Refined mechanical and mathematical model of an elastic half-plane




elastic foundation, half-plane, displacements, stresses, deformation, distribution function


Loads cause vertical movements of the foundations of all the structures. Their magnitude determines the building safe operation. A closed analytical solution to the problem of the linear elastic theory for the distribution of stresses and strains in a homogeneous isotropic elastic foundation has been presented. The article considers the calculation of the stress-strain state of an elastic half-plane by the method in displacement functions. The theory of calculating an elastic half-plane has been built. New formulas have been are found that determine displacements and stresses at any points of an elastic foundation. An example of calculating an elastic half-plane under the action of a normal and tangential loads has been given.


Soil mechanics, bases and foundations / B.I. Dalmatov. | Leningrad: Stroiizdat, 1988. — 415 p.

Soil mechanics / N.A. Tsytovich. | Moscow: Gosstroiizdat, 1963. — 636 p.

The calculation of the deformation of the buildings and structures / S.G. Kushner. | Zaporozh'e: PLC «IPOZaporozh'e», 2008. — 496 p.

Complex variable methods in plane elasticity / Lu Jian-ke. | World Scientific, 1995. — 237 p.

On complex variable method in finite elasticity / A. Akinola // Applied Math. — 2009. — Vol. 1. — P. 1–16.

Analytical Methods in Geomechanics / K.T. Chau. | CRC Press, 2012. — 424 p.

Methods of the theory of functions of a complex variable in problems of geomechanics / A.N. Bogomolov, A.N. Ushakov. | Volgograd: Peremena, 2014. — 227 p.

A complex variable solutions for a deforming circular tunnel in an elastic half-plane / A. Verruijt // Numerical and Analytical Methods in Geomechanics. — 1997. — Vol. 21, No. 2. — P. 77-89.<77::AID-NAG857>3.0.CO;2-M DOI:<77::AID-NAG857>3.0.CO;2-M

Stress-strain state of an elastic half-plane at a linear shift of a part of its boundary / A.N. Bogomolov, A.N. Ushakov // Vestnik MGSU. — 2017. — Vol. 12, No. 2. — P. 184–192. DOI:

Representation of incomplete contact problems by half-planes / H. Andresen, D. Hills, M. Moore // Eur. J. Mech. A Solids. — 2020. — 85:104138. DOI:

Closed-form solutions for tilted three-part piecewise-quadratic half-plane contacts / H. Andresen, D.A. Hills, J. Vazquez // Int. J. Mech. Sci. — 2019. — 150. — P. 127–134. DOI:

Half-plane partial slip contact problems with a constant normal load subject to a shear force and differential bulk tension / M. Moore, R. Ramesh, D. Hills, et al. // J. Mech. Phys. Solids. — 2018. — Vol. 118. — P. 245–253. DOI:

Explicit equations for the half-plane sub-surface stress field under a flat rounded contact / J. Vazquez, C. Navarro, J. Domınguez // J. Strain Anal. Eng. Des. — 2014.— Vol. 49, No. 8. — P. 562–570. DOI:

Asymptotic analysis of stresses in an isotropic linear elastic plane or half-plane weakened by a finite number of holes / J. Kratochvil, W. Becker // Arch. Appl. Mech.— 2012.— Vol. 82.— P. 743–754. DOI:

Development of a distributed dislocation dipole technique for the analysis of multiple straight, kinked and branched cracks in an elastic half plane / M.W. Hallback, N. Tofique // Int. J. Solids Struct. — 2014. — Vol. 51. — P. 2878–2892. DOI:

Stress intensity factor for the interaction between a straight crack and a curved crack in plane elasticity / M.R. Aridi, N.M.A. Nik Long, Z.K. Eshkuvatov / Acta Mech. Solida Sin. — 2016. — Vol. 29. — P. 407–41. DOI:

Evaluation of the T-stress for multiple cracks in an elastic half-plane using singular integral equation and Green’s function method / Y.Z. Chen // Appl. Math. Comput. — 2014. — Vol. 228. — P. 17–30. DOI:

Fundamentals of the elastic and plastic theory / A.V. Alexandrov, V.D. Potapov. | M.: High School, 1990. — 400 p.

Mechanics of flat and spatial structures / K.A. Tursunov. | Karaganda: KTU Publishing House, 2010. — 130 p.




How to Cite

Akhazhanov, S., Baltabai, D., & Nurlanova, B. (2022). Refined mechanical and mathematical model of an elastic half-plane. Technobius, 2(1), 0014.