De Beer, Geotechnique, 1970

8/13/2019 De Beer, Geotechnique, 1970

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DE BEER, E. E. 1970). otechnique 20, No. 4, 387-411.

EXPERIMENTAL DETERMINATION OF THE SHAPE FACTORSAND THE BEARING CAPACITY FACTORS OF SAND

E. E. DE BEER*SYNOPSIS

In order to determine the values of the shape fac-tors to be introduced in the formula of ultimatebearing capacity of shallow foundations, an exten-sive series of tests on small footings resting on finesand was performed at Ghent. To obtain a homo-geneous and given density the sand was placed in afully automatic way, under control of the output andthe height of fall of the sand. To eliminate thedepth effect the overburden pressure was realized byair pressure in inflated rubber bags covering the sur-face of the sand.From the tests formulae for the shape factors sq,s, and sy are deduced. It appears that the shapefactor sy in the weight term is independent of theangle of friction, while the shape factors sq and s, arenot.In the formulae given for sq and s, the influence ofthe state of strain is implicitly included; however,different formulae are given depending on whetherthe effect of the curvature of the intrinsic curve onthe introduced shear strength parameters has beenconsidered or not.

Afm de determiner les coefficients de forme aintroduire dans la formule de la capaciti: portantelimite sous des fondations directes, un t&s grandnombre dessais sur petites semelles reposant sur unsable fin ont CtC effect& a Gand. Afin dobtenirune densite donnee, homogbne, le sable a Cte placedune facon completement automatique, permettantde regler la hauteur de chute et le debit. Afindeliminer linfluence du coefficient de profondeur,la surcharge laterale a Cte rCalisCe au moyen dunepression dair dans des sacs en caoutchouc recouv-rant la surface du sable.A partir des essais des formules ont et.6 Btabliespour les coefficients de forme sq, s, et sy. On con-state que le coefficient de forme sy dans le terme dupoids-propre du sol est independant de langle defrottement interne, tandis que par centre les coeffi-cients sq et s, varient avec cet angle.Dans les formules donnees pour sp et s, linfluencede letat de deformation est imphcitement inclue.Toutefois les formules sont differentes, selon que le

parametre de cisail lement introduit tient compte ounon de la courbure de la couche intrinseque dumateriau.

INTRODUCTIONAs proposed by Brinch Hansen (1961) the ultimate bearing capacity under centrallyloaded footings can approximately be expressed by

~5, = N,d,s,i,q+ N,d,s,i,c + -2- syiyyk, i b . . . . .InIthe tests carried out only vertical loadings were considered. Therefore equation (1) can besimplified to

9, = N,d,s,q+ NC c+ 2 syyk, ib . . . . . * (2)Applying the theorem of the corresponding states of Caquot, it can be shown that

N, = (N,-l)cot+ . . . . . . . . (3)

Therefore even if the tests are performed on cohesionless materials, the values of s, and d,can be obtained from expressions (4) and (5).

* Professor, Universities of Ghent and Louvain; Director of the Belgian Institute of Soil Mechanics.387


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E. E. DE BEER

NOTATIONwidth of the footing, mcohesion, t/m2 Cdepth factor for the cohesiontermdepth factor for the overburden D=termdiameter of the particles corre- Eqsponding to the ordinate of10% of the grain size distribu- Htion diagramcoefficient of uniformity, d /dcoefficient of uniformity, d /dinclination factor for the cohesion N terminclination factor for the over- Nclburden term N*inclination factor for the weightterm Rlength of the footing, mpercentage of voids Ydmaximum percentage of voids Yk.1minimum percentage of voidsunit load on the footing, t/m2 6unit rupture load on the footing,t/m2 6,initial lateral overburden pres- (Tsure, t/m2 =rnlateral overburden pressure at the ugmoment of rupture, t/m2shape factor for the cohesion 0g.Mtermshape factor for the overburdenterm

points in the load settlementgraphs corresponding to therupture criterion of Christiaens(1966)relative density,(nmsx-4/@max-niZmin)modulus of elasticity of the solidparticlespoints in the load settlementgraphs corresponding to therupture criterion of BentHansen (1961)bearing capacity factor for thecohesion termbearing capacity factor for theoverburden termbearing capacity factor for theweight termhydraulic radius of the footing,4x1 mdry weight, t/m3effective volume weight of the soilunderneath the foundationlevel, t/m3shape ratio, b/lspecific weight of the grainmaterial, t/m3normal stress component, t/m2mean normal stress, t/m2normal stress on the sliding plane,t/m2mean value of the normal stresson the sliding surface, t/m2

angle of internal frictionsecant angle of internal frictionshape factor for the weight term angle of internal friction, varyingsettlement of the footing, m with the percentage of voids,and the mean normal stresssettlement of the footing at themoment of rupture, m along the rupture surfacedepth of the foundation under- VW angle of internal friction, varyingneath the soil surface, m only with the percentage ofvoidsdimensionless quantity defined x perimeter of the footing, mby the expression (10) w surface of the footing, m2


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND 389Table 1. Characteristics of the Mol sand

Mean grain sizeDegree of uniformity

Effective grain sizeSphericityRoundnessSpecific weight&linimum densityILZaximum density

d,, = 0.19 mmd 0.135g=cO=m= 0.66d 0.15g=z=w= 0.56

d,, = 5.135 mm0.790.408, = 2.65 t/m3ya = 1.391 t/m3, nmax = 47.5oj,yd = 1.696 t/m3, nmrn = 36%

I.45 I.50 I.55 I 60 I 65 I 70yk : t/dis is 44 43 42 41 4b is 1 37 36 8n:7.

0.1 0.2 0.3 Oh 0:5 .6 0.7 0.8 d.9 I 0 I.1Dr

Fig. 1The aim of the tests was to determine the expression of the shape factors sq and s,. Fromthe tests it was also possible to deduce the variation of the bearing capacity factors N, and N,in relation to the relative density.

SAND USEDAll tests were performed with dry Mol sand. Its principal characteristics are given inTable 1. It is a uniform fine sand, composed almost exclusively of quartz.The angles of shearing resistance as obtained in normal triaxial tests are given plotted

against relative density in Fig. 1 (de Beer and VesiC, 1958). They vary between 29 and 45.Under normal triaxial tests, tests are run with three different cell pressures, e.g. 5000, 10000and 15000 kg/m2, and a common tangent is drawn as nearly as possible through the origin.In such tests the discrepancies between the circles and the origin are levelled out althoughthey are often wrongly considered as testing errors.The intrinsic law of a cohesionless material for a given density is not a straight line, but acurve turning its concavity to the (5 axis. Thus for a given density the shearing strength of acohesionless material cannot be expressed by a unique value of the angle of shearing resistance.

In order to account for the influence of the normal stress on the shearing strength para-meter the value of the secant angle & is introduced, obtained by ,drawing the tangent to theMohr circle defining the limit state of equilibrium through the orlgm (Fig. 2).


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390 E. E. DE BEER

Fig. 2

/Top silo

Propulrionfthe trolley \ Propulrion of the vertk.Itranslation of the bottom place

Fig. 3. View on the sand container withrubber bags

600

1.44 I .48 I .52 I .56 I.60yJ : dry unit weight: f/m3

Fig. 4. Set-up of the test apparatus Fig. 5

Fig. 6 (right)

2 4T

6

IO After Verit (1963) footing


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND 391From triaxial tests performed with different relative densities and different cell pressures,Ladanyi (1960) determined for the Mol sand the variation of the secant angle. He publisheda diagram giving the variation of the secant angle against the relative density D , with theratio CT,..JE~s a parameter, where (T, is the mean normal pressure existing on the sample at

rupture and E, is the elasticity modulus of the quartz constituting the grains.THE TESTS

All the tests are small-scale. The width or diameter of the small footings varies between 36and 150 mm. The shape of the footings is characterized by the ratio 6 = b/ l ,where b s the widthand 1 is the length. Two kinds of footings were used, one with S=l (circular and squarefootings) and another with 6 = 4. The base of all footings was covered with a rough materialto give the case of a perfectly rough foundation.A series of tests was performed with footings at the surface of the sand, in order to deter-mine the factors N, and sY.A second series of tests was performed with various overburden pressures q. Most of thesetests were performed with overburden pressures of 1 and 3 t/m2. In order to separate thevariables the overburden pressure was obtained by covering the surface of the sand with airinflated rubber bags, covered by a steel plate (Fig. 3). In that way the overburden pressure isexerted by a material without shearing strength, and the influence of the approximate valueof the depth factor d, is therefore greatly reduced.As both the bearing capacity and the angle of shearing strength are extremely sensitivefunctions of the density, utmost care is needed to obtain a homogeneous density and to deter-mine as accurately as possible the actual density.In order to obtain a homogeneous density the set up of the apparatus was as shown inFig. 4. The sand ran from a silo into two lateral containers, which automatically filled a

spreader box with a split of 3.26 mm running with a constant speed back and forth over the testbox. The density obtained depends on the height of the fall and on the sand output. Inorder to maintain a constant height of fall while filling the test box, the bottom of the test boxis lowered automatically in proportion to the increase of the height of sand in the test box.The relationship between the density and the height of fall for a speed of translation of thespreader box of 130 mm/s is given in Fig. 5. By increasing or decreasing the speed of thespreader box, for a given height of fall, higher or lower densities can be obtained.In all tests the water content was below 0.1%. A total of 662 tests was performed, 350 atthe surface, and 312 with an overburden pressure.CRITERION OF RUPTURE

As is also stated by VesiC (1963), the rupture underneath a footing may be obtained by ageneral shear failure, by local shear or by punching. The appearance of one of these phenomenadepends on the relative density and on the overburden pressure. If the ratio q/ yk , R or z/ R isintroduced, where z is the depth of the foundation under the soil surface, yk,i is the effectivevolume weight of the material under the footing plane and R is the ratio of the surface w ofthe footing to its perimeter x, W/X; he zones of general shear, local shear and punching shearare located as shown in Fig. 6 (VesiC, 1963).As the overburden pressure increases, the zone of general shear has a tendency to disappear.For very heavy relative overburden pressures only the phenomenon of punching remains.In the case of general shear there is no difficulty in determining the rupture load, but in thecase of local shear and punching it becomes difficult to define clearly the value of the ultimatebearing capacity. When the footingissinkinggradually into the soil, the overburden pressureincreases and, in the case of very low densities, the density also increases. Both phenomena


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392 E. E. DE BEERn:t/ml

0 4 8 I2 16 -20 24 28 32 36

EPE 16i;f 220

14 \ ,242 230 244 248 239 246

32 ,OC. B

qo=o

Fig. 7 above andright)

_-TFooting 150 mm dia.%JzJ$Y$I I Fig. 8 left). Rupture criterion of Brinch Hansen 1963)

produce a gradual increase of the bearing capacity, and in a load-settlement diagram a con-tinuous increase of the load against settlement is found (Fig. 7).For loading tests at the surface of the sand Bent Hansen (1961) has given a method whicheliminates the influence of the increasing overburden pressure. However, for loading testswith an initial overburden pressure this method does not work quite satisfactorily.Brinch Hansen (1963) has defined the ultimate bearing capacity as the stress, for which thestrain is twice the strain at a 10% smaller stress (Fig. 8).If the settlements are represented by ZJAw 1-=-wr 2 for AP 1pr=TO

AP 1Aw-= -_5w, . . . . , . . . * (6;)


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SHAPE FACTORS AND THE BEARING CAPACITY FACTORS OF SAND 393This can be expressed more generally as

dp dw-_=a- w . . . . . . . . . (7)dlnp = adlnw . . . . . . . . . (8)

where a is a characteristic value depending on the relative density.In order to obtain quantities without dimension, this is expressed as

dlnL = WAy,, adln- b . . . . . . . 9)where, according to Bent Hansen, the factor A for the present tests is in a first approximation

A =(- +;)~,(1+035;)+~ . . . . . . 10)The expression of the factor A is based on the following assumptions of Bent Hansen andBrinch Hansen.

N,=NY . . . . . . . . . . . (11)s,=l-0.4; . . . . . . . . - (12)s,=1+0.2; . . . . . . . . * (13)d = 1+0.35; . . . . . . . . . (14)

Christiaens (1966), found that by drawing the values of w/b against plAy,,ib on a double logscale, a diagram is obtained which in many cases consists of an upper curved part and a lowerpart which is a straight line (Fig. 9). As the value of a is a measure of the slope angle of thediagram, the intersection of the curved part and the straight line can be considered as therupture point. The criterion defined by Christiaens is therefore in close relation with the

Footing: 38 mm dia.qo= I t/m2

Fig. 9. Rupturecriterion ofChristiaens1966)


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394 E. E. DE BEERcriterion defined by Brinch Hansen, the only difference being that a constant value for a is notintroduced, but a value varying with the relative density,For some tests it was impossible to deduce a well-defined intersection point. In suchcases an upper and lower limit for the ultimate bearing capacity is defined (points C, and C,on Fig. 9).With the criterion of Christiaens it is finally possible to deduce from the tests the corre-sponding ultimate bearing capacity 9,.

DEDUCTIOK FRO35 LOADINGTE STS AT SURFACE OF FIRST APPROXIhfATEVALUE OF N,s,/Z VERSUS $(a CT~,~)

The shearing strength of a cohesionless material for a given density cannot be defined bya unique value of the angle of shearing strength, as the intrinsic law is curved, and furthermoreanother intrinsic law is found for each state of strain (plane or triaxial).

From a great number of triaxial tests performed with different densities and different meannormal pressures, the secant angle of the Mol sand in relation to the relative density D, andthe mean normal stress (T, is the variation at disposal (Ladanyi, 1960). Fig. 10 shows thevariation of the secant angle 4 plotted against the relative density, and considers the valueof the normal stress (TV n the shearing plane.

Meyerhof (1950) has shown that the mean normal stress (T=,~ along the shearing planeunderneath a footing is about one tenth of the ultimate bearing capacity $J,. The value of(J=, M can also be expressed by~~,M=p*(l-sin+) . . . . . . .

Following Meyerhofs work the assumption is now made that the curved intrinsic law AMNmay be replaced by the straight law OMN defined by the secant angle +(n, Us, M) correspondingto the mean normal stress along the shearing surface (Fig. 11).This is a very rough approximation, but it eliminates intricate calculations based on thecurved shape of the intrinsic curve, and finds justification in the good correlation whichMeyerhof (1950) obtained between the values calculated with this approximation and histest results.

Further, the triaxial angle d(rt, a,, M) is used irrespective of the shape of the footing. Thismeans that the influence of the state of strain on the shearing characteristics is not directlytaken into consideration. Therefore this influence will be hidden in the experimental valuesof NPs, and N,s,.

44

Fig.10 Fig.11


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND 395The way in which the value of N,s, can be deduced from the tests without overburden pres-sure is now described.Equation (2) shows that at the start the problem is indeterminate. For loading testswithout overburden equation (2) can be written as

NYJ, = Nsdsss~lr, $@a/,+~ yk, 8 . . . . . . (16)which includes the unknowns N,s, and N,s,. Solving for s,,N, gives

N Pr-Y=-_y2 Yk, ib sNd 3qqqb * * . . (17)Solving for N, gives N, = P= 18)

yk, ib s,dq~ + S, &f 9In order to find a more approximate value of N, expressions (1 l)-(14) are introduced into thesecond term of equation (18). Introducing this value of N,, together with expressions (13)and (14) for s, and d, into (17) one obtains a first approximate value N$%$l) of Ngsy.

The value of ug, M can be determined from equation (15) and, from this value and thevalue of D,, one can determine the value of +(rt, a,,,).For each tested footing the values of Nkl)~$~)/2can be plotted against the secant angle+(% ug,M . On Fig. 12 the experimental points corresponding to rectangular footings (S=&)are shown by small rectangles, those corresponding to square footings by squares and thosecorresponding to circular footings (6= 1) by circles. The curve AB gives the mean curveobtained for all circular footings irrespective their size, and the curve CD the mean curve forall rectangular footings (6 = 6) irrespective their size.

From the ratio of the ordinates of the curves AB and CD it can be deduced that the shapefactor sp) can be expressed by$,l) = l-04b/ l . . . . . . . . (19)

if 0 IO.82.3 16.89.1 20.5&O:t=0 13.8S.9 ;::: 23.06540 IS.8 29.2 31.3::a 0 2257.6 4P94.9 46.07.543 34.5 53.6 57.544 44-4 69.0 7+045 60.6 94.2 101.046 85.2 132.5 I420I Is= I 6=& 6=0-___

Fig. 12


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396 E. E. DE BEERUsing this expression for 6 = 0 the values Nil)/2 of the curve EF as a function of #lz, Q, J areobtained.The ordinates corresponding to the same values of +(n, Up,M) are compared; thus the derivedshape factor does not directly give the ratio of the bearing capacity of two footings of differentshapes and with the same small dimension placed on a sand with a given relative density.The defined shape factor is related to the case of two footings with the same small dimen-sion placed on sand with the same straight substitution line of the intrinsic curve.DEDUCTION OF VARIATION OF FIRST APPROXIMATE VALUE OF Nbsb FROM LOADING

TESTS WITH OVERBURDEN PRESSUREFrom equation (2) $J~ may be given as

p, = N~s~d,(q, + yr&?$) +y s$lylc, J . . . . * (20)where q,, is the initial overburden pressure and d, = 1 +O-35w,/b. By introducing intoequation (20) the approximate values N$%il), an approximate value Ny)s $) for N,s, isobtained. As q,, is large compared with yk,,b the error introduced in the calculation ofN $)shl_) y introducing an approximate value N$ )s$~)of N,s, is rather small.From equation (15) Us,M is calculated and with this value and the value of D, from Fig. 10the value of +(n, CT,,~). For each footing and for each value of qo, Nhl_kh])can be plottedagainst #(n, up,M).

Fig. 13

II4.36

8

Il.712.814.216.018.3ii327.035.050.075.0120.0


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND 397All experimental points for circular footings are shown on Fig. 13. The same mean curveAR is drawn, based on the points for all circular footings (6 = 1) and for all values of q,,.On Fig. 14 are shown all experimental points for the rectangular footing 38 x 228 mm.Similar diagrams have been drawn for the other rectangular footings which were used. Based

on the points for all rectangular footings (6 = $J and for all values of q0 the same mean curve CDhas been drawn.The curves AB and CD are shown on Fig. 15. From the ratio of the ordinates of these twocurves it may be deduced thatbsql) = l+dtan+(rz,u,,,) . . . . . . . (21)

Based on this value, the curve EF for the values of NY) against +(Pz, TV,M) s obtained.The shape factor, expressed by equation (21) is not related to the case of a sand with a givenrelative density D,, but to the case of sands characterized by the same value of #(n, ug,M), andthus by the same straight substitution line of the intrinsic curve.DEDUCTION FROM LOADING TESTS WITHOUT OVERBURDEN PRESSURE OFVARIATION OF MORE EXACT VALUE OF N,s, IN FUNCTION OF $(n, (I%,~)

With equation (18) deduced from the general law (2) a more exact value of N, can beobtained by calculating the ratio syN,/2N, of the second member from the approximate

Fig. 15

Fi g. 16. Reg r e s s i o nanalysis : circularfootings, q0 = 0

1 : o 2 733 . 02, 4 0. 00 . 04 5' 6 3. 546 8 5. 047 I IS.0___- M


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398 E. E. DE BEERTable 2

Data regression analysis syN,/2 = F[$(n, Q, M)]Footing Limits 4(n, c~, M Numberof tests log syN,/2 =a+(%, og, M) -b Co;~daton

Fig. 16 37 55 41 45 33 0.064 1.210 0.73742 40 46 40 , 39 0.114 3.308 0.736

Fig. 17 1Rectangular 36 50 41 45 37 ) 0.060 0.874 0.789I footings 42 46 55 33 ~ 0.098 2.435 0.863

Data regression analysis ssiV, = F ;+(n, (tg,M)]

Footing

Fig. 19 Circularfootings6=1Fig. 20 RectangularfootingsS=&

Limits +(ti, ug, M Numberof tests

32 50/ 41 t:: 40 ;:- 33 45 39 7139 42 50 11443 46 55 i 34 I

lo ssiVs = a+, og, M) -b Correlationfactor0.0690.249

0.068 1.318 0.7670.091 2.220 0.7670.292 10.758 0.913

values s$~)N$~)and NY) already obtained. Further, sq can be calculated with the expression(21) giving

. . (22)A better approximation for syN, is obtained by introducing this value of N, and that of s, fromequation (21) into equation (17).

The experimental points obtained for all circular footings (S = 1) are shown on Fig. 16.The linear regression analysis could be applied on the obtained points. However, on a semi-log diagram the variation of s,N, is not exactly linear but is represented by a slight curve withits concavity turned to the large values of jyN,. Therefore the experimental points were notconsidered as a whole, but were divided into groups on which the linear regression analysiswas applied separately. From the linear relationship obtained a smooth curve a/3has beendeduced. The data of the linear regression analysis are given in Table 2.The experimental points obtained for all rectangular footings (6 = Q) are shown on Fig. 17.As for the circular footings the mean curve yS has been obtained. The data of the linearregression analysis are given in Table 2.The two curves a / 3 and yS are shown on Fig. 18. From these the value of

(syNy)d=1,6/(sYNy)6=1can be deduced. The ratio appears to be independent of the value of +(n, (TV,M). Its value is

[ sY) , =~, GI / I I s~) ~=~1. 476In order to be able to determine correctly the variation of s, in B = b/l, it is necessary to haveexperimental data for some other values of the shape ratio 6.


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND

20.923.025.529.2

-I4.740.348.759.073.893.7125.5169.7YS

Fig. 17. Regressionanalysis: rectangu-lar footings, qo=O,S=b f=1 6

Fig. 18

urves drawn ar sightAB CD EFIO.7 lb-7 18.012.0 18.9 20.213.8 21.4 23.015.9 24.7 26.5IS.8 29.2 31.3I2.5 34.9 37-527.6 42.9 46.034.5 53.6 57.544.4 69-O 74.060 6 94.2 101.005.2 132.5 142.0122.0 192.0 208.0-__s= I s=_6 6=0

With lack of such data, an approximate relationship between sy and S may be obtained inone of the following two ways.

First, based on previous test data (de Beer and Ladanyi, 1961) one could in a first approxi-mation assume that for S = 1 the value of sy is 0.6. I f this assumption is made, the experi-mental value of (s~)~=~,~ should be given by

(s&=~,~ = 1.476(s,),=, = 1.476x 0.6 = 0~3857Now two different expressions of sy

1+0*2SSY=m * . . . . . . . (23)s> = l-O.46 . . . . . . . - (19)


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400 E. E. DE BEERboth give sy= O-6 for S = 1. For S = 6 the first expression gives

The second expression givess,, 6=1,6 = 1 -y = 1 - 0.0667 = 0.9333

Therefore, if s,, d= r =0.6 expression (23) should correspond better with the experimental resultsthan expression (19).Second, however, it is possible to make no assumptions concerning the value of sy for S = 1,and to assume arbitrarily a linear relationship between s,, and 6. The tests gives,=I-ms . . . . . . . . . (24)

sy,6=1/6 = 1.476s,,,=,l-; = l-476(1 -m)

0.476m = - = O-3631.318s, = l-0.3636 . . . . . . . . (25)

For S= 1s, = 0.637

The experimental results are therefore also covered by expression (25), in which case for foot-tings S = 1, the shape factor s,, related to (n, (TV, ) s no longer 0.6 but 0.637.However, as long as experimental data for other shape ratios are not available, it is impos-sible to find out which of expressions (23) and (25) is the more exact. I t can be seen that thedifference between the results is not very great, and that for practical purposes one of theexpressions (19), (23) or (25) can be used.The following deductions use expression (23) giving s, = 0.6 for S = 1. Given the expressionof s, the values of NJ 2 can be calculated. The curve shown in Fig. 18 gives the variationof NJ 2 against the secant angle +(n, (T~,~. For comparison also shown are the curves AB,CD and EF corresponding to the first approximate values N$l)~$,~)/2f N,s,/2. I t can be seenthat the difference between the value of the first approximation and the more exact ones is

not negligible.DEDUCTION FROM LOADING TESTS WITH OVERBURDEN PRESSURE OFVARIATION OF MORE EXACT N,s, AS A FUNCTION OF + n, Q,~)

Knowing the more exact values of syN,/2 (curves c+Iand yS in Fig. 18), these values canbe introduced into equation (18) to give a more exact value of N,s,. . . . (26)

A more exact value of ssN, can now also be deduced from the tests without initial overburdenpressure (q,,=O). Equation (18) can be written asssN, = 2% . . . . . . (27)

In equation (27) are introduced the values of syN,/2 from the curves c@ or yS in Fig. 18, and


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SHAPE FACTORS AND THE BEARING CAPACITY FACTORS OF SAND 401the values of shl)Nhl) rom the curves AB or CD of Fig. 15, and thus a more exact value of s,N,is obtained.For all circular footings the points representing the values of ssN, in function of +(rt, ug, M)are shown in Fig. 19. All points are shown, irrespective of the diameter of the footing and theoverburden pressure. The open circles represent the points corresponding to the tests with aninitial overburden pressure, and the solid circles those corresponding to the tests without aninitial overburden pressure. The circles are quite well related.The experimental points are divided into two groups, and for each group the linear regres-sion analysis is applied. From the two straight lines shown in Fig. 19 the curve cls is deduced.The data of the linear regression analysis are given in Table 2. As all points are locatedaround the same curve, irrespective of the value of the overburden pressure q it is obviousthat if the values of ssN, are drawn against the secant angle +(n, ug, M) the values of sPN, areindependent of the overburden pressure.For all rectangular footings 6 = 8, irrespective of their width or the initial overburden pres-sure, the points giving ssN, against +(+z,Ok, are shown in Fig. 20. The open circles represent

17.05.221-49.024.68.542.04.472.03.0173.007.0308.0660.0da'

1 9 .ig. Regress ionanalysis: circularfootings

l-

36' 40'

% IO.2II-2:z* 12.53.9Zk IS.88.1:L 21.56.041 32.5

Fig. 20. Regressionanalysis:rectangu-h&oomgs: S= b / l

43.564.0102.0180.0380.0Y6


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E. E. DE BEER

Bt>DI4P I SqN,d(n. ~&Y.M) Regression analysis Curves drawn at sight

Fig. 21

IB16.217-6IV-521.925.028.834.243.657.282.825.5lO.O - I--6=l CD EFthe results of the test with initial overburden, and the solid circles those of the tests withoutinitial overburden pressure. The experimental points are divided into three groups, givingthree straight lines, from which the curve yS is deduced. The data of the regression analysisare given in Table 2.

The two curves c$=c(/? and y6 are shown together on Fig. 21. From the ratio of theordinates of these two curves, the following expression for sq is found

s, = l+ptan [+(n,a,,,)] . . . . . . .Given the values of s,, the values of N, can be calculated. The variation of N, againstd(V %.M) . gS iven by the curve 65 in Fig. 21.Also in Fig. 21 are given the curves AB, CD and EF corresponding to the first approxima-tion shl)N&l) of s N

Again the de&r&l shape factor does not give the ratio between the bearing capacities of twofootings of different shape but of the same small dimension, placed on a sand with a givendensity, but the defined shape factor is related to the same value of $(n, u=, M).

COMPARISOS OF THEORETICAL AND EXPERIMENTAL VALUES OF N,In Fig. 22 the angle of shearing strength 4 is plotted against the theoretical values of NJ2as given by Brinch Hansen (1961), Buisman (1940), Kerisel and Caquot (1956), Lundgren andMortensen (1953) and Meyerhof (1955). The curve represents the experimental values ofN, against the secant angle +(n, cr=,M). The experimental values are seen to be lower than the

values of Lundgren and Mortensen (1953).That the experimental values are lower than the theoretical ones can be explained easilyby the fact that the theoretical values are based on the assumption of a general shear failureof a material with constant volume. In fact at medium and low densities the failure is


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND 403

Fig. 22 Fig. 23

. .I6 &,, 32. 40

induced by local shear failures or by punching, phenomena which are not considered in thetheoretical deductions.

It can also be seen that at high relative densities the experimental curve has the tendencyto give higher values than the theoretical ones. Also at low densities the experimental valuesof NY/2 tend to become larger than the theoretical values. An explanation for this anomalycan be sought in the fact that in very loose soils as soon as the footing starts to penetrate intothe soil an increase in density occurs. Therefore to be correct the values of NY/2 should notbe related to the angle $(n, ug, J corresponding to the initial relative density, but to a highervalue of that density.Further, no absolute meaning must be attached to the relative situation of the curve ofthe experimental values of NJ2 with respect to the theoretical curves. The experimentalvalues have been obtained using a crude approximation based on the secant angle +(n, (T~,,Jand on a crude approximation of the mean normal stress along the shearing plane. Thereforethe values of NY/2 are to be considered rather as a tool which will enable the calculation of theultimate bearing from the classical formula than as a purely physical parameter.

COMPARISON OF THEORETICAL AND EXPERIMENTAL VALUES OF N,A curve has been drawn in Fig. 23 which gives the variation of the theoretical values of N,against the angle of shearing strength. The theoretical values are given by the equation

l&T, = eTtsn@ an2 ( )+g . . . . . . .The curve in Fig. 23 gives the experimental values of N, against the values of the secantangle +(n, 0 =, M). The experimental values are lower than the theoretical ones, except for veryhigh densities where they tend to become larger.In order to take the influence of local shear and punching into account Terzaghi (1943)suggested for low and medium dense sand calculating the ultimate bearing capacity by intro-ducing in the classical formula an arbitrary angle 4 given by

tan 4 = $ tan C#J . . . . . . . * (39)N, = ezn anN3 tan2

[ i+karctan(j jtan+)] . . . . .


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E. E. DE BEER

15617-O18-520.322.525.228.5:6:;42.048.757.066.578 092 0108.0GH

004

41Fig. 24. Regression Fig. 25. Regression 0analysis : circular analysis: rectangu- ::a0footings, q0 = 0 lar footings, q,=o,t j = b / l = 1 / 6 : : J

%0

24.126.228.831.935.439.244.550.557.566.577.090.0104-o121.0143.0169.0

30 15.631 17.018.5:;::t;.:32.136.842.040.757.066.578.092.0108.0

24.126 228 831.935-439.244.5

66.577.090.0104.0121.0143-O169-O

I-545 26.0l-541 28.3I.557 30.8I.571 33.8I.573 37.5I.556 42 0I.561 47.5I.568 53-7563 61.3I.583 70.0l-581 81.2I-579 95.0I.564 I IO.8I 55 I 130.0I.554 153.3I .565 180.0Fig. 26


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND 405The values of the bearing capacity factor N, calculated in this way are also given against theangle of shearing strength 4 in Fig. 23. The experimental curve ~5 gradually tends to becomethe Terzaghi curve.

VARIATION OF s,N,/2 AGAINST ANGLES OF SHEARING STRENGTH 4(n)DETERMINED IN NORMAL TRIAXIAL TESTS

Instead of drawing the experimental values of syN,12 against the secant angles $(n, ug, M)rit is possible to draw them against the angles C(S) as determined in conventional triaxial tests.These values are given as a function of the percentage of voids n in Fig. 1.For all circular footings, irrespective of their diameter, the points giving sVN,/2 against4(n) are shown in Fig. 24. They are divided into two groups, giving the straight lines, fromwhich the curve GH is deduced. The data of the linear regression analysis are given inTable 3.For all rectangular footings (a=&), irrespective of their width, the points giving s,N,/2against q%(n), re shown in Fig. 25. They are considered as one group, and by the method ofregression a straight line is obtained, which is replaced by the curve IJ. The data of theregression analysis are given in Table 3.

Data regression analysis s&,/2 = F[4(n)]Table 3

Footing Limits C(n) Numberof tests log s~N,/~ =a+(%) -b Correlation

a bFig. 24 Circularfootings 30 25 38 z: 0.0445 0.151 0.78438 10 45 05 0.0682 1.038 0.8756=1Fig. 25 Rectangular 29 15 45 05 70 0.057 0.373 0.949footingss=g

Data regression analysis s,N, = F[$(n)]

Footing Limits 4(n) Numberof tests log s,N, = a+(n) -b Correlation

a bFig. 27 Circular 0 30 24 37 39 34 0.094 1,547 0.942footings 38 09 45 06 38 0.151 3.721 0.972a=1 ___~___1 39 24 45 48 :: 0.080 1.416 0.90327 27 37 39 0.045 0.025 o+Q39-~

3 27 27 48 19 0.039 0.029 0.945~____ ____-Fig. 28 Rectangular 1.381 0.952footings 0 29 15 37 55 z: 0.08538 IO 41 05 0.144 3,531 0,964s=g -~~-_-- ---- ---

1 28 42 36 42 42 0.048 0.331 0.85437 39 45 54 31 0.062 0.904 0.895~___3 29 06 46 67 0.045 0.355 0.951


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406Curves GH

curves gives

E. E. DE BEERand IJ are shown together in Fig. 26. The ratio of the ordinates of both

sy = 1-0.4;Here the defined shape factor corresponds to the usual definition, giving the ratio betweenbearing capacities of two footings with different shape and same width, placed on a sand with

4(n)_-I Regression anal ysi s

MO16-3 13.8 I I.218.4 14-9 12.121.0 16.2 12.825.0 17.9 14.030.2 19.8 15.037.7 22.0 16-547.4 24-5 18.060.7 27.5 19.978.2 31.0 21.8IO-l.0 35.0 24. I140.0 39.9 26.5190.0 45.5 29.8270.0 52.0 33.2390.0 60.2 37.5576 0 71.0 42.6- 85 0 48-O

40=0 qo= I r/m:

Fig. 28

0 go=0. qo=ltlrn20 q0=3tlm2

__4fi__

lo= 3r/m*

J-

II -I

5-28

I 40=0

Fig. 27

0 go =3rlm2I

3236 $() 40e

44.


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SHAPE FACTORS ASD BEARING CAPACITY FACTORS OF SAND 407given relative density. From this expression of s, the values of NY/2 can be deduced. Theyare given as a function of d(s) by the curve KL.The curve KL is shown in Fig. 22 with the value 4(s) on the abscissa. On the same graphare also given the theoretical curves from other workers, and the curve ECgiving the observedvalues against the secant angle $(+z, +, M).

The curve KL which gives the experimental values of NY/2 against the conventional tri-axial angles d(n) is well above the theoretical values. The observed values are two to fourtimes larger than the theoretical values and this corresponds with the data found by otherworkers. The explanation of the difference between the observed and experimental values isin the curvature of the intrinsic curve, which plays an important role in case of very smallmodel footings tested without overburden pressure, and also in the influence of the state ofstrain.VARIATION OF sqN, AGAINST CONVENTIONAL TRIAXIAL ANGLES @z)

Instead of drawing the values of ssNs against the secant angles +(Pz,Q, M) they can also bedrawn against the angles C(N) deduced from conventional triaxial tests. The values s,N, ofthe tests made by a given initial overburden pressure qO, and with different footings butcharacterized by the same shape factor 6, are considered as a separate group.For each group a graph is made of the points representing s,N, as a function of $(s). Forcircular footings (6 = 1) the experimental points are shown for q0= 0, q0= 1 t/m2 and q,,= 3 t/m2in Fig. 27. For rectangular footings (S=&) the experimental points are shown for q,,=O,q = 1 t/m2 and q =3 t/m2 in Fig. 28.For each of the graphs corresponding to q. =0 and q = 1 t/m2 the experimental points aresubdivided into two groups and for each of these groups the linear regression analysis isapplied. For each of the graphs corresponding to q =3 t/m2 the experimental points are con-sidered as a whole, on which the linear regression analysis is applied. The straight linesobtained are shown in Figs 27 and 28. From these straight lines curves GH and I are de-duced. The data of the linear regression analysis are given in Table 3.All the curves in Figs 27 and 28 are shown together in F ig. 29. Each curve corresponds toa given value of the initial overburden pressure q and to a given value of the shape ratio 6.

4(n)

4142433::o1q(6= I]F;;(6= &(theor.)I.385I.395I.406I-416I ,426I.436I.446I.456I ?65I.475I ,484I.493I 502I.510I.519I.527 )- ii I.387I.397I-409I.408I 424I.432I.436I.466I.470I.471I 500I.500I.51 II.51 I1.528--_ __i 4a= I.t/mI.391I.396I.395I.413I -424I.436I.445I.455I.465I.475I.486I.497I.502I.51 II.518I 527

Fig. 29

40= 3,t/m?

(s,N&( &),I,

I.384I.388I.390I.414I.440I.442I.444I.452I.466I -477I.483I.493I.506I.509I.519I.527


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4081000

I-100s

I O

526-

E. E. DE BEER

I I I I32 36 40 44$ Cn)

*(n)

__

__

I

qa=oN,__-K&o15-l17019.322.827 634.342.955.671-293.9127.8172.1244 4350.8519.3800-O

40= I,t m

N,K,L

N,--LLI

12.8 10313.7 I I.114-8 I I.716.4 12.818.1 13920 I 15.122 0 16.425.0 18.028. I 19-E31.7 21-a36. I 23-940.6 26 946 8 30.054. I 33-763 6 38.276 0 42.9

Fig. 30

40=3,t m2

N 1. theor.

18 420 623-226.1::.:37-a42.948 955.964.273-985.499 0II53134.9

All the curves give the variation of sqNq against the angle 4(n). The solid lines GH corre-spond to circular footings (6 = 1) and the dashed lines (I J) to rectangular footings (6 = 4).Clearly, for a given shape of footing and a given relative density and thus a given value-ofd(n), the value of s,N, depends on the initial overburden pressure q,,. If the ratio of theordinates of two curves 8 = 1 and 6 = -k corresponding to the same value of q. is calculated,irrespective of the value of qO, he expression sq is given by

sq = l+Psin #z) . . . . .The value sq gives the ratio between the bearing capacities of two footings with same widthbut different shape placed with the same overburden on sand with the same relative density.However, N, is now a function of qo. Using expression (32), the values of N, given by thecurves KL in Fig. 30 are found. The solid lines were obtained directly and the dashed lineswere obtained by interpolation.It is seen that for a given relative density or a given angle 4(n) the values of N, decreasewhen the overburden pressure q. increases. Fig. 30 also shows the theoretical values of N,.For the extreme case of q. = 0 the experimental values of N, are lower than the theoreticalones in the case of low densities, and become much larger than the theoretical values for highdensities. On the other hand for very large overburden pressures the experimental valuesbecome smaller than the theoretical values, even for very high densities.The general trend of the curves agrees well with the fact that the zone of general shearfailure gradually disappears when the relative density decreases and the overburden pressureincreases. The curves therefore tend to flatten when q. increases. Further, this also showsthat the dependence of N, on q. is to be explained by the fact that for a given density or C(n),when gradually increasing values of q,, are considered, the rupture phenomenon graduallychanges from a general failure to a local failure, and finally to a punching failure. It is evi-dent that such different states of failure cannot be covered by a unique value of N,.


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND 409It must be stressed that the graph of Fig. 30 is not in accordance with the dimensionalanalysis, as 4(n) and N, are dimensionless quantities, where q. has the dimensions of a stress.Thus q. has to be replaced by a dimensionless ratio. Trials have shown that q. cannot bereplaced by qO/ylc,b as the experimental points are then scattered over the entire area.

WAYS OF CALCULATING THE ULTIMATE BEARING CAPACITYI t is now well known that the intrinsic curve of a sand with a given density is not a straightline but a curve. Further, it is known that the shearing strength characteristics depend on thestate of strain. When taking these two influences into account correctly the classical trinomeformula has to be abandoned and special tests have to be made to define the shearing strengthcharacteristics in plane strain.This postulates that the results of rather elaborate calculations based on a curved intrinsiclaw should become available and that the plane strain tests should become routine tests.When this is not the case, approximate calculation methods based on an assumed straightintrinsic law, obtained by conventional test equipment, have to be used. The triaxial

apparatus is such a piece of conventional test equipment.There are now two possibilities. The first is to run a large series of tests with the triaxialequipment in order to define the variation of the shearing strength as a function of the relativedensity, and the mean normal pressure om or the normal stress up n the shearing plane. Thusthe curved intrinsic law for triaxial strain for a given relative density is obtained. Accordingto the simplification introduced by Meyerhof (1950) this curved law is replaced by a straightlaw giving the same values as the curve law for Q M=p,/lO.The second possibility is to run only a series of conventional triaxial tests giving the varia-tion of the conventional angle of shearing strength against the relative density.Both these possibilities give only data about the shearing strength in triaxial strain. Inorder not to overlook the influence of the state of strain, it therefore becomes necessary to basecalculations not on the theoretical values of the bearing capacity factors but on the experi-mental values, in which the influence of the state of strain is implicit.Against the use of the secant angle $(n, (T=,M the objection can be made that it necessitatesa large number of triaxial tests. However, compensating for this disadvantage is the fact thatthe values of N, to be used are independent of the overburden pressure.The second method has the advantage of being based on a smaller number of conventionaltests. However, when the angles +(n) are used, it becomes essential to introduce values of N,which depend on the overburden pressure q,,. Further, in certain cases the method can giverise to misleading results.

I f properly used with the corresponding values of the bearing capacity factors and shapefactors, both methods will in normal cases lead to acceptable results.SHAPE FACTOR s, FOR COHESION TERM

According to the law of the corresponding states of Caquot, the shape factor s, for thecohesion term is given by equation (4). In this expression if the secant angle +(lz, ug,,J isused the experimental values of N, and the values sq as given in equation (23) have to beintroduced. This givesb N% = +rN,_1tan$(lz,a,,,) . . . . . .

The values of s, against +(n, +,J obtained from the experimental values of N, are given inFig. 31.The values of s, with the theoretical values of N, can also be determined, giving the dashedlines in Fig. 31. By using expression (33) and the method of 1Hospital it is found that for


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410 E. E. DE BEER


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SHAPE FACTORS AND BEARING CAPACITY FACTORS OF SAND 411$=O one obtains sC= 1.2; this is the experimental value found on stiff clays (Skempton, 1951).

Using the values N, corresponding to the reduced shearing angles after Terzaghi (com-pressible material) gives the full lines and s,= 1.3 for $=O. This is the value found manyyears ago in Delft on soft clays (Polder clays).

If instead of the secant angles 4(n, u,,J the conventional triaxial angles 4(n) are used,equation (32) for s, has to be introduced into equation (4) which givesb0 = +zN,_1sin C(n) . . . . . . .

However, a value for N, which depends on qo has to be introduced into equation (34).Thus theoretically for each value of qo, another value of s, should be obtained. This is shownby the experimental curves in Fig. 32. However, the calculations show that the influence ofq. on s, is small and can be neglected.Introducing the theoretical values of N, in equation (34) gives the dashed curves in Fig. 32,and introducing the Terzaghi values gives the solid curves. Again with the equation (34),using the rule of 1Hospital gives s, = 1.2 for 4 = 0, with the theoretical values of N,, and s, = I-3with the Terzaghi values.

It can be concluded that with the expressions given for sq, the experimental values s, = 1.2for $=O can theoretically be proven; this shows that the proposed formulae do not contradictexperimental evidence.

CLOSING REMARKThe Author found it worthwhile to give a description of the small-scale tests performed in

Ghent to show that as soon as knowledge concerning the bearing capacity of the cohesionlessmaterials is refined, the secondary parameters (curvature of the intrinsic curve, influence ofthe state of strain, incomplete development of the shearing surface) appear to complicate theproblem. However, these parameters have to be taken into account in interpreting correctlythe results of loading tests, especially those of a small scale.

REFERENCESBENT HANSEN (1961). The bearing capacity of sand, tested by loading circular plates. Froc. 5th Int. Conf.Soil M ech., Par is 1, 659-664.BRINCH, HANSEN J . (1961). A general formula for bearing capacity. Bull. geotek. Inst. No. 11.BRINCH HANSEN, J . (1963). Discussion on hyperbolic stress-strain response : cohesive soils. J . Soil Me&.Fdn s Div. Am. Sm . civ. En grs 89, SM 4, 242.BUISMAN, K. (1940). Grondmechanica. Delft.CHRISTI AENS 1966), Private communication.DE BEER, E. E. & LADANYI , B. (1961). Etude experimentale de la capacite portante du sable sous des fonda-tions circulaires Btablies en surface. C. Y. 5 Congr. I nt . M ec. Sols Trav. Fond., Paris 1.DE BEER, E. E. & VESIC, A. B. (1958). Etude experimentale de la capacite portante du sable sous desfondations directes etablies en surface. An Tr av. Publ., No. 3.KERISEL, J . & CAQUOTA. (1956). T r ai ts de mecani que des s s. Paris: Gauthier-Villars.LADANYI, B. (1960). Etude des relations entre les contraintes et les deformations lors du cisaillement dessols pulverulents. Annls Trav. publ . Be ., No. 3.LEUSSINK, H., BL INDE, A. & ABEL, P. G. (1966). Versuche tiber die Sohldruckverteilung unter starrenGrtindungskorpern auf kohasionslosem Sand. Verti fl . I nst. Boden-Mech. Felsmechan ik T echni schenH ochsch. Fredericiana, Karlsruhe 22.LUNDGREN, H. & MORTENSEN,K. (1953). Determination by the theory of plasticity of the bearing capacityof continuous footings on sand. Proc. 3rd Int. Conf. Soil Mech., Ziir ich 1.MEYERHOF, G. G. (1950). The bearing capacity of sand. Ph.D. thesis, University of London.MEYERHOF, G. G, (1955). Influence of roughness of base and ground-water conditions on the ultimate bear-ing capacity of foundations. Gdotechnique 5, No. 3, 227-242.SKEMPTON,A. W. (1951). The bearing capacity of clays. Proc. B Zd gs Res. Congr ., L ondon.TE RZAGHI, K. (1943). Th eoreti cal soil mechani cs. N ew York: Wiley.VESIc, A. B. (1963). Bearing capacity of deep foundations in sand. Soil Mechanics Laboratory Report.Georgia Institute of Technology.

Documents

De Beer, Geotechnique, 1970