DESIGN OF LIQUID MANURE TANK ROOF SLABS

J.C. Jofriet

School of Engineering, University of Guelph, Guelph, Ontario NIG 2WI

Received 12 April 1977

Jofriet, J.C. 1978. Design of liquid manure tank roof slabs. Can. Agric. Eng. 20: 38-44.

In many cases, it isdesirable to cover liquidmanuretanks. The roof slabwilloften besubjected to trafficloadsfrom tractors,wagons and other farm implements. When substantial wheel loads areto bedesigned for,a poured concrete flatslabveryoftenprovides an economical solution. This paper deals with the effectof concentrated loads on rectangular flat slab panels. It alsodeals withtheanalysis anddesign ofaroofslab forcircular tanksthatare too large forasingle central columnsupport. The resultsof the various analyses havebeen used to provide design recommendations that can be used directly by the designer if used inconjunction with the Canadian Farm Building Code and concrete design code, CSA A23.3.

INTRODUCTION

An increasing number of liquid manuretanks are being covered to satisfyenvironmental considerations. If the roof of

a liquid manure tank is at ground level,traffic loads should be allowed for in thedesign of such a roof. The Canadian FarmBuilding Code (1977) requires a minimumdesign uniform live load of 150 psf (7.18kN/m2) for farm machinery traffic areas. Ifit is anticipated that the area will be occupiedby loaded farm trucks or large tractors,larger design live loads have to be used.General traffic areas also have to be

investigated for a concentrated load of 5,000lb (240 kN).

Liquid manure tanks may be rectangularor circular. The design of roof slabs forrectangular tanks is adequately covered inSection 11 of CSA A23.3-1973 where it

concerns uniformly distributed loads.Design information for small circular tankroofs without columns and with a singlecentral column is contained in a Portland

Cement Association publication on circularconcrete tanks without prestressing.

Liquid manure tank roofs can beconstructed of cast-in-place or precastconcrete. The cast-in-place alternative hasan advantage if the area in question will besubjected to concentrated loads. Unlessprecast units are interlocked in some way,each unit has to be designed for theconcentrated design load or set of loads,whereas cast-in-place slabs have the abilityto distribute concentrated loads over largeareas; consequently the uniformly distributed design load will govern in most cases.

Many alternative types of cast-in-placeconcrete slab construction are available. Of

these, the flat slab is probably one of themost economical alternatives because of its

simple and economical formwork.This paper provides the results of some

finite element analyses of flat slabs withrectangular panels subjected to point loadsand of a flat slab suitable for circular tanks

of 50 - 80-ft (15.2- 24.4-m) diam. The resultsof the analyses have been used to providedesign recommendations.

POINT LOADS ON RECTANGULAR

FLAT PLATE PANELS

The finite element method using the

38

£ COL PANEL £I- 'x/2

1 /2• • nx

fciJ — •*A

J ioo

\ A A[A

A NODE 12

CM

_c

r \

ivNODE 7

rsi INODE 6- X

_j

z

NODE 1

Figure 1. Typical layout of finite elements in one quadrant of a flat plate panel.

Clough-Felippa quadrilateral bendingelement has been used for all results

presented herein (Clough and Felippa1968). Medium thick plate theory and alinearly elastic, homogeneous, isotropicmaterial is assumed.

The present analyses are restricted to aninterior panel that forms part of an infinitesystem of equally-sized, equally-loadedpanels. The assumption of equal-sized,equally-loaded panel allows the analysis ofone quadrant of a panel. Figure 1 shows atypical layout of finite elements as used foranlysis I. The influence of pattern loading onthe bending moments has been discussed insome depth by Jofriet and McNeice (1971)for flat slabs subjected to uniformlydistributed loads. Design recommendationsto take account of the effect of patternloading on concentrated load moments willbe made later.

The analyses were chosen to investigatethe influence on the distribution of bendingmoments from panel aspect ratio and fromlead position. Details regarding the variousanalyses are provided in Table I.

CSA A23.3-1973 has provisions for thedirect design of reinforced concrete flatplates providing the geometry and loadingfalls within certain limitations. Bendingmoments in middle and column strips areobtained from a specified division of thetotal panel moment, M0, which is defined asthe numerical sum of the positive andaverage negative moment, summed over thewidth of a panel, i.e. one full column stripplus two half middle strips. The sameapproach will be followed here in thepresentation of the results.

The Code (CSA A23.3-1973) providesfor an M0, that equals the total staticmoment in a panel assuming the total reaction to act at the face of the column. In all

analyses, the point load or point loads areapplied at midspan for the x span direction.The total panel moment, obtained from thesummation of numerical results, wascompared with the total static moment

M0-WPtn (1)

where: P is the total load on the panel and /n

CANADIAN AGRICULTURAL ENGINEERING, VOL. 20 NO.l, JUNE 1978

TABLE I. RECTANGULAR PANEL ANALYSES

Analysis lx\number (ft)t

I 22

II 18

III 14

VII

VIII

IX

X

22

22

22

14

lyX(ft)t

22 )

22 )

22 )

22

22

22

22

Loadingdetails

Unit point load at centerof panel (node 1)

Unit point load midwaybetween nodes 6 and 7

Unit point load on columncenter line (node 12)

Uniformly distributed loading

t 1 ft = 0.304 m.

%Ix, ly = span, center-to-center of columns, in x and y directions. Clear spans are, in all cases, 2 ft lessthan the center-to-center spans.

TABLE II. COMPARISON OF POSITIVE PANEL MOMENTS - POINT LOAD VERSUS

UNIFORMLY DISTRIBUTED LOAD

Analysis Mpos in% Analysis Mpos in %number of M0 number of M0

(point load) (point load) (UDL) (UDL)

I 54.6

VII 54.2 IX 39.0

VIII 54.5

III54.9

X40.6

54.4 39.0

TABLE III. PANEL MOMENTS FOR ANALYSES I, II AND III

/2f Mpos Mmid MmidAnalysis

— in% in% in%number /, of MQ of MpOS of Mneg

(1) (2) (3) (4)

I 1.00 54.6 65.2 42.2

II 1.22 54.8 73.3 50.5

0.82 54.6 63.4 41.8

III 1.57 54.9 80.4 51.7

0.64 54.4 61.1 40.0

t /, is the center-to-center span in the direction of the moments under consideration; /2 is the center-to-center span in the direction transverse to /,.

is the clear span. The numerical resultsranged from 96 to 104% of M0.

The distribution of M0 into the totalpositive moment, MpOS> and the totalnegative moment, Mneg, differs, of course,from that for a uniformly distributed load.Table II presents the total positive momentas a percentage of M0 for analyses I, VII,VIII and IX to illustrate the difference

between point load and uniformly distributed load positive moments in a square flatpanel. The results from analyses III and Xprovide a similar comparison for two of therectangular panels. In the case of the latter,there are two sets of results, one for each ofthe two bending directions. It appears thatthe total positive moment portion of thetotal static moment due to a center point

CANADIAN AGRICULTURAL ENGINEERING, VOL. 20 NO.l, JUNE 1978

load is 35 - 40% greater for a central pointload than for a uniformly distributed load.

The results from analyses, I, II and III forthe distribution of the total positive andnegative moments into middle and columnstrip moments are presented in Table III. Incolumn 3, the positive middle strip momentsare given as a percentage of the total positivemoments. Similarly, column 4 presented thenegative middle strip moments.

The distribution of the positive momentsacross the width of the slab is dependentmainly on the position of the concentratedload or loads. This is clearly demonstrated inFig. 2 where maximum positive momentsdue to a concentrated load are plotted versusthe transverse distance from the center lineof column. The distribution of positivemoments is shown for three positions ofpoint loads (analyses I, VII and VIII). Thedistribution of the same moments due to auniformly distributed load, q, is shown forcomparison (analysis IX).

The distribution of the negative momentsis affected by the load position to a lesserextent. This may be seen in Fig. 3 wheremaximum negative moments are plotted inthe same way.

The distribution of the individual

moments, as shown in Figs. 2 and 3, reflectsin the distribution of the total positivemoment, MpOS, and of the total negativemoments, Mneg> into middle and columnstrip moments. This is shown in Fig. 4 wherethe results from analyses I, VII and VIII areplotted versus load position. The positivemiddle strip moment increases linearly asthe load moves away from the center line ofcolumn. The negative middle strip momentbehaves similarly but at a much lesser rate.

Geometric changes of the panel alsoaffect the distribution of the positive andnegative panel moments. For the geometricvariations considered herein, the positivemiddle strip moments, in percent of thepositive panel moments, vary from 61.1 to80.4% (Table 3, column 3). For the negativemoments, they vary from 40.0 to 51.7%(Table 3, column 4).

CIRCULAR TANK FLAT PLATE ROOF

SLAB

Flat slab roof slabs of circular tanks of

50-80-ft (15.2 - 24.4-m) diam subjected toloads typical for the cover ofa liquid manuretank require interior column supports. Theoptimum arrangement of columns allows anarrangement of slab reinforcement in aradial and circumferential pattern so that alarge number of bars are of equal length.

The chosen layout of columns results inmaximum positive and average negativemoments that are approximately equal. Thelayout provides for six columns placedequidistant on a circle with a radius one-halfof the radius of the tank (see Fig. 5). Thecolumns were modelled as point supportswith linear rotational springs in bothbending directions to provide for thebending stiffness of the columns. The layout

39

TRANSVERSE DISTANCE FROM £ COLUMN

Figure 2. Maximum positive moments versus distance across plate

TRANSVERSE DISTANCE FROM %_ COLUMN

Figure 3. Maximum negative moments versus distance across plate

40

Analyses I, VII, VIII, IX.

Analyses I, VII, VIII, IX.

includes a circular opening in the center ofthe slab with a radius one-tenth of the radiusof the tank. The slab is assumed hingedalong the tank wall. Because of symmetry,only one quadrant of the circular slab wasanalyzed. A finite element layout of thequadrant is shown in Fig. 5.

The roof slab was subjected to variousconfigurations of uniformly distributedloading and to concentrated loads in anumber of locations. Details are provided inTable IV. In all cases of uniformlydistributed loading on areas inside thecolumn circle, the same loading wasassumed to act on the opening in the centerof the slab and transmitted to the slab as a

reaction along the edge. Because of theboundary conditions imposed on the sectionof the slab that was analyzed, all loadings aresymmetric about the center of the tank.

In the presentation of the results, bendingmoments due to uniformly distributed loadshave been non-dimensionalized by divisionby the intensity of loading, q and the squareof the tank radius,./?2. Those fromconcentrated loads were divided by themagnitude of that load. Radialmoments, Mr, are bending momentsassociated with radial bending stresses.Tangential moments, Mf, are the transversebending moments.

Radial Moments due to UniformlyDistributed Loading

Figure 6 presents radial moments along aradius through a column center (line AH)found from analyses XI, XII and XIII.Similar moment curves for radial moments

along a radius midway between two columns(line AN) are shown in Fig. 7.

The maximum negative radial momentoccurs over the columns. Between columns,the maximum negative moment occursalong the straight line connecting thecolumns. The variation of the maximum

negative radial moments along such a line isshown in Fig. 8. The positive radialmoments are fairly constant along circlesthrough the maximum positive momentposition.

From Figure 6 it is evident thatmaximum negative moments result from auniformly distributed load over the entireslab. Partial loading, on the other hand,causes greater positive moments. Radialmoments have been summed over a 60°

segment of the slab to provide total segmentmoments for both dead and live load. Their

magnitude and radial location are given inTable V.

The loading conditions of analyses XIV -XVII, inclusive, do not cause radialmoments that exceed those from analyses XI- XIII, inclusive, and are therefore notreported.

Tangential Moments due to UniformlyDistributed Loading

Tangential bending moments fromanalyses XI, XII and XIII are presented inFigs. 9 and 10. Moments along a radius

CANADIAN AGRICULTURAL ENGINEERING, VOL. 20 NO.l, JUNE 1978

- 60 2

£ COL. PANEL £ £ COL MEL £

LOAD POSITION LOAD POSITION

Figure 4. Positive and negative middle strip moment in percent of total positive and negative moment.

PLAN

Figure 5. Circular tank roof slab plan including finite element layout of quadrant.

through a column center (line AH) areshown in Fig. 9; those along a radius midwaybetween two columns (line AN) are shown inFig. 10. Tangential moments are plottedversus transverse direction; that is, alongradial lines.

Along the radii through column centers,the tangential moments are negative in theregion near the column. The only significantpositive moments occur in the region nearestto the center of the tank. The greatestnegative tangential moments occur when theentire roof slab is loaded. The total negativemoment equals 0.0125 qR3.

The tangential moments along radiimidway between columns are generallypositive, except near the center of the tank inanalysis XIII, in which a uniformlydistributed load is placed outside the columncircle only. The greatest positive tangentialmoments also occur in the area nearest to the

center of the circle. Loading conditions foranalysis XV, uniformly distributed loadover areas CDJI and FGML (Fig. 5), resultin larger positive tangential momentsbeyond 0.6/? from the center of the tank.These results are also shown in Fig. 10.Results from analyses XIV, XVI, and XVII

CANADIAN AGRICULTURAL ENGINEERING, VOL. 20 NO.l, JUNE 1978

did not provide governing values and havenot been shown.

Moments due to Point Loads

The variation of positive radial momentsfrom analyses XVIII and XIX along thecircumferential line through P and Q (at0.%R) is similar to that shown in Fig. 2 foranalyses I and VIII, respectively. In bothcases, the moment per unit width varies froma maximum of about 0.32P under the pointload to a minimum of about 0.03 P 30° from

the load point. The total radial positivemoment over a 60° segment is 0.093PR inanalysis XVIII and 0.106/7? in analysisXIX.

The maximum values of tangentialpositive moments per unit width fromanalyses XVIII - XX, inclusive, are of aboutthe same magnitude as the radial ones. Thepoint load at point Q causes some negativetangential moments at and near the column.The magnitude of the maximum negativetangential moment is about one-third of thepositive moment along the outer radial spanlength (0.5R) from analysis XVIII (pointload at node P, Fig. 5) and along the innerradial span length (OAR) for analysis XX(point load at node R) is 0.058P/?. In thecase of analysis XIX, it is less.

A point load at point Sand, of course, thepoint symmetrically opposite, leads tosomewhat larger tangential moments. Themaximum moment along the inner radialspan length (OAR) is 0.09PR.

Finally, it is worth mentioning that thecolumns carry 65% of the total load on a 60°segment of the roof slab, assuming fulluniform dead and live load over the entire

area. This value is required for thedetermination of shear stresses in the slab

and for the design of the column.

DESIGN RECOMMENDATIONS

a. Point Loads on Rectangular PanelsThe results of the present analyses and

those of earlier work (Jofriet 1971) havebeen used to provide design recommendation in the form of suggested designmoments for rectangular flat slab panelssubjected to concentrated loads in a panel.The present results indicate that the totalpanel moment, Mq, may be determined inthe same way as is suggested in CSA A23.3for a uniformly distributed load, i.e. byequation 1, M0 - 1/4 P In. The distributionof M0 into positive and negative momentsis, for an interior panel, virtuallyindependent of panel aspect ratio and ofload position in the transverse span direction. A good approximation of the numerical results may be obtained from

Mrpos' •0.55 Mn (2)

A simple design recommendation for thepositive and negative moments in columnand middle strips is difficult to provide sincethese moments are influenced by panelaspect ratio and load position. Using the

41

TABLE IV. CIRCULAR FLAT PLATE ANALYSES

Analysisnumber

XI

XII

XIII

XIV

XV

XVI

XVII

XVIII

XIX

XX

XXI

Loading details

Uniformly distributed loading (UDL) on entire slab

UDL on part of slab inside column circle

UDL on part of slab between column circle and wall

UDL on ACD and AFG (see Fig. 5)

UDL on CDJI and FGML

UDL on ACG and ADF

UDL on CGMI and DFLJ

Unit point load at node P (see Fig. 5)

Unit point load at node Q

Unit point load at node R

Unit point load at node S

-0.10

ii i i 1 1 ! 1 1

-

-008 - -

1 • ANALYSIS XI-

-006

_

1 • ANALYSIS XII'1 —♦— ANALYSIS XIII

_

-004 -

-0.02 -

-

0

O02

I i i I I l l l

y -

0 0.2 0.4 0.6 0.8 1.0

RADIAL DISTANCE FROM CENTRE,r/R

Figure 6. Radial bending moments along a radius through a column center (line AH).

-0.02

2 0.02

0.04,

ANALYSIS X

• ANALYSIS XII

ANALYSIS XIII

0.2 0.4 0.6 0.8

RADIAL DISTANCE FROM CENTRE, r/ R

1.0

Figure 7. Radial bending moments along a radius midway between column centers (line AN).

available values and assuming a moveableload, it is recommended that the totalnegative moment be proportioned intocolumn and middle strip moments using thepercentages given in Table VI. A similarrecommendation for the proportioning ofthe total positive moment is also containedin Table VI. Linear interpolation may beused between the values shown.

The proportioning suggested in Table VIprovides, in most cases, safe values for anyload position along the center line of paneland, therefore, the sum of middle andcolumn stip moments does not equal 100%.

All analyses were based on theassumption that all panels are loadedequally. The influence of pattern loading hasbeen investigated in some depth for flat slabssubjected to uniformly distributed loads(Jofriet and McNeice 1971). For such loads,pattern loading causes significant increasesin the positive moments. The influence onthe negative moments is insignificant. For acontinuous beam on point supports with aninfinite number of equal spans, the ratio ofthe maximum positive moment with patternloading to that with all spans loaded is 2.0for a uniformly distributed loading; for asingle point load at center span, this ratio is1.4. On this bais, it is recommended thatpositive moments due to moveable pointloads should be increased by a factor, /?,which is 75% of that recommended foruniformly distributed loads (Jofriet andMcNeice 1971).

This means that the total positive liveload design moment due to point loadsshould be multiplied by a factor which forsquare panels may be taken as

0= 1.5 -0.8 a*+0.3 a*2 (3)

in which a* =ac/(ac + 1) and ac is the ratioof the flexural stiffness of the column to

that of the slab, as defined in CSA A23.3.For rectangular panels, the reader shouldrefer to the original work on patternloading (Jofriet and McNeice 1971).

Liquid manure roof slabs have to bedesigned for either a uniformly distributedloading or a concentrated load placed in themost unfavorable position. A comparison ofthe strip moment from analysis I and VIIIwith those from analysis IX shows that if aslab has been designed for a uniformlydistributed loading, the positive middle stripmoment will be the critical area for bendingmoments due to a concentrated load. In this

area, the concentrated load case will start togovern when the concentrated load is greaterthan about 20% of the total panel live load.Increases in positive moment due to patternloading were not taken into account incalculating this percentage. It would tend toincrease the value somewhat.

In the case of a live load of 150 psf (7.18N / m2), the consideration of two wheel loadsof 5,000 lb (240 kN) each would govern forspans of about 18 ft (5.5 m) or less if thepanels are square. It should be noted thatthis governing value of concentrated loadwill change with panel aspect ratio.

42 CANADIAN AGRICULTURAL ENGINEERING, VOL. 20 NO.l, JUNE 1978

TABLE V. TOTAL RADIAL MOMENTS FOR A 60° SEGMENT

Location Total

moment

Distance

from center

Distribution

Column - Mid-segment

Neg. DL and LLf O.OI8O4R' Varies^ 65% - 35%Moments

Pos. DL moments 0.0165<7R3 0.8R 50% - 50%Outer span

Pos. LL moments 0.0205<?/?3 0.8R 50% - 50%Outer span

Pos. DL moments 0.0010(/fl3 0.2R 50% - 50%Inner span

Pos. LL moments 0.0025?/?3 0.2R 50% - 50%Inner span

tDL = dead load, LL = live load.iw • '• ,,,,., 0.05 0.10 0.15 0.20 0.25JMaximum negative moments occur along the chord connecting the column centers. DISTANCE FROM COl iimn

MEASURED ALONG CHORD, d/R

TABLE VI. SUGGESTED DESIGN MOMENTS IN COLUMN AND MIDDLE STRIPS FOR Figure 8. Variation of radial bending momentsRECTANGULAR FLAT PLATE PANELS SUBJECTED TO POINT LOADS along a chord connecting two column

______^ centers — Analysis XL

h /, 0.5 1.0 1.5

Middle strip

Column strip

Middle strip

Column strip

Negative moments

35% 40% 50%

75% 70%

Positive moments

60%

60% 65% 80%

75% 70% 60%

b. Circular Tank Roof Slab

The design recommendations providedherein are limited to circular tank roof slabs

that have support conditions that closelyresemble those shown in Fig. 5.

The design radial moments can best behandled by dividing a 60° segment of theroof slab into a column segment extending15° either side of the column and a 30° mid-

segment centered about a radius midwaybetween column centers.

The total design moments for live anddead loads are provided in Table V, togetherwith the position of the governing momentand the recommended division of the total

moment in column and mid-segmentmoment. In determining the governingmoment, the fact that radial reinforcing steelhas variable spacing has been taken intoaccount. It should be noted that the

maximum negative moments in the mid-segment occur along a straight lineconnecting the column centers and thepositioning of the top steel in this regionshould be symmetric about this line.

To enhance the serviceabilityperformance of the slab, it is recommendedthat the negative column segment be furtherdivided into two equal segments. The innersegment should be designed for two-thirds

of the total negative column segmentmoment, the outer two half segments forone-third. Such an arrangement places morereinforcement in the immediate vicinity ofthe column and should reduce flexural

cracking. The ultimate strength, of course,will remain approximately the same. Therecommended negative radial designmoments are shown in Fig. 8.

The circumferential top steel should bedesigned for a total tangential negativemoment of 0.0125 qRJ, about two-thirds ofthe radial negative moment. The reinforcingsteel should be distributed over a width of

about 0.4 R, or 0.2 R, either side of thecolumn. Again, the serviceability would beenhanced if the top steel were to be spacedcloser in the immediate vicinity of thecolumns, as recommended for the radial topsteel.

The circumferential bottom steel should

be designed for a bending moment of 0.015qR1 (moment per unit width), except in theregion near the center of the slab. A ring, 0.1R wide, should be reinforced more heavilyfor a design moment of 0.03 qR1. Thesedesign moments were based on a live to deadload ratio of one and one-half. It is

recommended that the circumferential

bottom steel be continuous.

CANADIAN AGRICULTURAL ENGINEERING, VOL. 20 NO.l, JUNE 1978

ICOL.1 1 1 " 1

-0.10-

-008- \-

_

-0.06--\ ] ^RECOMMENDED

y r DESIGN MOMENTS\ \l

>-

-*-0.04 |\

L N.-0.02 ^^-*^_____

- 1/2 COL. SEGMENT ^1/2 MIOSEGMENT •

1 1 1 1

It is more difficult to make

recommendations for positive designmoments for concentrated loads. Obviously,the maximum moments that will actuallyoccur depend very much on the contact areaover which the concentrated load is spread.In any case, it is impractical and unnecessaryto reinforce for such peak moments as is thecase with the peak negative moments thatoccur over the columns.

Point loads in the spans adjacent to thewall of the tank cause significant positiveradial and tangential moments. Areasonable positive radial design momentsover a 60° segment is 0.1 PR. One-half ofthis segment immediately under the pointload should be reinforced for 75% of thismoment. This means that if the point load ismoveable, the design moment per unit widthis:

0.075 PR

77:0.18P

X0.8tf

An appropriate total tangential positivedesign moment would be 0.06PR. If 75% ofthe steel is again concentrated over thecenter half of the span, the design momentper unit width is:

0.045 PR

0.25R= 0.1BP

Point loads within the column circle are

carried mainly by circumferential steel.Except near the center of the tank, a designmoment per unit width of 0.18P will beappropriate. The inner ring, OAR wide,should be reinforced for twice this moment.

As for rectangular panels, point loaddesign moments and uniformly distributedlive load design moments were compared for

43

-006

-0.04

-0.02

£ °

0.02ANALYSIS XI

ANALYSIS XII

ANALYSIS XIII

0.04-

0 0.2 0.4 0.6 0.8

RADIAL DISTANCE FROM CENTRE, r/R

Figure 9. Tangential bending moments along a radius through a column center (line AH).

1.0

-0.02

0.02

0.04-

0.06,Q2 0.4 0.6 0.8

RADIAL DISTANCE FROM CENTRE, r/R

Figure lO.Tangential bending moments along a radius midway between column centers (line AN).

a typical case of manure tank roof loads.Again, a live load of 150 psf (7.18 N/m2)and two wheel loads of 5,000 lb (240 kN)were selected. The radial moments, due tothe 150 psf loading, will equal those due tothe wheel loads if the tank radius is 22 ft (6.7m). This means that for the radii less than 22

ft (6.7 m), a point load design momentswould govern for the particular loadingschosen for the example. For tangentialmoments, the break-even radius is 28 ft (8.5m). This example illustrates that for mostpractical cases, concentrated loads will notgovern the structural design of the roof slab.

SUMMARY

The results of a number of finite element

bending analyses of manure tank roof slabshave been used to provide designrecommendations for concentrated loads on

rectangular flat slab panels. Also, aparticular circular roof slab configurationsuitable for tanks in the 50 - 80-ft (15.2- 24.4-m) diam range have been analyzed foruniformly distributed live and dead loadsand for concentrated loads. Designmoments are provided. The recommendedvalues for the circumferential positivemoments are based on a live to dead load

ratio of one and one-half.

ACKNOWLEDGMENTS

This work was carried out with the financial

assistance of the National Research Council of

Canada. The contribution made by Don Noakesis gratefully acknowledged.

ASSOCIATE COMMITTEE, NATIONAL

RESEARCH COUNCIL OF CANADA.

1977. Canadian Farm Building Code, 1977.National Research Council of Canada,

Ottawa, Ont. NRCC No. 15564.

CANADIAN STANDARDS ASSOCIATION.

1973. Code for the Design of ConcreteStructures for Buildings. C.S.A. StandardA23.3 — 1973, Canadian Standards

Association, Rexdale, Ont.

CLOUGH, R.W. and C.A. FELIPPA. 1968. A

refined quadrilateral element for analysis ofplate bending. Proceedings of the conferenceon matrix methods in structural mechanics.

Wright-Patterson Air Force Base, Ohio.JOFRIET, J.C. 1971. Analysis and design of

concrete flat plates. Ph.D. Thesis, Universityof Waterloo, Waterloo, Ont.

JOFRIET, J.C. and G.M. McNEICE. 1971.

Pattern loading on reinforced concrete flatplates. ACI Journal, Vol. 68, No. 12. pp. 968-972.

PORTLAND CEMENT ASSOCIATION.

Circular concrete tanks without prestressing.IS072.01D, PCA, Skokie, 111.

44 CANADIAN AGRICULTURAL ENGINEERING, VOL. 20 NO.l, JUNE 1978