Use of Mononobe-Okabe equations in seismic design of ... · PDF fileIn pseudo-static analysis, the Mononobe-Okabe (M-O) solution is typically applied ... chosen to be equivalent to

Chin, C.Y. & Kayser, C. (2013) Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils

Proc. 19th 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. [email protected] (Corresponding author)

C Kayser

URS New Zealand Ltd. [email protected]

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. (2008) and Al Atik & Sitar (2008, 2010) 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. (2008) described the effects of wave-scattering and propose height-dependent scaling factors to reduce

PGA to derive earth pressures. Al Atik & Sitar (2010) 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

1170.5:2004) of parts of the North Island (North A, Oyarzo-Vera et al., 2012) of New Zealand. Results of numerical analyses for cantilever walls using Quake/W & Sigma/W 2012, 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 ~0.7% – 5% of the exposed wall height, calculated maximum dynamic

active forces (∆PAE) had a reasonable match against M-O derived forces based on a seismic

coefficient equal to 65% of the free-field PGA up to 0.3g. When free-field PGA exceeds 0.3g,

the analyses suggest that M-O derived forces based on 65% of free-field PGA over-predict ∆PAE. 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 soil-structure 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. (2013)

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

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 > 0.1% (Greek

Regulatory Guide E39/93) to ∆h/H > 0.5% (Wood & Elms, 1990). 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 90% of active conditions are reached by outward movements as small as ∆h/H of 0.1% 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., 2003), reasonably matching (e.g., Seed &

Whitman, 1970 and Steedman & Zeng, 1990) or larger conservative estimates of dynamic earth

pressures (Gazetas et al., 2004, Psarropoulos et al., 2005, Anderson et al., 2008 and Al Atik & Sitar, 2010).

Anderson et al. (2008) 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, 2012), ground motion records suitable for shallow soils (Class C) in Zone North A (Table 1 and Figure 1 from

Oyarzo-Vera et al., 2012) 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., 2012)

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., (2012) meet the criteria in NZS 1170:2004:-



“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, 2006) based on one-dimensional (1D) equivalent

linear analyses using STRATA (2013). The deconvoluted signals at the base of the one-dimensional (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



Figure 3 – a) Original ground acceleration time history (Acc1) for the Delta, Imperial Valley record scaled to 0.206g and b) Comparison between Acc1 and 2D numerical model

ground acceleration time history (Acc3) for a time period between 10sec and 20sec

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 10m 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

-0.3

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X-A

cce

lera

tio

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Original Motion (Delta) (scaled to 0.206g)

RHS 2D Quake/W model

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Original Spectra (Grd Surface)

RHS 2D Quake/W model

a)

b)



STRATA based on a 10m 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 2012 (Sigma/W & Quake/W)

Table 2 – Summary of soil properties used in Strata and Quake/W & Sigma/W

Layer # Layer* Elevation

(m)

su

(kPa)

φ’

(deg)

c’

(kPa)

ν

(-)

γ

(kN/m3)

ko

(-)

1 EMB1 12.5 - 38 0 0.3 19 0.384 2 EMB2 11.5 - 38 0 0.3 19 0.384 3 EMB3 10.5 - 38 0 0.3 19 0.384 4 FC1 9.5 42 - - 0.49 18 1.000 5 FC2 8.5 46 - - 0.49 18 1.000 6 FC3 7.5 50 - - 0.49 18 1.000 7 FC4 6.5 54 - - 0.49 18 1.000 8 FC5 5.5 58 - - 0.49 18 1.000 9 FC6 4.5 62 - - 0.49 18 1.000

10 FC7 3.5 66 - - 0.49 18 1.000 11 FC8 2.5 70 - - 0.49 18 1.000 12 FC9 1.5 74 - - 0.49 18 1.000 13 FC10 0.5 78 - - 0.49 18 1.000

* EMB: Embankment, FC: Firm to Stiff Clay

The values for Maximum Shear Modulus (Gmax) were based on shear wave velocity values

obtained following the method by Ohta & Goto (1978). Variations in Gma x are plotted in Figure

6. Damping ratios and G/Gmax values were derived from Idriss (1990) and are plotted in Figure

7. The undrained shear strengths, su, for Firm to Stiff Clay (FC) were selected to vary between 42kPa to 78kPa. The 10m thick FC layer was modelled as 10 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 750mm 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 0.2m 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.



Figure 6 – Maximum Shear Modulus of soil used in Quake/W & Sigma/W

Figure 7 – Damping ratio and G/Gmax properties for (a) Granular embankment and (b) Firm to Stiff Clay

Table 3 - Free-field Peak Ground Accelerations (PGA) derived from Quake/W

Record Name PGA1* (g) PGA2* (g) PGA3* (g) El Centro 0.08 0.25 0.29

Delta Valley 0.11 0.20 0.25

Convict Creek 0.13 0.25 0.31

Bovino 0.09 0.21 0.30

Kalamata 0.14 0.34 0.39

Matahina Dam 0.13 0.41 -

KAU001 0.11 0.27 0.32

* Surface Free-field Peak Ground Acceleration

4 RESULTS

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 0.1sec intervals for the duration of the seismic event from Quake/W. The dynamic active force (∆PAE,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 (∆PAE,Quake/W) was selected for comparison against the dynamic active force determined using

the Mononobe-Okabe method (∆PAE,M-O).

The horizontal seismic coefficient, kh, used in the M-O equation was set to equal the free-field surface PGA (Table 3) to determine ∆PAE,M-O,100%PGA. The results comparing ∆PAE,M-O,100%PGA

against ∆PAE,Quake/W are shown in Figure 8a. These showed that the M-O method with a seismic

coefficient equal to 100% 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 (∆PAE,M-O,65%PGA) had a reasonable

0

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Gmax (MPa)

Embankment

Firm to stiff clay with embankment

Firm to Stiff Clay without embankment

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mp

ing

ra

tio

G/G

ma

x

Shear strain (%)

G/Gmax ratios

Damping ratios

0

0.05

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0.25

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mp

ing

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tio

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ma

x

Shear strain (%)

G/Gmax ratios

Damping ratios

a) b)



match against ∆PAE,Quake/W (Figure 8b). For surface PGA’s exceeding 0.3g, 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 > 0.3g and wall displacements

>300mm. 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, ∆PAE, has been discussed by many authors.

Pressure distribution diagrams associated with the calculated dynamic active forces (∆PAE-

Quake/W) were analysed for the location of this force. The average point of thrust was found to be 0.3 H, from the base of the exposed wall, with a standard deviation of 0.02 and a coefficient of

variation of 1.0%.

Figure 8 – Comparison of dynamic active forces determined by Quake/W (∆PAE-Quake/W)

against dynamic active forces calculated using M-O based on (a) seismic coefficient =

100% PGA (∆PAE,M-O,100%PGA) and (b) seismic coefficient = 65%PGA (∆PAE,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, 2012) (b) shallow soils (Class C, NZS 1170, where the fundamental period is

less than 0.6 seconds) and have been based on analyses for a 3m high cantilever wall which experienced maximum outward deflection ∆h/H ≥ 0.7%. 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, ∆PAE was found to apply at a point 0.3H 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. (2008). National Cooperative Highway

Research Program Report 611. Seismic analysis and design of retaining walls, buried structures, slopes and embankments.

Al Atik, L. & Sitar, N. (2008). Pacific Earthquake Engineering Research Center 2008/104. Experimental and analytical study of the seismic performance of retaining structures.

0

5

10

15

20

0 5 10 15 20

Dy

na

mic

act

ive

fo

rce

, Δ

PA

E,M

-O,

10

0%

PG

A(k

N/m

)

Dynamic active force, ΔPAE,Quake/W (kN/m)

45 degree line

Delta Valley

Matahina Dam

Kalamata

Bovino

El Centro

Convict Creek

KAU001

0

5

10

15

20

0 5 10 15 20

Dy

na

mic

act

ive

fo

rce

, Δ

PA

E,M

-O,

65

% P

GA

(kN

/m)

Dynamic active force, ΔPAE,Quake/W (kN/m)

45 degree line

Delta Valley

Matahina Dam

Kalamata

Bovino

El Centro

Convict Creek

KAU001

a) b)



Al Atik, L. & Sitar, N. (2010). Seismic earth pressures on cantilever retaining structures.

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Bolton, M.D. (1990). Geotechnical stress analysis for bridge abutment design. Transport and

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soil indexes. Earthquake Engineering & Structural Dynamics, 6 (2), pp. 167-187.

Okabe, S. (1926). General theory of earth pressures. J. Japan. Soc. Civil Eng., 12(1), pp. 123 –

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Oyarzo-Vera, C., McVerry, G.H. and Ingham, J.M. (2012). Seismic zonation and default suite

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Use of Mononobe-Okabe equations in seismic design of ... · PDF fileIn pseudo-static analysis, the Mononobe-Okabe (M-O) solution is typically applied ... chosen to be equivalent to