# Analysis of non-stationary reaction of elastic shell to impulse load

## Authors

• Olga Krivenko Kyiv National University of Civil Engineering and Architecture, Ukraine
• Yurij Vorona Kyiv National University of Civil Engineering and Architecture, Ukraine

## Keywords:

thin elastic shell, universal solid finite element, reduced model, impulse load

## Abstract

An effective numerical method for studying non-stationary vibrations of thin elastic shells is proposed. The method is based on the finite element model of a thin elastic inhomogeneous shell and the reduced model created on its basis for the dynamics problems.

The finite-element shell model is based on the relations of the three-dimensional theory of thermo-elasticity and is developed with the use a tensor calculus apparatus, a geometrically nonlinear formulation of the problem in increments and the application of the moment finite-element scheme. To develop the finite-element shell model we approximate a thin shell by one spatial finite element throughout the thickness which is an efficient approach. The structural elements of an inhomogeneous shell require the finite element to be universal: it should be eccentrically arranged relative to the mid-surfaces of the casing (of the shell’s sections without stepwise-variable thickness), it should be possible to vary the thickness of the lateral edges of the finite element; the lateral edges of the neighboring finite elements should be in continuous contact; and it should be possible to model sharp bends and the multilayer structure of the shell. The universal finite element is based on an isoparametric spatial finite element with polylinear shape functions for coordinates and displacements. Additional variable parameters are introduced to enhance the capabilities of the modified finite element. Two hypotheses are used to describe the features of the stress–strain state of a thin inhomogeneous shell. The first static hypothesis assumes that the compressive stresses in the fibers throughout the thickness are constant. The next is the nonclassical kinematic hypothesis of deformed straight line: though stretched or shortened during deformation, a straight segment along the thickness remains straight. This segment is not necessarily normal to the mid-surface of the shell.

The method for studying non-stationary vibrations of the shells under the action of short-term loads is based on the application of reduced models. The use of the basic nodes method allowed us to develop a simple and effective algorithm for solving this problem. We have transformed a system of coupled differential equations describing the motion of a shell to independent ones. The solution of obtained Cauchy problems is easily found by the well-developed Runge-Kutta numerical method.

The possibility of applying the developed method to assess the effect of short-term load on the behavior of a thin-walled structure is shown on the test problems. Convergence of solutions is investigated and a comparison with theoretical data and results obtained with the help of the SCAD software is made.

## Author Biographies

### Olga Krivenko, Kyiv National University of Civil Engineering and Architecture

кандидат технічних наук, старший науковий співробітник, провідний науковий співробітник НДІ будівельної механіки КНУБА

### Yurij Vorona, Kyiv National University of Civil Engineering and Architecture

кандидат технічних наук, доцент, професор кафедри будівельної механіки КНУБА

## References

Bazhenov V.A., Kryvenko O.P., Solovei M.O. Neliniine deformuvannia ta stiikist pruzhnykh obolonok neodnoridnoi struktury (Nonlinear deformation and stability of elastic shells with inhomogeneous structure). – K.: ZAT «Vipol», 2010. – 316 s.

Bazhenov V.A., Krivenko O.P., Solovey N.A. Nelineynoe deformirovanie i ustoychivost uprugih obolochek neodnorodnoy strukturyi: Modeli, metodyi, algoritmyi, maloizuchennyie i novyie zadachi (Nonlinear deformation and stability of elastic shells with inhomogeneous structures: Models, methods, algorithms, poorly-studied and new problems). – M.: Knizhnyiy dom «LIBRIKOM», 2013. – 336 s.

Bazhenov V., Krivenko O. Buckling and Natural Vibrations of Thin Elastic Inhomogeneous Shells. – LAP LAMBERT Academic Publishing. Saarbruken, Deutscland, 2018. – 97 p.

Kryvenko O.P., Lehostaiev A.D., Hrechukh N.A. Analiz vlasnykh kolyvan obolonok neodnoridnoi struktury z vykorystanniam redukovanykh skinchennoelementnykh modelei (Analysis of natural vibrations of shells with inhomogeneous structure using reduced finite element models)// Opir materialiv i teoriia sporud: nauk.-tekh. zbirn. – K.: KNUBA, 2017. – Vyp. 98. – S. 72-88.

Chybiriakov V.K., Kryvenko O.P., Lehostaiev A.D, Hrechukh N.A. Deformuvannia pruzhnykh neodnoridnykh obolonok pid diieiu nestatsionarnykh dynamichnykh navantazhen (Deformations of elastic inhomogeneous shells under the action of non-stationary dynamic loads) // Opir materialiv i teoriia sporud: nauk.-tekh. zbirn. – K.: KNUBA, 2017. – Vyp. 99. – S. 123-141.

Golovanov A.I., Tyuleneva O.N., Shigabutdinov A.F. Metod konechnyih elementov v statike i dinamike tonkostennyih konstruktsiy (Finite element method in statics and dynamics of thin-walled structures). – M.: FIZMATLIT, 2006. - 392 s.

ScadSoft: Svobodno opertaya balka s raspredelennoy massoy pod deystviem ravnomerno raspredelennogo mgnovennogo impulsa (udar balki o nepodvizhnyie oporyi) (A simply supported beam with a distributed mass under the action of a uniformly distributed instantaneous pulse (impact of a beam on fixed supports) [Elektroniy resurs]. – Rezhim dostupu http://scadsoft.com/tests_scad/index/test/68.

Rabinovich I.M., Sinitsyin A.P., Luzhin O.V., Terenin B.M. Raschet sooruzheniy na impulsivnyie vozdeystviya (Calculation of constructions at impulsive effects). – M.: Iz-vo po stroitelstvu, 1970. – 303 s.