3D-shell mathematical models and finite elements

2013 ◽  
pp. 21-25
Author(s):  
Dominique Chapelle
Keyword(s):  
Finite Elements ◽  
Mathematical Models

scholarly journals MIXED AND COMPATIBLE FINITE ELEMENTS IN THE ANALYSIS PROBLEM OF ELASTOPLASTIC STRUCTURES

1996 ◽  
Vol 2 (5) ◽  
pp. 29-43
Author(s):  
S. Kalanta
Keyword(s):  
Mathematical Model ◽  
Finite Elements ◽  
Mathematical Models ◽  
Mixed Finite Elements ◽  
Analysis Problem ◽  
Energy Principle ◽  
Complementary Energy Principle ◽  
Initial Strains ◽  
Discrete Mathematical Model ◽  
Yield Conditions

A problem of ideal elastoplastic structures stress-strain field determination is considered The dual general and discrete mathematical models of analysis problem are made on the basis of the extremal energy principles and finite element method. In these models the possible discontinuity of displacements and the dissipation of energy in the place of those discontinuities, also the different external effects (load, initial strains and support settlements) are estimated. At first, on the basis of the mixed functional and mixed finite elements the discrete expressions of fundamental relationships (geometric equations, yield conditions) and the discrete mathematical model of mixed formulation of the problem are made. This mathematical model corresponds to the minimum total energy principle for a kinematically admissible displacements. The dual static formulation of the problem is obtained by Lagrange's multipliers method; this corresponds to the minimum complementary energy principle. The kinematic formulation of the problem is obtained in the case of linear yield conditions. These mathematical models permit to determine the lower values of the stress and displacements of structures. It has shown that the approximation of geometric equations and yield conditions by Bubnov-Galiorkin's collocation method gives the more accurate results.

scholarly journals THE PROBLEMS OF LIMIT LOAD ANALYSIS AND OPTIMIZATION USING EQUILIBRIUM FINITE ELEMENTS

1996 ◽  
Vol 2 (7) ◽  
pp. 13-23
Author(s):  
Stanislovas Kalanta
Keyword(s):  
Energy Dissipation ◽  
Finite Elements ◽  
Mathematical Models ◽  
Limit Load ◽  
Bending Moments ◽  
Equilibrium Equations ◽  
External Power ◽  
Load Analysis ◽  
Equilibrium Finite Elements ◽  
Yield Conditions

The equilibrium dual discrete mathematical models of the problems of limit load analysis and optimization are investigated in the article. These models are presented in terms of static and kinematic formulation using equilibrium finite elements. In these mathematical models the possible discontinuities of displacement velocities are evaluated and the velocity of energy dissipation is estimated not only within the volume of finite elements, but at the plastic surfaces between elements. At first, on the basis of the energy principle of the maximum external power [1,2] the general mathematical models (3) and (7) of static formulation of limit load analysis and optimization problems are created. In these models the yield conditions are controlled not only within the volume, but also at the surfaces of finite elements. The equilibrium finite elements and interpolation functions of strains (9) are used for discretization of these models. The constancy of external power is taken as the optimum criterion. The discrete expressions of fundamental relationships—equilibrium and geometric equations, yield conditions (10)-(12) for finite element and (14)-(17) for the discrete model of a body are developed. The discrete expressions of yield conditions are given using the classic collocation methods: collocation at the point, collocation at the sphere (element) and Bubnov-Galiorkin's collocation method [11,12]. The equilibrium equations of discrete structure are developed on the basis of virtual displacement principle while geometrical equations are derived using virtual force principle. In contrast to the approach of other authors, yield conditions and geometrical equations are described not only within finite elements, but also at the surfaces between elements. That helps to design the dual discrete mathematical models of the problems (21)-(26), in which the discontinuities of displacement velocities and the velocity of energy dissipation in the place of those discontinuities are estimated. The mathematical models (22), (24), (26), (29) and (32) of kinematic problem formulation are developed from sensible static formulations by Lagrange's multiplier method. The modified mathematical models (27)-(29) are presented. In these models the equilibrium equations are eliminated or the geometrical equations are transformed into compatible equations of plastic stress velocities, in this way decreasing the number of equations and unknown values. The dependence of the numerical results (limit load) of the frame on the approximation degree of bending moments, as well as on the discretization method of yield conditions are illustrated. In table 1 the values of limit loading parameter F 0 and their error of calculation ΔF 0 (per cent, in comparison with the analytic solution F 0 = 0,4662M 0) are presented. They are given for the first and second order finite elements with linear and parabolic distribution of bending moments using different discrete yield conditions and a different number of finite elements. The numerical result shows, that the discretization of yield conditions by Bubnov-Galiorkin's method gives the best accuracy and stable solutions. By discretizing yield conditions using the point's collocation and collocation at the element, the accurancy of numerical results depends not only on the number of elements, but also on a more or less successful choice of finite elements net.

scholarly journals DUAL MATHEMATICAL MODELS OF LIMIT LOAD ANALYSIS PROBLEMS OF STRUCTURES BY MIXED FINITE ELEMENTS

1997 ◽  
Vol 3 (10) ◽  
pp. 43-51
Author(s):  
Stanislovas Kalanta
Keyword(s):  
Finite Elements ◽  
Mathematical Programming ◽  
Mathematical Models ◽  
Optimization Problems ◽  
Mixed Finite Elements ◽  
Limit Load ◽  
Parameter Analysis ◽  
Lagrangian Multipliers ◽  
Multipliers Method ◽  
Load Analysis

The general and discrete dual mathematical models of the limit load analysis and optimization problems of rigid-plastic body are created in the article. The discrete models are formulated by mixed finite elements and presented in terms of kinematic and static formulation. In these models the velocity of the energy dissipation is estimated not only within the volume of finite elements, but also at the plastic surfaces between elements, where the discontinuities of displacement velocities functions appear. The theory of plastic flow, the theory of duality and mathematical programming are applied. The mixed energy functional (1) and (3) of both problems are formulated using the general static formulations of these problems, presented in the article [10], and Lagrangian multipliers method. The mixed finite elements are used for their discretization. The discrete expressions (8), (9) and (13) of mixed functionals are given choosing the interpolation functions (7) for the stress, displacement velocities, plastic multipliers and external load. Stationary conditions are created by static variables (stress and load vectors) of theses functionals. The discrete expressions of the geometric compatibility equations and constraint of load power are received from them. Using them as preliminary conditions for the functionals (8) and (9), the mathematical models (14), (15) and (17) of kinematic formulation of limit load analysis and optimization problems are formulated. The model (20) with a smaller number of unknowns is formed by elimination the displacement velocities. Using Lagrangian multipliers method, the mathematical models (21)-(23) of static formulation for the limit load parameter analysis problem and the models (24)-(26) for the load optimization problem are derived. All of them are the problems of mathematical programming. The mathematical models of static formulation for engineering purposes are more important and fit better. They are easier solved (a smaller quantity of unknowns), besides, they allow to determine the optimum distribution of the load. The formulated mathematical models allow to determine upper values of limit load, stresses, displacement and plastic multipliers velocities. Together with equilibrium models of these problems, presented in the article [10], they allow to determine the lower and upper values of aforementioned parameters. So, a good possibility is created to check reliability and exactness of numerical calculation results and to establish, if the computing net density of finite elements is sufficient.

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