Numerical Evaluation of Dynamic Earth Lateral Pressure on Basement Walls in Cohesionless Soils (A Case Study of Iranian Earthquakes)

Various earth-retaining structures are extensively used in different civil engineering projects. Using the results of experimental and numerical modeling, the effect of the earthquake loading on the distribution of the earth lateral pressures on basement walls was investigated. The UBCHYST constitutive model was used in the performed numerical simulations, and using the results of 12 centrifuge dynamic model tests, a calibration process was conducted to obtain the proper input parameters for numerical models. Based on the results, some correlations between the dynamic increment coefficient of earth lateral pressure (ΔKae) and the free-field peak ground acceleration (PGAff) were established. The correlations are different for stiff and flexible basement walls. In the end, the results of the model calibration process were verified by the examination of 12 major earthquakes that occurred in the Iran plateau. The results showed that the proposed ΔKae-PGAff correlations are precise enough for the applied design projects.


Introduction
The retaining walls are traditionally categorized into two main groups [1]: Yielding walls can displace sufficiently to mobilize the minimum active pressure.In general, free gravity and cantilever walls are considered as yielding walls.In contrast, non-yielding walls do not satisfy the mentioned displacement condition.The building basement walls, which are restricted at the top and bottom, are often considered as non-yielding walls.
Various analytical methods have been presented for evaluating the dynamic performance of yielding and nonyielding retaining walls.Using a shaking table, Mononobe and Matsuo performed a series of fundamental experiments to evaluate the effects of seismic loading on retaining structures [2].Based on the results, Okabe [3] modified Coulomb's static earth pressure theory to dynamic loading conditions.Considering Coulomb's assumptions, he proposed the following equations for calculating the resultant active thrust on a retaining wall.These equations are currently known as Mononobe-Okabe (M-O) method [3]: where H is the wall height, k v is the coefficient of the vertical acceleration of soil wedge, k h is the coefficient of the horizontal acceleration of soil wedge, c is the unit weight of the backfill, / is the friction angle of the backfill, d is the friction angle at the wall-backfill interface, i is the backfill slope to horizontal, and b is the angle between the inner face of the wall and the vertical line.
Performing a parametric study, Seed and Whitman [4] noted that the total dynamic lateral earth pressure (P ae ) could be divided into two separate ingredients: initial static pressure (P a ) and the dynamic increment due to the seismic loading (DP ae ): Therefore: where K ae is the total dynamic coefficient of the earth lateral pressure, K a is the static coefficient of the earth lateral pressure, and DK ae is the dynamic increment coefficient of the earth lateral pressure [4].Based on the results of parametric sensitivity analyses, Seed and Whitman [4] suggested that for practical applications: where k h is the earth's horizontal acceleration, which is expressed as a fraction of gravitational acceleration.
Based on the experimental works of Mononobe and Matsuo [2] and Prakash [5], Seed and Whitman [4] suggested the thrust line of active dynamic force at 0.6H on the top of the wall base.Mylonakis et al. presented a new alternative for the M-O method in plastic soils.They developed a closed-form plastic solution for the earth's gravitational and seismically induced pressures on retaining walls [6].
For the non-yielding walls, Wood (1973) used the theory of elasticity and elastic wave propagation for an elastic soil layer resting on a stiff base.He determined that the observed dynamic component of thrust is probably twice the amount resulting from the M-O method.Wood developed the dynamic force (DP ae ) acting on the face of a smooth non-yielding wall as follows [7]: The F value is approximately equal to unity [8].This leads to the following equation for a rigid wall resting on a stiff base: Generally, the performed experimental studies on the analyses of dynamic loads on retaining structures are categorized into two main groups: scaled studies on 1-g shaking tables and dynamic centrifuge tests.The first group of experiments has been performed in tiny scales [2,[9][10][11][12][13][14][15][16].
The obtained results have exhibited different levels of agreement with those of theoretical solutions [17].These experiments generally show that the M-O method results in the magnitude of the total resultant force with an acceptable level of precision.Moreover, depending on the wall movement, the incremental dynamic earth pressure acts on a location between 0.45 and 0.55H from the base [18].
The shaking table models are more common and easier to access.However, the 1-g tests suffer the lack of similarity between the prevailing stress levels of small-scaled models and actual full-scaled structures.Since the vital static and dynamic properties of soil material depend on the stress level, therefore, the applicability of small-scaled 1-g shaking tables is limited for quantitative evaluating.However, the results are still helpful for qualitative assessments [19].
Centrifuge dynamic tests have been performed on retaining wall models with non-cohesive dry and saturated backfill materials [19][20][21][22][23][24][25].The dynamic centrifuge tests are more recent but less common.The results of some centrifuge tests exhibit fair agreement with the predictions of the M-O method [20,21].Based on the results of a series of geotechnical centrifuge model studies with different types of structures backfilled by cohesionless and cohesive materials, Wagner and Sitar (2016) and Wagner et al. (2017) showed that the M-O method can provide an upper bound for the response of stiff retaining structures.However, the measured loads on flexible retaining structures are significantly smaller than those predicted by the M-O method [26,27].Using centrifuge dynamic models, Mikola et al. (2016) modeled three different types of prototype retaining structures: a non-displacing cross-braced (basement) structure, a non-displacing U-shaped cantilever structure, and a free-standing, cantilever retaining wall.The overall results showed that the maximum dynamic earth pressure increases with depth.Moreover, a triangular distribution can be used to approximate the earth's pressure, reasonably [28].The results reported by Candia et al. (2016) revealed that with cohesive backfill, the seismic earth pressure on walls increases linearly with the free-field PGA.Moreover, they showed that the resultant acts near 0.33H above the footing as opposed to 0.5-0.6H,which is suggested by most current design methods [29].
The numerical simulation of wall-backfill interaction, if the models are well calibrated, provides extra valuable insight.Alampalli and Elgamal proposed a numerical model based on the wall-backfill shapes compatibility [30].Using a model including a flexible cantilever wall and a semi-infinite viscoelastic medium, Veletsos and Younan revealed that the displacement magnitude and distribution as well as the pressure could be very sensitive to the flexibility of the wall and foundation [31].Using a kinematic model, Richards et al. found that the location of the application of earth resultant dynamic pressure differs in the different wall motions [32].Al-Homoud and Whitman reported good agreement between the results of numerical models and three-dimensional centrifuge tests for wall displacements [33].Using the finite difference FLAC code, Green and Ebeling showed that under very low acceleration levels, the seismic load on the stem of a cantilever retaining wall conforms to the predictions of the M-O method.However, when the acceleration level increases, the obtained earth seismic pressure becomes larger than that of the M-O method [34].After a series of numerical modeling, Gazetas et al. showed that the actual effects such as wall flexibility, failure of soil materials in the backfill, and separation and sliding between soil and wall tend to reduce the effects of dynamic excitations on retaining structures [35].Ostadan used the concept of a single degree of freedom and presented a simplified method for predicting the peak earth seismic load on the walls constructed on firm bases [36].Pathmanathan (2006) used finite element numerical models for understanding the dynamic behavior of different retaining structures.He concluded that if the level of shaking events is small, the induced dynamic pressures on the wall show good correlations with the results of the M-O method.However, when the level of shaking events increases, the calculated earth dynamic pressure is smaller than that predicted by the M-O method.Moreover, the acting point of the dynamic incremental pressure will exhibit significant fluctuation around 0.6H proposed by Seed and Whitman [4,37].Jung and Bobet found that rotational, bending, and transitional flexibility significantly change the dynamic pressure magnitude and distribution [38].Comparing the results of dynamic centrifuge experimental models and those of a nonlinear two-dimensional finite element model, Al Atik and Sitar (2008) concluded that the finite element model could simulate the major response characteristics of retaining wall-backfill system [39].Argyroudis et al. (2013) proposed a numerical approach for constructing fragility curves.They used these curves to assess the response of bridge abutments to increasing levels of seismic intensity.In addition, they examined the effect of backfill material on the overall response of the abutment wall [40].Osouli and Zamiran (2017) used numerical simulations to investigate the seismic response of cantilever retaining walls backfilled with c-/ materials.They used centrifuge test results to validate their numerical dynamic analyses.According to the results of fully dynamic numerical simulations, they provided some design hints for cantilever retaining walls with cohesive sandy backfills [41].Bakr and Ahmad (2018) used finite element models to develop relationships between the dynamic active and passive earth pressure values and the movement modes of a rigid retaining wall.The results of existing centrifuge model tests were used to validate the outcomes of numerical modeling [42].Using calibrated finite difference models, Zamiran and Osouli (2018) developed nonlinear correlations between free-field peak ground acceleration (PGA ff ) and maximum relative displacement of the retaining wall.They also showed the effect of the cohesion of the backfill materials on the displacement of the retaining wall and the probability of its damage under seismic conditions [43].Annapareddy and Pain (2021) used finite element simulations to evaluate the performance of conventional rigid retaining walls under harmonic and transient ground motions.They found that earth pressure distributions corresponding to the peak ground acceleration are nonlinear and unique in shape and are strongly related to the rotational movement of the wall.Based on their numerical results, they concluded that existing force-based methods for rigid retaining walls overestimate the earth's pressure under seismic conditions [44].
As one of the most recent investigations, Ke et al. proposed a closed-form solution for soil-flexible cantilever retaining wall under seismic conditions in a layered soil profile [45].Compared to the analytical solutions, the adaptability of numerical (finite element and finite difference) methods allows the study of more practical conditions.For example, the inhomogeneity of supporting soil, the flexibility of the retaining system, and the stiffness of the foundation can be included in the simulation process.
A review on the basement walls performance in the past earthquakes shows that the failure of basements or deep excavation walls during earthquakes is rare [46], even if the structures have not explicitly been designed for the dynamic loading condition.The failure of other retaining structures is reported rare as well [8,17,47].The observed collapses usually include more complicated situations such as the existence of a weak layer (e.g., liquefiable) [35,39] or sloping ground either in the upper hand or lower hand or both [48].Due to the lack of observed failures in modern retaining structures during recent earthquakes, the evaluation of current design approaches, which are mainly based on the works of Mononobe-Okabe, can be interesting.Therefore, this research paper aims to assess the dynamic behavior of basement walls with medium-dense dry sand backfill.For this purpose, a model calibration procedure was accomplished based on the results of some dynamic centrifuge tests.Then, the generalization of the calibrated model was examined using the records of a series of Iranian earthquakes.

Model Calibration Soil Parameters Calibration
The UBCHYST constitutive model was used in the performed dynamic numerical simulations.The model has been developed at the University of British Colombia and is used for the dynamic analyses of soil materials subjected to earthquake loading [49].This constitutive model allows the simulation of the hysteretic behavior of soil including damping, material softening, and shear modulus reduction with an increase in strain during dynamic analysis.Based on this hysteretic model, the tangent shear modulus (G t ) is a function of peak shear modulus (G max ) multiplied by a reduction factor that is a function of the developed stress ratio and the change in stress ratio to reach failure.This function is expressed as Eq. ( 10) and illustrated in Fig. 1.
where g 1 is stress ratio g since last reversal (g -g max ), g is stress ratio (= s xy / r' v ), g max is maximum stress ratio (g) at last reversal, g 1f is a change in stress ratio to reach failure envelope in the direction of loading (g f -g max ), g f- = (sin(u f ) ?Cohesion 9 cos(u f ) / r' v ), s xy is developed shear stress in the horizontal plane, r' v is effective vertical stress, u f is peak friction angle, mod1 is a reduction factor for first-time or virgin loading (typically 0.6 to 0.8), mod2 is an optional function to account for permanent modulus reduction with large strain (= 1 À g 1 g 1f n 1 Â dfac > 0:1), mod3 is an optional function to account for cyclic degradation of modulus with strain or number of cycles, etc., and n 1 , R f , and n are calibration parameters with suggested default values 1, 1, and 2, respectively.Equation ( 11) defines the peak shear modulus of soil: where G ref is the reference shear modulus of soil, P' is the mean effective stress, P A is the atmosphere pressure (= 100 kPa), and n e is the elastic bulk modulus exponent.A C?? user-defined constitutive model, compiled as a DLL file, has been used to implement the UBCHYST model into the FLAC 8.00 [51] code [52].Moreover, a FISH function has been prepared to calculate G max based on Eq. (11).
A model calibration process was performed to obtain the UBCHYST model parameters.For this purpose, the results of 12 dynamic centrifuge experiments [52] were used.Based on the applied and measured acceleration time series, respectively, at the base and the top of the centrifuge models, 1D wave propagation simulations were repeated for the selected events to obtain the calibrated parameters.Figure 2 shows the configuration of the calibration model.The model is composed of a 1 m 9 19.8 m column with a zone size of 0.5 m 9 0.495 m.According to Kuhlemeyer and Lysmer (1973), the accurate representation of wave transmission through a model requires a spatial element size of about one-tenth of the wavelength associated with the highest frequency component of the input wave [53].Considering the minimum shear wave velocity of 200 m/s [54,55] in cohesionless materials and the maximum frequency of 25 Hz for the shaking events, the minimum possible wavelength of studied earthquakes will be 8 m.Therefore, the considered zone size satisfies the criterion stated by Kuhlemeyer and Lysmer.In the first step, the sides of the model were fixed in the X (horizontal) direction and the bottom was fixed in the Y (vertical) direction and the model was solved to initiate an in situ stress state in the model zones.Then, the static boundary conditions were removed from the model and the given earthquake event was applied as an acceleration time history at the bottom of the model.Based on the results of numerical computations, the horizontal acceleration time history was obtained at the top of the model (history point 3 in Fig. 2) from the numerical calculations.The time series of 12 different events were used in the centrifuge and numerical simulations.The free-field parameters were measured at the top of the centrifuge models and computed at the top of the numerical ones.The events and their relevant ground motion parameters obtained from the centrifuge and numerical simulations are summarized in Table 1.The table includes PGA (peak ground acceleration), PGV (peak ground velocity), and PGD (peak ground acceleration) values applied at the bottom of the centrifuge and numerical models as well as the measured values at the surface of the centrifuge and numerical models.Ideally, the measured parameters at the surface of both the experimental and Fig. 1 UBCHYST model key variables [50] numerical models should be the same.In the last three columns of Table 1, the percentage of error between centrifuge models and relevant numerical models is reported.The error percentage was calculated using the following equation: Where P C and P N are the measured parameters (PGA, PGV, and PGD), respectively, in the centrifuge and numerical models.The obtained errors between the freefield measured PGA values of calibrated numerical and experimental models are less than 4%.The obtained error for the PGV parameter is less than 6%.However, the error for the PGD parameter is very large in some cases.Considering the following points about the displacement, the high error values of PGD in the numerical models can be overlooked: 1.The cases with high errors are relevant to relatively small earthquake events.Therefore, despite the error percentage being high, the absolute difference is not so great (less than about 5, 4, and 2 cm for the three events of kocaeli ypt330-2, kocaeli ypt060-2, and kocaeli ypt060-1, respectively).2. In numerical calculations, no parameter is computed from the displacement values.Therefore, the relevant errors will not affect the other simulation parameters.
Considering the error values between experimental and numerical models along with the aforementioned points, the numerical models can be supposed as calibrated models with the same seismic behavior as centrifuge ones.Table 2 summarizes the obtained calibrated soil parameters.Some parameters of Table 2 are selected based on Jones [56].
In addition to hysteretic damping, Rayleigh damping is also essential to obtain reasonable responses in dynamic numerical simulations.For geological materials, the recommended damping is in the range of 2-5% critical [57].In this study, the Rayleigh damping fraction of 5% was used in numerical modeling.This fraction was applied to the predominant frequencies of the different exciting acceleration time histories at the bedrock.Predominant frequency is defined as the frequency at which the amplitude spectrum is maximum.The predominant frequencies of different exciting earthquake events were obtained using   3 summarizes the obtained dominant frequencies for different shaking events.
Figure 3 shows the applied exciting acceleration time history at the bedrock of centrifuge and numerical models.As can be seen from the figure, the same time history is applied at the base of both models.Figure 4 compares the free-field acceleration time histories of the centrifuge and numerical models.As the graphs reveal, the general behavior of the measured acceleration time histories at the free field of the centrifuge and numerical models is reasonably similar.Figure 5 compares the pseudo-acceleration response spectra at the base (bedrock) and top (free field) of the FLAC models and centrifuge data.Good fits to the data from centrifuge tests were obtained with the calibrated parameters for numerical models.

Earth Pressure Calculations
Two-dimensional plane stress numerical modeling by FLAC 8.00 software [51] was used to calculate the 123 distribution of earth lateral pressure on the basement walls under dynamic loading conditions.The calculations were separately performed for two different kinds of stiff and flexible basements.It should be mentioned that the relative flexibility of the wall and retained soil can be calculated as follows [58]: where D w is the flexural rigidity per unit length of the wall: In the above equations, H is the height of the wall, G is the shear modulus of the backfilled soil material and t w , E w , and t w are the thickness, modulus of elasticity, and Poison's ratio of wall material, respectively.For d w \ 1 and d w [ 5, the wall is considered rigid and flexible, respectively [58].Table 4 summarizes the structural parameters of the modeled stiff and flexible basements [52].Linear elastic beam elements were used to model the structural basement elements.Moreover, the soil-structure interaction was simulated using interface elements with the Mohr-Coulomb failure criterion.Itasca has proposed the following equation as a good rule of thumb to calculate the shear and normal stiffness of interface elements [51]: where k n and k s are normal and shear stiffness, respectively, K and G are bulk and shear modulus, respectively, and Dz min is the smallest width of neighbor zones in the normal direction of the interfacing surface.As discussed in the previous section, the values of K and G parameters change as a function of the stress state.Therefore, the greatest ones at the bottom of the basement structure were used to calculate the relevant normal and shear stiffness values.Table 5 summarizes the basic input data of the interface.The interface friction angle is selected based on Mikola and Sitar [52].The other model input parameters were selected based on the results of the soil parameters calibration procedure discussed in the previous section.Similar to the model calibration process, the horizontal acceleration time histories of selected events were applied at the model base (Fig. 6), and the normal stress component on the soil-structure interface elements (r x ) was calculated by numerical modeling.To monitor the earth lateral stress distribution on the basement walls during a shaking event, 21 history points were established on each sidewall.The interface normal stress was recorded in these history points during dynamic simulations.Figures 7 and 8 show the time history of earth lateral stress values at the different elevations of stiff basement walls, respectively, at the left side and right sidewalls.Similar plots were obtained for flexible basement walls as well.However, to keep the paper concise, the plots are not presented.
Comparing the plots of Figs. 7 and 8 reveals that although the same acceleration is applied on both walls, the histories of earth lateral stress values on the two sidewalls seem quite different, especially in severe earthquakes.The main reason is that the acceleration time history acts in the different directions on the left and right walls.As Fig. 6 shows, on the right side, the applied acceleration acts in the outward direction to the ground.However, on the left side, the direction of the applied acceleration is inward to the ground.This effect does not been addressed in pseudostatic or analytic solutions.
The resultant force on the basement walls can be calculated by the integration of the applied stress values at different elevations along the wall height: where P is the resultant force on the basement wall in unit length, H is the basement depth, and (r x ) z is the lateral earth pressure at elevation z.Equation ( 16) can be calculated at any time of the dynamic loading interval.However, the most critical resultant force is the largest one, which is applied at a determined time during a shaking event.Obtaining the time at which the applied resultant force is maximum from the plots of Figs. 7 and 8 is not straightforward.Therefore, the following equation was used to access the maximum resultant force on the basement walls and the relevant time at which it is applied:  for the shaking events of g kocaeli ypt330-1, h kocaeli ypt330-2 i loma prieta wvc270-1, j kocaeli ypt330-3, k kobe tak090-2, and l loma prieta wvc270-2 Fig. 4 The obtained acceleration time histories at the free field of the centrifuge and numerical models for the shaking events of a kobe tak090-1, b loma prieta sc, c kocaeli ypt060-2, d kocaeli ypt330-4, e kocaeli ypt060-1, and f kocaeli ypt060-3.The obtained acceleration time histories at the free field of the centrifuge and numerical models for the shaking events g kocaeli ypt330-1, h kocaeli ypt330-2, i loma prieta wvc270-1, j kocaeli ypt330-3, k kobe tak090-2, and l loma prieta wvc270-2 Fig. 5 The response accelerations for the bedrock and the free field of the centrifuge and numerical models for the shaking events of a kobe tak090-1, b loma prieta sc, c kocaeli ypt060-2, d kocaeli ypt330-4 e kocaeli ypt060-1, and f kocaeli ypt060-3.The response accelerations for the bedrock and the free field of the centrifuge and numerical models for the shaking events of g kocaeli ypt330-1, h kocaeli ypt330-2, i loma prieta wvc270-1, j kocaeli ypt330-3, k kobe tak090-2, and l loma prieta wvc270-2 where P max is the maximum resultant force on the basement wall in unit length, t is the time interval of the shaking event, T is the time at which the maximum      Maximum lateral earth stress profiles computed by FLAC before and after the earthquake on stiff basement walls for the shaking events of g kocaeli ypt330-1, h kocaeli ypt330-2 i loma prieta wvc270-1, j kocaeli ypt330-3, k kobe tak090-2, and l loma prieta wvc270-2 resultant force is applied on the basement wall, and (r xt ) z is the dynamic earth lateral pressure at elevation z at the time of t of the shaking event.In other words, Eq. ( 17) calculates Eq. ( 16) on all times of the applied shaking event and gives the maximum value of P and the time at which it is applied on the basement wall.Maximum lateral earth stress profiles computed by FLAC before and after the earthquake on flexible basement walls for the shaking events of g kocaeli ypt330-1, h kocaeli ypt330-2 i loma prieta wvc270-1, j kocaeli ypt330-3, k kobe tak090-2, and l loma prieta wvc270-2 Fig. 10 continued Using a simple MATLAB code, Eqs. ( 16) and ( 17) were calculated on both left and right walls.The outcome is the distribution of the largest lateral stress values on the basement walls.Figure 9 plots the obtained critical distributions of earth lateral stress on the stiff basement walls.Figure 10 shows the same plots for flexible basement walls.
Similar to Figs. 7 and 8, the plots of Figs. 9 and 10 again show that in the severe earthquake events, the distribution of critical earth lateral stress on the left and right walls is different.It is not surprising that the relevant times, which these maximum stress distributions act, are different as well.Moreover, the plots reveal that in the same shaking event, the stress distribution on stiff and flexible basements is quite different.The difference is apparent not only on the dynamic loading conditions but also on the static loading condition.Therefore, in addition to the applied earthquake specifications and the earth-retaining structure type, the stiffness of the retaining structure (basement structure here) itself has an important effect on the earth-retaining structure yield and hence the lateral stress distribution profile.The final goal of this research paper is to develop practical correlations, which can be used in the dynamic design of future basement walls.To do this, the calculated earth lateral distributions by FLAC numerical models were compared with the traditional dynamic earth lateral pressure theories, and the results were used to develop recommendations on the dynamic design of basement walls.
According to Seed and Whitman [4], the dynamic earth lateral resultant force (P ae ) is the sum of the earth lateral resultant force at static condition (P a ) and the dynamic increment of DP ae (Eqs.4 and 5).During a design process of a retaining structure, K a in Eq. ( 5) is calculated easily using traditional theories.Therefore, knowing DK ae , the designer can generalize the design to dynamic loading conditions.The different earth stress coefficient values are calculated assuming a triangular distribution for the earth lateral stress values.For static conditions [1]: or: Fig. 12 The elevation of the application point of the incremental dynamic thrust from the wall base (h) against free-field PGA values for a stiff basement walls, b flexible basement walls (H is the wall height) where P a is the resultant lateral earth force, c is the unit weight of backfilling soil, and H is the depth of retaining structure.Knowing the distribution of lateral earth stress on the retaining structure, P a can be calculated by integrating the earth stress distribution on wall height as discussed before (Eq.( 16)).
where (r x ) z is the lateral earth pressure at elevation z.
Similarly, the earth stress coefficient in dynamic conditions can be calculated using Eq.(20).However, since the earth lateral stress distribution changes during a shaking event, it is conventional (and at the same time conservative) to consider the greatest lateral earth stress distribution when calculating K ae : where (r xt ) z is the dynamic earth lateral pressure at elevation z at the time of t of the shaking event.
Combining Eqs. ( 5), (20), and (21) will result in the following equation for DK ae : Equation ( 22) was used to calculate the values of DK ae from the results of FLAC numerical models.The values were calculated separately for the left and right walls of the stiff and flexible basement walls.The relevant free-field PGA values were also obtained from the corresponding numerical models.Figure 11a and b shows the variations of DK ae against free-field PGA values for stiff and flexible basement walls, respectively.These plots provide an effective tool for the dynamic design of non-displacing basement walls.The mean and upper bound fitted trend lines along with the trend proposed by Seed and Whitman [4] are plotted on the graphs as well.The obtained correlation coefficient values are quite significant.Comparing the plots of Fig. 11a and b reveals that the DK ae values for stiff and flexible basement walls exhibit different trends.In other words, the stiffness of retaining structures is an important factor even for the same type of retaining structures.Based on the obtained trend lines, for the stiff basement walls: and for the flexible basement walls: where DK ae mean is the mean calculated value of the dynamic component of the earth stress coefficient, DK ae UB is the upper bound calculated value of the dynamic component of the earth stress coefficient, and PGA ff is the relevant free-field PGA value.
It should be noted that for the flexible basement walls, the mean fitted trend line and the trend proposed by Seed and Whitman [4] are the same.Moreover, based on the results of centrifuge modeling, Geraili Mikola [52] proposed the following equations for non-displacing basement walls: Equations ( 27) and ( 28) include both stiff and flexible basement walls in the results reported by Geraili Mikola [52].However, comparing Eqs. ( 23) and (24) to Eqs. ( 25) and (26) shows that the results of numerical models for the stiff and flexible basement walls are different.It seems that Based on the results of numerical modeling, the point of the application of dynamic incremental thrust above the base of the basement walls (h) was calculated for different shaking events.Figure 12a and b shows the variation of h/ Fig. 15 continued H (H is the wall height) against free-field PGA values, respectively, for stiff and flexible basement walls.For stiff basement walls, the h/H values change between 0.2 and 0.6.As Fig. 12a shows, there is a modest linear correlation between the elevations of resultant dynamic incremental thrust and free-field PGA values, and the higher the freefield PGA values, the higher the elevation of the applied dynamic incremental thrust from the wall base.The obtained data points for flexible basement walls (Fig. 12b) are more scattered.It seems that when the free-field PGA values are between 0.4 and 0.6 g, the relevant elevations of dynamic incremental thrust exhibit a wide range of fluctuation between 0.4 and 0.7 H.However, when the freefield PGA values are smaller than 0.4 g or larger than 0.6 g, the mean elevation of about 0.4H is a good approximation for the application point of the incremental dynamic thrust.

Case Study of Iranian Earthquakes
To investigate the generalization of the proposed correlations in Eqs.(23) to (26), the time history of 12 major Iranian earthquakes was examined.These earthquakes are not included in the dynamic centrifuge simulation program, and therefore, only the base acceleration time histories are available for this group of events.Table 6 summarizes the ground motion parameters relevant to the bedrock for these events.The predominant frequencies of the events resulting from the corresponding Fourier transmissions are included in Table 6 as well.
The horizontal acceleration time histories of these events were applied at the model base (Fig. 6), and the normal stress component on the soil-structure interface elements (r x ) was calculated by numerical modeling.Figure 13 shows the applied acceleration time histories at the bedrock of this series of numerical models.Similar to Figs. 7 and 8, the time history of earth lateral stress values at the different history points located at the different wall elevations was obtained for both stiff and flexible basement walls.However, to keep the paper concise, the plots are not presented.
Considering the recorded time histories of earth lateral stress values for the Iranian earthquakes, Eq. ( 17) was used to calculate the maximum resultant force on the basement walls (P max ) and the relevant time at which the maximum force is applied.Knowing the time at which the applied resultant force is the highest, one can obtain the critical distribution of earth lateral stress values, making the highest resultant force level.Figures 14 and 15 plot the critical earth lateral stress distributions on the stiff and flexible basement walls, respectively.
Using the lateral earth stress profiles of Figs. 14 and 15, the values of the dynamic increment coefficient of earth lateral pressure (DK ae ) were calculated by Eq. ( 22) for the 12 time history of Iranian earthquakes.The relevant freefield PGA values (PGA ff ) were also obtained from dynamic numerical models.Similar to Fig. 11a and b, the variations of DK ae against free-field PGA values were plotted for the selected events.Figure 16a and b shows the plots obtained for stiff and flexible basement walls, respectively.The fitted trend lines on the data points of Fig. 16a and b reveal that there is a correlation between DK ae and relevant freefield PGA values.The obtained correlation coefficient values are quite significant, and the following equations can be used to predict the dynamic increment coefficient of earth lateral pressure from PGA ff with an acceptable level of accuracy: Similar to Eqs. ( 23) and ( 25), Eqs. ( 29) and ( 30) can be used as an effective tool for the dynamic design of basement walls.However, comparing Figs.11 and 16 reveals a more general conclusion.In the plots of Fig. 16, the mean and upper bound lines resulted from the calibrated models (Fig. 11) are plotted as well.It is clear that for both stiff and flexible basement walls, the fitted trend lines on the data points of the case study fall between the mean and upper bound lines of the calibrated models.Therefore, it is concluded that the proposed range between Eqs. (23) and (24) for stiff basement walls and Eqs.(25) and (26) for flexible basement walls can be considered as the general range of DK ae in the practical design projects.Moreover, based on Fig. 16a and b, it seems that, at least for the considered case study, DK ae values are closer to the mean line for stiff walls and closer to the upper bound line for flexible walls.
Figure 17a and b shows the variations of h/H against free-field PGA values for stiff and flexible basement walls, respectively.Here, h is the point of applying dynamic incremental thrust above the base of the basement walls, and H is the wall height.The relevant fitted trend lines from the calibrated models (Fig. 12a and b) are plotted on the figures as well.Despite the results for DK ae , there is not any correlation between the data points of the case study on the one hand, and there is not any similarity between the behavior of the data points in Figs. 12 and 17 on the other hand.The only conclusion from Figs. 12 and 17 is that the prediction of the thrust line for the resultant dynamic force on basement walls is not an easy task and will be a challenge for practical design projects.

Conclusions
In the present paper, numerical models were used to calculate the distribution of earth lateral pressure on the basement walls under dynamic loading conditions.The results of the model calibration process showed that the UBCHYST constitutive model could be effectively used to simulate the effect of seismic loading on the retaining structures.Based on the obtained results: 1.The direction of the dynamic loading in the same basement structure affects the distribution of the earth's lateral pressure.The effect is more significant in the more severe shaking events, which have greater PGA values.2. The flexibility of the retaining structure has a considerable effect on the dynamic earth pressure distributions.The effect is important even in the same types of retaining structures with the same yielding modes.3. Based on the results of numerical modeling, some correlations were proposed between the dynamic increment coefficient of earth lateral pressure (DK ae ) and recorded free-field peak ground acceleration (PGA ff ).The correlations were presented for mean DK ae values as well as for the upper bound ones.The obtained correlation coefficient values are significant.Moreover, the obtained trends for stiff basement walls are the same as the trends from centrifuge tests.4. Comparing the DK ae -PGA ff trends for flexible and stiff basement walls reveals that under the same free-field acceleration levels, the DK ae values in flexible basement walls are greater than the DK ae values in stiff ones. 5.The results of a case study on the earthquake events in Iran revealed that the range between the mean and upper bound lines of the calibrated models could be used as a general tool to predict the DK ae values of different earthquake events.6.The thrust line of the resultant dynamic incremental force was calculated from the results of numerical simulations.For the stiff basement walls, a fair linear correlation was obtained between the elevation of the thrust line and the recorded free-field PGA.It seems that when the free-field PGA increases from zero to about 1.2 g, the elevation of the thrust line from the base of the wall increases from about 0.2H to about 0.5H.7.For flexible basement walls, a meaningful correlation between the elevation of the thrust line and the recorded free-field PGA values was not obtained.However, the results show that in the free-field PGA range of 0.4-0.6 g, the elevation of the resultant thrust line changes in a range between about 0.4H and about 0.7H.However, when the free-field PGA values are smaller than 0.4 g or greater than 0.6 g, the value of 0.4H can be considered as a reasonably good estimation for the elevation of the resultant thrust line.8.For the results of the case study, a meaningful correlation between the elevation of the thrust line and the recorded free-field PGA values was obtained neither for stiff basement walls nor for flexible ones.It seems that predicting the location of the thrust line is not as straightforward as predicting DK ae .

Fig. 2
Fig.2The configuration of the calibration model

Fig. 3
Fig. 3 Acceleration time histories applied at the base of the centrifuge and numerical models for the shaking events of a kobe tak090-1, b loma prieta sc, c kocaeli ypt060-2, d kocaeli ypt330-4 e kocaeli ypt060-1, and f kocaeli ypt060-3.Acceleration time histories applied at the base of the centrifuge and numerical models

FLAC
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B BB B

Fig. 6
Fig. 6 Configuration of the basement numerical model

Fig. 11
Fig. 11 Calculated dynamic earth stress coefficients at the time of maximum dynamic earth stress on the nondisplacing basement wall as a function of peak ground acceleration measured at top of the soil in the free field a stiff basement walls, b flexible basement walls

Fig. 13
Fig. 13 Acceleration time histories applied at the base of numerical models for the shaking events of a Ahar-Varzeghan-1, b Ahar-Varzeghan-2, c Bam, d Caldiran, e Kor e Bas, and f Manjil.Acceleration time histories applied at the base of numerical models

Fig. 16
Fig. 16 Calculated dynamic earth stress coefficients at the time of maximum dynamic earth stress on the nondisplacing basement wall as a function of peak ground acceleration measured at top of the soil in the free field a stiff basement walls, b flexible basement walls (case study)

Fig. 17
Fig.17The elevation of the application point of the incremental dynamic thrust from the wall base (h) against free-field PGA values for a stiff basement walls, b flexible basement walls (H is the wall height) (case study)

Table 3
Predominant frequencies of different events used in Ray-

Table 5
The input basic data of the interface elements

Table 6
The ground motion parameters of Iranian examined earthquakes (bedrock)