# Mathematical modeling of unsteady elastic stress waves in a console with a base (half-plane) under fundamental seismic action

## Abstract

The aim of the work is to consider the problems of numerical modeling of seismic safety of the console with the base in the form of an elastic half-plane under unsteady wave influences. Stress waves of different nature, propagating in the deformed body interact with each other. After three or four times the passage and reflection of stress waves in the body, the process of propagation of disturbances becomes steady, the body is in oscillatory motion. The problem of modeling problems of the transition period is an actual fundamental and applied scientific problem. Methods. The finite element method in displacements is used to solve the two-dimensional plane dynamic problem of elasticity theory with initial and boundary conditions. On the basis of the finite element method in displacements, an algorithm and a set of programs for solving linear plane two-dimensional problems have been developed, which allow solving problems with non-stationary wave effects on complex systems. The algorithmic language “Fortran-90” was used in the development of the complex of programs. The study area is divided by spatial variables into finite elements of the first order. According to the time variable, the study area is also divided into finite elements of the first order. Results. The problem of the influence of a plane longitudinal elastic wave in the form of a Heaviside function on a console with a base (the ratio of width to height is one to ten) is considered. The initial conditions are taken as zero. The system of equations from 16 016 084 unknowns is solved. Contour stresses and stress tensor components are obtained in characteristic areas of the problem. On the basis of the conducted researches it is possible to draw the following conclusions: the console (the ratio of width to height one to ten) is modeled with the elastic basis in the form of an elastic half-plane; the elastic contour stresses on the faces of the console are almost a mirror image of one another, that is, antisymmetric; the console under seismic action works as a rod of variable cross-section, that is, if there are tensile stresses on one face, then compressive stresses on the other; on the contours of the console under seismic action, bending waves mainly prevail.

## Keywords

### Vyacheslav K. Musayev

Russian University of Transport; Mingachevir State University

Author for correspondence.
Email: musayev-vk@yandex.ru

Doctor of Technical Sciences, Professor, Professor of the Department of Technosphere Safety of the RUT (MIIT)

9 Obraztsova St., bldg. 9, Moscow, 127994, Russian Federation; Dilyara Alieva St., Mingachevir, AZ4500, Republic of Azerbaijan

## References

1. Kolsky G. (1955). Volny napryazhenij v tverdyh telah [Stress waves in solids]. Moscow, Inostrannaya literatura Publ. (In Russ.)
2. Zenkevich O. (1975). Metod konechnyh ehlementov v tekhnike [Finite element method in engineering]. Moscow, Mir Publ. (In Russ.)
3. Timoshenko S.P., Gudyer D. (1975). Teoriya uprugosti [Theory of elasticity]. Moscow, Nauka Publ. (In Russ.)
4. Musaev V.K. (2009). O modelirovanii sejsmicheskoj volny parallel'noj svobodnoj poverhnosti uprugoj poluploskosti [On modeling of a seismic wave parallel to the free surface of an elastic half plane]. Structural Mechanics of Engineering Constructions and Buildings, (4), 61–64. (In Russ.)
5. Spiridonov V.P. (2105). Opredelenie nekotoryh zakonomernostej volnovogo napryazhennogo sostoyaniya v geoob"ektah s pomoshch'yu chislennogo metoda, algoritma i kompleksa programm Musaeva V.K. [The definition of some patterns of wave stress in geoobject using numerical method, algorithm and program complex of Musayev V.K.]. Sovremennye naukoemkie tekhnologii, (12–5), 832–835. (In Russ.)
6. Dikova Ye.V. (2016). Dostovernost' chislennogo metoda, algoritma i kompleksa programm Musaeva V.K. pri reshenii zadachi o rasprostranenii ploskih prodol'nyh uprugih voln (voskhodyashchaya chast' – linejnaya, niskhodyashchaya chast' – chetvert' kruga) v poluploskosti [Reliability of the numerical method, algorithm and software complex of Musayev V.K. in solving the problem of propagation of plane longitudinal elastic waves (the ascending part is linear, the descending part is a quarter of a circle) in a half-plane]. Mezhdunarodnyj zhurnal ehksperimental'nogo obrazovaniya, (12–3), 354–357. (In Russ.)
7. Starodubtsev V.V., Akatyev S.V., Musayev A.V., Shiyanov S.M., Kurantsov O.V. (2017). Modelirovanie uprugih voln v vide impul'snogo vozdejstviya (voskhodyashchaya chast' – chetvert' kruga, niskhodyashchaya chast' – chetvert' kruga) v poluploskosti s pomoshch'yu chislennogo metoda Musaeva V.K. [Modeling of elastic waves in the form of impulse action (ascending part – a quarter of a circle, descending part – a quarter of a circle) in a half-plane by means of the numerical method of Musayev V.K.]. Problemy bezopasnosti rossijskogo obshchestva, (1), 36–40. (In Russ.)
8. Starodubtsev V.V., Akatyev S.V., Musayev A.V., Shiyanov S.M., Kurantsov O.V. (2017). Modelirovanie s pomoshch'yu chislennogo metoda Musaeva V.K. nestacionarnyh uprugih voln v vide impul'snogo vozdejstviya (voskhodyashchaya chast' – chetvert' kruga, srednyaya – gorizontal'naya, niskhodyashchaya chast' – linejnaya) v sploshnoj deformiruemoj srede [Modeling of unsteady elastic waves in the form of pulse action (ascending part – a quarter of a circle, the middle part – horizontal, the descending part – linear) in a continuous deformable medium using the Musayev V.K. numerical method]. Problemy bezopasnosti rossijskogo obshchestva, (1), 63–68. (In Russ.)
9. Kurantsov V.A., Starodubtsev V.V., Musayev A.V., Samoylov S.N., Kuznetsov M.E. (2017). Modelirovanie impul'sa (pervaya vetv': voskhodyashchaya chast' – chetvert' kruga, niskhodyashchaya chast' – linejnaya; vtoraya vetv': treugol'nik) v uprugoj poluploskosti s pomoshch'yu chislennogo metoda Musaeva V.K. [Modeling of momentum (the first branch: the ascending part – a quarter of a circle, the descending part – linear; the second branch: a triangle) in an elastic half-plane using the numerical method of Musayev V.K.]. Problemy bezopasnosti rossijskogo obshchestva, (2), 51–55. (In Russ.)
10. Musaev V.K. (2017). Primenenie volnovoj teorii sejsmicheskogo vozdejstviya dlya modelirovaniya uprugih napryazhenij v Kurpsajskoj plotine s gruntovym osnovaniem pri nezapolnennom vodohranilishche [Application of the wave theory of seismic action for modeling elastic stresses in the Kurpsay dam with a soil base in an unfilled reservoir]. Geologiya i geofizika Yuga Rossii, (2), 98–105. (In Russ.)
11. Musayev V.K. (2019). Mathematical modeling of non-stationary elastic waves stresses under a concentrated vertical exposure in the form of delta functions on the surface of the half-plane (Lamb problem). International Journal for Computational Civil and Structural Engineering, 15(2), 111–124.