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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 load- ing 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 an- isotropy occured with repititions of isotropic load. ( 3 ) Rowe (1962) has shown that the ratio of num- ber 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 be- comes 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 al- most 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.
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