Some specificities of the finite difference approximation of the boundary conditions of conjugation of elements of complex structures for the solution of nonlinear initial-boundary value problems

Аuthors

Dmitriev V. G.^*, Egorova O. V.^**, Rabinsky L. N.^***, Roffe A. I.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: vgd2105@mail.ru
**e-mail: janus_olga@mail.ru
***e-mail: f9_dec@mai.ru

Abstract

Some new methods of construction of conservative finite difference schemas and iteration procedures are presented. The mentioned methods are developed for the numerical simulation of static and dynamic deforming of compound cage structures that are widely used as supporting frames for aerospace complexes, for instance in launchers. It is supposed that the frames are made from the reinforced concrete; this fact does not reduce the applicability domain of the developed mathematical models. The Timoshenko’s beam model is used as a base element of the compound structure, therefore the effect of the low shear rigidity that is common for various structural materials (fibrous composites, reinforced concrete, e t. c.) can be efficiently accounted. The mathematical model is based on the nonlinear relationships of the middle flexure beams theory, the hypothesis of elastic behavior for the concrete, and the one of elastic-plastic behavior for the concrete reinforcement. The deformation plasticity theory is used to consider the elastic-plastic strain state of the reinforcement; the plastic strains initiation is modeled by the Mises yielding condition while the unloading is elastic. In the same time the initiation of cracks in the concrete is considered as well as their closure.

The Lagrange principle is used to derive the equilibrium equations for the compound structures, and the dynamics equation are constructed using the Hamilton’s variation principle. Various types of structural element conjugation are considered and the corresponding formulations of the boundary conditions are introduced.

To construct the discrete analog of the stated problem the finite difference approach is used for the spatial and time variables. The conservatism of the resulting schema and the convergence of solutions to exact ones are secured by the variation difference method. The accurate finite difference approximations for different conjugations of structural elements and the initial conditions are presented. The use of the quasi-dynamic formulation of the pseudoviscosity method for the static strain state of the compound structures allows one to construct the single-type iteration procedure for solution of both linear and nonlinear problems. The estimation of the optimum parameters of the iteration procedure are obtained and the convergence acceleration algorithm for the pseudoviscosity method is proposed for nonlinear static problems. In the same time the approximation of the second order time derivatives in the grid equations of motion by the finite difference operators of the second order of accuracy allows one to construct the unified finite difference schema for static and dynamic problems. This property of the developed method is useful for the computation of reinforced concrete structures of launchers subjected to the dynamic loads because of the possibility of efficient accounting of the initial static stress state induced by the gravitation.

The proposed algorithm of numerical solution of nonlinear initial-boundary value problems allows one to estimate the residual bearing capacity of compound frame structures after intensive dynamic loading of various type including the seismic ones.

Keywords:

mathematical modeling, finite difference method, pseudoviscosity method, nonlinear deformation, composite reinforced concrete structures, approximation, stability, convergence

References

1. Dmitriev V.G. Materialy IV Mezhdunarodnogo seminara «Tekhnologicheskie problemy prochnosti». Podol’sk, MGOU, 1997. pp. 57-67.

2. Dmitriev V.G., Egorova O.V., Rabinsky LN., Roffe A.I. Mekhanika kompozitsionnykh materialov i konstruktsii, 2014, Vol. 20, no.3, pp. 364 — 374.

3. Dmitriev V. G., Sudyin A. A. Deformation of reinforced concrete spherical dome with cutouts on the damped foundation beds. — Int. Journal for Computational Civil and Structural Engineering. 2009. (1&2), no. 5, pp. 13-22.

4. Dmitriev V. G. Mathematical Modeling of Non-Linear Deformation Process for Frame-Type Building Structures Under Seismic Loads. — Int. Journal for Computational Civil and Structural Engineering. 2012. Volume 8, Issue 2, pp. 13-29.

5. Dmitriev V.G., Roffe A.I., Sudyin A.A. Materialy XIX Mezhdunarodnogo seminara «Tekhnologicheskie problemy prochnosti», Podolsk, MGOU, 2012, pp. 37-44.

6. Ilyushin A.A. Mekhanika sploshnoy sredy (Mechanics of Solids), Moscow, MGU, 1990, 310 p.

7. Bakhvalov N.S., Zhidkov N.P., Kobelkov G.M. Chislennye methody (Numerical Methods), Moscow, Fizmatlit, 2001, 632 p.

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