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1、Thin-Walled Structures 31 (1998) 203220Postbuckling strength design of open thin-walled cylindrical tanks under wind loadH. Schmidt*, B. Binder, H. LangeDepartment of Civil Engineering, University of Essen, D-45117 Essen, GermanyAbstractOpen cylindrical steel tanks for the storage of liquids tend to
2、 be rather thin-walled becausethey are primarily designed for the circumferential tensile stresses from hydrostatic internalpressure. It is desirable to take advantage of the postbuckling strength of the thin cylindricalwall when designing the empty tank against wind load. The theoretical external p
3、ressure carry-ing behaviour of edge-stiffened thin-walled cantilever shells is discussed with regard to bothbifurcation and postbuckling phenomena. For experimental verification, a series of postbuck-ling tests on cylindrical PVC and steel specimens with large radius/thickness ratio (r/t 5 2500)unde
4、r internal underpressure and under a “wind-like” load arrangement has been carried out.Based on the numerical and experimental results, recommendations are put forward for aneconomic postbuckling strength design strategy. They are compared with existing design rulesin tank codes. 1998 Elsevier Scien
5、ce Ltd. All rights reserved.Keywords: Shell buckling; Shell stability; Postbuckling; Tanks; Wind load1. IntroductionOpen cylindrical steel tanks are used for permanent storage of various liquidsranging from water to livestock slurry, or as a catch basin for emergency storage ofenvironmentally danger
6、ous liquids. The tank walls are primarily designed for thehoop stresses from hydrostatic internal pressure; no significant axial compressiveforces that would create a severe shell buckling case are present. As a consequence,* Corresponding author. Tel: 1 49 201 183 2766; Fax: 1 49 201 183 2710.0263-
7、8231/98/$see front matter 1998 Elsevier Science Ltd. All rights reserved.PII: S0263-8231(98)00009-3204H. Schmidt et al./Thin-Walled Structures 31 (1998) 203220the walls are in many tanks are extremely thin; radius/thickness-ratios r/t of 3000 5000 in the upper sections are not unusual.However, all t
8、hese cylindrical tanks are susceptible to wind when empty (Fig. 1).In order to prevent overall failure of the empty tank under wind load, a ring stiffenerat the top of the tank is obligatory. This has unvoluntarily been demonstrated byseveral failures in practise, and it has convincingly been underl
9、ined by wind tunneltests on unstiffened model tanks at the University of Leeds 15,16. With regard tothe wind buckling design of the wall itself presuming a sufficiently stiffenedupper edge , two strategies are thinkable, depending on the safety requirements.1. Either the buckling strength of the wal
10、l is verified by means of “classical” buck-ling design rules for externally pressurized circular cylinders 9,11. This approachresults in walls which are buckling-resistant against both local inter-ring bucklingand overall buckling including the rings. However, the wall wil either be consider-ably th
11、icker than necessary for the tensile hoop stresses, or a number of rathernarrow-spaced wind stiffeners between the bottom of the tank and the top stiffenerwill be necessary. This approach should, in the authors opinion, be used as aprecaution in cases of permanent storage of liquids with high risk p
12、otential.2. Or one takes advantage of the well-known postbuckling strength of end-supportedshort cylinders under external pressure by making the tank buckling-resistant onlyagainst overall buckling. This approach results in more economic walls, with onlya few or even no intermediate wind stiffener a
13、nd no significant increase in thick-ness compared to the tensile hoop stress design. Of course, this tank wall wouldunder strong wind eventually buckle between the wind stiffeners (or between thetop stiffener and the bottom if no additional wind stiffeners were present), butwould jump back into its
14、proper shape after the strong wind, if appropriatelydesigned. This approach could be used in cases of liquids with lower risk poten-tial, e.g. water or livestock slurry, or in the case of catch basins for emergencystorage of environmentally dangerous liquids.Design strategy 1 may be applied rather s
15、traightforwardly. The necessary knowledgeFig. 1.Open cylindrical tank under wind load.205H. Schmidt et al./Thin-Walled Structures 31 (1998) 203220about the critical (eigenvalue) buckling wind loads and about the handling of imper-fection influences is at the designers disposal.Design strategy 2 is,
16、as far as the state of knowledge is concerned, more vague.In fact, the postbuckling strength of the cylindrical wall has implicitly been utilizedto a certain amount for many years in the design of petroleum and agricultural tanks2,5,6,9, but to the authors knowledge on a merely empirical basis.In or
17、der to provide a more rational basis for the economic design of strategy 2,a comprehensive program of numerical and experimental investigations into thebuckling and postbuckling behaviour of open thin-walled cylinders under externalpressure has been carried out at the University of Essen 3,4,13. Thi
18、s paper presentssome results of these investigations.2. Numerical investigations on edge-stiffened cylindrical cantilever shellsunder external pressure2.1. Theoretical bifurcation behaviour2.1.1. Critical buckling behaviourIf the top stiffener is strong enough, the cylindrical wall buckles in a mode
19、 whichleaves the stiffened edge practically circular (Fig. 2a). This buckling mode is called“local”. It has one half-wave in an axial direction with maximum radial deformationw at midheight of cylinder and a certain number of waves in a circumferential direc-tion within the externally pressurized wi
20、ndward region of the cylinder (see Fig. 1).If the top stiffener is too weak, the cylindrical wall buckles with a mode whichincludes column-like buckling of the stiffener as a circular ring. This buckling modeis called “global”. It has one quarter-wave in axial direction with maximum radialdeformatio
21、n w near or at the edge and a certain number of waves in circumferentialFig. 2.Critical buckling modes of cylindrical cantilever shells under wind loading, (a) localbuckling/strong top stiffener, (b) global buckling/no top stiffener.206H. Schmidt et al./Thin-Walled Structures 31 (1998) 203220directi
22、on, similar to the local mode, but with greater wave lengths. The global buck-ling mode resembles the one of a cylinder with an unstiffened free edge (Fig. 2b).The critical buckling pressure qcr,lunder which local buckling occurs may approxi-mately be determined assuming the stiffened edge as simply
23、 supported; relevant for-mulas may be found in the codes, e.g. in reference 8. In order to define the limitbetween a “too weak” top stiffener and a “strong enough” top stiffener from theviewpoint of bifurcation theory, a minimum second moment of area min Its(bendingwithin the plane of the top circle
24、) can be defined, above which local bucklingbecomes the critical buckling mode. Ansourian 1 found for a cylinder with constantwall thickness the approximate formulamin Its5 0.05t3h.(1)This top stiffener delivers at least 95% of the critical wind buckling pressure ofthe simply edge-supported cylinder
25、. Furthermore, it may be concluded from relevanteigenvalue analyses of tanks with stepwise variable wall thickness, that Eq. (1) yieldsa conservative value for min Itswhen introducing the average wall thickness. A topstiffener with min Itsis called “bifurcation-optimized” within this paper.2.1.2. Co
26、mplete bifurcation behaviourNow the question arises whether min Itsis of any practical use if the local post-buckling strength of the wall shall be utilized for the wind buckling design strategy 2.In order to answer this question, a deeper understanding of the complete bifurcationbehaviour is necess
27、ary. Fig. 3 shows for a shell example under axisymmetric uniformexternal pressure (its basic behaviour is comparable to the windward side of wind-loaded cylinders 13) complete sets of eigenmode-eigenvalue pairs as a function ofthe top stiffener cross section. In Fig. 3 the following cases are marked
28、 by thick lines:“Very strong” stiffener with Its 3103min Its(bts5 10 mm): All eigenmodesaround the critical one with ncr5 38 are of purely local type. The critical eigenmoderepresents a well-defined minimum pressure qcr,lwhich is virtually identical with theone for a simply supported edge.“Very weak
29、” stiffener with Its 3103min Its(bts5 0.1 mm): All eigenmodesaround the critical one with ncr5 27 are of the purely global type. The criticaleigenmode represents a well-defined minimum pressure qcr,gwhich is virtually ident-ical with the one for a free edge.“Bifurcation-optimized” stiffener with Its
30、 min Itsaccording to Eq. (1) (bts50.7 mm): The eigenmodes around the critical one with ncr5 35 include local andglobal modes and all types of mixtures of them. The critical eigenmode representsa rather poorly defined minimum pressure qcr 0.95qcr,l.The conclusion from Fig. 3 is that it can not be rec
31、ommended to use Eq. (1) asa design tool even not for a classical full buckling strength design according tostrategy 1, i.e. with a safety factor of about 1.5 against local buckling of the cylindri-cal wall. It should be noted that in Fig. 3 the case with bts5 0.6 mm i.e. withonly little less stiffne
32、ss than min Its has many eigenvalues with rather differenteigenmodes within a narrow scatter band of 6 5%. Having in mind that structural207H. Schmidt et al./Thin-Walled Structures 31 (1998) 203220Fig. 3.Complete bifurcation behaviour of differently edge-stiffened cylindrical cantilever shells under
33、uniform external pressure (test specimen I in Table 1).systems with such “multi-mode” bifurcation behaviour tend to be highly imperfec-tion-sensitive, the normal knock-down factora5 0.65 0.80 for external pressurebuckling would certainly be inadequate. In the authors opinion, the top stiffenershould
34、, for a classical buckling design according to strategy 1, have a second momentof area of at leastIts$ 10 min Its5 0.5t3h.(2)With regard to postbuckling design, see Section 5.2.2.2. Theoretical postbuckling behaviourThe second author has carried out a great number of numerical postbuckling calcu-lat
35、ions for perfect elastic cylinders with r/t 5 2500 under uniform external pressure3. A shell analysis computer program by M. Esslinger and her co-workers has beenused 11. By means of this program, it is possible to trace the postbuckling load-deformation-curve for a given number n of buckles around
36、the circumference; theprogram does not automatically detect secondary bifurcations into other modes. Thusthe complete postbuckling path including secondary cascade buckling has to be com-posed as lower festoon curve of various computer runs. Fig. 4 shows an example;it is the same test shell as for F
37、ig. 3, with the top stiffener being “very strong” (bts5 10 mm).208H. Schmidt et al./Thin-Walled Structures 31 (1998) 203220Fig. 4.Numerically determined postbuckling behaviour of a cylinder with “very strong” top stiffener(test specimen I with bts5 10 mm, see Table 1).Fig. 4 illustrates the well-kno
38、wn theoretical postbuckling behaviour of a shortcylindrical shell which buckles locally: After bifurcation at qcr5 qcr,lwith n 5 ncr5 38 the load-deformation-path decreases while successively changing the mode(cascade buckling) until finally at the well-defined postbuckling minimum min q 0.7 qcr,lwi
39、th n 5 31 and w qcr, (c,d) latepostbuckling state: q 3qcr.Fig. 13.Load deformation-curves of PVC specimen III without ring stiffeners.4.2. Design-relevant conclusionsThe following conclusions summarize those findings which are relevant fordesign aspects:1. If sufficiently stiff ring stiffeners are p
40、rovided, elastic cylindrical tank walls exhi-bit under external pressure the expected good-natured local postbuckling behav-iour up to several times the critical buckling pressure.2. For steel walls with r/t-ratios of 2500, the postbuckling behaviour remains elastic216H. Schmidt et al./Thin-Walled S
41、tructures 31 (1998) 203220(and may therefore be utilized in the design) up to at least 1.5 times the criticalbuckling pressure. This postbuckling range may properly be described by avail-able numerical tools.3. Unsufficiently strong ring stiffeners fail after the tank wall has primarily dis-played l
42、ocal buckling followed by good-natured postbuckling under increasingpressure suddenly secondarily in a catastrophic manner which should be reliablyexcluded by the desgin. For the time being, this postbuckling behaviour can notbe described by available numerical tools.4. Determining the minimum cross
43、 section of ring stiffeners from a classic eigen-value buckling analysis (“bifurcation-optimized” stiffening) results in stiffenerswhich are far too weak with regard to their resistance against secondary buckling.5. Design recommendations5.1. Tank wall thicknessThe question in which cases a full buc
44、kling strength design according to strategy1 should be performed for the wall of an open cylindrical tank (e.g. in case ofpermanent storage of high risk liquids), is not being discussed here. If a postbucklingstrength design according to strategy 2 shall be performed 3, gives the followingrecommenda
45、tion for determining the characteristic pressure resistance qRkof a thin-walled cylindrical shell section of length l between two radially restrained edges(two stiffeners or bottom edge and stiffener):qRk5a*qcrlwitha*5 1.71 2 0.71l/r $ 0.65(3)The lower limit 0.65 in Eq. (3) is the elastic imperfecti
46、on factor (knock-downfactor) for external pressure buckling cases in the German code DIN 18800-4. Eq.(3) may be considered as being experimentally verified by the present postbucklingtest series for r/t $ 2500 and 0.3 # l/r # 1.0. However, the authours believe that itcould be applied for structural
47、steel (or other steels with similar strngth characteristicsrespectively) without risk to r/t $ 2000 and 0.05 # l/r # 2. Some check-up testsin order to verify the extended application range are underway.Introducing “normal” DIN wind design conditions 50 year wind speed Vw528.3 m/s; pressure coefficie
48、nts cpo1 cpi5 1.0 1 0.6 (see Fig. 7); partial safetyfactorsgF5 1.5 andgM5 1.1 and the DIN formula for qcr, recommendation(3) may be plotted as a design curve for the minimum wall thickness min t as afunction of l/r (Fig. 14). For l/r # 1.0 it may be approximated by the followingsimple formula:min t/
49、r 5 0.5103(l/r)0.5for 0.05 # l/r # 2.(4)Again, the authors believe that this formula although for l/r 1 lying below217H. Schmidt et al./Thin-Walled Structures 31 (1998) 203220Fig. 14.Minimum thickness of open tank walls under “normal” German wind load conditions.Binders original proposal could be ap
50、plied up to l/r # 2. However, for austeniticsteel the thickness has to be increased by the factor (Estruct/Eaust)0.4. In reference 9a similar formulamin t/r 5 0.55103(l/r)0.667(5)is given for agricultural tanks. It is empirically based on manufacturers experienceand on two full scale tests 14. It se
51、ems to be unconservative for short cylinderswith l/r , 0.3 (see Fig. 14).In the internationally widely used tank standards API 650 2 and BS 2654 5rules for the maximum stiffening distance max l are given. They may be rearranged3 as min t formula which reads introducing the same DIN wind design con-d
52、itions as follows:min t/r 5 0.45103(l/r)04.(6)This equation yields (see Fig. 14) virtually the same thickness values as Eq. (4)for the main range of interest, i.e. 0.1 # l/r # 0.5. However, for longer cylindersthe API/BS tank rules tend to overestimate the postbuckling strength. This should,in the a
53、uthors opinion, not be made light of, because the API/BS tank rules are notonly applied to open tanks but also to closed tanks where the roof loads create axialcompressive stresses in addition to the wind hoop compressive stresses. It is well-known that this interaction reduces the postbuckling stre
54、ngth.218H. Schmidt et al./Thin-Walled Structures 31 (1998) 203220Full buckling strength according to design strategy 1 would again formulatedwith the above DIN wind design conditions approximately be provided bymin t/r 5 0.6103(l/r)04.(7)Eq. (7) is, for comparison, also plotted in Fig. 14. Looking a
55、t Fig. 14, it shouldbe emphasized that the design curves are only valid for the above described “normal”German DIN wind load conditions with their relatively low characteristic wind speed.Higher design wind speeds lead of course to thicker walls.5.2. Top ring stiffenerIn buckling-endangered stiffene
56、d plate and shell structures, the stiffener dimen-sions are often derived from eigenvalue calculations by a so-called “minimum stiff-ness” criterion (called “bifurcation-optimized stiffening” within this paper). Such anapproach would, as outlined in this paper, for the present case yield far too wea
57、kstiffeners 3. The 1 3 2 mm top stiffener of test specimen VI (see Table 1) failed,although fulfilling this criterion, by overall buckling at 0.7 qcr, triggered by unusallyearly local buckling at 0.4 qcr. Such a stiffener would not even be strong enoughfor design strategy 1, see Section 2.1.2.For de
58、sign strategy 2 the second moment of area Itsof the stiffener (which controlsthe eigenvalue analysis) is definitely not the relevant cross section property. Theinward buckled wall when being loaded by wind pressure from outside causes con-siderable radial forces at the stiffener which are varying al
59、ong the circumference.This has to be accounted for by a second order bending design of the stiffener.Reference 10 gives rules for such a top stiffener design which may be written asthe following elementary beam-column equation:sts5 Nts/Ats1 Mts/Zts# fy/gMwithNts5 qdrh/2 and Mts5 Ntsw01/(1 2 qd/qcr,t
60、s).(8)In Eq. (8) qdis the design wind pressure at the stagnation meridian (outside press-ure plus inside suction) includinggF; h is the total tank height; w0is an appropriatevalue for the radial imperfection; the bracket term is the moment amplification factorwith qcr,tsdescribing the bifurcation bu
61、ckling of the top stiffener as an isolatedring column:qcrts5 3EIts/(0.5r3h).(9)Eq. (8) may be rearranged as formula for the minimum section modulus min Ztsof the top stiffener:219H. Schmidt et al./Thin-Walled Structures 31 (1998) 203220min Zts5 C(2r)2h(10)with C 5 0.125(w0/r)qd/(fy/gM2 Nts/Ats)1/(1
62、2 qd/qcrts).(11)Eq. (10) corresponds to the well-known and worldwide approved design formulain many tank codes 2,5,7, where for a 50 year wind speed Vw5 45 m/s the constantin Eq. (10) is given asC 5 0.058106.(12)This particular value for C would be delivered by Eq. (11) if, for instance, thefollowin
63、g reasonable set of data would be introduced:w0/r 5 0.01;qd/(fy/gM2 Nts/Ats) 5 (2103N/mm2)/(200N/mm2) 5 105;(13)qd/qcrts5 0.78.It turns out that Eq. (8) and the approved tank code formula describe virtually thesame bending carrying capacity requirement for the top stiffener of an open tank.Top stiff
64、eners which fulfill this requirement have in the investigations presentedin this paper proved to be strong enough for design strategy 2.5.3. Intermediate ring stiffenersThe rules in reference 10 for externally pressurized ring-stiffened cylinders fromwhich Eqs. (8) and (9) for the top stiffener have
65、 been extracted contain a similardesign procedure for intermediate stiffeners. It is rather complicated, and it has notyet been verified by the authors to cover design strategy 2. For the time being, theminimum stiffener cross sections specified in various tank codes should be used,though carefully
66、observing their application limits. The authors are presently tryingto develop more general postbuckling design rules for intermediate stiffeners of thintank walls.AcknowledgementsThe experimental postbuckling investigations mentioned in Section 3 have beenfinancially supported by the Deutsche Forsc
67、hungsgemeinschaft (DFG) under the pro-ject No. Schm 385/8. The authors are thankful for this support. Beyond that, theauthors express their great respect and gratitude to Mrs. Prof. Maria Esslinger whohad the basic idea for the test set-up in Fig. 6 and who developed the computerprogram mentioned in
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