Elastic predictions of pressures in conical silo hoppers J. Y. Ooi and J. M. Rotter
Department of Ovil Engineering, University of Edinburgh, UK (Received to be supplied)
Theoretical techniques for predicting the pressures on the wails of conical silo hoppers have generally depended on the assumption that the mass of material within the hopper is in a plastic state of stress. The fact that this imposes considerable restrictions on the stress history and strain state has almost always been ignored. Moreover, few investigators have undertaken checks to verify the assumption.
In this study, the alternative simple assumption is made that the material stored within the hopper is in an elastic state. The study is heuristic in character, as it is not claimed that an elastic state =does exist within the hopper, though the assumption is not unreasonable for the initial filling condition. Nevertheless, a number of interesting discoveries are made about the stress distribution within the stored mass, and concerning the influence of various para~te rs on the hop- per wall pressures. The analysis is conducted using a finite e l e ~ n t analysis which includes the effects of hopper wall friction and hopper wall flexibility, and these influences are investigated.
Keywords: bulk solids, hoppers, loads, pressures, silos, theory
Many attempts have been made to produce theories which delrme the distribution of pressure on the walls of conical silo hoppers. The majority of these have adopted the assumption that the mass of material within the hop- per is in a plastic state of stress. This assumption is so regularly made that it is sometimes not even stated, but the use of failure properties, such as the angle of internal friction, indicates that the assumption is being made.
The fact that this plastic assumption imposes con- siderable restrictions on the stress history and deforma- tions of the mass has almost always been ignored. Moreover, few investigators have undertaken checks to verify the assumption. The properties of dry granular solids under stress states which do not involve failure have also been explored only rarely and incidentally. These are poor reasons for assuming that the material within the hopper is in a plastic state, especially for the initial filling condition, which has been shown to be the critical state for the structural design of mass flow steel hoppers ~.
In this study, the alternative simple assumption is made that the material stored within the hopper is in an elastic state. The study is hcm'istic in character. It is not claimed that an elastic state does exist within the hopper, or that the initial strains of different parts of the solid are unimportant. The study is undertaken to explore the stress states which occur in a homogeneous linear elastic mass of material without initial strains, subject to self
weight, and stored in a conical hopper with frictional sliding contact on the wall. The findings of this study form an important but simple result, which can be used as a reference when the predictions of more complex models are being examined. It should also be noted that the assumption of linear elastic stress states within the hopper has been made before 2, but additional assump- tions were always made, so that the calculations were not rigorous.
The present analysis is conducted using a finite ele- merit analysis which includes the effects of hopper wall friction and hopper wall flexibility, and these influences are investigated.
It is claimed that the findings of the study have relevance to hopper pressures for a number of reasons. Firstly, the study clarifies the pattern of pressures which can be expected when a very different constitutive model from the plastic assumption is used. This suggests that the major feature of known pressure distributions is not the assumption of a plastic state, but the predominance of equilibrium requirements on the basic form of the distribution which arises from an assuml~on of unifor- mity for the solid.
Second, it is shown that if gravity is applied to the whole elastic mass at once, the initial filling cot~t ion is far from plastic. This same conclusion could be drawn from the work of Waiters 3, in which an atmrapt to use a plastic assumption for the filling condition led to
Elastic predictions of pressures in~onical~si~t~oppers: J. Y. Ooi and J. M. Rotter
pressures which were clearly wrong 4. By contras~;41~ widely-used initial filling theory of Walker 5 assumed only that the principal stresses were in the vertical and horizontal directions, together with sliding on the hop- per surface, and frustum slice equilibrium. The first of these assumptions leads to an elastic state of stress for all but very rough steep hoppers.
Third, if it is accepted that the initial ffiling state is not plastic, then theories like those of Walker s , or the more general version of M c ~ 6 might be adopted. Both of these imply an elastic state of stress, but both make the assumption that the ratio of normal wall pressure to mean vertical pressure is constant throughout the hopper. The analysis is thus not rigorous, since this assumption is not open to verification. In the present investigation, the uniformity of the lateral pressure ratio will also be examined.
Fourth, the critical pressure distribution for hopper structural design is generally initial filling. Under these Conditions, the elastic solution for hopper pressures pro- vides a useful comparison with the results of ex- periments and a useful alternative to Walker's 5 theory, which is commonly used for this purpose. It shows that a rigoro-_s satisfaction of equilibrium throughout the stored mass leads to slightly different pressure distribu- tions from those of the simple theory.
Finite element formulation
The conical hopper and the contained stored solid are each treated as homogeneous isotropic linear elastic materials without initial strains. This very elementary and certainly invalid assumption is for heuristic pur- poses only. The stored solid and the silo wall are modelled as axisymmetric bodies using the finite ele- ment method 7. The twelve noded cubic isoparametric element with nine point Gaussian integration is used.
A contact element is introduced between the stored solid and the hopper wall to model the wall friction characteristic. It allows frictional sliding to occur only tangentially along the surface between the stored solid and the wall. This finite element has been described fully elsewhere s. The calculation is nonlinear and is con- ducted iterafively until the frictional shear at every point is less than or within a close tolerance (typically 1%) of the maximum sustainable friction.
The characteristic geometry and a typical mesh representing a radial section through the hopper are shown in Figure Ia and Ib.
Classical t lumries on hopper pressu res
Many theories have been proposed for the distribution of pressures in conical silo hoppers. A brief description is given here of the more commonly quoted theories for hopper pressures and their underlying assumptions. More complete descriptions may be found elsewhere 4,9.
Walker theory ~
The differential slice method, originally used by Janssen l° for .cylindrical silos, was also adopted by Walker s, Waiters 3 and Enstad H to derive theories to describe presmte distributions in conical hoppers. Under static or initial filling conditions, Walker 5
Ozt
R . . . . . . .
Stored Solid, Density
Z
_i_ (a)
(b)
Figure I Characteristic geometry, a, hopper geometry; b, typical finite element mesh
Eng. Struct. 1991, Vol. 13, January 3
Elastic predictions of pressures in conical silo hoppers: J. Y. Ooi and J. M. Rotter
assumed that the major principal stress is vertical. This assumption leads to a hydrostatic distribution of vertical pressures
a: = 3,(H - z) + a,, (1)
in which a z is the mean vertical stress at height z above the hopper apex, H is the hopper height and 3' is the stored solid density. The mean vertical stress in the stored solid at the cylinder/hopper transition is given by OZI .
Assuming that the wall friction is fully mobilized, the normal wall pressure distribution is given by
Pi =Fioz (2)
in which
F , - 1 + # c o t B
(3)
Here, /z is the coefficient of wall friction and/3 is the hopper half angle (i.e. the angle between the hopper wall and the vertical). This stress distribution in the solid is often referred to as a 'peaked' stress field.
Under flow conditions, Walker 5 examined a cylin- drical horizontal elemental slice and adopted the assumption that material adjacent to the hopper wall is at passive failure. This leads to
a. - - + azt (4) " n - 1
in which
2BD n = - - (5)
t a n B
B = sin ~ sin(2B + 2ey) (6) 1 - sin ~ cos(2B + 2~/)
2e/= ---r + tan- '# - c ° s - l ( - # ) 2 sin ~x/(1 + #2)
(7)
in which B is the ratio of vertical shear stress to vertical direct stress at the wall. The distribution factor D, which is the ratio of the vertical stress at the wall to the mean vertical stress at any given level, is assumed by Walker to be unity. The predicted normal wall pressure is found as
py = F:Do~ (8)
in which
F: = 1 + sin ~ cos 2~: (9) 1 - sin ~ cos(2B + 2e/)
The resulting stress distribution is often referred to as an 'arched' stress field.
Generalized version of/Walker theory
The two above theories of Walker 5 for filling and flow appear to be very different. It also appears that the filling condition is independent of the failure properties of the material, whilst that for flow appears to depend on the angle of internal friction. This difference clouds the truth that the filling theory is a limiting case of the flow theory, and that the relation for pressures derived for flow conditions has more general application.
If it is assumed that the ratio of wall pressure p to mean vertical stress az is constant at a single value of FD throughout the hopper, then equilibrium of the material in the hopper leads to equation (4). If the total equilibrium of the hopper is also considered, then it is found that the pressures on the hopper wall must satisfy the condition
n = 2 { FDI~otB + FD - I } (lO)
Equations (5) and (6) are then no longer needed, and the only assumptions are equilibrium and the value of the wall pressure ratio, FD. The latter may be derived as by Walker's flow theory (equations (7) and (9)) for the material in a state of plastic failure (thus involving the angle of internal friction), or by assuming that no shear occurs on vertical planes within the solid, which leads to equation (3), which in turn gives n = 0 through equation (10), leading to the Walker filling solution of equa- tions (1) and (2). Other means may be employed to obtain relations for FD, so these are described here as modifications of Walker's theory.
Waiters theory and other modifications o f Walker's treatment
Waiters 3 extended Walker's analysis to allow for the non,uniformity of the vertical stresses on a horizontal slice of the stored solid, assuming that the entire mass is in a state of plastic failure. He assumed that an active stress state pertains on initial filling and a passive stress state during flow. To simplify the equations, it was also necessary to assume that the vertical shear varies linearly with radius. These assumptions lead to the Walker equations (4), (9) and (t0) but With D given by
D = cos~(1 + sin2~o) 4- 2x/(sin2~o - sin2~) (11)
cos~[(1 + sin2~o) 4- 2ysin~o]
in which
2 Y = 3cc (1 - (1 - c ) 3/2) (12)
tan T/~2 c = (13)
\ t a n ~ /
~7=tan_t[ s in(2e+2B)s in~ ] (14) 1 + cos(2e + 2fl) sin ,p
Elastic predictions of pressures M conical silo hoppers: J. Y. Ooi and J. M. Rotter
The positive signs above refer to the initial , ~ g (active) condition and the negative sign~ to the fl~W (passive) state. The angle e for the passive condition is the same as ~( in Walker theory.
If Waiters Itheory is expressed instead in terms of the parameter B, then it should be noted that equation (5) must be adjusted to
2( BD ) n = ~tan/3 + D - 1 (16)
It has been noted 4 that Waiters' solution yields unrealistically high stresses for the initial filling condi- tion unless the hopper half angle is very small: that is, unless
~ s u r e s m hoppers by considering the equilibrium of a fall[tufa Of the hopper bounded by two non-concentric circular arcs. The stresses within the arch are assumed to vary only in direction, not in magnitude. Expressed in terms of the mean stress in the arch a, Enstad's solu- tion for pressures during flow is given by
- I "rYs + a, - (20)
° = X - 1 X - 1
in which s is the meridional distance up the cone from the apex, s, is the meridional distance measured to the top of the hopper and at is the mean vertical arch stress acting at the transition due to cylindrical surcharge. The terms X and Y are given by
1 /3 < -=- (T - 12e;) (17)
2
in which ei is the value of e for initial filling. This limitation implies that the active solution is only useful for very steep hoppers. The passive solution, which is assumed to occur during discharge, is identical to that of Walker if the distribution factor D is taken as 1.0. Unfortunately, the assumption that the shear stress on any horizontal plane increases linearly with radius can be shown n to be invalid if the entire mass is in a passive state of failure and the hopper half angle/3 is greater than
/ 3~ = ~ - tan-l/~ + c°s-~ sin~x/(1 + #2)
(18)
To overcome this limitation, Home and Neddermn._n 12 suggested that, for hoppers shallower than /3m,x, the value of D should be taken as the value for a bopper of half angle /3u"
More recently, McLean ~ suggested that equation (4), which Walker derived as the general solution of the hopper equifibrium equation when the lateral pressure ratio is comnant, should be used to represent the pres- sure distribotion for initial filling. Based on the Jenike and Johamon 2 radial stress field solution, he recom- mended that a value of the parameter n should be deduced as
2/~ n = ~ (19)
tan/3
which corresponds to FD = 1. He suggested that n should be about 3 for typical conical hoppers, though this simplification is not adopted here. The value of n given by equation (19) differs significantly from the value n = 0 which leads to m e Walker filling equations (equations ( 1 ) - ( 3 ) ) .
Enstad theory
Enstad II derived another approximate theory for
sin(2ef +/3) ] 2 sin ~ 1 + (21) X = 1 - sin ~0 sinB
y =
2 s in/3 [ 1 - cos(e I +/3)] + sine/sin2(ef +/3) (1 - sin ~o)sin3(e/+
(22)
in which e! is given by the passive version of equation (15). The normal wall pressures on the hopper can be expressed in terms of the mean arch stress a as in equa- tion (2).
p = (1 + sin ~o cos 2¢f)a (23)
k should be noted that Enstad's analytical expression is for a cylinder-hopper combination. For a hopper without a cylindrical surcharge, it has the uncertain disadvantage that the top boundary surface has an unrealistic spherical form.
Jenike and Johanson theory
The most commonly quoted theory for the pressure distribution in a hopper is probably that proposed by Jenike and his co-workers 2'~-~6. Based on the assump- tion that the pressure distribution near the apex of the hopper is in the form of a radial stress field, Jenike et al solved the equilibrium equations without considering the top boundary condition.
For a hopper without a cylindrical silo surcharge above it, a triangular pressure distribution is assumed such that the radial stress field is cut off by a second linear variation of stress with magr.itudes decreasing towards the free surface at the hopper top. For the static (or initial filling) condition, this radial stress field depends on the hopper half angle/3, the wall friction coefficient ~t and the elastic properties of the stored solid. For the flow (or discharge) condition, the radial stress field depends on the hopper half angle/3, the wall friction c6efficient ~, and the effective angle of internal friction of the stored solid ~,. Vertical equilibrium of the hopper requires that the triangular pressure distribution has a unique peak pressure, pp.
Eng. Struct. 1991, Vol. 13, January 5
Elastic predictions of pressures in conical silo hoppers: J. Y. Ooi and J. M. Rotter
3,R Pp = (24)
(1 + z/H)(tan/3 +/~)
in which the peak pressure is taken to occur at height z, and the whole hopper is of height H.
More recent treatments
Other methods of analysis of stresses in hoppers have also been used. The method of characteristics17 is one way of eliminating assumptions concerning the stress distribution. It was used by Johanson 13, Home and Nedderman 12 and Wilms TM. Arnold et al 4'19"2° have pro- posed modified versions of Jenike's treatment which are easier to use. Finite element methods have also been developed to predict the hopper pressure distribution. Haussler and Eibl 2~ and Link and Elwi 22 have described finite element analyses of mass flow hoppers during discharge, using complex nonlinear material characteri- zations.
By contrast, this paper presents a relatively simple finite element analysis in which the wall and the stored solid are both treated as linear elastic materials, con- nected by a frictional sliding contact surface.
Comparison for a typical mass flow rigid hopper after initial tilting
t3
,w=0.3
Example hopper
An analysis of a typical rigid mass flow hopper with gravity loading alone (no cylindrical surcharge) is presented here first. The hopper wall was built-in at the top and a horizontal top surface was used for the stored solid.
Young's modulus for the hopper wall was taken as 2.0 x 105 MPa and that for the stored solid as 50 MPa, with a Poisson's ratio of 0.3 for both materials. A wall friction coefficient of 0.3 was adopted. The radius at the top of the hopper R was related to the wall thickness t, as R/t = 100. This rather thick wall ensures that wall deformation is minimal and the case resembles a rigid hopper. These values of the governing parameters are taken as the reference values in the later parametric study.
Elastic and plastic stress states
Some existing theories 3'23 are based on the assumption that a state of active failure pertains on initial filling, with the major principal stress trajectories approxi- mately vertical. The hypothesis that the material is at failure was investigated first in the present study using a simple Mohr-Coulomb failure criterion.
The zones in the stored solid in which the stresses are predicted by the present analysis to exceed the yield criterion are shown in Figure 2. Two different values of the effective angle of internal friction ~ (30 ° and 40 °) are used. Only a very small portion of the stored solid is found to exceed the yidd condition, especially if the effective angle of internal friction is large. It should be noted that most bulk solids have an effective angle of in- temal friction in excess of 30 .24. Thus the present analysis is seen to be valid for the larger part of the stored solid.
Figure2 Y i e l d e d z o n e s f o r ~ = 3 0 ° a n d • = 4 0 °
The predicted principal stress field is shown in Figure 3. Near the apex, the major principal stress trajectories are close to being vertical and approximate the pattern of a 'peaked' stress field, as suggested by most writers. However, the present elastic analysis also suggests that there is sufficient eonstdidation during filling for the major principal stress trajectories to approach an 'arched' stress field in much of the hopper.
Normally this would only be expected during flow 2'3'5. Thus, the present finding is in contrast to the assumption used by most previous theoretical studies. Nevertheless, there is considerable exper~nental evidence for pressure distributions of the form predicted here occurring on initial filling 25'~. These experiments lend credence to the present analysis. Thus, existing solutions which characterize the entire bulk solid as in an active or a plastic state (e.g. Walker's flow treatment, Walt=rs' analyses), or with a hydrostatic vertical stress field (Walker's static treatmemt) cannot be expected to give accurate predictions of the wall pressures for initial filling.
The phenomenon of a changing stress field was also investigated by examining the ratio of wall pressure to mean vertical stress, FD, at different heights in the
6 Eng. Struct. 1991, Vol 13, January
X
Elastic predictions of pressures in conical sik) hoppers: J. Y. Ooi and J. M. Rotter
0.8
J: 0 6 -
"E
==
02
,I t , t 0.6 O.B 1.0 1.2 1/, 1.6
Latera( Pressure Ratio (p/o=)
Figure 4 Latera l p ressu re ra t io d i s t r i b u t i o n f o r t h e e x a m p l e h o p p e r
hopper. The result is presented in Figure 4. Near the bottom, the value of FD is as low as 0.87. It then rises steadily for much of the hopper, reaching 1.0 at mid- height, but increasing rapidly towards the top, where 1.6 is achieved. This variation compares with the constant values adopted in applications of Waiker theory: FD ffi 0.55 for Walker filling theory, 1.29 for Walker flow theory, 2.07 for Waiters flow theory, and 1.0 for McLean's proposai.
Pressure distributions
In Figure 5 normal wail pressures deduced from the wail membrane stresses using the membrane theory of shells are compared with the normal wail pressures in the con- tact elements. The two pre~*.,lre distributions should be shr, ilar except aear the top boundary where shell bend- ing phemmmm affect the wall deformations and wall s ~ , which in turn affect the derived wall pressures. The close match between the two pressure distributions indicates that equilibrium is being satisfied properly.
1.0
0.8
£ O.6
c5
0.2
o Normal WaU Pressures ' ~ in Eontact Etement
~ a U Pressures deduced ~o
I I 0.2 O~ 0.6 Dimensionless Normal Wall Pressure (P/1R)
Figure 5 C o m p a r i s o n o f no rma l we l l p ressu res
0,8
The pressure distributions predicted by the present finite element formulation appear to be less susceptible to spurious variations within each element than those of similar studies (Compare, for example pressure distribu- tions calculated by Mahmoud and Abdel Sayed 27, and Link and Elwi22).
The wall pressure distribution obtained from the pre- sent analysis is compared in Figure 6 with several pre- vious solutions, assuming the effective angle of internal friction ~, is 30 °. The present predictions lie near McLean's theory near the hopper bottom, but closer to Walker's flow theory towards the hopper top. None of the existing analytical solutions provides a really close fit to the elastic prediction, but the distributions are very similar in form. This same pattern can be observed for a wide range of hopper angles, wail frictions and hopper flexibilities.
Towards the hopper top (zlH > 0.9), the present results are quite close to the Walker and Waiters flow theories. This should be expected, as the stress field in the stored solid is not far from the 'arched' form
Eng. Struct. 1991, Vol. 13, January 7
Elastic predictions of pressures in conical silo hoppers: d. Y. Ooi and d. M. Rotter
Figure 6 Wall pressures for a typical rigid hopper
' l ~ I - - - - - 7 - " ~ - - ' - - ~ , '
. . szs - " i
- . -o-- Present Study l
1 ~ " / / - - - - w,,te,s Fi0~ i
, ¢ " ~ I I i 1 I 01 0.2 0.3 04 05 0 6
Dlmensionless Ctrcumferentla[ Stress (NB/IcR 2)
Figure 7 Circumferential wall stresses in a typical rigid hopper
throughout much of the hopper (Figure 3), though it is far from fully plastic.
Jenike and Johanson's 2 triangular pressure distribu- tion is also shown in Figure 6 for comparison. The peak pressure in this treatment is dependent on the wall fric- tion coefficient #, the hopper angle/9 and the elastic pro- perties of the stored solid. The locus of peak pressures is marked in Figure 6 to indicate the range of possible distributions. Two alternative values of Jenike and Johanson's parameter k are used here to indicate alter- native distributions which might be deduced from their analysis based on values they propose. It is interesting to note that the radial stress field elastic solution using k = 0.9 is closely asymptotic to the present elastic pre- diction. However, only at the very bottom of the hopper is this radial stress field valid. If the maximum pressure is chosen to occur at the bottom of the hopper, Jenike and Johanson's solution matches Walker theory exactly for initial filling conditions.
Waiters' solution for the initial filling condition is not included here because the hopper half angle B exceeds the limiting value give by equation (17), with the im- plication that the vertical stresses in the solid exceed hydrostatic values, which seems unlikely. Enstad's solu- tion predicts slightly larger wall pressures because it assumes a spherical top surface,
Wall stress distributions
The normal pressure and the frictional shear exerted by the stored solid on the wall generate a meridional stress and a circumferential stress in the hopper wall. Indeed, the purpose of predicting hopper pressures is primarily to predict hopper wall stresses for structural design. It is therefore useful to examine the wall stresses, and to compare the stresses predicted by different theories. The wall stresses are shown in Figures 7 and 8. Results are shown for two different effective angles of internal friction @ (30 ° and 40°). A= _;~ expected from the mem- brane theory of shells, the hopper circumferential stress distribmion peaks further from the apex than the n o ~ wall pressure distribution (Figure 6). Near the top of the hopper, .where a structural discontinuity exists, shell bending phenomena influence the circumferential mere-
02 OL 06 08 10 Dimensionless Mer~dmnat Stress (6N®sinl3/l(R z)
Figure 8 Meridional wall stresses in a typical rigid hopper
t 2
brane stress signifmantly. This effect is seen in the pre- sent analysis, but cannot be modelled by classical lmoper pressure theories. Shell bending effects on the meri- dional stress are minimal, as expected 2s.
The meridional stress depends on both the normal pressures and the frictional tractions. At the top of the hopper all theories must give the same value for the meridional stress, as it is governed by the static global equilibrium of the hopper and its contents. The bopper top meridional stress is given by
,yR 2 N . , - - - ( 2 5 )
6 sin B
Also shown in Figures 7 and 8 are the corresponding wall s t r~s predictions of t h e Walker, WaRers and McLean theories coupled with the ~ theory of shells. It should be noted that shell membrane theory accurately reflects the real ~ in the body of the hopper, but the support ~ ~ adjacent to the hopper topL Walker s static solution does not
8 Eng. Struct. 1991, Vol 13, January
Elastic predictions of pressures in conical .silo!hoppers: J. Y. Ooi and J. M. Rotter
change with the internal friction angle. The two flow theories and McLean lead to patterns of hopper wall stress which are very similar to that of the present analysis, though, for the flow theories, the differences increase With increasing internal friction.
It is clear that the differences between the predicted hopper pressure patterns (Figure 6) are larger than the differences between the resulting wall stresses (Figures 7 and 8). The question of the 'right' hopper pressure distribution should therefore not be treated as too serious a matter. Walker flow theory and McLean give the closest comparison with the present analysis for both hopper wall stress distributions.
Parametric study of mass flow hoppers
The reference hopper parameters
In the following parametric study of mass flow hoppers without cylindrical surcharge, the effects of varying the wall friction coefficient /~, the hopper half angle ~, the bulk solid elastic modulus E~ and Poisson's ratio J,~ are explored. The hopper analysed in the previous section of this paper is taken as a reference ex- ample in this parametric study. Pressures from Walker flow theory (~, = 30 °) and McLean are also plotted for comparison.
Effect of the wall friction coefficient
The coefficient of wall friction /~ was varied to in- vestigate its effect on hopper pressures. The dimen- sionless nornud wall pressure is shown in Figure 9 for three different values of the wall friction coefficient. The present elastic analysis predicts that the normal wall pressures decrease throughout the hopper as the wall friction coefficient increases. This effect is also seen in the Walker and McLean theories.
The maximum normal wall pressure is found to occur closer to the bopper bottom as the wall friction coeffi- cient is reduced. For very smooth hoppers (~, = 0.1), the maximum pressure is near the bottom, giving a pressure distribution more like that of Walker filling theory. McLean's theory moves the locus of this maximum in a
Figure 9 Norma l wa l l p ressures fo r va r y i ng wa l l f r i c t ion coe f f i c i en ts
similar manner, though the peak value is different. By contrast, Walker flow theory suggests that the locus of the maximum rises when the wall friction is reduced.
The corresponding dimensionless circumferential and mcridion~ wall stress resultants are shown in Figures I0 and 11. These stress resultants are important because they are the real goal of hopper pressure predictions. The Circumferential stresses (Figure 10) are well related to the pattern of the normal wall pressures, in the manner expected from shell membrane theory. With in- creasing wall friction, the meridional stress (Figure 11) is predicted to decrease throughout most of the hopper except near the top where a slight increase occurs.
By contrast with this finding, Walker flow theory predicts that the meridional stress should increase throughout the hopper as the wall friction coefficient in- creases. Waiters flow theory matches the present finding in this matter, even though these two classical theories only differ in the distribution factor D.
Eng. Struct. 1991, Vol. 13, January 9
Elastic predictions of pressures in conical silo hoppers: J. Y. Ooi and J. M. Rotter
10
= 0 6 . . . . . . . . McLean z~ ~ P r e s e n t Study \
~04 7 ~ / / /
02
0 2 Or, 06
Dimensionless Normal Wail Pressure (p/l~R)
Figure 12 ra t io
08
N o r m a l w a l l p ressures f o r v a r y i n g bulk so l id Po i sson ' s
o
c
c3
, I I 02 O~ 06 08 Dimensionless Normal Watl Pressure (p/l"Ri
Figure 13 N o r m a l wa l l p ressures f o r v a r y i n g bu lk so l i d -wa l l s t i f f n e s s e s
Effect of Poisson's ratio of stored solid
The Poisson's ratio of the stored solid might be expected to affect the hopper pressures. Dimensionless normal wall pressure distributions are presented in Figure 12 for varying values of the bulk solid Poisson's ratio ~,s. The effect of Poisson's ratio is seen to be very small, but the maximum normal wall pressure increases slightly as Poisson's ratio increases, and the normal wall pressures near the top and near the bottom of the hopper decrease. McLean's prediction again gives a good fit to the present results, giving the appearance of being valid for 1, = 0.5.
Effect of stored solid and hopper stiffnesses
Jenike and Johanson 2 suggested that the stored solid compressibility should alter the pressure distribution significantly. McLean 6 also believed this, and added that a flexible hopper wall would have the same result. The effect of the relative stiffness of the stored solid and the hopper wall were explored by varying the modular ratio of the stored solid and the hopper wall EJE,.. Both a thick-walled hopper (R/t = 100) and a thin- walled hopper (R/t = 1000) were examined so that both stiff and flexible hoppers could be investigated.
There is little experimental data or agreement on appropriate precise values for the elastic modulus for most stored solids. However, it has been suggested 23"29 that the elastic modulus of the stored solid Es may be estimated as
E, = Kva~ (26)
in which Kv is termed the modulus contiguity coefficient 29, and is a constant varying between about 70 for dry grains and about 100 for loose dry sand but up to 200 for dense-packed hard granules. Here the practical range of the bulk solid modulus Es is taken as 5 MPa to 500 MPa.
The dimensionless normal wall pressure distributions are shown in Figure 13. As would be expected from the theory of elasticity, it was found that the wall pressure distributions are almost identical for fixed values of the
parameter
E~ R o~ - - - ( 2 7 )
Ew t
This has been widely verified, and is indicated in Figure 13 by the three results shown for c~ = 0.25, which derive from the three combinations E, = 500 MPa, E w = 2 × 1 0 sMPa, R/t=lO0; E , = 5 0 M P a , Ew= 2 x 1 0 S M P a , RIt=lO00 and E , = 5 0 M P a , E, ,= 2 x 10 4 MPa, /tlt = 100. As the value of (z increases from zero, very little change in the calculated pressures occurs until ~ reaches about 0.1. Thereafter, the peak of the distribution begins to occur at lower points in the hopper, and to increase very slightly. Pressures near the top and bottom of the hopper (z/H < 0.4, z/H > 0.8) correspondingly change slightly.
These changes in the pressure distribution occur only in very thin-walled hoppers containing very stiff solids. The highest practical value of o~ is found for a very thin steel hopper for which R/t might reach 1000. For c~ to be as high as 0.1, the bulk solid elastic modulus must then exceed 20 MPa. Adopting equation (26) with a value of ~v = 100, the vertical stress must exceed 200 kPa for the effects of bulk solid compressibility and hopper flexibility to make any difference to the wall pressure distribution. Such a b.igh value of the required vertical stress in a stiff solid with a very light hopper in- dicates that the effects of material compressibility and hopper wall flexibility should be negligible in virtually all practical situations.
Another conclusion may also be drawn from the insen- sitivity of the pressures to the assumed stored solid modulus. It indicates that equilibrium (and not kine- matic) considerations probably dornirmte the pattern of initial fitlmg hopper pressures. The use of a nonlinear constitutive model for the bulk solid may therefore make little diffea-ence to the calculated hopper pressures.
Effect of hopper half angle
Dimensionless normal wall pressure distributions for a
10 Eng. Struct. 1991, Vol 13, January
Elastic predictions of pressures in conical silo hoppers: J. Y. Ooi and J. M. Rotter
10
08
= OL,
E
02
\ \ \% \ ~ , \ "~/\./[ ,,' \ \ \"~ k ~o~,? °" V / r .Z.,-"
~=60o~ ~ z
/ -- ~ - - - ~ McLean
02 0, 06 0.8 ,0 Dimensionless Normal Wall Pressure (p/'llR)
Figure 14 Normal wal l pressures for varying hopper half angles
~ r s , the present work suggests that material failure ~ y l i O t ~ :~s~ntial in satisfactory hopper pressure predictions for the initial filling state. However, it is not suggested here that material failure properties are not vital in predictions relating to bulk solids flow.
The parametric studies presented here give some in- sight into the characteristics of these pressure distribu- tions. An increase in the wall friction coefficient decreases the wall pressures. Changes in the assumed elastic modulus and Poisson's ratio of the stored solid have very minor effects on the wall pressures. In very shallow hoppers, the maximum pressure tends to occur towards the bottom. As the hopper is made steeper, the maximum pressure occurs progressively higher up the hopper.
More complex material characteristics can easily be introduced into the present model, but the present study suggests that these may not alter the predictions substan- tially in many practical cases.
range of hopper half angles # are shown in Figure 14. In these calculations, the radius at the top of the hopper has been kept constant, so that the hopper becomes shallower as the hopper half angle increases. This results in a smaller total load in the hopper, and consequently reduced hopper pressures.
The present elastic predictions are remarkably similar in form to those of the Walker flow and McLean theories for a wide range of hopper half angles. This indicates again that equilibrium considerations dominate the pat- tern of hopper pressures.
Conclusions
Theoretical predictions for the pressures on the walls of conical hoppers have been presented. These were obtained using a finite element analysis in which the stored solid was treated as a homogeneous linear elastic body with frictional sliding against the hopper wall. This heuristic analysis was undertaken to explore the nature of the pressure distributions arising from an elastic assumption, but it has been shown to be able to give many satisfactory predictions of wall pressures for the initial filling state. The initial filling state has previously been shown to be the critical loading case for mass flow steel binsL
Elastic pressure distributions were obtained which relate closely to the modification of Walker theory by McLean s, and slightly less closely to the flow theories of Walked and Waiters 3, although all these theories have a very different basis from the present analysis. The pressure distributions were shown to be quite insen- sitive to the assumed elastic properties of the solid, so that uncertainty about the proper value for these parameters is seen to be unimportant. These considera- tions suggest that the form of hopper pressure distribu- tions is dominated by equilibrium and the assumption of homogeneity, and that const;tutive laws may play a lesser role.
It was found that stress states in most of the stored bulk solid are not near the Mohr-Coulomb failure sur- face. Although much of the literature suggests that material failure properties determine wall pressures in
Notation
B
D
F
k n
N,,N
P R S
t
X , Y z
ratio of vertical shear stress to vertical direct stress at the wall ratio of vertical stress at the wall to the mean vertical stress in the solid Young's modulus of stored solid, hopper wall ratio of normal pressure to vertical stress at the wall Jenike radial stress field parameter pressure relation exponent circumferential, meridional membrane stress resultant normal wall pressure radius of hopper at the top surface (transition) meridional distance up the inclined wall (origin at apex) thickness of the hopper wall terms used in Enstad's theory (equation (20)) vertical coordinate (origin at apex)
ot
-y Kv ~t vs, Vw oz 0
hopper-bulk solid relative stiffness parameter (equation (27)) hopper half angle unit weight of stored solid modulus contiguity coefficient (equation (26)) wall friction coefficient Poisson's ratio for bulk solid, hopper wall mean vertical stress in stored solid mean stress in arch (Enstad's analysis) effective angle of internal friction
Subscripts
f during flow i after initial filling or storing t at the cylinder/hopper transition
References
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Eng. Struct. 1991, Vol. 13, January 11
Elastic predictions of pressures in conical silo hoppers: d. Y. Ooi and d. M. Rotter
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