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:-
Chin, C.Y. & Kayser, C. (2013)
Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils
“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
Chin, C.Y. & Kayser, C. (2013)
Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils
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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-0.1
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0 10 20 30 40 50 60 70
X-A
cce
lera
tio
n (
g)
Time(sec)
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10 12 14 16 18 20
X-A
cce
lera
tio
n (
g)
Time(sec)
Original Motion (Delta) (scaled to 0.206g)
RHS 2D Quake/W model
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ctra
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ele
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(g
)
Period (sec)
Original Spectra (Grd Surface)
RHS 2D Quake/W model
a)
b)
Chin, C.Y. & Kayser, C. (2013)
Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils
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.
Chin, C.Y. & Kayser, C. (2013)
Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils
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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0 20 40 60 80
Ele
vati
on
(m
)
Gmax (MPa)
Embankment
Firm to stiff clay with embankment
Firm to Stiff Clay without embankment
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0.0001 0.001 0.01 0.1 1 10
Da
mp
ing
ra
tio
G/G
ma
x
Shear strain (%)
G/Gmax ratios
Damping ratios
0
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mp
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G/G
ma
x
Shear strain (%)
G/Gmax ratios
Damping ratios
a) b)
Chin, C.Y. & Kayser, C. (2013)
Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils
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)
Chin, C.Y. & Kayser, C. (2013)
Use of Mononobe-Okabe equations in seismic design of retaining walls in shallow soils
Al Atik, L. & Sitar, N. (2010). Seismic earth pressures on cantilever retaining structures.
Journal of Geotechnical and Geoenvironmental Engineering ASCE, 136 (10), pp. 1324 –1333.
Bolton, M.D. (1990). Geotechnical stress analysis for bridge abutment design. Transport and
Road Research Laboratory, Department of Transport. Report 270.
Gazetas, G., Psarropoulos, P.N., Anastasopoulos, I., and Gerolymos, N. (2004). Seismic behaviour of flexible retaining systems subjected to short-duration moderately strong excitation.
Soil. Dyn. Earthquake Eng., 24, pp. 537 – 550.
Greek Regulatory Guide E39/93 (1998). Regulatory guide E39/93 for the seismic analysis of
bridges (Ministry of Public Works). Bulletin of Greek Technical Chamber, No. 2040, 1998.
Green, R.A., Olgun, C.G., Ebeling, R.M. and Cameron, W.I. (2003). Seismically induced lateral earth pressures on a cantilever retaining wall. Proc. 6
th US Conf. and Workshop on
Lifeline Earthquake Engineering, Long Beach, Calif.
Idriss. I. M. (1990). Response of soft soils sites during earthquakes. Proceedings of the
Memorial Symposium to Honor Professor Harry Bolton Seed. University of California, Berkley.
Kramer, S.L. (1996). Geotechnical Earthquake Engineering. Prentice-Hall
Mejia, L.H. & Dawson, E.M. (2006). Earthquake deconvolution for FLAC. 4th
International
FLAC symposium on numerical modelling in Geomechanics. Hart & Varona (eds.).
Mononobe, N. and Matsuo, M. (1929). On the determination of earth pressures during
earthquakes. Proc. World Eng. Congress, 9, pp. 179 – 187.
NZS 1170.5:2004. New Zealand Standard. Structural design actions Part 5: Earthquake actions
– New Zealand.
Ohta, Y. & Goto, N. (1978). Empriocal shear wave velocity equations in terms of characteristic
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 –
134.
Oyarzo-Vera, C., McVerry, G.H. and Ingham, J.M. (2012). Seismic zonation and default suite
of ground-motion records for time-history analysis in the North Island of New Zealand.
Earthquake Spectra, 28(2), pp. 667 – 688
Psarropoulos, P.N., Klonaris, G., and Gazetas, G. (2005). Seismic earth pressures on rigid and
flexible retaining walls. Int. J. Soil Dyn. Earthquake Eng., 25, pp. 795 – 809.
Steedman, R.S. (1997). Seismically induced displacement of retaining walls. Seismic
behaviour of ground and geotechnical structures, Seco e Pinto (ed), Balkema, Rotterdam, pp.
351 – 360.
Steedman, R.S. & Zeng, X. (1990). The influence of phase on the calculation of pseudo-static
earth pressure on a retaining wall. Geotechnique 40(1), 103 – 112.
STRATA (2013). Rathje, E.M. & Kottke, A. https://nees.org/resources/strata
Wood, J.H. & Elms, D.G. (1984). Volume 2: Seismic design of bridge abutments and retaining walls. Transit New Zealand RRU Bulletin 84.
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