A Study on Seismic Coefficient of the Mononobe-Okabe Equation and Dynamic Pressure Distribution for External Stability Analysis of a MSE Wall in Auckland Region

Abstract

This paper describes a study on seismic coefficient and dynamic earth pressure distribution of a geo-grid reinforced Mechanically Stabilized Earth (MSE) retaining wall. Dynamic finite element modeling is implemented on the wall to investigate its dynamic responses under selected historic ground motions for a typical geological condition of subsoil Class C within Auckland area. Results from the analysis are discussed and compared against estimations from conventional Mononobe-Okabe equation that is being applied for external stability assessment of the wall. Horizontal seismic coefficient kh is found to be ranging from 0.32 to 0.52 given that vertical seismic coefficient kv is ignored. It is also revealed from this study that conventionally adopted dynamic soil pressure distribution overestimates dynamic soil pressure within the upper half of the wall but underestimate dynamic soil pressure within the lower half of the wall. A modified profile is thus proposed based on observed increasing dynamic pressure rather than decreasing as assumed in the traditional assumptions, to improve accuracy of predicted dynamic soil pressure. Amplification of horizontal acceleration activated on the interface of wall mass and retained soil is also discussed in the context. 

1 Introduction

Traditional limit equilibrium pseudo-static approach has been extensively applied for seismic design of earth retaining structures due to its theoretical simplicity and readily implementation. Within this framework, the Mononobe-Okabe theory (1926) is the most popular and typical approach for estimation of dynamic earth pressure under a ground motion defined by a horizontal seismic coefficient a_h and a vertical seismic coefficient a_v. To date, only limited research has been made aimed at improving its practical application for the design of geosynthetic reinforced retaining soil structures. Cai and Bathurst (1995) reported that conservative total dynamic forces were calculated using Mononobe-Okabe method compared to results yielded from finite element analysis. However, in their study, no quantitative assessment was made to describe this. Their study also explored small variations in the peak accelerations along the height of the wall selected for the analysis, which can be considered as an evidence of taking a single acceleration coefficient in Mononobe-Okabe method. Another comprehensive evaluation was published in the paper by Cai and Bathurst (1996) in which the theoretical critical acceleration coefficient k_c was derived from the factor of safety against both external sliding failure and internal sliding failure equations. Berg et al. (2009) recommended a maximum acceleration k_max for dynamic earth pressure calculation in Mononobe-Okabe expression and k_max can be determined taking peak ground acceleration by a site-specific factor F_PGA. 

In New Zealand, how to better utilize the Mononobe-Okabe equation by appropriately adjusting its inputs incorporating specific geological conditions has attracted major attention from academic researchers and practicing engineers. Overseas design guidelines in combination with New Zealand experiences and research are well summarized in the Guidelines for Design & Construction of Geosynthetic-reinforced Soil Structures in New Zealand (Murashev, 2003). In this document, the horizontal seismic coefficient is taken as 60% of Peak Ground Acceleration (PGA) for calculation of dynamic soil pressure by Mononobe-Okabe method. Chin and Kayser (2013) investigated dynamic active earth pressure that is produced in a well-established 3m high cantilever wall and drew a conclusion that 65% of the free-free PGS up to 0.3g yields a reasonable match against Mononobe-Okabe derived forces. 

In this paper, one typical Mechanically Stabilized Earth Retaining Structure (MSE Wall) is examined for the purpose of looking into the seismic coefficient which is a critical parameter in the Mononobe-Okabe equation to derive dynamic active earth pressure for external stability check. A representative sub-surface profile encountered in Auckland area, typical soil material properties and design criteria are considered in the analysis with an attempt to provide generalized findings and recommendations. Dynamic finite element modeling by using an interactive process of Sigma/W and Quake/W in GeoStudio 2020, is conducted to compare against theoretical seismic earth pressures given by the Mononobe-Okabe equation. The findings obtained in this study provides a useful reference to designers with regards to how to select a reasonably realistic seismic coefficient and also an insight into the pattern of dynamic pressure distribution in relation to a similar MSE wall in Auckland region. It shall be emphasized that this study is limited to insights into external stability of the wall and, thus, only seismic responses on the vertical
face of back of the reinforced mass are discussed in
the context.

2 Engineering Case

In New Zealand design practice, the Bridge Manual published by New Zealand Transport Agency (2018) is a prevailing guideline for the design of significant geotechnical works for projects not limited to transport infrastructural projects, for instance, embankment slopes, retaining structures and pile foundations etc. Among these engineering works, Mechanically Stabilized Earth Retaining Structures (MSE Wall) are almost always required to accommodate a grade separation for its incomparable advantages over other types of retaining structures including gravity wall, bored pile wall or others.

In response to such an extensive application of MSE walls in New Zealand, there is a necessity to pay continuous attention and interests in the study on its design and performance. In this study, a trial has been made with an intention of creating general knowledge on a MSE wall of typical scale with regards to selection of a proper design ground acceleration in the hypothesized pseudo-static analysis of MSE walls. 

The model of interest and to be studied in this context is shown in the Figure 1. Assumed dimensions of the retaining wall and design criteria are presented in Table 1 and the wall in combination with the sub-surface conditions and its geometry is depicted in Figure 1. Note in this context no surcharge on the retaining wall is included.

Figure 1. Typical MSE wall and assumed shear velocity with depth

3 Subsurface Conditions and Parameters

Typical soil parameters are taken based on the author’s knowledge and local experiences in Auckland area. A 16m-thick shallow cohesive soil layer and 10m-thick underlying hard stratum are defined in the subsurface profile, which are considered representative of Tauranga Group (TG) and ECBF (East Coast Bay Formation) extensively encountered in the Auckland region. The base of model is considered as unweathered (intact) rock having a shear velocity of 800m/s and it can be assumed to be rigid. Groundwater table is taken to be at the level of toe of the wall. Table 2 and Figure 2 give material parameters and variations of several critical parameters (dynamic shear modulus and damping ratio).

Figure 2. Variations of dynamic parameters (shear modulus and damping ratio)

Shear modulus is estimated based on the classic correlation with shear velocity,

G_max = ρV_s2 (1)

Where, ρ is the unit density of soil, and V_s is the shear velocity.

Degrading pattern of dynamic shear modulus developed by Ishibashi and Zhang (1993) is taken into account in the study. It has been recognized that TGA falls among CI, CH and CV, exhibiting intermediate plasticity, high plasticity further to very high plasticity with a Plasticity Index PI value changing significantly (20 ≤ PI ≤ 60) (BRANZ, 2002). For the purpose of this investigation and also for simplicity, a figure of PI=35 is selected considering it can be deemed as representative of major part of TGA of CH. PI=35 is also used for estimation of damping ratio function proposed by Kramer (1996).

4 Dynamic Finite Element Analysis

4.1. Experimental Finite Element Model and Assumptions 

To capture the dynamic earth pressure more accurately and its variations, dynamic analysis is made on the generic case study shown in Figure 1. Accordingly, a finite element model for a plan-strain case, is developed in Quake/W and is presented in Figure 2. Since the main interest of this study lies in the dynamic responses of the as-built retaining wall, the procedure in the analysis is simplified to be as follows:

• Stage 1: Initial stress to define the initial stress state in Sigma/W
• Stage 2: Excavate from +26.0m to +20.0m and build the MSE wall in Sigma/W
• Stage 3: Dynamic analysis in Quake/W

In total 1840 quadrilateral elements are created with minimum size of 0.5m in order to fit the model geometry. Vertical boundaries are defined to be 30m away from the face of wall. Both horizontal and vertical moments are restricted throughout the analysis at the bottom of model. On two vertical boundaries, only vertical movements are allowed in Stage 1 while only horizontal movements are permitted in Stage 3. In the Stage 3, cumulative movements from Stage 1 and Stage 2 are excluded.

In GeoStudio Sigma/W, geo-grids can be simulated by structural bar elements and their tensile stiffness are set to be EA=880kN/m. Keystone facing blocks are modeled as a chain for 10 continuous structural beams and they are connected by rigid joints. The bending stiffness of the beams is taken to be EI =1.13×10⁴kNm² with a cross-section area of 0.3m. Both tension and compression are allowed in the analysis. Interface elements are set up at the interface between MSE gravel fill and its adjacent TG material. The intention of this is to reflect the impact of friction along the interface on the calculated earth pressure that is likely to be mobilized by movements of the wall when subject to a ground motion.

4.2. Acceleration Time History 

15 time history records are selected in the analysis. Basic features of them are summarized in the Table 3. To reduce the data storage, vibrating noise at the end of the recorded acceleration time history, which represents very little ground motion, has been ignored in the dynamic analysis. These seismic records are scaled in DEEPSOIL 7.0 (2016, 2020) for the first step to match the selected specific seismic case of interest with the design free field PGA a_m=0.28g.

4.3. Deconvolution of Acceleration-time records

Deconvolution of chosen recorded surface acceleration records was implemented using DEEPSOIL 7.0 (2016, 2020) by running an equivalent linear analysis on one-dimensional soil column model. The model geometry and material parameters are kept the same as given in Figure 3 and Table 2. Deconvoluted time history records at the top of rock (the bottom of the numerical model) were directly input into Quake/W models to assess responses of the retaining wall. Calculated PGAs and spectral responses of dynamic motions at ground surface obtained from Quake/W modeling are compared with those derived from DEEPSOIL analysis. Table 3 gives a comparison of PGAs. As it shows, some discrepancies in PGAs and spectral responses are discovered and several seismic records (No. 6, 7, 12, 13 and 15 named in the table) yield much higher PGAs than designed PGA=0.28g. To ensure the feasibility and appropriateness of the Quake/W modeling, their corresponding time histories are eliminated in the analysis and accordingly their results from Quake/W are excluded in the following discussions. Figure 4 illustrate a typical comparative result for horizontal spectral acceleration for No.2 seismic record. Despite there are differences in cyclic peaks they generally appear comparable with each other.

Figure 3. Finite element mesh for dynamic analysis

Figure 4. A typical comparison of spectral acceleration at the ground surface between DEEPSOIL and Quake/W for seismic record of New Zealand 02

4.4. Calibration of Model Under Pre-Shaking Static State

The initial stress conditions under both K_o static and the K_A stress state are investigated to calibrate the finite element model. Here, K_o is the static earth pressure coefficient and can be written as K_o=1-sinφ. K_A is Coulomb’s active earth pressure coefficient. Comparisons of calculated horizontal stresses from the model and theoretical estimations are plotted in Figure 5. As can be seen in the figure, there is basically a good agreement between theoretical derivations and Quake/W results under both stress conditions. This, therefore, well demonstrates the reliability and rationality of the model adopted in this study.

Figure 5. Calibration of static state prior to dynamic shaking

5 Results of Analysis and Discussions

5.1. Triggered Soil-Wall Seismic Motion

Amplification of horizontal acceleration throughout the height of wall has been previously revealed by a number of researchers (e.g. Segrestin and Bastick, 1988; Cai and Bathurst, 1995). However, the level of amplification relies on a variety of influencing factors, such as the peak ground acceleration, height of wall, stiffness of wall mass and retained soil and etc. Therefore, for this reason, it is recommended to carry out a specific analysis to assess amplification effect for the particular case of concern. Triggered horizontal peak accelerations on the interface of MSE wall mass and retained soil are extracted from the output of analysis, and are plotted in Figure 6. Data in this figure show that variations in the peak acceleration on the selected face of wall are moderately large. On average, it seems that the peak acceleration on the wall top is about 1.5 times that at the wall bottom for most of prescribed time history records. Results also imply that at the half of wall, the peak acceleration approximately equal to designed peak ground acceleration a_m=0.28g.

Figure 6. Horizontal peak accelerations k_hg triggered at different levels along the height of back of reinforced mass

5.2. Induced Vertical Peak Acceleration k_vg at Back of MSE Mass

Vertical accelerations can be taken from calculations and, therefore, this makes it possible to review vertical seismic coefficients produced on the back of wall in every dynamic analysis. The ratios of k_v/k_h along the height of the wall at the vertical plane of X=25.5m are given in Figure 7. It suggests that the magnitude of k_v/k_h changes from 0.1 to 0.7. This observation is well consistent with the estimation of Stewart et al. (1994) who recommended an estimated k_v=2/3k_h based on seismic data in Los Angeles area. No doubt k_v=(0.67 to 0.7)k_h is a conservative estimation and it shall be deemed as an extreme case. For the engineering design, it seems that taking k_v=(0.2 to 0.3)k_h would be more realistic with a moderate conservatism.

Figure 7. Ratios of k_v/k_h along the height of the back of wall mass

5.3. Seismic Coefficient kh

Under ULS case, defined by PGA a_m=0.28g, the Mononobe-Okabe equation can be applied to estimate the seismic active earth pressure coefficient K_AE and furthermore to derive the total active pressure to be generated on the back of reinforced MSE wall block in the external stability assessment. This methodology is being adopted in New Zealand as recommended in NZTA BM 3rd Amendment 3 (2018) and NZTA Research Report No. 239 (2003). The equation is written as follows,

(2)

Furthermore, an increment in the active pressure as result of a ground motion is written as:

(3)

Where, K_A refers to the Coulomb’s active earth pressure coefficient. 

The dynamic analysis performed in this study also gives the dynamic horizontal total active stress acting on the back of wall mass and, therefore, the total active earth pressure can be determined by integrating it over the entire height of the wall. No doubt the maximum pressure (denoted as F_AE-Q) over the duration of ground motion is considered to be critical and is of our concern. By using it, active pressure increment ΔF_AE-Q can be subsequently derived by subtracting static total force on the active side of the wall computed in the step before shaking. The ratio of ΔF_AE-Q/ΔF_AE-MO is plotted in Figure 8 for all the ten seismic records and it illustrates the ratio changes primarily changes between 0.26 and 0.50 in case that vertical seismic coefficient k_v is zero. In this case, k_h can be calibrated based on this ratio and is calculated to be varying from 0.32 to 0.52 provided k_v=0 is adopted, to match the obtained value of ΔF_AE-Q/ΔF_AE-MO. On the other hand, in the event that a vertical acceleration is taken by using 10% to 70% of horizontal acceleration, k_h can be calculated to vary from 0.32 to 0.58. These data indicate that k_v is not sensitive to the derivation of k_h and a generalized range of k_h=0.32 to 0.58 may be appropriate to include two different cases. Cai and Bathurst (1996) proposed a theoretical solution to the critical acceleration coefficient previously given that the factor of safety against base sliding failure is expected to be 1.0. By applying their proposed method and expressions, the critical acceleration coefficient k_h can be approximately calculated as 0.35. This implies for the specific case discussed in the context, Cai and Bathurst (1996)’s recommended value is likely to be the lower bound of horizontal coefficient. The author notes NZTA Research Report No. 239 (2003) recommends k_h = 0.6a_m for external stability analysis. Findings obtained in this study demonstrates this recommendation is reasonable and acceptable although it is slightly conservative. 

Figure 8. Derived ratios of ΔF_AE-Q/ΔF_AE-MO in this study

5.4. Distribution of the Dynamic Earth Pressure

The horizontal total stress mobilized on the back of MSE wall mass obtained from the analysis is presented in Figure 9. It is obvious that its distribution remains close to a triangular shape with a resultant thrust force acting on the level of 1/3H. Traditionally accepted dynamic earth pressure profile proposed by Bathurst and Cai (1995) suggests the pressure at the top of wall can be assumed as 0.8ΔK_AEγH and (K_A+0.2ΔK_AE) γH at the toe of the wall and this forms an approximate inverted triangular. For a comparison, this distribution is also plotted in Figure 6. As we can see from the figure, Bathurst and Cai (1995)’s hypothesized distribution overestimates the dynamic earth pressure increment approximately within the higher half of the wall but underestimates the dynamic earth pressure increment within the lower half of the wall. 

Figure 9. Variations of earth pressures corresponding to maximum total dynamic active earth pressure and their distributions

Results of dynamic analysis performed in the present study enables us to examine the pressure distribution and compare to the assumed one by Bathurst and Cai (1995). The line named as the modified dynamic earth pressure profile in Figure 9, is produced based on the present results and it appears as the envelop of induced dynamic earth pressure. The distribution of the dynamic earth pressure increment proposed by Bathurst and Cai (1995) exhibits a decreasing pattern from the top of the wall to the bottom of the wall. This hypothesized profile was developed on purpose (to some degree) to accommodate the probable amplification effect of horizontal acceleration. Nevertheless, this study gives an entirely different distribution mode, which indicates an increasing profile rather than decreasing. This observation implies that the current model does not present a noticeable amplification overt the full height of the wall. The key reasons for this may be attributed to: 1) comparable dynamic rigidity of MSE fill gravel and its retained TG soil, and 2) relatively low ground accelerations. By doing simple calculations, the ratio of dynamic shear modulus between MSE fill and TG soil is found to fall within a range of 4.7 to 2.1 (from the top to the bottom of the wall). These comparable stiffnesses results in compatible soil-wall interactions and subsequent movements, which is supposed to largely suppress the mobilization of amplification effect. This type of decreasing distribution of dynamic soil pressure was also reported by Bathurst and Hatamai (1988).

Figure 10 shows a comparison between two pressure distributions in more details. The proposed incremental dynamic earth pressure has a lower pressure of 0.4ΔK_AEγH on the level of top of wall and 0.6ΔK_AEγH at the level of bottom of the wall. Its resultant force acts at approximately the middle of the wall. Combing with static earth pressure, the resultant total earth pressure forms a different trapezoid and the total active force acts at the level of one third to a half of the wall. 

5.5. Maximum Lateral Displacement

The predicted maximum outward lateral displacement of the wall takes place at the top of wall face (at Point (30, 26) in the mesh) and it falls in the range of (0.7% to 2.5%)H. Movements of this order are expected to be a sign that the wall has deformed enough so that Mononobe-Okabe theory and the method remain applicable.

6 Conclusions and Recommendations

This study has been conducted for the purpose of analyzing the seismic coefficient and dynamic earth pressure distribution for a geo-grid reinforced MSE retaining wall sitting on a representative subsoil Class C in Auckland area under representative seismic ground motions. Major findings and recommendations are summarized as follows:

1. The seismic amplification of horizontal acceleration along the height of wall back is observed at the top of the wall. The vertical acceleration on the back of the wall mass is also driven on the back of the wall mass and in general it has a maximum value of 0.7 horizontal acceleration.
2. The dynamic active earth pressure well matches with predicted values from the conventional Mononobe-Okabe approach when the horizontal seismic coefficient is set to be 0.32 to 0.52 if the vertical seismic coefficient is disregarded. While a vertical acceleration is included by a ratio k_v/k_h ranges from 0.1 to 0.7, the horizontal seismic coefficient is calculated between 0.32 and 0.58. This implies that the recommended k_h = 0.6a_m for external stability analysis by NZTA Research Report No. 239 (2003) is considered conservative and close to the upper bound.
3. The traditional distribution of dynamic earth pressure proposed by Bathurst and Cai (1995) is found conservative in particular within two third of the wall. A modified distribution mode (as shown in Figure 10) is proposed, which appears better fit the calculated results from the analysis. Compared to the traditional distribution, the proposed one displays an increasing dynamic pressure from the top to the bottom of the wall rather than decreasing. 

It is important to note that the findings and recommendations presented in this study may be only for those cases in association with similar ground conditions, comparable geometry and design criteria etc. A specific dynamic finite element analysis is always a comprehensive and reliable approach compared to pseudo-static method to produce best solutions for a MSE wall. 

Figure 10. Vertical dynamic earth pressure distribution under seismic loading for external stability

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