Kul-49.45 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 )sinφ+ 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
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.
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. Solution A compact representation of gradient operator in terms of directed derivatives di = e i, gradient of the basis vectors in terms of Christoffel symbols Γ ijk = e i ( e j ) e k, and summation convention aid in writing the component forms of the equilibrium and constitutive equations from the coordinate system invariant forms. Notice that curvature and Christoffel symbols are related by κ = Γ jniee i j ). For example ( Γ = e ( e ) e de = Γ e ) ijk i j k i j ijk k N= ( ed) ( N ee ) = e ( dn ) ee + e N ( de ) e + e N e ( de ) i i jk j k i i jk j k i jk i j k i jk j i k N = ( d N +Γ N +Γ N ) e. j ji kjk ji jki jk i The non-ero quantities of the spherical coordinate system corresponding to the midsurface n = are Γ φφn =Γ θθn =, Γ φθφ = cot θ, dφ = φ, sinθ dθ = θ, and d n = n By considering each component at a time (notice that the components of the stress resultants do not depend on n, components with n vanish, and only the 6 of the Christoffel symbols are non-eros ) i = φ : ( N b) eφ = d jn jφ +Γ kjk N jφ +Γ jkφn jk + bφ = dφ Nφφ + dθ Nθφ +Γ φθφ Nθφ +Γ φθφ Nφθ + bφ = Nφφ, φ + Nθφ, θ + cot θ ( Nθφ + Nφθ ) + bφ =. sinθ i = θ : ( N b) eθ = d jn jθ +Γ kjk N jθ +Γ jkθn jk + bθ = dφ Nφθ + dθ Nθθ + Γ φθφ Nθθ +Γφφθ Nφφ + bθ = Nφθ, φ + Nθθ, θ + cot θ ( Nθθ Nφφ ) + bθ =. sinθ
i = n: ( N b) e = d N +Γ N +Γ N + b = n j jn kjk jn jkn jk n Γ φφnnφφ +Γ θθ nnθθ + bn = Nφφ + Nθθ + b n =.
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 Solution The membrane equations of the cylindrical coordinate system are (formulae collection) Nφ, φ + N, + b =, Nφ, + Nφφ, φ + bφ =, and Nφφ + b n =. Definition of the external distributed force b = f dn + t take into account the volume forces acting on the body and given traction acting on the outer and inner surfaces. In the present case f = and the traction part is due to the hydrostatic pressure of the liquid inside the cylinder: p p gx = ρ, t = pen where x= cosφ in which p is the pressure difference between inner and outer surfaces and p is constant. Let us denote the pressure difference between the outer surface and center of the container by p (positive when the inner pressure is larger) to get b =, b φ =, and b = p + ρgx = p + ρg cosφ. n The equations to be solved become (notice that integration by a partial differential equation involves unknown functions instead of constants) N p + g cos = φφ ρ φ Nφφ = p ρg cosφ, N + N = Nφ, + ρgsinφ = Nφ = ρg sin φ + A( φ ), φ, φφ, φ N, N, b φ φ + + N, = N, φ φ N, = ρ g cos φ A ( ) φ N = ρ g cos A( ) B( ) φ φ + φ. In the solution, F( φ ) and G( φ ) are arbitrary functions subjected to A( φ) = A( π + φ ) and B( φ) = B( π + φ ) φ (periodicy) as the domain is closed in the φ direction. According to the assumption, force component in the direction of the axis vanishes at the ends. Therefore
N = ρ g cos A( ) B( ) φ φ + φ = {, L} B ( φ ) = and ρ φ φ φ g L cos L A( ) B( ) + = A ( φ) = ρg Lcosφ A( φ) = ρg Lsinφ + A. Solution to force resultants becomes Nφφ = p ρg cosφ, Nφ = ρg( L )sinφ + A, ( )cos N = ρg L φ in which pressure difference p and integration constant A cannot be solved with the information given.
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 Solution The mapping defining the geometry and the relationship between the basis vectors of the Cartesian and curvilinear coordinate system are (here r ( ) = tanα ) r(, φ, n) = [tanαcosφi + tanαsin φj + k ] + nen giving i sinαcosφ sinφ cosαcosφ e j= sinαsinφ cosφ cosαsinφ eφ k cosα sinα en k = e cosα + e sinα. n If geometry, loading etc. are assumed rotation symmetric, the equilibrium equations of membrane model boil down to ordinary differential equations (given in lecture notes or obtained with Membrane.nb) ( N, r+ Nr Nφφr ) + b T e r + r eφ ( Nφ, r+ Nφr + Nφr ) + bφ =. r + r en [ N ( r ) N ] 3/ φφ + rr + bn r( + r ) External distributed force is due to gravity and there is no contribution from the inner and outer surfaces i.e. t = : b = f dn + t = f dn = ρgkdn = tρgk = tρg( ecosα + ensinα ) b = tρgcosα, b φ =, and b = tρgsinα. n In the present case r ( ) = tanα, r ( ) = tanα, r ( ) =, equilibrium equations simplify to + tan α = / cosα and the
r Nφ, + Nφ = Nφ, + Nφ = N φ =, r rr / Nφφ N + br( ) n + r = ( + r ) Nφφ + tρg tan α = Nφφ = tρgtan α, r N, + ( N Nφφ ) + + r b = r ( N ) Nφφ + tρg = ( ) tan ) N + tρg( + α = N ( tan ) A = tρ g + α +. At the free end, stress resultant vanishes i.e. N ( ) = tan A = tρ g( + α) N = tρg( + tan α) ( ).
Kul-49.45 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)
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
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
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
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 )
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 64 4 4 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 )
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