constants of partial differential equations are not constants but arbitrary functions of φ. Denote the function B( φ ) here) N, = N = 6. By assumption, there are no axial forces at the ends {, L}. Therefore N (, φ ) = = N ( L, φ ) = = giving A( φ ) = B( φ ) = 7. Solutions to the force resultants are

6 N (, φ ) = gt cos ( L) ρ φ Nφ (, φ ) = ρgtsin φ ( L ) + A Nφφ (, φ ) = ρgtcos φ NOTICE. Solutions to quite similar demo problems will be discussed in detail during the exercise session on Thu -4! The compulsory home problems are published in the course homepage on Fridays and the deadline for answers is the next weeks Friday eturn 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.

7 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 GENEAL 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 CYLINDICAL rφ SYSTEM r = r cosφi + r sinφ j + k er cφ sφ i er er eφ = sφ cφ j eφ= eφ φ e k e e = er + eφ + e r r φ SPHEICAL θφr SYSTEM r( θφ,, r) = r(s θ c φ i + s θ s φ j + c θ k)

8 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 FO PLANA BEAMS r(, s n) = r () s + ne () s es r, s / r, s r, s = = e n ess, / ess, ess, = es + en n s n n es en / = s en es / OTHONOMAL CUVILINEA COODINATES eα i α x, α y, α, α x x eβ = [ F] j β = x, β y, β, β y= [ H] y en k x, y, γ γ γ, γ 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 EPESENTATIONS 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 GEOMETY ( rφ n) r ( r, φ, n) = [ ir cosφ+ jr sin φ ] + nen Γ ijk = D ir D rjk

9 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 GEOMETY ( 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 = CYLINDICAL SHELL GEOMETY ( φ n) r (, φ, n) = [ i cosφ+ jsin φ + k] + nen e i e eφ = sinφ cosφ j eφ = en φ en cos φ sinφ k en eφ d = φ = ( ) φ d n = n d n Γ φφn = Γ φnφ = ( n) dv = ( n ) dn( dφ ) d = ( n ) dndω LINEA ISOTOPIC ELASTICITY σ = E: ε = E: u (minor and major symmetries of the elasticity dyad assumed) ε = [ u + ( u )] c

10 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 ) PINCIPLE OF VITUAL WOK 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 ( xy ) E = Eii + Gjj + Gkk N + bx Q y + by= Q + b T + cx M y Q + cy= M + Qy + c

11 N EAu ESψ + ES yθ Qy= GA( v ψ) GS yφ Q GA( w + θ) + GSφ TIMOSHENKO BEAM ( snb ) T GS y( v ψ) + GS( w + θ) + GIrrφ M y = ES yu EIyψ + EI yyθ M ESu + EIψ EI yθ 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 d = iin + ijn + jin + jjn + ( ki + ik ) Q + ( kj + jk ) Q xx xy yx yy x y M = σ d = iim + ijm + jim + jjm + ( ki + ik ) + ( kj + jk ) xx xy yx yy x y EISSNE-MINDLIN PLATE ( xy ) Nxx, x + Nyx, y + b x = Nyy, y + Nxy, x + by Qxx, + Qyy, + b 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 KICHHOFF PLATE ( xy )

12 Nxx, x + Nyx, y + b x = Nyy, y + Nxy, x + by Mxx, xx + Mxy, xy + Myy, yy + b ( 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 EISSNE-MINDLIN PLATE ( rφ ) 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 + b [( 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 OTATION SYMMETIC KICHHOFF PLATE D w+ b = d d = ( r ) r dr dr 4 r r ( r ) b ( ) r wr = + a ln + b + cln r+ d D MEMBANE EQUATIONS IN CYLINDICAL GEOMETY ( φ n) Nφ, φ + N, b Nφ, + Nφφ, φ + bφ = b n Nφφ te [ u, + ν ( u φφ, u n)] N ν te Nφφ = [ ( u φ, φ un) + νu, ] ν Nφ tg( u, φ + uφ, ) MEMBANE EQUATIONS IN SPHEICAL GEOMETY ( φθ n )

13 cscθnφφ, φ + Nθφ, θ + cot θ( Nθφ + Nφθ ) bφ csc θnφθ, φ + Nθθ, θ + cot θ ( Nθθ Nφφ ) + bθ = 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] ν Nφθ tg( cscθuθφ, co tθuφ + uφθ, ) SHELL EQUATIONS IN CYLINDICAL GEOMETY ( φ n) κ Nφ, φ + N, + b Nφ, + κnφφ, φ κqφ + bφ = κqφ, φ + Q, + κnφφ + bn Mφ, + κmφφ, φ κmφn Qφ + cφ M + κm Q + c =, φ, φ N u, + νκ( uφφ, un) Et Nφφ = u ν, + κ( uφφ, un) ν Nφ ( ν)( uφ, + κu, φ) / M ω, + κνωφφ, κu, Mφφ νω, + κωφφ, + κ ( uφφ, un) M φ D ( ν )( ωφ, κω, φ κuφ, ) / = + Mφ ( ν)( ωφ, + κω, φ + κ u, φ) / M ( νκκ ) ( u + κu + ω) / φn n, φ φ φ Q un, + ω = tg Q ω + κ( u + u ) φ φ n, φ φ ω θ φ = ωφ θ