Journal of Constructional Steel Research 67 (2011) 1545 –1553
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Journal of Constructional Steel Research
Steel silos with different aspect ratios: II
—
behaviour under eccentric discharge
A.J. Sadowski ⁎, J.M. Rotter The University of Edinburgh, Scotland, UK
a r t i c l e
i n f o
Article history: Received 13 December 2010 Accepted 28 March 2011 Available Avail able online 4 May 2011 Keywords: Eccentric discharge Silos, aspect ratio Shell buckling Finite element analysis Shells under unsymmetrical pressures
a b s t r a c t
The phen phenome omenon non of ecce eccentr ntric ic dis discha charge rge is wid widely ely reco recogni gnised sed as the mos mostt dan dangero gerous us con condit dition ion forthinforthin-wal walled led metal silos and the cause of many catastrophic buckling failures. A realistic pressure model for this condition appear app earss in a regu regulat latingstanda ingstandard rd forthe firsttime in EN 199 1991-4 1-4 (20 (2006)on 06)on Act Action ionss on Sil Silos os andTanks.Howev andTanks.However er the str structu uctural ral con conseq sequen uences ces of itsapplic itsapplicati ation on arecurren arecurrently tly lar largel gely y unk unknow nown. n. The beh behavi aviourof ourof a sil silo o sub subject jected ed to these pressures is certainly very dependent on the aspect ratio of the silo, the granular solid properties and the discharge channel geometry. This paper explores the behaviour of four thin-walled cylindrical silos with stepwise-varying wall thickness and aspect ratios varying from intermediate to very slender, subject to the codified EN 1991-4 eccentric discharge pressures. It is shown that a silo design that was found to be very safe under the EN 1991-4 concent con centric ric dis discha charge rge pres pressur sures es beco becomesvery mesvery uns unsafeunder afeunder ecce eccentr ntric ic dis discha charge.Furthe rge.Further, r, as it is kno known wn tha thatt the aspect ratio has an important effect on the flow pattern in discharging granular solids, and that slender silos exhibit very different flow patterns from squat silos, it is currently not certain whether a suitable range of aspect ratio over which the codi fied eccentric discharge model is to be applied has been prescribed in the standard. This paper is the second of a pair. In the first, the behaviour of a set of example silos under the EN 1991-4 concentric discharge conditio condition n was studied. The same example silos silos are are studied here under eccentric eccentric discharge. discharge. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Thephenomenonof Thephenome nonof ec eccen centri tricc dis disch charg arge e is wid widelyunder elyundersto stood od to be the most serious loading condition for a thin-walled metal silo, and the cause of many catastrophi catastrophicc buck buckling ling failures failures in the past. The associated assoc iated patterns of normal pressures pressures and frict frictional ional tractions exerted exer ted by the ecce eccentri ntricall cally y flowing granular solid are known to produce produ ce very unsymmetri unsymmetrical cal patterns of stre stresses sses in the silo wall, which precipitate early buckling failure [1–5] 5].. With recent advances in the power of both computers and the finite element method, it is now possible possible to unde undertake rtake nonlinear nonlinear analyses analyses of thin metal silos under complex load patterns that were extremely dif ficult only a decade ago. In thecompa thecompanio nion n pap paper,it er,it wasstate wasstated d tha thatt the cla classi ssificat cation ion of sil silos os in the European standard standard EN 1991-4 [6] on Actio Actions ns on Silos and Tank Tankss is made on the basis of their aspect ratio (height to diameter, H /D), which whic h grea greatly tly influenc uences es the dist distribut ribution ion of axis axisymme ymmetric tric pres pressure suress in the silo under mass flow. The aspect ratio is also known to have an important influence on the possible patterns of mixed and pipe flow (Fig Fig.. 1), wit with h squ squat at sil silos os hav having ing si signi gnificant cantly ly diff differen erentt pipe flow regimes from slender ones [2,6,7] [2,6,7].. The flow pattern in turn in fluences thepressure thepress uress exe exerte rted d by bot both h thestati thestaticc and flowin owing g solid compo component nentss
⁎
Corresponding author. Tel.: +44 131 650 6781. E-mail address:
[email protected] (A.J. Sadowski).
0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.03.027 doi: 10.1016/j.jcsr.2011.03.027
on the silo wall, and thus the structural behaviour of the silo. The numero num erous us stu studie diess tha thatt tri tried ed to pre predic dictt the pre pressu ssures res in the sil silo o numerically based on an assumption of a particular flow pattern (e.g. [8–11] 11])) have above all demonstrated the great dif ficulty involved in doing so reliably. 2. The EN 1991-4 eccentric discharge pressure model
Therecent Therece nt Eur Europe opean an sta standa ndard rd EN 199 1991-4 1-4[6] [6] provi provides des a reas reasonab onably ly realistic pres realistic pressure sure dist distribut ribution ion for silos under ecce eccentric ntric disc discharg harge e (Fig. 2). 2). This model assumes a parallel-sided flow channel (No. 4 in Fig. 1) 1) with a truncated circular cross-section cross-section forming against the silo wall throughout the height of the silo, termed eccentric parallel pipe flow [2] [2].. It is prescribed for application as a separate load case on all but the very squattest of silos ( H /D N 0.4) if the capacity is large enough or where large filling or discharge eccentricities eccentricities are expected. fi The size of the channel is de ned in terms of the ratio of the flow channel radius to full silo radius, kc = r c/R. EN 1991-4 recommends that three specific channel sizes be investigated, kc =0.25, 0.40 and 0.60, though a National Annex may prescribe different values. In the EN 1991-4 model, the ‘flowing’ solid exerts a low normal pressure on the silo wall at the centre of the flow channel ( phce) and a high normal pressure at the ‘edges’ of the flow channel ( phae), based on a si simp mpli lified inter interpret pretation ation of expe experime rimental ntal obser observatio vations ns [1,12–14] 14].. The high edge edge pressures pressures were chosen chosen so that the integral integral of the rise in pressure pres sure is equal equal to that of the fall fall in in pressur pressure. e. The ‘static’ solid is then
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A.J. Sadowski, J.M. Rotter / Journal of Constructional Steel Research 67 (2011) 1545 –1553
2
Key: 1) Internal (taper) pipe flow 2) Mixed flow 3) Eccentric taper pipe flow 4) Eccentric parallel pipe flow
3
Effective transition Flow channel boundary
1
Flow channel boundary
Flowing
Flowing
Flowing
Flow channel boundary
4
Effective transition Flow channel boundary
Effective hopper
Stationary
Stationary
1 Stationary
Stationary
Stationary
Retaining silo
Stationary
Stationary
Squat silo
Stationary
Slender silo
Very slender EN 1991-4 silo eccentric discharge flow model
Fig. 1. Aspect ratio effects in different flow patterns, after EN 1991-4 [6].
assumed to follow either the axisymmetric Janssen or modified Reimbert pressure distribution ( phse), implying that the Janssen equilibrium condition must be met irrespective of the non-uniformity of the pressures. This makes the pressures dependent on the silo aspect ratio but independent of the size of the flow channel. The EN 1991-4 model is a simplified version of the original derivation of Rotter [1,3], where the static solid pressure was instead derived rigorously by mechanics, making it a function of the flow channel size. Further, for each normalpressurecomponent there is a corresponding frictional traction following the relation pw = μ ph (i.e. pwce, pwae and pwse) where μ is the fully-developed wall friction, taken as the lower characteristic value in EN 1991-4 to emphasise the unsymmetrical nature of the normal pressure component. As noted above, the EN 1991-4 eccentric discharge pressure pattern is based on a parallel-sided flow channel throughout the height of thesilo. This is actually unlikely to occur because thechannel size must reduce as it approaches the outlet, and also usually spreads out somewhat near the surface [2]. Nonetheless, the channel has been defined with straight vertical sides (No. 4 in Fig. 1) in EN 1991-4 to produce a simple model for design calculations. In slender silos, the effects of this error are con fined to a small part of the structure, but in squatter silos this error covers a significant part of the silo and results in quite unrealistic imposed pressure patterns, especially when combined with themodified Reimbert distribution for static pressures ( phse).
The structural response of silo structures of different geometry to this pressure pattern remains largely unknown because the pattern was only recently codi fied and very few experiments have ever been conducted to explore this failure mode. It is probable that the computational prediction of the structural behaviour depends on the geometry of the silo, the size and position of the flow channel, the assumed material properties of the granular solid and the type of computational analysis. The present study is an investigation of the most influential of these factors: the silo aspect ratio. The first known studies of the EN 1991-4 eccentric discharge model are those of Sadowski & Rotter [4,5], who performed a full set of computational analyses according to EN 1993-1-6 [15]: LA, LBA, MNA, GNA, GMNA, GNIA andGMNIA, thede finitions of which may befound in the companion paper. They studied only metal silos with slender aspect ratios (H /D N 2) with stepped walls and always assuming the largest recommended flow channel size of kc = r c/R =0.60. They showed that the mechanics of behaviour of the shell under this load pattern is highly complex and that the predicted buckling modes correspond well to those observable in the field. Rather surprisingly, a slender silo that is subject to the EN 1991-4 eccentric discharge pressures was found to exhibit a higher buckling strength when analysed with a geometrically nonlinear analysis (GNA) than with a linear bifurcation analysis (LBA). On average, the lowest GNA load proportionality factor wasfound to be approximately 44% higher than the lowest LBA linear buckling eigenvalue. However, these explorations were clearly very limited in Increased channel edge pressures, phae
Static Janssen/Reimbert pressures, phse
Decreased flow channel pressures, phce R
ec r c
ψ θ c
Section modelled with FEA
θ c
R – silo radius ec – eccentricity of flow channel r c – flow channel radius θ c – wall contact angle w.r.t silo centre ψ – wall contact angle w.r.t channel centre k c = r c / R – flow channel size
Fig. 2. Circumferential cross-section of eccentric flow channel horizontal pressures, after EN 1991-4 [6].
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A.J. Sadowski, J.M. Rotter / J ournal of Constructional Steel Research 67 (2011) 1545 –1553 Table 1 Summary of flow channel properties for any of the four design silos. kc = r c/R ec/R θc/π (%) Ac /Atot (%)
Dimensionless flow channel size Dime nsionles s eccen tricity Percentage perimeter contact Percentage channel area
0.25 0.808 5.7 5.8
0.40 0.688 9.7 14.8
0.60 0.517 16.0 33.4
thatthey studied only a single flow channel size andtwo aspect ratios in the slender range which, as will be shown in this paper, are far from representing the fullrangeof possible behavioursunderthis particularly complex load condition.
slender silo. This issue was not considered in the authors' earlier work [4,5] and was not discussed by the code drafting committee for EN 1991-4. The geometryof theEN 1991-4eccentric discharge pressure model is shown in Fig. 2. The flow channel wall contact angles θc and ψ (Eqs. (1)and(2)), thearearatio Ac/ Atot (Eq. (3)) and the dimensionless eccentricity ec/R (Eq. (4)) are each a function of the assumed dimensionless size of the channel kc = r c/R and the friction properties of the solid and the silo wall only. The general definition of the flow channel geometry is thus independent of the aspect ratio of the silo. 1 R e 2 1−kc + c 2 ec R
−1
θc = cos
3. Scope of the present study
The companion paper introduced the design of five example silos with varying aspect ratios in the range 0.65 ≤ H /D ≤ 5.20, linked by a common storage capacity of 510 m3. The silos were assigned identi fication acronyms based on their slenderness category according to EN 1991-4 [6]: ‘squat’ SiloQ(H /D =0.65), ‘intermediate’ SiloI(H /D =1.47), ‘boundary’ Silo B (H /D =2.06), ‘slender’ Silo S (H /D =3.00) and ‘very slender’ Silo VS (H /D =5.20). The reader may consult the companion paper to find full details of the structural design, modelling procedure andsubsequent analysis, which areemployedagainin thepresent study of eccentric discharge. The aspect ratios of all but the squattest of the example silos were chosen to be in a range where an eccentric pipe flow pattern might be physically possible (Nos 3 and 4 in Fig. 1). Eccentric pipe flow is known to occur in slender silos storing densely packed or slightly cohesive solids [2,16], but it is no longer credible in squat silos which exhibit fully internal and mixed flow patterns where the channel spreads progressively outwards from the outlet(Nos 1 and 2 inFig. 1). Consequently, the very squat Silo Q (H /D =0.65) was omitted from the present study. Nonetheless, it will be shown here that the structural behaviour of a silo of low aspect ratio under the EN 1991-4 eccentric pipe flow model is significantly different from that of a Concentric, k c = 0.00
k c
−1
ψ = sin
ð1Þ
1 sinθc kc
ð2Þ
The angle ψ approaches 90° when the wall is very smooth [1]. Ac = Atot =
ec = R =
(
1 π
n
o
2
ðπ−ψÞkc + θc −kc sinðψ−θc Þ
μ lower ð1−kc Þ + tanϕi;upper
1−
μ lower tanϕi;upper
ð3Þ
!q ffi ffi ffi ffi ffi ffi )
k c
= 0.40
k c
= 0.60 ec
ec ec R
r c
r c
θ c ψ
r c
Fig. 3. Comparison of the geometry of four different EN 1991-4 discharge conditions (independent of aspect ratio), drawn to scale.
Critical location #1 : Axial membrane stress resultant High (but usually not Channel highest) compression causes edge a ‘midheight’ buckle across the flow channel; predominantly elastic, well reported in service Critical location #2 : Opposite the Very high compression channel may cause a buckle adjacent to the ‘edge’ of the flow channel at the bottom of the silo; predominantly plastic, no known observations Compressive
ð4Þ
where kc = r Rc . In the present study, Silos VS, S, I and B were analysed under each of the three flow channel sizes recommended by EN 1991-4: kc = r c/R =0.25, 0.40 and 0.60. The values of the dimensionless geometric parameters are summarised in Table 1 and Fig. 3, assuming the same granular solid properties as those initially used in design. For the smallest flow channel ( kc = 0.25), the region of low pressure covers less than 6% of the wall perimeter, while for the largest flow channel (kc =0.60) it rises only to 16%. Thus all three recommended channel sizes have a relatively small wall contact perimeter.
= 0.25
Centre
1−kc
Stationary solid (high pressure) Flowing solid (low pressure) Flow channel centreline & axis of symmetry
Channel centre
Tensile
Fig. 4. Illustration of the critical locations under eccentric pipe
flow
in slender silos.
