Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils

Chin, C.Y. & Kayser, C. (213) Proc. 19 th NZGS Geotechnical Symposium. Ed. CY Chin, Queenstown Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils C Y Chin URS New Zealand Ltd. cy.chin@urs.com (Corresponding author) C Kayser URS New Zealand Ltd. claudia.kayser@urs.com Keywords: seismic design, retaining walls, Mononobe-Okabe ABSTRACT In pseudo-static analysis, the Mononobe-Okabe (M-O) solution is typically applied to determine seismic earth pressures acting on retaining walls where resulting displacements are relatively large. These equations require the input of a horizontal seismic coefficient which is frequently chosen to be equivalent to the free-field Peak Ground Acceleration (PGA). Recent work by Anderson et al. (28) and Al Atik & Sitar (28, 21) have highlighted the conservatism of derived earth pressures when applying PGA to the M-O method. Based on dynamic numerical analysis using US-centric time histories, Anderson et al. (28) described the effects of wave-scattering and propose height-dependent scaling factors to reduce PGA to derive earth pressures. Al Atik & Sitar (21) studied earth pressure responses on cantilever walls using centrifuge model testing and numerical analysis based on a number of different acceleration time histories. They propose amongst other recommendations that for both stiff and flexible walls, using 65% of the PGA with the M-O method provide a good agreement with measured and calculated pressures. This paper describes the analysis of cantilever retaining walls using deconvoluted acceleration traces of 7 acceleration time histories appropriate for the shallow soils (Class C, NZS 117.5:24) of parts of the North Island (North A, Oyarzo-Vera et al., 212) of New Zealand. Results of numerical analyses for cantilever walls using Quake/W & Sigma/W 212, based on these deconvoluted traces, are presented. The calculated seismic earth pressures are compared to the M-O method. It is shown that where maximum outward wall displacements at the top of the wall fall between ~.7% 5% of the exposed wall height, calculated maximum dynamic active forces ( P AE ) had a reasonable match against M-O derived forces based on a seismic coefficient equal to 65% of the free-field PGA up to.3g. When free-field PGA exceeds.3g, the analyses suggest that M-O derived forces based on 65% of free-field PGA over-predict P AE. It is noted that these are geographic- and soil-specific recommendations, based on a modelled wall height of 3m. 1 INTRODUCTION The determination of seismic earth pressures acting against retaining walls is a complex soilstructure interaction problem. Factors which affect these earth pressures include the nature of the input motions (including amplitude, frequency, directivity and duration), the response of the soil behind & underlying the wall, and the characteristics of the wall (including the strength and bending stiffness). One approach to determining the magnitude and distribution of earth pressure acting on a retaining wall is to consider the magnitude of permanent wall displacements that will occur as a result of combined gravity and earthquake earth pressures acting on the wall. This process is iterative with the underlying logic being that a wall which does not yield will provide a significant reaction to soil inertial loads with correspondingly large earth pressures. Conversely, a wall that is relatively flexible will provide a reduced reaction to soil inertial loads. The Mononobe-Okabe (M-O) solution (Okabe, 1926 and Mononobe & Matsuo, 1929), assumes that sufficient wall movement will occur to allow active conditions to develop, provides a

Chin, C.Y. & Kayser, C. (213) convenient method of determining seismic earth pressures acting on retaining walls. Various publications differ on the magnitude of outward wall deformations ( h) to allow the use of the M-O solution. These are expressed as ratios of h to the exposed wall height (H); h/h. The range of h/h, which the M-O solution is said to apply, varies from h/h >.1% (Greek Regulatory Guide E39/93) to h/h >.5% (Wood & Elms, 199). The amount of soil shear strains that need to develop before active soil conditions are reached have been quoted by Steedman (1997) based on Bolton (1991) indicating that some 9% of active conditions are reached by outward movements as small as h/h of.1% in dense sands, and somewhat more in looser sands. There are differing views as to whether the application of free-field PGA in the M-O solution results in smaller unconservative (Green et al., 23), reasonably matching (e.g., Seed & Whitman, 197 and Steedman & Zeng, 199) or larger conservative estimates of dynamic earth pressures (Gazetas et al., 24, Psarropoulos et al., 25, Anderson et al., 28 and Al Atik & Sitar, 21). Anderson et al. (28) described the effects of wave-scattering and propose height-dependent scaling factors to reduce free-field PGAs to be used in M-O solutions for deriving earth pressures. They use US-centric acceleration motions and demonstrate differences in these scaling factors as a function of location within the United States (Western, Central or Eastern US). Using centrifuge model testing and numerical analysis of cantilever walls, Al Atik & Sitar propose amongst other recommendations that for both stiff and flexible walls, using 65% of the PGA with the M-O method provides a good agreement with measured and calculated pressures. As the seismic events used by the above authors have unique seismic signatures which may not apply to New Zealand, it was decided to carry out dynamic numerical analyses based on acceleration records applicable to New Zealand. 2 SELECTION OF GROUND MOTIONS Based on the recommendation of McVerry (Personal communication, 212), ground motion records suitable for shallow soils (Class C) in Zone North A (Table 1 and Figure 1 from Oyarzo-Vera et al., 212) were used for dynamic analyses. Figure 1 Seismic hazard zonation for North Island of New Zealand proposed for the selection of suites of ground-motion records (Oyarzo-Vera et al., 212) Characteristics of seismic motions (including PGA, frequency content, directivity and duration) are known to influence the response of soil and acceleration time-records selected by Oyarzo- Vera et al., (212) meet the criteria in NZS 117:24:-

Chin, C.Y. & Kayser, C. (213) actual records that have a seismological signature (i.e., magnitude, source characteristic (including fault mechanism), and source-to-site distance) the same as (or reasonably consistent with) the signature of the events that significantly contributed to the target design spectra of the site over the period range of interest. The ground motion is to have been recorded by an instrument located at a site, the soil conditions of which are the same as (or reasonably consistent with) the soil conditions at the site. 2.1 Deconvolution of Acceleration-time records The acceleration-time histories are ground surface motions (referred to as Acc1, Figure 2). As the acceleration in the numerical model needs to be input at the base of the model, time histories were deconvoluted (e.g., Meija & Dawson, 26) based on one-dimensional (1D) equivalent linear analyses using STRATA (213). The deconvoluted signals at the base of the onedimensional (1D) column (Acc2) were subsequently applied at the base of a two-dimensional (2D) numerical model in Quake/W and transmitted accelerations at the ground surface corresponding to the free-field (Acc3) were subsequently compared against the original ground motion (Acc1, Figure 2). Although there are some differences in the cyclic peaks, both surface acceleration time-histories and acceleration spectra were found to be comparable (Figure 3 and Figure 4). This therefore confirmed the appropriateness of the 2D Quake/W model as far as the free field ground motion at depth is concerned. Figure 2 1D Seismic deconvolution and appropriateness of 2D numerical model accelerations

Chin, C.Y. & Kayser, C. (213) X-Acceleration (g).3.2.1 -.1 -.2 -.3 1 2 3 4 5 6 7 Time(sec) RHS 2D Quake/W model.3 b).2.1 -.1 -.2 -.3 1 12 14 16 18 2 Time(sec) Figure 3 a) Original ground acceleration time history (Acc1) for the Delta, Imperial Valley record scaled to.26g and b) Comparison between Acc1 and 2D numerical model ground acceleration time history (Acc3) for a time period between 1sec and 2sec X-Acceleration (g) a) Original Motion (Delta) (scaled to.26g) X-Spectra Acceleration (g).9.8.7.6.5.4.3.2.1 Original Spectra (Grd Surface) RHS 2D Quake/W model.1.1 Period (sec) 1 1 Figure 4 Comparison between original ground acceleration spectra (Acc1) and 2D numerical model ground acceleration spectra (Acc3) for 5% damping 3 NUMERICAL MODELLING In order to simulate Class C shallow soil conditions, a 1m deep layer of firm to stiff clay was modelled overlying bedrock. A 3m high cantilever retaining wall supporting compacted granular backfill was modelled. Acceleration histories from Table 1 were amplitude-scaled (by multiplying accelerations in a given trace by a constant multiplier) and deconvoluted using

Chin, C.Y. & Kayser, C. (213) STRATA based on a 1m thick soil layer to emulate site conditions prior to any retaining wall construction. The deconvoluted histories were subsequently applied at the base of a 2D Quake/W model (Figure 5). This enabled acceleration time-histories to retain seismic frequency characteristics and allowed a range of free-field PGAs to be developed. Free-field PGAs at the top of the granular backfill were determined and used in subsequent M-O calculations. Figure 5 2D model set-up in GeoStudio 212 (Sigma/W & Quake/W) Table 2 Summary of soil properties used in Strata and Quake/W & Sigma/W Layer # Layer* Elevation s u φ c ν γ k o (m) (kpa) (deg) (kpa) (-) (kn/m 3 ) (-) 1 EMB1 12.5-38.3 19.384 2 EMB2 11.5-38.3 19.384 3 EMB3 1.5-38.3 19.384 4 FC1 9.5 42 - -.49 18 1. 5 FC2 8.5 46 - -.49 18 1. 6 FC3 7.5 5 - -.49 18 1. 7 FC4 6.5 54 - -.49 18 1. 8 FC5 5.5 58 - -.49 18 1. 9 FC6 4.5 62 - -.49 18 1. 1 FC7 3.5 66 - -.49 18 1. 11 FC8 2.5 7 - -.49 18 1. 12 FC9 1.5 74 - -.49 18 1. 13 FC1.5 78 - -.49 18 1. * EMB: Embankment, FC: Firm to Stiff Clay The values for Maximum Shear Modulus (G max ) were based on shear wave velocity values obtained following the method by Ohta & Goto (1978). Variations in G ma x are plotted in Figure 6. Damping ratios and G/G max values were derived from Idriss (199) and are plotted in Figure 7. The undrained shear strengths, s u, for Firm to Stiff Clay (FC) were selected to vary between 42kPa to 78kPa. The 1m thick FC layer was modelled as 1 one metre thick layers with constant properties within each 1m thick layer. All soil parameters are presented in Table 2. The 3m high cantilever retaining wall comprising 75mm diameter concrete piles with 2.25m spacing and a Young s Modulus of 27.8GPa was modelled in Sigma/W & Quake/W. To model the interaction between wall and soil, a.2m thick interface layer was generated. In this case the interface layer was taken to have the properties of the surrounding soil with an angle of wall friction δ equal to 2 3 φ. For these analyses, the wall height was kept constant and a total of 21 acceleration records were input to the base of the numerical model. Each of the 7 acceleration records (from Table 1) was amplitude-scaled in order for approximately 3 records to be generated from every original record.

Chin, C.Y. & Kayser, C. (213) Elevation (m) 2 18 16 14 12 1 8 6 4 2 Embankment Firm to stiff clay with embankment Firm to Stiff Clay without embankment 2 4 6 8 Gmax (MPa) Figure 6 Maximum Shear Modulus of soil used in Quake/W & Sigma/W G/Gmax 1.25.9.8.2.7.6 G/Gmax ratios.15.5 Damping ratios.4.1.3.2.5.1.1.1.1.1 1 1 Shear strain (%) a) b) Figure 7 Damping ratio and G/G max properties for (a) Granular embankment and (b) Firm to Stiff Clay Table 3 - Free-field Peak Ground Accelerations (PGA) derived from Quake/W 4 RESULTS Record Name PGA1* (g) PGA2* (g) PGA3* (g) El Centro.8.25.29 Delta Valley.11.2.25 Convict Creek.13.25.31 Bovino.9.21.3 Kalamata.14.34.39 Matahina Dam.13.41 - KAU1.11.27.32 * Surface Free-field Peak Ground Acceleration Damping ratio 1.25.9.8.2.7.6 G/Gmax ratios.15.5 Damping ratios.4.1.3.2.5.1.1.1.1.1 1 1 Shear strain (%) The maximum total active force was determined by assessing discrete total active forces derived from integrating total pressures over the height of the active side of the wall at.1sec intervals for the duration of the seismic event from Quake/W. The dynamic active force ( P AE,Quake/W ) was subsequently determined by subtracting the static total force on the active side of the wall (derived from Sigma/W) from the maximum total active force. This dynamic active force ( P AE,Quake/W ) was selected for comparison against the dynamic active force determined using the Mononobe-Okabe method ( P AE,M-O ). The horizontal seismic coefficient, k h, used in the M-O equation was set to equal the free-field surface PGA (Table 3) to determine P AE,M-O,1%PGA. The results comparing P AE,M-O,1%PGA against P AE,Quake/W are shown in Figure 8a. These showed that the M-O method with a seismic coefficient equal to 1% of free-field surface PGA overestimates the dynamic active force. For a moderately conservative outcome, the calculated dynamic active force using a seismic coefficient set to 65% of free-field PGA in the M-O method ( P AE,M-O,65%PGA ) had a reasonable G/Gmax Damping ratio

Chin, C.Y. & Kayser, C. (213) match against P AE,Quake/W (Figure 8b). For surface PGA s exceeding.3g, the M-O equation over-predicts the dynamic active forces. At larger displacements and PGA s, the authors are conscious that modelling inaccuracies will increase for such finite element analyses. Hence, results shown in Figures 8a & b exclude free-field PGAs >.3g and wall displacements >3mm. Whilst the Quake/W analyses accounts for the stiffness of the wall, inertial effects of the wall are not included. Hence, wall design should separately consider wall inertial effects. The point of thrust of the dynamic active force, P AE, has been discussed by many authors. Pressure distribution diagrams associated with the calculated dynamic active forces ( P AE- Quake/W) were analysed for the location of this force. The average point of thrust was found to be.3 H, from the base of the exposed wall, with a standard deviation of.2 and a coefficient of variation of 1.%. Dynamic active force, ΔP AE,M-O, 1% PGA (kn/m) 2 15 1 5 a) b) 45 degree line Delta Valley Matahina Dam Kalamata Bovino El Centro Convict Creek KAU1 5 1 15 2 Dynamic active force, ΔP AE,Quake/W (kn/m) Dynamic active force, ΔP AE,M-O, 65% PGA (kn/m) 2 15 1 5 45 degree line Delta Valley Matahina Dam Kalamata Bovino El Centro Convict Creek KAU1 5 1 15 2 Dynamic active force, ΔP AE,Quake/W (kn/m) Figure 8 Comparison of dynamic active forces determined by Quake/W ( P AE-Quake/W ) against dynamic active forces calculated using M-O based on (a) seismic coefficient = 1% PGA ( P AE,M-O,1%PGA ) and (b) seismic coefficient = 65%PGA ( P AE,M-O,65%PGA ) 5 RECOMMENDATIONS The above results are specific for (a) parts of the North Island of New Zealand (North A, Oyarzo-Vera et al, 212) (b) shallow soils (Class C, NZS 117, where the fundamental period is less than.6 seconds) and have been based on analyses for a 3m high cantilever wall which experienced maximum outward deflection h/h.7%. For such relatively flexible and low cantilever walls, a seismic coefficient equal to the 65% of surface free-field PGA used in the Mononobe-Okabe equations was found to reasonably match results from dynamic numerical analyses. The location of the dynamic active force, P AE was found to apply at a point.3h above the base of the exposed wall. Wall inertial effects should be separately assessed and considered in wall design. Conservatively, wall inertial effects should be assumed to act concurrently and in-phase with M-O pressures. Further work for other wall configurations, soil classes and for other parts of New Zealand form part of on-going research for the seismic design guidelines for retaining walls to be published by the New Zealand Geotechnical Society. REFERENCES Anderson, D.G., Martin, G.R., Lam, I. and Wang, J.N. (28). National Cooperative Highway Research Program Report 611. Seismic analysis and design of retaining walls, buried structures, slopes and embankments. Al Atik, L. & Sitar, N. (28). Pacific Earthquake Engineering Research Center 28/14. Experimental and analytical study of the seismic performance of retaining structures.

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