STAVE SILO HOOP DESIGN,HOOP TENSION AND HOOP TENSION LOSSES

H.S. Kleywegt and J.C. JofrietSchool of Engineering, University of Guelph, Guelph, Ontario NIG 2W1

Received 31 July 1978

Kleywegt, H.S. and J.C.Jofriet. 1979. Stave silo hoop design, hoop tension and hoop tension losses. Can. Agric. Eng. 21:91-96.

Hoops of stave silos have two main functions. They resist the internal wall pressure exerted by the contents of the silo andprovide integrity to thestructure.Thissecondfunctionrequires the hoopsto betensionedsuchthat enoughstressremainsafteralllosses from friction, elastic shortening, creep and shrinkage to maintain an integral stave assembly. Otherwise, hoops becomeloose and the silo may fail when subjected to high wind loading or asymmetric silage pressures. Tests were carried out on a stavesilo to determine its structural behavior when subjected to the prestressing operation and to quantify the various prestress losses.Each of the losses is discussed in detail and suggested design values are provided. Recommendations are made for tensioninghoops to a predetermined value by means of the rotation angle of the lugs.

INTRODUCTION

The concrete stave farm silo is a

prestressed concrete structure in which thehoops are tensioned to precompress thestave wall circumferentially. However,current construction procedures, thebuilding code (Canadian Farm BuildingCode 1977), and other standards (OntarioSilo Association 1974, National Silo

Association) governing the design of thesestructures do not adequately treat the stavesilo as a prestressed concrete structure. Insimilar prestressed concrete structures suchas standpipes, much greater emphasis isplaced on ensuring that proper tension existsin the steel prestressing tendons during theexpected service life of the structure, and onminimizing the inevitable tension losses. It isbelieved that several wind-induced failures

of concrete stave farm silos can be attributed

to improper regard of initial hoop tensionrequirements and the ensuing hoop tensionlosses.

The subject of hoop tension and hooptension losses has been examined

perfunctorily by only a few investigatorssince the first concrete stave silo was built

circa 1910. The American Concrete Institute

(ACI) Committee 714 (1946) suggested thatsilo hoops should be tensioned in three steps:(I) the hoops were to be uniformly tightenedto 50% of the design stress, (2) after jointgrouting the hoops were to be tensionedfurther to full stress; and (3) the hoops of thelower two thirds of the silo were to be

retightened prior to filling the silo. Johnsonet. al. (1971) instrumented a number ofhoops on a newly erected 6.1-m diametersilo. They determined that after tightening,the tensions in six 14.3-mm diameter hoopsranged between 9.4 and 21.4 kN, andaveraged 12.9 kN. This constitutes a tensilestress of only 80 MPa. Sadler (1972) statesthat only 10-40%of the capacity of the hoopis developed when kinking at the lug occurs.Kinking refers to the formation of a plastichinge in a hoop at the lug during tensioning.Many contractors apply only sufficienttension for this kinking to commence.

Sadler (1972) also discusses briefly theimportance of hoop tension losses caused byfriction, elastic shortening, creep and

Figure 1. The fully instrumented test silo prior to experimentation.

shrinkage, and by temperature differencesbetween the concrete staves and steel

hooping. ACI Committee 313 (1977)provides regulations for concrete staveindustrial silos used in the storage ofgranular bulk material, but not silage. Itrecommends that 90% of the dryingshrinkage must take place in a stave before itcan be used in a finished silo structure. With

regard to tensioning of the hoops, it requiresthat "stave silo hoops shall be tensioned suchthat enough stress remains after all lossesfrom shrinkage, creep, elastic shortening,and temperature changes to maintain therequired vertical and circular strength andstiffness of the stave assembly." The ACIstandard also presents test procedures fordetermining the vertical and horizontalstiffness of stave silo walls.

Kleywegt (1978) carried out acomprehensive structural investigation of a4.9-m diameter concrete stave farm silo

situated in a laboratory. This paper presentsthose findings that provide necessary datafor an engineered hoop tensioning system.Specifically, the relationship between hooptension and the torque applied at the lug is

CANADIAN AGRICULTURAL ENGINEERING, VOL. 21, NO. 2, DECEMBER 1979

TABLE I. COMPARISON OF HOOP TEN

SIONS AFTER APPLYING AN

EQUAL TORQUE OF 100 N-m

Tension in Tension in

exp. 1 exp. 11Hoop station kN kN

3-32 7.98 27.72

4-32 6.64 30.94

5-32 5.88 22.99

6-32 12.39 40.90

7-32 6.18 32.15

8-32 9.29 19.59

9-32 5.34 35.52

10-32 13.07 29.34

11-32 7.24 32.95

examined. A correlation is shown to exist

between the hoop tension at the lug at thetime of tensioning, and the lug rotationangle. Experimentally observed hooptension losses resulting from friction, elasticshortening, and creep and shrinkage areanalyzed. Recommendations for determining and minimizing hoop tension lossesare provided.

91

TEST SILO

A 4.9 m x 3.8 m high (five stave lengths)concrete stave farm silo was constructed in

the Research and Development Laboratory,School of Engineering, University ofGuelph. The staves were layed up with athree-step-break. Fifteen 14.3-mm diametergalvanized steel hoops were evenly spaced at254 mm, corresponding to the horizontaljoints of the staves. The hooping wasdetermined by test to have a yield strength of292 MPa. The hoop segments were coupledby a standard lug with a length of 84 mm andan eccentricity of 32 mm. At the time ofconstruction the hoops were tensioned onlynominally.

Nine of the hoops were instrumentedwith electrical resistance strain gauges. Thestrain gauge circuitry and positioning wasdesigned to minimize the effect of bendingstrains in the hoops. All strain gaugelocations were aligned with a verticalcolumn of staves. Similarly, all hoops wereadjusted to have one lug on each hoopaligned a distance of four stave widths fromthe strain gauge locations. Hoop no. 7(counted from the bottom) had 15 straingauge stations spaced around half itscircumference so that the effect of friction

losses could be monitored. Figure 1 shows aview of the instrumented test silo.

In all experiments, the hoops weretightened sequentially starting from thebottom. All lugs were tightened on eachhoop before proceeding to the next one.Only the upper nut of each lug was used fortensioning. The lug nuts on the 15.9-mmraised rolled thread of the hoop ends wereuniformly torqued to 110 N-m. This torquewas considered to be a practical upper limitfor actual construction practice. Hooptensioning was accomplished without thebenefit of lubrication on the hoop endthreads, except in the last test.

TORQUE AT THE LUGAND HOOP TENSION

The hoops were tensioned on theassumption that a reasonable relationshipwould exist between torque applied to thenut and hoop tension. The correlation ofthese two parameters was examined forunlubricated and lubricated threads. Thehoop tensions were those achievedimmediately after the first lug (the lugnearest the hoop instrumentation) of thehoop under consideration was fullytensioned to a torque of 110 N-m. Theresults of the two tests (nos. 1 and 11) arerecorded in Table 1.

In the first test, the nine instrumentedhoops were found to have tensions rangingfrom 5.34 to 13.1 kN. After lubricating witha machine oil and retensioning the hoops,the hoop tensions varied between 19.6 and40.9 kN. It is evident that the torque appliedto tension the hoop is a poor indicator ofhoop tension. It is further evident thatlubrication of the hoop threads allowed asignificant increase in hoop tension for the

92

Figure 2. Kinking of a hoop at the lug.

50-

40-

rB

I 30-sc«

I-

S 20-o

I

10-

) ' 2 • 4 ' 6Lug Rotation Angle (degrees)

Figure 3. Hoop tensionat the lugversus lugrotation angle. Comparisonofexperimentaland analyticalresults.

o Exp't. I

o Exp't. II

O Ancillary

— Model

same applied torque. However, it did notimprove the poor correlation betweentorque and hoop tension.

It is not well understood why there wassuch a large variance in hoop tensions foridentical torques, despite the carefullycontrolled laboratory conditions. Pooruniformity of the raised rolled threads of thehoops is suspected to be a contributingfactor. Friction between the lug and thehoop may also be a factor.

LUG ROTATION ANGLE AND

HOOP TENSION

The eccentric method of coupling thehoop segments causes the lug to rotate as thehoop is tensioned. At a certain angle ofrotation, the combined bending and tensionstresses in the hoop causes local plasticyielding of the hoop steel where the hoopexits from the lug. This yielding is observed

as a kink in the hoop at the lug (Fig. 2).Hoop kinking at the lug is frequently

used by the silo contractors as an indicationof sufficient tension in the hoop. Hence, anexperimental investigation was performedto determine a relationship between the lugrotational angle (measured from thehorizontal) and the hoop tension at the lug.It was ensured that hoop kinking occurredonly in the non-threaded portion of the rod.The results for all instrumented hoops arepresented in Fig. 3.

At zero hoop tension the lug rotationangle is about two degrees. This is becausethe holes in a lug are slightly larger than thenominal hoop diameter. The small scatter ofthe results suggests that the lug rotationangle is a reliable indicator of hoop tension.Hoop kinking at the lug initially occurredafter the lug rotation angle exceeded about6°, at a hoop tension of about 7 kN. Thisobservation is in agreement with the findings

CANADIAN AGRICULTURAL ENGINEERING, VOL. 21, NO. 2, DECEMBER 1979

EXP'T. I

Observed Tension History

Generated Tension History

81EXP'T. 6

6-

-i 1 1 1 1 1 1 1 1

Time

Figure 4. Hoop tension histories showing elastic shortening loss of tension. Comparison ofexperimental and analytical results.

of Sadler (1972).An analytical model was developed for

the relationship between the lug rotationangle and hoop tension (Kleywegt 1978).The analytical results are compared with theexperimental data in Figure 3. Goodagreement between the experimental andanalytical results is evident for the range oflug rotation angles between 7° and 18°. Forangles less than 7° there is no agreementbecause the analytical model does notaccount for elastic bending of the hoop.

A parametric study was conducted withthe analytical model to determine therelative effect of the parameters involved.

This study showed that:1. increasing the yield strength of the hoop

steel leads to an increase in hooptension per unit of angle,

2. increasing the lug length does notsignificantly alter the hoop tension atthe onset of hoop kinking, but doesdecrease the total angle a lug can bend;longer lug lengths would probablymake it easier to tighten the hoops tohigher tensions because less work isrequired in plastic bending of the hoopat the lug,

3. decreasing the eccentricity of the lugpostpones the onset of hoop kinkingand reduces the total angle the lug canturn; it is desirable to keep theeccentricity small.

Judging from field observations of lugrotational angles on concrete stave silos withhoops and lugs similar to that of the test silo,it is evident that very few silos areconstructed with hoop tensions exceeding 14kN. This tension is only about 30% of the

hoop's yield strength for 290 MPa yieldstress steel. The Canadian Farm BuildingCode (1977) requires that hoops betensioned to 60% of their yield strength.

PRESTRESS LOSSES

The tension in the hoops applied at thetime of construction will reduce with time.

This results in a loss of circumferential

precompression in the stave wall. In somecases the prestress losses are sufficientlylarge to cause completely slack hoops withina year of constructing the silo. The majorlosses of hoop tension are from:1. friction between hoops and staves,2. elastic shortening of the concrete stave

wall,

3. creep and shrinkage of the staves.Each of these three categories of losses willbe examined separately.

Friction Losses

It is well established that friction between

the hooping and staves of the silo decreaseshoop tension along the circumference of thehoop, away from the point of tightening.This unequal distribution of stress in thehoops disappears almost entirely within acouple of months. The resulting averagestress in the hoop is less than the initialtension at the lug. Friction, therefore, mustbe considered as one of the causes of loss ofprestress.

The relationship between tension at thepoint of tightening (T0) and the tension (7)at some point Q rad from the point oftightening, may be given by an equation ofthe form:

T(d) =T0e-Ve (1)

CANADIAN AGRICULTURAL ENGINEERING, VOL. 21, NO. 2, DECEMBER 1979

TABLE II. AVERAGE HOOP TENSION

LOSS DUE TO FRICTION?

No. lugsj %loss§ T/T0

1 35.3 0.647

2 20.3 0.797

3 14.2 0.858

4 10.9 0.891

5 8.9 0.911

6 7.5 0.925

7 6.4 0.936

8 5.7 0.943

t For a coefficient of friction of 0.3.I Number of lugs equally spaced around silo.§ Loss refers to the tension at the lug, when all

lugs are tightened to a uniform tension.

where e is the base of natural logarithm andjU is the coefficient of friction. The coefficientof friction was determined by monitoringthe tension gradient in the fullyinstrumented hoop (no. 7) of the test silo. Afriction coefficient of 0.3 was found for hooptensions under 20 kN. At higher hooptensions, local plastic yielding of the hoopsteel at the vertical joints between stavescaused an apparent friction coefficient of 1.1(Kleywegt 1978). _

The average tension (7) in a hoop afterfriction losses can be given in terms of thetension in the hoop applied equally at eachlug, and the number (TV) of equally spacedlugs per hoop:

r=^(l_e-^a);a =7r/TVIda

.(2)

Average tensions for values of TV from 1 to 8are tabulated in Table II. With four lugs perhoop equally spaced, the average tension inthe hoop is 0.89 T0 for a coefficient offriction of 0.3. Note that the diameter of the

silo does not affect the friction losses.

Elastic Shortening LossesElastic shortening is defined here as the

effect the tensioning of a hoop has on thetension in adjacent hoops of the silo becauseof the resulting radial deflection of the silowall. Elastic shortening was monitoredexperimentally for three cases of hoopspacing; 254, 508 and 762 mm.

Figure 4 shows the tension histories ofseveral hoops as the test silo was prestressed.The hoop spacing was 254 mm. The firstpoint of each curve denotes the initialtension in the hoop. Each subsequent pointindicates the tension after the next hoop istightened. All curves have a similar shape.At 254-mm hoop spacing, the cumulativeelastic shortening loss was 11 - 35% of theinitial average hoop tension. At 508-mmhoop spacing this loss ranged between 4 and14%, and at 762-mm hoop spacing the losswas of the order of 1 - 5%.

A model was developed, based on theassumption that the behavior of the jointedstave structure would be similar to a linear

elastic homogeneous cylindrical shell.Figure 5 shows the elastic curve of a

93

(a)Schematic of Ring Loaded Cylinder

iP

IT

3?44 1

(b) Deflection Profile

Under Ring Load

(c) Bending Moment

Diagram

Figure 5. Deflection profile and bending moment diagram for an axisymmetric ring-loadedcylindrical shell.

TABLE III. ELASTIC SHORTENING LOSS OF HOOP TENSION IN PERCENT OF INITIALTENSION+

Hoop

Elastic shortening loss in percent of initial tension

Hoop Initial hoop ension Initial hoop ension

diam spacing 10 kN 25 kN

(mm) (mm)

Silo diam Silo diam Silo diam Silo diam

5 m 7 m 5 m 7 m

10.0 100 21.2 18.9 18.8 16.8

10.0 250 9.1 8.1 8.0 7.1

10.0 500 3.5 3.1 3.1 2.7

10.0 750 1.3 1.2 1.2 1.0

14.3 100 100.0 47.0 45.5 31.9

14.3 250 15.9 14.4 14.3 12.8

14.3 500 6.5 5.8 5.4 5.1

14.3 750 2.5 2.2 2.2 1.9

t Based on an analytical model for the elastic shortening loss of prestress in the hoops of the testsilo.

•( T)2x 101

(4)

cylindrical shell subjected to a ring load(Timoshenko and Woinowsky-Krieger1959). The shape of the elastic curve is,among other things, a function of ageometric parameter, /?. It is defined by:

(3)

in which v is Poisson's ratio, r is the meanradius of the cylinder and l is the thickness ofthe cylindrical shell.

The shape of the bending moment curvedue to the radial loading is also a functionof /? (see Fig. 5). The first point of zeromoment is 0.25 7t/j3, and the second1.25 7T//3 from the ring load position.

where o is the average circumferential stressin the stave wall in Pa.

In a vertical column of staves there is no

continuity between adjacent staves althoughthere is some continuity through adjacentcolumns of staves because the horizontal

joints are at different levels. There appearsto be a tendency for the shape of the momentcurve to be influenced by the horizontal jointlocations. Working on the assumption thatthis will tend to decrease (5 such that0.25 Tl/P equals one-half stave length, themodified cylindrical shell model predicts theexperimentally found elastic shorteninglosses very well.

Elastic shortening histories from modelanalyses have been included in Fig. 4 forcomparison with experimental observations. Since the computer model proved tobe satisfactory it was used to generate typicalelastic shortening losses for two silos, one 5and the other 7 m in diameter, hooped with10-mm and 14.3-mm diameter rods, andtightened to an average tension of 10 and 25kN. Four hoop spacings are examined: 100,250, 500 and 750 mm. The losses expressedin percent of the initial tension are listed inTable 111. The results indicate that:1. elastic shortening loss in terms of force

can be reduced significantly by usinghigher stressed, smaller diameterhoops,

2. elastic shortening loss is not verysensitive to silo diameter,

3. elastic shortening loss may be 100% ofthe original hoop tension at hoopspacings less than 100 mm,

4. higher hoop tensions lead to a slightlylower elastic shortening loss,

5. elastic shortening loss is less than 3% ofthe initial average hoop tension at hoopspacings greater than 750 mm.

The model can predict the effect ofmultipass tensioning on elastic shorteningloss. By sequentially re-tightening the hoopsto their design tension in a second pass, theelastic shortening loss can be reduced to lessthan 3% for hoop spacings greater than 250mm.

CREEP AND SHRINKAGE LOSSES

The time-dependent deformation ofconcrete in the direction of a sustained stress

is referred to as creep. The concrete staves ofa silo are susceptible to creep caused by thecircumferential compressive stress in the silowall. The resulting creep strain causes agradual loss of hoop tension. The creepstrain is generally given as a multiple of theinitial elastic strain of the material. Forconcrete, the total creep strains may be 1.3 -4.2 times the instantaneous elastic strain. Anaverage value for the ultimate creep straincoefficient is 2.35 (ACI Committee 2091971).

Shrinkage of concrete is also a time-dependent deformation. It is caused by thegradual release of excess water from the

A stave silo is not a homogeneousstructure. This causes two deviations fromthe behavior of a homogeneous cylindricalshell. The first relates to the presence ofmany vertical joints, the second to thediscontinuities caused by the staggeredhorizontal joints.

The vertical joints were assumed to closein a non-linear elastic way with increasingcircumferential stress, thus adding to thelinear elastic deformation of the concrete

between joints. An empirical expression forthe additional circumferential deformationper vertical joint was derived from theexperimental results. It expresses thecircumferential displacement x (in m) as:

94CANADIAN AGRICULTURAL ENGINEERING, VOL. 21, NO. 2. DECEMBER 1979

6001

500

400-

Mic restrain

300-

200-

100

0 I i i i i i i i i i i i i0 2 4 6 8 10 12

Months

Wet Cured

—i 1 I I I14 16 18

creep calculation. Creep in the strain gaugecement could also have been a contributingfactor. It is evident that further investigationof these creep losses is warranted. Untilfurther research is conducted, it isrecommended that a creep coefficient of 4.0be used to estimate hoop tension losses dueto creep. Based on this recommendation, thecreep strain loss can be determined from:

Ar>cr

s2>0ToAs (5)st

Figure 6. Average shrinkage strain remaining in staves versus time after casting.

in which Tcr - creep loss, As - crosssectional area of a hoop, s = hoop spacingand t - stave thickeness.

The shrinkage strain loss may becalculated using the remaining shrinkagestrain for the age of the stave (Fig. 6).

CONCLUSIONS AND

RECOMMENDATIONS

Hoops of stave silos have to be tensionedsufficiently to provide an integral cylindricalstructure. It has been shown by experimentand by analysis that friction between stavesand hoops, elastic shortening of the stavesduring tensioning, and creep and shrinkageof the concrete staves reduce the tension in

the hoops. To provide a structurally soundstructure it is necessary (a) to design therequired hoop tension after all losses, (b) toestimate fairly accurately the total loss ofprestress that will take place during the lifeof the structure, and (c) to be able to build astave silo with the required initial hooptension.

Results from the various tensioning testscarried out on the test silo have shown that

the angle of rotation of the lugs is the bestavailable indicator in the field for the initial

hoop tension at the lug. Figure 3 providesthe relationship between angle of rotationand initial hoop tension for 14.3-mmdiameter hoops, a yield strength of 290 MPaand 83-mm lugs with 32 mm eccentricity.Kleywegt (1978) provides similar relationships for other values of hoop diameter,yield strength and lugs.

In general, tension losses in the hoops ofa stave silo are large when viewed relative tothe initial hoop tension, because of the lowyield strength of the hooping steel andthe jointed nature of the concrete stavestructure. This paper provides the necessaryinformation to estimate the various losses of

hoop tension.In view of the high percentage of hoop

tension it is recommended that they beminimized as much as possible. Shrinkagelosses can be reduced drastically by usingstaves that are 1 yr old. Creep losses arereduced also if this recommendation is

followed. Elastic shortening losses can bealmost eliminated by multipass tensioning.The use of smaller diameter higher stressedhoops will reduce the percentage loss.

Even if these recommendations are

followed, the total loss of hoop tension maybe calculated to be in the order of 23 - 26%for one 14.3-mm diameter hoop per stave,

22

20-

18-

16-

14

12Hoop Tension

Loss (KN) 10-

8

6-

4 -

2-

0

h oD Do °

Hoop 7 oOther Hoops - o

T i i I i i I I I i I I 1 I I I I18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

Initial Hoop Tension (kN)

Figure 7. Time-dependent hoop tension loss over 59-day period versus initial hoop tension.

concrete. Average total shrinkage strains are800 x 10~ 6for moist cured concrete and 730x 10~6 for steam-cured concrete (ACICommittee 209 1971). The approximaterelationship between remaining shrinkageand time for a typical concrete stave isplotted in Fig. 6. About 80% of the totalshrinkage strain takes place within about 6mo after the concrete is cast and cured;about 90% takes places within a year. Theshrinkage strain is partially recovered whenthe concrete absorbs water.

Hoop strains in the test silo weremonitored over a 59-day period. At thebeginning of the 59-day test period thehoops were tensioned to an average tensionof 24 kN. In terms of stress this is 150 MPa,just over 50% of the measured yield strengthof the hoop steel. At the end of the period anaverage hoop tension loss of 5.3 kN wasrecorded. In Fig. 7 the hoop tension lossesfor the 59-day test period measured at allstrain gauge locations are plotted versus theinitial hoop tension. The experimental

results show a considerable scatter but there

is an upward trend with increase in hooptension. This might be expected because thecreep strain component is a function of thestress level of the concrete.

The observed creep and shrinkage losseswere compared to those predicted by theempirical methods suggested by ACICommittee 209 (1971). It was estimated thatthe average hoop tension loss over the 59-day period due to creep should have been1.26 kN, assuming an average value of 2.35for the creep coefficient and an average hooptension of 24 kN. The hoop tension loss dueto shrinkage in the staves was estimated at0.67 kN. This estimated loss is about one

third of the average observed loss and is lessthan the minimum recorded loss of 2.7 kN.

The discrepancy between the observedand calculated hoop tension losses due tocreep and shrinkage was attributed to highcontact stresses in the vertical joints betweenthe staves of the silo. These contact stressescould, of course, not be accounted for in the

CANADIAN AGRICULTURAL ENGINEERING, VOL. 21, NO. 2, DECEMBER 1979 95

about 38% for three hoops per stave andabout 62% when there are six hoops perstave.

ACKNOWLEDGMENTS

This work was carried out with the financial

assistance of Agriculture Canada. The test silowas provided by the Ontario Silo Association.

AMERICAN CONCRETE INSTITUTE

COMMITTEE 714. 1946. Recommended

practice for the construction of concrete farmsilos (ACI 714-46). Amer. Concrete Inst. J.18(2): 148-164.

AMERICAN CONCRETE INSTITUTE COM

96

MITTEE 209. 1971. Prediction of creep,shrinkage, and temperature effects in concretestructures. ACI Publ. SP-27. ACI, Detroit,Michigan, pp. 51-93.

AMERICAN CONCRETE INSTITUTE COMMITTEE 313. 1977. Recommended practicefor design and construction of concrete bins,silos, and bunkers for storing granularmaterial (ACI 313-77) and commentary.ACI Standard 313-77. ACI, Detroit,Michigan: 40 pp.

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JOHNSON, J.E., W.H. MACEMON, and B.STANEK, 1971. Conditions affecting design

of concrete stave silos. Proc. of the NationalSilo Association International SilageResearch Conference, Chicago, pp. 276-295.

KLEYWEGT, H. 1978. Structural aspects relatedto the prestressing of concrete stave silos.M.Sc Thesis, School of Engineering,University of Guelph, Guelph, Ont. 154 pp.

ONTARIO SILO ASSOCIATION. 1974.Standards for concrete tower siloconstruction. 19 pp.

SADLER, J.E. 1972. Above ground silo designconsideration. Proc. of the 60th Annual

National Silo Association Conference, CedarFall, Iowa. pp. 39-61.

TIMOSHENKO, S.P. and S. WOINOWSKY-KRIEGER. 1959. Theory of plates and shells.McGraw-Hill Book Company Inc., NewYork, N.Y. 580 pp.

CANADIAN AGRICULTURAL ENGINEERING, VOL. 21, NO. 2, DECEMBER 1979