DESIGN OF INTERMEDIATE RING STIFFENERS FOR COLUMN-SUPPORTED CYLINDRICAL STEEL SHELLS

Transcription

1 DESIGN OF INTERMEDIATE RING STIFFENERS FOR COLUMN-SUPPORTED CYLINDRICAL STEEL SHELLS Öze Zeybek Reseach Assistant Depatment of Civil Engineeing, Middle East Technical Univesity, Ankaa, Tukey Cem Topkaya Pofesso Depatment of Civil Engineeing, Middle East Technical Univesity, Ankaa, Tukey J. Michael Rotte Pofesso Institute fo Infastuctue and Envionment, Univesity of Edinbugh, Scotland, UK 1. ABSTRACT A combination of a ing beam and an intemediate ing stiffene can be used fo lage silos to edistibute the stesses fom the local suppot into unifom stesses in the shell. This pape eploes the design equiements fo intemediate ing stiffenes placed at o below the ideal location. Pusuant to this goal, the cylindical shell below the intemediate ing stiffene is analyzed using the membane theoy of shells and the eactions poduced by the stiffene on the shell ae identified. Futhemoe, the displacements imposed by the shell on the intemediate ing stiffene ae obtained. These foce and displacement bounday conditions ae then applied to the intemediate ing stiffene to deive closed fom epessions fo the vaiation of the stess esultants aound the cicumfeence to obtain a stength design citeion fo the stiffene. These analytical studies ae then compaed with complementay finite element analyses to veify closed-fom design equations fo ing stiffenes.. INTRODUCTION Discete suppots in cylindical metal silos intoduce local foces into the shell and poduce a cicumfeential non-unifomity in the aial membane stesses in the silo, which must be taken into account in assessing the stability of the shell. A combination of a ing beam and an intemediate ing stiffene can be used fo lage silos to edistibute the stesses fom the local suppot into unifom stesses in the shell as shown in Fig. 1. Geine [1] and Öy and Reimedes [] showed that an intemediate ing stiffene can be vey effective in educing the cicumfeential non-unifomity of aial stesses in the shell.

2 Studies conducted by these eseaches identified the vaiation of the aial membane stess distibutions up the height of the shell. It was shown that an intemediate ing stiffene can achieve a damatic decease in the peak aial membane stess, poducing a moe unifom stess state above the intemediate ing. Recently Topkaya and Rotte [] showed that thee is an ideal location fo an intemediate ing stiffene, such that the aial membane stess above this ing is cicumfeentially completely unifom. The ideal location is identified by the height HI above the ing beam, defined as the vetical distance between the top of the ing beam and the cente of the intemediate ing stiffene as shown in Fig. 1. This was detemined analytically and is epessed (fo the case whee. ) in tems of basic geometic vaiables as follows: 4 H I 1(1 ).95 (1) n n n whee n = numbe of unifomly spaced column suppots; = middle suface adius; and ν = Poisson s atio. Conical oof Cylindical shell Simplified shell design (Constant aial stess) Ring stiffene H I Ring beam Hoppe t U Column t L Fig. 1: Typical cicula planfom silo In cases whee a shell with lage adius ests on a few suppots, the ideal location can be quite high and the option of placing the intemediate ing stiffene below the ideal height may povide a viable solution [4]. This pape eploes the stength equiements fo intemediate stiffenes placed at ideal location o below this location, whee the foce tansfe and displacement bounday conditions diffe fom those fo a ing at the ideal location. A geneal shell and ing combination is studied using the membane theoy of shells to identify the membane shea foces induced in the shell by the ing. These foces ae then consideed as loads applied to the intemediate ing stiffene. Vlasov s cuved beam theoy is used to deive closed fom epessions fo the vaiation of the stess esultants aound the cicumfeence to obtain a suitable stength design citeion fo the stiffene.. STRESS AND DISPLACEMENT TRANSFER INTO INTERMEDIATE RING STIFFENERS Topkaya and Rotte [] detemined the ideal location fo an intemediate ing stiffene using the membane theoy of shells [5-7]. The loading on intemediate ing stiffenes can be obtained by solving fo the eactions on the shell poduced by a stiffene. All defomations, loading and stess esultants can be epessed in tems of a hamonic seies aound the cicumfeence [6, 8] in ode to solve the govening diffeential equations. In the case of discete suppots, the apid decay in the effect of highe tems [9] means that

3 the fundamental hamonic tem of the column suppot foce is sufficient to study the equiements of the ing stiffene, so the suppot foce can be epesented by P Pncosn () whee P = etenal distibuted aial line load applied to the base of the shell; Pn = Fouie coefficient fo the n th hamonic of aial line load; and = cicumfeential coodinate. H L Uppe shell Lowe shell P = P n cosnθ u u θ = u θ = N =κ P n cosnθ u θ = H L u θ = P = P n cosnθ p n u displacements and applied foces p u p u N membane foces N N Fig. : Bounday conditions used in closed fom solution, loading, displacements, and stess esultants in an element of the cylindical shell The cylindical shell is hee sepaated into two pats: an uppe shell and a lowe shell with the intemediate ing at thei junction, as shown in Fig., which also indicates the chosen bounday conditions. The lowe shell, of height HL, is subjected to the fundamental hamonic of the column suppot. The uppe shell is assumed to be unloaded on its uppe bounday and estained against cicumfeential displacements by a ing. Topkaya and Rotte [] demonstated that the inteface between the lowe shell and uppe shell will be fee of both aial stess and aial displacements if an intemediate ing is placed at the ideal location. When the intemediate ing is placed below the ideal location some level of aial stess non-unifomity is pesent in the uppe shell segment. In addition, the aial displacements no longe vanish, so the nonzeo aial displacements can also be found at this inteface. Consideing the cylindical shell element shown in Fig., the equilibium equations ae: N 1 N N 1 N p p N p n () whee N, N, N = aial, cicumfeential and shea membane stess esultants espectively; and p, p, pn = etenal distibuted pessues in the aial, cicumfeential and adial diections espectively. The stain, displacement and constitutive elationships can be witten as: u 1 1 u u 1 N N N N (4) Et Et 1 u u 1 (1 ) N N (5) Gt Et whee u, uθ, u = displacements in the aial, cicumfeential and adial diections espectively; ε, ε = stains in the aial and cicumfeential diections espectively; γ = shea stain; ν = Poisson s atio; E = modulus of elasticity; G = shea modulus; and t = thickness of the shell. In the lowe shell, eq. () may be solved sequentially by integating in the diection to obtain: N 1 N p n N p d f1( ) 1 N ( ) (6) N p d f

4 whee f1(θ), f(θ) = unknown functions of to be detemined fom two bounday conditions. The geneal solution fo the displacements of the shell may then be found as: Etu N N d f ( ) u Etu Et N N (7) Et u Etu (1 ) Nd d f4( ) (8) whee f(θ), f4(θ) = additional functions to satisfy the bounday conditions on the edges = constant. Whee thee is no suface loading on the shell p p pn, eq. (6) give: N N f1 ( ) 1 d d N f1( ) d f( ) f1( ) f( ) (9) d d At the base, =, the aial membane stess esultant is chosen as the fundamental hamonic of the discete suppot, N P cosn (eq. ()), leading to: n f ( ) Pn cos n (1) When the ing is placed at the ideal location, the aial stess vanishes at this height ( N ), but when the ing is placed below the ideal location non-unifom aial stesses will still be pesent. As shown in Fig., a cetain popotion of the applied aial membane stess esultant is assumed to be pesent at the inteface. The atio of the aial membane stess esultant at the inteface to the applied fundamental hamonic of the column suppot is hee temed. Topkaya and Rotte [] eploed the magnitudes of aial membane stess esultants that emain at this inteface using many linea finite element analyses. The location of the intemediate ing, shell adius, numbe of suppots and shell thickness atios (g= tu/tl whee tu and tl ae the thicknesses of the uppe and lowe shells espectively) wee consideed as the pimay vaiables. Fig. shows the vaiation of the atio of aial membane stess esultants fo the case of g =.5. Fig. : Vaiation of aial stess esultant fo vaious intemediate ing heights with uppe to lowe shell thickness atio g =.5 The following convenient lowe and uppe bound epessions can be developed to epesent the data points: m N HL HL 1g 1g (11) Pn HI HI

5 with m.5g. 4 g fo uppe bound m.4g. 4 g fo lowe bound (1) Consideing an aial stess esultant N Pncos n at the ing ( = HL) leads to the following: f1 1 Pn sinn (1) HLn Inseting eqs (7) and (9) into eq. (8) yields the cicumfeential displacement as: d d 1 d Etu f1( ) f( ) 1 f1( ) f( ) f4( ) 6 d d d (14) At = and = HL, the bounday condition of zeo cicumfeential displacements, u, yields the two esults: H 1 1 f 4 ( ) and L f P cos n n (15) 6 HLn Inseting eqs (9) and (15) into eq. (7) yields the aial displacements as: P cos n n HL u (16) Et H L 6 HLn When the intemediate ing is placed at the ideal height the aial displacements and aial stess esultants at the inteface vanish. The condition of = with u at HL HI leads to the ideal location of the intemediate ing stiffene, peviously epessed in eq. (1). The eactions in the shell at this bounday can be teated as the loading eeted on the intemediate ing. Combining eqs (9) and (1) gives the following epession fo the shea membane stess esultant and the displacements at the inteface ( = HL) can be found fom eq. (16) as: 11 1 HLn 1 N 1 Pnsinn u P cos n n (17) HLn 6EtH Ln 4. STRESS AND DISPLACEMENT TRANSFER INTO INTERMEDIATE RING STIFFENERS 4.1 Deivation of stess esultants In plane behaviou The si basic equilibium equations fo the cuved beam element shown in Fig. 4 can be epessed using the Vlasov s diffeential equations [1-11] as follows: 1 dq 1 dq 1 dq Q q q d d Q q (18) d 1 dm 1 dm 1 dt T Q m m Q d d M m (19) d whee = adius of the ing beam centoid; M = bending moment in the ing about a adial ais; M = bending moment in the ing about a tansvese ais; T = tosional moment in the ing; q, q, q = distibuted line loads pe unit length in the tansvese, cicumfeential and adial diections espectively; m, m, m = distibuted applied toques pe unit cicumfeence about the tansvese, cicumfeential and adial diections espectively; Q = cicumfeential tensile foce in the ing; Q, Q = shea foces in the ing in tansvese and adial diections espectively.

6 The si basic equilibium equations can be educed to thee diffeential elationships fo the case whee the only loading is q = N (i.e. q = q = m = m = m = ) as: 1 dt 1 d M dt 1 d M dm M d d d q () d d Solution of eq. () and substituting obtained stess esultants into equilibium equations eveal the following elationships fo in-plane loading: Pn M T Q Q 1 cosn H L ( n 1) (1) Pn Pn M 1 cosn Q 1 sin H L n ( n 1) n H L nn ( 1) () These equations give epessions identical to those deived by Zeybek et al. [1] fo the case whee HL = HI and. Q Q M T d Q +dq M +dm T +dt Q +dq u u u d q m q m m q Q Q +dq Loading and displacements Fig. 4: Diffeential cuved beam element and sign conventions 4. Deivation of stess esultants Out-of-plane behaviou The following epession define the foce-defomation elationship [11]: EI d u M () d whee u, uθ, u = displacements in the vetical, cicumfeential, and adial diections espectively; = otation; I, I= bending moment of inetia of the ing stiffene about adial and vetical aes espectively; Iw = waping constant of the ing stiffene; KT = unifom tosional constant of the ing stiffene; A = coss sectional aea of the ing stiffene; and G = shea modulus. Inseting eq. (17) into eq. (), substituting into equilibium equations and consideing =, and substituting obtained stess esultant into equilibium equations eveal the following elationships: I 111 HLn 1 M P cos n n (4) 6HL t I 111 HLn 1 T P sin n n (5) 6HLn t I n HLn 1 Q P sin n n (6) 6H n t Eqs (4), (5) and (6) esult in, T, Q and. M Foces L M +dm M fo the case of HL = HI

7 Consideing typical atios of the stong ais to weak ais elastic section moduli of olled I sections, it is hee ecommended that the atio of out-of-plane moment to in-plane moment should be limited to 1% (M/M<.1). Neglecting the contibution of (i.e. = ), the section can be selected by using the following epession fo the second moment of aea about the adial ais (I): t I (7) H L 1 n n 1 1 H I Having esticted the out-of-plane bending moment using eq. (7), the intemediate ing can be designed to esist out-of-plane moments of only 1% of the maimum in-plane bending moment. 5. COMPUTATIONAL VERIFICATION OF THE ABOVE EQUATIONS The commecial finite element pogam, ANSYS v1.1 [1], was used to pefom the numeical analysis. The computational time was educed by modelling only a segment coveing the angle /n. Fou-node shell elements (shell6) wee employed to model the cylindical shell. The intemediate ing stiffene was modelled using two-node beam elements (beam4). The modulus of elasticity was taken as GPa and Poisson s atio as.. The cylinde base was subjected to loading in hamonic 4 (eq. ()), coesponding to the numbe of equally spaced discete suppots. The loading at the bottom of the shell was chosen to give a maimum aial membane stess of 1 MPa above the suppot. The silo stuctue analyzed hee had a cylinde adius of mm and a height of 1 mm. The lowe and uppe shell thicknesses wee 6 mm and mm espectively. A fleible intemediate ing whee the out-of-plane second moment of aea is defined by the uppe limit given in eq. (7) was placed at half of the ideal height (HI/). Ring Bending Moment Ring Shea Foce M (kn-m) Q (kn) Ring Cicumfeential Tension Q (kn) Closed Fom Solution Numeical Solution Cicumfeential Coodinate (deg) Closed Fom Solution Numeical Solution Cicumfeential Coodinate (deg) Closed Fom Solution Numeical Solution Cicumfeential Coodinate (deg) Ring Bending Moment Fig. 5: Compaison of closed fom solution with numeical solution fo a fleible intemediate ing M (kn-m) Ring Tosional Moment T (kn-m) Ring Shea Foce Q (kn) Closed Fom Solution Numeical Solution Cicumfeential Coodinate (deg) Closed Fom Solution Numeical Solution Cicumfeential Coodinate (deg) Closed Fom Solution Numeical Solution Cicumfeential Coodinate (deg)

8 The calculated vaiations of M, Q, Q, M, T and Q aound the cicumfeence, togethe with the pedictions of the closed fom solutions (eqs (1), (), (4), (5) and (6)) fo the intemediate ing ae shown in Fig. 5. The lowe bound epession (eq. (1)) was used in closed fom solutions. The compaisons show that the above equations povide acceptably accuate solutions, with the lagest diffeences being 4.8%, 5.1%, 4.8%, 18.%,.% and 18.4%, fo maimum M, Q, Q, M, T and Q espectively 6. SUMMARY AND CONCLUSIONS This pape has developed a citeia fo the stength of intemediate ing stiffenes used in cylindical silo shells esting on column-suppoted ing beams. The closed fom epessions eveal that the intemediate ing stiffene is subjected to out-of-plane intenal foces and bending moments in addition to the in-plane stess esultants when the ing is placed below the ideal height. The developed epessions wee compaed with numeical solutions and a good ageement has been demonstated. The closed fom solutions indicate that the out-of-plane intenal foces and bending moments depend on the out-of-plane bending stiffness of the ing. Fo economical designs, it is poposed that the out-of-plane bending moment should be kept below 1% of the in-plane bending moment. 7. REFERENCES [1] GREINER R., Zu Laengskafteinleitung in stehende zylindische Behaelte aus Stahl, Stahlbau, 5(7), 1984, [] ÖRY H. and REIMERDES H.G., Stesses in and Stability of Thin Walled Shells unde Non-ideal Load Distibution, Poc. Int. Colloq. Stability Plate & Shell Stucts, Gent, ECCS, 1987, pp [] TOPKAYA C., ROTTER J.M., Ideal location of intemediate ing stiffenes on discetely suppoted cylindical shells, ASCE Jounal of Engineeing Mechanics, 14(4), 14, pp [4] ZEYBEK Ö., TOPKAYA C. and ROTTER J.M., Requiements fo intemediate ing stiffenes placed below the ideal location on discetely suppoted shells, Thin Walled Stuctues, 115, 17, pp. 1-. [5] VENTSEL E., and KRAUTHAMMER T., Thin plates and shells: theoy, analysis, and applications, Macel Dekke, NY, 1. [6] TIMOSHENKO, S.P. and WOINOWSKY-KRIEGER, S., Theoy of Plates and Shells, McGaw-Hill, New Yok, [7] FLÜGGE, W., Stesses in Shells, Spinge-Velag, Belin, 197. [8] CALLADINE, C.R., Theoy of shell stuctues, Cambidge Univesity Pess, U.K., 198. [9] ROTTER, J.M., Bending Theoy of Shells fo Bins and Silos, Tans. Mech. Engg, IEAust, ME1 (), 1987, pp [1] VLASOV, V.Z. Thin-walled elastic beams, National Science Foundation, Washington, D.C, [11] HEINS, C.P. Bending and tosional design in stuctual membes, Leington Books, Leington, Massachusetts, [1] ZEYBEK Ö., TOPKAYA C. and ROTTER J.M., Stength and stiffness equiements fo intemediate ing stiffenes on discetely suppoted cylindical shells, Thin Walled Stuctues, 96, 15, pp [1] ANSYS, Vesion 1.1 On-Line Use s manual, 1.