Modeling the responses of saturated sand under dynamic loads.

Kim, Soo-Il ; Park, Keun-Bo ; Park, Seong-Yong 等

The main objective of this paper is to develop an improved method for the analysis of liquefaction potential and to verify the modified model that can simulate behavior of saturated sand under dynamic loading conditions. The model is developed through modification of the DSC model. For the development of the model, general formulation for the DSC constitutive model is modified and calibrated for more realistic description of dynamic responses of saturated sand using the energy dissipation approach. The research is focused on evaluation of the disturbance in the DSC which is defined and applied by dissipated energy induced by stress. This paper presents the basic theory and verification of the modified model based on results from triaxial tests for saturated sand with various input motions including sinusoidal, linear increasing, and real earthquake motions.

INTRODUCTION

Over the last 30 years or so, there have been several soil models for describing undrained behaviors of fully saturated sands under dynamic loading conditions. Recently, the energy-based approach (Figueroa et al. 1994) and the disturbed state concept (DSC) (Park and Desai 2000) to evaluate the liquefaction potential have received great interests because these approaches provide more detailed information for dissipated energy degradation and cumulated deformations. While those models have been successfully verified for many geotechnical dynamic problems, they have been primarily for cyclic loading test results. For the liquefaction analysis, it is known that the sinusoidal type of cyclic loading does not realistically represent actual irregular earthquake motions.

Therefore, effects of the actual irregular earthquake motions should also be taken into account for more detailed and advanced liquefaction analysis. In this paper, the general formulation of the DSC constitutive model is modified and calibrated using the energy dissipation approach for better numerical prediction with various loading types. Especially, the disturbance caused by an applied force in the DSC model is defined by dissipated energy. A procedure for back-calculation is developed based on the incremental integration method as well. In order to verify the modified model, dynamic tests are performed and compared with results obtained from the model.

BASIC THEORY

Dissipated Energy Approach

Dissipated energy in dynamic loading is an important measure to define progressive changes of undrained behavior for sands. The dissipated energy can be represented by the area of the hysteresis loop given the stress-strain curve under dynamic loading conditions. The expression for dissipated energy W is given by:

W = [integral][sigma]d[[epsilon].sup.p] [approximately equal to] [n.summation over (i=l)][sigma], [DELTA][[epsilon].sup.p.sub.i]

where [sigma] = stress vector; [DELTA][[epsilon].sup.] = plastic strain increment vector; t = time; and n = total number of increments or cycles.

Disturbed State Concept

According to the DSC proposed by Park and Desai (2000), external forces cause changes and disturbances in the microstructure system of a material. The stress-strain response of the material at a certain loading condition is then determined from a degree of the disturbance caused by the external force. There are two reference states defined in the DSC model: relative intact (RI) and fully adjusted (FA) states. A material in the RI state upon loading modifies continuously through a process of natural self-adjustment, and a part of it approaches the FA state at randomly disturbed locations in the material. As a result, observed responses of the material can be determined from responses of RI and FA states in terms of the disturbance D. The RI state is defined with continuum soil models, such as the elasto-plastic stress-strain models, while the FA state is defined as a response of a material at the ultimate state. In the original DSC model, the RI state is determined by Hierarchical Single Surface model with the isotropic hardening and associated flow rule whereas the FA state is given by the critical state model (Roscoe et al. 1958).

In DSC model, the yield surface in the HiSS model for the RI state is given by:

F = [J.sub.2D]/[P.sup.2.sub.a]-[-a [(J.sub.1]/[P.sub.a]).sup.n] + [[gamma].sub.u] [(J.sub.1] / [P.sub.a].sup.2]][(1 - [beta][S.sub.r]).sup.-0.5] = 0

where [J.sub.1] = the first invariant of the stress tensor; [J.sub.2D] = the second invariant of the deviatoric stress tensor; [p.sub.a] = atmospheric pressure in the same unit as the stress tensor; n, [beta], and [[gamma].sub.u] = model parameters; a = hardening function. The hardening function a can be defined in terms of the dissipated energy [i.e., Eq. (1)] for undrained conditions as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [h.sub.1] and [h.sub.2] = hardening function parameters. The FA state, on the other hand, can be represented by:

[square root of [J.sup.c.sub.2D]] = m[J.sup.c.sub.1] (4)

where [J.sup.c.sub.1] the first invariant of stress tensor at critical state; [J.sup.c.sub.2D] = the second invariant of deviatoric stress tensor at critical state; and m = slope of the critical state line.

The disturbance D in the DSC model determines the stress-strain response and shear resistance at a certain loading stage. From stress-strain responses of RI, FA, and observed states, the D and effective stresses in the DSC model are given by:

D = ([[sigma]'sup.i.sub.ij] - [[sigma]'sup.a.sub.ij])/ ([[sigma]'sup.i.sub.ij] - [[sigma]'sup.c.sub.ij]) (5)

[[sigma]'.sup.a.sub.ij] = (1 - D)[[sigma]'sup.i.sub.ij] + D[[sigma]'sup.c.sub.ij]) (7)

where D = disturbance at current observed state; and ([[sigma]'sup.i.sub.ij], [[sigma]'sup.c.sub.ij]), and ([[sigma]'sup.a.sub.ij] effective stresses at RI, FA, and observed states, respectively. Differentiation of Eq. (6) leads to incremental equation as follows:

d[[sigma]'sup.a.sub.ij] = d(1 - D) ([[sigma]'sup.i.sub.ij] + D (d[[sigma]'sup.c.sub.ij] (7)

Integrating Eq. (7), the incremental dissipated energy dW is given by:

dW = [integral] [[sigma]'sup.a.sub.ij]d[[epsilon].sup.a.sub.ij]dV = [integral](1 - D) [[sigma]'sup.i.sub.ij] d[[epsilon].sup.i.sub.ij]dV + [integral]D[[sigma]'sup.c.sub.ij]d[[epsilons].sup.c.sub.ij]dV (8)

Where d[[epsilon].sup.i.sub.ij], d[[epsilon].sup.c.sub.ij], d[[epsilon].sup.a.sub.ij] are increments of the deviatoric plastic strain tensor at RI, FA, and observed states under undrained conditions, respectively. Integrating the Eq. (8), D can be defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = dissipated energy in total RI state; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = dissipated energy in total observed response; [W.sup.i.sub.1] and [W.sup.c.sub.1] = dissipated energy at RI and critical states, respectively.

ANALYSIS

Experimental Program and Test Results

The saturated Jumunjin sand, a representative silica sand in Korea, was tested using a cyclic triaxial device under the stress-controlled loading condition with three different wave shapes including the sinusoidal, linear increasing, and actual earthquake motions. The Jumunjin sand is a clean quartz sand and defined as SP according to the Unified Soil Classification System. Diameters of the sand particles range from 0.1 to 0.84 mm. The coefficient of uniformity [C.sub.u] is 1.47, and the mean grain size [D.sub.50] is 0.52 mm. The maximum and minimum void ratios [e.sub.max] and [e.sub.min] are 0.89 and 0.63, respectively. Its specific gravity [G.sub.s] is 2.61.

Based on test results, various dynamic soil responses, including the deviatoric stresses, pore water pressures (PWP), and deviatoric stress-strain relationship, were analyzed. Figure 1 shows input motions and corresponding mobilization of the PWP with time for different types of dynamic loads. Test results with sinusoidal loading are given in Figure 1(a). As shown in Figure 1(a), the excess pore pressure continuously builds up until a certain number of loading cycles and then a rapid increase and oscillation of the excess pore pressure at a value approximately equal to the initial confining stress are observed. In Figure 1(b) for the linear increasing load, it is observed that the PWP builds up gradually until a certain time [i.e., t = 8 sec in Figure 1(b)] and eventually reaches a value equal to the initially applied confining pressure. For the actual earthquake motion shown in Figure 1(c), the PWP curve with time appears to be quite different from those for sinusoidal and linear increasing loads. As shown in Figure 1(c), it is seen that the pore water pressure does not increase in earlier stages, but suddenly jumps up when the maximum deviatoric stress is reached, whereupon liquefaction sets in.

[FIGURE 1 OMITTED]

Comparison of Disturbances

In order to compare disturbances D in earthquake loading types, disturbances D using the original DSC method (method 1) and modified method (method 2) were adopted. Model parameters for calculation of D were obtained from Kim (2004).

Figures 2 and 3 show values of D for sinusoidal load and earthquake motion tests with the plastic strain trajectory? and dissipated energy W for method 1 and 2, respectively. Figure 2 shows sinusoidal loading test results. As shown in Figure 2, D curves from the two methods show good agreement. Figure 3 shows earthquake motion test results. Since strains are remarkably unsystematic in comparison with sinusoidal loading test results, D curve with [zeta] appears to be of irregular shape. This indicates that D function corresponding to the initial liquefaction in DSC cannot be properly estimated. D curve from method 2 based on the dissipated energy, on the other hand, is observed to be well defined, and thus superior in the liquefaction analysis for the determination of D function.

[FIGURE 2-3 OMITTED]

Comparison of Stress-Strain Response

In order to verify the modified model based on the dissipated energy, dynamic tests were performed. Test results are shown in Figure 4. Figure 4 shows the deviatoric stress versus axial strain curves for the different types of dynamic loading tests. As shown in Figure 4, significant differences in stress-strain curves are observed in the tests for sinusoidal, linear increasing, and earthquake loading tests.

[FIGURE 4 OMITTED]

A numerical code of the modified model based on the incremental integral scheme for modified DSC model was developed. The modified model is used to calculate the entire stress-strain responses of sands for cyclic triaxial tests. This is achieved by numerically integrating Eq. (8) based on the procedure given by Park and Desai (2000). Necessary input DSC parameters were re-calculated for static triaxial tests as presented in Kim (2004). Parameters E, [upsilon], [gamma], [beta], m, n, [h.sub.1], [h.sub.2], [bar.m], [lambda], [e.sup.c.sub.o], [D.sub.u], Z, and A are 210,000 kPa, 0.380, 0.250, 0, -0.5, 2.667, 0.1515, 0.0922, 0.5, 0.045, 0.0636, 0.99, 11.64, and 0.35, respectively.

Figure 4 also compares calculated and tested cyclic stress-strain responses. As shown in Figure 4, stress-strain curves obtained from the modified model represent overall agreement with tested results. As the stress-strain curves arrived at the initial liquefaction stage, it is seen that the difference between calculated and tested results becomes slightly pronounced at compression or extension region. This is because actual soils under dynamic loading conditions behave as a composite material of liquid and solid and thus produce large deformations with significant reductions of elastic modulus. The discrepancy is not considered to be severe and may be seen due to experimental and/or computational errors. From Figure 4, it is demonstrated that the modified model provides highly satisfactory correlations with the tested laboratory behavior of saturated sands under linear increasing loading and earthquake motions.

CONCLUSION

For the analysis of dynamic soil behavior under undrained loading conditions, a model was developed for better simulation of dynamic responses for saturated sands based on the DSC. A constitutive model of the original DSC was modified and calibrated using the dissipated energy approach. In order to verify the modified model, results of the numerical program based on the incremental solution of integral scheme developed in this study were compared with those from experimental tests. Back-calculation results showed good agreement with tested results until the initial liquefaction. From back-calculation results, it is found that the modified model describes reasonably well the stress-strain responses for various loading types of both regular and irregular motions.

REFERENCES

Figueroa, J.L., Saada, A.S., Liang, L., and Dahisaria, M.N. (1994). "Evaluation of soil liquefaction by energy principles", Journal of geotechnical engineering, ASCE, Vol. 120, No. 9, September, 1554-1569.

Kim, S.I. (2004). "Liquefaction potential in moderate earthquake regions", 12th Asian Regional Conf. on Soil Mechanics & Geotechnical Engineering, Leung et al. (eds), World Scientific Publishing Co., Vol. 2, 1109-1138.

Park I.J. and Desai C.S. (2000). "Disturbed state modeling for dynamic and liquefaction analysis", Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 126, No. 9, 834-846.

Roscoe, K.H., Scofield, A., and Wroth, C.P. (1958). "On yielding of soils", Geotechnique, Vol. 8, 22-53.

KIM, SOO-IL

Department of Civil Engr., Yonsei University, 134 Shinchon-Dong, Seodaemoon-Gu Seoul, 120-749, Korea

PARK, KEUN-BO

Department of Civil Engr., Yonsei University, 134 Shinchon-Dong, Seodaemoon-Gu Seoul, 120-749, Korea

PARK, SEONG-YONG

Department of Civil Engr., Yonsei University, 134 Shinchon-Dong, Seodaemoon-Gu Seoul, 120-749, Korea

PARK, INN-JOON

Department of Civil Engr., Hanseo University, 360 Daegok-Ri, Haemi-Myeon Chungnam, 356-706, Korea