SILAGE PRESSURES IN TOWER SILOS. PART 1.THEORETICAL AND DESIGN CONSIDERATIONS

S.C. Negi and J.R. Ogilvie

Agricultural Engineering Department, Mactlonaltl College of McGill University, Ste. Anne De Bellevue, Quebec

Received 23 December 1975

Negi, S.C. and J.R. Ogilvie. 1977. Silage pressures in tower silos. Part I. Theoretical and design considerations. Can. Agric. Eng.19: 92-97.

An analysis of the active and passive pressure fields developed by silage materials in cylindrical tower silos is presented in thisstudy. The solutions are obtained in closed form and consider the variation of silage density with depth of fill, internal friction ofsilage, wall friction, diameter and height of the silo. The practical application ofthe results in predicting silo wall loadings in lightof the unloader systems is discussed. Finally, the implications for the structural design of tower silos are examined.

INTRODUCTION

The estimation of pressures exerted bysilage materials on the containing silostructure continues to be a critical task fordesign engineers (Mohsenin 1970). Theavailable design formulas are empirical andtake no account ofthe physical properties ofthe contained material.

In allied fields, the Janssen theory (Jan-ssen 1895) is most widely used by silodesigners. In his theory, Janssen made twoassumptions: first, that stresses are independent of the horizontal coordinates and,second, that the ratio of lateral to verticalpressures, A", is a constant relative to depthof fill. The apparent weakness of predictionsbased on the Janssen equation lies mainly inthe selection of a proper value for thepressure-ratio K.

The bulk density ofthe stored material isalso assumed constant in the Janssen for

mula. Thus the validity of the Janssenequation for compressible materials likesilage is questionable because experimentalobservations (Otis and Pomroy 1957; Al-drich 1963) indicate that silage density variessignificantly with the depth of storage.

It is the purpose of this study to considerthe pertinent parameters and set out arational procedure for determining pressures developed by silage materials in cylindrical tower silos.

NOMENCLATURE

a - parameter: variation of unit weightb - parameter: distribution of vertical pressurec - subscript: conditions at silo axisD= diameter of silo/ = frictional stressM- moisture contentn = parameter: n = 1 for active case, n = -1 for

passive casep = lateral pressureq = vertical pressureR = radius of silor = radial coordinate5 = circumferential pressurew - subscript: conditions at silo wallz = depth coordinatefj - coefficient: unit weight7 = unit weighty„= initial unit weightfi = wall friction angle

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Figure 1. Variation of pressure on a volume element in r and r directions.

CANADIAN AGRICULTURAL ENGINEERING. VOL. 19 NO. 2. DECEMBER 1977

1

•

8 - circumferential coordinate0 = effective angle of internal friction0 = angle between major principal pressure axis

and vertical axis.

TERMINOLOGY

Definitions of terms as given by Jenike et al.(1973) are: "Active pressure field — field in whichthe major pressure is vertical and /or close tovertical. Passive pressure field —field in which themajor pressure is horizontal and/or close tohorizontal. Mass flow — flow pattern in which allsolid in a bin is in motion whenever any of it isdrawn out. Funnel flow — flow pattern in whichsolid flows in a channel formed within stagnantsolid."

This nomenclature and terminology will beused in the companion papers.

THEORETICAL CONSIDERATIONS

Pressure Equilibrium Equations

Consider a volume element of silage in atower silo, bounded by two concentriccylindrical surfaces, two horizontal surfacesand two axial planes. Figure 1 shows thevariation of pressures acting on the sides ofthe volume element that contribute to itsequilibrium in the r and z directions asdefined in cylindrical coordinates. The silagematerial is assumed to be compressible,frictional, cohesive, isotropic and treated asa continuum. The pressure components areconsidered to be independent of the circumferential coordinate 6. The unit weightof the silage material y is assumed to be afunction of the depth coordinate z alone.

Bysumming up the forces on the volumeelement in the z and r directions, making thesmall angle approximations and neglectinghigher-order terms, the following equationsof equilibrium are obtained.

h + dI +L= y (1)9z 3>

= 7

9/ dp p= 0 (2)

wherep, q, s denote the lateral, vertical andcircumferential pressure components, respectively, and/is the frictional stress.

Variation of Unit Weight of Silage withDepth

A great deal of experimental work on thevariation of unit weight of silage materials intower silos was done by Perkins et al. (1953)Otis and Pomroy (1957) Shepherd andWoodward (1941) and Eckles et al. (1919)and culminated in the work of Aldrich(1963), who reduced the available data to auseful form. A suitable expression wasdeveloped by the present authors to describethe experimental curve obtained from the"unit weight — silage depth" data compiledby Aldrich.

The proposed equation is an exponentialfunction of the form

7(z) = 70 + a(l .."*> (3)

n

» 10

e

UJ12

14

16

ASAE DATA \ ASAE D252

( A.E. YEARBOOK )

r= r0 + a ( i •

529.7

a =516.2

/9= 0.181

20-

22200 400 600 800

UNIT WEIGHT Y , Kg per cu meter

Figure 2. Unit weight of silage in tower silos.

1000 1200

The coefficients y„, a and )8 were determined from the experimental data with theaid of a nonlinear least-squares curve fittingcomputer program. The silage unit weightscomputed from the proposed function areplotted in Fig. 2along with the experimentalcurve.

Relationships Between PressureComponents

AI silo axis

The Mohr circle representing the stressconditions for an active pressure field isshown in Fig. 3. Along the axis of symmetryof a tower silo the frictional stress is zero andthe major principal pressure acts in thevertical direction. For a silage material in thestate of plastic equilibrium, the stress circlemust touch the effective yield locus (Jenikeand Johanson 1969).

From the Mohr stress circle in Fig. 3, theangle between the direction of the majorprincipal pressure and the vertical axis isi//c = 0, and the ratio of lateral to vertical

pressure is given by

Pc _ 1

He '

sin <p

where </> is the effective angle of internalfriction to be used for cohesive solids likesilage materials (Jenike 1964), and thesubscript c refersto conditions at the axis ofsymmetry.

In a passive state of pressure, the majorprincipal pressure acts in a horizontaldirection at the silo axis and thus the majorpressure angle i/ft = it12. The Mohr circle inFig. 4 represents the stressescorrespondingto the passive condition. The ratio of lateralto vertical pressure is now expressed by

1 + sin 0Pc

1c 1 - sin 0

The circumferential pressure components can be related to the lateral pressure byutilizing the Haar-von Karman hypothesis(Haar and von Karman 1909). The hypothesis states that in axial symmetry the

CANADIAN AGRICULTURAL ENGINEERING. VOL. 19 NO. 2, DECEMBER 197793

94

STRESSES AT THE

AXIS OF SYMMETRY

STRESSES NEAR

SILO WALL

COMPRESSIVE STRESS

Figure 3. Mohr circles representing the active state of pressure in a tower silo.

COMPRESSIVE STRESS

Figure 4. Mohr circles representing the passive state of pressure in a tower silo.

CANADIAN AGRICULTURAL ENGINEERING, VOL. 19 NO. 2, DECEMBER 1977

circumferential stress is equal to either themajor or the minor stress of the meridianplane. Accordingly, s is the minor principalpressure for the active case, and the majorprincipal pressure for the passive case.

Therefore,

V = Pc = Qc ( t) f°r tne active case, and1 + sin 0

, 1 + sin 0. r .sr -pc -qc ( J) for the passive case.

1 - sin 0

The relationships between the lateral,vertical and circumferential pressure components for active and passive conditionsalong the axis of symmetry can be conveniently written in a single expression:

A - n sin0 (A.p = s = q ( —) (4)L L L 1 + n sin 0

in which

n - 1corresponds to the active pressure fieldn - -1 corresponds to the passive pressurefield.

At silo wall

At the wall of a silo, the vertical andlateral pressures are no longer the principalpressures because wall friction introduces avertical shear stress which distorts thepressure field. The stresses in the materialadjacent to the silo wall must therefore berepresented by an additional Mohr circle.From the Mohr circles representing thestresses near the wall for active and passiveconditions in Figs. 3 and 4, respectively, thefollowing relationships between the pressurecomponents can be obtained.

(1 - sin 0 cos 2 ^yy) ...Pw =Qw (1 +sin0cos2*w) Pi

_ sin 0 sin 2 tyw (f-.Jw =Qw (l+sin0cos2*w) W

(1 -n sin0) m5vv =^u' (1 +sin0cos2^w) U)where the subscript w refers to conditions atthe wall.

The angle between the direction of themajor principal pressure and the silo wall isgiven by,1, ,1 -wx7T n r . -l.sin 5 X1 ,0.*»-<—>2 +2tan l*$-nd]-lS)where 6 is the angle of friction between thesilage material and the silo wall.

Method of Integral Relations

The method of integral relations, stemming from the work of Dorodnitsyn (1962),has been widely applied to solve a variety ofgas dynamics problems as elucidated inBelotserkovskii and Chushkin (1965), aswell as to analyze boundary layers in fluidmechanics. Savage and Yong (1970) werethe first to use the integral method for thedetermination of stresses developed by co-hesionless granular materials during slowcontinuous mass flow in wedge-shaped andparallel wall bins.

In this method, the integration of initial

differential equations containing partial derivatives reduces to systems of ordinarydifferential equations. This is accomplishedby subdividing the region of integration intostrips and integrating the set of partialdifferential equations across these strips.After this, the unknown functions occurringin the integrands are replaced by interpolation expressions of the most generalform.

Consider the governing pressure equilibrium equations 1 and 2 for the tower siloproblem. In view of axial symmetry it isadequate to study only the half-region0 < r ^ R, where R is the radius ofthe silo.Multiply each of the equations 1and 2 ofthesystem by r and integrate with respect to theindependent variable r from the axis ofsymmetry r = 0 to the silo wall r = R.

0/*(^+4/+')d'=*z>T (9)0 oz or 2

JR (r¥ +rfy +p-s)dr =0 (10)° oz or

In order to eliminate the partial derivatives with respect to one variable, theapproximations in the variation of pressuresare constructed with respect to the independent variable r. The distributions ofpressurecomponentsqand s (evenfunctionsof r), and / (odd function of r) across theradius of the tower silo are assumed to bepolynomials(Belotserkovskii and Chushkin1965; Savage and Yong 1970) of the type

q=qc(z) +b(z){£)2 (ID

,,Sc +(!w^)(I) (12)

/ =/w(z)(£) <13>Substituting equations 11, 12, 13 and the

empirical relation of unit weight of silage —equation 3 — into equations 9 and 10, andintegrating:

R dqc R db _Rr _e-fc)l1 tfz" +4 dz+fw~2[Jo ail >J

(14)

i^-!*-^**-0 (15)Solution of Equations

At the silo wall r - R, and the verticalpressure at the siloaxis from equation 11 isconsequently

qc = q*-b(z) (16)Insert equations 6 and 16into equation 14toobtain

Rdqv sin 0 sin 2 ^H2 dz (1 +sin 0 cos2 Vw)'w 4 dzflw

Rdb

the following equation is obtained

4R sin 0 sin 2 ^w (1 + n sin 0) dqw9(1 + sin 0 cos 2 ^w)(l - n sin 0) dz

{n - cos 2^w)(n sin2 0+7 sin0)3(1 - n sin0)(1 + sin0 cos 2 ^w)

qw + b = 0

.(18)

Differentiation of equation 18 with respectto z yields

4R sin 0 sin 2 tyw (1 + n sin 0) d2qw9(1 + sin0 cos 2 ^w)0 - n sin0) dz2

(n - cos 2 tyw)(nsin 0 + 7sin0) dqw db ~3(1 - n sin0)(1 +sin0cos 2 ^w) dz dz

(19)

Elimination of db/dz between equations17 and 19 leads to a nonhomogeneous linearsecond-order differential equation with constant coefficients.

dz2

where

dz

C/ =

.(20)

2R(1 + n sin 0) sin 0 sin 2 ^w9(1 - n sin0) (1 + sin0 cos 2 ^w)

(n - cos 2^w) in sin2 0 +7 sin 0)6(1 - n sin0) (1 + sin0 cos 2 ^w)

K=l +

It»_ 2sin 0sin 2^w (21)R (1 + sin 0 cos 2 ^w)

Byapplying the method of undeterminedcoefficients to equation 20 and using theprinciple of superposition, a closed-formsolution is obtained.

ae

where

^1,2

-0* + (7o+g) (22)(UP2 - K0+HO W

In the cases of practical interest the rootsk\ and k2 are real and distinct. The arbitraryconstants C, and C2are determined from thefollowing boundary conditions.

For a horizontal pressure-free top surface:

(i) At z = 0; qw = 0(ii) From equations 16 and 18

Atz =0;fc=0;^ =0dz

The vertical pressure at the silo wall is nowgiven by

Qw =C\eklZ +^2^2Z +C3e~Pz +C4 ... (23)

W 2V V

_ C3 (k2 + g) + C4fc2

(*|-*2)

= C\ Qcx +P) +C4kx(*2-*l)

4UW,

=*[7o+fl(1_e-^)] (17) Ci

Bysubstituting equations 4, 5, 6, 7, 16inequation 15 and with some rearrangement,

CANADIAN AGRICULTURAL ENGINEERING, VOL. 19 NO. 2, DECEMBER 1977 95

3 'W&1- vp +w)

r _ (7o+a)

The vertical pressure at the axis ofsymmetry of the tower silo is obtained byintroducing equation 18 into equation 16.

4R(1 +n sin0) sin0 sin 2 ^w dqwq° ~9(1 - nsin 0)(1 +sin 0cos 2^w)~dz~

r (n - cos 2 ^w)(n sin 20 +7sin 0 ),+L +3 (1 -Aisin0)(l +sin0cos2 Vw) *Qw

(24)

Rearrangement of equation 24 yields

qc =2U^ +(2V-l)qw...dz

The rate of change of vertical wall pressurewith depth is given by

^=Cxkxekiz+C2k2ek*z + *fe~f—dz ll 22 (U(l2-Vp+W)

(26)

(25)

The complete pressure field within atower silo can thus be established by the aidof equations 4, 5, 6, 7, 23, 25, and 26.

The tedious numerical procedure required to obtain the complete pressure fieldwithin a tower silo suggests the use of adigital computer. Therefore, a computerprogram was developed to calculate all thesilage pressure components for the activeand passive pressure fields from the equations derived in the preceding sections. Thecomplete computer program and numericalsolutions for specific sets of system parameters are available in Negi (1975).

DESIGN CONSIDERATIONS

From the silo designer's standpoint, thepractical question at the present time is:what pressures should the silo walls bedesigned for? It has been elucidated that twotypes of pressure fields can develop in a silo.During filling and subsequent state of rest,an active pressure field develops, providedthat the silage is charged into the silowithout significant impact (Jenike et al.1973). The pressure condition in a siloduring withdrawal of silage is highly dependent on the location and the design oftheunloading device. When discharging is carried out by a silo unloader located on top ofthe stored material, there is essentially nomotion of the silage mass in the silo. It isbelieved that with such a top unloadersystem, an active state of pressure prevailsthroughout the storage period. Accordingly,the order of the pressures which will bedeveloped in silage silos equipped with topunloaders can be computed using the proposed theory for the active case.

The bottom unloading of a silo generatesmass or funnel flow of silage, depending onthe design of the unloader system. Duringmass flow, the motion of the containedmaterial is sufficiently slow and close to

96

steady state for the inertial forces to benegligible (Jenike and Johanson 1968).Consequently, the stress state is similar tothe active case except that the kinematicangle of wall friction should be used toaccount for the motion of the silage mass.The use of this parameter will result insomewhat higher wall pressures than thoseencountered in the static state.

In the event of funnel flow, the overpressures at the plane of transition frompeaked to arched conditions are subdued bythe nonflowing mass (Jenike and Johanson1969), and thus the resulting wall pressuresare attenuated. In this instance, a stresscondition somewhat in between the activecase and the passive case may develop.However, it should be noted that silage iscohesive and not sufficiently free-flowingmaterial and is thus capable of forming aself-supporting obstruction to flow. It can beconceived that when a stable arch of silagecollapses it will cause prominent dynamiceffects and impose momentary overpressures on the silo. Therefore, under theseconditions, silage pressures can be calculated using the present theory for the passivecase. In passing, it is noted that the activeand passive cases are synonymous with thestatic and dynamic conditions, respectively.

In addition to silage pressures, the following loads caused by the multiplicity offactors must be taken into account in thedetail design of silos:

1. Extraneous loads imposed by the filling,spreading and unloading equipment.

2. Reduction of the wall friction effect and

the attendant increase in lateral pressuresas a consequence of vibrations from theequipment mentioned above and/orfrom vehicular usage in the vicinity ofthesilo.

3. Axial compression and bending moments induced by the wind pressure in theleeward walls, which are of importancewhen the silo is empty.

4. Effects of temperature variations, particularly in geographic locations wherethere are marked temperature fluctuations. For instance, in the province ofQuebec, the temperature varies typicallybetween -30 C and 30 C. Thus, thermalstresses would be of critical significance.

5. Loads applied during and after tensioning of steel hoops around the outside of aconcrete stave silo.

6. Secondary stresses due to shrinkage ofthe concrete and creep due to sustainedloading. The amount of shrinkage depends on the age of the concrete and onthe method of curing. Creep is affected byambient conditions, properties of concrete and its matrix, loading history andespecially duration of sustained load. Theformation and propagation of cracks inreinforced concrete silo walls are believedto be caused by these so-called "secondary" factors.

7. Effects of the discontinuity at the top of

the wall with the roof of the silo.8. Effects of slight imperfections in the

shape or finish ofcylindrical silos, such asledges at the level of girth seams in awelded steel silo, or inevitable deviationsfrom a perfect cylinder in a slip-formconcrete silo. It has been reported thatboundary layers developed during flow atthese distortions cause pressure oscillations and impose sharp overpressure onthe walls (Jenike et al. 1973).Footings for tower silos must be designed

to minimize differential settlement of supports, and to prevent excessive settlementand tilting of the silo. The self-weight of thestructure, storage load and aforementionedexternal forces must be considered in deter

mining the size of, and the amount ofreinforcement in the footing. In addition,the maximum pressure must not exceed thesafe bearing capacity of the soil. Commentsregarding problems associated with foundations for tower silos as well as allowablebearing capacity of clay soils are to be foundin a recent publication by Bozozuk (1974).

Finally, the design engineer must keep inmind that, while the silo is an integral part ofthe farming system, the choice of size may begoverned as much by the vertical load on thesilo wall as by the required depth to beremoved per day.

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BELOTSERKOVSKII, O.M. and P.I. CHUSHKIN. 1965. The numerical solution of problems in gas dynamics. Pages 34-89 in M. Holt,ed. Basic developments in fluid dynamics.Academic Press, New York, N.Y.

BOZOZUK, M. 1974. Bearing capacity of claysfor tower silos. Can. Agric. Eng. 16(1): 13-17.

DORODNITSYN, A. A. 1962. General method ofintegral relations and its application to boundary layer theory. Pages 207-219 in T. vonKarman et al., eds. Advances in aeronauticalsciences, 3. Pergamon Press, New York, N.Y.

ECKLES, C.H., O.E. REED, and J.B. FITCH.1919. Capacities of silos and weights of silage.Kansas Agric. Exp. Sta. Bull. 222.

HAAR, A. and T. von KARMAN. 1909. ZurTheorie der Spannungszustaende in plasti-schen und sandartigen Medien. Nachr. Ges.Wiss. Goettingen, Math.-Phys. Kl. pp. 204-208.

JANSSEN, H.A. 1985. Versuche liber Getreider-druck in Silozellen. Z. VDI. 39: 1045-1049.

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JENIKE, A.W. and J.R. JOHANSON. 1968. Binloads. Proc. Amer. Soc. Civil Eng., J. Structural Div. 94(ST4): 1011-1041.

JENIKE, A.W. and J.R. JOHANSON. 1969. Onthe theory of bin loads. Trans. Amer. Soc.Mech. Eng., J. Eng. Industry 91(2): 339-344.

JENIKE, A.W., J.R. JOHANSON, and J.W.CARSON. 1973. Bin loads — part 2: concepts.Trans. Amer. Soc. Mech. Eng., J. Eng.Industry 95(1): 1-5.

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MOHSENIN, N.N. 1970. Physical properties of OTIS, C.K. and J.H. POMROY. 1957. Density: a SAVAGE, S.B. and R.N. YONG. 1970. Stressesplant and animal materials. Gordon and tool in silo research. Agric. Eng. 38(11): 806- developed by cohesionless granular materialsBreach Science Publishers, New York, N.Y. 807. in bins. Int. J. Mech. Sci. 12: 675-693.

NEGI, S.C. 1975. Pressures developed by silage PERKINS, A.E., A.D. PRATT, and C.F. ROG-materials in cylindrical tower silos. Unpub- ERS. 1953. Silage densities and losses as SHEPHERD, J.B. and T.E. WOODWARD.lished Ph.D. Thesis, McGill University, Mon- found in laboratory silos. Ohio Agric. Exp. 1941. Estimating the quality of settled corntreal, Que. Sta. Res. Circ. 18. silage ina silo. U.S. Dep. Agric. Circ. No. 603.

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