Next Article in Journal
Optimizing Road Safety Inspections on Rural Roads
Previous Article in Journal
Optimization of Surface Preparation and Painting Processes for Railway and Automotive Steel Sheets
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Predictive Stress Modeling of Resilient Modulus in Sandy Subgrade Soils

by
Tadas Tamošiūnas
* and
Šarūnas Skuodis
Department of Reinforced Concrete Structures and Geotechnics, Vilnius Gediminas Technical University, 10223 Vilnius, Lithuania
*
Author to whom correspondence should be addressed.
Infrastructures 2023, 8(2), 29; https://doi.org/10.3390/infrastructures8020029
Submission received: 26 January 2023 / Revised: 2 February 2023 / Accepted: 6 February 2023 / Published: 8 February 2023

Abstract

:
The mechanical properties of pavement materials are crucial to the design and performance of flexible pavements. One of the most commonly used measures of these properties is the resilient modulus (Er). Many different models were developed to predict the resilient modulus of coarse soils, which are based on the states of stresses and the physical and mechanical properties of the soil. The unconsolidated unsaturated drained cyclic triaxial tests were performed for three variously graded and three well-graded sand specimens to determine the resilient modulus, and to perform predictive modeling using the K-θ, Rahim and George, Uzan, and Universal Witczak models. Obtained Er values directly depended on the confining pressure and deviatoric stress values used during the test. The Octahedral Shear Stress (OSS) model, proposed by the authors of the paper, predicts the resilient modulus with a coefficient of determination (R2) ranging from 0.85 to 0.99. The advantage of the model is the use of small-scale data tables, meaning fixed K1 and K2 regression coefficients, and it can be assigned to a specific specimen type without the need to determine them using the specific deviatoric and confining stresses.

1. Introduction

The mechanical properties of pavement materials are crucial to the design and performance of flexible pavements. One of the most commonly used measures of these properties is the resilient modulus (Er) [1,2,3]. It can be determined for sandy subgrade [4,5,6] and clayey subgrade [7,8,9].
The resilient modulus (Er) is a measure of the ability of a material to resist deformation under repeated loading. It is commonly used in the design of flexible pavements, as it provides a measure of the stiffness of the pavement material and its ability to resist rutting and fatigue [10,11].
The resilient modulus (Er) of a material is affected by several factors, such as the type and amount of loading, the moisture content of the material, and the temperature [12,13].
To determine the resilient modulus, cyclic triaxial pressure tests are performed in the laboratory. It is difficult to perform these tests since qualified specialists are needed. Also, the tests take quite a long time and the equipment for performing such tests is expensive [14]. For these and other reasons, resilient modulus predictive models have been developed which allow modulus values to be determined without cyclic triaxial testing [15,16,17].
The purpose of this research work is to verify the accuracy of commonly used resilient modulus prediction models by relating their results to the experimentally determined resilient modulus of sandy soils, and also, to propose a more accurate model for forecasting sandy soils modulus values.

2. Materials and Methods

2.1. Test Procedure

To determine the resilient modulus (Er), the isotropic unconsolidated unsaturated drained cyclic triaxial tests were performed using a Wille Geotechnik dynamic triaxial apparatus, according to the low-stress test program (method B) provided in EN 13286-7:2004 [2], which was slightly adjusted using confining stress from 20 kPa to 70 kPa (Figure 1), and the minimum value of the deviator was fixed at the limit of 10 kPa due to the limitations of the test apparatus. The maximum deviator stress and the number of cycles for a particular state of specimen loading are provided in Figure 1. The loadings of the specimen were performed at a frequency of 1 Hz with data recording intervals ranging from 100 to 150 times per second. The dimensions of the samples were 100 mm in diameter and 200 mm in height.
At the beginning of the tests, conditioning of specimens was performed with 20,000 periodic cyclic loadings at the same frequency of 1 Hz with variable stress deviator ranging from 10 to 200 kPa.
The resilient modulus was determined according to [18] and the formula given in EN 13286-7:2004 [2]:
E r = σ 1 r 2 σ 1 r σ 3 r 2 σ 3 r 2 σ 1 r ε 1 r + σ 3 r ε 1 r 2 σ 3 r ε 3 r
where σ1r is residual axial stress, σ3r is residual radial stress, ε1r is residual or restored axial relative deformation determined using displacement values from two vertical linear variable differential transformers (LVDTs), and ε3r—residual or restored radial relative deformation determined using displacement value from radial LVDT (Figure 2).
Resilient or recovered axial strain (ε1r) is determined by dividing resilient axial displacement at cycle N, defined as the displacement during the unloading part of the cycle (between the point where the applied stresses are maximum and the end of the cycle) from the gauge length for axial displacement (Displacement 1 and 2, see Figure 2). Resilient or recovered radial strain is determined by dividing resilient radial displacement at cycle N, defined as the displacement during the unloading part of the cycle, from the gauge length for radial displacement (Displacement 3, see Figure 2). Axial displacement was determined using two LVDTs, radial—one LVDT (Figure 2). The example determination of values of displacements at the maximum deviator stress and the last values of displacement during one cycle can be seen in Figure 3.

2.2. Materials

A total of six different samples were tested and classified according to the LST 1331:2022 [19] standard: three as variously graded sands (SP) and three as well-graded sands (SG). Particle size distribution curves are provided in Figure 4.
Classification of all samples according to LST 1331:2022 [19] and the Unified Soil Classification System (USCS) [20] is presented in Table 1, as well as uniformity coefficients (Cu) and coefficient of curvature (Cc). The density of each specimen was controlled at 100±5% dry density.

2.3. Models

Many different models were developed to predict the resilient modulus of coarse soils, which were based on the states of stresses and the physical and mechanical properties of the soil [21,22,23]. In this study, the authors have used four main models which are based on the stress state of the soil.
One of the most well-known models is a model developed by Hicks and Monismith [24,25,26], also known as the K-θ model, presented below:
Er = K1 (θ)K2
where K1 and K2 are fitting parameters or regression coefficients, and θ is bulk stress which equals the sum of major σ1, minor σ3, and intermediate σ2 stresses.
Another widely used model is that developed by Rahim and George [27,28], which incorporates deviatoric stress (σ1) and atmospheric pressure (Pa) into the equation:
E r = K 1 P a Θ ( σ d + 1 ) + 1 K 2
Uzan [29,30] proposed a model that incorporates octahedral shear stress (τoct):
E r = K 1 P a Θ P a K 2 τ oct P a K 3
octahedral shear stress (τoct) equal to (√2/3)(σ1 − σ3), and K3, same as K1 and K2, fitting parameters or regression coefficients.
Similar to Uzan’s model is the Universal Witczak [31,32] model:
E r = K 1 P a Θ P a K 2 τ oct P a + 1 K 3

3. Results and Discussion

3.1. Test Results

After performing unconsolidated unsaturated drained cyclic triaxial tests, resilient modulus (Er) values were obtained, which are shown in Figure 5. Received values directly depended on the confining pressure and deviatoric stress values used during the test, which are presented in the test program in Figure 1.

3.2. Modeling Results

After obtaining the resilient modulus (Er) values, using the previously listed models in formulae 2–5 to predict the resilient modulus, modeling was performed using the nonlinear generalized reduced gradient method to determine the average regression coefficients (Kn) and coefficients of determination (R2) for every 100 cycles. The results of modeling for every specimen are provided in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7.
After the predictive modeling, it can be seen in Table 2 that all used models for specimen SP1—variously graded sand almost perfectly predicted the resilience modulus values. The coefficient of determination (R2) values of all models were higher than 0.98. The regression coefficient K1 used in the K-θ model (2) varied from 0.79 to 1.05 and regression coefficient K2 varied from 1.07 to 1.25. The regression coefficient K1 used in the Rahim and George model (3) varied from 0.40 to 1.77 and regression coefficient K2 varied from 0.46 to 1.90. The regression coefficient K1 used in the Uzan model (4) varied from 1.20 to 3.08, regression coefficient K2 varied from 0.76 to 1.89, and regression coefficient K3 varied from 0.03 to 1.19. The regression coefficient K1 used in the Universal Witczak model (5) varied from 0.97 to 2.29, regression coefficient K2 varied from 0.94 to 1.46, and regression coefficient K3 varied from 0.99 to 1.43.
A similar situation is seen with specimen SP2—variously graded sand (Table 3), all used models almost perfectly predicted the resilience modulus values, and the coefficient of determination (R2) values of all models were higher than 0.99. The regression coefficient K1 used in the K-θ model (2) varied from 1.01 to 1.05 and regression coefficient K2 varied from 1.08 to 1.25. The regression coefficient K1 used in the Rahim and George model (3) varied from 0.54 to 1.76 and regression coefficient K2 varied from 0.50 to 1.87. The regression coefficient K1 used in the Uzan model (4) varied from 1.18 to 3.04, regression coefficient K2 varied from 0.76 to 1.89, and regression coefficient K3 varied from 0.01 to 1.17. The regression coefficient K1 used in the Universal Witczak model (5) varied from 1.15 to 2.19, regression coefficient K2 varied from 0.90 to 1.46, and regression coefficient K3 varied from 1.04 to 1.42.
For the specimen SP3—variously graded sand (Table 4), the coefficient of determination (R2) values of all models were higher than 0.99 also. The regression coefficient K1 used in the K-θ model (2) varied from 1.00 to 1.05 and regression coefficient K2 varied from 0.98 to 1.24. The regression coefficient K1 used in the Rahim and George model (3) varied from 0.38 to 1.76 and regression coefficient K2 varied from 0.39 to 1.88. The regression coefficient K1 used in the Uzan model (4) varied from 1.18 to 2.98, regression coefficient K2 varied from 0.89 to 1.89, and regression coefficient K3 varied from 0 to 1.17. The regression coefficient K1 used in the Universal Witczak model (5) varied from 0.84 to 2.14, regression coefficient K2 varied from 0.95 to 1.46, and regression coefficient K3 varied from 0.99 to 1.39.
For the specimen SG1—well-graded sand (Table 5), the coefficient of determination (R2) value using the K-θ model (2) is 0.24. Other used models almost perfectly predicted the resilience modulus values and the value coefficients of determination (R2) for the rest of the models were higher than 0.99. The regression coefficient K1 used in the K-θ model (2) varied from 0.65 to 9.72 and regression coefficient K2 varied from 1.04 to 1.24. The regression coefficient K1 used in the Rahim and George model (3) varied from 0.31 to 1.76 and regression coefficient K2 varied from 0.41 to 1.87. The regression coefficient K1 used in the Uzan model (4) varied from 1.10 to 3.00, regression coefficient K2 varied from 0.78 to 1.89, and regression coefficient K3 varied from 0.05 to 1.17. The regression coefficient K1 used in the Universal Witczak model (5) varied from 0.72 to 2.14, regression coefficient K2 varied from 0.96 to 1.47, and regression coefficient K3 varied from 0.97 to 1.39.
For the specimen SG2—well-graded sand (Table 6), the coefficient of determination (R2) value using the K-θ model (2) is 0.22. Other used models almost perfectly predicted the resilience modulus values. The value coefficients of determination (R2) for the rest of the models were higher than 0.99. The regression coefficient K1 used in the K-θ model (2) varied from 0.78 to 9.13 and regression coefficient K2 varied from 1.09 to 1.26. The regression coefficient K1 used in the Rahim and George model (3) varied from 0.43 to 1.77 and regression coefficient K2 varied from 0.47 to 1.90. The regression coefficient K1 used in the Uzan model (4) varied from 1.19 to 3.12, regression coefficient K2 varied from 0.74 to 1.90, and regression coefficient K3 varied from 0.01 to 1.18. The regression coefficient K1 used in the Universal Witczak model (5) varied from 1.07 to 2.24, regression coefficient K2 varied from 0.95 to 1.48, and regression coefficient K3 varied from 1.00 to 1.45.
For the specimen SG3—well-graded sand (Table 7), the coefficient of determination (R2) value using the K-θ model (2) is 0.22. The regression coefficient K1 varied from 0.61 to 9.72 and regression coefficient K2 varied from 0.99 to 1.23. The coefficient of determination (R2) value used in the Rahim and George model (3) is 0.80, the regression coefficient K1 varied from 0.32 to 1.59, and regression coefficient K2 varied from 0.41 to 1.79. The coefficient of determination (R2) value used in the Uzan model (4) is 0.80, the regression coefficient K1 varied from 1.07 to 2.81, the regression coefficient K2 varied from 0.82 to 1.86, and the regression coefficient K3 varied from 0.12 to 1.17. The coefficient of determination (R2) value used in the Universal Witczak model (5) is 0.80, the regression coefficient K1 varied from 0.73 to 1.95, regression coefficient K2 varied from 0.98 to 1.45, and the regression coefficient K3 varied from 0.97 to 1.37.

3.3. Proposed Model

The values of regression coefficients (Kn) for all used predictive models strongly fluctuate at different deviatoric stresses and confining stresses (Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7). Therefore, to accurately predict the resilient modulus of test specimens, large-scale data tables should be used, meaning that regression coefficients (Kn) need to be determined using specific deviatoric and confining stresses. To avoid this, a new, simpler model was searched for, which would help make resilient modulus predictions with sufficient accuracy.
The models for predicting the resilient modulus presented in the Materials and Methods chapter were based on three main stress variables—deviatoric stress, bulk stresses, and octahedral stress. The power dependence of the resilient modulus on the bulk stress is presented in Figure 6. Power dependence, compared to linear, exponential, logarithmic, and 2nd order polynomials, best predicted the fit of values for the dependence of the resilient modulus on bulk stress.
The linear dependence of the resilient modulus on octahedral stress is presented in Figure 7. Linear dependence, compared to power, exponential, logarithmic, and 2nd order polynomial, best predicted the fit of values for the dependence of the resilient modulus on octahedral stress.
As can be seen from Figure 6 and Figure 7, the linear dependence of the resilient modulus on octahedral stress had a coefficient of determination from 0.85 to 0.99, while the power dependence of the resilient modulus on the bulk stress had a coefficient of determination only from 0.68 to 0.83. Based on these results, the authors of the paper propose the following Octahedral Shear Stress (OSS) prediction model:
Er = K1 τoct − K2
where K1 and K2 are regression coefficients which are provided for every tested specimen separately in Table 8.

4. Conclusions

After the determination of the resilient modulus values of variously graded sands (SP1, SP2, SP3) and well-graded sand (SG1, SG2, SG3), and predictive modeling using the models reviewed in the Materials and Methods chapter, the following conclusions can be drawn:
-
Using the K-θ model developed by Hicks and Monismith to predict resilient modulus, the coefficient of determination (R2) value using determined regression coefficients provided in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 ranges from 0.22 to 0.99;
-
Using the Rahim and George model to predict resilient modulus, the coefficient of determination (R2) value using determined regression coefficients provided in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 ranges from 0.80 to 0.99;
-
Using the Uzan model to predict resilient modulus, the coefficient of determination (R2) value using determined regression coefficients provided in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 ranges from 0.80 to 0.99;
-
Using the Universal Witczak model to predict resilient modulus, the coefficient of determination (R2) value using determined regression coefficients provided in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 ranges from 0.80 to 0.99;
-
The Octahedral Shear Stress model, proposed by the authors of the paper, predicts the resilient modulus with a coefficient of determination (R2) ranging from 0.85 to 0.99, using regression coefficients provided in Table 8. The advantage of the model is the use of small-scale data tables, meaning that fixed K1 and K2 regression coefficients can be assigned to a specific specimen type without the need to determine them using specific deviatoric and confining stresses. Additional investigation of the regression coefficient must be performed, separately taking into account different stress states of specimen to avoid overfitting as much as possible.
Authors hope that more advanced laboratory testing such as cyclic triaxial testing will be performed to derive more accurate prediction models or to calibrate existing models for the determination of the resilient modulus. Wider use of a resilient modulus determined by prediction models could potentially help to design road structures more accurately while saving time and expenditures.

Author Contributions

Conceptualization and methodology, T.T. and Š.S.; formal analysis, investigation, and data curation, T.T. and Š.S.; writing—original draft preparation T.T. and Š.S.; writing—review and editing, T.T. and Š.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data presented in this study are available on request.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Transportation Officials. Mechanistic-Empirical Pavement Design Guide: A Manual of Practice; AASHTO: Washington, DC, USA, 2008. [Google Scholar]
  2. EN 13286-7; Unbound and Hydraulically Bound Mixtures–Cyclic Load Triaxial Test for Unbound Mixtures. CEN: Brussels, Belgium, 2004.
  3. Transportation Officials. AASHTO Guide for Design of Pavement Structures; AASHTO: Washington, DC, USA, 1993. [Google Scholar]
  4. Kumar, P.; Puppala, A.J.; Tingle, J.S.; Chakraborty, S.; Sarat Chandra Congress, S. Resilient Characteristics of Polymer Emulsion-Treated Sandy Soil. Transp. Res. Rec. 2022, 2676, 526–538. [Google Scholar] [CrossRef]
  5. Kim, S.S.; Pahno, S.; Durham, S.A.; Yang, J.; Chorzepa, M.G. Prediction of Resilient Modulus from the Laboratory Testing of Sandy Soils; Georgia Department of Transportation, Office of Performance-Based Management and Research: Atlanta, GA, USA, 2019. [Google Scholar]
  6. Liu, X.; Zhang, X.; Wang, H.; Jiang, B. Laboratory testing and analysis of dynamic and static resilient modulus of subgrade soil under various influencing factors. Constr. Build. Mater. 2019, 195, 178–186. [Google Scholar] [CrossRef]
  7. Ackah, F.S.; Zhuochen, N.; Huaiping, F. Effect of wetting and drying on the resilient modulus and permanent strain of a sandy clay by RLTT. Int. J. Pavement Res. Technol. 2021, 14, 366–377. [Google Scholar] [CrossRef]
  8. Yaghoubi, E.; Yaghoubi, M.; Guerrieri, M.; Sudarsanan, N. Improving expansive clay subgrades using recycled glass: Resilient modulus characteristics and pavement performance. Constr. Build. Mater. 2021, 302, 124384. [Google Scholar] [CrossRef]
  9. Tuan, N.A.; Chieu, P.Q. The Effect of Moisture and Fine Grain Content on the Resilient Modulus of Sandy Clay Embankment Roadbed. Eng. Technol. Appl. Sci. Res. 2021, 11, 7118–7124. [Google Scholar] [CrossRef]
  10. Behiry, A.E.A.E.M. Fatigue and rutting lives in flexible pavement. Ain Shams Eng. J. 2012, 3, 367–374. [Google Scholar] [CrossRef]
  11. Sas, W.; Głuchowski, A.; Soból, E.; Bąkowski, J.; Szymański, A. Analysis of the multistage cyclic loading test on resilient modulus value. Ann. Wars. Univ. Life Sci.–SGGW-Land Reclam 2016, 48, 53–65. [Google Scholar] [CrossRef]
  12. Jin, M.S.; Lee, K.W.; Kovacs, W.D. Seasonal variation of resilient modulus of subgrade soils. J. Transp. Eng. 1994, 120, 603–616. [Google Scholar] [CrossRef]
  13. Filotenkovas, V.; Vaitkus, A. Influence of the aggregate shape and resistance to fragmentation on unbound base layer resilient modulus. Balt. J. Road Bridge Eng. 2022, 17, 104–119. [Google Scholar] [CrossRef]
  14. Titi, H.H.; Elias, M.B.; Helwany, S. Determination of Typical Resilient Modulus Values for Selected Soils in Wisconsin; Wisconsin Highway Research Program: Madison, WI, USA, 2006. [Google Scholar]
  15. Tamošiūnas, T.; Žaržojus, G.; Skuodis, Š. Indirect determination of soil Young’s modulus in Lithuania using cone penetration test data. Balt. J. Road Brodge Eng. 2022, 17, 1–24. [Google Scholar] [CrossRef]
  16. Fathi, A.; Tirado, C.; Rocha, S.; Mazari, M.; Nazarian, S. A Machine-Learning Approach for Extracting Modulus of Compacted Unbound Aggregate Base and Subgrade Materials Using Intelligent Compaction Technology. Infrastructures 2021, 6, 142. [Google Scholar] [CrossRef]
  17. Pahno, S.; Yang, J.J.; Kim, S.S. Use of machine learning algorithms to predict subgrade resilient modulus. Infrastructures 2021, 6, 78. [Google Scholar] [CrossRef]
  18. Skuodis, Š.; Karpis, R.; Zakarka, M.; Gedvilas, M.; Raginis, V.; Orlova, K.; Katauskas, M. Grunto, veikiamo periodinėmis apkrovomis, elgsenos tyrimai. Geol. Geogr. 2018, 4, 159–167. [Google Scholar] [CrossRef]
  19. LST 1331; Automobilių Kelių Gruntai. Klasifikacija [Soil for roads. Classification]. LSD: Vilnius, Lithuania, 2022.
  20. Daryati, D.; Widiasanti, I.; Septiandini, E.; Ramadhan, M.A.; Sambowo, K.A.; Purnomo, A. Soil characteristics analysis based on the unified soil classification system. J. Phys. Conf. Ser. 2019, 1402, 022028. [Google Scholar] [CrossRef]
  21. Zhang, J.; Peng, J.; Zeng, L.; Li, J.; Li, F. Rapid estimation of resilient modulus of subgrade soils using performance-related soil properties. Int. J. Pavement Eng. 2021, 22, 732–739. [Google Scholar] [CrossRef]
  22. Zhang, J.; Peng, J.; Liu, W.; Lu, W. Predicting resilient modulus of fine-grained subgrade soils considering relative compaction and matric suction. Road Mater. Pavement Des. 2021, 22, 703–715. [Google Scholar] [CrossRef]
  23. Hanandeh, S.; Ardah, A.; Abu-Farsakh, M. Using artificial neural network and genetics algorithm to estimate the resilient modulus for stabilized subgrade and propose new empirical formula. Transp. Geotech. 2020, 24, 100358. [Google Scholar] [CrossRef]
  24. Hicks, R.G. Factors Influencing the Resilient Properties of Granular Materials; University of California, Berkeley: Berkeley, CA, USA, 1970. [Google Scholar]
  25. Adomako, S.; Engelsen, C.J.; Thorstensen, R.T.; Barbieri, D.M. Repeated load triaxial testing of recycled excavation materials blended with recycled phyllite materials. Materials 2022, 15, 621. [Google Scholar] [CrossRef]
  26. Chowdhury, S.R.M. Evaluation of resilient modulus constitutive equations for unbound coarse materials. Constr. Build. Mater. 2021, 296, 123688. [Google Scholar] [CrossRef]
  27. Rahim, A.M.; George, K.P. Models to estimate subgrade resilient modulus for pavement design. Int. J. Pavement Eng. 2005, 6, 89–96. [Google Scholar] [CrossRef]
  28. Fedakar, H.I. Developing New Empirical Formulae for the Resilient Modulus of Fine-Grained Subgrade Soils Using a Large Long-Term Pavement Performance Dataset and Artificial Neural Network Approach. Transp. Res. Rec. 2022, 2676, 58–75. [Google Scholar] [CrossRef]
  29. Uzan, J. Characterization of granular material. Transp. Res. Rec. 1985, 1022, 52–59. [Google Scholar]
  30. Ni, B.; Hopkins, T.C.; Sun, L.; Beckham, T.L. Modeling the resilient modulus of soils. In Bearing Capacity of Roads, Railways and Airfields; CRC Press: Boca Raton, FL, USA, 2020; pp. 1131–1142. [Google Scholar]
  31. Yoder, E.J.; Witczak, M.W. Chapter 2: Stresses in flexible pavements. In Principles of Pavement Design; John Wiley and Sons Inc.: New York, NY, USA, 1975; pp. 24–76. [Google Scholar]
  32. Al-Dulaimi, Y.F.; Awed, A.M.; Gabr, A.R.; El-Badawy, S.M. Predicting Resilient Modulus of Unbound Granular Base/Subbase Material.(Dept. C). MEJ. Mansoura Eng. J. 2022, 47, 1–10. [Google Scholar] [CrossRef]
Figure 1. Test program—stress levels for the determinations of resilient behavior of soil specimens.
Figure 1. Test program—stress levels for the determinations of resilient behavior of soil specimens.
Infrastructures 08 00029 g001
Figure 2. Specimen with attached mounted LVDTs before the start of cyclic triaxial test.
Figure 2. Specimen with attached mounted LVDTs before the start of cyclic triaxial test.
Infrastructures 08 00029 g002
Figure 3. Fragment of results in graphs from a cyclic triaxial test of soil specimens.
Figure 3. Fragment of results in graphs from a cyclic triaxial test of soil specimens.
Infrastructures 08 00029 g003
Figure 4. Particle size distribution curves of soils under discussion.
Figure 4. Particle size distribution curves of soils under discussion.
Infrastructures 08 00029 g004
Figure 5. Obtained values of resilient modulus at a certain number of cycles.
Figure 5. Obtained values of resilient modulus at a certain number of cycles.
Infrastructures 08 00029 g005
Figure 6. Resilient modulus correlation with bulk stress.
Figure 6. Resilient modulus correlation with bulk stress.
Infrastructures 08 00029 g006
Figure 7. Resilient modulus correlation with octahedral shear stress.
Figure 7. Resilient modulus correlation with octahedral shear stress.
Infrastructures 08 00029 g007
Table 1. Classification of specimens according to LST 1331:2022 and USCS.
Table 1. Classification of specimens according to LST 1331:2022 and USCS.
Name of SpecimenCu CcSoil Classification
LST 1331:2022USCS
SP14.900.72Variously graded sand (SP)Silty sand (SM)
SP24.740.75Variously graded sand (SP)Poorly graded sand (SP)
SP34.600.99Variously graded sand (SP)Silty sand (SM)
SG18.291.21Well-graded sand (SG)Well-graded sand (SW)
SG217.691.39Well-graded sand (SG)Well-graded sand (SW)
SG36.231.07Well-graded sand (SG)Well-graded sand (SW)
Table 2. Determination of models’ average regression coefficients for specimen SP1.
Table 2. Determination of models’ average regression coefficients for specimen SP1.
Start CycleNo. of Cyclesσdσ3K-θRahim and GeorgeUzanUniversal Witczak
K1K2K1K2K1K2K3K1K2K3
20,10010020.2120.130.791.070.400.461.330.760.180.971.020.99
20,20010034.9820.410.961.170.730.762.160.860.031.780.941.12
20,30010049.8121.061.031.231.011.013.021.210.032.291.101.30
20,40010069.1322.131.051.251.231.253.081.670.042.211.361.43
20,50010035.0135.410.911.120.730.731.691.270.091.441.101.05
20,60010049.9135.450.991.160.920.921.951.500.111.611.221.11
20,70010069.6135.631.041.201.141.182.211.820.391.681.411.22
20,80010089.2235.791.041.221.301.372.131.890.811.631.451.25
20,900100116.9938.561.041.241.521.611.871.721.141.541.461.26
21,00010049.8950.300.991.090.850.811.631.490.471.191.121.04
21,10010069.8450.071.021.131.021.031.721.580.711.281.221.08
21,20010089.6149.981.031.161.181.231.641.630.901.331.291.12
21,300100118.9850.111.031.201.401.491.531.561.091.331.351.16
21,400100155.9553.271.021.231.601.801.381.471.191.301.381.19
21,50010070.0269.801.021.090.980.971.451.460.821.121.121.03
21,60010089.7070.441.021.121.111.161.421.470.941.181.191.06
21,700100119.5369.811.021.141.281.371.321.381.051.181.211.08
21,800100158.7569.911.031.191.511.681.251.341.121.201.271.12
21,900100196.4971.601.041.211.771.901.201.291.141.201.291.14
R20.9838350.9849070.9846690.984817
Table 3. Determination of models’ average regression coefficients for specimen SP2.
Table 3. Determination of models’ average regression coefficients for specimen SP2.
Start CycleNo. of Cyclesσdσ3K-θRahim and GeorgeUzanUniversal Witczak
K1K2K1K2K1K2K3K1K2K3
20,10010019.6720.701.021.080.540.501.750.760.131.430.901.04
20,20010034.4220.571.021.160.730.762.160.880.021.800.931.13
20,30010049.2520.581.041.210.970.972.821.180.032.161.081.24
20,40010069.3920.731.051.251.221.243.041.650.012.191.351.42
20,50010034.7435.711.031.130.810.751.861.290.021.601.181.09
20,60010049.6635.761.031.170.970.952.061.520.071.711.291.15
20,70010069.7335.761.041.201.151.192.221.830.391.691.421.22
20,80010089.5535.781.041.221.311.382.131.890.811.631.461.25
20,900100119.1935.731.041.241.521.611.851.691.141.531.451.26
21,00010049.8850.751.011.100.880.831.651.510.451.231.151.05
21,10010069.8151.111.031.151.061.081.781.650.671.341.281.10
21,20010089.7050.751.031.171.201.261.671.670.901.351.321.13
21,300100119.4451.211.041.201.421.521.541.581.091.351.371.17
21,400100159.0050.891.041.221.631.791.381.421.171.301.361.18
21,50010069.9270.741.021.101.000.991.471.500.811.151.151.04
21,60010089.8170.911.021.131.121.181.441.500.941.191.211.07
21,700100119.7971.201.031.161.321.421.361.431.061.211.261.10
21,800100159.5370.921.031.191.511.691.251.341.121.201.271.12
21,900100198.9271.311.031.201.761.871.181.261.121.181.261.12
R20.9955570.9955650.9955250.995596
Table 4. Determination of models’ average regression coefficients for specimen SP3.
Table 4. Determination of models’ average regression coefficients for specimen SP3.
Start CycleNo. of Cyclesσdσ3K-θRahim and GeorgeUzanUniversal Witczak
K1K2K1K2K1K2K3K1K2K3
20,10010020.0619.921.000.980.380.391.400.890.280.841.040.99
20,20010034.7420.061.031.140.710.731.960.910.001.670.951.10
20,30010049.8420.281.041.210.970.972.741.160.002.131.081.23
20,40010069.5320.201.051.241.211.222.981.610.022.141.331.39
20,50010034.9135.681.031.130.810.761.851.280.021.601.181.09
20,60010049.8335.741.031.170.970.962.071.520.061.711.291.15
20,70010069.6035.681.041.201.151.192.221.830.381.691.421.22
20,80010089.5535.681.041.221.311.382.131.890.811.631.461.25
20,900100119.5535.561.041.241.531.611.861.701.141.541.461.26
21,00010049.8850.901.021.100.890.831.661.520.441.241.161.05
21,10010069.8650.881.031.141.051.071.771.640.681.331.271.10
21,20010089.7051.161.031.171.211.281.681.690.891.371.331.14
21,300100119.7450.791.041.201.411.511.531.571.091.341.361.16
21,400100159.5150.801.041.221.631.791.381.421.171.301.361.18
21,50010069.9570.691.021.100.990.991.471.500.811.151.151.04
21,60010089.9271.231.021.131.131.191.441.500.941.201.221.07
21,700100119.8470.901.031.161.311.411.361.421.061.211.251.09
21,800100159.5571.211.031.191.521.691.251.341.121.211.271.12
21,900100199.6270.941.031.211.761.881.181.261.121.181.261.12
R20.9976920.9976660.9976710.997683
Table 5. Determination of models’ average regression coefficients for specimen SG1.
Table 5. Determination of models’ average regression coefficients for specimen SG1.
Start CycleNo. of Cyclesσdσ3K-θRahim and GeorgeUzanUniversal Witczak
K1K2K1K2K1K2K3K1K2K3
20,10010020.1120.040.651.040.310.411.100.780.290.721.100.97
20,20010034.7120.290.871.180.680.742.010.870.051.650.961.08
20,30010050.0120.250.961.220.970.972.841.180.062.131.071.22
20,40010069.9220.291.041.241.201.233.001.630.062.141.331.39
20,50010035.0735.363.071.120.650.721.511.270.081.281.051.03
20,60010050.0635.380.951.160.880.911.861.500.111.551.181.09
20,70010070.0035.641.031.191.121.162.171.790.421.631.391.21
20,80010089.8435.681.041.221.311.382.131.890.811.631.461.25
20,900100119.8535.681.041.241.541.631.871.721.141.551.471.27
21,00010050.1650.910.991.110.890.841.661.530.431.251.171.05
21,10010070.0950.951.021.141.051.081.761.640.681.331.271.10
21,20010089.9250.291.031.161.191.241.651.640.901.331.301.12
21,300100119.8850.511.031.201.401.501.531.561.091.331.351.16
21,400100159.8650.911.011.231.631.791.381.421.171.291.371.18
21,50010070.1470.580.991.070.960.971.411.460.831.111.111.03
21,60010090.0871.131.021.131.131.191.441.500.941.201.211.07
21,700100119.9370.989.721.151.311.421.361.431.061.271.241.09
21,800100160.0270.421.031.181.511.671.241.321.111.201.261.11
21,900100199.6270.711.031.201.761.871.181.251.121.181.261.12
R20.2447360.9917750.9918050.991622
Table 6. Determination of models’ average regression coefficients for specimen SG2.
Table 6. Determination of models’ average regression coefficients for specimen SG2.
Start CycleNo. of Cyclesσdσ3K-θRahim and GeorgeUzanUniversal Witczak
K1K2K1K2K1K2K3K1K2K3
20,10010020.2320.320.781.090.430.471.390.740.161.071.011.00
20,20010034.9020.380.921.180.730.762.160.870.041.770.951.11
20,30010049.9820.530.991.230.990.992.921.200.042.211.091.26
20,40010069.8221.031.051.261.251.273.121.710.012.241.381.45
20,50010035.1735.503.341.130.740.741.701.280.061.451.091.05
20,60010050.0135.510.981.170.940.932.001.500.101.641.231.12
20,70010069.9035.691.041.201.151.182.221.830.391.681.421.22
20,80010089.8235.741.041.221.311.392.131.900.811.641.461.25
20,900100119.5136.041.041.241.541.631.871.721.141.551.481.27
21,00010050.0250.270.951.100.840.811.611.480.491.171.111.03
21,10010069.8850.441.021.141.031.051.751.610.701.301.241.09
21,20010089.8150.381.031.171.191.251.661.650.901.341.301.12
21,300100119.7550.371.031.191.401.501.521.551.091.331.341.16
21,400100159.3851.401.021.241.651.831.401.461.181.311.401.20
21,50010069.8870.031.011.090.970.961.441.460.831.111.111.03
21,60010089.9470.341.021.121.111.161.411.460.941.171.181.06
21,700100119.8270.489.131.151.301.401.351.411.051.251.221.08
21,800100159.6470.571.031.191.511.681.241.331.111.201.261.11
21,900100199.3470.841.041.211.771.901.191.271.131.201.281.13
R20.2346740.9945740.9945930.994436
Table 7. Determination of models’ average regression coefficients for specimen SG3.
Table 7. Determination of models’ average regression coefficients for specimen SG3.
Start CycleNo. of Cyclesσdσ3K-θRahim and GeorgeUzanUniversal Witczak
K1K2K1K2K1K2K3K1K2K3
20,10010021.1819.940.611.010.320.411.130.820.510.731.090.97
20,20010034.9120.400.761.160.580.711.790.880.151.390.981.03
20,30010049.6520.320.831.200.810.932.441.170.191.771.061.17
20,40010069.7020.650.961.231.091.222.811.560.201.951.311.37
20,50010035.4935.623.491.120.680.711.581.260.121.321.081.04
20,60010050.5135.630.831.140.780.911.701.460.191.391.181.09
20,70010070.0535.700.911.180.981.151.941.700.481.421.371.20
20,80010089.5735.750.991.211.221.371.981.860.821.511.451.24
20,900100119.0735.770.991.231.421.581.721.681.141.431.431.24
21,00010050.3749.760.781.010.680.731.341.320.650.951.011.00
21,10010070.1750.610.921.090.911.011.531.540.741.151.171.06
21,20010090.0150.740.951.141.081.221.501.600.911.221.271.11
21,300100119.8050.870.991.191.321.501.431.561.091.261.341.16
21,400100159.5650.861.001.221.581.781.341.411.171.251.361.18
21,50010070.2370.540.870.990.830.891.231.330.880.961.021.00
21,60010090.1271.261.001.131.091.181.391.490.941.161.211.07
21,700100120.0971.329.721.051.141.281.181.291.041.111.111.04
21,800100159.6170.990.971.181.461.681.211.321.111.161.261.11
21,900100199.7871.170.931.171.591.791.071.201.091.061.211.10
R20.2214600.7995310.8005320.802060
Table 8. The OSS models’ regression coefficients for specimens.
Table 8. The OSS models’ regression coefficients for specimens.
Name of SpecimenSoil ClassificationOSS Model (ER = K1 τoct − K2)
LST 1331:2022USCSK1 K2R2
SP1Variously graded sand (SP)Silty sand (SM)16.0853.900.98
SP2Variously graded sand (SP)Poorly graded sand (SP)15.6019.460.99
SP3Variously graded sand (SP)Silty sand (SM)15.9137.240.85
SG1Well-graded sand (SG)Well-graded sand (SW)16.0956.290.99
SG2Well-graded sand (SG)Well-graded sand (SW)16.2455.740.99
SG3Well-graded sand (SG)Well-graded sand (SW)15.5877.730.85
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Tamošiūnas, T.; Skuodis, Š. Predictive Stress Modeling of Resilient Modulus in Sandy Subgrade Soils. Infrastructures 2023, 8, 29. https://doi.org/10.3390/infrastructures8020029

AMA Style

Tamošiūnas T, Skuodis Š. Predictive Stress Modeling of Resilient Modulus in Sandy Subgrade Soils. Infrastructures. 2023; 8(2):29. https://doi.org/10.3390/infrastructures8020029

Chicago/Turabian Style

Tamošiūnas, Tadas, and Šarūnas Skuodis. 2023. "Predictive Stress Modeling of Resilient Modulus in Sandy Subgrade Soils" Infrastructures 8, no. 2: 29. https://doi.org/10.3390/infrastructures8020029

APA Style

Tamošiūnas, T., & Skuodis, Š. (2023). Predictive Stress Modeling of Resilient Modulus in Sandy Subgrade Soils. Infrastructures, 8(2), 29. https://doi.org/10.3390/infrastructures8020029

Article Metrics

Back to TopTop