Evaluation of Seismic Displacements of Quay Walls.
A new simplified dynamic analysis method is proposed to predict the seismic sliding displacement of quay walls by considering the variation of wall thrust, which is influenced by the excess pore pressure developed in the backfill during earthquakes. The method uses the Newmark sliding block concept and the variable yield acceleration, which varies according to the wall thrust, to calculate the quay wall displacement. A series of 1g shaking table tests was executed to verify the applicability of the proposed method, and a parametric study was performed. The shaking table tests verified that the proposed method properly predicts the wall displacement, and the parametric study showed that the evaluation of a realistic wall displacement is as important as the analysis of liquefaction potential for judging the stability of quay walls.
The simplified dynamic analyses based on the Newmark sliding block concept are widely used for preliminary designs of quay walls because they can easily evaluate the wall displacement by using basic design parameters such as the weight of the wall, the internal friction angle of the backfill, and the frictional coefficient at the bottom of the wall. Richard and Elms (1979) and Whitman and Liao (1985) proposed simplified dynamic analyses based on the Newmark sliding block concept to evaluate the seismic displacement of a quay wall. In their methods, yield acceleration is defined as the wall acceleration, when the factor of safety of the wall for sliding becomes 1.0, and the wall displacement is presumed to occur if the ground acceleration exceeds the yield acceleration. However, these analyses does not consider the variation of wall thrust due to the development of excess pore pressure in the backfill when they determine the yield acceleration; therefore, previous analyses are inappropriate for the design of quay walls with saturated backfill soils, where high excess pore pressure can develop during earthquakes.
Several researchers suggested degrading yield acceleration models, in which yield acceleration decreased as a function of shear deformation for geosynthetic cover analyses (Matasovic et al., 1998) or as a function of the magnitude of the excess pore pressure for saturated slope analyses (Giovanni et al., 2001). To account for this excess pore pressure in the calculation of wall sliding displacement, we propose a new simplified dynamic analysis method, which still utilizes the Newmark concept but varies the yield acceleration according to the varying wall thrust.
A parametric study is performed to analyze the effect of the input parameters of this new method on the seismic wall displacement, and the proposed method is verified by comparing the predicted displacements with the results of a series of shaking table tests.
DEVELOPMENT OF NEW DISPLACEMENT CALCULATION METHOD
The following assumptions were used in the proposed method.
(1) Quay walls always fail in a sliding mode. This assumption is valid only for dense foundation soils.
(2) Wall displacement occurs in a forward direction only. This should be a reasonable assumption for most cases since the wall can hardly move toward the backfill soils during shaking.
Newmark sliding block method
The Newmark sliding block method defines the yield acceleration as the amplitude of the block acceleration when the factor of safety for sliding becomes 1.0, and evaluates the block displacement by double integration of the ground acceleration, which exceeds the yield acceleration. The integration method by Wilson and Keeper (1983) was used to calculate the block displacement by integration of the ground acceleration.
Determination of yield acceleration
The method proposed in this study evaluates the yield acceleration according to the varying wall thrust and therefore, this method is distinct from the previous ones. The wall thrust on a quay wall ([F.sub.TH]) is the resultant of diverse force components, as shown in Figure 1 (Kim et al., 2004): static water forces acting on the back and front of the wall, inertia force of the wall ([F.sub.I]), dynamic water force on the front of the wall ([F.sub.FWD]), static thrust on the back of the wall before shaking ([F.sub.ST]), and dynamic thrust on the back of the wall, which develops during shaking ([F.sub.DY]). In this research, we assumed that the water levels on both sides of the wall were the same, and thus, the static water forces acting on the both sides of the wall were not considered. The wall thrust [F.sub.TH] can be obtained by summing the various force components, as shown in Equation (1). The resisting force ([F.sub.g]) of the wall, which comes from the frictional force between the bottom of the wall and the foundation soil, can be calculated by Equation (2).
[FIGURE 1 OMITTED]
FTH = FI + FFWD + FST + FDY FR = cB*L + W*tan [phi]B
where, L = length of contact surface between bottom of wall and foundation, W = weight of wall per running unit length, [c.sub.B] = cohesive stress between bottom of wall and foundation, and [[phi].sub.B] = interface friction angle between bottom of wall and foundation
The wall displacement begins to occur when the wall thrust [F.sub.TH] exceeds the resisting force [F.sub.R] (Equation (3a)). The inertia force of the wall ([F.sub.I]) at this point can be obtained by multiplying the mass of the wall (M) by the yield acceleration of the wall ([a.sub.y]) (Equation (3b)). Finally, ay is obtained by Equation (3c). However, ay is also needed to calculate [F.sub.FWD] and [F.sub.DY] on the right side of Equation (3c). Therefore, the yield acceleration can only be determined by an iterative calculation.
[F.sub.TH] (= [F.sub.I] + [F.sub.FWD] + [F.sub.ST] + [F.sub.DY]) [greater than or equal to] [F.sub.R] (3a) [F.sub.I] = [F.sub.R]--([F.sub.FWD] + [F.sub.ST] + [F.sub.DY]) = M x [a.sub.y] (3b) [a.sub.y] = [F.sub.R]-([F.sub.FWD]+[F.sub.ST]+[F.sub.DY]) / M (3c)
where, M = mass of wall
Determination of force components acting on wall
The time histories of force components acting on walls have to be evaluated to determine the time history of the yield acceleration, as was shown in Equation (3). The evaluation methods of force component are summarized in Table 1, as suggested by Kim et al. (2004). Kim's method requires the time histories of the wall acceleration and the excess pore pressure ratio, ru in the backfill as input for the evaluation of the force components. The latter can be evaluated by either a simple empirical formula like Equation (4) or a dynamic analyses or a laboratory test such as a shaking table test.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where, [N.sub.L] = number of loading cycles required to produce initial liquefaction corresponding to the cyclic stress ratio, CSR (=cyclic stress required to initiate liquefaction / vertical effective stress in soil) and [theta] = constant representing soil properties and test condition (commonly, [theta]=0.7)
The CSR can be calculated by Equation (5) (Seed and Idriss, 1971).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where, [[tau].cyc] = cyclic shear stress, [a.sub.max] = maximum amplitude of ground acceleration, [[sigma].sub.v], [[sigma].sub.v]' = total stress and effective stress in backfill, respectively, and [r.sub.d] = stress reduction factor
A parametric study was performed to analyze the wall behavior under various combinations of input parameters, using the proposed method. Figure 2 shows an example quay wall, which was used for the parametric study. The backfill was made with sand, whose effective grain size was 0.10 mm. The coefficient of permeability was 1.0x[10.sup.-4] m/sec. Relative densities of the backfill ([D.sub.r]) were varied to be 40 %, 60 %, and 70%. The interface friction angles between the bottom of the wall and the foundation soil ([[phi].sub.B]) were varied to 25, 30, 35 and 40 degrees. Input acceleration was a 1 Hz sine wave. The number of loading cycles of the input acceleration was set to 10, which corresponds to the design earthquake magnitude of 6.5 in Korea. The amplitude of the input acceleration ([a.sub.max]) was varied to 0.072 g, 0.10 g, 0.12 g, 0.14 g, 0.15 g and 0.20 g. The 0.072 g was obtained by converting the amplitude of irregular earthquake wave of 0.11 g into the amplitude of regular sine wave. To obtain the excess pore pressure ratio, [r.sub.u] which is one of the input parameters, Equation (4) was used. Figure 3 shows the cyclic strength curves, which was obtained by the cyclic triaxial tests for the backfill sands of various relative densities. The CSR (Cyclic Stress Ratio) at mid-depth of the backfill (depth=7.5 m) was calculated by Equation (5) by inputting 1.0 for [r.sub.d]. Figure 3 also shows how [N.sub.L] is obtained after CSR is calculated by Equation (5).
[FIGURE 2 OMITTED]
Figure 4 shows the time histories of the calculated excess pore pressure ratios, using Equation (5) and Figure 3, for various relative densities of backfill at [a.sub.max]=0.10 g. Liquefaction occurred for [D.sub.r]=40 % and [D.sub.r]=60 %, whereas, the liquefaction did not occur for [D.sub.r]=70%.
[FIGURES 3-4 OMITTED]
Results of parametric study
Figure 5 shows the time histories of the input acceleration and the yield acceleration at [a.sub.max]= 0.10 g. As cyclic loading continues, the yield acceleration decreases and finally becomes smaller than the input acceleration at 3.3 sec for [D.sub.r]=40 % and at 7.3 sec for [D.sub.r]=60 % after the backfill soil liquefies. If the effect of excess pore water pressure is not considered in the determination of the yield acceleration as is in the existing methods, the yield acceleration does not change from the initial value as is shown in the same figure for [D.sub.r]=40 % and [r.sub.u]=0. Thus, previous methods will predict zero wall displacement for the example quay wall, which is obviously an erroneous result.
[FIGURE 5 OMITTED]
Figure 6 shows the final displacement of the wall for various combinations of input parameters. Even if the backfill liquefied at [a.sub.max]= 0.072 g with [D.sub.r]=40 %, the final displacement of the wall was only 9.6 cm at [[phi].sub.B]=35 degrees and 0.7 cm at [[phi].sub.B]=40 degrees. On the other hand, the wall displacement of 59 cm occurred at [a.sub.max]= 0.12 g with [D.sub.r]=70 % and [[phi].sub.B]=25 degrees, where liquefaction did not occur in the backfill. In addition, the displacements calculated by the proposed model were very sensitive to the interface friction angle. Therefore, the frictional resistance between a wall and foundation must be properly evaluated.
[FIGURE 6 OMITTED]
VERIFICATION OF PROPOSED METHOD
Test set-up and procedure
The proposed method was verified by comparing the calculated wall displacements with the results of 1g shaking table tests. The dimension of the soil box was 194 cm long, 44 cm wide, and 60 cm high, and the model wall was 17.5 cm long, 42.0 cm wide, and 26.4 cm high. Figure 7 shows the test section and the instrumentation. The amplitude of the sinusoidal input motion at 5 Hz was increased linearly up to 0.2 g during the initial 5 sec, and the final amplitude was maintained for the next 5 sec.
[FIGURE 7 OMITTED]
The model soil was Joomoonjin sand, whose average particle size was 0.55 mm and uniformity coefficient was 1.37. The maximum and minimum dry unit weights of the sand were 16.7 kN/[m.sup.3] and 13.9 kN/[m.sup.3], respectively. The loose backfill of 20 % relative density was prepared by the water sedimentation method. The internal friction angle and the saturated unit weight of the backfill soil were about 30 degrees and 18.9 kN/[m.sup.3], respectively. The permeability coefficient of the backfill soil was measured to be 4.1x[10.sup.-4] m/sec by the constant head permeability test. A dense foundation layer was made by preshaking the foundation soil. The relative density and the internal friction angle of the foundation layer were about 90 % and 40 degrees, respectively. The interface friction angle between the foundation soil and the bottom of the wall was estimated by pulling tests. The interface friction angle increased with the wall movement velocity. The average value of the interface friction angle for the velocity range of the wall movement in the shaking table tests was about 28 degrees. The shaking table tests were performed for two walls of identical geometry but of different unit weights, 23.0 kN/[m.sup.3] and 25.7 kN/[m.sup.3]. In the latter, water was situated in the space where load cells were installed (Figure 7).
Comparison with results of shaking table tests
The time history of the horizontal seismic coefficient [k.sub.h] was obtained by non-dimensionalizing the time histories of the input acceleration. The average excess pore pressure, which is the average value of the excess pore pressures measured from the two water pressure transducers (P2 and P3) installed on the back side of the wall, was used to obtain the time history of the excess pore pressure ratio in the backfill [r.sub.u]. The time histories of [k.sub.h] and [r.sub.u] measured during the tests were used to predict the displacements of the wall.
Figure 8 shows the time histories of the excess pore pressure ratio. Figure 9 shows the comparisons between the measured and the predicted displacements of the walls of two different unit weights. The measured wall displacement started to occur at around 4 sec which is the time when the excess pore pressure ratio increased rapidly and reached almost its maximum value (Figure 8). The final differences of the horizontal displacements at the top and the bottom of the wall were about 0.4 cm (0.9 degrees to vertical) for the wall of unit weight of 25.7 kN/[m.sup.3] and 1.0 cm (2 degrees to vertical) for the wall of unit weight of 23.0 kN/[m.sup.3], which are small compared with the final horizontal displacements of 5.5 cm and 8.3 cm, respectively. Therefore, this observation satisfies the assumption of the proposed model in that only the sliding failure of walls occurs. The calculated final displacements of the walls compared very well with the measured values : 5.2 cm for the wall of unit weight of 25.7 kN/[m.sup.3] and 7.6 cm for the wall of unit weight of 23.0 kN/[m.sup.3].
[FIGURE 8 OMITTED]
Thus, it is believed that the proposed method predicted the quay wall behavior properly in terms of its cumulative displacement with time.
(a) wall with unit weight of 25.7 kN/[m.sup.3] (b) wall with unit weight of 23.0 kN/[m.sup.3]
Figure 9. Comparisons between measured and predicted wall displacements
[FIGURE 9 OMITTED]
The conclusions of this study are as follows.
1. A new displacement calculation method was proposed which considers the effect of the excess pore pressure developed in backfill. This method basically uses the Newmark sliding block concept but varies the yield acceleration according to the varying wall thrust.
2. The parametric study showed that the evaluation of realistic wall displacements under earthquakes is as important as the analysis of liquefaction potential for judging the stability of quay walls.
3. It was verified from a series of 1g shaking table tests that the proposed method properly predicts the wall displacement.
The financial support from the Ministry of Maritime Affairs and Fisheries (MOMAF) in support of this work is gratefully acknowledged.
Giovanni B, Ernesto C, Michele M. Seismic response of submerged cohesionless slopes. In: Proceedings of the 4th International Conference on Recent Advances in Geotechnical Earthquake Engineering, San Diego, California, 2001, Paper No. 7.07.
Matasovic N, Kavazanjian E Jr, Giroud JP. Newmark seismic deformation analysis for geosynthetic interfaces. Geosynthetics International 1998, Special Issue on Geosynthetics in Earthquake Engineering; 5(1-2):237-264.
Kim SR, Kwon OS, and Kim MM. Evaluation of force components acting on gravity type quay walls during earthquakes. Soil Dynamics and Earthquake Engineering 2004; 24(11):853-866.
Richards R, Elms D. Seismic behavior of gravity retaining walls. Journal of the Geotechnical Engineering Division 1979;105(4):449-464.
Whitman RV, Liao S. Seismic design of retaining walls. Miscellaneous Paper GL-85-1, U.S.Army Engineer Waterways Experiment Station, Vicksburg, Mississippi. 1985.
Wilson RC, Keefer DK. Dynamic analysis of a slope failure from the 6 August 1979 Coyote Lake, California, earthquake. Bulletin of the Seismological Society of America 1983;73(3):863-877.
Dept. of Civil Engineering, DongA University, #840 Hadan 2-dong, Saha-gu, Busan, Korea
Korea Ocean Research and Development Institute, Sa2-dong, Sanglok-gu, Ansan-si, Kyoungi-do, Korea
School of Civil, Urban and Geosystem Engineering, Seoul National University, San 56-1, Shinlim-dong, Gwanak-gu, Seoul, Korea
School of Civil, Urban and Geosystem Engineering, Seoul National University, San 56-1, Shinlim-dong, Gwanak-gu, Seoul, Korea
Table 1. Evaluation of force components acting on quay walls (Kim et al., 2004) Force component Evaluation method Inertia force of wall mass of wall x time history of input ([F.sub.1]) acceleration Dynamic water force acting on front side of wall ([F.sub.FWD] Westergaard equation Static thrust ([F.sub.ST]) Coulomb method Dynamic Fluctuating [F.sub.D] = [F.sub.DI] x (1-[r.sub.u] + thrust component [F.sub.DF] x [r.sub.u] ([F.sub.DY] = ([F.sub.D] [F.sub.DI] = [F.sub.WD] + [F.sub.ED] - [F.sub.D] + ([F.sub.I] + [F.sub.FWD]) [F.sub.\s] [F.sub.DF] = 7 / 12 [k.sub.h] [gamma] [sub.sat][H.sup.2] where, [F.sub.DI]: fluctuating component of dynamic thrust without excess pore pressure [F.sub.DF]: fluctuating component of dynamic thrust after liquefaction [F.sub.ED]: fluctuating component of dynamic earth force acting on back side of wall [F.sub.WD]: fluctuating component of dynamic water force acting on back side of wall [r.sub.u]: excess pore pressure ration in the backfill H: wall height [k.sub.h]: horizontal seismic coefficient [[gamma].sub.sat]:the saturated unit weight of soil Non- [F.sub.S] = -1 / 2 [[gamma].sub.sub] fluctuating [K.sub.AS] [H.sup.2] [r.sub.u] + component 1 / 2 [[gamma].sub.sub] [H.sup.2] ([F.sub.S]) [r.sub.u] Where, [[gamma].sub.sub]: submerged unit weight of backfill soil [K.sub.AS]: static active earth pressure coefficient calculated by Coulomb method
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|Author:||Kim, Sungryul; Jang, Insung; Chung, Choongki; Kim, Myoungmo|
|Publication:||Geotechnical Engineering for Disaster Mitigation and Rehabilitation|
|Article Type:||Conference news|
|Date:||Jan 1, 2005|
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