Qx-Qy2 cbmo=arctanv ax=-arctanzzxylay=ax+ r am-(ax+a) RS-1ksineccosImo2S(sin-sinmo)2 RT-ks sin bsincbmosin -sin qs mo Rc-1sinsinImodam2sin-sinoUm dys=

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1 MODELLING SOIL BEHAVIOR UNDER ARBITRARY LOADING AND ITS APPLICATION TO LIQUEFACTION ANALYSIS By Kiyoshi FUKUTAKE and Akira OHTSUKI In order to simulate the ground motion under arbitrary loading including rotation of the principal stress axes, the authors incorporate a proper constitutive equation in dynamic analyses. In the present paper, a parameter is proposed to consider the effect of stress reversal subjected to earthquake loading. The parameter is substituted in the Matsuoka's constitutive model (1986) that can evaluate the yield due to rotation of the principal stress axes. Simulations of simple shear tests, shaking table tests and liquefaction analysis of irregular grounds are carried out. Those numerical studies indicate that Matsuoka's model can be applied to the seismic response analysis of the ground under the arbitrary loading. Keywords: constitutive equation, repeated shearing, liquefaction, arbitrary loading, dynamic analysis

2 Qx-Qy2 cbmo=arctanv ax=-arctanzzxylay=ax+ r am-(ax+a) RS-1ksineccosImo2S(sin-sinmo)2 RT-ks sin bsincbmosin -sin qs mo Rc-1sinsinImodam2sin-sinoUm dys=rssin2a 1dyr=Rr'sin2(a+S) Zdye=Rsin2a d=rssa+rscos2a der=rtsa+r1cos2(a+s) den=nrsd+rcos2a+d Fig. 1 Three stress parameters, 16mu, a and Cm used in Matsuoka's model.

3 d-1+e n for dqm<0 de=0.4343cce0 for dam>0 Rowe: Sd=K+Q/Q3 Fig. 2 Consolidation terms expressed by Mohr's circle. Fig. 3 Stress-strain matrix: [D2]-1

4 ms s-k1+k2 1-sin q5-sin(0.9 O)Z-sin-sinq sinb+sin (oa-arctan 2u) ai dy'moct<0 or x+by<90. dmucj>0 and Nx+e9yZ90 Fig. 4 Schematic diagram of parameter for stress reversal Fig. 5 Example of judgement of shearing direction.

5 k=0.441+ea Table 1 Parameters of Toyoura sand. (a) clockwise (b) counterclockwise Fig. 6 State of strain increments in rotational shear. Fig. 7 Stress paths, (c-r) and (aj-r).

6 Fig. 8 Calculated stress-strain relationship. Fig. 11 Measured effective stress path and stress-strain relation in undrained simple shear test (after Matsuoka et al.).

Fig. 12 Stress path in stress-deviation field Txy-2(ox-ay) Fig. 13 Effective stress path and stress-strain relation in undrained simple shear by response analysis.

7 Fig. 12 Stress path in stress-deviation field Txy-2(ox-ay) Fig. 13 Effective stress path and stress-strain relation in undrained simple shear by response analysis. Fig. 14 Ground model for shaking table test (after Iwasaki et al.). Fig. 15 Accelerations liquefaction and pore water pressures obtained by test (after Iwasaki et al.).

8 Fig, 20 Effective stress path. Fig. 17 Analytical model for embankment. Fig. 18 Distribution of maximum shear strain (%) in nonlinear zone. NOD 461 Max=120 gal Fig. 21 Stress-strain relation. NOD 335 Max=111 gal NOD 185 Max=159 gal NOD 115 Max=190 gal NOD 203 Max=166 gal Fig. 19 Computed accelerations. Fig. 22 Computed normal stresses and shear stresses.

2) Ishihara, K. and Towhata, I.: Sand response to cyclic rotation of principal stress directions as induced by wave loads, Soils and Foundations, Vol. 23, No. 4, pp. 11-26, 26, 1983. 4) Matsuoka, H.

9 2) Ishihara, K. and Towhata, I.: Sand response to cyclic rotation of principal stress directions as induced by wave loads, Soils and Foundations, Vol. 23, No. 4, pp , 26, ) Matsuoka, H., Iwata, Y. and Sakakibara, K.: A constitutive model of sands and clays for evaluating the influence of rotation of the principal stress axes, Proc. of 2nd Int. Conf. on Numerical Models in Geomechanics, Ghent, Belgium, pp , ) Oka, F. and Washizu, H.: Constitutive equations for sands and overconsolidated based on elasto-plasticity, clays under dynamic loads Proc. of Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, 1, pp , ) Towhata, I. and Ishihara, K.: Modelling soil behavior under principal stress axes rotation, Fifth International Conference on Numerical Methods in Geomechanics, pp , ) Tobita, Y. and Yanagisawa, E.: The applicability and limitation of the anisotropic hardening model to cyclic deformation behavior of soils, Seventh Japan Earthquake Engineering Symposium, pp , ) Ohtsuki, A, and Itoh, T.: Two dimensional effective stress analysis of liquefaction including soil-structure interaction, Earthquake Eng. Struct. Dyn., Vol. 15, No. 3, pp , ) Matsuoka, H.: Stress-strain relationship of sands based on the mobilized plane, Soils and Foundations, Vol. 14, No. 2, pp , ) Rowe, P. W.: Theoretical meaning and observed values of deformation parameters for soil, Proc. of Roscoe Memorical Symposium, pp , 1971.

10 20) Ishihara, K. and Li, S.: Liquefaction of saturated sand in triaxial torsion shear test, Vol. 12, No. 2, pp , Soils and Foundations, pp , ) Lysmer, J. and Kuhlemeyer, K. L.: Finite dynamic model for infinite media, J. Eng. Mech. Div., ASCE, Vol. 95, No. EM 4, pp , ) Kunar, R. R, and Rodriguez-Ovejero, L.: A model with non-reflecting boundaries for use in explicit soilstructure interaction analyses, Dyn., Vol. 8, pp , Earthquake Eng. Struct.

144 福武松岡 Fig. 2 Definition of stresses and strain increments, Photo 1 Fig. 3 The resultant strain shear strain T and cumulative Multi-directional s

144 福武松岡 Fig. 2 Definition of stresses and strain increments, Photo 1 Fig. 3 The resultant strain shear strain T and cumulative Multi-directional s A UNIFIED LAW FOR DILATANCY UNDER MULTI-DIRECTIONAL SIMPLE SHEARING By Kiyoshi FUKUTAKE and Hajime MATSUOKA Actual earthquakes generate complicated patterns of motions involving changes not only in amplitude