The Structural Design of Light Gauge Silo Hoppers

Missouri University of Science and Technology

Scholars' Mine International Specialty Conference on ColdFormed Steel Structures

(1988) - 9th International Specialty Conference on Cold-Formed Steel Structures

Nov 8th

The Structural Design of Light Gauge Silo Hoppers John Michael Rotter

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Ninth International Specialty Conference on Cold-Formed Steel Structures St. Louis, Missouri, U.S.A., November S-9, 1988

THE STRUCTURAL DESIGN OF LIGHT GAUGE SILO HOPPERS by J. Michael Rotterl

Synopsis Elevated light gauge silos usually have a conical discharge hopper at the bottom. Although this hopper often carries much of the total weight of the stored solids within the silo, it can often be cold-formed from thin steel sheet because the structural form is very efficient. However, guidance on the design of light gauge hopper structures is rare. The term "light gauge" is used here to describe the class of cold-formed silo structure which is not restricted by a nominal minimum plate thickness requirement (eg 1/4" or 6 mm). This paper addresses several aspects of the design of light gauge hoppers. Current proposals concerning hopper loads are discussed first, and recommendations are made. Appropriate structural analysis is then presented. The potential failure modes of the hopper are identified, and corresponding strength checks described. 1.

INTRODUCTION

Silos made from cold-formed steel sheets are widely used in agriculture throughout the world. Failures in silos are common, and improvements in silo technology are clearly desirable. This paper is concerned with the design of light gauge metal silo hoppers, and those aspects of the rings and column support conditions which are intimately related to the hopper (Fig. 1). It relates only to silos of circular planform. Conical discharge hoppers are generally subjected to only symmetrical stored solids loading. However, the pressures on hoppers are less well understood than those on vertical silo walls; the structural action is a little more complex, and little attention has been paid to hopper design in the past. These factors underlie the present review paper. Cold-formed steel silo hoppers are susceptible to more modes of failure than the larger industrial silo hoppers, because they often have bolted joints of limited strength. The design of these joints requires a more careful assessment of hopper loading patterns, so a significant part of this paper is devoted to the definition of hopper loads. Farm silos often differ considerably from industrial silos, and much of the available design advice is concerned either with industrial and mmmg applications (Wozniak, 1979; Trahair et aI, 1983; Gaylord and Gaylord, 1984; Rotter, 1985) or with the cylindrical walls of light gauge silos (Trahair et aI, 1983; Abdel-Sayed et aI, 1985; Rotter, 1986b). 1 Senior Lecturer Wales, Australia.

in

Civil

Engineering,

529

University

of

Sydney,

New

South

530

2. 2.1

LOADS ON HOPPER WALLS Introduction

The chief loading on conical discharge hoppers derives from symmetrically placed stored solids. However, the assessment of these loads involves both the cylinder and the hopper. A detailed argument concerning appropriate pressures for light gauge hoppers is presented elsewhere (Rotter, 1988), and only the recommendations are given here. The most commonly used theories for pressures in hoppers are those of Walker (1966), Walters (1973) and Jenike et al (1973). J'vIost codes of practice (American Concrete Institute, 1977; DIN 1055, 1986; Gorenc et aI, 1986; BJ'vIHB, 1987) specify either a constant pressure within the hopper or a linearly varying pressure. J'vIost include a local high "s,,,itch" pressure near the hopper/cylinder junction (the transition) to account for flow conditions, but the details of the pressure distribution in the body of the hopper are often thought to be relatively unimportant. However, light-gauge bolted steel hoppers require a more careful assessment of pressure distributions. 2.2

Defining the Total Load on the Hopper

In elevated silos, the hopper supports the majority of the total weight of stored material. The total load on the hopper is defined by the hopper volume and the mean vertical stress in the stored material at the transition (hopper/cylinder junction) (Fig. 2a). The latter depends on the height of stored material above this point, and the proportion which is supported by friction on the cylindrical walls. Cylindrical wall pressures are therefore needed to define the loading on the hopper. The pressures on the walls of the cylinder p, wall frictional tractions v, and vertical stress in the stored solid q (Fig. 3a) are most easily assessed using Janssen's equation (1895) p

= Po (1 - e-Y/Yo)

(1)

v

= /.Lp

(2 ) (3 )

in which Po = yR/2/.L, % = yR/2/.Lk, Zo = R/2/.Lk, R is the silo radius, y is the distance below the effective surface of the solid, y is the bulk solid density, /.L is the wall friction coefficient, and k is the lateral pressure ratio (ratio between horizontal and mean vertical stresses in the stored solid). 2.3

J'vIaximising the Loading due to Bulk Solids

J'vIost silos are used to store a range of materials, so that the properties may vary significantly from time to time. Other changes may occur as the silo becomes polished or roughened by stored solids. The silo should therefore be designed for a variety of different values of Y, k and /.L in Eqs 1-3. All loads are maximised when the value of y values of wall pressure occur when k is at its minimum value. The maximum vertical loads on and /.L take their minimum values. The smallest by the simple Rankine pressure ratio

is maximised. The largest maximum value and /.L at its hoppers occur when both k possible value of k is given

531

k

=

sin



+ sin

-



(4 )

in which is the effective angle of internal friction of the stored solid (usually in the range 28-33 0 for grains). More realistic values of k can be derived from the relation first advanced by Walker (1966) and since adopted by many others

k

=

1 + sin 2 <1> - 2/(sin 2 <1>-g2cos 2 <1» 4g2 + cos 2 <1>

(5)

However, the values obtained from Eqs 4 and 5 differ only slightly unless the wall is very rough. For steel silos, it therefore is arguable that Eq. 4 should be used to determine k for the cylinder when the total hopper load is being determined. Corrugated silos provide the one possible exception to this proposal (Rotter, 1988). 2.4

Initial Filling Loads on Hopper Walls

The simplest theory of hopper filling pressures is that of Walker (1966). It assumes that the stored material in the hopper carries no shear stresses. The maximum pressure occurs at the outlet (Fig. 4b). This pattern is often the worst pressure distribution for welded hoppers without strong transitional ring support (Rotter, 1986a) as it places the maximum load as far from the support as possible. However, it may be unduly conservative. Walker filling theory is used in some codes (Gorenc et aI, 1986; BMHB, 1987), but not in others (DIN 1055, 1986; American Concrete Institute, 1977). It should be noted that the frictional traction v has sometimes been omitted in drafting code rules based on Walker theory (BMHB, 1987) leading to an unsafe definition, since the hopper is deemed to carry less than the total load on it. Walker (1966) also presented a theory for discharge conditions. This theory involves the general solution of the hopper equilibrium equation subject to the condition (similar to the Janssen assumption) that the ratio of wall pressure to mean vertical stress in the solid is invariant with depth (Figs 4b and 4c). Walker also proposed one method of determining this ratio. Many writers have made modifications to this basic theory by determining the ratio in different ways. McLean's (1985) modification of the Walker theory is supported by some experimental evidence (Motzkus, 1974; Hofmeyr, 1986) and the finite element calculations of Ooi and Rotter (1987). The general Walker theory leads to the pressure distribution (Figs 4b and 4c) p v

=F

q

(6 )

g p

(7) (8 )

in which p is the normal wall pressure, v is the frictional traction, q is the vertical stress in the stored solid at height z above the apex, qt is the value of q at the transition and H is the height of the hopper. Equilibrium of the complete hopper requires that

532

n

=

2(Fg

cot~

+

F

1)

(9 )

McLean (1985) suggests that F = 1.0 leads to reasonable estimates of hopper initial filling pressures, so Eq. 9 reduces to n

= 2g

cot~

( 10)

This theory is recommended here to define hopper 2.5

i~tial

filling pressures.

Mass Flow and Funnel Flow Pressures

The pattern of solids flow from a silo is known to affect both the pattern and the magnitude of the pressures. Two simple forms of flow pattern have been widely accepted (Jenike et aI, 1973) and are known as the mass flow and funnel flow modes (Fig. 3b). The hopper pressures are normally defined with one or other of these flow modes in mind. Most published theories are concerned with the pressures during mass flow. widely recognised that the pressures at the outlet decrease during discharge, as only then can flow of the solids occur. A local high pressure also develops at the transition (Fig. 4c), and most design guides recommend that this "switch" pressure be considered. The magnitude of the transition switch pressure only becomes really large when a very steep hopper is used. Most cold-formed silos have quite shallow hoppers (-45 0 ), so this switch pressure is not a very significant item.

It is

In cold-formed bolted hoppers, the critical point is usually a short distance from the transition, and this distance is defined by the hopper structural behaviour. In these circumstances, it is difficult to decide between the different theories and codified rules, but the original Walker (1966) discharge theory may provide reasonable values. It is certainly preferable to uniform pressures when cold-formed hoppers with bolted joints are being designed. It is given by Eqs 6-9 with

F =

1 + sin<\> cosE: 1 - sin<\> cos(2~-E:) .

Sl.n

-1

( 11)

[1 -/ [l+g2 g2 ]] sin<\>

in which g is the hopper wall friction angle of internal friction.

coefficient and

( 12 ) <\> is the effective

Funnel flow pressures are less well understood. Gaylord and Gaylord (1984) and Rotter (1988) have presented strong criticisms of the commonly assumed pressure distributions. The best current treatment is probably to assume that all hoppers are subject to mass flow hopper pressures irrespective of their flow patterns (Eqs 6-9, 11-12).

3. 3.1

STRESSES IN HOPPER WALLS Introduction

When the silo is continuously supported (not supported on columns), the stresses developing in the greater part of a steel hopper can be determined using the membrane theory of shells. Bending stresses in the hopper body

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occur only at changes of plate thickness and are generally small. However the stresses in the shell and ring at the transition are strongly affected by the discontinuity there, are much larger, and require more computational effort to establish precisely. 3.2

Hopper stresses under Symmetrical Filling, Discharge and Support

For symmetrical loading, the membrane theory of shells (Rotter, 1987b) leads to two equilibrium equations in terms of the meridional stress resultant Nell (force per unit circumference) and the circumferential stress resultant Ne (force per unit width) for conical shells (Fig. 2b)

=

sec~

tan~

d dz (z N Cil ) = z

sec~

NCil

P z

( 13) (p

tan~

These may be solved for the distribution (Eqs 6-9), to yield

(14 )

+ v) general

form

of

the

hopper

pressure

( 15)

[

yH [z]2 + (n+2) 1 [ qt - n-l yH ] [z]n+l] II FHsec~(tan~+g) 3(n-l) II

(16 )

should be noted that, in general, the highest pressure on the hopper does not coincide with the highest membrane stresses in the hopper wall (Fig. 5).

It

The hopper is in biaxial tension. For welded hoppers, which can sustain combined stresses in two directions, the meridional and circumferential stress resultants should be combined to find the effective stress resultant N vm using von Mises yield criterion ( 17) Cold-formed hoppers usually have meddional bolted joints, so it is difficult to exploit this additional strength. Moreover, the joints usually have lower strengths than the sheets which are joined, so the circumferential stress resultant (Eq. 15) must be compared with the joint strength. The circumferential tension varies up the hopper, but to avoid changes of bolt spacing, it is sometimes useful to work to the maximum value of circumferential tension. If the silo has a significant cylindrical section, so that (18 ) the maximum occurs at the top of the membrane stress resultant is Ne

= qt F H

sec~

tan~

hopper,

where the circumferential

(19)

However, if the hopper occupies much of the silo, then Eq. 18 will not be satisfied, and the maximum value of the circumferential force per unit length

534

Ne occurs within the hopper and can be determined from Eq. 15 with z

II

1 n-1

=

(20)

Finally, it is useful to note that the simpler assumption (American Concrete Institute, 1977) of uniform pressure in the hopper with no friction on the hopper wall always overestimates the maximum circumferential tension, unless Eq. 18 is not satisfied. The meridional membrane stress resultant often reaches its maximum value at the top of the hopper, where a bolted joint may be made to connect with the transition ring. The membrane stress resultant here is also needed in design of the ring, and is given by overall hopper equilibrium (21 ) When very light rings are used, the meridional stress resultant falls below the value given by Eq. 21, but a conservative design of the hopper top and transition ring is obtained if this effect is ignored. 3.3

Stresses in Symmetrically Loaded and Supported Transition Rings

The transition is a point of greater complexity. The top of the hopper is in meridional tension, which applies a radial force to the transition (Fig. 6). This radial force induces a circumferential compression in the transition, which can lead to plastic collapse or buckling. The radial force per unit circumference on the ring can be derived from Eq. 21, and induces a circumferential force in the transition junction of P

= NcI>

R sinf3

(22)

When the silo is continuously supported, the circumferential compressive stresses may be deduced quite accurately using hand methods, for example, as derived by Rotter (1983, 1985). The phenomena are demonstrated here by means of an example silo. In large industrial silos, the form of the ring is relatively well defined. By contrast, a considerable yariety of different types of ring are used by different cold-formed silo manufacturers. The example chosen here is intended to illustrate typical stress patterns, without describing the product of anyone manufacturer in detail. The example silo is not proposed as an adequate design. An example silo, with its hopper and supporting ring, is shown in Fig. 7. The ring consists of a cold-formed channel, which is welded to the top of the hopper. The cylinder wall is supported from a cold-formed angle, so that the two components can be assembled separately. The silo is supported on twelve columns, equally spaced around the circumference. The stored solid is assumed to have a density of 9.0 kN/m 3 , an effective angle of internal friction of 28 0 , and a wall friction coefficient of 0.364. The hopper was subjected to a Walker (1966) discharge pressure distribution (Eqs 6-9, 11, 12). The silo was analysed using the linear finite element program LEASH of the FELASH suite developed at the University of Sydney (Rotter, 1982; 1987c). The corrugated wall was treated as an orthotropic shell with properties determined using the relationships of Trahair et al (1983). The stresses are all taken as tensile positive. Two versions of the

535

design were studied: one with a cylinder of simple rolled sheet steel (Design A), and the other made from corrugated sheeting (Design B). The choice of cylinder wall type will be seen to have a bearing on the design of the hopper. The structure was analysed first as if it were continuously supported all around the circumference. This illustrates the pattern of stresses arising in and near the ring as a consequence of the ring compression. The results are virtually independent of the form of the cylindrical wall. The circumferential membrane stresses are shown in Fig. 8a. The compression varies considerably through the ring, being only sensibly constant in the annular plate element of the channel. Hand methods of analysis are chiefly aimed at determining the value in this element alone. It is also clear that circumferential compressive stresses arise in the hopper and cylindrical wall, that these differ in magnitude from the value in the ring, and that they decline in a non-linear but rapid manner away from the ring. Thus, a calculation based on an effective section must be interpreted with care, as the stress distribution is very different from those found in steel frame members which do not distort. The meridional membrane stresses near the top of the hopper are shown in Fig. 8b. When very small rings are used and the hopper is relatively thick, the meridional tension falls below the value defined by Eq. 21, because some of the hopper load is supported by transverse shearing in the hopper (Rotter, 1987c). The reduced tension leads to a slightly reduced compression in the ring, but very high bending stresses occur at the top of the hopper. The meridional bending stresses are shown in Fig. 8c, where the very high very local maximum at the transition junction is clearly seen. Alternative hand methods of determining these bending stresses were advanced by Barthelmes (1977), Fuchssteiner and Olsen (1979), Gaylord and Gaylord (1984) and Rotter (1985). Some authors (e.g. Gaylord and Gaylord, 1984) have argued that the bending (discontinuity) stresses should be determined and allowed for in the design. However, unless the silo is to be loaded and unloaded so many times that a fatigue failure is possible (Rotter, 1985), the bending stresses are not directly implicated in a definable failure mode, as noted below. The tedious calculation to determine these stresses is therefore not normally warranted. 3.4

Stresses in Hoppers of Column-Supported Silos

The top of the hopper in a column-supported silo is subject to a stress state which is closely related to that of the transition ring or junction. The hopper must therefore be designed with the ring and support condition in mind. Column-supported silos present a much more difficult problem than continuously supported silos. They were first described by Ketchum (1907), and the procedure which he suggested, although often in serious error, was not re-examined until the 1980s. Column-supported silos have been the subject of a number of recent studies (see Rotter, 1988) and several hand methods for estimating the stresses in both the transition ring and the cylindrical shell have been advanced. Many of these hand methods are both complicated and give rather inaccurate results. More reliable stresses are obtained from finite element calculation (Rotter, 1982). The problem of the column-supported silo is essentially three-fold: first the

536

ring at the transition must be examined for bending or combined bending and torsion: second, non-uniform axial forces develop in the cylindrical wall above the ring, and these may lead to buckling of the cylinder: and thirdly, the hopper is in non-uniform meridional tension, and this non-uniformity may lead to rupture of the hopper. This paper deals only with the last of these three. The meridional membrane stresses in the hopper of Design A are shown in Fig. 9a. The non-uniformity of hopper meridional tensions extends approximately half-way down the hopper. The meridian of the column support is much more highly stressed than that of the midspan. The corresponding stresses for Design B are shown in Fig. 9b. These stresses are significantly higher than those for Design A. This is because the cylindrical wall of Design A plays an important role in redistributing the forces from the columns, but the corrugated wall of Design B is very flexible in vertical deformation, so it cannot fulfil this role. As a result the ring in Design B is more highly stressed, sustains larger deformations and leads to greater non-uniformity in the hopper meridional stresses. The column-support condition affects the circumferential stresses in only a small zone at the top of the hopper, and most of the hopper is stressed as defined by Eq. 15. The circumferential variation of meridional membrane stresses in the top of the hopper is shown in Fig. lOa for the three conditions of continuous support (C), Design A and Design B. The corresponding variation of circumferential stresses is shown in Fig. lOb. The column support causes a major change in hopper meridional tension, and a significant change in circumferential stress at the top of the hopper. However, the difference between a rolled steel and corrugated cylinder wall causes a less easily anticipated but large difference in meridional tension (40%). Safe hopper design clearly depends on more than the loads acting on the hopper itself.

4. 4.1

CRITICAL STRENGTH ASSESSMENTS FOR THE HOPPER Introduction

The conical hopper of a silo is susceptible to several different failure modes, including plastic collapse, meridional seam rupture, and transition joint rupture. The transition junction, which is intimately related to the hopper, is susceptible to plastic collapse and buckling, and may initiate either hopper rupture or cylinder compression buckling. The transition junction is referred to here only insofar as it affects the hopper design. 4.2

Plastic Collapse of Hoppers

Because the hopper is in biaxial tension, its resistance exceeds the value determined by simply restricting the effective membrane stress to the yield stress, provided the hopper seams are strong enough. A clear distinction must therefore be made between fully welded hoppers and bolted hoppers. The strengths of fully welded hoppers have been the subject of study (Teng and Rotter, 1988), in which it was shown that the estimated using the results of membrane theory underestimates strength by about 10%. A typical collapse mode at the top of the shown in Fig. lla. The plastic collapse design strength Nevm for

a recent strength the real hopper is a hopper

537

strake of thickness t with top edge radius R may be closely approximated (Teng and Rotter, 1988) by R [0.91 + 0.136/(~+1.5)l tFy R 2.4/(Rt/cos~) sin~

(23)

in which Ncvm may be compared with the strake top edge von Mises effective stress resultant under working loads Nvm (Eq. 17) and Fy is the steel yield stress. Because the structure displays a strongly stiffening plastic behaviour, it is not easy to define the collapse load. Larger strengths may therefore be assumed if very large deformations of the hopper are permitted. The critical segment of the hopper is generally the one with the largest upper-edge radius-to-thickness ratio. For continuously supported silos, the pressures during discharge are likely to provide the critical design loading. The high bending stresses which may be calculated at the transition junction under elastic conditions lead to early yielding. This yielding has only a small effect on the load-deflection curve, and a plastic hinge soon forms, which moves down the hopper as the yielding spreads. Under very large deformations, the plastic hinge moves far from the junction. High transition bending stresses are therefore thought to be important only in silos which may fail in fatigue. The strength of bolted hoppers is also restricted by plastic collapse, but the full plastic collapse strength may not be achieved because of premature joint rupture. No reported failures of hoppers by plastic collapse are known. The simple explanation is that the hopper becomes steadily stronger and stronger under large deformations until a secondary failure occurs by joint rupture. By then the hopper may be badly deformed, but many users are unperturbed by the deformations of a hopper local plastic collapse mechanism. Nevertheless, it seems desirable that hoppers should be designed to prevent plastic collapse. 4.3

Hopper Meridional Seam Rupture

A light gauge hopper is often fabricated from a number of pie-shaped segments, which may be field-bolted together. The joints run down the hopper meridian. This joint must transmit tensile forces arising from circumferential tension, and should be designed (Fisher and Struik, 1974) for the highest local value of circumferential tension which can occur in the hopper. This is given by Eqs 15, 19 and 20. A few failures have been reported in which the meridional seam in a hopper ruptured, leading to loss of the silo contents. 4.4

Hopper Circumferential Seam Rupture

Bolted hoppers are sometimes built up from plates which do not extend over the full height. Joints between these strakes run circumferentially around the hopper. Each joint should be designed for the maximum meridional tension which may be transmitted through it. For a continuously supported silo, this tension can be determined from Eq. 16, using the appropriate height z. These joints will usually be most highly stressed under initial filling conditions.

538

1.5

Hopper Transition Joint Rupture

Both "'elded and bolted silos are susceptible to rupture of the hopper transition joint under the meridional forces which must be transmitted to the supporting ring. Several failures of this kind have occurred. In most silos, the majority of the weight of stored solids rests on the hopper, and the whole of this force must pass through the hopper transition joint. The transition junction joints are particularly difficult to fabricate, as a conical shell must be joined to one or two cylindrical shells, and perhaps a ring. Where the circumferential joint at the transition is given a strength equal to that of the thinner joined plate, it can be shown that transition joint rupture is most likely to precede hopper plastic collapse in steep hoppers with rough walls. Light gauge silos seldom satisfy these conditions, but they also rarely have joints of strength equal to the joined plate. The transition joint is usually the most critical design detail for the hopper. Hopper plate thicknesses are often increased near the transition. The reason given is generally that high bending stresses occur at this point in the structure. As noted above, the high bending stresses do not lead to failure, but the increased plate thicknesses do make it easier to design a strong joint. Thus it is possible that a satisfactory remedy for transition joint rupture is in common use, though the reason given is in error. 4.6

Transition Junction Plastic Collapse

The hopper/cylinder junction (transition junction) is in circumferential compression. The compressive stresses are high because only a small part of the shell carries the large force required to equilibrate the radial component of the hopper meridional tension. Whilst buckling failures are possible if a wide thin ring is used, the most likely failure mode at the ring is by plastic collapse (Rotter, 1987a). The plastic collapse strength of the ring and junction under continuous support conditions was defined by Rotter (1985). Plastic hinges form in the hopper, cylinder, and skirt (if the design includes a skirt), whilst the ring itself is quite uniformly yielded under circumferential stresses (Fig. llb). As the loading rises, the hopper plastic hinge moves from being adjacent to the junction down into the hopper, redistributing the high elastic bending stress at the top of the hopper. Plastic collapse of the junction is affected by the hopper thickness, but the plastic collapse modes of the hopper and ring do not interact significantly (Teng and Rotter, 1988). A small transition ring, or no ring at all, generally means that junction plastic collapse will control the silo strength. Where the ring is made larger, hopper plastic collapse is likely to control. The plastic collapse strength of the junction, with or without a be closely approximated by

ring, may

(24 )

in which Pc is the circumferential force in the junction at collapse (cf the working value derived from Eq. 22), BT is the area of the ring at the

539

junction, t c ' ts and th are the thicknesses of the cylinder, sltirt and hopper respectively, and the effecth-e lengths of adjacent shell segments leh may be assessed as lec = 0.975/(Rt c )' les = 0.975/ (Rt s ) and 0.975/ (Rth/cosf3).

=

For continuously supported silos, traditional simple transition junction design techniques (Wozniak, 1979; Trahair et ai, 1983; Gaylord and Gaylord, 19841 are generally conservath-e, but sometimes they are "e17 conservative. Useful savings may thel'efore be made by designing each component to its real strength. Nevertheless, it should be realised that the transition ring is ineffective if placed only a short distance above the hopper/cylinder intersection (Rotter, 1985). 4.7

The Column-Supported Silo and its Transition Ring

One of the most difficult tasks in silo design is to achieve an economic solution to the problem of an elevated silo on columns. The column supports introduce high local \-ertical compressive forces into the shell. These lead to high vertical compressions in the cylinder, and high local meridional tensions in the hopper. In addition, the transition ring is subjected to axial compression, bending about two axes and torsion. The problem of stress analysis of these components was mentioned above, but very little work has been undertaken to establish rational failure criteria for any of them. has been shown above that the meridional stresses are locally much silos than in those of higher in the hoppers of column-supported continuously-supported silos. No previous study appears to have discussed failure criteria for the hopper when the silo is supported on columns. In particular, no rigorous calculations are known.

It

Three criteria are therefore proposed here: one for ring plastic collapse, one for hopper plastic collapse and the third for hopper l'upture. It is proposed that the ring plastic collapse strength should be assessed as the same as that of the continuously supported structure (Eq. 24). This proposal is based on the observation that the junction collapse mode involves large bending deformations of the shell elements meeting at the tl'ansition, and that significant redistribution may therefore be possible. It is also proposed that hopper plastic collapse should

be assessed as if the hopper were continuously supported (Eq. 23). The reasons are similar to those for junction collapse, but the additional observations may be made that the hopper plastic collapse occurs within the hopper, away from the region most influenced by the column supports. Further, the non-uniformity of the meridional membrane stress extends further into the hopper than that of the circumferential membrane stress. The strength of most designs (shallo\-:, smooth walled hoppers) should not be affected very much by locally elevated meridional membrane stresses because the shape of the biaxial yield criterion indicates insensitivity to this parameter. By contrast with the two above criteria, it is proposed that hopper transition joint rupture and circumf~rential joint rupture should be expected when the maximum meridional membrane stress (Fig. lOa) attains the yield stress or the strength of the joint. This proposal recognises that local rupture of the hopper near the column could lead to complete rupture of the hopper. It is also based on the observation that there is little scope

540

for redistrib:,tion bec~use the support condition (ring) is in bending, whilst the. h.opper IS stret?hmg. The scope for redistribution of the high hopper merIdIonal stresses IS thus very limited. A joint of high strength is most easily achieyed by using thicker steel sheeting for the hopper. Unfor·tunately, no simple hand method of predicting the local high meridional tensions in the hopper is known, so a finite element analysis is currently required ""hen designing this joint to the proposed criteria. The rings on column-supported silos require separate and careful analysis (Rotter, 1982, 1984, 1985), which is beyond the scope of the present paper. A number of practical matters, relating to the use of steep hoppers, column braeing and ground-supported skirts are discussed elsewhere (Rotter, 1988). 5.

SUJ'vlJ'vlARY

In this paper, a review of design advice for light-gauge cold-formed steel silos has been presented. The review has made specific recommendations on the pressures which should be used in design and has described appropriate stress analysis of the structure. The failure modes which control the design have been defined, and simple rules for some of these have been presented. For column-supported silos, it has been shown that the hopper must be thicker than it is for continuously supported silos. It has also been shown that the form of the cylindrical silo wall can affect hopper stresses markedly. Design criteria for the hoppers of column-supported silos have been proposed. No comparable existing criteria are known. J'vlore detailed information is given in a longer report (Rotter, 1988).

6.

ACKNOWLEDGEJ'vlENTS

This paper forms part of a major research program into the loading, behaviour, analysis and design of silo and tank structures being undertaken at the University of Sydney. Support for this program from the Australian Research Grants Scheme, the University of Sydney and cooperating commercial organisations is gratefully acknowledged. APPENDIX.- REFERENCES Abdel-Sayed, G., J'vlonasa, F. and Siddall, W. (1985) "Cold-Formed Steel Farm Structures Part I: Grain Bins''', Jnl of Structural Engineering, ASCE, Vol. 111, No. STlO, Oct., pp 2065-2089. American Concrete Institute (1977) "Recommended Practice for Design and Construction of Concrete Bins, Silos and Bunkers for Storing Granular J'vlaterials", ACI 313-77, Detroit (revised 1983). Barthelmes, W. (1977) "Ermittlung del' Schnittafte in kreiszylindrischen Silos mit kegelrmigem Boden", Bauingenieur, Vol. 52, pp 423-435. BJ'vlHB (1987) "Silos: Draft Design Code for Silos, Bins, Bunkers and Hoppers", British Materials Handling Board and British Standards Institution, London.

541

DIN 1055 (1986) "Design Loads Standard, Sheet 6, September.

for

Fisher, J.W. and Struik, J.H.A. (1974) and Rivetted Joints", Wiley, New York. Fuchssteiner, W. and Olsen, O.W. Bauingenieur, Vol. 54, pp 17-21.

Buildings;

Loads

on

Silos",

German

"Guide to Design Criteria for Bolted

(1979)

"Ein Problem der

Stahlblechsilos",

Gaylord, E.H. and Gaylord, C.N. (1984) Design of Steel Bins for Bulk Solids, Prentice Hall, Englewood Cliffs, New Jersey.

Storage of

Gorenc, B.E., Hogan, T.J. and Rotter, J.M. (eds) (1986) "Guidelines for the Assessment of Loads on Bulk Solids Containers", Institution of Engineers, Australia. Hofmeyr, A.G.S. (1986) "Pressures in Bins", MSc(Eng) Witwatersrand, Johannesburg.

Thesis, University of

Janssen, H.A. (1895) "Versuche uber Getreidedruck in Silozellen", Zeitschrift des Vereines Deutscher Ingenieure, Vol. 39, No. 35, pp 1045-1049. Jenike, A.W., Johanson, J.R. and Carson, J.W. (1973) "Bin Loads- Part 2: Concepts; Part 3: Mass Flow Bins", Jnl of Engng for Industry, ASME, Vol. 95, Series B, No.1, Feb., pp 1-12. Ketchum, M.S. (1907) "Design of Walls, Bins and Grain Elevators", 1st edition, McGraw-Hill, New York (2nd edn 1911, 3rd edn 1919). McLean, A.G. (1985) "Initial Stress Solids Handling, Vol 5, No 2, April.

Fields

in

Converging

Channels",

Bulk

Motzkus, u. (1974) "Belastung von Siloboden und Auslauftrichtern durch kornige Schuttguter", Dr.-Ing Dissertation, Technical University of Braunschweig. Ooi, J.Y. and Rotter, J.M. (1987) "Elastic Predictions of Pressures in Conical Silo Hoppers", Research Report R555, School of Civil and Mining Engng, Univ. of Sydney, Oct. Rotter, J.M. (1982) "Analysis of Ringbeams in Column-Supported Bins", Eighth Australasian Conference on the Mechanics of Structures and Materials, University of Newcastle, Aug. Rotter, J.M. (1983) "Effective Cross-sections of Ringbeams and Stiffeners for Bins", Proc., International Conference on Bulk Materials Storage Handling and Transportation, Institution of Engineers, Australia, Newcastle, Aug., pp 329-334. Rotter, J.M. (1984) "Elastic Behaviour of Isolated Column-Supported Ringbeams", Journal of Constructional Steel Research, Vol 4, 1984, pp 235252. Rotter, J.M. (1985) "Analysis and Design of Ringbeams", in Design of Steel Bins for the Storage of Bulk Solids, edited by J.M. Rotter, Univ. Sydney,

542

March, pp 164-183. Rotter, J.M. (1986a) "On the Significance of Switch Pressures at the Transition in Elevated Steel Bins", Proc., Second International Conference on Bulk Materials Storage Handling and Transportation, Institution of Engineers, Australia, Wollongong, July, pp 82-88. Rotter, J.M. (1986b) "Recent Studies of the Stability of Light Gauge Steel Silo Structures", Proc., Eighth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Missouri, Nov., pp 543-562. Rotter, J.M. (1987a) "The Buckling and Plastic Collapse of Ring Stiffeners at Cone/Cylinder Junctions", Proc., International Colloquium on Stability of Plate and Shell Structures, Ghent, April, pp 449-456. Rotter, J.M. (1987b) "Membrane Theory of Shells for Bins and Silos", Transactions of Mechanical Engineering, Institution of Engineers, Australia, Vol. ME12 No.3 Sept., pp 135-147. Rotter, J.M. (1987c) "Bending Theory of Shells for Bins and and Silos", Transactions of Mechanical Engineering, Institution of Engineers, Australia, Vol. ME12 No.3 Sept., pp 148-159. Rotter, J.M. (1988) "The Research Report, School Sydney, May.

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Pressure and Arching in

Walters, J.K. (1973) "A Theoretical Analysis of Stresses in Axially-Symmetric Hoppers and Bunkers", Chern. Engng Sci., Vol. 28, No.3, March, pp 779-89. Wozniak, R.S. (1979) "Steel Tanks" in Structural Engineering Handbook, 2nd edn, Section 23, Eds. E.H. and C.N. Gaylord, McGraw-Hill.

543

Roof Ring Cylinder Ring Skirt Conical Hopper

FIG.1

TYPICAL COLD-FORMED ELEVATED SILO

'\

R

FIG.2

z

Me

(b) Axisymmetric Stress Resultants

Ncp

HOPPER GEOMETRY, LOADING AND NOTATION

(a) Equilibrium and Notation

Supporting Meridional Tension

......c:n

545

~ 0 LL QJ

c: c:

:J LL VI QJ

"'C 0

L ~ 0 LL.

.c

~

....J

u...

~

LL

....J c(

VI VI ItI

~

UJ

~ 0

~ 0

--- ----

Vl 0 0

9:: w z

L

~

a.. 0

z

c(

>

Vl

~c..

UJ

------

a::

::::>

Vl Vl UJ

a::

a..

---

a:: UJ

-

D Z

c-

--

:::J >w LQJ

"'C

c: >-

~

W

c: VI QJ L-

:J VI VI

>-

..

QJ L-

a..

ItI

,.,.. l.:i u...

546

qt

~~~~~~~~~~ ~=20o

f.l=0.4 cD=30 o H

Walker Filling

o

(al Hopper FIG.4

0.2 0.4 p/'6'H (bl Pressure from Hopper Contents

0.2 (e)

0.4

0.6

p/qt Pressure from Surcharge

THEORETICAL HOPPER PRESSURE DISTRIBUTIONS

0.8

1.0

~

""'-

=30 0

o

E

QJ

~

o

c::

---

FIG.5

(a) Pressure Distribution

~'

)

'''.

0.5

, 0.4 0.6 0.8 1.0 1.2 Meridional S.R. (6N~sin~/oR2) (c) Meridional Membrane S.R.

0.2

/,/

/,,/'

/,/

TYPICAL STRESS RESULT ANT DISTRIBUTIONS (HOPPER ONL Yl

0.1 0.2 0.3 0.4 Circumferential S.R. (N e/ oR2) (b) Circumferential Membrane S.R.

_/-/_/-~.

,

'"

0.2 0.4 0.6 0.8 1.0 Wall Pressure (p/oR)

o

0.2

0.6

.

---..:..

~:::::C=:::~~:;l--=:::::r-::::::=:~'-I--'---r--'-'----..

~ ~ 0.4)

0.8

I 1.0 "'N

;

Cl1

"'" -l

548

Ring

~

Skirt

! /1

Cylinder VertiCal Junction~ Compression - - -.....~ Radial Force provided by Cone Skirt Ring Compression Meridional Tension

tsupport /cone JI' Meridional Tension (a) Junction Local Geometry FIG.6

(b) Static Equilibrium at the Junction

TRANSITION JUNCTION EQUILIBRIUM: CONTINUOUS SUPPORT

549

r 100x100x6 3500 o

o=9kN/m 3
o

L.f'l (Y'l

,

1mm Smooth Wall (A) 1mm Corrugated Wall (8)

50x50x6

o o

L

L

1.6mm Smooth Wall (A) 1.6mm Corrugated Wall (8)

L.f'l

See detail

~

(a) Structure

Wall

(b) Ring Detail FIG.7

EXAMPLE SILO

FIG.8

(a) Circumferential Membrane Stress

L-........J

100MPa

T

L...-....-...J

300MPa

Stress (Hopper only)

(e) Meridional Bending

STRESSES AT HOPPER TOP (CONTINUOUS SUPPORT)

(b) Meridional Membrane Stress

100MPa

c

T: tension C: compression

c.n c.n o

551

X

X

T

T

Support Position

Support Position 200MPa

200MPa L--............

(a) Design A FIG.9

I

(b) Design B

MERIDIONAL MEMBRANE STRESSES IN HOPPERS OF COLUMN-SUPPORTED SILOS

I

200

-50 I -15

oI

V) 100

QJ L-

VI VI

L

~

300

-5

0

I

5

10

:::::0"""

FIG.10

Design B

Compression

Tension

-5

o

5

10 Circumferential Coordinate (degrees) (b) Circumferential Membrane Stress

-10

\

II

15

..L...-_ _....L..._ _---'-_ _ _ _I . -_ _......

L.'_ _- - - ' -_ _ _ _

-150 15 -15 I

--I

-50

-100

V1

.....

QJ L-

VI VI

I

CIRCUMFERENTIAL VARIATIONS OF STRESS AT TOP OF HOPPER

Circumferential Coordinate (degrees) (a) Meridional Membrane Stress

-10

,........--

ro

L

a..

oI

50,,--~----r---'---~----r---,

""

01 01

553

(a) Collapse at Top of Hopper FIG.11

(b) Collapse Mode of Junction

PLASTIC COLLAPSE MODES OF HOPPER AND TRANSITION