DOI 10.1140/epje/i2010-10581-7

Regular Article

Eur. Phys. J. E 31, 311–314 (2010) THE EUROPEANPHYSICAL JOURNAL E

Effect of the ratios of diameter of silo to bead on the pressurescreening in granular columns

A. Qadir1,a, H. Guo1, X. Liang1, Q. Shi1,b, and G. Sun2,c

1 Department of Physics, Beijing Institute of Technology, Beijing 100081, China2 Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

Received 18 August 2009 and Received in final form 21 December 2009Published online: 23 March 2010 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2010

Abstract. We present the apparent mass measurements at the bottom of granular packings for differentbead and silo sizes. The redirection parameter K in Janssen theory is found to increase with the ratios ofthe diameter of the silo to the bead. We attribute this feature to the friction between the beads and theconfining wall of silo; it is the role of friction that leads to variations in the shielding of vertical stresses aswell as pressure screening.

1 Introduction

The mechanical behavior exhibited by granular media isusually different from other forms of matter such as liquid,solid and gas due to the complex force distribution amonggrains. The study of force distribution in dry, cohesionlessgranular media has attracted much attention both fromscientific and engineering community [1–4]. However, thehighly heterogeneous and anisotropic nature of force net-work in static granular media hinders in presenting a rig-orous macroscopic theory of granular mechanics. Now, thestatic granular media such as confining granules in cylin-drical column have emerged as an important system toinvestigate the propagation of force and also deformationof force chains in disordered material. Any advancementin better understanding the distribution of forces chainsspecifically in response to forces that are exerted at thesystem boundaries is bound to have vast applications incivil engineering and geophysics [5]. The pioneering workof Janssen [6,7] has demonstrated that, unlike the hydro-static case, the mass measured below a granular columndoes not increase linearly with the height, on the contraryit reaches exponentially a limit. The apparent mass is onlya part of the total mass of the grains in the column, whilethe rest is supported by the side walls. This is due to theshielding effect redirecting most of the weight toward theconfining walls rather than to the base of the silo. Sincethen, many experiments have been conducted in silos toexplore the suitability of Janssen model under variousconditions, such as using fixed or movable side walls [8,9].

a On leave from: Govt. Degree College Tando Jan Moham-mad, Sindh, Pakistan; e-mail: sujjad2000@yahoo.com

b e-mail: qfshi123@bit.edu.cnc e-mail: gsun@aphy.iphy.ac.cn

In particular, Vanel et al. [10] have devised a robust andnovel experimental set-up, and checked different theoriesfor stress propagation in granular media. The Janssen lawwas checked against the theory of Incipient Failure Every-where (IFE) [11] and the Oriented Stress Linearity (OSL)model [11]. The conclusion was in favor of OSL model.

In previous work [12] by fixing the diameter of grainsand varying the width of the container, some of us have ex-perimentally verified that the saturation mass in Janssenmodel obeys the three power function of the diameter ofthe silo. However the more important parameter K, whichis the ratio between horizontal and vertical stresses, wasnot explored. In this paper we focus on the deflection ofvertical stresses by varying both the diameter of grainsand silos, and investigate the dependence of K on variousratios of diameter of the silo to the bead.

2 Experimental set-up

The schematic diagram of the experimental set-up isshown in fig. 1. A mass M of grains is uniformly pouredby a funnel placed at the top of the silo. The bottom partof tube is formed by a cylindrical piston, that is so alignedthat during motion neither touches the wall of tube nor al-lows the leakage of grains. The bottom end of the piston isscrewed onto a pressure sensor allowing one to determinethe force of the beads on the upper horizontal surface ofthe piston. In such a case the average static pressure is en-tirely transmitted to the sensor, and a computer recordsthe data throughout the process. The corresponding forceexerted by the beads on the sensor will be referred to asapparent mass Mt, since its value is different from thereal filled mass Mf . An electric motor controls the de-scending speed of the piston. For each experimental run

312 The European Physical Journal E

Fig. 1. Sketch of the experimental apparatus and fillingmethod.

the piston is first inserted inside a vertical cylinder to acertain height and then a mass M of grains is poured uni-formly via a funnel placed at the top of cylinder. A briefrelaxation time (normally 30 to 60 seconds) is allowed af-ter the grains are filled into the cylindrical column, so thatthe elastic energy accumulated by the sensor may be re-laxed. When the cylinder is filled with granules the pistonis allowed to descend at a slow velocity (0.02mm/s) ona total distance of 20mm. The slow downward motion ofthe piston could fully mobilize the friction force betweenthe granular column and the cylinder wall.

3 Measurement procedure

In the present work seven silos and six beads of differentdiameters are used separately. Two types of experimentshave been carried out. In the first experimental trial dryand mono-disperse glass spheres of diameter d = 2mm,3mm, 4mm, 6mm, 7mm and 8mm are uniformly pouredin a silo having internal diameter D = 43.6mm. Figure 2displays the typical plots of the apparent mass Mt as afunction of time for different bead diameters. In the nextexperimental trial for the same bead diameter d = 3 mm,silo sizes D are varied from 25.4 mm, 29.5 mm, 33.1 mm,39.4mm, 43.6mm, 46.3mm and 53.8mm. For such casesthe apparent mass Mt scaled with D3 as a function oftime is illustrated in fig. 3. It can be observed that at theonset of tube the apparent mass decreases abruptly andeventually a relatively steady state is attained with smallerfluctuations. Here, we take the statistical average of 500data points in the sequence of time series as a measuringvalue of Ma. To assure the reproducibility and also reducethe error, the experiment is done 5 times and then theaverage value of these is taken as a data point Ma.

4 Results and discussion

Figure 4 depicts the dependence of the apparent mass Ma

on the filling mass Mf . The saturation curves follow quali-

Fig. 2. Time evolution of the effective mass for various beadsizes.

Fig. 3. Time evolution of the scaled apparent mass for a seriesof silos.

tatively the predictions of Janssen’s model for static pack-ing. The scatters are experimental results, and the linesrepresent a fitting from all the data for different diame-ters of granules. It can be observed that for low values offilling mass, Ma is approximately equal to Mf which sug-gests the hydrostatic behavior of granular material in silo.As the filling mass is increased, Ma seems to illustrate asaturated state implying that the screening effect is nowdominant, while the state has some fluctuations. We ob-serve that the form exhibited by each stress saturationcurve is almost similar, however, they branch off with theincrease in grain diameter. This phenomenon inspires usfor further investigations because the stress transmissionin granular media is affected by the bead size. In orderto reveal the reason of splitting of data lines, we use theJanssen model which is still used in designing silos despiteof its simplicity.

The model is based on the assumptions that in a silohorizontal stresses are proportional to vertical stresses, thefrictional forces are in yield criterion and the medium is

A. Qadir et al.: Effect of the ratios of diameter of silo to bead on the pressure screening in granular columns 313

Fig. 4. Apparent mass versus filling mass for different grainsizes.

considered as continuous. Janssen model describes thatthe mean pressure at the bottom does not increases lin-early with the height of filling. Instead, it saturates to avalue depending on silo radius and friction between thewall and grains. The relationship between the filling massMf and the apparent mass Ma is the following:

Ma = Ms(1 − e−Mf /Ms), (1)

whereMs = ρπ(D/2)3/(2μK), (2)

ρ is the density of the material, μ is the coefficient of fric-tion between grains and wall, H is the height of the gran-ular column, and K is a redirection parameter. It maybe noted that eq. (2) does not provide any relationshipbetween the diameter of granules and saturation mass.However, we can use eq. (1) to simulate the data pointsline and then the values of Ms can be determined. Thefitting lines of this procedure are not shown here. Follow-ing this procedure Ms for different d is further separatelysimulated and decided, so that the variation of Ms with dcould be ascertained. Furthermore, considering that all theobtained Ms should change with the diameter of granulesaccording to a uniform law, we fit them so as to get themodified saturation mass M ′

s, and from the fitting curvewe obtain the value of M ′

s. Taking all modified M ′s into

eq. (1) again, the modified Ma can be calculated and thefitting lines could be easily obtained as shown in fig. 4,which are more accurate than separately simulated witheq. (1) for every Ms. The relationship between M ′

s and dis depicted in the inset of fig. 5. It is revealed that M ′

s isan increasing function of d.

What is the mechanism that results in the variationof Ms with d? According to eq. (2), Ms varies only ifK adopts different values because the other parametersare constant. Using eq. (2), we can determine the valuesof μK, so as to get the variation of K with the ratio ofdiameters of the silo to the bead D/d, as shown in fig. 5.Since μ is a constant, so it does not affect the behavior

Fig. 5. μK as a function of D/d.

exhibited by K. In fig. 5 the down triangles represent thecase where, for the same diameter of silo D = 43.6mm,the diameters of granules are varied from 2mm, 3mm,4mm 6mm, 7mm, to 8mm. The up triangles illustratethe case where the bead diameter d = 3mm is kept thesame, while the silo diameter D has been changed from25.4mm, 29.5mm, 33.1mm, 39.4mm, 43.6mm, 46.3mmto 53.8 mm.

From fig. 5 it can be observed that μK increase withD/d till a point is reached where the two data lines coin-cide. Beyond this point down triangles seem to saturatesomewhat asymptotically, while the up triangles still tendto increase. These observations indicate that the redirec-tion parameter K works well in converting vertical stressesinto horizontal ones for the cases of larger D/d ratios thanthe smaller ones.

Also note that the larger values of the ratio D/d repre-sent a higher number of grains while the lower ones meansa lower number of grains in a silo. The larger ratio D/dalso exhibits higher μK as illustrated in fig. 5. The pres-ence of a larger number of beads in a cylinder will havemore contact points with the confining walls due to whichthe conversion of vertical stress into horizontal one is rein-forced. In such cases the screening effect becomes stronger,consequently μK exhibits increasing values. Generally, itis assumed that the Janssen effect is a consequence of arch-ing [13]. Hence, one might expect that the force shieldingshould be reinforced as the ratio of the particle size to thediameter of the silo is reduced. However, our results re-veal that not only this is not true but an opposite trend isobserved. The data obtained illustrate that the shieldingeffect is a macroscopic one and it should not be attributedonly to the arching, but also to the friction between grainsand the confining wall of the silo.

5 Conclusion

The apparent mass for various bead and silo sizes atthe bottom of granular columns has been systematicallyinvestigated. The results indicate that the redirection

314 The European Physical Journal E

parameter K in Janssen theory reveals an increase withthe ratio of diameter of the granular columns to the beadsD/d. It is evidenced from this study that the friction be-tween the beads and the confining wall causes a varia-tion in the shielding of vertical stresses as well as pressurescreening. The presence of a large number of beads in asilo enhances the deflection of vertical stresses into hori-zontal ones. Our experimental results may be helpful inbetter understanding the complex force network in a silo.

This work is supported by the Chinese National Science Foun-dation, Project Nos 10675018 and 10674157.

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