Engineering method for determining pile settlement taking into account the permissible elastic-plastic behaviour of the soil

Introduction. The feasibility of increasing the accuracy and generality of simplified solutions for calculating the settlement of single piles is shown. Such solutions are recommended by the building codes of the Russian Federation for assessing the design settlement of piles and are also used to determine the parameters of contact models when developing pile foundation structures. The limitations of existing methods for calculating single piles are noted.

Materials and methods. An analysis was carried out and an improvement was developed for an engineering methodology in which the settlement of a single pile is calculated using elastic dependencies, according to the data of a normative assessment of the load perceived by the friction of the soil against the lateral surface of the pile shaft, and the load transferred to the soil by the fifth pile under the working load on the pile. The ratio of these loads reflects the influence of the elastic-plastic work of the soil, in the range of the design load allowed on the pile.

Results. Based on the results of the presented studies, the field of application of the analytical calculation method based on the approximation of the pile behaviour in an elastic half-space is indicated — assessment of its initial linear resistance. Alternative engineering solutions to the problem of determining the pile settlement are considered taking into account the practical elastic-plastic work of the soil, in the range of the permissible design load on the pile. A refined solution to this problem has been developed, providing more practical results of pile settlement calculations that meet the concepts and empirical basis of the current design standards of the Russian Federation. Comparisons of the calculation results by various methods are presented. The previously developed analytical elastic and elastic-plastic engineering solutions substantiated in this paper’ describe the transformation of the initial linear resistance of the soil to its elastic-plastic resistance under the permissible design load on the pile. The combined application of these solutions ensures the development of a reliable and rational pile foundation design through calculation using a nonlinear contact model.

Conclusions. Simplified solutions for calculating the settlement of single piles are in demand in construction practice; increasing the accuracy and generality of such solutions ensures the determination of adequate parameters of contact models for developing reliable and rational designs of pile foundations. The developed engineering solution for determining pile settlement taking into account the practical elastic-plastic behaviour of the soil is advisable to use for the design assessment of pile settlement from the permissible working load on it and for determining the parameters of linear and nonlinear contact models. The adequacy of the developed solution is ensured by the use of a well-proven in geotechnical practice consideration of stress and strain concentration in soil areas adjacent to the load transfer point.

1. Alekseev A.G., Bezvolev S.G. Updating the linear telescopic pile settlement model. Vestnik MGSU [Monthly Journal on Construction and Architecture]. 2025; 20(5):718-733. DOI: 10.22227/1997-0935.2025.5.718-733. EDN WIOGQO.(rus.).

2. Fedorovsky V.G., Levachev S.N., Kurillo S.V., Kolesnikov Yu.M. Piles in hydraulic engineering: tutorial. Moscow, ASV Publishing House, 2003; 240. EDN QNKHAN. (rus.).

3. Bartolomey A.A., Omelchak I.M., Yushkov B.S. Forecast of settlement of pile foundations. Moscow, Stroyizdat, 1994; 384. (rus.).

4. Poulos H.G., Davis E.H. Pile Foundation Analysis and Design. Sydney, Rainhom Bridge Book Co., 1980; 398.

5. Fleming K., Weltman A., Randolph M., Elson K. Piling Engineering. 3rd ed. Boca Raton, FL, CRC Press Taylor & Francis Group, 2020; 406.

6. Randolph M.F., Wroth C.P. Analysis of deformation of vertically loaded piles. Journal of the Geotechnical Engineering Division. 1978; 104(12):1465-1488. DOI: 10.1061/ajgeb6.0000729

7. Randolph M.F. Design Methods for Pile Groups and Piled Rafts. Proceeding of the 13th International Conference on Soil Mechanics and Foundation Engineering (ICSMFE). 1994; 5:61-82.

8. Mylonakis G., Gazetas G. Settlement and additional internal forces of grouped piles in layered soil. Geotechnique. 1998; 48(1):55-72. DOI: 10.1680/geot.1998.48.1.55

9. Schleicher F. Zur Theorie des Baugrundes. Zeitschrift für Angewandte Mathematik und Mechanik. 1925. Bd. 5. S. 199; Bauingenieur. 1926. Bd. 7. Heft 48. S. 931-935. Heft 49. S. 949-952.

10. Giroud J.P. Tables pour le calcul des fondations. T.1. Tassement. Paris, Dunod, 1972; 383.

11. Ter-Martirosyan Z.G., Ter-Martirosyan A.Z., Ermoshina L.Yu. Settlement and long-term bearing capacity of the pile. Industrial and Civil Engineering. 2021; 9:18-23. DOI: 10.33622/0869-7019.2021.09.18-23. EDN HEIXNL.(rus.)

12. Bezvolev S.G. Analytical solution for calculating additional settlements of pile fields. Structural Mechanics and Calculation of Structures. 2012; 4(243):58-65. EDN PCOHRJ. (rus.).

13. Fedorovsky V.G., Bezvolev S.G. Method of calculation of pile fields and other vertical reinforced soil masses. Soil Mechanics and Foundation Engineering. 1994; 3:11-15. EDN KSYPMF. (rus.).

14. Bezvolev S.G. General procedure for analysis of the stress-strain state of a soil mass with strengthening or weakening elements. Soil Mechanics and Foundation Engineering. 2008; 3:2-8. EDN SKFDDP. (rus.).

15. Bokov I.A., Fedorovskii V.G. Taking into account the soil depth inhomogeneity in calculation of the piles settlement. Soil Mechanics and Foundation Engineering. 2021; 4:2-6. EDN QIVGYU. (rus.).

16. Banerjee P.K., Davies T.G. The behaviour of axially and laterally loaded single piles embedded in nonhomogeneous soils. Geotechnique. 1978; 28(3):309-326. DOI: 10.1680/geot.1978.28.3.309

17. Butterfield R., Banerjee P.K. The elastic analysis of compressible piles and pile groups. Geotechnique. 1971; 21(1):43-60. DOI: 10.1680/geot.1971.21.1.43

18. Slepak M.E. Rainfall and the VAT of plastic and frozen soil in the basis of the central loaded pile. Soil rheology and engineering permafrost science. 1982; 154-158. (rus.).

19. Rajapakse R.K.N.D. Response of an axially loaded elastic pile in a Gibson soil. Geotechnique. 1990; 40(2):237-249. DOI: 10.1680/geot.1990.40.2.237

20. Lee C.Y., Small J.C. Finite-Layer Analysis of Axially Loaded Piles. Journal of Geotechnical Engineering. 1991; 117(11):1706-1722. DOI: 10.1061/(asce)0733-9410(1991)117:11(1706)

21. Gotman N.Z. Determination of the shear strength of soil along the lateral surface of driven piles in numerical investigations. Soil Mechanics and Foundation Engineering. 2018; 6:8-13. EDN YWZHML. (rus.).

22. Sharafutdinov R.F., Razvodovskii D.E., Zakatov D.S. Single bored pile settlement prediction taking into account the elastic-plastic behavior of the soil. Soil Mechanics and Foundation Engineering. 2024; 3:7-15. EDN OFOAVE. (rus.).

23. Golubkov V.N. Issues of studying pile foundations and designing based on deformations : abstract of dissertation of Doctor of Technical Sciences. Moscow, 1969; 35. (rus.).

24. Tomlinson M., Woodward J. Pile Design and Construction Practice. 6-th edition. London, CRC Press Taylor & Francis Group, 2015. DOI: 10.1201/b17526

25. Poulos H.G. Tall Building Foundation Design. London, CRC Press Taylor & Francis Group, 2017. DOI: 10.1201/9781315156071

26. Barvashov V.A., Ekimyan N.B., Arshba E.T. Methods for assessing the bearing capacity of piles under vertical load. Moscow, VNIIIS, 1986; 69. (rus.).

27. Gotman A.L., Gavrikov M.D. Calculation of long bored piles for vertical loading. Soil Mechanics and Foundation Engineering. 2024; 4:2-8. EDN MSBJDN. (rus.).

28. Boussinesq J. Applications des Potentiels a l’Etitude de l’Equilibre et du Mouvement des Solides Elastiques. Paris, Gauthier-Villiars, Imprimeur-Libraire, 1885.

29. Silin K.S., Glotov N.M., Zavriev K.S., Luga A.A. Pile foundations. Moscow, Transport, 1975; 432. (rus.).

30. Geddes J.D. Boussinesq-Based Approximations to the Vertical Stresses Caused by Pile-Type Subsurface Loadings. Geotechnique. 1969; 19(4):509-514. DOI: 10.1680/geot.1969.19.4.509

31. Geddes J.D. Stresses in foundation soils due to vertical subsurface loading. Geotechnique. 1966; 16(3):231-255. DOI: 10.1680/geot.1966.16.3.231

32. Mindlin R.D. Force at a Point in the Interior of a Semi-Infinite Solid. Physics. 1936; 7(5):195-202. DOI: 10.1063/1.1745385

33. Fedorovskii V.G., Bezvolev S.G. Prediction of shallow-foundation settlements and selection of bed models for slab analysis. Soil Mechanics and Foundation Engineering. 2000; 4:10-18. EDN WESCFN. (rus.).

34. Efremov M.G., Konovalov P.A., Mikheev V.V. On the distribution of layer-by-layer soil deformations in the compressible thickness of clay and sandy foundations. Soil Mechanics and Foundation Engineering. 1963; 6:5-7. EDN XYWMZF. (rus.).

35. Florin V.A. Fundamentals of Soil Mechanics. Vol. 1: General dependencies and stress state of foundations of structures. Leningrad; Moscow, Gosstroyizdat, 1959; 357. (rus.).

36. Nishida Y. Vertical stress and vertical deformation of ground under a deep circular uniform pressure in the semi-infinite. Proceedings of the 1st International Society for Rock Mechanics Congress. 1966; 2:493-498

37. Smolyanskiy M.L. Tables of indefinite integrals. Moscow, Fizmatgiz, 1963; 112. (rus.).

38. Harr M.E., Lovell C.W. Vertical stresses under certain axisymmetrical loading. Highway Research Record. 1963; 39:68-81.

39. Bezvolev S.G. Method of accounting for the deformability of an inhomogeneous elastoplastic bed in analyzing foundation slabs. Soil Mechanics and Foundation Engineering. 2002; 5:8-14. EDN VZFWXG. (rus.).

40. Fedorovskiy V.G., Bezvolev S.G., Dunayeva O.M. Methodology for calculating foundation slabs on a nonlinearly deformable base. Nonlinear soil mechanics : proc. 4th Russian Conf. St. Petersburg, 1993; 81-86. (rus.).

41. Ladd C.C., Germaine J.T., Lancellotta R., Jamiolkowski M.B. New developments in Field and Laboratory Testing of Soils. 11th International Conference on Soil Mechanics and Foundation Engineering. 1985; 1:57-154.

42. Mossakovsky V.I. The main mixed problem of elasticity theory for a half-space with a circular line of separation of boundary conditions. Applied Mathematics and Mechanics. 1954; 18(2):187-196. (rus.).

43. Savinov A.V. Application of piles driven by pressing in the reconstruction of historical urban development : abstract of dis. … Doctor of Technical Sciences. Volgograd, 2008; 34. (rus.).

44. Klepikov S.N. Calculation of structures on an elastic foundation. Kyiv, Budivelnik, 1967; 184. (rus.).

45. Klepikov S.N. Calculation of structures on a deformable foundation. Kyiv, Research Institute of Building Structures, 1996; 203. (rus.).

46. Bezvolev S.G. Problems of designing and computing foundations when large groups of piles and other vertical elements of a ground massif transformation are used. Geotechnics. 2011; 3:30-67. EDN OWGMPZ. (rus.).