(2013) buckling design of large circular steel silos subjeted to wind pressure

Buckling design of large circular steel silos subject to wind pressure

Yang Zhao, Qing-shuai Cao, Liang Su n

Space Structures Research Center, Zhejiang University, Hangzhou 310058, China

a r t i c l e i n f o

Article history:Received 27 May 2013Accepted 20 August 2013Available online 27 September 2013

Keywords:Steel siloWind pressureBucklingGeometrical nonlinearityMaterial nonlinearityWeld imperfection

a b s t r a c t

Large steel silos are typical kinds of thin-walled structure which are widely used for storing hugequantities of granular solids in industry and agriculture. In the present analyses, buckling design of largesteel silo subject to wind pressure is demonstrated in accordance with Eurocode (EN1990, 1991, 1993)and the proposed combinational Load Case WE (wind and empty silo) and WF (wind and full silo). Thefinite element model is established by using the commercial general purpose computer package ANSYS.Five types of buckling analyses are carried out for the geometrically perfect and imperfect models withand without the consideration of the material plasticity, which are designated as LBA, GNA, GMNA, GNIA,and GMNIA in EN 1993 Part 16. The geometrical imperfections are known to have a large impact on thebuckling behavior of steel silo structure, in which the magnitude and distribution of the weld depressionduring construction process is adopted to account for fabrication quality. The buckling behavior of areference silo with a diameter of 40 m and an aspect ratio of 0.9 is first investigated, which shows thatthe buckling behaviors from Load Case WE and WF are much different. The material nonlinearity showslittle influence on buckling resistance in Load Case WE, while the buckling resistance and buckling modeis much sensitive to weld imperfection. In Load Case WF, both material nonlinearity and geometricalnonlinearity effect is strong and detrimental to buckling behavior of steel silos, resulting in decrease ofbuckling resistance. The buckling deformation corresponding to the critical point in Load Case WE isgoverned by the circumferential compression which is generated in the windward region of the shellslocalized at the top part of silo wall. The buckling mode in Load Case WF takes the form of the well-known elephant-foot deformation at the bottom part of the shell wall, which is induced by themeridional compressive stress. It is also indicated from the parametric analyses that the bucklingresistance of steel silo is closely correlative with the loading conditions involving the wind velocity, thepatch load, and the geometrical parameters including the aspect ratio, the radius-to-thickness ratio, thetype of wall thickness, and the wall openings.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Large steel silos widely used for storing huge quantities ofgranular solids in industry and agriculture are typical kinds ofthin-walled structure which usually consist of a rigid flat bottom, acylindrical shell with stepped wall thickness, a conical or domeroof on the top and above affiliated buildings. The silo wall may besupported in two fundamental forms: the discrete support and theground support. The former type is applicable for the silo withsmall diameter which is constructed using a local bracket orcolumn, giving a limited number of narrow supports around thesilo circumference, and the latter is appropriate to the silo withlarge diameter (such as diameter larger than 15 m) which issupported exactly on bottom plate with the underpass beneaththe flat bottom [1,2]. With the development of domestic economy,

the request for storage of some strategic materials in china, such asthe grain, the coal, the cement, etc, becomes larger and larger. Boththe number and capacity of steel silo has been increasing forrecent decades. The diameters of some steel silos are even largerthan 100 m, with the contents more than 100,000 m3. Fig. 1 showsa practical ground-supported circular steel silo.

Large steel silo is usually squat with a large diameter tothickness ratio, which is particularly vulnerable to buckling underwind pressure when the silo is either empty or partially filled. Thestructural design of circular steel silos subject to wind pressure iswell known to be usually controlled by considerations of bucklingof silo wall with the buckling deformation as the potential failuremode. The design is concerned with a complicated process, whichwould be demonstrated in accordance with the current Europeanstandard: EN1990, 1991 and 1993 [18] in the following presenta-tion. The buckling design of steel silo structure is dominated by theresistance of shell wall to buckling failure in service, which issupposed to take two categories of nonuniform pressures intoaccount [1,2,5]: the wind pressure on external surface of silo wall

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/tws

Thin-Walled Structures

0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.tws.2013.08.015

n Corresponding author. Tel.:86 571 88208752.E-mail address: [email protected] (L. Su).

Thin-Walled Structures 73 (2013) 337349
www.sciencedirect.com/science/journal/02638231www.elsevier.com/locate/twshttp://dx.doi.org/10.1016/j.tws.2013.08.015http://dx.doi.org/10.1016/j.tws.2013.08.015http://dx.doi.org/10.1016/j.tws.2013.08.015http://crossmark.crossref.org/dialog/?doi=10.1016/j.tws.2013.08.015&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.tws.2013.08.015&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.tws.2013.08.015&domain=pdfmailto:[email protected]://dx.doi.org/10.1016/j.tws.2013.08.015

and the wall pressure exerted by the stored granular solids. Whilstthe buckling behavior of cylindrical shells subject to wind pres-sures have been extensively explored, few studies have evaluatethe effects of wind pressure on steel silo when it is empty or full,and the buckling design of steel silos considering combined windload and bulk solids pressure has received even less attention.

The wind pressure on external surface of cylindrical shells as silos,tanks and bins have held the interest of many researchers [913],who try to determinate the distribution and magnitude of windpressure by wind tunnel test [9,12,13] or numerical simulation. Windpressures and buckling of cylindrical steel tanks with a conical anddome roof are presented by Portela and Godoy [12,13], which showsthat buckling occurs in the form of deflections in the cylindrical shelland the buckling mode is localized in the windward region. Thestudy of anchored stocky and intermediate length cylindrical shellsof uniform thickness under wind pressure is presented by Chen et al.[14], which indicates that both linear and nonlinear analyses predictthe circumferential compression buckling mode in stocky cylinders.The buckling behavior of cylindrical shells with stepwise wallthickness under uniform external pressure is also explored by Chenet al. [15], who make predictions for a wide range of geometries ofsilos and tanks with both anchored and unanchored base bound-aries. The buckling of cylindrical steel tanks under wind pressure isevaluated for conical roof and open top tanks by Sosa and Godoy[16], to compute a lower-bound for critical wind pressures and theresults are compared with the static nonlinear analysis carried out onthe same models. Load-bearing capacity of slender wind-loadedcylindrical shells is also investigated by Schneider and Zahlten [17]with consideration of geometrical and material nonlinerity, whichshows that the slender shells do not behave as beams under windloading in stress state, failure loads and failure modes. Bucklingresults could also be found in other investigations [1821] fordiversity of geometries and load conditions of shells. The postbuck-ling behavior is also evaluated by Schmidt et al. [22] throughnumerical analyses and experimental verification, and some usefulrecommendations are put forward for an economic postbucklingstrength design strategy.

The effect of imperfections on wind-pressurized cylindricalshells is evaluated by Greiner and Derler [23], while the imperfec-tion sensitivity to elastic buckling of wind loaded open cylindricaltanks is reported by Godoy and Flores [24]. Pircher [25] exploresthe influence of a weld-induced axisymmetric imperfection on thebuckling of a medium-length silo under wind loading, who foundthat the position of the weld along the height of the thin-walledcylinder has a great influence on the buckling strength underwind-loading and weld-induced residual stress fields reduce thebuckling resistance by a small amount.

For recent decade, the buckling failures of wind-pressurizedsteel cylindrical containers, such as silos and tanks, have beenfrequently reported [2629] during their normal service or duringconstruction. It is confirmed by the large numbers of engineeringpractice that the wind load or the bulk solids pressure oncylindrical shell wall are the main causes which would lead tobuckling failure of such kind of structures.

This paper investigates the buckling behavior of steel silothrough great deal of numerical analyses, which aims to improvethe understanding of large circular steel silos subject to windpressure, when the silos are either empty or full. The layout isorganized as follows: Section 2 summarizes the load cases relatingto wind pressure proposed by Eurocode [17], together with thewall pressure exerted by bulk solids and the wind pressure onexternal surface of silo; Section 3 then expatiates upon the finiteelement model and the types of buckling analyses; the numericalresults of the reference silo with the diameter of 40 m and aspectratio h/d of 0.9 are presented for both the perfect and imperfectmodels in all proposed buckling analyses in Section 4 and theassessments of the different consequences of different load casesare also made for the buckling resistance and buckling mode; theparametric studies of the wind velocity ranging from 20 m/s to60 m/s, the patch load, and the geometry parameter including theaspect ratio h/d ranging from 0.5 to 1.1, the radius-to-thicknessratio r/teq ranging from 500 to 1000, the type of wall section(uniform, stepped, or tapered wall), the wall openings, are spe-cially undertaken in Section 5 and some valuable conclusionsexpected to benefit the understanding of the buckling behavior ofsteel silos subject to wind pressures are obtained in Section 6.

2. Load cases

Silos are different to many other structures because they maybe subjected to the full loads from the particulate solids for mostof their life, accordingly silos should be designed taking account ofactions exerted by the particulate solids and combinations withother actions, such as wind pressure on the one hand. On the otherhand, when emptying of solids in the silo is complete (empty silo),the wind pressure on external surface of silo wall would becomedominant in verifying the ultimate limit state of buckling. Thefollowing actions are supposed to be considered in the ultimatelimit state design of the silo: wind pressure when the silo is eitherfull or empty, wall pressure during filling and storage of particulatesolids, permanent action (self-weight), which would be introducedin brief and determined in accordance with Eurocode [17].

In addition, buckling analyses for exploring the ultimate limitstate of silos depends strongly on the following working condi-tions: the action assessment class, the fabrication tolerance qualityclass, the consequence class, which takes account of the influenceof the silo capacity, the supporting patterns, and the eccentricityduring filling and discharge, fabrication imperfection, etc., asappropriate. In this paper, large diameter of steel silo is taken asthe reference example whose contents are usually more than50,000 t in normal service life. The associated requirements of theabove design conditions for buckling assessment are specified inEurocodes [1,2,7], the appropriate design conditions are adoptedand listed in Table 1.

2.1. Wind pressure

The wind action has been an integral part of structural design.The distribution and magnitude of wind pressure on silo structureare to be calculated in accordance with EN 1991-1-4 [5], whichgives guidance on the determination of natural wind actionsfor the structural design of building and civil engineering works.

Fig. 1. A practical ground-supported circular steel silo.

Y. Zhao et al. / Thin-Walled Structures 73 (2013) 337349338

The distribution of wind pressures around a squat circular silo canbe important to the assessment of wind buckling resistance. Thepressure variation around an isolated silo may be defined in termsof the circumferential coordinate with its origin at the windwardgenerator (see Fig. 2). The circumferential variation of the pressuredistribution cp (positive inward) on an isolated closed roof silo isrecommended by EN 1993-4-1 [2] in following expression:

Cp 0:540:16 dh 0:280:04dh

cos 1:040:2 d

h

cos 2

0:360:05 dh

cos 3 0:140:05 d

h

cos 4 1

The variation of wind pressure coefficient cp around halfcircumference is plotted in Fig. 2 for various aspect ratios h/d(0.5, 0.7, 0.9 and 1.1), in which wind pressure is positive directedtowards the external surface in the adjacent zone of the origin atthe windward generator, and becomes negative directed awayfrom the external surface beyond a certain critical angle fromwindward generator origin. The critical circumferential angle crwhich distinguish the windward area (wind pressure) from lee-ward area (wind suction) is about 421, 401, 391 and 381 for siloaspect ratio h/d of 0.5, 0.7, 0.9 and 1.1, respectively. The peakpositive pressure coefficient is equal to 1.0 obtained on the wind-ward meridian, while the peak negative suction coefficient isfound to be 0.93, 1.17, 1.30 and 1.38 for silo aspect ratioh/d of 0.5, 0.7, 0.9 and 1.1 at an angle of about 831 from thewindward generator origin.

As steel silos are constructed mostly in the open countryterrain, the terrain category II [5] is assumed and applied to thefollowing numerical analyses. The basic wind velocity vb ispostulated as 20 m/s corresponding to wind grade Gale inBeaufort scale.

2.2. Bulk solid pressure

The distribution patterns and magnitudes of pressures on thesilo wall exerted by the stored granular solid have been a crucialproblem to be solved for design of silos. In the recently published

Eurocode: EN 1991-4 [1], principles for the determination ofpressures on the vertical walls of silos are definitely provided thatthe wall pressures induced by stored bulk solids during the fillingand discharge shall be evaluated according to the slenderness (theaspect ratio) of the silo. The pressures on vertical wall of siloinduced by the stored bulk solids are composed of two parts: thesymmetrical component and the non-symmetrical componentreferred to as the patch load in EN 1991-4 [1], which shall betaken to act simultaneously on the wall of silo. The symmetricalcomponent consists of horizontal pressures ph and vertical fric-tional pressures pw on the wall. For the squat silos, the fillingpressures phf and pwf are given by Eq. (2) and Eq. (3), and thedistributions of solid pressures along the depth are shown in Fig. 3.

phf z Kz0 1zh0z0h0

1

n 2

pwf z phf z 3

in which, and K are the unit weight and lateral pressure ratio ofthe particulate solid, respectively; is the wall friction coefficientfor solid sliding on the vertical wall; z0r/(2K) for circular silos,

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 45 90 135 180

Cp

h/d=0.5 h/d=0.9

h/d=0.7 h/d=1.1

cr (o)

d

wind

Fig. 2. Wind pressure variation around half circumference for various aspect ratio h/d.

d

s=0.

2d

equivalent surface

h ph

pw h0

pp

h p

Fig. 3. Schematic distribution of solid pressure on silo wall.

Table 1Design conditions assumed for the numerical analyses.

Designconditions

Actionassessmentclass

Fabrication tolerancequality class

Consequenceclass

Wallsurfaceclass

Grade AAC3 Class B high: qualityparameter Q25.

CC3 D1

Y. Zhao et al. / Thin-Walled Structures 73 (2013) 337349 339

and r is the radius of silo. h0 is the depth below the equivalentsurface to the lowest point on the wall that is not in contact withthe stored solid of the top pile, while h0 r tan r=3 for asymmetrically filled circular silo with top pile. n1tan r1h0=z0, and r is the angle of repose of the particulatesolid of conical pile.

Before pressures on vertical wall of silo can be evaluated, therelevant properties of the stored particulate solid have to be madeclear first. For commonly encountered bulk solids in industry suchas cement clinker, the characteristic value of material propertiesare: 18 kN/m3, K0.381.310.498, 0.461.070.492,and r471 which are excerpted from Table C.2, EN 1991-4 [1].The characteristic values K, are determined by multiplying ordividing the corresponding mean value by the conversion factorsa: KaKKm, am. When a limit state verification is sensitive tothe variability of a material property, upper and lower character-istic values of the material property should be taken into accountaccording to EN 1991 [1] and EN 1990 [3]. As a result, K, arechosen the upper values resulting in maximum frictional pressureswith corresponding horizontal wall pressures.

2.3. Permanent action

Permanent action is taken into account in combinations ofactions as a single action, which includes the total self-weight ofstructural and non-structural members. The cylindrical wall of siloprovides structural stiffness for resisting the meridional stress andtransverse shear, which is treated as structural element. The non-structural members of silos includes mainly the roofing system,such as the roof supporting members, the top building, thesurfacing and coverings, fixed services, etc., which is comprisedby a overall uniform load 1.5 kN/m2 acting on the roof in thegravitational direction. The self-weight of silo wall as structuralelement is accounted for automatically by the finite elementprogram, and the self-weight of roofing system as non-structuralmembers is converted to edge load uniformly acting on the topedge of silo wall, as the roof system is excluded from the numericalmodel (see Fig. 4).

2.4. Load cases

Partial factor method is implemented for limit state verifica-tions of structural equilibrium subject to combinations of actions,which is stipulated by EN 1990 [3]. When using the partial factor

method, it is verified in EN 1990 [3] that in all relevant designsituations, no relevant limit state is exceeded when design valuesfor actions or effects of actions and resistances are used in thedesign models.

Two working stages of silo are selected as representative ofload cases relating to wind pressures, which agree well with theempty silo before filling and the full silos during storage in theirservice life. The Load Cases used for numerical verification of limitstate and also proposed by EN1991-4 [1] are summarized inTable 2 together with the recommended partial factor f andcombination factor 0. The dominant action and the permanentaction are taken at their full value in each Load Case, but theaccompanying actions is reduced by the combination factor 0 toaccount for the reduced probability of simultaneous occurrence inaccordance with EN 1990 [3].

In above action combinations, the self-weight of structure isregarded as unfavorable for the full silo and favorable for theempty silo, thus the partial factor for self-weight is taken as1.1 and 0.9 for the full and empty silos respectively. The windpressure on silo external surface and solid filling pressures for fullsilos are deemed as unfavorable, thus the partial factor are bothtaken as 1.5 (EN 1990 [3]).

The partial safety factor M1 for the resistance of shell wall tostability is recommended for silos as 1.1 by EN 1993-4-1 [2], givingan overall safety amplification factor applied to the design valuesof the combination of actions for the relevant Load Case. Thissafety factor of 1.1 provides a valuable reference for load propor-tionality factor against which nonlinear numerical analyses maybe estimated to obtain a realistic measure of safety of the designedstructure.

3. Numerical analysis procedure

3.1. Finite element model

The buckling behavior of steel silos subject to wind pressure isinvestigated using the commercial general purpose finite elementcomputer package ANSYS [30]. The 8-node quadratic isopara-metric shell element SHELL93 involving both bending and mem-brane effect is used to discretize the shell wall. Each finite elementnode has six degrees of freedom (DOF), including translations inthe nodal x, y, and z directions and rotations about the nodal x, y,and z axes. The FE model is defined in the cylindrical coordinatesystem (i.e., x-radial direction, y-circumferential direction, z-mer-idional direction), including the definition of silo geometries andoutput results, except if it is specifically stated. The results arefrom top layer of the shell element, in which the meridionaldisplacement u, the circumferential displacement v are specifiedas positive in the coordinate positive direction z, y and radialdisplacement w as positive outwardly normal to the shell surface.In the light of EN1991-1-4 [5], the wind pressure distributionaround the circumferential direction possess only one axis ofsymmetry, so FE half-model of the silo structure is introduced inthe following numerical analyses, which is obtained by cutting the

Uy,

Rot

x,z

=0

Ux,y =0

Ux,y,z =0; Rotx,y,z =0

Uy,

Rot

x,z

=0

w v

u

in cylindrical CS

Fig. 4. FE half-model of circular steel silo.

Table 2Summary of combinations of actions to be considered (by EN1991-4 and EN1990).

Load Cases Dominantaction

Permanentaction

Accompanyingaction

Wind (f0) Self-weight (f0) Solid filling (f0)

WE (wind andempty silo)

1.51.0 0.91.0

WF (wind andfull silo)

1.51.0 1.11.0 1.51.0


shell wall vertically across a diametrical plane defined by theorigin of the windward generator of wind pressure. The half-model is as good as the complete model for structures whosedeformations are at least symmetric about one diametrical plane,but exhibiting more computational efficiency, as the half-modeland complete model are evidenced to show the identical structuralperformances by FE results from both models.

The base strake of silo wall is commonly anchored to the rigidbottom plate in most practical engineering, the bottom plate isherein not included in the numerical model for simplification, andall translational and rotational degrees of freedom (Ux,y,z, Rotx,y,z0) at bottom edge of silo is restrained as boundary conditions.The roof and wind girder at the top of silo are meanwhile omittedin the numerical model, so only a cylindrical shell is modeled. Theradial and tangential translational restrains (Ux,y 0) are appliedto the silo top to simulate the influence of the roof and the windgirder, as they provide realistic restraints of out-of-round defor-mation for silo top. The symmetric boundary condition (Uy, Rotx,z0) is correspondingly imposed in both the meridional cut edgesof the FE model. The finite element meshes and restrains conditionfor boundary edges of the silo wall are shown in Fig. 4.

The material of the silo wall is assumed to be isotropic elastic-perfectly plastic with properties typical of steel (grade S355): anelastic modulus E of 2.1105 N/mm2, a Poisson's ratio of 0.3 anda characteristic value of yield strength fyk of 355 N/mm2 for wallthickness no more than 40 mm (EN 1993-1-1 [6]).

3.2. Types of buckling analysis

The limit state of buckling should be taken as the condition inwhich all or part of the structure suddenly develops largedisplacements normal to the shell surface, caused by loss ofstability under compressive membrane or shear membrane stres-ses in the shell wall, leading to inability to sustain any increase inthe stress resultants, possibly causing total collapse of the struc-ture. For the determination of load carrying capacities whenchecking the buckling limit state, the following five types ofbuckling analysis are carried out for the numerical investigationwhich is also recommended by EN 1993-1-6 [7].

(1) LBAlinear elastic bifurcation analysis of the perfect silo;(2) GNAgeometrically nonlinear elastic analysis of the perfect

silo;(3) GMNAgeometrically and materially nonlinear analysis of the

perfect silo;(4) GNIAgeometrically nonlinear elastic analysis of the imper-

fect silo;(5) GMNIAgeometrically and materially nonlinear analysis of the

imperfect silo.

4. Numerical results and discussions of the reference silo

4.1. Geometry of the reference silo

In the present analyses, buckling behavior is firstly investigatedfor a flat-bottomed steel silo with tapered wall thickness. Forvarious reasons, silo walls are constructed in a diversity of formsknown as the uniform thickness, the stepwise thickness, and thetapered thickness. The thickness forms and distribution over wallheight has been proved to have significant influence on bucklingbehavior of silos under axial compression and external pressures.Previous research has been focused on walls of uniform thickness,and the results accordingly are of limited application. The taperedwall thickness for the reference silo is preferred because of itsbetter structural performance and economy in cost, which would

bring about an effectively continuous taper and avoid discontinu-ity of wall stiffness compared to the stepped wall thickness. Thereference silo to be analyzed subsequently has a diameter d of40 m, a height of silo wall h of 36 m, radius to equivalent thicknessratio r/teq of 800 and thickness ratio tratio of 0.25 for Load Case WEand Load Case WF, respectively. As a matter of convenience, theconical pile of solid at the top is not considered here, i.e. h0 isassumed to be zero in the following analyses. The aspect ratio ofthe reference silo could then be obtained: h/d0.9. The equivalentthickness teq is determined by the wall thickness at the uppermostttop and at the bottom most tbase, and in the form of teq(ttoptbase)/2, while the thickness ratio is defined by tratiottop/tbase.

Another geometrical parameter which has been proved to havemuch effect on buckling behavior of circular steel silo is the welddepression imperfection arising from the fabrication process. Inthe present analyses, the assumed shapes of weld depression areadopted and introduced to the FE model, which are proposed byRotter and Teng [31] and given by Eqs. (4) and (5).

Type A : 0 expxlc

cos

xlc sin x

lc

4

Type B : 0 expxlc

cos

xlc

5

in which, 0 is the amplitude of the weld depression, assumed forthe following analysis to be equal to wok defined by EN1993-4 [2],wok

ffiffiffiffirt

p=Q and lc

ffiffiffiffirt

p=3121=4 2:44 ffiffiffiffirtp , is the linear

meridional bending half wavelength, while t is equal to theequivalent thickness of silo wall teq.

The shape defined by Eq. (5) (Type B) is modeled in the finiteelement analysis throughout the height of silo to account for animperfect shell. The height of a normal strake is assumed to be3.0 m. For the reference silo (h36 m), eleven circumferentialweld depression imperfections along the wall height are alto-gether modeled, in which areas in the vicinity of the welddepression is discretized with denser meshes. The finite elementmeshes for the reference silo with weld imperfections are plottedin Fig. 5.

Fig. 5. FE meshes for reference silo with weld imperfection.


4.2. General buckling behavior of the reference silo

Both the linear bifurcation and nonlinear buckling analysis iscarried out for the reference silo in this section. The buckling loadfactor is determined by an amplification factor applied to thedesign values of the combination of actions for Load Case WE andLoad Case WF. The linear buckling analysis predicts the theoreticalbuckling strength (the bifurcation point) of an ideal linear elasticstructure, while in the nonlinear buckling analysis, the arc-lengthsolution technique is used to follow the structural loaddisplace-ment full course responses for both geometrically perfect andimperfect models. The loaddisplacement curves are plotted inFigs. 6 and 7 for various analysis types of the reference silo whichis calculated by selecting the monitoring node in the windwardmeridian at the critical buckling point. The vertical axis is the loadfactor being applied to the design value of combinational actions,while the horizontal axis represents the out-of-plane radial dis-placement normalized by the equivalent thickness teq of the silowall. It could be observed that the loaddisplacement curvespredict a distinct maximum load followed by a descending path,in which the maximum load is taken as the critical buckling pointcr for the equilibrium path.

In the Load Case of WE, the load factor initially increaseslinearly with the increasing radial displacement in the range ofapproximate 0.1teq before the critical point cr is reached, and theequilibrium paths of the two groups of FE model for perfect (seeFig. 6, GNA and GMNA) and imperfect shell wall (see Fig. 6, GNIAand GMNIA) are almost coincident in each plotting. Once the

critical buckling point cr is got across, the monitoring pointsuddenly turns into a large radial deflection indicating the shift ofthe equilibrium path to the postbuckling stage. The equilibriumpath subsequently descends slightly and reaches a plateau. The loadfactors from the perfect (GNA and GMNA) or imperfect (GNIA andGMNIA) shell are almost identical in each group when account istaken for the material nonlinearity. It is indicated that the bucklingresistance is almost independent of the material nonlinearity inLoad Case WE. The equilibrium path of imperfect shell (GNIA andGMNIA) in Fig. 6 shows a 21% decrease of the buckling resistancefactor R in comparison to that of the perfect shell (GNA and GMNA)in the same plotting, which means the buckling resistance, is muchsensitive to weld imperfection in Load Case WE.

When it comes to Load Case WF, the buckling behavior showedby Fig. 7 is much different. The equilibrium paths display a muchwider range of radial displacement of about 1.0teq than that in LoadCase WE, within which the loaddisplacement curve keeps linear. Itsuggests that an evident increase of strength to sustain radialdeflection gains. Both the material nonlinearity and geometricalnonlinearity effect is strong and detrimental to buckling behavior ofsteel silos, resulting in decrease of buckling resistance. The equili-brium paths of imperfect shell (GNIA and GMNIA) in Fig. 7 shows a12.5% and 7.5% respective decrease of the critical buckling point crin comparison to that of the perfect shell (GNA and GMNA) in thesame plotting. When account is taken of the material nonlinerity,the critical buckling point plotted in Fig. 7 shows a decrease of 46%and 43% respectively (GMNA and GMNIA) in contrast with that fromthe linear elastic model (GNA and GNIA).

The load carrying capacity of buckling is characterized by thebuckling resistance factor R, which is proposed by EN 1993-1-6[7], when checking the buckling limit state. For clarity of termi-nology, the buckling resistance factor R is referred to the criticalbuckling point cr which distinguishes the prebuckling stage fromthe postbuckling stage determined from the loaddisplacementequilibrium path. The buckling resistance factor R is rearrangedand plotted in Fig. 8 for various analysis types of the reference silo(h/d0.9, r/teq800) in Load Case WE and WF. It is indicated thatthe buckling resistance factor R in the Load Case WE for thereference wind velocity is much higher than that in Load Case WFfor the same buckling analysis. This means that the bucklingdesign is most likely to be governed by Load Case WF in thepresence of the wind velocity of 20 m/s for the reference silo. Onecould also discover from Fig. 8 that the GMNIA analysis gives outthe least results in both Load Case WE and WF from all proposedbuckling analysis types. The minimum buckling resistance factor isabout 1.12 from GMNIA in Load Case WF, which exactly satisfy theoverall safety factor M1 proposed by EN1993-4-1 [2]. In addition,the load factor from LBA is also presented as the critical bucklingpoint in Fig. 8 which is obtained from the lowest Eigenvalue. The

0.0

1.0

2.0

3.0

4.0

5.0

-2.0-1.5-1.0-0.50.0

GNA

GMNA

LBA

GNIA

GMNIA

w/teq

Fig. 6. Typical loaddisplacement curves for various analysis types in Load Case WE.

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0

GNA

GMNA

LBA

GNIA

GMNIA

w/teq

Fig. 7. Typical loaddisplacement curves for various analysis types in Load Case WF.

0.0

1.0

2.0

3.0

4.0

5.0

WEWF

R

LBA GNIAGNA GMNA GMNIA

Fig. 8. Buckling resistance factor R of the reference silo for various analysis types.


buckling resistance factor from GNA analysis is approximatelyequal to that from LBA analysis in both Load Cases, which indicatesthat large deformation makes a trivial change to the geometry ofshell and results in little loss of the buckling strength.

The buckling deformations shown in Figs. 9 and 10 correspondto the critical buckling points of the loaddisplacement curvesplotted in Figs. 6 and 7 for Load Case WE and WF, where diffe-rent scale factors are used in plotting the buckling modes forbetter visualization of the deformation. The buckling deformation

corresponding to the critical point cr in Load Case WE is governedby the meridional and circumferential compression which is gene-rated in the windward region of the shells having well conspic-uous circumferential waves localized at the top part of silo wall.The nonlinear buckling deformations agree well with the bucklingmode predicted by the linear bifurcation result.

The buckling modes in Load Case WF shift into the well-knownelephant-foot deformation at the bottom part of the shell wall,which is induced by the meridional compressive stress. In such

LBA GNA GNIA

GMNA GMNIA

Fig. 9. Buckling modes of the reference silo for Load Case WE.

LBA GNA GNIA

GMNA GMNIA

Fig. 10. Buckling modes of the reference silo for Load Case WF.


case, the vertical frictional pressure acting on the internal surfaceof silo wall becomes predominant to the buckling deformation.The buckling modes predicted by GMNA is slightly different fromthat predicted by other types of buckling analyses, which ischaracterized by nothing more than one meridional wave atthe shell bottom, while several meridional waves with assortedmagnitudes can be found in other types of buckling analyses. Theunsymmetrical wind pressure seems to have little impact on thebuckling deformation for a full silo, in which the buckling mode isapproximately axisymmetrical induced by the symmetrical pres-sure exerted by the stored bulk solid. The wind pressure acting onthe external surface of silo wall is much smaller compared to thepressures exerted by granular solids acting on internal surface of silowall. The positive wind pressure in the windward region is counter-balanced by part of the solid pressures and the negative windsuction is integrated with the solid pressures.

5. Parametric analyses

The buckling behavior discussed in the above section is derivedfor the reference silo with given geometry and specified loadingconditions which could be deemed as the representative of thesquat silos. Nevertheless, other relevant parameters, such as theloading conditions including the wind velocity, the patch load, andthe geometry parameters including the aspect ratio, the radius-to-thickness ratio, the wall section (uniform, stepped, or taperedwall), the wall openings etc, are actually variant and validated tohave significant impact on buckling behavior of steel silos from theviewpoint of practical design. Since the linear bifurcation analysismakes no allowance for geometrical or material nonlinearity towhich buckling behavior of steel silo is confirmed to be quitesensitive, the results from the LBA analysis could not be justified asa reliable evaluation for the practical design of silo structures. Theparametric studies in nonlinear buckling analyses (GNA, GNIA,GMNA, and GMNIA) are consequently carried out for the referencesilo from the perspective of practicality.

5.1. Wind velocity

Buckling design in case of an empty silo is generally dominatedby the wind pressure acting on external surface of silo wall. In thepreceding analyses, wind pressure for the reference silo is deter-mined by assuming a basic wind velocity vb of 20 m/s, correspond-ing to a maximum wind pressure 0.806 kN/m2 acting on the topand at the windward generator of silo wall. In most practical designof silo, the basic wind velocity vb is nevertheless far beyond 20 m/scompared with meteorological data from site survey [2628]. Thebuckling resistance factor R is plotted in Fig. 11 for various basicwind velocities in both Load Case WE and WF. It is shown that thevariations of basic wind velocity, in the Load Case WE, have muchinfluence on buckling resistance factor R and would dominate thebuckling design. The buckling resistance factor R from the GMNIA(or GNIA) analysis decreases from 3.25 to 1.11 (reduced by 65.8%)when the basic wind velocity vb increases from 20 m/s to 40 m/s,and the factor R for vb of 60 m/s continually drops and reduced by76.9% by contrast with that from vb of 20 m/s. When Load Case WFfor a full silo is considered, the buckling resistance factor R isreduced by 5.3% and 7.1% for the basic wind velocity vb 40 m/s and60 m/s respectively, in comparison to that from vb of 20 m/s, as isshown in Fig. 11. It is indicated that the wind pressure on externalsurface of silo has limited influence on buckling behavior when thesilo is full filled with bulk solids.

It is also indicated in Fig. 11 that the GMNIA results give out theleast buckling resistance factor in both Load Cases WE and WFfrom all proposed buckling analysis types. The buckling design of

the reference silo is demonstrated by the previous studies to begoverned by Load Case WF for a basic wind velocity vb of 20 m/s.But the buckling design of silo may be governed by Load Case WEfor an empty silo, if the basic wind velocity vb is continuouslyincreased. The buckling resistance factor R from the GMNIAanalysis and in Load Case WE for a basic wind velocity vb of40 m/s is about 1.11 which is approximately equal to that from theGMNIA in Load Case WF. It is worth notice that the factor R fromthe GMNIA analysis and in Load Case WE for a basic wind velocityvb of 60 m/s is far below that in Load Case WF. It could beconcluded that buckling design in Load Case WE and WF dependsgreatly on the wind velocity. A critical basic wind velocity vb,cr isfound to be about 40 m/s for the reference silo, which would tellthe buckling design from the dominant Load Case WF to dominantLoad Case WE. That is to say, the buckling design would begoverned by Load Case WE for a basic wind velocity vb greaterthan vb,cr of 40 m/s and would be governed by Load Case WF for vbless than vb,cr.

5.2. Aspect ratio

Silos with large capacities are usually squat and stocky. Thebuckling behavior of steel silos with aspect ratio h/d ranging from

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

GNA GNIAGMNA GMNIA

R

20 40 60 m/s wind velocity vb

0.0

0.5

1.0

1.5

2.0

2.5

3.0

GNA GNIAGMNA GMNIA

R

20 40 60 m/s wind velocity vb

Fig. 11. Buckling resistance factor for various wind velocities (h/d0.9, r/teq800,Tratio0.25). (a, b) Load case.


0.5 to 1.1 is investigated in Load Case WE and WF in this section, asis shown in Fig. 12. The buckling resistance factor R decreasesgreatly with the increasing aspect ratio h/d in each analysis type,which shows great influence of aspect ratio on the bucklingbehavior of steel silos.

In Load Case WE, the GMNIA analysis for aspect ratio h/d of0.5 presents a buckling resistance factor R which is about 5.3 timesthat of the reference silo (h/d0.9), but the results for h/d of1.1 shows a reduction about 8.5% compared to that of the referencesilo. This suggests that the buckling resistance of an empty silounder wind pressure increases greatly with the decrease ofthe aspect ratio h/d. The beneficial effect originates mainly fromthe following two factors: the distribution of wind pressure onexternal surface of silo wall is specified by the reference height ze[5] which is regarded as the maximum height of the designed steelsilos, thus the positive wind pressure on the windward meridiandecreases from 0.806 kN/m2 to 0.703 kN/m2 on the one hand, onthe other hand, the lateral stiffness to wind pressure multiples forsilo structure with an invariant radius-to-thickness ratio when theaspect ratio decreases from 0.9 to 0.5.

In Load Case WF, the buckling resistance factor R for aspectratio h/d of 0.5 from the GMNIA analysis is about 55% higher thanthat obtained from the reference silo (h/d0.9), but the results forh/d of 1.1 shows a reduction about 21.5% compared to that of thereference silo. This suggests that the buckling resistance of a fullsilo under wind pressure decreases greatly with the increase of theaspect ratio h/d. The unfavorable effect originates mainly from the

following fact that the meridional compressive stress in the silowall results with the depth from the accumulation of the frictionalpressure exerted from the stored granular solid, so that thebuckling stress is substantially increased towards the base of thesilo when the height of silo wall increases.

5.3. Radius-to-thickness ratio

The effects of radius-to-thickness ratio r/teq ranging from 500 to1000 on buckling resistance factor R are explored, as is shown inFig. 13. The buckling resistance factor R decreases greatly with theincreasing r/teq ratio in each analysis type, which shows greatinfluence of r/teq ratio on the buckling behavior of steel silos. InLoad Case WE, the factor R in both GNA and GNIA analysis for r/teqratio of 500 is about 3.1 times that of the reference silo (r/teq800), but the result for r/teq ratio of 1000 is reduced by about43% in comparison to that of the reference silo. It is also revealedin the Fig. 13 that the results from GNA (or GNIA) analysis arealmost identical to that from GMNA (or GMNIA) analysis implyinglittle influences of material nonlinearity on buckling resistance ofsilo for diversity of r/teq ratios. In Load Case WF, both thegeometrical and material nonlinearity have detrimental influenceon buckling resistance of silo, where the factor R from the GMNIA

0

5

10

15

20

GNA GNIAGMN A GMNIA

R

0.5 0.7 0.9 1.1 h/d

0.0

1.5

3.0

4.5

6.0

GNA GNIAGMN A GMNI A

R

0.5 0.7 0.9 1.1h/d

Fig. 12. Buckling resistance factor for various aspect ratios (r/teq800, Tratio0.25).(a, b) Load case.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

GNA GNIAGMNA GMNIA

R

500 800 1000r/teq

0.0

1.0

2.0

3.0

4.0

5.0

6.0

GNA GNIAGMNA GMNIA

R

500 800 1000r/teq

Fig. 13. Buckling resistance factor for various r/teq ratios (h/d0.9, Tratio0.25).(a, b) Load case.


analysis has the least value in each r/teq ratio. The factor R inGMNIA analysis for r/teq ratio of 500 is about 1.7 times that of thereference silo, while the result from r/teq ratio of 1000 is reducedby about 28% with respect to that of the reference silo.

5.4. Stepped and uniform wall thickness

The preceding investigation is carried out by assuming that thewall thickness of steel silo is tapered from the bottom to the top.This assumption is theoretically ideal, whereas silo strakes withdifferent thicknesses are joined with welds to construct thecomplete structure in most practical engineering, forming theso-called stepped wall thickness. So that it is of great significanceto perform buckling analysis of silo structures with stepped wallthickness. The buckling behavior of steel silo with stepped wallthickness is evaluated in this section. The results will also becompared with that from the tapered and uniform wall thicknessunder the circumstances that the three types of wall section sharethe same equivalent thickness teq equal to 25 mm and have thesame aspect ratio h/d of 0.9. In the case of the stepped wall silo, theequivalent thickness teq is defined in the following form (Eq. (6)),which is deemed as an economic index meaning equal steelconsumption in the three types of wall thickness.

teq hitihi6

in which, hi and ti is the height and thickness of the ith wall strake.For the convenience of comparison, two groups of thickness forstepped wall are designated, as are noted Group A and B in Table 3with thickness ranging from top strake t1 to bottom strake t12.

The effects of different types of silo wall thickness on bucklingresistance factor R are demonstrated in Fig. 14 for diverse bucklinganalysis types. In Load Case WE, the factor R from the uniformwall thickness has the most numerical value among them, which isapproximate 2.0 times that from the least results of the steppedwall thickness (Group B) in the GNA and GNIA analysis. It is alsoindicated that material nonlinearity has little impact on thebuckling behavior of the reference silo in various types of wallthickness. Whereas it is reversed in Load Case WF, the factor Rfrom the uniform wall thickness has the least numerical value,which is only 0.52 time that from the stepped wall thickness(Group A) in the GMNIA analysis (reduced by 48%). Furtherscrutinizing Fig. 14a and b show that the buckling resistance factorfor stepped wall thickness (Group A) obtains the largest value inLoad Case WF and meanwhile satisfies the design requirement inLoad Case WE. As a result, the thickness arrangement scheme:Group A is the optimal one of all.

The buckling deformation in Load Case WE for an empty siloprimarily arises at the windward meridian of the upper region (seeFig. 9), while it arises at the lower region in the Load Case WF (seeFig. 10). The uniform wall thickness of 25 mm is the largest at theupper region and the smallest in the lower region among the threethickness types. This could explain why the buckling resistancefactor R for a uniform thickness silo is the largest in Load Case WEbut the smallest in Load Case WF. Some structural measures couldbe taken, such as thicken the wall section in the upper or lowerregion, in order to enhance the buckling resistance to the windpressurized empty or full filled silos. Under the condition of the

same material consumption, different schemes for wall thicknessarrangement is suggested to be lay out and selections are sup-posed to be made in buckling design of steel silos to achieve theoptimal buckling strength.

5.5. Patch load

Large steel silo has often been advised to be concentricallyfilled and discharged to avoid the unsymmetric patch load on thewall if at all possible. However, patch load resulting from theeccentric filling and discharge is difficult to avoid due to manypractical considerations, such as the opening, multiple outlets, thesegregation of stored solids, the ease of access, etc. The effects ofpatch loads have been found to be very detrimental to bucklingstrength of slender silos, but their effects on squat silos arecommonly deemed to be quite small or even ignored by EN1991-4 [1]. In spite of the above existing regulations, the quanti-tative analysis has not been carried out, on the one hand, todiscover the extent to which the influence of patch load on bulkingbehavior of large squat silos could be left out; on the other handthe unsymmetrical patch load on the large steel silo is inevitableduring filling or discharging of the stored bulk solids, whichshould be considered to act on any part of the silo wall. In theviewpoint of safe design, the effects of patch load on bucklingresistance factor R are investigated for various patch load posi-tions hp (the height from the base of the silo wall to the center ofthe patch load, see Fig. 3) for the reference silo in this section.Buckling analyses in form of four types (GNA, GNIA, GMNA,GMNIA) are undertaken by comparing two groups of wall

Table 3Groups of stepped wall thickness.

Thickness t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 teq

Group A 8 10 12 16 20 24 28 30 36 36 40 40 25Group B 6 8 10 16 24 28 28 32 32 36 40 40 25

0.0

2.0

4.0

6.0

8.0Taper Group AGroup B Uniform

R

GNIA GNA GMNA GMNIA

0.0

0.5

1.0

1.5

2.0

2.5

Taper Group AGroup B Uniform

R

GNIA GNA GMNA GMNIA

Fig. 14. Buckling resistance factor for various types of wall thicknesses (h/d0.9,r/teq800, teq25 mm). (a, b) Load case.


pressures: one for the reference silo subject to the symmetricalsolids pressure (defined by Eqs. (2) and (3)) plus the eccentricfilling patch load acting in several typical height hp (hp0.2h, 0.4h,0.6h and 0.8h) and the other for the reference silo subject to onlythe symmetrical solids pressure. The non-symmetrical patch loadppfs takes the form of cosine relation with the circumferentialangle , which is taken to act over a height s, but to extend from amaximum outward pressure on one side of pp to an inwardpressure pp on the opposite side, as is plotted in Fig. 3. The fillingpath load ppfs is assumed to be 0.2phfcos() proposed by EN 1991-4 [8], in which is the patch load magnifier 14.0e/d, and e isthe maximum filling eccentricity e0.25d. In the above expression

of ppfs, the patch loads around the circumferential direction alsopossess only one axis of symmetry, so the patch load is assumed toshare the same symmetric axis with the wind pressure in the FEanalyses.

The results are illustrated in Fig. 15 which shows that the effectof patch load on buckling behavior of the reference squat silo isrelatively small. It is shown that the buckling resistance factor R inGNA and GNIA analysis makes not much discrepancy for hpranging from 0.2h to 0.8h, which is reduced by a ratio less than4% in comparison to that of silo subject to only symmetricalpressure. The GMNIA analysis for patch load position hp of 0.2hpresents the least buckling resistance factor R with a reduction of18.7% with respect to that of silo without the patch load. Inaddition, the patch load position hp has certain effect on the factorR, which is a little larger for patch load acting on the higher heightof silo (e.g., hp0.8h, with a reduction of 9.8% in GMNIA analysis)than that on the lower position (e.g., hp0.2h). The bulking resultsconfirms the validity of the design concept proposed by EN 1991-4[1] that For squat silos in all Action Assessment Classes, the fillingpatch load need not be considered..

The deformations corresponding to the critical buckling pointare plotted in Fig. 16 for the reference silo with patch load heighthp of 0.6h. The buckling mode in each analysis type is similar tothat for silo without patch load shown in Fig. 10 except that theunsymmetrical deformation in the position which the patch loadacts is distinctly observed, which is concave in the positive patchload region and convex in the negative patch load region.

5.6. Opening

Squat silos with large capacities are often ground-supportedwith a great deal of outlets designed at the flat bottom for thedischarge of the granular solids. During the discharge, some storedsolids in the corner of silo which are far enough from the outletcould not be discharged in the normal way, giving rise to the

hp

0.4h0.0

0.5

1.0

1.5

2.0

2.5GNA GNIA GMNA GMNIA

R

Withoutpatch load

0.2h 0.8h0.6h

Fig. 15. Buckling resistance factor for various patch load positions in Load Case WF(h/d0.9, r/teq800, Tratio0.25).

GNA GNIA

GMNA GMNIA

Fig. 16. Buckling modes with patch load height hp of 0.6h for various analysis types (Load Case: WF, h/d0.9, r/teq800, Tratio0.25).


obsolete material at the silo bottom. In order to clear out theobsolete material, the silo wall, as a rule, is equipped withrectangular openings at the bottom for the convenience of theentrance and exit of the clearing machine.

In this section, attention is focused on the effects of largeopenings in the wall on buckling resistance of silo. The opening inthe wall makes the distribution of the stress and deformationdiscontinue, resulting in the stress concentration at the openingcorner, and the decrease of the buckling resistance of the silo wall.Usually, Openings in the wall of the silo are reinforced by thevertical and horizontal stiffeners adjacent to the opening whichcould decrease the corner concentrated stress on the one hand,and enhance the buckling strength of silo to avoid local instabilityof silo wall on the other hand. The opening in the analysis isassumed to be rectangular with a common dimension of 4.0 mwidth and 6.0 m height. The schematic drawing of stiffenerarrangement adjacent to the opening is shown in Fig. 17, wherethe Box section B30020014 is used for longitudinal stiff-eners and the Box section B20015010 is used for transversestiffeners.

The results are presented in Fig. 18a and b for Load Case WEand WF. In Load Case WE, the buckling resistance factor R for silowall with opening and without stiffener is reduced by about 3%with respect to that for silo wall without opening in GMNIAanalysis, while the factor is almost equal to that of silo withopening and stiffener. It is indicated that the bottom opening showlittle influence on the buckling behavior of wind pressurizedempty silo, by reason that the buckling deformation in Load CaseWE arise mainly in the upper area of the windward meridian, as isshown in Fig. 9. In Load Case WF, the buckling resistance factor Rfor silo wall with opening and stiffener is approximately equal tothat of silo without opening, while the fact for silo wall withopening and without stiffener is reduced by about 25% withrespect to that for silo wall without opening in GMNIA analysis.Similar conclusions could also be drawn from other types ofbuckling analyses. It is indicated that the bottom opening showmuch influence on the buckling behavior of a full silo, by reasonthat the buckling deformation in Load Case WF arise mainly in thelower area of the silo wall, as is shown in Fig. 10. The lost of

buckling resistance induced by the bottom opening could becompensated for by the reinforcement of stiffeners adjacent tothe opening.

6. Conclusions

This paper has presented a comprehensive study of the buck-ling behavior of large circular steel silos subject to wind pressure.The analyses are demonstrated in accordance with Eurocode, byconsidering two working stages of silos in their service life whichis closely relating to wind pressures: Load Case WE (wind andempty silo) and Load Case WF (wind and full silo). The followingconclusions can be drawn from these investigations:

(1) The buckling behavior in Load Case WE, is almost indepen-dent of the material nonlinearity, while the buckling resis-tance and buckling mode is much sensitive to weldimperfection. In Load Case WF, both the material nonlinearityand geometrical nonlinearity effect is strong and detrimentalto buckling behavior of steel silos, resulting in decrease ofbuckling resistance.

(2) The buckling deformation corresponding to the critical pointin Load Case WE is governed by the meridional and circum-ferential compression which is generated in the windwardregion of the shells having conspicuous circumferential

1.5 1.54.0

opening

(4.0x6.0)

2.0

2.0

2.0

2.0

transverse stiffener

long

itudi

nal s

tiffe

ner

Fig. 17. Schematic drawing of stiffener arrangement for bottom opening (unit: m).

0.0

0.5

1.0

1.5

2.0

2.5

GNA GNIA

GMNA GMNIA

Without openning

Opening without stiffener

Opening with stiffener

R

Without opening

Opening without stiffener

Opening with stiffener

0.0

1.0

2.0

3.0

4.0

5.0

GNA GNIA GMNA GMNIA

R

Fig. 18. Buckling resistance factor for silos with bottom opening (h/d0.9, r/teq800, Tratio0.25). (a, b) Load case.


waves localized at the top part of silo wall. The bucklingmode in Load Case WF takes the form of the well-knownelephant-foot deformation at the bottom part of the shellwall, which is induced by the meridional compressive stress.

(3) Buckling design of steel silo considering Load Case WE andWF depends greatly on the wind velocity. The bucklingdesign of steel silo is governed by Load Case WF for arelatively low wind velocity, while it may be governed byLoad Case WE for an empty silo, if the wind velocity iscontinuously increased.

(4) Buckling behavior of steel silo subject to wind pressure variesgreatly with the variation of aspect ratio h/d and the radius-to-thickness ratio r/teq. The buckling resistance of steel silounder wind pressure increases greatly with the decrease ofthe aspect ratio h/d and the ratio r/teq.

(5) The type of wall thickness arrangement plays an importantrole in buckling behavior of steel silo. Under the condition ofthe same material consumption, the reasonable thicknessarrangement is recommended to be laid out and selectionsare supposed to be taken in buckling design of steel silos toachieve the optimal buckling strength.

(6) The bottom opening shows little influence on the bucklingbehavior of an empty silo subject to wind pressure. Whereas,the effects of bottom opening on a full filled silo is fairlyobvious, and the lost of buckling resistance induced by theopening could be compensated for by the reinforcement ofstiffeners adjacent to the opening.

Acknowledgments

The authors gratefully acknowledge the support of the NationalNatural Science Foundation of China (No. 51378459 and No.50778159) and the Key Science and Technology Innovation TeamProgram of Zhejiang Province, China (No. 2010R50034).

References

[1] EN 1991-4. Actions on structuresPart 4: Silos and tanks. European Standard;2006.

[2] EN 1993-4-1. Design of steel structuresPart 4-1: silos. European Standard;2007.

[3] EN 1990. Basis of structural design. European Standard; 2002.[4] EN 1991-1-1. Actions on structures-Part 1-1: general actions: densities,

self-weight, imposed loads for buildings. European Standard; 2002.[5] EN 1991-1-4. Actions on structuresPart 14: general actions-Part 1-4, wind

actions. European Standard; 2005.[6] EN 1993-1-1. Design of steel structuresPart 1-1: general rules and rules for

buildings. European Standard; 2005.

[7] EN 1993-1-6. Design of steel structures-Part 16: strength and stability ofshell structures. European standard; 2007.

[8] EN 1991-4. Basis of design and actions on structures-Part 4: actions in silosand tanks. European standard; 1996.

[9] Macdonald PA, Kwok KCS, Holmes JD. Wind loads on circular storage bins,silos and tanks: I. Point pressure measurements on isolated structures. Journalof Wind Engineering and Industrial Aerodynamics 1988;31:16588.

[10] Purdy DM, Maher PE, Frederick D. Model studies of wind loads on flat-topcylinders. Journal of the Structural Division, ASCE 1967;93:37995.

[11] Sabransky IJ, Melbourne WH. Design pressure distribution on circular siloswith conical roofs. Journal of Wind Engineering and Industrial Aerodynamics1987;26:6584.

[12] Portela G, Godoy LA. Wind pressures and buckling of cylindrical steel tankswith a conical roof. Journal of Construction Steel Research 2005;61(6):786807.

[13] Portela G, Godoy LA. Wind pressures and buckling of cylindrical steel tankswith a dome roof. Journal of Construction Steel Research 2005;61(6):80824.

[14] Chen Lei, Michael Rotter J. Buckling of anchored cylindrical shells of uniformthickness under wind load. Engineering Structures 2012;41:199208.

[15] Chen L, Rotter JM, Doerich C. Buckling behaviour of cylindrical shells ofstepwise wall thickness under uniform external pressure. Engineering Struc-tures 2011;33(12):35708.

[16] Sosa Eduardo M, Godoy Luis A. Challenges in the computation of lower-boundbuckling loads for tanks under wind pressures. Thin-walled Structures 2010;48:93545.

[17] Schneider W, Zahlten W. Load-bearing behaviour and structural analysis ofslender ring-stiffened cylindrical shells under quasi-static wind load. Journalof Construction Steel Research 2004;60:12546.

[18] Kundurpi PS, Savamedam G, Johns DJ. Stability of cantilever shells underwind loads. Journal of the Engineering Mechanics Division, ASCE 1975;10(5):51730.

[19] Johns DJ. Wind-induced static instability of cylindrical shells. Journal of WindEngineering and Industrial Aerodynamics 1983;13:26170.

[20] Uematsu Y, Uchiyama K. Deflection and buckling behavior of thin, circularcylindrical shells under wind loads. Journal of Wind Engineering andIndustrial Aerodynamics 1985;18:24561.

[21] Jerath S, Sadid H. Buckling of orthotropic cylinders due to wind load. Journal ofEngineering Mechanics, ASCE 1985;111(5):61022.

[22] Schmidt H, Binder B, Lange H. Postbuckling strength design of open thin-walled cylindrical tanks under wind load. Thin-walled Structures 1998;31:20320.

[23] Greiner R, Derler P. Effect of imperfections on wind-loaded cylindrical shells.Thin-walled Structures 1995;23:27181.

[24] Godoy LA, Flores FG. Imperfection sensitivity to elastic buckling of wind loadedopen cylindrical tanks. Structural Engineering and Mechanics 2002;13(5):53342.

[25] Pircher M. The influence of a weld-induced axi-symmetric imperfection onthe buckling of a medium-length silo under wind loading. Journal of Solidsand Structures 2004;41:5595610.

[26] Jaca Rossana C, Godoy Luis A. Wind buckling of metal tanks during theirconstruction. Thin-walled Structures 2010;48:4539.

[27] Flores FG, Godoy LA. Buckling of short tanks due to hurricanes. EngineeringStructures 1998;20(8):75260.

[28] Godoy LA. Performance of storage tanks in oil facilities following HurricanesKatrina and Rita. Journal of Performance of Constructed Facilities, ASCE2007;21(6):4419.

[29] Pavlovic P, et al. The testing and repair of steel silo. Construction and BuildingMaterials 1997;11:35363.

[30] ANSYS. ANSYS user's manual. Houston: Swanson Analysis Systems Inc.; 2008.[31] Rotter JM, Teng JG. Elastic stability of cylindrical shells with weld depressions.

Journal of Structural Engineering, ASCE 1989;115(5):124463.

http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref1http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref1http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref2http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref2http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref3http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref4http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref4http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref5http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref5http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref6http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref6http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref7http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref7http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref8http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref8http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref9http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref9http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref9http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref10http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref10http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref11http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref11http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref11http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref12http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref12http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref12http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref13http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref13http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref14http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref14http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref15http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref15http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref15http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref16http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref16http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref16http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref17http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref17http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref17http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref18http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref18http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref18http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref19http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref19http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref20http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref20http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref20http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref21http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref21http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref22http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref22http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref22http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref23http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref23http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref24http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref24http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref24http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref25http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref25http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref25http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref26http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref26http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref27http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref27http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref28http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref28http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref28http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref29http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref29http://refhub.elsevier.com/S0263-8231(13)00209-7/othref0005http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref30http://refhub.elsevier.com/S0263-8231(13)00209-7/sbref30Buckling design of large circular steel silos subject to wind pressureIntroductionLoad casesWind pressureBulk solid pressurePermanent actionLoad casesNumerical analysis procedureFinite element modelTypes of buckling analysisNumerical results and discussions of the reference siloGeometry of the reference siloGeneral buckling behavior of the reference siloParametric analysesWind velocityAspect ratioRadius-to-thickness ratioStepped and uniform wall thicknessPatch loadOpeningConclusionsAcknowledgmentsReferences

(2013) buckling design of large circular steel silos subjeted to wind pressure

Documents

silo wall

buckling deformation

buckling mode

load case wf

silo withsmall

types of buckling analyses

behavior of steel silos

wf wind