Stability of Shell-Stiffened and Axisymmetrically Loaded Annular Plates

Transcription

1 TECHNISCHE MECHANIK,,, ), 8 sumitted: January 8, Staility of Sell-Stiffened and Axisymmetrically Loaded Annular Plates Dániel Burmeister Te present paper is concerned wit te staility prolems of a solid circular plate and some annular plates eac stiffened y a cylindrical sell on te external oundary. Assuming an axisymmetric dead load and axisymmetric deformations we determine te critical load in order to clarify wat effect te stiffening sell as on te critical load. Introduction In engineering practice we often meet structural elements loaded in teir own plane like rods or plates. Because of teir importance, staility prolems of plates loaded in teir own planes are an especially significant issue. As regards te staility prolems of circular plates, we mention tat te first paper devoted to tis question was pulised in 89 Bryan, 89). It is also wort citing an article y Nádai 95), wo investigated some fundamental staility issues. Since ten a numer of papers ave een devoted to te staility prolems of circular plates ut ere we lay an empasis only on tose wic deal wit te influence of structural stiffening. A structure can e stiffened in various ways. A solid circular plate or an annular one) can e made more resistant to uckling y te use of a corrugation or y applying a stiffening along te diameter of te plate or y attacing a ring to te outer oundary of te plate or y making tem more rigid wit a cylindrical sell attaced to te outer oundary of te plate. Seeking in te scientific literature for papers devoted to te staility prolem of stiffened annular plates we ave come to te conclusion tat tere are only a few works dealing wit tis issue. A ring stiffened circular plate is investigated in a paper y Turvey and Der Avanessien 989). Te paper cited is concerned among oters wit experimental results. However, te staility issues are left out of consideration. A furter paper y Turvey and Salei 8) deals wit an annular plate stiffened y a single diameter stiffener. Te staility prolem is, owever, again left out of consideration. Te influence of a stiffening ring attaced etween te two oundaries of te middle surface of circular plates on te staility is investigated y Frostig and Simitses 988). It turns out from te references tat te autors did not take into account te corresponding results of Szilassy 97, 976). Szilassy investigated some staility prolems of circular plates stiffened y a cylindrical sell on te outer oundary in Szilassy, 97, 976). Te main ojective of is researc was to clarify wat effect te stiffening sell as on te critical load. If te stiffening significantly increases te uckling load tan tin plates can e made more resistant against uckling y te application of a stiffening sell. Te prolems considered tere include a oundary value prolem for a solid circular plate and a few oundary value prolems for annular plates. Te autor assumes tat i) te load is an in-plane axisymmetric dead one, and ii) te deformations of te annular plate and te cylindrical sell are also axisymmetric. For te circular plates te autor uses te solution of a differential equation set up for te rotation field tis coice excludes tose oundary value prolems from te set of solvale ones were a oundary condition is imposed on te deflection. As regards te cylindrical sell te solution is ased on te teory of tin sells. After te investigations of Szilassy te following questions can e raised: a) is it wort using a differential equation set up for te deflection in order to expand te range of solvale oundary value prolems te numer of solvale prolems can e increased in tis way); ) wat is te influence of te sell eigt on te critical load; c) is it wort investigating te case wen te load is axisymmetric ut te deformations due to te load are not; d) wat appens if te load is not axisymmetric.

2 Te main ojective of te present paper is to solve tose oundary value prolems wic require te use of te differential equation set up for te deflection of te plate, i.e., were tere is a oundary condition imposed on te deflection. Te paper is organized in seven sections. Section sortly outlines te pysical prolem to e solved. Section presents te governing equations ot for te circular plate and for te cylindrical sell. Te staility prolem of a solid circular plate is solved in Section. Tere we present ot te non-linear equation wic provides te critical load and te critical load for various sell eigts. Section 5 is devoted to annular plates. First we sortly review te numerical metod we use. Ten we solve four different oundary value prolems. Section 6 is a summary of te results. Appendix A includes some longer transformations. As regards te possile applications we want to make te following remark. Assume tat te inner space of a pressure vessel a cylindrical sell) is separated into parts y annular plates. If te distance of te plates is aove a certain limit and te uckling prolem of te plates arises ten te prolem to e solved may coincide wit te staility prolem of a sell-stiffened annular plate provided tat te sell eigt tends to infinity. Prolem Formulation Te cross section of te sell-stiffened structure we are concerned wit is sown in Fig.. Te structure consists of eiter a solid circular plate or an annular one te latter is sown in Fig. and a cylindrical sell, wic stiffens te plate on its external oundary. Te inner radius of te plate is denoted yr i, te radius of te intersection line of te middle surfaces of te plate and te sell y. We sall assume tat coincides wit te external radius of te plate. Te ticknesses of te plate and sell are denoted y p and s, respectively. Te sell is symmetric wit respect to te middle plane of te plate. Its eigt is. Te structure is loaded y radial distriuted forces wit a constant intensityf o acting in te middle plane of te plate. Te load is a dead one. f o p R i s R ζ z Figure : Te structure and its load We assume tat te plate and te sell are tin, consequently we can use te Kircoff teory of plates and sells. It is also assumed tat te prolem is linear wit regard to te kinematic equations and material law. Heat effects are not taken into account. Te plate and te sell are made of te same omogeneous isotropic material for wic E = E p = E s and ν = ν p = ν s are Young s modulus and Poisson s ratio. ξ p a. f M o R i u w f M o R z. f o p M o f f M o u ζ f o ζ Figure : Free ody diagram for plate and sell ξ

3 Under te assumption of small, axisymmetric and linearly elastic deformations we determine a) te critical load of te structure and ) te effect of te stiffening sell on te critical load. In order to solve te prolem raised we separate te sell and plate from eac oter mentally. In accordance to tis Figs. a and sow te annular plate and te cylindrical sell one y one. Te cylindrical coordinate system R,ϕ,z) is used for te equations of te plate te plane z = coincides wit te middle surface of te plate. Fig. a sows te corresponding coordinate curves on te circle wit radius. Te displacements on te middle surface in te directionsrandz are denoted yuandw, respectively. Te inner forcesf and te ending moment M o exerted y te cylindrical sell on te annular plate are also sown in Fig..a. For te cylindrical sell te coordinate systemζ,ϕ,ξ) is applied. Te coordinate surfaceξ = z, ζ = coincides wit te middle surface of te sell wit radius. Te polar angle ϕ is te same in te two coordinate systems due to te axisymmetry it plays, owever, no role in te investigations). Te coordinate curves on te middle surface of te sell are sown in Fig.. u ρ = R u ζ ζ v ϕ w z ϕ a.. Figure : Coordinate curves in te coordinate systems u ϕ uξ ξ Assume tat te deformations are axisymmetric and tere is no load in te direction ξ on te sell. u ζ = u ζ ξ) is te only displacement component on te middle surface wic is different from zero. Ten It is also ovious tat u = ur),w = wr) and u ζ = u ζ ξ). Fig. sows te inner forces f and te ending moment M o exerted y te plate on te cylindrical sell. Governing Equations. Governing Equations for te Cylindrical Sell It is known tat under te condition of axisymmetric deformations, te radial displacement u ζ sould satisfy te following differential equation Timosenko and Woinowski-Krieger, 987, Capter 5, p. 68) d u ζ dξ +β u ζ = p ν N ) ξ ) I s E s were p is te constant radial load exerted on te middle surface of te sell its value is zero in te present case), N ξ is te inner force in direction ξ its value is zero as well), ν s and E s are Young s modulus and Poisson s ratio, respectively. In addition te following notations are introduced ν o = νs), β = ν o, s a) I s = s/, E s = E s / νs). ) Te sell sown in Fig. is sujected to te line loadsf o andf as it as already een mentionedp = ). Tere is also no load in te direction ξ on te sell. Consequently N ξ =. Te solution of equation ) in te interval ξ,] takes te form u ζ ξ) = a i V i βξ)+u ζ p ; u ζ p = p/β I s E s =, ) i=

4 were V i βξ)i =,...,) denote te Krylov-functions teir definitions and derivatives are presented in te Appendix see equations 5a) and 5) for details. Te sear force and ending moment in te sell can e given in terms of u ζ Q ζ = I s E s d u ζ dξ and M ξ = I s E s d u ζ dξ. ) Te solution for u ζ is a superposition of te solutions we determine for te following two partial loads Load. Te sell is sujected to te line loadsf o andf sown in Fig. a Te corresponding oundary conditions are as follows Q ζ ξ= = f o f du ζ, dξ =, 5a) ξ= Q ζ ξ= =, M ξ ξ= =. 5) Since u ζ ξ) = u ζ ξ) due to te load, te rotation aout te axis ϕ is zero cf. equation 5a). Te oter oundary conditions are ovious. Load. Te sell is sujected to te couple system M o sown in Fig. Now we ave te following oundary conditions uζ) ξ= =, M ξ ξ= = M o, 5c) Q ζ ξ= =, M ξ ξ= =. 5d) Oserve tat u ζ ξ) = u ζ ξ) for tis partial load. Consequently te displacement in te direction ζ sould e zero at ξ =. Te oter oundary conditions are again ovious. Q ζ ξ ) M ξ ξ ) f f o ζ M o ζ Q ζ ξ +) M ξ ξ +) ξ a.. Figure : Partial loads of te sell It follows from te symmetry of te prolem tat it is sufficient to determine te solution for te sell in te interval ξ,]. Te solutions for te partial loads include te distriuted force f and te ending moment M o as unknown parameters. Teoretically, tese quantities can e calculated from te continuity conditions. We prescrie te continuity conditions on te intersection line of te middle surfaces of te plate and te sell. Since R = and ξ = on te intersection line, te kinematic quantities sould satisfy te following continuity conditions ξ and u R= = u o = u ζo = u ξ= ζ ϑ o = dw dr = du ζ R= dξ. ξ= 6a) 6)

5 After some and made calculations see Sections.. and. of te Appendix for details, in wic use as een made of te definitions of te Krylov-functions, we otain and ϑ o = du ζ dξ = ν o ξ= E u ζo = u ζ ξ= = ν o E s ) cosβ +cosβ + sinβ sinβ } {{ } κ s ) cosβ +cosβ + sinβ +sinβ } {{ } α were α and κ are defined y te aove relations. M o = κm o. It follows from te continuity conditions 6) and relations 7) tat te equations s 7a) f o f) = αf o f), 7) dw dr = κm o and u o = αf o f) 8) are satisfied. We sall see later tat equation 8) provides te value of f as non-linear equation and equation 8) is to e used for calculating f o.. Deformation of te Annular Plate, Governing Equations for te In-Plane Load Under te assumption of axisymmetric deformations, te inner forces in te plate we use te cylindrical coordinate system R,ϕ,z) due to te in-plane load exerted on te outer oundary are as follows N R = A+ B R, N ϕ = A B R and N Rϕ = N ϕr = 9) were te constants A and B depend on te oundary conditions. It follows from te axisymmetry tat N Rϕ vanises. Let ρ = R/ e a dimensionless coordinate. Furter let ρ i = R i /. It is clear tat ρ i = for a solid circular plate. If te inner oundary is free and f is te line load on te outer oundary see Fig. a ten te constants are as follows A = f ρ i and B = f R i ρ i If te radial displacement vanises on te inner oundary ten. ) +ν Ri A = f +ν +ρ i ν) and B = f ν) +ν +ρ. ) i ν) If te plate is a solid one ten A = f and B =. ) Te radial displacement on te inner oundary can e calculated using te relations u o = K p f E, were te constant K depends on te oundary condition ) K = ν) ρ i ) +ν +ρ i ν) if N R ρ=ρi =, ) K = ν)+ρ i +ν) +ρ i if u o ρ=ρi =, 5) K = ν if ρ i =. 6) 5

6 . Deformation of te Annular Plate, Equations for te Displacement Field after Staility Loss Let us introduce te notations A = A f ; B = B f, 7) I p = p/ and E p = E p / ν p). 8) Furter let w e te displacement of te middle plane of te plate in te direction z see Figure. Making use of te notations introduced one can sow tat w satisfies te differential equation ] H H w = f A+ B ) d w I p E p R dr + A B ) dw R, 9a) R dr H = d dr + d R dr if te plate is an annular one. Wit regard to equations ) ] H H w = f d w I p E p dr + dw R dr 9) is te differential equation for w if te plate is solid. Te rotation, te sear force and te ending moment can all e given in terms of te solution forw as follows ϑ = dw dr, a) d w M R = I p E p dr +ν ) dw, ) RdR d d w Q R = I p E p dr dr + ) dw dw N R RdR dr. c) Staility of te Sell-Stiffened Solid Circular Plate. Nonlinear Equation for te Critical Load Introducing te notations F = f I p E p and R e = H = d dρ + ) d ρ dρ a) in equation 9) yields te differential equation w+f w =. ) Its solution takes te form wρ) = c Z +c Z +c Z +c Z, Z =, Z = lnρ, Z = J o Fρ), Z = Y o Fρ) ) were c i, i =,...,) are undetermined constants of integration wile Z i denote te linearly independent particular solutions in wic J n Fρ) and Y n Fρ), n =,,,,... are te Bessel functions of integer order. For small values of Fρ te following asymptotic relation olds: Y o Fρ) π ln Fρ). In addition Z ) = J o ) =. Consequently te solution forw is limited if a) c = c /π. ) 6

7 Accordingly we get wρ) = c +c Y o Fρ) ] π ln Fρ) +c J o Fρ). ) By making use of te relations x J x) = J x)+j o x) and x Y x) = Y x)+y o x), 5) te derivatives of Z i from equations 7) and relation ) te latter provides us te sear force we otain after some manipulations tat Rk Q R = d ] +F w = c I E dρ π F ρ. 6) Since te sear force Q R is zero on te circle wit radius R = ρ ifρ, te previous equation yields πrq R = c f =. 7) Consequently c =. Terefore te solution for w is of te form wρ) = c +c J o Fρ). 8) Te remaining integration constants can e calculated from te oundary conditions prescried at ρ i =. In wat follows we assume tat te ticknesses and te material of te cylindrical sell and te circular plate are te same. Tus E = E s = E p,ν s = ν p, s = p = and I = I s = I p. According to equation 6) ϑ o = κm o. Te rotation atr = is of te form ϑ o = dw dr = c FJ F), R= 9) a) were we ave utilized relations a) and 7c). Considering equation 9), te properties of te solution and utilizing relations 7), wic provides te derivatives of te Bessel functions, we otain for te ending moment d w M o = I E dr +ν dw I E = c RdR] R= Comparing equations 9) and ) we get a non-linear equation for te critical load ν) FJ F) FJ o ] F). ) FJ F) κ I E ν) FJ F) FJ o ] F) =. ) Tis equation can e furter transformed if we make use of relations a) and ) te structural elements are of te same material and tickness) togeter wit te definition of κ in equation 7a) FJ F) νo cosβ +cosβ + ν ν) J F) FJ o ] F) = ) sinβ sinβ F If we now take into account tat β = ν o we can conclude tat te dimensionless critical load F cr te corresponding value of f is denoted y f cr depends only on te dimensionless variales / and / for a fixed value of ν. In te sequel we assume tat we ave solved te aove equation, i.e. we know te values of F cr and f cr. 7

8 . sults for a Solid Plate In accordance wit te notations introduced letf ocr e te critical value off o. A comparison of equations 7) and ) yields ν) Consequently f cr E = αf ocr f cr ). f ocr f cr = F cr = + ν ν o cosβ +cosβ + sinβ sinβ f ocr, =. ) I p E p Clearly te quotient focr f cr = Focr F cr depends also on te dimensionless variales/ and/ for a fixed value of ν. Tis function is presented in Fig. 5. F cr a. ν =.5. F cr. ν = c. ν =.5 F cr.. =. ; =.75 ; =.5 ; =.5 Figure 5: Quotient /F cr against / for some values of/ and ν We introduce te following notations for te critical load and te dimensionless critical load if tere is no stiffening sell f ocr = ) = f of cr and = ) = F of cr. ) It is again ovious tat te quotient of te critical loads or wic is te same tat of te dimensionless critical loads) for a given ν f ocr f of cr = F of cr 5) depends only on / and /, wic, in te sequel, are referred to as dimensionless eigt and tickness. 8

9 A code as een written in Fortran 9 to solve te non-linear equation ) for F cr and compute f ocr, f of cr, and, F of cr. Te computational results are presented in Fig. 6 for ν =.5,. and.5. It is clear from te graps tat te eigt of te plate does not affect te critical load if te eigt is larger tan a certain value. F a. ν =. ocr 5 F of cr F of cr. ν = F of cr c. ν = =. ; =.75 ; =.5 ; =.5 Figure 6: f ocr /f ocr against / for some / ;ν =.5,.,.5 It follows from equations 7) tat te stiffening sell can e replaced y a tension spring and a torsion spring on its outer diameter. Te corresponding arrangement is sown in Fig. 7. f o D κ f o D α Figure 7: Spring model Hooke s law for tese springs tension and torsion) takes te form f o f = D α u ζo, M o = D κ ϑ o 6) were D α = α and D κ = κ are te spring constants. If te eigt of te sell tends to infinity te terms α and κ are limited. Consequently te critical load is limited 9

10 as well. Te limits of te critical loads elong to te values lim α = lim ν o E lim κ = lim νo E s s ) cosβ +cosβ + = ν o sinβ +sinβ E ) cosβ +cosβ + sinβ sinβ s = ν o E s Tese limits are presented wit orizontal lines in te upper grap in Fig Staility of te Sell-Stiffened Annular Plates s ), 7) ) s. 8) 5. A Numerical Procedure to Determine te Critical Load Equation 9a) as no closed form solution. For tis reason we develop a numerical algoritm for te solution of te eigenvalue prolem defined y equation 9a) and te corresponding oundary conditions. First we rewrite equation 9a) in te form d w d dρ = F w dρ, d w dρ, dw ) dρ,w,f,ρ = = d w ρ dρ + d w ρ dρ dw ρ dρ +F ) A+ B d w ρ dρ + A B ρ and ten we replace tis ordinary differential equation of order four y a system of differential equations of order one. To tis end we introduce appropriately defined new intermediate variales p, q and s y te use of wic we otain were dw dρ = p, dp dρ = q, dq dρ = s, ds dρ = F s,q,p,w,f,ρ). ) ρ ] dw dρ F = s ρ + q ρ p ) ) ) ρ +F A+ B ρ q + A B p ρ. ) ρ 9) a) Variales w,p, q and s constitute a column matrix u T ρ) = w p q s ] ) were te superscript T stands for te transpose of a matrix. It is clear tat u fulfills te matrix equation du dρ = p q q F s,q,p,w,f,ρ) ] T. ) We seek solutions in te interval ρ ρ i,]. It is easy to see tat te particular solutions u k = u k u k u k u k ] T k =,,,) are linearly independent if tey satisfy te following initial conditions u ρi =, u ρi =, u ρi = Consequently te solution of equation ) assumes te form, u ρi = ) u =C u +C u +C u +C u, ) were C,...,C are constants of integration. Differential equations ) are associated wit omogenous oundary conditions on te inner oundary ρ = ρ i and also on outer oundary ρ = ρ e =. We sall detail te oundary conditions wen we present te solution for te various support arrangements. At tis point we remark tat te differential equation ) and te omogenous

11 oundary conditions togeter define te eigenvalue prolem wic provides te dimensionless eigenvalue F for a given support arrangement of te structure consisting of a sell and an annular plate. Every pysical quantity can e given in terms of te particular solutionsu,...,u. On te asis of relations ) and ) we otain te displacement field wρ) = C u +C u +C u +C u, te rotation field ϑ = dw dρ = C u +C u +C u +C u ), te ending moment M R = I E R k u + ν ) ρ u C + u + ν ) ρ u C + + u + ν ) ρ u C + u + ν ] ρ u )C 5a) 5), 5c) and te sear force Q R = I E R e { u + ρ u + F ρ ] )u u + uρ)+ + C + F ρ )u ] C + u + ρ u + F ρ ] )u C + u + ρ u ρ)+ F ρ } )u ]C. 5d) Finally we remark tat te solutionsu,...,u are computed using an adaptive fourt-order Runge-Kutta metod. 5. Comparison wit te Solution Valid for te Solid Plate We ave tested te numerical algoritm descried aove y solving te prolem of te solid circular plate again. Te corresponding oundary conditions are in principle te same as efore lim w = finite, dw ρ dρ =, 6) ρ= dw dρ = ν o cosβ +cosβ + d w ρ= ν sinβ sinβ dρ + υ ) dw 7) ρ dρ ρ= d d dρ dρ + ] d ρ dρ +F w =. 8) ρ=+ε Te displacement of te plate at ρ = is u = C. It does not violate generality if we set tis value to zero it is actually te rigid ody motion of te structure in te vertical direction. If C = te tree oundary conditions left yield te following omogenous linear system of equations C u )+C u )+C u ) = C κ I E u )+νu )) u ) ]+C κ I ] E u )+νu )) u ) + R k +C κ I ] E u )+νu )) u ) = 9) {C u ρ)+ ρ u ρ)+ F ρ ) ] u ρ) + +C u ρ)+ ρ u ρ)+ F ρ ) ] u ρ) + + C u ρ)+ ρ u ρ)+ F ρ ) u ρ)]} ρ=+ε 9a) =, 9c)

12 wit C, C and C as unknowns. Tis equation system as a non-trivial solution if te determinant of te coefficient matrix vanises DF) = u ) κ IE R k u )+νu ) { u )) u ρ)+ ρ u ρ) + ) + F ρ u ρ) } ρ=+ε u ) u ) κ IE R k u )+νu )) { u ) u ρ)+ ρ u ρ) + ) + F ρ u ρ) } ρ=+ε κ IE R k u )+νu )) { u ) u ρ)+ ρ u ρ) + ) + F ρ u ρ) } ρ=+ε = 5) Fig. 8. sows te critical load otained y using te two metods: a) solution of equation ) ) solution y te use of te Runge-Kutta procedure. ρ i = 5 5 closed form solution numerical metod Figure 8: analytic-numeric comparison We ave carried out te computations under te assumption tat / =.5, E =. 5 N/mm and ν =.. It is clear from Fig. 8 tat te two curves coincide asically wit eac oter. It is wort noting again tat f te dimensionless load F as well as te quotient ocr f ocr depend only on / and / ot for te sold circular plate and for te annular plates ifν and ρ i ave fixed values. 5. sults for Annular Plates In te present section we determine te critical loads for four different support arrangements. Te solutions are presented in te susections 5..., 5..., 5... and Simple Supported Plate First we sall investigate a sell-stiffened annular plate wit a simple support on its inner oundary. Since te radial displacement is not prescried on te inner oundary it follows tat te oundary condition N R ρ=ρi = must e satisfied tere. Te wole structure is presented in Fig. 9. We remark tat te values of A and B in equation 9), wic provides te critical value of F, are computed from relations ) and 7). Furtermore we also remark tat K can e otained from relation ) wen we calculate u o using equation ).

13 Figure 9: Simple support on te inner oundary F of cr a. ρ i =.5 5 F of cr. ρ i =.5 8 / =. / =.75 / =.5 / =.5 ] F of cr c. ρ i =.75 6 / =. / =.75 / =.5 / = d. / =. / =. / =.75 / =.5.9 / =.5.5 ρ i Focr.5 Figure a-c: Quotient of te dimensionless critical loads as a function of te dimensionless eigt and tickness for te structure sown in Fig. 9 ifρ =.5, ρ =.5, ρ =.75 d: Dimensionless critical load as a function of ρ i and te dimensionless sell eigt Te solution w of te plate equation 9a) or te solutions of te differential equations )) sould fulfill te following oundary conditions w ρ=ρi =, d w dρ + ν ρ dw dρ = νo ρ= ν d d dρ dρ + ρ ] d dρ +F w =. ρ= dw dρ = ρ=ρi cosβ +cosβ + d w sinβ sinβ dρ + υ ρ ) dw dρ ρ= Te computational results are sown in Fig. a-c for tree different values of ρ i. Fig. d sows te critical load of a specific structure E =. 5 N/mm, ν =.) against ρ i and / in a tree-dimensional grap for te fixed value / =.. Te axis is a logaritmic one in tis part of te figure. 5a) 5) 5c) 5.. Annular Plate wit Clamped Inner Boundary If a) te radial displacement is not prescried on te inner oundary ut ) te rotation is zero in contrast to te previous prolem) see Fig. for furter details

14 Figure : Sell and plate clamped on te inner oundary F of cr a. ρ i =.5 F of cr. ρ i =.5 / =. / =. /R e =.75 /R e =.75 / =.5 / =.5 / =.5 / = d. / =. c. ρ F i =.75 of cr / =. / =.75 / =.5.9 / =.5.5 ρ i Focr.5 Figure a-c: Quotient of te dimensionless critical loads as a function of te dimensionless eigt and tickness for te structure sown in Fig. ifρ =.5, ρ =.5, ρ =.75 d: Dimensionless critical load as a function of ρ i and te dimensionless sell eigt ten te equation dw dρ =, ρ=ρi 5) togeter wit equations 5a), 5) and 5c) are te oundary conditions. Te effect of te stiffening to te structure sown in Fig. can e seen in Fig. a-c. We remark tat Fig. d sows again te critical load against ρ i and / in a tree-dimensional grap for a specific structure te parameters are te same as tose of te structure in Fig. 9). Te vertical axis is logaritmic as well. 5.. Axially and Radially Supported Annular Plate If a) te radial displacement on te inner oundary of te plate is zero, i.e. u o ρ=ρi = ten te values of A and B in equation 9), wic provides te critical value of F, sould e computed from relations ) and 7). We also remark tat K can e otained from relation 5) wen we calculate u o using equation ). Furtermore if te plate is simply supported on its inner oundary ten te corresponding oundary conditions are te same as tose given y equation 5).

15 Figure : Simple supported plate wit no radial displacement on te inner oundary Fig. 5 a sows again te critical load against ρ i and / in a tree-dimensional grap for a specific structure te parameters are te same as tose of te previous two prolems). Te vertical axis is logaritmic as well. 5.. Annular Plate Fixed on te Inner Boundary If te displacements and te rotation are zero on te inner oundary te structure is sown in Fig. ten te only difference etween tis prolem and te previous one is tat oundary condition 5) sould e applied instead of oundary condition 5a). Figure : Plate fixed on te inner oundary In accordance wit te Figures tat ave een already presented efore Fig. 5 sows te critical load against ρ i and / in a tree-dimensional grap for a specific structure te parameters of wic are also te same as earlier. Te vertical axis is again a logaritmic one. a. / =.. / =. Focr Focr.9 ρ i ρ i.5..5 Figure 5 a-: Dimensionless critical load as a function of ρ i and te dimensionless sell eigt for te structures sown in Fig. and Fig. 6 Concluding marks In accordance wit te ojectives in te introduction we ave summed up te differential equations for te staility prolem of solid circular plates and annular plates under te assumption of axisymmetric deformations. In addition we ave clarified wat te continuity conditions are etween te plate and te cylindrical sell wic stiffens te plate. We ave estalised te nonlinear equations wic provide te critical load. We ave also developed a procedure for computing numerical solutions. One of tese can e used for te solid circular plate te oter for annular plates. We ave coded te procedure in Fortran 9 and made computations in order to determine te critical load. Solutions are provided for a solid circular plate and for four support types concerning te annular plates. According to te results otained te stiffening sell increases te value of critical load significantly if te quotient/ does not exceed a limit te value of wic depends on / and ρ i. 5

16 calling te introduction we finally remark tat i) it is an interesting question wat appens if te load is axisymmetric ut te deformations due to it are not, and ii) wat appens if te load is also non-axisymmetric. Partial results for prolem i) are presented in a conference proceeding Burmeister, ). 7 Acknowledgement Tis researc was carried out as part of te TAMOP-...B-//KONV-- project wit support y te European Union, co-financed y te European Social Fund. Te autor is also grateful to te unknown reviewer for is/er critical remarks. A Solution of Partial Prolems. Krylov-Functions For completeness we present te Krylov-functions togeter wit teir derivatives wit respect to te coordinate ξ elow: V = cosβξcosβξ, V = cosβξsinβξ +sinβzcosβξ), V = sinβξsinβξ, V = 5a) cosβξsinβξ sinβzcosβξ). V = βv, V = βv, V = βv, V = βv, V = β V, V = β V, V = β V, V = β V, V = β V, V = β V, V = β V, V = β V. 5). Solution for te First Partial Load On te asis of equations ) and 5) we take into account in our calculations tat te following relations are valid for te solutions of differential equation ) and its derivatives u ζ = a V +a V +a V +a V u ζ = βa V +βa V +βa V +βa V u ζ = β a V β a V +β a V +β a V u ζ = β a V β a V β a V +β a V. For te sake of our later considerations it is wort introducing te notations V = V ξ = ) = cosβcosβ V = V ξ = ) = cosβsinβ+sinβcosβ) V = V ξ = ) = sinβsinβ 5) 55) V = V ξ = ) = cosβsinβ sinβcosβ). Boundary condition 5a) yields u ζ ξ= = βa = terefore a =. 56) Similarly we otain from a comparison of te oundary condition 5a) and relation ) I s E s u ζ ξ= = I s E s β a = f o f, terefore a = f o f On te asis of oundary conditions 5), and ) we otain te linear system of equations u ζ ξ= = β a V β a V β f o f I s E s β V =, u ζ ξ= = β a V β a V +β f o f I s E s β V = 6 I s E s β. 57) 58)

17 to calculate te two integration constants left. Sustituting te solution ) ] V +V V a = f o f V V +6V V a I s E s β V V V 59) V V +V V and te oter two integration constants into 5) we ave u ζ = f o f V +V V I s E s β V + V V ) V V V, 6) V V +6V V V V V V from were u ζ ξ = ) = f o f cosβ +cosβ + I s E s β sinβ +sinβ if te definitions of te notations 55) and te values of te Krylov functions are also inserted. Tis equation can e transformed furter if we sustituteβ togeter wite on te asis of a) and ) u ζ ξ = ) = u ζ o = f o f cosβ +cosβ + I s E ) = s R ν e 8 sinβ +sinβ o s. Solution for te Second Partial Load = ν o E s s 6) ) cosβ +cosβ + f o f). 6) sinβ +sinβ Comparing oundary condition 5c) and relation 5) we get u ζ ξ) ξ= = a =. Using oundary condition 5c) and formula ) for te sear force in a similar manner we can write I s E s u ζ ξ= = I s E s β a = M o, i.e. a = M o I s E s β. 6) Te oter two oundary conditions 5d), yield te following system of linear equations u ζ ξ= = β a V β M o I s E s β V +β a V =, u ζ ξ= = β a V +β M o I s E s β V +β a V =. Te solution are ] V +V V a M o = a I s E s β V V V V V +V 66) V. V V V V Making use of te solutions otained for te integration constants 6), 6) and 65) we ave from 5) tat M o V u ζ = +V V I s E s β V +V + V +V ) V V. 67) V V V V V V V V Its derivative wit respect toξ is du ζ dξ = M o V I s E s β β +V V V +βv +β V +V ) V V. 68) V V V V V V V V Sustituting te definitions of te notations 55) and tose of te Krylov functions togeter wit te constants β and E on te asis of a), ) we otain te final form of te derivative du ζ M o V dξ = +V V = cosβ +cosβ + M o = ξ= I s E s β V V V V I s E s β sinβ sinβ = ν o E s ) cosβ +cosβ + M o sinβ sinβ s 6) 65). 69) 7

18 . Solution and Derivatives for Equation ) Here we present te linearly independent solutions Z,...,Z of te differential equation ) togeter wit teir derivatives wit respect toρ Z o) = Z =, Z Z o) = Z = ln Fρ), Z = =, Z =, Z = ; 7a) F Z = F Fρ Fρ, Z = F/ F / ρ ; 7) Z o) = Z = J o Fρ), Z ] Z = F J Fρ) J Fρ), Z = F/ Z o) = Z = Y o Fρ), Z ] Z = F Y Fρ) Y Fρ), Z = F/ = FJ Fρ), 7c) J Fρ) J ] Fρ) ; 7d) = FY Fρ), 7e) Y Fρ) Y ] Fρ). 7f) ferences Bryan, G. H.: On te staility of a plate under trust in its own plane wit applications to te uckling of te sides of a sip. Proceedings of te London Matematical Society, s-), 89), Burmeister, D.: Staility of a circular plate stiffened y a cylindrical sell under te assumption of nonaxisymmetrical deformations. In: Proceedings of te t International Conference Computational Mecanics and Virtual Engineering ), -5. Collatz, L.: Eigenwertaufgaen mit tecniscen Anwendungen. Nauka, Moskow 968), Russian Edition. Erdélyi, A.: Higer Transcendental Functions, vol.. Nauka, Moscow 97), Russian Edition. Frostig, Y.; Simitses, G. J.: Buckling of ring-stiffened multi-annular plates. Computers & Structures, 9,, 988), Kozák, I.; Béda, G.: Mecanics of elastic odies. Műszaki Könyvkiadó, Budapest 987), in Hungarian). Nádai, A.: Üer das Auseulen von kreisförmigen Platten. Zeitscrift des Vereines deutscer Ingenieure, 59,, 95),. Pardoen, G.: Viration and uckling analysis of axisymmetric polar ortotropic circular plates. Computers & Structures,, 97), Szilassy, I.: Staility of circular plates wit a ole loded on teir outer oundary. P.D. tesis, University of Miskolc 97), in Hungarian). Szilassy, I.: Staility of an annular disc loaded on its external flange y an aritrary force system. Pul. Tecn. Univ. Heavy Industry. Ser. D. Natural Sciences,, 976), 55. Timosenko, S.; Woinowski-Krieger, S.: Teory of Plates and Sells. McGraw-Hill, nd edn. 987). Turvey, G. J.; Der Avanessien, N. G. V.: Axisymmetric elasto-plastic large deflection response of ring siffened circular plates. International Journal of Mecanical Sciences, -), 989), Turvey, G. J.; Salei, M.: Elasto-plastic large deflection response of pressure loaded circular plates stiffened y a single diameter stiffener. Tin-Walled Structures, 6, 8), 996. Address: Dániel Burmeister, Department of Mecanics, University of Miskolc, 55 Miskolc-Egyetemváros, Hungary daniel.urmeister@uni-miskolc.u 8