Kul Models for beam, plate and shell structures, 07/2016

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1 Kul-9.5 Models fo beam, plate and shell stuctues, 7/6 Demo poblems. Deive the component foms of the elastic isotopic Kichhoff plate constitutive equations (just bending) in the pola coodinate system. Use the invaiant foms 3 t M = E: w, ee ee ee + ee ee + ee ν φ φ φ φ E E ee = φ φ ν ee + ek+ ke G ek+ ke φ φ φ φ φ φ ν kk kk ke + ek ke + ek and conside the case ν = to shoten the expessions. Answe M w, 3 Et Mφφ = w, / + w, φφ / M ( w / ) φ, φ,. A simply suppoted cicula plate of adius is loaded by a point foce acting at the midpoint as shown in the figue. Detemine the displacement of the plate at the midpoint by using the Kichhoff plate model in the pola coodinate system. Poblem paametes E, ν and t ae constants. Assume that the solution depends on the adial coodinate only. F Answe 3 F ( ν)(3 + ν) w() = 3 Eπt 3. A simply suppoted cicula plate of adius is loaded by its own weight as shown in the figue. Detemine the displacement of the plate at the midpoint by using the Kichhoff plate model in the pola coodinate system. Poblem paametes E, ν, ρ and t ae constants. Assume that the solution depends on the adial coodinate only. g Answe 3 g ( ν)(5 + ν) ρ w() = 6Et he demo poblems ae published in the couse homepage on Fidays. he poblems ae elated to the topic of the next weeks lectue (Wed.5-. hall K3 8). Solutions to the poblems ae explained in the weekly execise sessions (hu.5-. hall K3 8) and will also be available

2 in the home page of the couse. Please, notice that the poblems of the midtems and the final exam ae of this type.

3 Deive the component foms of the elastic isotopic Kichhoff plate constitutive equations (just bending) in the pola coodinate system. Use the invaiant foms 3 t M = E: w, ee ee ee + ee ee + ee ν φ φ φ φ E E ee = φ φ ν ee + ek+ ke G ek+ ke φ φ φ φ φ φ ν kk kk ke + ek ke + ek and conside the case ν = to shoten the expessions. Solution he (mid-plane) gadient opeato of the pola coodinate system is given by = e + eφ φ and deivatives of the basis vectos e / φ = eφ and eφ / φ = e. Deivatives with espect to the adial coodinate vanish. heefoe w w w= e + eφ φ and w w w= ( e + eφ )( e + eφ ) φ φ w ( w ) w w ee ee w w = + w φ + ee φ φ + ee φ ee φ + ee φ φ φ φ φ φ w w w w w= ee + ee ( ) ( ) ( ) φ + ee φ w + ee φ φ +. φ φ φ By using the invaiant fom of the constitutive equation with the elasticity dyad of isotopic mateial (fomulae collection) 3 t M = E: w and ee ee ee + ee ee + ee ν φ φ φ φ E E ee = φ φ ν ee + ek+ ke G ek+ ke φ φ φ φ φ φ ν kk kk ke + ek ke + ek ee ( / ) 3 ν w, ee φ + ee φ w, φ t E M ( ee = φ φ ν ( w, / + w, )/ + ek+ ke G φφ φ φ ν kk ke + ek

4 3 t ee E ν w, = ( ( ) ( / ), ee + + ( w, / + w, )/ φ φ φ φ φ ν φφ M ee ee G w ν ee 3 ν w, Et M= ee φ φ ν ( w, / + w, )/ ( ) φφ. ν ee + ee ν φ φ ( w/ ), φ If Poisson s atio ν =, expession simplifies to ee w 3, Et M= ee φ φ ( w, φφ / + w, )/. ee φ + ee φ ( w/ ), φ

5 A simply suppoted cicula plate of adius is loaded by a point foce acting at the midpoint as shown in the figue. Detemine the displacement of the plate at the midpoint by using the Kichhoff plate model in the pola coodinate system. Poblem paametes E, ν and t ae constants. Assume that the solution depends on the adial coodinate only. F Solution Kichhoff plate equations follow fom the eissne-mindlin bending equations [( 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 when otations ae eliminated thee with the Kichhoff constaints (obtained fom the constitutive equations fo the tansvese shea foces). hen shea foces become constaint foces to be solved fom the moment equilibium equations. Afte elimination of the shea foces, foce equilibium equations in the tansvese diection becomes the equilibium equation of the Kichhoff plate model. In a otation symmetic case, all deivatives with espect to φ vanish and θ. Assuming that cφ = c =, moment esultants and equilibium equation take the foms ( θ φ = w, and θ = w, φ / ) = M = D( θφ, + ν θ φ) d d M = D( ) w d +ν d, M = D( νθ + θ ) φφ φ, φ d d Mφφ = D( ν + ) w, d d Q = ( M ), Mφφ d d d Q = D ( ) w d d d, d ( Q ) + b z = d d d d d b ( )( ) w= z. d d d d D Assuming that the distibuted foce is constant, solution to the equilibium equation is obtained by epeated integations

6 d d d d b ( )( ) w= z d d d d D d d d d b ( w) = z d d d d D d d d b ( ) z a w = + d d d D 3 d d bz w = + a ln + b d d D d d d bz w= + a d d d D d d bz w= + aln + b d d D d bz w= + a( + ln ) + b + c d D 6 3 d bz w= + a( + ln ) + b + c d D 6 bz w= + a ( ln ) + b + cln + d. D 6 he geneic solution contains paametes abcd,,, to be detemined fom the bounday conditions. It is notewothy that ln is not bounded when wheeas ln and ln ae (limit values ae zeos). heefoe if oigin belongs to the solution domain, a bounded solution equies that c =. o find the solution to plate unde concentated foce without distibuted foce i.e. b z =, one may conside an annulus with the inne adius = ε and oute adius =. he plate is simply suppoted at the oute edge i.e. W( ) = M( ) =. At the inne edge, the plate is loaded by shea foce Q( ε ) = F / ( πε ) of esultant F. Solution needs to be bounded when ε and theefoe c =. nq Q = d ( d )( d ) w+ F = d d d D πε a F ε + D πε = a = D F π, M ( ) = d w dw + = d d a [3 + ln + b +ν a [ + ] + ν b = ν ( )] ( ln ) a 3+ ν F 3+ ν b= ( + ln ) = ( + ln ), + ν D π + ν w ( ) = a ( ln ) + b + d = d = a ( ln ) b d = F 3 3 ( ln ) F ) ( + ln F + D ν + D + = ν π π ν D π + ν. Displacement at the centepoint F 3 + ν 3 F w() = d = = ( 3+ ν )( ν ). D π + ν 3 Et π

7 A simply suppoted cicula plate of adius is loaded by its own weight as shown in the figue. Detemine the displacement of the plate at the midpoint by using the Kichhoff plate model in the pola coodinate system. Poblem paametes E, ν, ρ and t ae constants. Assume that the solution depends on the adial coodinate only. g Solution he geneic solution to Kichhoff plate model equilibium equation in pola coodinates assuming otation symmety bz w= + a ( ln ) + b + cln + d D 6 contains paametes abcd,,, to be detemined fom the bounday conditions. As oigin belongs to the solution domain and only the distibuted load is acting on the plate, all deivatives should be continuous at the oigin which implies that a = c =. he plate is simply suppoted at the oute edge giving w ( ) = M ( ) =. z b ( ) z b M = 3 b b D ν D 6 + ν = 3 bz + ν b = D 8 + ν, b ( ) z w = + b + d= D 6 d = 5 bz + ν D 6 + ν. Displacement at the centepoint with bz = ρ gt b 5 3 () z + ν ρg w = = (5 + ν)( ν). D 6 + ν 6 Et

8 Kul-9.5 Models fo beam, plate and shell stuctues INDEX NOAION (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 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 IDENIIES a ( b c) = ( a b) c a ( b c) = bac ( ) cab ( ) a:( b) = ( a b) ( a) b c CYLINDICAL φ z SYSEM = cosφi + sinφ j + zk e cφ sφ i e e eφ = sφ cφ j eφ= eφ φ ez k ez ez = e + eφ + ez φ z SPHEICAL θφ SYSEM ( 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 φ e sθeφ eθ e eφ =, θ e eθ = eθ + eφ + e θ sinθ φ HIN BODY snb SYSEM FO PLANA BEAMS (, s n) = () s + ne () s es, s /, s, s = = e n ess, / ess, ess, = es + en n s n n es en / = s en es / OHONOMAL CUVILINEA COODINAES eα eα eα i e = ( i[ F])[ F] e = [ D] () i e e = D e en en en β β β i j ijk k eα α eα α = e F H = e D e e 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 a= ( a) = dda i i +Γjijda i Γ ijk = D i D jk PLAE GEOMEY ( φ n) (, φ, n) = [ i cosφ+ j sin φ ] + nen e cosφ sinφ i eφ = sinφ cosφ j en k e eφ eφ = e φ e n d = d = d = φ φ n n

10 Γ = Γ = φφ φφ dv = dndω BEAM GEOMEY ( snb ) ( s, n, b) = [ ( s)] + nen + beb 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 = CYLINDICAL SHELL GEOMEY ( zφ n) ( z, φ, n) = [ i cosφ+ jsin 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 n Γ φφn = Γ φnφ = ( n) dv = ( n ) dn( dφ ) dz = ( n ) dndω LINEA ISOOPIC ELASICIY σ = E: ε = E: u (mino and majo symmeties of the elasticity dyad assumed) ε = [ u + ( u )] c 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 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

11 ii E ii ij + ji G ij + ji E = jj jj + jk + kj G jk + kj (beam) kk kk ki + ik G ki + ik 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 ii E ii ij + ji ij + ji E = jj jj + jk + kj jk + kj (uni-axial) kk kk ki + ik ki + ik G = E ( +ν ) 3 Et D = ( ν ) PINCIPLE OF VIUAL 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 BEAM EQUAIONS 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 IMOSHENKO BEAM ( xyz ) E = Eii + Gjj + Gkk N + bx Q y + by= Qz + bz + cx M y Qz + cy= Mz + Qy + cz N EAu ESzψ + ES yθ Qy= GA( v ψ) GS yφ Q z GA( w + θ) + GSzφ IMOSHENKO BEAM ( snb ) GS y( v ψ) + GSz( w + θ) + GIφ M y = ES yu EIzyψ + EI yyθ M z ESzu + EIzzψ EI yzθ

12 N Qnκ b + bs Qn + Nκb Qbκs + bn= Qb + Qnκ s + bb Mnκb + cs Mn + κb 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) 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) PLAE EQUAIONS 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 ) + ( kj + jk ) xx xy yx yy x y EISSNE-MINDLIN PLAE ( 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 KICHHOFF PLAE ( 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 )

13 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 EISSNE-MINDLIN PLAE ( 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, + θ φ = Gt Q w θ φ, φ / OAION SYMMEIC KICHHOFF PLAE D w+ b z = d d = ( ) d d ( ) b ( ) z w = + a ln + b + cln + d D 6 MEMBANE EQUAIONS IN CYLINDICAL GEOMEY ( zφ n) Nφz, φ + Nzz, z bz Nzφ, z + Nφφ, φ + bφ = b n Nφφ te [ u zz, + ν ( u φφ, u n)] Nzz ν te Nφφ = [ ( u φ, φ un) + νuzz, ] ν Nzφ tg( uz, φ + uφ, z) MEMBANE EQUAIONS IN SPHEICAL GEOMEY ( 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

14 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 EQUAIONS IN CYLINDICAL GEOMEY ( 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