About Seismic Acceleration Coefficient for Pseudo-Static Analyses_Rev3

Prévia do material em texto

1 
About Seismic Acceleration Coefficient for Pseudo-
Static Analyses 
Felipe V. A. S. Cruz1*, Gabriel O. Sant’Ana1, Marina P. Secco1 and Vinícius R. F. Queiroga1 
1. Geotecnia, DF+ Engenharia, Brazil 
ABSTRACT (PALATINO LINOTYPE, 11, BOLD, CAPITAL LETTERS) 
Seismicity in Brazil, although less frequent and intense compared to more tectonically active regions, 
cannot be overlooked. Concentrated in the Southwest Atlantic Seismic Zone, resulting from the 
interaction between the South American Plate and the Nazca Plate, this activity is the result of 
intraplate faults and demands careful assessment of associated risks. This study presents a 
comparative analysis of the methodologies proposed by Papadimitriou et al. (2014) and Bray and 
Macedo (2019) for sizing the seismic acceleration coefficient, where it was found that both analyzed 
methodologies vary in being more conservative, maintaining a consistent order of magnitude in the 
obtained values. However, in the methodology proposed by Bray and Macedo (2019), a point of 
inflection was identified where the sensitivity of the approach is compromised, and the values 
diverge significantly, increasing abruptly. This abrupt change occurs for tolerable permanent 
displacements considered low, ranging between 1.5 and 5,.0 cm, but still within the range estimated 
by the methodology. 
The study emphasizes that the estimation of tolerable displacement is a fundamental point for the 
use of the analysed methodologies, providing guidelines for the rational estimation of tolerable 
displacement. 
 
INTRODUCTION 
The seismicity in Brazil, although less frequent and intense compared to tectonically more active 
regions, is a prevalent phenomenon. Located on the South American Plate, predominantly composed 
of continental crust, the country experiences seismic events due to the complex interaction with the 
Nazca Plate, especially in the Southwestern Atlantic Seismic Zone. Despite most earthquakes being 
of low magnitude, notable events like the 1955 earthquake in Mato Grosso (magnitude 6.2) 
underscore the importance of understanding seismicity patterns and causes for risk assessment. 
Seismic risk analysis requires consideration of local geology, geodynamic structures, and intraplate 
faults. Seismological research, focused on mapping and identifying these faults, is crucial for 
understanding Brazilian seismicity. 
The implementation of seismic building codes and continuous monitoring by institutions such as the 
Seismological Observatory of the University of Brasília (Obsis/UnB) and the Seismology Center of 
the University of São Paulo (IAG/USP), along with independent stations, is essential for minimizing 
 
 2 
damage and preserving lives. These initiatives aim to understand patterns, identify potential threats, 
and improve seismic prediction capabilities. In the Brazilian geotechnical context, the evaluation of 
structure stability considering seismic loading effects has gained prominence due to the ABNT NBR 
13.028:2017 standard, which requires the analysis of pseudo-static conditions in tailings dams. This 
standard stipulates the need for seismic studies in the implementation area of these structures, even 
during the initial project phases, to assess potential seismicity in the project region. 
Although ABNT NBR 13028:2017 indicates the determination of the Maximum Credible Earthquake 
(MCE) through deterministic analyses based on geological considerations, seismological data, and 
detailed mapping of active faults, this methodology is currently outdated, especially in low-
seismicity continental regions, due to the scarcity of seismic data in these areas. Thus, seismic threat 
assessment has been predominantly carried out through the Probabilistic Seismic Hazard Analysis 
(PSHA) methodology, which provides the acceleration spectrum for a probable earthquake, allowing 
for the estimation of peak ground acceleration (PGA) for different return periods. PSHA is crucial for 
determining the seismic acceleration coefficients used in pseudo-static analyses, becoming a 
normative requirement. Pseudo-static analyses represent a simplified approach that excludes intense 
dynamic effects, inertial behavior, and does not evaluate the deformation levels necessary to trigger 
failure. Despite this limitation, it is a simple method, widely used in Brazil, and incorporated into 
computer programs for static stability analysis of slopes by limit equilibrium. However, the accuracy 
of the results depends on the value of the horizontal seismic acceleration coefficient (kh) used to 
estimate the inertia force. In this sense, the use of an appropriate methodology for determining kh is 
essential. Thus, this study presents a comparison of results obtained with the adoption of two widely 
disseminated methodologies in the literature, proposed by Bray and Macedo (2019) and by 
Papadimitriou et al. (2014), for the determination of kh in seventeen analyses of data from eight 
Brazilian dams. 
METHODOLOGY 
To estimate the horizontal seismic acceleration coefficient (kh) used in pseudo-static analyses, this 
study adopted two methodologies, Bray and Macedo (2019) and Papadimitriou et al (2014), which 
have a more precisely defined rational analysis, considering specific characteristics of the structure 
under analysis and the variable tolerable permanent displacement, depending on the structure and 
location. These methodologies, in addition to verifying the tolerable permanent displacement, also 
consider the amplification of peak ground acceleration (PGA), spectral acceleration (Sa), structure 
geometry, and failure surface. The results of kh obtained by applying the aforementioned methods 
were evaluated in 08 structures of Brazilian mining dams, with close locations and distinct 
characteristics, based on varied allowable displacements (Dall). According to Duncan et al. (2014), 
the most recent methodologies for estimating permanent seismic displacement are more intricate and 
require careful study before implementation. 
 
 
 3 
Bray and Macedo (2019) 
The methodology proposed by Bray and Macedo (2019) aims primarily to estimate the tolerable 
permanent displacement, with the estimation of the seismic acceleration coefficient being a secondary 
aspect of this analysis. This approach is grounded in spectral acceleration (Sa), which exhibits less 
dispersion compared to Peak Ground Acceleration (PGA) and considers the acceleration distribution 
across different frequencies. This provides detailed insights into how different frequencies affect the 
structure. Furthermore, this methodology conducts a comprehensive data analysis, as 6711 data sets 
were evaluated. The methodology relies on analyzing the structure's tolerable displacement and its 
dynamic response to the seismic event to determine the pseudo-static acceleration coefficient (k). This 
model was designed for events where the Peak Ground Velocity (PGV) value is less than 115 cm/s. 
The authors describe that the first step in defining k is establishing the structure's tolerable 
displacement (Da), which should be determined collaboratively between the responsible engineer 
and the structure's owner, taking into account the implications of unsatisfactory performance if such 
a value is exceeded. This is estimated by the methodology, along with an exceedance probability of 
50% and 16%, wherein a lower probability resulting in higher values of the coefficient k. 
Resonance effects are captured by the relationship between the structure's natural vibration period, 
represented by the variable Ts, and the excitation frequency of the seismic demand, represented by 
the spectral acceleration of the event. Bray and Macedo (2019) assert that the value of k increases as 
Ts grows from 0 s to resonance and decreases progressively as it moves away from this region. They 
conclude that stiffer structures, with shorter vibration periods, tend to displace more than more 
flexible structures dueto resonance effects with previously recorded seismic events. 
 
Papadimitriou et al. (2014) 
 The methodology proposed by Papadimitriou et al. (2014) was developed in 5 steps: 
Step 1: Estimation of Peak Ground Acceleration (PGA) and the fundamental period of seismic action. 
Step 2: Estimation of the nonlinear fundamental vibration period of the dam. 
Step 3: Estimation of peak acceleration at the crest of the dam (PGAcrest). 
Step 4: Estimation of the horizontal seismic acceleration coefficient (𝑘ℎ) based on PGAcrest. 
Step 5: Estimation of the effective horizontal seismic acceleration coefficient (𝑘ℎ) based on tolerable 
displacement. 
The first step of the methodology determines the value of Peak Ground Acceleration (PGA), which 
is derived from the Peak Ground Acceleration obtained in the rock (PGArock), obtained through PSHA 
analysis. Additionally, the PGA at the foundation is influenced by the fundamental period of seismic 
action (Te). The second step involves dimensioning the fundamental period (Te), which can be 
estimated through correlations using the maximum ground velocity peak (PGV), considering the 
excitation frequency, as proposed by Fajfar et al. (1992). For the analysis of the nonlinear fundamental 
 
 4 
vibration period of the foundation soil layer (T), the foundation conditions of the structure are taken 
into account through the shear wave velocity of the foundation soil layers (Vb). In cases where the 
structure under analysis has one or more soil layers overlying the bedrock, the methodology 
considers that this material will cause an amplification of PGArock due to its lower stiffness compared 
to the rock. It is emphasized that in cases where the structure is supported on the bedrock, the value 
of PGA is considered equal to PGArock. The third step of the methodology would be to dimension for 
the determination of PGA at the structure's crest (PGAcrest), but for this, it is necessary to estimate 
the nonlinear fundamental period (T0). In the fourth step, the maximum seismic acceleration 
coefficient is dimensioned based on coefficients that input the conditions of the structure's geometry, 
rupture surface, and foundation. Papadimitriou et al. (2014) emphasize that designing structures 
through pseudo-static analyses using the maximum khmax factor would be excessively conservative. 
Therefore, in the fifth and final step, the author suggests adopting an effective value (khe) as a 
percentage of this maximum value, where the sliding factor q (≥1) is the factor correlated with the 
allowable displacements (Dall) of the structure under analysis. 
 
RESULTS AND DISCUSSION 
The Table 1 presents the database used in the studies. 
Table 1 Database used in the studies. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
Structure Cross-section Vb Hb H ky vc t w z h 
Ba) 1 700 26,43 15,99 0,21 350 5,75 38,134 15,33 6,06 
Ba) 2 670 21,41 15,23 0,22 400 7,04 43,63 16,48 16,28 
Bb) 1 400 15,05 12,12 0,15 250 8,93 36,72 11,8 8,25 
Bb) 2 440 15,77 11,84 0,17 250 2,26 13,16 6,86 6,69 
Bc) 1 280 10 16,3 0,27 200 13,8 91,3 16,48 14,61 
Bd) 1 500 21 14,5 0,19 300 10,85 60,25 13,64 11,95 
Bd) 2 500 18 14,5 0,34 300 2,12 13,16 6,51 6,51 
Be) 1 275 41,1 25,8 0,5 180 7,4 52,1 11 10,3 
Bf) 1 280 27,5 104 0,19 200 28 282 100 100 
Bg) 1 370 13,1 97,02 0,22 300 17,85 215,75 85,48 97,02 
Bh) 1 370 20 74 0,006 373 37 180 48 44 
Bh) 2 370 30 78 0,006 373 37 180 48 44 
Bi) 1 400 5,3 31,8 0,37 300 11,01 99,552 31,8 30,74 
Bj) 1 370 3,93 14,65 0,23 200 5,54 33,9 14,65 14,65 
Bj) 2 370 16 20,85 0,21 200 6,18 35,54 15,02 15,02 
Bj) 3 370 15,12 21,81 0,21 200 5,8 35 14,98 14,98 
Bj) 4 370 11,2 20,03 0,2 200 5,25 32,12 14,94 14,94 
Table nomenclature - Vb – Shear wave velocity in the foundation soil layer (m/s); Hb – Thickness of 
the soil layer at the foundation (m); H - Height of the structure (m); ky - Seismic yeld coeficiente (g), 
vc - Shear wave velocity in the failure wedge (m/s); t - Thickness of the failure wedge (m); w - 
Horizontal projection of the failure wedge (m); z - Depth of the failure wedge relative to the structure 
crest (m); h - Height of the failure wedge (m). 
In Figure 1, the results of kh varying as a function of Dall for all analyzed structures (dams in the Iron 
Quadrangle in Brazil) are depicted. It is notable that the two methodologies used are influenced by 
Sthe selection of the tolerable permanent displacement. The authors of this study acknowledge the 
importance of evaluating the tolerable permanent displacement on an individualized basis, in order 
to establish a rational analytical approach that takes into account the specificities of each structure. 
The use of methodologies adopting a predefined tolerable permanent displacement is not advisable 
without a specific evaluation for the structure in question, to determine if this imposed displacement 
truly satisfies the conditions of the structure, as suggested by Hynes-Griffin and Franklin (1984). The 
authors suggest that determining the tolerable permanent displacement can be approached through 
three main focuses: operability, functionality, and mechanical behavior of the material. 
The definition of the approach to be employed depends directly on factors such as the rupture 
surface, the design of internal drainage (considering the materials and thicknesses of the layers used 
in the transitions), the water table, and the type of material under analysis. According to 
Papadimitriou et al. (2014), the difference between the rational and conservative approaches results 
in a variation ranging from 1.3 to 2.0, depending on the admitted tolerable displacement. 
 
 6 
 
Figure 1 – Obtaining the kh as a function of Dall for the methodologies Bray and Macedo (2019) and 
Papadimitriou et al. (2014). 
CONCLUSION 
In this study, it was observed that both methodologies exhibit variation varying with inferred 
tolerable displacement. For the methodology proposed by Bray and Macedo (2019), a point of 
inflection was identified where the sensitivity of the methodology is lost and the values disperse, 
increasing abruptly. This inflection occurs for tolerable permanent displacements considered low, 
between 2.0 and 5.0 cm, but still within the range estimated by the methodology. It is noticeable that, 
in general, the methodology of Bray and Macedo (2019) provides higher values for low tolerable 
displacements, approaching the methodology proposed by Papadimitriou et al. (2014) with the 
increase of calculated tolerable displacement. 
The main challenge in sizing the seismic acceleration coefficient has been estimating the tolerable 
displacement, where the smaller this displacement, the higher the value of Kh obtained. 
Due to space limitations imposed for this publication, the authors could not delve deeper into the 
studies presented in this work, but plan to address these issues in future publications for specialized 
journals. 
 
ACKNOWLEDGEMENTS 
 The authors would like to thank DF+ Engineering for their support of this research and study. 
 
 7 
 
REFERENCES 
Associação Brasileira de Normas Técnicas (2017) Mining ― Preparation and presentation of design of tailings, 
sediments and/or water dams ― Requirements. ABNT NBR 13028, Rio de Janeiro. 
Bray, J. D., Macedo, J. (2019) ‘Shear-induced seismic slope displacement estimates for shallow crustal earthquakes’, 
International Society for Soil Mechanics and Geotechnical Engineering, 7th International Conference on Earthquake 
Geotechnical Engineering, Italy, 17-20 June 2019. 
Duncan, SG Wright, TL Brandon (2014) Soil Strength and Slope Stability, 2rd ed., Wiley, Hoboken, New Jersey, 
USA. 
Fajfar P, Vidic T, Fischinger M. (1992) ‘Nonlinear seismic analysis and Design of Reinforced Concrete Buildings’, Bled, 
Slovenia 13-16,July 1992. 
Hynes-Griffin M E, Franklin A G. (1984) Rationalizing the seismic coeficient method. US Army Corps of Engineers, 
Department of the Army, Waterways Experiment Station, Miscellaneous paper GL-84-13. 
Papadimitriou A, Bouckovalas G, Andrianopoulos K. (2014) Methodology for estimating seismic coefficients for 
performance-based design of earthdams and tall embankments. Soil Dynamics and Earthquake Engineering, 56, 57-
73.