FundamentalConsiderationsOnShearStrengthOfSoil_Bjerrum

8/12/2019 FundamentalConsiderationsOnShearStrengthOfSoil_Bjerrum

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210 LAURITS BJ ERRUMHvorslev (1937) expressed shear strength as a function of water content and effectivestresses, and application therefore requires an analysis o f the deform ation properties.In 1948, Skem pton made an attempt to analyse the results of undrained triaxial testsbased on Hvo rslevs results. Thro ugh the introdu ction of the coefficients of elasticity forcompression and for expansion, he succeeded in obtaining expressions for the effective stressesand the strength. Skem ptons h-theory is a most valuable method in research work ; itis, however, rather com plicated in practical application, and the determination of the intro-duc ed coefficient is quite difficult (Ske mp ton, 1948 u--.A new method of considering the shear strength of soil is reported in a review of theAm erican Triaxial Shear Research Program (U.S. War Department, 1947). This review statestwo important facts : first, that the shear strength found by tests can be interpreted as beingdependent only on the water content ; and, secondly, that the water content is a functionof the major principal stress only.App lying tho se two observa tions to the results of Hvorslevs investigations, it is possibleto develop a simple working hy pothesis for the analysis of the shear strength of normallyconsolidated saturated homogeneous soil. In a treatise which has not yet been published,the Author has made an attempt to do so ; and in this Paper some of the fundam ental findingsare summarised.The object of this Paper is, first and foremost, to try to make a contribution to a betterunderstanding of the fundamental strength properties of cohesive soils. For this purpose,shear streng th is divided into cohesion and frictional resistance ; these tw o properties areconsidered in relation to the factors w hich respectively determine their magnitude. Sincethe terms cohesion and angle of internal friction are often loosely used, an attempthas been made to define these terms clearly.

THE SHEAR STRENGTH OF A SOILOn any plane a, passing through a point in a stressed soil mass, a direct stress u, normalto the plane a nd a shear stress T. in the plane will, in general, be acting. A failure will occu rin any plan e if the shear stress 7,_ exceeds the shear strength s,,. No w, if the strength on allplanes through the same point were equal, failure would take place in the two planes withmaximu m shearing stresses-that is to say, on the two planes which intersect the directionsof the principal stresses at an angle of 45 degrees. Com pression tests on soil samples, how ever,show first that the angle between the failure planes generally deviates from 90 degrees, andsecond ly th at this angle is consta nt for any one soil. Therefore it may be considered a factthat the shear strengths n the different planes through a point in a stressed soil mass are

not equal b ut d epend on the no rmal stress a,. which acts on the planes.In general, the strength s, can be expressed by the Coulom b failure criterion :-s, =c +cr,tan+, . . . . . . . . . . (1)

It is kno wn tha t, for sand, this expression is valid for c = 0. In 193 7, Hvorslev stated thatequation (1). could also be applied to clay. The angle +I can, with sufficient accuracy, beconsidered con stant for a clay, but Hvorslevs tests showed that the magnitude of c variedwith the water content. The Coulomb-H vorslev failure criterion (equation (1)) may in thisway be accepted as a basis for a strength analysis of all types of saturated clays. It is assumedhere that the soil is homog eneous and isotropic (Casagrande and Carillo, 1944).Expressed in words, equation (I) states that the strength in a plane through a point in asoi l mass is composed of two essentially different parts.

The first comp onent, c, is characterized by the fact that its magnitude is the same in allplanes through the considered point. This part of the shear strength is &led the cohesion.The m agnitude of the cohesion depends, for saturated soils, merely on the water content at


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FUNDAMENTALS OF SHEAR STRENGTH OF SOILS 2x1the point under consideration ; in a soil mass the cohesion therefore varies from p oint topoint with the water content.The second comp onent is the frictional resistance, the magnitude of which is proportionalto the effective normal stresses on the planes through the point under consideration. 4,devote s the angle of internal friction, being a characteristic consta nt of the soil in ques tion.If, at the considered point, an excess hydrostatic pore-water pressure exists, the efe t i v estress will determ ine the frictional resistance (Terzagh i, 1938 ; Bishop and Eldin, 1950).It is most important to keep clearly in mind these definitions of cohesion and frictionalresistance ; in this respect, it cannot be too strongly emphasized that the true cohesion hasnothing to do with the shear strength of the unloaded sample.

THE SHEAR STRENGTH DIAGRAMIn the shear-strength diagram the Mohr stress-circle enables a graphical determ ination tobe made of the stresses in the different planes through a point in a stressed body. On the

same diagram the shear strength alone the different planes through the considered p oint canbe plotted as a function of-the not&alstress on the plane s, resulting in a straightline (equation (1)).As stated above, however, it is notpossible to determine the shear strengthof a cohesive soil merely from statical con-siderations. The cohesion is a function ofthe water content, and it therefore appearsto be an obvious course to establish therelationship between cohesion and watercontent in the strength diagram. If theresults of a standard consolidation test areplotted above the shear-strength diagram,a curve will result which gives the watercontent as a function of the consolidationpressure, as shown in Fig. 1. Now , acertain water content corresponds to anypoint on the consolidation curve, and, thecohesion being a function merely of thewater conte nt, there is also a certain co-hesion corresponding to this point on the

slkss : KILOGRAUSCRo. c-w -Shear mwqth diagram Wiener Togo1 : after

Hvor8lev, 1937)consolidation curve. It is therefore suggested that, in the shear-strength diagram, themagn itude of the corresponding cohesion be plotted below each point on the consolidationcurve. A curve then appears which, as desired, establishes the relation between w ater contentand cohesion.Fig. 1 shows the cohesion curve for the tile clay investigated by Hvorslev. If, at any pointin this clay, the w ater content is WO, the m agnitude of the cohesion can be found from thecohesion curve below that po int on the consolidation curve which corresponds to the watercontent Wo as illustrated in Fig. 1. On any arbitrary plane through the considered point,the total sh ear strength wiIl be :

s,=c+u,tan+ . . . . . . . . . (1)in which a, is the normal stress on the considered plane. This equation gives the comm onfailure line in Fig. 1.At an adjoining point in the soil mass the water content is, perhaps, smaller, and the


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212 LAURITS BJERRUMcohesion correspondingly greater. For this point the failure line (showing the shear strengthin the different planes through the point) runs parallel to the first failure line, but interceptsthe vertical axis at anoth er v alue of c. For a cohesive soil, conseq uently, the failure criterioncannot be illustrated by a single line on the strength diagram. The failure lines, expressedby equation (l), form a system of parallel lines with the cohesion as parameter.For clays normally consolidated from the liquid limit, the cohesion curve will be a straightline, often passing through the co-ordinate origin, with the inclination K (known from H vorslev(1937) and Skem pton (1943~4)). As will be shown below, it may be assumed for non-precon-solidated clays that the water content depends only on the effective major principal stress, al.For such soils the cohesion curve gives the cohesion directly as a function of ul.

DETERMINATION OF WATER CONTENT AND EFFECTIVE STRESSESIt has been shown above that a determination of shear strength can be mad e from themagnitude of the water content and the effective stresses. An analysis of shear strengthconsequently requires a knowledge of the deformation properties.In a review of the Am erican Triaxial Shear Research Program (U.S. War Department,1947) two fundam ental test results are reported.First, it was observed that, if the water content of a clay were plotted against the effectivemajor principal stress, all points conform ed to a single curve, independent of the magnitudesof the minor and intermediate principal stresses.Secondly, if the compressive strength, determined in the triaxial apparatus, were plottedagainst the water co ntent at failure, all points wo uld fall on or near a single curve, indepen dentof metho d of test, pore-water pressures, and so on.Wh en those two test results are interpreted it is found that an interdependence mu stexist between the effective major principal stress and the water content. This inter-dependence can be used as the basis for the developmen t of a working hypothesis w hichenables the water content and the effective stresses to be determined from the external totalpressures. This working hypothesis can be summ arised by the following points :-

(1) The water content of a clay element is dependent solely on the magnitude of theeffective major principal stress.(2) If, by a shear, the total external stresses are changed so quickly that the watercontent of the clay element rem ains constant, an excess hydrostatic pore-waterpressure will arise, the m agnitude of which will adapt itself in such a way that theeffective major principal stress remains unaltered.(3) Provide d th at the pore wate r can escape , the effective intergranu lar stresses will,after a certain time, be equal to the applied external pressures.This working hypothesis (which, in terms of Skem ptons work, can b e called a h = 0theory) is not valid for all soils ; it is know n, for instance, that it cannot be applied to non-saturated or preconsolidated clays. It can, however, be shown that, if the points resultingfrom plotting strength against w ater content at failure fall on a single curve, independent ofthe method of test, pore-water pressures and so on, then the working hypothesis outlinedabove may be applied to the soils in question.

DRAINED AND CONSOLIDATED UNDRAINED TRIAXIAL SHEAR TESTSThe application of the strength diagram described above and of the proposed workinghyp othesis will be illustrated by considering the triaxial test.Figs 2 show diagrammatically three stages of a drained triaxial shear test carried out on acohesive sample, normally consolidated under the pressure ue. For each stage the shearstresses and the normal stresses in the different planes through the sample are indicated bythe M ohr circles, and the shear strengths in the planes are given by the failure lines. The


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FUNDAMENTALS OF SHEAR STRENGTH OF SOILS 213diagram shows how the water content decreases as the major principal stress increases ;conseq uently, the cohesion w ill increase and the failure line will rise. During the drainedtriaxial test the strength will conseq uently increase as the axial load increases. The shearstresses, how ever, ex hibit a mo re rapid increase, and, whe n the axial load has reached a certainvalue, a failure will occur.

Figs 3 illustrate, in the same way , three stages of a consolidated -undraine d triaxialshear test. During this test the water content remains unaltered and, consequently, thecohesion and the failure line will be the same for all stages of the test. As the normal loadincreases, how ever, an excess pore-w ater p ressure will arise, and the effective intergran ularstresses will therefore decrease during the test. Using the working hypothesis describedabove, it can be assumed that, for the type of clay under consideration, the effective majorprincipal stress remains unaltered during the consolidated-undrained test, because the watercontent remains constant. The M ohr circles for the external total stresses, as well as for theinternal effective stresses, are shown in Figs 3 for the three stages in question .

STRENGTH/WATER CONTENT RELATIONSHIPAnalysis of the two types of triaxial test (Figs 2 and 3) shows the relation between shearstrength and water content. On comparison of the two diagrams it is seen that, for equalwater contents at failure for both types of tests, cohesions , effective major p rincipal stresses,and failure lines are equal. It m ay now be concluded that the M ohr stress-circles will be


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Fig. 4

A Resutisol consdidatid-undromrd CU) tests

0 Results of consolidated-constant-vokme@XV)

STRESSKILOGRAMSER SO. CW.Shear otrongth/water content relation&p.slammary of reoulto o dir+ ohear toot0with romoul e Uetlibag day oonoolidatodf&m an initial water oontant = liqui imit= 63porcant

Consolidation cows

Mobrm circloa for two mamplsawith equal water coatant atfailure, but conmolidrted onadiflaront initial water contont8tw

Fig. 6

Shoor--/water content rolationohip. Sum-mary o remalb of two oerhs of direat ohaar toot0camiod out with remoulded tile clay, Zurioh, oonsol-idat@d from ini tial water con-t8 of 44 per coat N(liquid limit) and 31 per cent mqmtively z


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F U N DAMEN T AL S OF SH EAR ST R EN GT H OF SOI L S 2x7the initial wa ter content. In Fig. 6 such a curve is shown ; it is seen to run approxim atelyparallel to the strength/wa ter-content curve relating to samples consolidated from the liquidlimit.

COHESION, /WATER-CONTENT CURVESThe magnitude of the cohesion of a saturated intact clay is a function of the wa ter content

only, and the best impression of this relationship is obtained by plotting the logarithm of thecohesion against the wa ter content. The resultant curve is characteris tic for the soil inquestion, being independent of all factors whic h otherw ise affect the total shear strength.

Figs 4 and 6 each show such a curve, and in Fig. 7 the cohesion/water-content curves areplotted for ten different clays. It may be observed that the magnitude of the cohesionshow s a logarithmic increase for decreasing water content, the curves being straight orslightly curved lines. Moreov er, the inclinations of the curves sho w a regular decre ase fr omthe colloidal clays at the top of the diagram to the silty and sand y clays at the bottom.

If the cohesion curves are compared with the corresponding Atterberg limits, a certainregularity may be observe d. In Fig. 8 the curves from Fig. 7 are re-plotted, the wat er contentin this case being exp ressed in terms of the consistency index tl -:L. The zero ordinate-corresp onds in this grap h to the liquid limit and the lOO -per-cent. ordinate to the plastic limit.The diagram shows that the cohesion reaches a magnitude of 03-3 0 kilograms per squarecentimetre at the plastic limit, the highes t values relating to the mo re colloidal c lays. Half-way between the liquid limit and the plastic limit the cohesion amo untsto O 03 0 1 kilogramper square centimetre. No test results are available for water contents higher than S O percent. For normal clays the cohesion is negligible above this value ; for the more colloidalclays, however, the thixotropic effect will be more pronounced for water contents close tothe liquid limit.

A C K N O W L E D G E M E N T SThe Author is indebted to Professor Dr h.c. E. Meyer-Peter, Director of the Federal

Institute of Hydraulics and Soil Mechanics, Zurich, and to Professor Dr R. Haefeli for per-mission to carry out the shear tes ts described above and to publish the results.

R E F E R E N C E SBI SHO P, A. W.. and E L DI N, G., 1950. Undrained triaxial tests on saturated sands and their significancein the general theory of shear strength. Gktechnique. 2 : 13-32.CASAGRANDE, ., 1939. U eber die Scherfestigkeit von Boeden. (On the shear strength of soils.) Bodenme-

chanik und neuzeitl icher Strassenbau-Zweite F olge. (Soil mechani cs and modern road-constr uction.-Second series.) Volk und Reich, Berlin.CASAGRANDE A., and CARILLO,N., 1944. Shear failur e of anisotropic material s. J . Boston Sot. Civ.

Eng. 31 : 74-87.COOPERATIVE ESEARCHON STRE SS-DEFOR MATIONND STRI XNGTH H ARAC TE RI STIC S F SOILS, 1944.Seventh Progress Report.COU L OM B, . A., 1776. E ssai sur une application des regles de maximis et min imi s a quelques pmblbmesde statique. (Essay on the application of maxima-and-min ima ru les to some statics problems.) M noivesAcadmie Royale des Sciences, Pari s. 7 : 353A-353C.H AE FE L I, R ., 1948. Sheari ng strength and water content, a complement to the sheari ng theory. Pvoc.Second Int. Coltf. Soil Mech. 3 : 38-44.H AE FE L I. R .. and SCH AE RE R . .. 1946. Der Triaxialanparat. (The triaxial apparatus.) Schweizer ischeBa ei g. 128 : 51-53; 65-67, 81-84. __ __

HANSEN. T.B.. and GIBSON.R . E .. 1949. Undrained shear stieneths of anisotronicallv consolidated clavs.GLloieihnique. 1 : 189-iO4. _ -

HVORSLEV,M. J .. 1937. Ueber die Festigkeitseigenschaftengestoerter bind iger Boeden. (On the strengthproperties of disturbed cohesive soils.) I ngeniavvidenskabelige Skvifter. A. No. 45. 159 pp.J UR GE NSON, ., 1934. The shearin g resistance of soils. J. Boston Sot. Civ. Eng. 21 : 242-275.PWNIRCIOGLU H ., 1939. U eber die Scherfestigkeit bindi ger B oeden. (On the shear strength of cohesive5oils.J Degebo No. 7. Springer, Berl in ..w?.


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218 B J ERRUM FUNDAMENTALS OF SHEAR STRENGTH OF SOILSK E M P T ON , A. .W., 194&a. A study of the immediate tri axial test on cohesive soils. P roc. S econd Znr.ConJ Soil M ech . 1 : 192-196.SK BM P T ON , A. W., 19486. T he effective stresses in saturated clays strained at constant volume. P roc.

Sev en t h Z n t . G ong . A pp l . M ech . 1 : 378 392 .SKEMPT ON ,A. W., 1948c. Th e I &= 0 analysis of stability and its theoretical basis. P Y OC .Second Zn t . Con f .So i l M ech . 1 : 72 78 .

SK E ;PT ~~_N ~A.W., I 948d. A study of the geotechni cal properties of some post-glacial clays. Gto t ec hn i qne .SK E M ~T ON , A. W., 194& . Vane tests in the alluvial plain of the Ri ver F orth near Gr angemouth. Gdo-

t e c hn i que . 1 : 111-124.SK E M P T ON , A. W., 1948f. A possible r elationship between true cohesion and the mineralogy of clays.P roc . Second Zn t . Con f . So i l M ech . 7 : 45 46 .T E RZA GH I , K .. 1936. T he shearing r esistance of saturated soils and the angle between the planes of shear.PYOC . n t . C on f . S o i l M ech . 1 : 54-56.T E RZA GH I , K .. 1938. E influss des P orenwasserdruckes auf dem Scherwi derstand der Tone. (In fluence ofpore-water pressure on the shearing resistance of clays.) D en t s che Wa s sc tw i r t s cha f t . 9 pp .

U . S . WAR DE PA RTM EN T, 1947. T ri axial shear research and pressure distr ibuti on studies on soils. U .S .Wa t e vw . E x p t . S t a . 3 3 2 pp.

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FundamentalConsiderationsOnShearStrengthOfSoil_Bjerrum