Anisotropy and Stress Distribution in Sand

121

PROC. OF JSCE, No. 212, APRIL, 1973

ANISOTROPY AND STRESS DISTRIBUTION IN SAND

By Nobuchika MOROTO*

ation of the stress in sand from that in the ideal SYNOPSIS elastic media has been explained by pointing out the following main reasons, (1) the sand adjacent Sand has both inherent and induced anisotropy. to the loaded area at the surface tends to yield The former is produced by sedimentation and even by low stress, (2) the rigidity of sand in- the latter occurs in shear. Degree of the an- creases with depth. The author considers, how- isotropy is evaluated by means of the deforma- ever, the anisotropies of sand may also contribute tion modulus ratio n = Ex/Ez , and the case n <1 to the measured stresses and displacement in is treated. sand. Barden (1963) has made a short remark Deriving the solutions of stress and displace- on an effect of anisotropy on the behaviour of ment in a cross-anisotropic soil, effect of n on sand mass. the stress and surface settlement in sand is ex- Nonlinearity and anisotropy in the behaviour amined. The results show that the stress and of sand are evident. Hasegawa and Sawada (1971) surface settlement tend to increase near the load show, in their paper of stress solutions in non- in case of n<1. This stress increase in plane linear and heterogeneous elastic half space, that strain case is smaller than that in axial sym- problem of material nonlinearity can be treated metry. It can be known that the author's stress as a kind of problem where rigidity varies with solution is not coincidence with Wolf's. depth. Then, to evaluate the anisotropic effect From the evidence that a radial stress field is on stress distribution in sand in a simple manner, formed in the plane strain case, it can be con- in this paper, the linear stress-strain laws are sidered that the value n will be partly responsible applied. for the measured concentration index in Frolich's formula. 2. SOME ANISOTROPIC FEATURES OF Usefulness of the anisotropic stress-strain laws SAND for expressing the dilatancy of soil is emphasized. Kallstenius and Bergan (1961) showed that even 1. INTRODUCTION spherical particle tended to pack anisotropically in deposition. Oda (1970) fixed sand particles with Two types of anisotropy have been observed polyester resin to examine the initial fabrics of in sand, namely, an inherent anisotropy and an sand sample prepared by sedimentation in water. induced anisotropy. The inherent anisotropy is He stated that the particles deposited in this way produced by sedimentation through water or to adopt a preferred orientation, with the maxi- pouring in air. The induced anisotropy occurs mum dimension alined in horizontal plane. Parkin in successive shearing process. Such anisotropies et al. (1968) reported that the lateral strain of will have an influence on stresses in sand and the triaxial sample of sand was much greater displacement of foundation resting on it. It has than the vertical strain in isotropic compression been known that the vertical stress in loaded as shown in Fig. 1. Their tests were carried out sand mass exceeds the indicated value by Bous- on medium sand deposited in air. El-Shoby con- sinesq's equation. Investigators have studied this ducted a series of constant stress ratio tests on problem theoretically (Ohde 1936, Borowicka 1943) fine sand. His figure shows the greater lateral and in experiments (Ichihara 1955, Curtis & deformability. Richard 1955, Turnbull et al. 1961). The devi- Moroto (1972) carried out the repetitional loading tests on a river sand under an isotropic stress * Grad. Student, Department of Civil Engi- condition. He reported that the triaxial sand neering Tohoku University, Sendai specimen became almost isotropic. Karst et al. 122 Nobuchika MOROTO

strain laws including a term of the volume change due to shear have been therefore required. Isotropic formulae are inappropriate for the pur- pose. Considering the experimental facts that sand shows anisotropic natures in deformation, let anisotropic stress-strain relations be applied here.

Axial symmetric case

Write the stress-strain relations of triaxial Fig . 1 Variation of Strain Ratio in Isotropic sample as follows in a conventional form, Compression (After Parkin, et al.). ( 2 ) (1961) also noted that a reduction of initial anisotropy occured with repititions of isotropic load. ( 3 ) Rowe (1962) has shown that the ratio of number of points of contact per unit horizontal area It has been justified that mechanical behaviour to that of vertical area depends on the ratio of of soil are better expressed by the spherical and applied stress and is greater than unity. Oda deviatric components of stress tensor. Then, (1972) stated in his study on the fabrics of sand rewrite the above formulae as sample in loading that a preferred re-orientation ( 4 ) of the initial fablics occurs with increase of axial strain, and that the long axis of grains tended ( 5 ) to aline in the direction perpendicular to the axial where stress. According to this the sand sample becomes anisotropic. The anisotropy in shear were ( 6 ) related to the dilatancy as follow by Oda, ( 7 ) ( 1 ) and C, D, A, B are constants. where Sz and Sx are summations of the projected From Eqs. (2), (3) and (4), isotropic compres- area of contact surface at the contacting point sion gives the constant C as, of sand particles on the horizontal and vertical planes respectively. dv is the volumetric strain ( 8 ) increment and dea is the axial strain increment. k3 and k4 are constants determined from test. Substituting Eq. (8) into (4), the constant D can The ratio SzlSz is reported less than unity in the be obtained paper of Oda. Such structures of sand sample should evidently be conductive to the greater lateral deformability. Barden (1963) wrote, " Current unpublished work ( 9 ) at Manchester University, both theoretical and The constants C and D are relating to the com- experimental, indicates that for sands n (the ratio pressional behaviour and dilatancy respectively. of lateral stiffness to vertical stiffness) can be Comparing Eqs. (4), (5) with (2), (3), a set of considerably less than unity." constants in Eq. (4), (5) can be given, In principle, the case n>1 can be considered and probably exists. As has been seen, however, (10) it can be recognized that the case n<1 is com- monly encountered for sand. The case n<1 is (11) therefore interesting in this paper. (12) 3. STRESS-STRAIN LAWS FOR DILA- TANT ANISOTROPIC SOIL (13)

Granular material dilates in shear. Stress- It is known that the conventional triaxial test is Anisotropy and Stress Distribution in Sand 123 not sufficient to determine uniquely all the con- sand. stants. It can be known in Eqs. (9), (21) that the con- Plane strain case (ey=O) stants on the dilatancy, D, D, can be expressed by the deformation moduli Ei and Poisson's ratios Neglecting the second order terms with respect vii . This shows the usefulness of the anisotropic to the Poisson's ratios, write the stress-strain formulae for dilatant anisotropic soil. Tests on relations as, sand necessarily bring errors and variations in (14) the tested values of the moduli. In addition, accurate determination of Poisson's ratio is almost impossible at present. The attention shall (15) be therefore focussed mainly on the deformation

Putting moduli Ea, Er or Ex , Ez and their ratio n = Er/Ea or n = Ex/Ez . (16) Pickering (1970) discussed parameters in the linear elastic soil with a cross anisotropy. He (17) used the relation vzx/Ez=vxz/Ex. In this case, from rewrite Eqs. (14), (15) as, Eqs. (9) and (21), the constants D, D become as (18) follows, (19) (28) where C, D, A, B are constants corresponding to C, D, A, B respectively. (29) From Eqs. (18) and (14), (15), isotropic com- Then, it can be known that the dilatancy is ex- pression gives the constant C as, pressed mainly by the deformation moduli Ea (20) and Er in axial symmetric case and only by Ez and Ex in plane strain case. There is an ob- Substituting Eq. (20) into (18), the constant D be- jectionable matter given by relation (26) or (27) comes in the use of linear elasticity, though the linear elasticity is very interesting for soils engineering, (21) because of its simplicity and convenience. Comparing Eqs. (18), (19) with (14), (15), a set of constants in Eqs. (14), (15) can be written 4. STRESS AND DISPLACEMENT IN A CROSS ANISOTROPIC SOIL (22) In order to evaluate the anisotropic effect on the stress and displacement in sand, a simple (23) case of anisotropy is treated herein. Wolf (1935) is the first to consider the anisotropy of soil (to (24) author's knowledge). However, the author has some questions about his solution in plane strain (25) problem and the neglection of Poisson's ratios in In Eqs. (11), (12) and (23), (24), the constants A, D, A, D do not take naught for anisotropic and dilatant material. Hence,

(26)

(27)

From these things, it is supposed that the re- striction of var/Ea or vzx/Er in the

theory of elasticity does not hold in general for Fig . 2 Unit Line Load and Coordinates. 124 Nobuchika MOROTO three dimensional case. (37) irst, stress and displacement solutions in a cross anisotropic soil shall be derived under plane strain condition. A system of Cartesian coordi- where nates is taken and the z- and x-directions are Let the function be assumed as follow defined vertical and horizontal respectively. Con- sider the case of the principal axis of the ma- (38) terial being coincidence with the coordinates axis. where m is a positive parameter. Substituting Eq. (38) into Eq. (37) gives

(39)

The solution of the differential equation (39) becomes (40) where i and 7)2 are available characteristics roots which are given by

(41) Fig . 3 Uniformly Distributed Load and Coordinates. and B1, B2 are constants. Substituting Eq. (40) into Eq. (33) gives The stress-strain relations are written as follow by neglecting the second order terms with respect to Poisson's ratios, (42)

(30)

(31) (43)

(32) (44) The stresses can be expressed by Airy's stress In the case that unit load is applied vertically function F as, at the origin of the coordinates, the boundary conditions are

(45) (33) The strain compatibility equation is (46) where 5(x) is Dirac's d-function. (34) Substituting Eq. (45) into Eq. (42) gives

Substituting Eq. (33) into Eqs. (30)•`32) and (47) then into Eq. (34) gives Substituting Eq. (46) into Eq. (44) gives (48) Determining the constants B1 and B2 from Eqs. (35) (47) and (48), and substituting them into Eqs. (42), (43) and (44) gives Assuming vzx/Ez-vxz/Ex and using the Barden's as- sumption of (49)

(50) (36) Substituting the relations of 221=k and 7)2=1 the compatibility equation reduces to into Eqs. (49) and (50) gives Anisotropy and Stress Distribution in Sand 125

(51)

(52) (60) On the other hand, Wolf's solution is It is easily recognized that Eqs. (58)~(60) will. (53) be identical with those derived by Sobotoka Solution (51) does not coincide with Eq. (53) ex- (1964). cept only in case of unit k. The author con- The surface settlement can be obtained by siders that Wolf would make a mistake in deter- multiplying Eq. (57) by q -dx and integrating from. mining the constant of his formula. The values (1-x)to -l x) as, of k(1 k) and 2k are calculated in Table 1. This table shows the deviation between the author's stress solution and Wolf's.

Table 1 Values of k(1 + k) and 2k. (61) The vertical stress azc along the center line of the load and the surface settlement woc at the center of the load may be more interesting for engineering use. Putting x=0 in Eq. (58) gives The vertical displacement under unit line load can be obtained by integrating the following equation, (62) (54) Putting z=0 in Eq. (61) gives

as (63)

Figs. 4 and 5 show the effect of k on the- (55) vertical stress and the settlement due to strip load. It can be noted that an n decreases so where Co is a constant of integration. does the load spreading capacity of the medium,. The surface settlement can be obtained by and both the vertical stress and the surface settle- putting z=0 as,

(56)

Putting the value of wo being zero at x = xo gives

(57)

The stress and settlement formulae for uniformly distributed load can be obtained by integrating the solutions for line load. Multiplying Eq. (51) by q-dx and integrating from (1-x) to -(1+ x) the stress formulae under uniformly distributed load are given by

(58)

Fig . 4 Effect of Anisotropy on Stress (59) Distributions. 126 Nobuchika MOROTO

ment increase. The similar results have been the surface settlement at the center of circular shown by Barden for the three dimensional case. loaded area can be readily calculated as

(67) where

(68) Fig . 5 Effect of Anisotropy on Surface Settlements.

(69) and A, C, G, F are defined by the following relations,

(70-a)

(70-b)

(70-c)

Fig . 6 Coordinates. (70-d)

Barden modified Michell's solutions for a homo- (70-e) geneous cross-anisotropic elastic half space. With r, 0, z defined as in Fig. 6, he gave the stresses 5. VERTICAL STRESS IN TWO AND and settldments as follows, THREE DIMENSIONS For a vertical unit point load: the vertical stress az Compare the vertical stress in case of the plane strain with that in the axial symmetry. (64) Putting x=0 in Eq. (51), the vertical stress the surface settlement wo along the loading line for unit line load becomes

(71)

(65) Putting r=0 in Eq. (64), the vertical stress For a circular uniform load of radius L with along the loading axis for unit point load becomes intensity 13: (72) the vertical stress along the loading center line

σzc (73) Neglecting the second order terms with respect to Poisson's ratios, the constants in Eq. (70-a)- Eq. (70-d) reduce to

(66) (74-a) (74-b) Anisotropy and Stress Distribution in Sand 127

(74-c) (77) (74-d)

Substituting Eqs. (74) into (69), and then into Considering the deviations of real sand mass Eq. (73) gives from the ideal elastic solid, Frohlich wrote Eq. (77) as follow by replacing the exponent of the (75) factor cos3 a by an arbitraly exponent v as

Then, Eq. (72) becomes (78)

The value v has been called the concentration (76) index (or factor) because it determines the inten- For the value of unit k, Eq. (71) and Eq. (76) sity of the pressure on horizontal sections beneath are known to be identical with isotropic solu- a given point load. tions which are usually made use of in stress For a vertical unit line load at surface, the calculation in soil. These formulae give aniso- radial stress can be obtained by the integration tropic effect on the stress increase in a simpler of Eq. (78) as, manner, in term of k. (79) Table 2 Ratio of Vertical Stress Along Load- ing Line in Anisotropic Solution to that in Isotropic Solution for Two (80) and Three Dimensional Cases.

Ohde introduced the elastic soil mass in which the stiffness E varies in the manner of

(81) where K is a constant. He obtained the following relation when the soil behaves as the work done becomes minimum,

(82)

He also showed that the Frohlich's formula satisfied the strain compatibility equation only in For both the two and three dimensional cases, case of the ratio of the vertical stress along the loading (83) line in the anisotropic solution to that in isotropic solution are calculated in Table 2. From where v is the isotropic Poisson's ratio. It is the table, it is known that the influence of k on is known that for example, in the case that IC= the stress increase is larger for the axial sym- 1, v =1/3, v =4 can be assumed for a soil, the metric case than for the plane strain case, and Frolich's formula becomes the exact stress solu- Poisson's ratio gives an effect to increase the tion in elastic media. load spreading capacity of the media in the Now, express the stresses in the cross-aniso- former case. tropic mass by the polor coordinates (Fig. 6). Eq. (51) can be transformed to 6. CONSIDERATIONS ON FROHLICH'S STRESS FORMULA As is well known, the major principal stress

plane strain condition. For the three dimensional ( 7 ) The well known FrOhlich's stress formula case, Barden stated that the radial distribution can be applied to stress problems in the cross- of stress can be reasonably assumed, as an engi- anisotropic media, because the cross anisotropic. neering approximation, for relatively low degree media forms a radial stress field. of anisotropy (0.5

REFERENCES The meaning of the concentration index is still 1) Arther, J. R. F. and B. K. Menzies (1972): " Inherent anisotropy in a sand ambiguous. Because the stress distribution fea- ," Geot., Vol. ture in sand can be considered being affected not 22, No. 1, pp. 115-128. only by the increase of stiffness with depth, the 2) Barden, L. (1963): " Stress and displacement anisotropic nature of sand but also by a number in a cross anisotropic soil," Geot., Vol. 12, of factors such as elastic-plastic behaviours of No. 3, pp. 198-210. sand, rigidity of footing, surface roughness of 3) Borowicka, H. (1943): " Die Druckausbrei footing, loading intensity, geometry of load, etc. tung im Halbraum bei linear zunehmenden_ Elastizitatsmodul," Ing. Archiv., Vol. 14, No. 7. CONCLUSIONS 2, pp. 75-82. 4) Curtis, A. J. and F. E. Richard (1955): ( 1 ) It is supposed that for sand the deforma- " Photoelastic analogy for nonhomogeneous tion modulus ratio n will be less than unity in foundation," Trans., ASCE, Vol. 120, No. 35. common case. 5) El-Shoby, M. A. (1969): " Deformation of ( 2 ) The availability of the anisotropic stress- sands under constant stress ratios, Proc., strain laws for dilatant soil becomes clear. 7th., ICSM, Mexico, pp. 111-119. ( 3 ) The stress increase near the load in sand 6) Hasegawa, T. and T. Sawada (1971): " An mass can be also explained in terms of the cross approximate solution of stress distribution. anisotropy of the type given by n<1. The in a heterogeneous and non-linear foundation author considers that the anisotropy is not likely and its characteristics," Trans. JSIDRE Feb., to be the entire explanation for the problem, pp. 61-70. however, it plays an important role for accerelat- 7) Ichihara, M. (1955): " On the distribution ing the increase of stress and also settlement of vertical pressure through sand fills caused near the load. by rigid loads," Jour., JSCE, Vol. 40, No. 4 ( 4 ) The author's stress solution in plane strain (in Japanese). case can be known not to be coincidence with 8) Karst, H., J. Lwgrad, Le Tirant, J. P. Sada Wolf's. and J. Weber (1961): " Contribution a l'Etude ( 5 ) The ratio n is predominantly important de la Mechanique des Milieux Granulaires," among the momuli in the present problems. Proc. 6th. ICSM, Vol. 1, pp. 259-263. ( 6 ) The axial symmetric case has larger effect 9) Moroto, N. (1972): " Recoverable deforma of the anisotropy than the plane strain case in tion of sand," Proc., JSSM, Vol. 12, No. 3,. the vertical stress formula. Sept., pp. 65-74 (in Japanese). Anisotropy and Stress Distribution in Sand 129

10) Oda, M. and H. Kazama (1970): " Funda- homogeneous soil masses," Proc., 5th., ICSM, mental study on anisotropy of sand," Jour., Paris, pp. 337-345. JSSM, Vol. 18, No. 9, pp. 15-21 (in Japa- 18) Wolf, K. (1935): " Ausbreitung der kraft in nese). der halbene und im halbraum bei anisotropen 11) Oda, M. (1972): "The mechanism of fabric material," Zeit. angew. Math. u. Mech., 15, changes during compressional deformation pp. 249-254. of sand," Proc., JSSM, Vol. 12, No. 2, June, pp. 1-17. MAIN NOTATIONS 12) Ohde, J. (1936): " Zur Theorie der Druck- A, B, C: verteilung im Baugrund," Der Bauing., Vol. constants, 25, August, pp. 451-459. A, 173,C: 13) Parkin, A. K., C. M. Gerrard, D. R. Wil- D, D : constants on dilatancy, loughly (1968): " Deformation of sand in Ei : deformation modulus in the i-direction, hydrostatic compression," discussion by εi : strain, Parkin, et al., Jour., SM Div., ASCE, Vol. k: 94, SM1, pp. 336-340. L : radius of circular load, 14) Pickering, D. J. (1970): " Anisotropic elastic 2/ : width of strip load, parameters for soil," Geot., Vol. 20, No. 3, νij: Poisson's ratio, ν:the concentration index (factor), pp. 271-276. 15) Rowe, P. W. (1962): " The stress-dilatancy n : deformation modulus ratio (Ex/Ez), relation for static equilibrium of an assem- P : mean pressure, bly of particles in contact," Proc., Roy., Soc., 23 : intensity of circular load, London, A. 269, pp. 500-527. q : deviator stress, 16) Sobotoka, Z. (1964): " Rheologie des pro- q : intensity of strip load, σi : stress, blemes de deformation plane des milieux ω : vertical displacement, continus," IUTAM Symposium on Rheology x : taken to be the horizontal direction, and Soil Mech., Grenoble, pp. 218-230. z : taken to be the vertical direction. 17) Turnbul, W. J., A. A. Maxwell and R. G. Ahlvin (1961): " Stress and deflections in (Received Oct. 7, 1972)