Characteristics of subgrade soil resilient modulus considering wide-range stress state

Abstract

Existing models for predicting the resilient modulus of subgrade soils can accurately characterize the dynamic stress zone. However, they suffer from two primary limitations: 1) an inability to predict modulus values within the nodynamic stress zone, and 2) the inaccessibility of these modulus values through conventional dynamic triaxial tests. To address these shortcomings, this paper introduces a comprehensive methodology for determining the resilient modulus of subgrade soil under a wide range of stress states. The no-dynamic stress zone is conceptualized as comprising two distinct zones: the no-stress zone and the self-weight stress zone. First, an integrated testing framework encompassing the entire subgrade domain is proposed. For the dynamic action zone, a testing method accounting for the combined influence of dynamic and static stresses is employed. For the area devoid of dynamic action, a novel testing method based on the dynamic triaxial apparatus is presented to measure the soil modulus in the no-dynamic stress zone. This method determines the modulus by maintaining constant confining and contact stresses while progressively reducing the cyclic stress. Subsequently, utilizing Gene Expression Programming (GEP), a unified prediction model integrating both the traditional resilient modulus and the modulus under no-dynamic conditions is developed, incorporating four fitting parameters. When the cyclic stress is below 10 kPa, the dynamic resilient modulus of the subgrade soil increases with increasing cyclic stress; when the cyclic stress exceeds 10 kPa, the modulus decreases with increasing cyclic stress. In contrast, under relatively high cyclic stresses, the dynamic resilient modulus decreases as the cyclic stress increases. Compared to the classical NCHRP 1-28A, Uzan, Ni, and K-θ models, the prediction error under small cyclic stresses is reduced from 15.49%, 16.15%, 18.85%, and 22.77%, respectively, to 4.68% using the proposed model. Furthermore, the model maintains high prediction accuracy across a total of 144 working conditions tested on 7 soil samples.

Keywords

subgrade engineering / resilient modulus / wide-range stress state / extremely small deviator stress / inherent modulus / GEP algorithm

References

[1] Du Y, Cui X, Jin Q, et al. Test equipment and prediction model for the service performance of subgrades under long-term traffic loads: A review [J]. Transportation Geotechnics, 2026, 56: 101812.

[2] Ng C W W, Zhou C, Yuan Q, et al. Resilient modulus of unsaturated subgrade soil: Experimental and theoretical investigations [J]. Canadian Geotechnical Journal, 2013, 502223232.

[3] Zhang J, Fan H, Zhang S, et al. Back-calculation of elastic modulus of high liquid limit clay subgrades based on viscoelastic theory and multipopulation genetic algorithm [J]. International Journal of Geomechanics, 2020, 201004020194.

[4] Zhang J, Shang H, Fan H. Resilient modulus characteristics of subgrade soil considering dynamic and static effects [J]. International Journal of Pavement Engineering, 2025, 26: 2546522.

[5] Zhang J, Peng J, Liu W, et al. Predicting resilient modulus of fine-grained subgrade soils considering relative compaction and matric suction [J]. Road Materials and Pavement Design, 2021, 223703715.

[6] Seed H B, Mitry F G, Monosmith C L, et al. Prediction of pavement deflection from laboratory repeated load tests [C], 1967224335

[7] Loach S. Repeated loading of fine grained soils for pavement design [D], 1987, Nottingham: University of Nottingham

[8] Zhang Xiaoning, Cui Xinzhuang, Ma Xiongying, et al. Hydromechanical responses of bentonite-based superabsorbent polymer improved silt under freeze-thaw cycles [J]. Journal of Rock Mechanics and Geotechnical Engineering, 2026. DOI: https://doi.org/10.1016/j.jrmge.2026.03.010. (in Press)

[9] Liao G. Numerical implementation of suction-dependent resilient modulus constitutive model for subgrade granular material [J]. Journal of Southeast University (English Edition), 2018, 342251258

[10] Zhao Y, Lu Z, Yao H, et al. Experimental study of dynamic resilient modulus of subgrade soils under coupling of freeze-thaw cycles and dynamic load [J]. Journal of Central South University, 2020, 27720432053.

[11] Zhang J, Peng J, Zheng J, et al. Characterisation of stress and moisture-dependent resilient behaviour for compacted clays in South China [J]. Road Materials and Pavement Design, 2020, 211262275.

[12] Nie R, Sun B, Leng W, et al. Resilient modulus of coarse-grained subgrade soil for heavy-haul railway: An experimental study [J]. Soil Dynamics and Earthquake Engineering, 2021, 150: 106959.

[13] Abdollahi M, Vahedifard F. Predicting resilient modulus of unsaturated subgrade soils considering effects of water content, temperature, and hydraulic hysteresis [J]. International Journal of Geomechanics, 2022, 22: 04021259.

[14] Qian J, Chen X, Ling J, et al. Estimating the contribution of suction on the resilient modulus of subgrade soils using capillary saturation [J]. Journal of Transportation Engineering, Part B: Pavements, 2021, 147104020081

[15] Luan X, Han L. Prediction model of dynamic resilient modulus of unsaturated modified subgrade under multi-factor combination [J]. Applied Sciences, 2022, 12189185.

[16] Xu X, Feng X, Lv G, et al. Dynamic behavior and numerical simulation of granitic residual soils under cyclic loading [J]. Geotechnical and Geological Engineering, 2025, 437336.

[17] Xu F, Chen Y, Zhang Q, et al. Characterization of resilient behavior of a fine-grained subgrade filler under graded variable confining pressure [J]. Case Studies in Construction Materials, 2025, 23: e04960.

[18] Ebrahimi A B, Mokhtari M, Aghaei A A. Behavior of fresh and recycled slag ballast under cyclic and post-cyclic monotonic loadings using large-scale triaxial tests [J]. Soil Dynamics and Earthquake Engineering, 2024, 182: 108720.

[19] Fan H, Zhang J, Zheng J. Back-calculation of nonlinear properties of subgrade using a new in situ nondestructive testing approach based on loading sequence [J]. International Journal of Geomechanics, 2024, 241004024235.

[20] Guzzarlapudi S D, Adigopula V K, Kumar R. Comparative studies of lightweight deflectometer and Benkelman beam deflectometer in low volume roads [J]. Journal of Traffic and Transportation Engineering (English Edition), 2016, 35438447.

[21] Yan F, Huang X. Linag R Y, Qian J, Tao J. Dynamic and static mechanical properties of loess subgrade in Shanxi [C]. Advances in Soil Dynamics and Foundation Engineering, 2014, Shanghai, China. Reston: ASCE, 39-44.

[22] Adam C, Adam D, Kopf F, et al. Computational validation of static and dynamic plate load testing [J]. Acta Geotechnica, 2009, 413555.

[23] Yan Q, Zhang F, Fang K, et al. Experimental investigation on static and dynamic resilient moduli of compacted fine soil [J]. Sciences in Cold and Arid Regions, 2017, 93297306

[24] Chen C, Zheng J. Relationship between dynamic modulus and static detection index in red sandstone subgrade [J]. Applied Mechanics and Materials, 2013, 353–356: 1112-1115.

[25] Chindaprasirt P, Sriyoratch A, Arngbunta A, et al. Estimation of modulus of elasticity of compacted loess soil and lateritic-loess soil from laboratory plate bearing test [J]. Case Studies in Construction Materials, 2022, 16: e00837.

[26] White D J, Vennapusa P K R. In situ resilient modulus for geogrid-stabilized aggregate layer: A case study using automated plate load testing [J]. Transportation Geotechnics, 2017, 11: 120-132.

[27] Naveen B P. Measurement of static and dynamic properties of municipal solid waste at Mavallipura landfill site, India [J]. International Journal of Geo-Engineering, 2018, 9: 26.

[28] Zhang N, Wang P, Xia C, et al. Evaluating subgrade dynamic and static resilience modulus through enhanced testing techniques [J]. Case Studies in Construction Materials, 2025, 22: e04159.

[29] Jiang H, Chen S, Sun H, et al. Dynamic and static resilient modulus and their prediction models of medium-high liquid limit clay in Yellow River flooded areas [J]. China Journal of Highway and Transport, 2021, 343103112(in Chinese)

[30] Liu X, Zhang X, Wang H, et al. Laboratory testing and analysis of dynamic and static resilient modulus of subgrade soil under various influencing factors [J]. Construction and Building Materials, 2019, 195: 178-186.

[31] Elliott R P. Selection of subgrade modules for aashto flexible pavement design[C]. Transportation Research Record, 19923944

[32] AASHTOGuide for mechanistic-empirical design of new and rehabilitated pavement structures, 2004(in Chinese)

[33] Chinese Standards InstitutionTest Methods of Soils for Highway Engineering, 2020(in Chinese)

[34] Uzan J. Permanent deformation of a granular base material [J]. Transportation Research Record: Journal of the Transportation Research Board, 1999, 167318994.

[35] Ni B, Hopkins T C, Sun L, et al. Modeling the resilient modulus of soils [M]. Bearing Capacity of Roads, Railways and Airfields, 2020, Boca Raton: CRC Press, 1131-1142.

[36] Monismith C L, Seed H B, Mitry F G, et al. Prediction of pavement deflections from laboratory tests [C]// Proceedings of the Second International Conference on the Structural Design of Asphalt Pavements. Ann Arbor: University of Michigan, 1967: 109−140.

[37] Ferreira C. Gene expression programming in problem solving [M]. Soft Computing and Industry: Recent Applications, 2002, London: Springer, 635-653.

[38] Koza J. Genetic programming as a means for programming computers by natural selection [J]. Statistics and Computing, 1994, 4287112.

[39] Hao J, Cui X, Bao Z, et al. Dynamic resilient modulus of heavy-haul subgrade silt subjected to freeze-thaw cycles: Experimental investigation and evolution analysis [J]. Soil Dynamics and Earthquake Engineering, 2023, 173: 108092.

[40] Wang y p, Zhou L. Experimental study of dynamic resilient modulus of cohesive soils in Henan [J]. Advanced Materials Research, 2014, 971–973: 2092-2095

[41] Kim D, Kim J R. Resilient behavior of compacted subgrade soils under the repeated triaxial test [J]. Construction and Building Materials, 2007, 21714701479.

[42] Liang R Y, Rabab’ah S, Khasawneh M. Predicting moisture-dependent resilient modulus of cohesive soils using soil suction concept [J]. Journal of Transportation Engineering, 2008, 13413440.

Rights & permissions

Central South University