a stress-strain theory for cohesion less soil with applications to earth pressures at rest and...

A STRESS-STRAIN APPLICATIONS

THEORY FOR COHESIONLESS. SOIL WITH TO EARTH PRESSURES AT REST AND

MOVING WALLS

by P. W. ROWE, PH.D.

SYNOPSIS

Experiments with cohesionless soils in the shear- box and triaxial compression machines are described in which the influence of sample thickness, soil type, length of slip line, type of test, confining pressure, soil density, direction of loading, and strain history of the soil, has been studied. The results sub- stantiate the hypothesis, that the degree of mobili- zation of internal friction depends upon the degree of granular interlocking, which is a function of the movement of the grains per unit length of the slip line.

A new type of confined compression test is described, the results being used to predict the variation of the “ at rest ” earth pressure coefficient with depth. The variation in earth-pressure distribution on stiff walls moving into, and away from, the soil by sliding and tilting motions is calculated.

General agreement is reached with observations of Terzaghi on a large retaining wall.

Des essais faites avec des sols non coherent dans des appareillages de compression a boite de cisaiie- ment et tiaxiale font l’objet d’une description, oh l’influence de l’epaisseur d’dchantillon, type de sol, longueur de ligne de glissement, type d’essai, pression de resserrement, direction de charge et historique des deformations subis par le sol est Btudi6e. Les r6sultats viennent a l’appui de l’hypothese selon laquelle le degre de mobilisation de frottement inteme depend du degre d’enchevetre- ment granulaire qui est fonction du mouvement des grains par unite de longueur de la linge de glissement.

Un nouveau type d’essai de compression dans une enceinte rigide est Bgalement d&it ; ses resultats servant a predire les variations du coefficient de poussee de la terre ” au repos ” par la profondeur. Les variations de distribution de poussee de la terre sur les murs rigides se rapprochant et s’kcartant du sol par mouvements de glissement et de basculement sont calculees. Un accord general se fait avec les observations de Terzaghi sur un grand inur de soutknement.

INTRODUCTION

Many structures are in contact with sand under conditions where the sand*is not failing, for example, tunnels, culverts, flexible walls, and foundations. Elasticity theories cannot be

applied to these problems, since the sand-mass is composed of individual particles, incapable of taking tension, dependent upon confining pressure in a non-linear manner for its strength, and having properties which vary with its stress history.

There are available, however, satisfactory theories of the states of stress in a soil-mass which require only a knowledge of the angle of friction for their solution. At failure this value has been known but, for soil that is not failing, it is first necessary to determine a relation between the value of the friction angle mobilized and the soil deformation ; then to relate the deformation to the boundary movements imposed by the structure.

The purpose of this Paper is to attempt to define the nature of the stress-strain function which it is required to measure, both by the correlation of its approximate measurement from soil tests, and by the results of simple applications to known results.

NOTATION

e denotes relative movement of shear box. H h, ::

wall height. head of dry sand equivalent to confining pressure.

K >> earth pressure coefficient. 70

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A STRESS-STRAIN THEORY FOR SAND

mean value of K over a wall. at rest pressure coefficient for normally loaded soil.

l&&h of Gp line. 8, I, preconsojidated soil.

soil stiffness modulus. angular rotation of wall. earth density. maximum value of wall friction angle. intermediate value of wall friction angle. confined compression strain. e/l for soil on soil. e/l for soil on steel. the ratio : horizontal wall movement to wall height.

the ratio : horizontal soil movement due to conjugate slip

wall height the ratio : depth below surface to wall height. maximum value of the angle of friction. intermediate value of the angle of friction.

SHEARING PROPERTIES

71

Hypothesis (1)

The degree of mbilization of the angle of friction +, and the angle of wall friction 6, depends @on the degree of interlocking of the soil grains, which in turn depends upon the fractional movement of the length of the shear planes, to be known as the slip straiu.

Hypothesis (1) is a radical departure from shear strain as defined by the theory of elasticity, and in order to test its validity the variation of 4, 6, and the volume change with slip strains E# and E’# has been measured subject to the influence of the following factors :-

(1) Rate of loading. (5) Confining pressure. (2) Sample thickness. (6) Soil density. (3) soil type. (7) Direction of loading. (4) Length of slip line. (8) Strain history.

Fig. 1. Modifloations to shear box


72 ROWE

30. t Three soil states were used throughont, as follows : the loose state, y = 96 lb. per cubic foot ; the dense state, y = 112 lb. per cubic foot ; and the pm-consolidated dense state, y = 113 lb. per cubic foot. The loose and dense states were obtained by methods corresponding closely to those

0-Sunph volume recommended by Ko1busrewski.l For the

0 pm-consolidated dense state the sample 0 ocQ5 001

‘a” MC.4 ; c, was reduced to the dense state and the loading plate P, in Fig. 1, was then placed

FQ.2. V~~~~'~dueto in position and hammered down. The shear box was then submerged in sand again, and the mass was vibrated with an

electric hand vibrator for 5 minutes. The box was then set up in the shearing apparatus and a vertical pressure of 50 lb. per square inch was applied. This was reduced to a Iower value for the test.

A dial gauge reading to OGlO1 of an inch was mounted directly between the two halves of the box on the plan centre line. provision was made to allow the direction of shearing to be

reversed. In all the tests the value 4 = tan-1 applied shear load

normal load was plotted against the

slip strain c# = box relative movement

inside length of box * Rate of Eoading.-The strain was applied in increments to mobilire values of 4 in Z+degree

intervals. For the main tests, the samples were left for a few minutes only before recording the strain. Rapid loading tests gave d; values l-14 degrees higher, while samples left stressed for 24-hour periods gave + values 2-3 degrees lower.

Sumplc thicks.-Tests were made with sample thickness varying from O-75 to @I times the box length for normal pressures between 40 feet and 5 feet of sand. To interpret the results it was necessaq to consider the influence of boundary restraint, disturbance due to shearing to size, and tra&fer of normal pressure to the box sides..

If increase in 4 is accompanied by interlocking of all the grains, the grains should be as free to rotate and slide as they are in a soil-mass. The grains next to the loading plates are restrained from rotation, however, and as the sample thiclmess was decreased, the restraint decreased the movements for a given value of 4, for both loose sand, and dense sand when no shearing to size was allowed. Shearing to size loosened the top layer of the sample, and the movements increased with decrease in sample thickness.

Increase in thickness led to a transfer of normal load to the box sides by friction, decreasing the recorded 4 values below the true values. The effect on the curve was, therefore, similar to that of decreasing end restraint, so that it was possible only to estimate the point where the one effect became small and the other began. A thiclcness~h ratio of O-4 was decided upon, with no shearing to size, and with an estimated possible error in # of 3 degrees for dense sand and 2 degrees for loose sand.

The slip strain does not increase in proportion to thickness, as would be expected if + depended on the elastic type shear strain. D. W. Taylor* showed that the latter type was about 3 per cent of the box movement divided by the sample thickness. Such elastic shear strain may, therefore, be neglected.

In contrast, the percentage thickness chauge during shear was found to be approximately proportional to slip strain and independent of thickness. This proves that the grains roll

~Thereferencesaregivenonp.88.


A &TRESS-STRAIN THEORY FOR SAND 73

together throughout the whok sample and not merely in a zone around the box centre line. If Hypothesis (1) is correct, interlocking will occur on the conjugate slip lines at angle 4 to the vertical, and the dilatancy effect during the initial stages of shear will be composed mainly of the vertical component of the interlocking part of the slip strain. Fig. 2 shows quite reasonable agreement, and illustrates the interlocking and slip components of the slip strain.

Soil typC.-Tests were made with rounded, sub-angular and angular grains. The greater the angularity the steeper the #/c, curves, although this influence was much less than that of density or confining pressure. A sub-angular sand was mainly used.

Lnrgur of slip line and type of test.-Details of the types of tests made and sources of information are given in Table 1.

TabhI.-LeqthoafsliplineandQpeofte~t

T&type Size Length of slip line Soiltype Investigator

square shear box

10 in. dia. I

31.4 in. Submgular sand I

Author and P. D. Arthur Torsional shear box

Triaxia

4 in. dia. x 8 in. long

l)in.dia. x 3 in. long

9) in.

3f in.

Sub-angular sand Author

Sand Lig-Sheng Chen 4

The &cm. square shear box was reduced to an inside length of 3 cm. by filling the two halves of the box with wood blocks, 6 cm. wide x 3 cm. long. The torsional box had a mean radius of 5 inches, the sample being I inch wide by 1 inch thick, so that greater side restraint may have

been present. The slip strain is given by r, = &, where 6 is the relative angular rotation of

the two halves of the box. In the triaxial test, the slip strain on the inclined slip and conjugate slip planes is, by

geometry, equal to the longitudinal strain of the specimen. It may be noted that this is not the same as the change in length between the ends of the slip line. A vertical line contracts and a horizontal line expands and some inclined line will not change in length, just as the horizontal lines above the rupture plane in the shear box do not change in length. Never- theless, the lines have been moved and have transmitted shear forces. The triaxial test results are, therefore, plotted in terms of 4 and the longitudinal strain.

All the results are plotted in Figs 3, where the curves are grouped into regions of confining pressure denoted by the equivalent head of sand of 100 lb. per cubic foot density. The shear box tests showed slip strains only after a value of 4 of 5-7 degrees had been mobilized, whereas the triaxial compression strains commenced from zero. The explanation is that the initial uniform triaxial compression induced no slip strain, whereas the application of normal pressure in the shear box test induced an initial strain before test.

The curves in Figs 3 are influenced by varying degrees of boedary restraint, by differ- ences of soil type, and by the scatter of individual test results from the mean. Nevertheless, for similar soil state and confining pressure, the close grouping of the curves obtained from widely varying lengths of slip line supports the general nature of Hypothesis (II.


74 ROWE


A STRESS-STRAIN THE-ORY FOR SAND 75

Confining pressure.-The variation of @with pressure in Figs 3 is well known,e but it is of interest to connect this with Hypothesis (1). An artificial dense pack-

’ ing is shown in Fig. 4 (a), all the spheres being separated by minute particles. The centre lines of any two adjacent spheres on adjacent planes lie at 54 degrees to the horizontal. If the box containing the spheres is tipped through an angle, Fig. 4 (b), no movement will occur until the box

Figs 4. Mechanism of pressure influence lies at 54 degrees to the horizontal. The next movement causes the spheres to roll

out down plane XX. The d/c8 curve is a rectangle, and corresponds to a normal pressure equal to zero.

The application of normal pressure to the sample is equivalent to a triaxial pressure pr, Fig. 4 (c), and a vertical pressureps which shears the sample, Fig. 4 (d). The displacements of the spheres in these two stages decrease the inclination of plane XX, decreasing 0. The spacing between the spheres on the horizontal plane is increased, so that the interlocking movements will be larger. The initial interlock will increase with pressure at a decreasing rate after the shape of the confined compression curve, Figs 8, so that, if 7. is the initial shear stress

necessary to overcome this interlocking at normal stress ao, then tan #o = - will decrease 70

with increase in pressure. Thus the decrease in slope and maximum values of?he curves with increase in pressure supports Hypothesis (1).

Soil density.-The wide variation in initial slope of the curves between loose and dense states is well known. The slope of the curves for the pre-consolidated samples was much steeper than for dense sand, see Fig. 6 (c).

Loading direction.-A structure which applies “ passive ” pressure to a soil Will, in general, strain the soil in the opposite direction to that in which it has sheared due to confined compression, and the +/c8 curves on unloading and reversed loading are required. The triaxial machine could not be adapted for reversed loading and the tests were confined to the shear box. The sampIe was sheared to a selected value of 4, such as point a, Fig. 5 (a). The straining handle was then reversed and readings were taken until maximum negative shear. The volume change due to conjugate slip remained negative until final shear, following the shape of the +/es curve. The additional movement on reversal increased the density of the loose sand and decreased that of the dense sand. It is important to note that the slope of the unloading curve decreases as the initial positive 4 value increases. The original +/Q curves are re-plotted as +/Q tan (45 - @!) curves for convenience of application to practical problems.

Strain history of the soil.-A pre-consolidated or tamped soil will have been strained to some point a, Fig. 6 (a), and unloaded to some point b. On re-loading, the initial slope is seen to be almost vertical until the previous pre-consolidation strain is reached. Each unloading and re-loading cycle led to a small residual’ strain, which increased in value as 4 approached @.

CONFINED COMPRESSION PROPERTIES

The existing compression apparatus for clays is unsuitable for, sands. The stiff loading platform does not give a uniform pressure to the surface, and side friction reduces the compression. The one-inch sample allows rather small deflexions subject to comparatively large errors due to loosening of the surface layer when it is cut to size, and to “ bedding down ” effects of the loading plate.


A STRESS-STRAIN THEORY FOR SAND 77

I2

% 4s

fW h (Cl plvcondld4ud dens4

blbwboxbmts-unloadingandre-loading

is shown in Fig. 7. Air pressure is applied at the surface of a A new type of apparatus flexible membrane A, O-017 inch thick. The movement of the soil over the centre section is measured by a dial gauge in contact with the soil surface by means of a metal disc B and plunger C. The loss of load under the disc due to the area of the plunger is compensated by placing small weights on the platform D. The internal diameter of the apparatus is 10 inches, and the depth of sample varies from 2 to 5 inches. In order to allow for strain of the apparatus, a calibration was made in which a block of steel was screwed to the inside of the base, with the steel surface level with the underside ‘of the rubber ring E. The remaining space was filled with sand and a test was made. The small compressions recorded were


78 ROWE

4 4 (u) Loose sand (I) Dense sand

Figr, 8. Confined compremsion teete

subtracted from those measured in ah tests before calculation of the strain. The results are shown in Figs 8.

Let a circular element of soil ABCD of unit dimensions, Fig. 4 (c), be compressed by triaxial pressure $r to a strain e,.

Then the new volume A ‘B’C’D’ ti 7r/4( 1 - 3eI).

Let an additional vertical pressureps be applied on face A’B’ to shear the soil to A”B”C”D” with an additional vertical strain es, so as to maintain the original lateral boundary line. If the volume change during shear is dV, equating volumes we have,

r/4(1 - 3er) = r/4(1 - er - ea) + dV.

The confmed strain is fC = er + ez = 3er + dV/V.

Writing W/V = fle2,

The value of n varies4 from $ to zero over the range of strains 0.001 to 0.02, and it is sufficient for the present purpose to write :

e2-gc . . . . . . ; . . . (1)

Hypothesis (2) EARTH PRESSURE THEORIES

The soluiions to the states of stress within a soil mass, and adjacelzt a rigid boundary, hitherto apPlied to faiJure problems usi% maximum values of friction angles @ and A, apply also where the soil is mx? failing and where the friction angles are 4 and 6.

Following large-scale earth pressure tests, s Terzaghi published a paper7 in which his first conclusion was as follows : “ The fundamental assumptions of Rankine’s earth pressure theory are incompatible with the known relation between stress and strain in soils, including sand. Therefore, the use of this theory should be discontinued.”

Quite reasonable agreement has been found between the experimental results by Terzaghi and the ideas presented here, and it is of value to re-read Terzaghi’s paper.7 Nevertheless, the above conclusion requires examination if Hypothesis (2) is to be substantiated.

The simple mathematics giving the lateral pressure coefficient within a soil mass as :

K = tans (45 - +/2) . . . . . . . . . (2)

is based upon the assumption that 4 is given by the relation

tanrj=+ . . . . . . . . . . . (3)



where 7 and D are the shear and normal stresses acting on the plane which makes their ratio a maximum. If this coefficient K does not give a value in agreement with experimental observation, as for example under “ at rest ” conditions, then the reason must be that the value of 4 inserted into equation (2) does not satisfy equation (3). Terzaghi points out that soil in an infinite mass cannot fail as in a shear test, so that the maximum value Qi cannot occur. This is true, and inserting 4 < @ into equation (2) the value of K becomes greater than the minimum value. Further, the Rankine formula no longer implies a hydrostatic type pressure distribution since + is not necessarily constant with depth.

Similar remarks apply to the equilibrium equations of a soil wedge adjacent a wall, only here the values of 4 and 6 will vary with depth over the length of the boundaries. For the present purpose the values mobilized at two thirds the depth of the wedge will be assumed to be constant over the boundary.

APPLICATIONS TO PRACTICAL PROBLEMS

I. Calculation of the at rest earth prczsure, coeficient

(a) No~muZly loaded soils. (i) Iu soil&ass.-The slip strain induced in a soil at depth h below the surface under pressure rh, is obtained from the & curves, Figs 8. The corresponding value of 4 is obtained from Figs 9 which have been plotted from the mean of the curves in Figs 3. An example of the calculation for depth 20 feet is given in Figs 8 (a) and 9 (a). The value of 4 is first estimated assuming the confining pressure is equal to yh, point a, and the approximate value of. K,(l + sin 4) given in Fig. 9 may then be used to estimate the normal pressure on the slip plane and a more accurate value of 4, point b. To allow for relaxation, the values of + are decreased by 3 degrees for loose sand at + equal to 25 degrees, and 1 degree for dense sand at I+ equal to 20 degrees.

The results of calculations at various depths are plotted in Fig. 10, together with the results of loose sand cell tests by Gersevanoff. s No results are available for dense sand as distinct from pre-consolidated dense sand.

(is] Against a steel wall.-Wall friction is developed by the vertical compression of the soil as the level is raised, and it will depend upon the compressibility of the subsoil. Considering

?‘ig8 3. Mean 410 curves estimated from Figs 3

F


80 ROWE

a rigid subsoil, the pressure ‘yk at depth k causes a movement of the layer relative to the wall of (H - IJ)Q,, which decreases to zero at the wall base. The average interlocking of the grains adjacent the wall will be approximately the same as if the whole layer of depth (H - h) had moved distance +(H - h)~ relative to the wall length (I3 - h). Hence, we may write,

r: -h +c. The values of l # and E: are calculated from the vertical pressure acting at two thirds of the

depth considered. The corresponding values of 4 and 8 determine the values of Ko, using earth pressure tables for active pressure. @ The results are included in Fig. IO.

Theboundary restraints in the shear test@ would tend to make the stress-strain curves too steep and lead to a minimum K,, value on the low side. Nevertheless, approximate agreement is reached that K,, is constant at the greater depths at which tests have usually been made. The theoretical results show, however, that in the model range O-2 feet for loose sand and 0s feet for dense sand, K. varies rapidly with depth. This may have an important influence on the extrapolation of model results to field behaviour.

(b) Pre-consd~ sod. In s&mass 01 against a r&L--In the compression and shear tests, unloading results in relatively small strain reversals. Thus, after tamping a fill behind a wall, the lateral pressure will be almost as great as the value which acted under the pre- consolidation pressure. On this assumption it can be shown that the fir& pressure coeffi- cientis:

.

K’O = . . . . . .

where h,, is the surcharge head removed, and h the final head. Equation (4) agrees with that derived by Wilson,‘@ and with experimental evidence by Janson, Wicker? and Renkert,rr and the Author.12 It is clear that KIO may have any value between the maximum passive value and the K,, value for dense sand. It may even fall below this if loose sand is tamped. Thus most of Wilson’s “ at rest ” values lie above the minimum theoretical curve, Fig. 10. For a particular site, h,, will be constant so that K’,, will decrease with depth after the manner of records of Meyerhof.13

The values of Kb by Gersevanoff* became constant at 0.53 at a depth of 41 feet, so that it may be assumed that the impact during compaction gave a confining pressure equivalent’to



that depth. The corresponding value of + is 18 degrees. Assuming linear unloading and re-loading stress-strain curves, the + value at a confining pressure equivalent to depth d, where d < 41 feet, is given by :

and the K’,, value is obtained from equation (2). Very reasonable agreement results, see Fig. 10.

Thus, laboratory tests on dense sand compacted by ramming yield no specific information until the influence of the weight and size of ram, the number of blows and the system of application is known.

II. Calculation of th lateral pressure o+t moving wa8.s

(a) Wdl moving away from soil. (i) Sliding mot&n. -Let the wall, Fig. 11, move outwards a distance F;H. The component of movement in the direction of a slip plane AB

The length of the slip plane F -$$-.

The planes AB and CD move together partly as if the whole mass were sliding on the limiting plane XX. Nevertheless, each layer of sand above XX slides down independently, due to the forces acting above the layer, and interlocking occurs over the planes AB and CD to a degree which depends upon the distance through which the plane is moved relative to its length. The mean slip strain is : Fig. il. snip strains induced by an

outward sliding wall

Neglecting the small influence of wall friction on the slip line length, we have that

5 7, = Ed tan (45 - d/2) . . . . . . . . . (5)

A correction to equation (5) is necessary to allow for conjugate slip strain on planes such as EF (volume change during shear). Consolidation along EF during shear moves the grains away from the wall, so that larger slips on planes AB are required to maintain the boundary line. For simplicity, the conjugate slip strain is taken equal to 4, although error results at failure, see Fig. 2. The horizontal component away from the wall equals:

[‘H = ~J’Q tan 8dv . . . . . . . . . (6) rl

For the purpose of applying equation (6) as a correction to equation (5), it is sufficiently accurate to substitute 4 equation (5) into equation (6) and integrate, whence


82 ROWE

The equivalent outward wall movement is now (5 + r)H and the strain induced is given by

(5 + 5’)H = WH tan (45 - 412) Substituting for r, we have

f(4) = c tan (45 - +/2) = i(l + 1.15 log,,~ )

. . . . (7)

The term in the brackets, equation (7), is the correction to equation (5) to allow for conjugate slip.

The vertical component of slip is 6 + 5’

( ) s H.

The main part of this, c, is constant over the wall height, so we may write

6’. -h (6 + 5’) cot 0 = 14 . . . . . . . . (8) (izJ Tilting motion abozlt the toe.-The outward movement at depth -qH due to a rotation I

radians is ~(1 - v)H = CH and proceeding as before it is found that :

f(+) = c, tan (45 - +/2) = $1 - 7 + 2.3 log,, l/v) . . . . (9)

To facilitate the calculations it is convenient to plot + as a function of Q tan (45 - $/2) = f(4), as in Figs 5. Similarly, 6 is plotted against c’~ tan (45 - 6/2) =f(s). For simplicity, this is taken to be equal to equation (9) assuming + = 6.

Before a wall yields, values of + and 6 are already mobilized by confined compression, and the correspondingf(4), andj(6) values must be added algebraically to those due to wall movement. The steps in the calculation are, therefore, as follows :-

(1) Tabulatef(4) andf(s), given by equation (7) or (9) and equation (8) for chosen values of 7 and 5 or r.

(2) Determine the “ at rest ” values of + and S at two thirds the depths considered, and read the corresponding initial values of f(4) and f(S) from Figs 12 (mean of triaxial and shear box) or from Fig. 5 (shear box only). Add these values to those

(3) Rezfhe(lf!nal values of 4 and S from Figs 12 or 5, corresponding to the total values

off(+) andf(8). (4) Read the value of K for each depth from earth pressure tables. The pressure

distribution is then given by % = qK.

(b) Wall moving ido the so&-It is generally taught that the first movements of a wall into soil induce “ passive ” pressure. In the sense that the earth resists this movement, this

I I I I I 1

I I I

M 5h,. ,y,-L 40%

1 1,’ \w SkC- w

b

ICJ

IL*_ lQ*-

10



is so. However, with the exception of dense sand behind a model wall, the soil will be in a state of active slip due to confined compression before the wall is moved. Shear will have occurred up to a point such as u, Fig. 5 (a). The first wall movement will decrease + to zero, point b. Between points u and b, K will be less than unity and the Mohr’s circle of stress will have the appearance of the active state. It is proposed that the term “ active ” pressure be used where + is positive and K < 1.

(2) Slidiltg motion.-For inward movements the slip lines are curved and somewhat longer than the straight line type. The influence of this on the calculated pressures is allowed for by using earth pressure tables, but the slip strains have been estimated on the straight line lengths for simplicity. Thus the wall movement calculated to mobilize a given resistance will be less than the correct value by an amount in proportion to the lengths of the two types of slip line, but the error in the estimated soil stiffness is negligible compared with the wide variations with small changes in soil density.

The conjugate slip strain continues in the same direction after reversed loading, decreasing the main strain. Hence, equation (7) becomes :

* f(#) = E# tan (45 - $/2) = $1 - 1.15 log,, l/q) . . . . (10)

andE:==r/c, . . . . . . . . . . (11) (ii) Tilting abow! the toe.-The values of f(4) are given by :

The steps in the calculations are similar to those for outward movements using Figs 5. The values off(+) due to wall movement must be subtracted from the initial values in step (2) above, and the unloading curve appropriate to the initial + value must be used in step (3).

Calculations have been made for Sinch model walk and the 5-foot wall used by Terzaghi. For the active case, pressure distributions are obtained which are not quite triangular. Terzaghi equated the observed moment about the toe to the theoretical valu.r, assuming a triangular distribution, and recorded his results in terms of a mean coefficient K. Following this procedure, and writing Kl - K, for values of K at 7 equal to 0.2, 0.4, 0.6, 0.8, and 1.0, we have that

K = 1.2[0*157K, + OC24(K2 + K3) + O.l6K, + O.OSK,] . . (13)

Terzaghi’s tilting motion was a combination of tilting and sliding, and the movement was recorded in terms of the ratio of the movement of the centre of the fill height to the fill height. Denoting this by b, the values of 5 at height 77 are given by :

5 = 1*19(134 - 7)[0 . . . . . . . . . (14)

Substituting c in equation (14) into equations (7) and (lo), the K values down the wall are obtained and inserted into equation (13). The corrections for conjugate slip should be ignored for 7 < O-3, since for large strains near the surface the conjugate strain is not equal to +

The results for the outward movements are shown in Figs 13. The slight difference in results between sliding and tilting motions observed by Terzaghi is detected’by the theoretical curves, and the observed curves relating to tamped sand might be expected to lie between those calculated for dense and pre-consolidated dense sands. Good agreement was reached with the variation in the height of the centre of pressure.

For inward movements the following cases were calculated :- (1) Wall pushed into pre-consolidated sand before outward yield. (2) Wall pushed into dense sand after outward yield, & = 06045. (3) Wall pushed into loose sand after the same outward yield.


ROWE

Left, Fige 13. Variation of mean hydroetatic ratio with wall movement- 5-foot wall after Ter- zagbi-outward yield

All the motions were tilting and sliding combined. The results in Figs 14 show fair agreement in two cases. The upward curve of curve (2) seems to suggest that the change in the direction of the slip line on reversal of loading leaves the slip line consisting partly of loosened sand and partly of sand which has not previously been loaded. For the calculation, the whole of the backfill was treated as if it had been subject to the strain resulting from the outward yield.

The theoretical distribution for a wall rotating about the toe was found to be approximately parabolic, with

Left and above, Figs 14. Comparison with passive movement tests by Terzaghi



IOTAnoN *nob-l THE Tof ( r”)

Fig. 15. Tbeoreticd and observed momeat/aaguhr rotatioa curvea

the pressure centre 04SH from the toe. Observed and calculated moment/rotation curves are compared in Fig. 15.

Terzaghi explained his observations in terms of the hypothesis that the lateral pressure on the wall was a function of the ratio of the lateral expansion or contraction of the sand to the width of the failing wedge of soil at the particular depth. This is similar in form to Hypothesis (1). and the above simple treatment of moving walls differs from Terzaghi’s only in that the movements are resolved in the directions of slip.

(c) Thtz freely embedded stiff @le.-By assuming a point of rotation, the pressure distribution may be calculated as for stiff pinned walls. If the resultant overturning moment and force do not balance that.applied, a new rotation point must be tried. The final point will be found to be close to that given by the parabolic type distribution. The theoretical distributions for a equal to 0.7 for loose and dense sand are shown in Figs 16 and moment/rotation curves are compared in Fig. 17.

In agreement with Rifaat,i* an equivalent parabolic pressure distribution can be sub- stituted for the theoretical curves, within the range of rotations investigated, without depart- ing unduly from the degree of accuracy of the calculations. The parabolic distribution is given by :

P _- vn =y?)(?; -7) . . . . . .


ROiVE

PC

Fig. i?. Variation of resisting moment with rotation and sand density



where ;io is the depth to the point of rotation, and m is a measure of the soil stiffness. Equat- ing moments about the toe M. we find that

where a is defined in Fig. 17. By fitting equation (15) to Figs 16, or by measuring the straight line slope to a point on a curve in Fig. 17, the value of m is obtained for a given degree of rotation. The results of a large number of tests are summarized in Fig. 18, which shows that logl,,m is approximately independent of a and is a function of the degree of angular rotation and of the relative density of the sand. The theoretical values show good agreement at the loose and dense limits. The graph is also of value for the analysis of sheet pile walls, since the product of the soil stiffness, m, and the wall flexibility determines the maximum stress in the wa.lI. The range of angular rotations of flexible walls in given types of cohesionless subsoil is limited to known regions so that for a given subsoil the average soil stiffness modulus may be estimated to an accuracy possibly exceeding that at which the soil can be sampled.

Extrapolation of model observations to behaviour of jield structures

With increasing confining pressures, the +/Q curves become less steep. In addition, the “ at rest ” C+ values are higher the larger the scale, and the +/Q curves on reversal are again less steep. For both these reasons, the rate of change of 4 with wall or footing movement into the soil decreases with increase in scale ; that is to say, the soil behaves as if it were less stiff. Therefore, all model research connected with deformation problems of cohesionless soil should, preferably, be conducted on as large a scale as possible.

GENERAL DISCUSSION

The civil engineer generally requires only approximate answers to deformation problems. He wants to know whether a foundation will settle &, 1 or 3 inches, or tihether the stiffness of a subsoil into which a sheet pile is driven is similar to a dense, medium, or loose sand, or a loose silt. For this reason the accuracy of the present calculations may already be sufficient for practical application, as for example the determination of the soil stiffness modulus for sheet piling or the “ at rest ” Parabola y-ii - p f+j(+T)

pressure likely to be acting in a pre- consolidated sand deposit.

However, the simple treatment presented requires refinement, which should folIow development in the technique of measuring the stress-strain curve. ‘Apparatus developed for this purposed ~6 should :-

Fig. 18. Theoretical and observed values of soil stSness modulus

(1) Give ,minimum boundary disturbance.

(2) Be able to apply reverse loading. (3) Be able to measure initial strains

of the order of 10-s. (4) Yield consistent results at con-

fining pressures of 6 inches of sand.

In addition, a fatigue testing machine capable of applying cyclic loading within


88 ROWE: A STRiSS-STRAIN THEORY FOR SAND

any given range of 4 values might yield results which would correlate with the deformations of flexible pavements subject to cyclic loading.

SUMMARY

The angle of friction in a cohesionless soil which is not failing is mainly a function of the movement of unit length of the planes subject to maximum shear. The function varies largely with confining pressure, density, strain history and strain direction, and, to a lesser degree, with soil grading and rate of strain.

The triaxial compression test provides a sounder basis for the measurement of the function than the shear box, although the latter would appear to be necessary for wall friction tests. Both methods suffer from boundary disturbance, and the triaxial apparatus requires re- design to allow reversed loading tests. Nevertheless, results from the two types of test give stress-strain curves of the same order.

A new type of compression test apparatus for cohesionless soil is shown to be satisfactory and provides the basic compression characteristics.

Applications of the stress-strain function to the existing solutions for the states of stress in a soil-mass, and to pressures on walls where zone ruptures occur, yield satisfactory results and suggest a new approach to the solution of a number of deformation problems of interest to the civil engineer.

ACKNOWLEDGEMENTS

The work was carried out in the Engineering Department, Manchester University. The Author is indebted to Professor J. A. L. Matheson, M.I.C.E., for providing the facilities

for this work. The torsional shear box tests were made by Mr P. D. Arthur at University College, Dundee, St Andrew’s University.

RhFERENCES

1. KOLBIJSZEWSKI, J. J., 1948. An Experimental Study of the Maximum and Minimum Porosities of Sands. Proc. 2nd Int. Con. Soil Mech., Rotterdam. 1 : 158.

2. TAYLOR, D. W., 1939. A Comparison of Results of Direct Shear and Cylindrical Compression Tests. Proc. A.S.T.M., 39: 1058.

3. BISHOP. A. W.. 1948. A Large Shear Box for Testing Sand and Gravels. Proc. 2nd Znt. Cox. Soil Mech., Rot&dam, 1 : 207. -

4. CHEN. L. S.. 1948. An Investination of Stress-Strain and Strength Characteristics of Cohesionlees Soils

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a stress-strain theory for cohesion less soil with applications to earth pressures at rest and...

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