Keywords: 3D, liquefaction, pore fluid, saturated soil, Biot equation, finite element.

In order to investigate the coupled interaction between saturated soil skeletons and pore fluid, a 3D finite element program DYNE3WAC was produced. The program was based on the same numerical scheme as the 2D version DYNE2WAC and adopted the subroutine library provided by Smith and Griffith [1]. Theoretically, DYNE3WAC is based on the fully coupled dynamic Biot equation [2]. In order to avoid the arising of the large algebraic equation systems, we adopted the simplified form u-p equation sets (see for details in Reference [3]) by neglecting the convective and the relative fluid acceleration terms. To discretize the equation in time, Generalized Newmark (GNpj) method was adopted in the program.

In order to verify and validate the program, series of tests were carried out and the numerical results were compared with physical, analytical results and numerical results obtained by other programs. In this paper, the liquefaction test performed by Zienkiewicz et al. was analyzed using DYNE3WAC under 3D condition.

This test is to simulate a saturated sand bed subjected to horizontal earthquake. In this test, the E-W component of the El Centro earthquake of May 1940 is used as the horizontal input motion and the soil layer is simplified into a 3D artificial saturated soil column with the restriction in y-direction, while the horizontal earthquake input motion is applied in x-direction. The drainage is restricted to the top of the layer and the bottom of the layer is impervious. The analyses are carried out using 8-noded linear elements and all degrees of freedom for u and w of the nodes at the same level are tied. The constitutive model namely Pastor-Zienkiewicz mark III Model was adopted in this analysis.

To obtain the initial effective stresses and initial pore pressures needed for the nonlinear analysis under horizontal earthquake, a linear elastic static draining analysis was performed on the column. To make further investigation on the soil behavior after the earthquake terminated, a consolidation test adopting Pastor-Zienkiewicz mark III Model was performed after the earthquake analysis. For the verification of DYNE3WAC, all the tests above were performed by DYNE2WAC under 2D conditions and DYNE3WAC in 3D conditions simultaneously so that the comparison can be performed.

For the convenience of data analysis, excess pore pressures were plot versus time and depth respectively for the earthquake analysis, while excess pore pressures were only plotted versus time for the 'post earthquake' consolidation test. For the comparing between 2D and 3D, both results obtained using DYNE2WAC and DYNE3WAC were plotted.

As virtually the same results were obtained by obtained using DYNE2WAC and DYNE3WAC, conclusion can be made that DYNE3WAC can work properly in static, consolidation, earthquake and material nonlinear analyses. Although in this test example, the 2D analysis looks more straightforward and the 3D analysis more or less seems unnecessary, in practical engineering, most of problems are three dimensional so that it can only be analyzed in 3D condition. When dealing such kinds of problems, the development of the 3D analysis program DYNE3WAC become very necessary and valuable.

From the curve of excess pore pressure versus time in 'post earthquake' consolidation analysis, it can be seen that high excess pore pressure can persist in the soil layer for a relatively long period after the earthquake terminates. Furthermore, the settlement obtained from the 400 seconds 'post earthquake' consolidation show that, this long duration of high excess pore pressure in soil layer can sometimes be more catastrophic, since comparing with the relatively small settlement appears during the 10 seconds earthquake, this much bigger vertical settlement is more likely to bring damages to the superstructure. So the soil behaviour after the earthquake excitation is also important in analysis earthquake problems.

Furthermore, the excess pore pressure-depth curve in earthquake analysis showed that the 'liquefaction' develops from the surface and progresses to the deeper layer of soil during the earthquake, while the excess pore pressure-time curve in 'post earthquake' analysis showed that the 'sedimentation' front begins from the bottom surface and progresses upward. This phenomenon implies that the deep foundation may be less influenced in earthquake in comparison to the shallow ones.

1

Smith, I.M. and Griffiths, D.V. (1997) "Programming the Finite Element Method", 3rd edn, John Wiley and Sons, Ltd

2

Biot M.A. (1956), "Theory of propagation of elastic waves in a fluid-saturated porous solid, part I-low-frequency range, doi:10.1121/1.1908239, part II-Higher frequency range", doi:10.1121/1.1908241, J. Aoust. Sco. Am, Vol. 28, No.9, 168-191

3

Zienkiewicz O.C., Chan A.H.C., Pastor, M, Schrefler, B A. and Shiomi T. (1999), "Computational Geomechanics with special reference to Earthquake Engineering", John Wiley and Sons, Ltd., Chichester

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