A New Frequency Domain Framework of Inverse Ground Response Analysis for the Determination of Dynamic Soil Properties in a Two-Layered System
Publication: International Journal of Geomechanics
Volume 23, Issue 11
Abstract
Ground response analysis (GRA) requires the use of dynamic soil property curves (DSPCs). However, because regional DSPCs are not always available, globally recorded DSPCs are utilized in regional GRA the majority of the time. For those regions where regional DSPCs, based on laboratory estimations, exist, the DSPCs are affected in a variety of ways. For this reason, inverse GRA (IGRA), performed on instrumented earthquake (EQ) data, appears to be a realistic alternative for determining in situ DSPCs. Existing IGRA studies, based on downhole EQ data from around the world, show constraints, particularly in determining damping ratio (β) properties. Furthermore, previous frequency domain IGRA studies have only detected DSPCs for the surficial layer. Whereas a few of the frequency domain IGRA studies have determined the shear modulus degradation (G/Gmax) curve for multiple soil layers, they have been unable to determine the β curve for those layers. In the current study, we propose a novel frequency domain framework for determining the G/Gmax and β curves for a two-layered soil system positioned between successive accelerometer levels. The proposed framework was applied to 28 EQ records from the Lotung downhole array in Taiwan to determine the G/Gmax, β, and shear strain (γ) values for the top two layers from the ground surface. In addition, average DSPCs for the two layers are proposed. The effectiveness of the proposed DSPCs was tested by using them in a GRA and comparing them with recorded ground motions.
Practical Applications
It has been noted that, at present, there are limited downhole array records available. Therefore, there is ambiguity involved in applying the proposed framework presented in this paper to a larger number of records from downhole arrays. However, physical modeling of the wave propagation mechanism through layered media can be done on a smaller scale to determine dynamic soil properties in the future. Several attempts have been made to estimate the G/Gmax and β based on centrifuge testing using recorded acceleration time histories at different levels in the centrifuge model. In addition, the one-dimensional propagation of shear waves could be determined in those studies by taking special measures. However, these studies were based on a stress–strain imaging method in the time domain, which has its own limitations. However, recently, a frequency domain method was used to determine the G/Gmax and β from centrifuge modeling for a single soil layer. In that study, the β values obtained were scattered. Thus, the frequency domain framework proposed herein could be a viable alternative for determining the G/Gmax and β of a multilayered soil profile (here, two layers) when used in centrifuge testing.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors offer their thanks to the authors of the soilquake.net website from which the Lotung downhole array data were downloaded. The dataset was originally provided by the US Electric Power Research Institute (EPRI) under the direction of H. T. Tang and J. Carl Stepp.
Notation
The following symbols are used in this paper:
- Ar and Br
- displacement coefficients in the rock layer;
- As and Bs
- displacement coefficients in the soil layer;
- Β
- damping ratio;
- f
- fundamental natural frequency;
- G
- shear modulus;
- Gmax
- maximum shear modulus;
- G/Gmax
- shear modulus ratio;
- H(ω)
- transfer function;
- H(ω)CR
- theoretical transfer function;
- H(ω)CR,6–11
- theoretical transfer function between 6 and 11 m depth;
- H(ω)RR
- empirical transfer function;
- H(ω)RR,6–11
- empirical transfer function between 6 and 11 m depth
- H(ω)r
- transfer function in the rock layer;
- H1
- soil layer thickness;
- H2
- rock layer thickness;
- complex wave number for rock;
- complex wave number for soil;
- Mw
- moment magnitude;
- T
- natural time period;
- Tc
- combined natural time period of the first and second layers;
- T1 and T2
- natural time period of the first and second layers, respectively;
- tp
- P-wave arrival time;
- ur
- displacement in the rock layer;
- us
- displacement in the soil layer;
- Vs
- shear wave velocity;
- maximum shear wave velocity;
- complex shear wave velocity in rock;
- complex shear wave velocity in soil;
- zr
- depth coordinate from the top of the rock layer;
- zs
- depth coordinate from the top of the soil layer;
- complex impedance ratio;
- γ
- shear strain;
- γeff
- effective shear strain;
- γmax
- maximum shear strain;
- γref
- reference shear strain.
- γs
- shear strain in the soil layer;
- ρ
- density; and
- ω
- angular frequency of loading.
References
Afacan, K. B., S. J. Brandenberg, and J. P. Stewart. 2014. “Centrifuge modeling studies of site response in soft clay over wide strain range.” J. Geotech. Geoenviron. Eng. 140 (2): 04013003. https://doi.org/10.1061/(asce)gt.1943-5606.0001014.
Alemu, B. E., A. Worku, G. M. Wassie, and G. T. Habtesellasie. 2018. “Ground response analysis of representative sites of Hawassa city.” In Geotechnical Earthquake Engineering and Soil Dynamics V: Seismic Hazard Analysis, Earthquake Ground Motions, and Regional-Scale Assessment, edited by S. J. Brandenberg and M. T. Manzari, 422–434. Reston, VA: ASCE.
Ameri, G., A. Oth, M. Pilz, D. Bindi, S. Parolai, L. Luzi, and G. Cultrera. 2011. “Separation of source and site effects by generalized inversion technique using the aftershock recordings of the 2009 L’Aquila earthquake.” Bull. Earthquake Eng. 9 (3): 717–739. https://doi.org/10.1007/s10518-011-9248-4.
Anbazhagan, P., T. G. Sitharam, and C. Divya. 2007. “Site response analyses based on site specific soil properties using geotechnical and geophysical tests: Correlations between Vs30, Gmax and N60.” In Proc., 4th Int. Conf. on Earthquake Geotechnical Engineering. Berlin, Germany: Springer.
Anderson, D. G., and Y. K. Tang. 1989. Vol. 2 of Summary of soil characterization program for the Lotung large-scale seismic experiment. Rep. No. NP-6I54. Palo Alto, CA: Electric Power Research Institute.
Archuleta, R. J., J. H. Steidl, and L. F. Bonilla. 2000. “Engineering insights from data recorded on vertical arrays.” In Proc., 12th World Conf. on Earthquake Engineering, 1–7. Silverstream, New Zealand: New Zealand Society for Earthquake Engineering.
Banerjee, S., and A. Kumar. 2018. “Determination of seismic wave attenuation in Delhi, India, towards quantification of regional seismic hazard American Standard Code for Information Interchange.” Nat. Hazard. 92 (2): 1039–1064. https://doi.org/10.1007/s11069-018-3238-7.
Beresnev, I. A., G. M. Atkinson, P. A. Johnson, and E. H. Field. 1998. “Stochastic finite-fault modeling of ground motions from the 1994 Northridge, California, earthquake. II. Widespread nonlinear response at soil sites.” Bull. Seismol. Soc. Am. 88 (6): 1402–1410. https://doi.org/10.1785/BSSA0880061402.
Bindi, D., F. Pacor, L. Luzi, M. Massa, and G. Ameri. 2009. “The Mw 6.3, 2009 L’Aquila earthquake: Source, path and site effects from spectral analysis of strong motion data.” Geophys. J. Int. 179 (3): 1573–1579. https://doi.org/10.1111/j.1365-246X.2009.04392.x.
Campillo, M., J. C. Gariel, K. Aki, and F. J. Sánchez-Sesma. 1989. “Destructive strong ground motion in Mexico City: Source, path, and site effects during great 1985 Michoacán earthquake.” Bull. Seismol. Soc. Am. 79: 1718–1735. https://doi.org/10.1785/BSSA0790061718.
Chandra, J., P. Guéguen, J. H. Steidl, and L. F. Bonilla. 2015. “In situ assessment of the g–γ curve for characterizing the nonlinear response of soil: Application to the Garner Valley downhole array and the wildlife liquefaction array.” Bull. Seismol. Soc. Am. 105 (2): 993–1010. https://doi.org/10.1785/0120140209.
Chang, C. Y., C. M. Mok, A. Memebers, and H. T. Tang. 1996. “Inference of dynamic shear modulus from Lotung downhole data.” J. Geotech. Eng. 122 (8): 657–665. https://doi.org/10.1061/(ASCE)0733-9410(1996)122:8(657).
Chang, C. Y., Y. K. Tang, C. M. Mok, H. T. Tang, and M. S. Power. 1991. “Development of shear modulus reduction curves based on Lotung downhole ground motion data.” In Vol. 16 of Proc., 2nd Int. Conf. Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics. Rolla, MO: Missouri Univ. of Science and Technology.
Chin, B., and K. Aki. 1991. “Simultaneous study of the source, path, and site effects on strong ground motion during the 1989 Loma Prieta earthquake: A preliminary result on pervasive nonlinear site effects.” Bull. Seismol. Soc. Am. 81: 1859–1884.
Elgamal, A., T. Lai, Z. Yang, and L. He. 2001. “Dynamic soil properties, seismic downhole arrays and applications in practice (CD-ROM).” In Proc., 4th Int. Conf. Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, edited by S. Prakash, 26–31. Rolla, MO: Missouri Univ. of Science and Technology.
Elgamal, A. W., M. Zeghal, H. T. Tang, and J. C. Stepp. 1995. “Lotung downhole array. I: Evaluation of site dynamic properties.” J. Geotech. Eng. 121 (4): 350–362. https://doi.org/10.1061/(ASCE)0733-9410(1995)121:4(350).
EPRI (Electric Power Research Institute). 1993. “Guidelines for site specific ground motions.” Palo Alto, CA: EPRI.
Eseller-Bayat, E. E., and M. Ada. 2021. “A methodology for estimation of site-specific nonlinear dynamic soil behaviour using vertical downhole arrays.” Eur. J. Environ. Civ. Eng. 25 (10): 1810–1832. https://doi.org/10.1080/19648189.2019.1603122.
Gazetas, G., K. Fan, T. Tazoh, M. Kavadas, and N. Makris. 1992. “Seismic pile-group-structure interaction.” In Piles under dynamic loads, Geotechnical Special Publication 34, edited by S. Prakash, 56–94. Reston, VA: ASCE.
Ghayamghamian, M. R., and H. Kawakami. 1996. “On the characteristics of non-linear soil response and dynamic soil properties using vertical array data in Japan.” Earthquake Eng. Struct. Dyn. 25 (8): 857–870. https://doi.org/10.1002/(SICI)1096-9845(199608)25:8%3C857::AID-EQE592%3E3.0.CO;2-R.
Ghayamghamian, M. R., and H. Kawakami. 2000. “On site nonlinear hysteretic curves and dynamic soil properties.” J. Geotech. Geoenviron. Eng. 2 (June): 543–555. https://doi.org/10.1061/(ASCE)1090-0241(2000)126:6(543).
Ghayamghamian, M. R., and M. Matosaka. 2001. “Identification of dynamic soil properties using vertical array recordings.” In Proc. Int. Conf. Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics. Rolla, MO: Missouri Univ. of Science and Technology.
Glaser, S. D., and L. G. Baise. 2000. “System identification estimation of soil properties at the Lotung site.” Soil Dyn. Earthquake Eng. 19 (7): 521–531. https://doi.org/10.1016/S0267-7261(00)00026-9.
Glaser, S. D., and A. L. Leeds. 1996. Estimation of system damping at the Lotung site by application of system identification. Rep. No. NIST GCR 96-700. Golden, CO: Colorado School of Mines.
Hadjian, A. H. 2002. “Fundamental period and mode shape of layered soil profiles.” Soil Dyn. Earthquake Eng. 22 (9–12): 885–891. https://doi.org/10.1016/S0267-7261(02)00111-2.
Han, L., L. Wang, X. Ding, H. Wen, X. Yuan, and W. Zhang. 2022a. “Similarity quantification of soil parametric data and sites using confidence ellipses.” Geosci. Front. 13 (1): 101280. https://doi.org/10.1016/j.gsf.2021.101280.
Han, L., L. Wang, and W. Zhang. 2020. “Quantification of statistical uncertainties of rock strength parameters using Bayesian-based Markov chain Monte Carlo method.” IOP Conf. Ser.: Earth Environ. Sci. 570 (3): 032051. https://doi.org/10.1088/1755-1315/570/3/032051.
Han, L., L. Wang, W. Zhang, B. Geng, and S. Li. 2022b. “Rockhead profile simulation using an improved generation method of conditional random field.” J. Rock Mech. Geotech. Eng. 14 (3): 896–908. https://doi.org/10.1016/j.jrmge.2021.09.007.
Hardin, B. O., and V. P. Drnevich. 1972. “Shear modulus and damping in soils: Design equation and curves.” J. Soil Mech. Found. Eng. Div. 98: 667–691. https://doi.org/10.1061/JSFEAQ.0001760.
Honjo, Y., S. Iwamoto, M. Sugimoto, S. Onimaru, and M. Yoshizawa. 1998. “Inverse analysis of dynamic soil properties based on seismometer array records using the extended Bayesian method.” Soil Sci. Soc. Am. J. 38 (1): 131–143.
Huang, H. C., C. S. Shieh, and H. C. Chiu. 2001. “Linear and nonlinear behaviors of soft soil layers using Lotung downhole array in Taiwan.” Terr. Atmos. Ocean. Sci. 12 (3): 503–524. https://doi.org/10.3319/TAO.2001.12.3.503(T).
Ishihara, K. 1996. Soil behaviour in earthquake geotechnics. Oxford, UK: Clarendon Press.
Kaklamanos, J., L. Dorfmann, and L. G. Baise. 2014. “Modeling dynamic site response using the overlay concept.” In Geo-Congress 2014: Geo-Characterization and Modeling for Sustainability, Geotechnical Special Publication 234, edited by A. J. Puppala, P. Bandini, and T. C. Sheahan, 1167–1176. Reston, VA: ASCE.
Koga, Y., and O. Matsuo. 1990. “Shaking table test of embankments resting on liquefiable sandy ground.” Soils Found. 30: 162–174. https://doi.org/10.3208/sandf1972.30.4_162.
Kokusho, T., T. Aoyagi, and A. Wakunami. 2005. “In situ soil-specific nonlinear properties back-calculated from vertical array records during 1995 Kobe earthquake.” J. Geotech. Geoenviron. Eng. 131 (12): 1509–1521. https://doi.org/10.1061/(ASCE)1090-0241(2005)131:12(1509).
Kokusho, T., K. Sato, and M. Matsumoto. 1996. “Nonlinear dynamic soil properties back-calculated from strong seismic motions during Hyogoken-Nanbu earthquake.” In Proc., 11th World Conf. on Earthquake Engineering, Paper No. 2080. Oxford, UK: Pergamon.
Kondner, R. L. 1963. “Hyperbolic stress-strain response: Cohesive soils.” J. Soil Mech. Found. Div. 89 (1): 115143. https://doi.org/10.1061/JSFEAQ.000047.
Kramer, S. L. 1996. Geotechnical earthquake engineering. Prentice-Hall International Series in Civil Engineering and Engineering Mechanics. Upper Saddle River, NJ: Prentice-Hall.
Kumar, A., O. Baro, and N. H. Harinarayan. 2016. “Obtaining the surface PGA from site response analyses based on globally recorded ground motions and matching with the codal values.” Nat. Hazard. 81 (1): 543–572. https://doi.org/10.1007/s11069-015-2095-x.
Kumar, A., N. H. Harinarayan, and O. Baro. 2017. “Nonlinear soil response to ground motions during different earthquakes in Nepal, to arrive at surface response spectra.” Nat. Hazard. 87 (1): 13–33. https://doi.org/10.1007/s11069-017-2751-4.
Kumar, A., and J. K. Mondal. 2017. “Newly developed MATLAB based code for equivalent linear site response analysis.” Geotech. Geol. Eng. 35 (5): 2303–2325. https://doi.org/10.1007/s10706-017-0246-4.
Lin, J. S. 1994. “Extraction of dynamic soil properties using extended Kalman filter.” J. Geotech. Eng. 120 (12): 2100–2117. https://doi.org/10.1061/(ASCE)0733-9410(1994)120:12(2100).
Madera, G. A. 1970. Fundamental period and amplification of peak acceleration in layered systems. Cambridge, MA: Dept. of Civil Engineering, M.I.T.
Markham, C. S., J. D. Bray, J. Macedo, and R. Luque. 2016. “Evaluating nonlinear effective stress site response analyses using records from the Canterbury earthquake sequence.” Soil Dyn. Earthquake Eng. 82: 84–98. https://doi.org/10.1016/j.soildyn.2015.12.007.
Mercado, V., W. El-Sekelly, M. Zeghal, and T. Abdoun. 2015. “Identification of soil dynamic properties through an optimization analysis.” Comput. Geotech. 65: 175–186. https://doi.org/10.1016/j.compgeo.2014.11.009.
Midorikawa, S., and H. Miura. 2008. “Nonlinear behavior of soil response observed in strong-motion records from recent Japanese earthquakes.” In Vol. 3 of World Conf. on Earthquake Engineering. Tokyo, Japan: International Association for Earthquake Engineering.
Mondal, J. K., and A. Kumar. 2021. “A new frequency domain based inverse ground response analysis framework for the determination of dynamic soil properties.” Int. J. Geomech. 21 (5): 1–23. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001973.
Mondal, J. K., and A. Kumar. 2022. “A systematic review on inverse GRA methodologies developed for the determination of dynamic soil properties using downhole seismic array records.” Indian Geotech. J. 52: 400–415. https://doi.org/10.1007/s40098-021-00571-2.
Nampally, S., S. Padhy, S. Trupti, P. Prabhakar Prasad, and T. Seshunarayana. 2018. “Evaluation of site effects on ground motions based on equivalent linear site response analysis and liquefaction potential in Chennai, south India.” J. Seismol. 22: 1075–1093. https://doi.org/10.1007/s10950-018-9751-z.
Ng, C. W. W., and G. T. K. Lee. 2002. “A three-dimensional parametric study of the use of soil nails for stabilising tunnel faces.” Comput. Geotech. 29 (8): 673–697. https://doi.org/10.1016/S0266-352X(02)00012-5.
Nihon. 2011. Liquefaction induced damages caused by the M 9.0 East Japan mega earthquake on March 11, 2011. Koriyama, Japan: Tokyo Metropolitan Univ., Hisataka Tano, Nihon Univ.
Oskay, C., and M. Zeghal. 2011. “A survey of geotechnical system identification techniques.” Soil Dyn. Earthquake Eng. 31 (4): 568–582. https://doi.org/10.1016/j.soildyn.2010.11.011.
Phanikanth, V. S., D. Choudhury, and G. R. Reddy. 2011. “Equivalent-linear seismic ground response analysis of some typical sites in Mumbai.” Geotech. Geol. Eng. 29 (6): 1109–1126. https://doi.org/10.1007/s10706-011-9443-8.
Puri, N., A. Jain, P. Mohanty, and S. Bhattacharya. 2018. “Earthquake response analysis of sites in state of Haryana using DEEPSOIL software.” Procedia Comput. Sci. 125: 357–366. https://doi.org/10.1016/j.procs.2017.12.047.
Pyke, R. 1979. “Nonlinear soil models for irregular cyclic loadings.” J. Geotech. Eng. Div. 105 (6): 715725. https://doi.org/10.1061/AJGEB6.000082.
Ranjan, R. 2005. Seismic response analysis of Dehradun city, India. Enschede, Netherlands: International Institute for Geo-Information Science and Earth Observations.
Schnabel, P. B. 1973. Effects of local geology and distance from source on earthquake ground motions. Berkeley, CA: Univ. of California.
Seed, H., and I. Idriss. 1970. Soil moduli and damping factors for dynamic response analysis. Berkeley, CA: Univ. of California.
Seed, H. B., I. M. Idriss, and K. Tokimatsu. 1986. “Moduli and damping factors for dynamic analyses of cohesionless soils.” J. Geotech. Eng. 112 (11): 1016–1032. https://doi.org/10.1061/(ASCE)0733-9410(1986)112:11(1016).
Shen, C. K., Z. Wang, and X. S. Li. 1991. “Pore pressure response during 1986 Lotung earthquakes.” In Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics. Rolla, MO: Univ. of Missouri.
Stanko, D., Z. Gülerce, S. Markušić, and R. Šalić. 2019. “Evaluation of the site amplification factors estimated by equivalent linear site response analysis using time series and random vibration theory based approaches.” Soil Dyn. Earthquake Eng. 117 (October 2018): 16–29. https://doi.org/10.1016/j.soildyn.2018.11.007.
Stanko, D., S. Markušić, S. Strelec, and M. Gazdek. 2017. “Equivalent-linear site response analysis on the site of the historical Trakošćan Castle, Croatia, using HVSR method.” Environ. Earth Sci. 76 (18): 642. https://doi.org/10.1007/s12665-017-6971-4.
Sun, J. I., R. Golesorkhi, and H. B. Seed. 1988. Dynamic moduli and damping ratios for cohesive soils. Rep. No. EERC 88-15. Washington, DC: National Science Foundation.
Taboada-Urtuzuastegui, V., H. Martinez, M. Romo, and C. Ardila. 2000. “Identification of Mexico City clay dynamic properties.” In Proc., 12th World Conf. on Earthquake Engineering. Paper No. 1–8. Upper Hutt, New Zealand: New Zealand Society for Earthquake Engineering.
Tang, H. T., Y. K. Tang, J. C. Stepp, I. B. Wall, E. Lin, S. C. Cheng, and S. K. Lee. 1989. “A large-scale soil-structure interaction experiment: Design and construction.” Nucl. Eng. Des. 111 (3): 371–379. https://doi.org/10.1016/0029-5493(89)90248-3.
Tokimatsu, K., and S. Midorikawa. 1981. “Nonlinear soil properties estimated from strong motion accelerograms.” In Proc., 1st Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, 117–122. Rolla, MO: Missouri Univ. of Science and Technology.
Tokimatsu, K., S. Midorikawa, and Y. Yoshimi. 1989. “Dynamic soil properties obtained from strong motion records.” In Proc., 12th Int. Conf. on Soil Mechanics and Foundation Engineering, 2015–2018. London: International Society for Soil Mechanics and Geotechnical Engineering.
Tokimatsu, K., T. Sekiguchi, H. Miura, and S. Midorikawa. 2006. “Nonlinear dynamic properties of surface soils estimated from strong motion accelerograms at K-NET and JMA stations in Ojiya.” J. Struct. Constr. Eng. 600: 43–59. https://doi.org/10.3130/aijs.71.43_2.
Tsai, C., and Y. M. A. Hashash. 2007. “An inverse analysis approach to extract dynamic nonlinear soil behavior from downhole array data.” In Proc., 4th Int. Conf. on Earthquake Geotechnical Engineering. Berlin, Germany: Springer.
Tsai, C. C., and Y. M. A. Hashash. 2009. “Learning of dynamic soil behavior from downhole arrays.” J. Geotech. Geoenviron. Eng. 135 (6): 745–757. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000050.
Uthayakumar, U. M., and E. Naesgaard. 2004. “Ground response analysis for seismic design in Fraser River Delta, British Columbia.” In Proc., 13th World Conf. on Earthquake Engineering. Vancouver, BC: WCEE Secretariat.
Vucetic, M., and R. Dobry. 1991. “Effect of soil plasticity on cyclic response.” J. Geotech. Eng. 117 (1): 89–107. https://doi.org/10.1061/(ASCE)0733-9410(1991)117:1(89).
Wang, H. Y., W. P. Jiang, S. Y. Wang, and Y. Miao. 2019. “In situ assessment of soil dynamic parameters for characterizing nonlinear seismic site response using KiK-net vertical array data.” Bull. Earthquake Eng. 17 (5): 2331–2360. https://doi.org/10.1007/s10518-018-00539-3.
Wen, Y. K. 1976. “Method for random vibration of hysteretic system.” J. Eng. Mech. Div. 102 (2): 249263. https://doi.org/10.1061/JMCEA3.0002106.
Wen, K.-L. 1994. “Non-linear soil response in ground motions.” Earthquake Eng. Struct. Dyn. 23 (6): 599–608. https://doi.org/10.1002/eqe.4290230603.
Yangisawa, E., and M. Kazama. 1996. “Nonlinear dynamic behavior of the ground inferred from strong motion array records at Kobe Port Island during the 1995 Hygo-Ken Nanbu earthquake.” In Proc., 11th World Conf. on Earthquake Engineering. Oxford, UK: Pergamon.
Zeghal, M., H. T. Tang, and J. C. Stepp. 1995. “Soil nonlinear properties.” J. Geotech. Eng. 121 (4): 363–378. https://doi.org/10.1061/(ASCE)0733-9410(1995)121:4(363).
Zhang, L., Q. Ou, and S. Zhou. 2020. “Analytical study of the dynamic response of a double-beam model for a geosynthetic-reinforced embankment under traffic loads.” Comput. Geotech. 118 (August 2019): 103330. https://doi.org/10.1016/j.compgeo.2019.103330.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 23 • Issue 11 • November 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Oct 21, 2022
Accepted: May 13, 2023
Published online: Aug 31, 2023
Published in print: Nov 1, 2023
Discussion open until: Jan 31, 2024
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.