MIXED FINITE ELEMENT FOR THE DYNAMIC ANALYSIS OF ORTHOTROPIC FLEXIBLE SHALLOW SHELLS

A fi nite-element methodology for studying the forced oscillations of orthotropic fl exible shallow shells relative to the initial deformed state defi ned on the basis of a geometrically nonlinear deformation theory is proposed. To derive the fi nite-element equations, the Galerkin method is used in combination with the mixed formulation of the problem. The fi nal fi nite-element equations have a simple structure and numerical integration is not required for calculating the matrices and vectors of fi nite elements. The accuracy and convergence of the mixed fi nite element is analyzed. Based on the developed methodology, the infl uence of geometric nonlinearity on the process of shell oscillations is studied.


INTRODUCTION
Finite element methodology for fl exible shallow shells is proposed based on orthotropic material model. In the obtained fi nite-element equations it is possible to set independently of each other the tensile-compressive stiffness and the bending stiffness of the element. This makes it possible to use this fi nite element to calculate ribbed, multilayer and reinforced concrete shells after calculating their reduced characteristics. The derivation of fi nite-element equations in mixed formulation is carried out using by the Galerkin method in the weak form [1][2][3][4]. This method allows to do without the construction procedure of the functional of problem and to solve equations of problem in the form in which they are written. Using mixed fi nite-element formulation, there is a decrease of the condition number of the matrix of the fi nite element and reducing the accumulation of rounding errors, compared with the fi nite-element in displacements formulation [05]. All this allows the use of linear shape functions to construct the fi nite element and to obtain the matrix and vectors of the fi nite element in explicit form. This allows us not to use numerical integration, which improves the accuracy of calculations.

DESCRIPTION OF THE FINITE ELEMENT CALCULATION METHODOLOGY
Equations of the forced oscillations of the orthotropic shell based on the quadratic nonlinear theory of shallow shells [01, 02] have a form: We introduce the dimensionless variables: where C -a characteristic value of one of the coeffi cients C 1 ; and -dimensionless radial coordinate and dimensionless time; -stress function; -function of the angles of rotation of shell's generating curve; -stress function; w -defl ection function; q -projection of the distributed loading on the z-axis coordinate; -generatrix function for shell of revolution; r -radial coordinate; t -time; g -acceleration of gravity; h -thickness of the shell; E i -elastic modulus, D i -bending stiffness of shell, i -Poisson ratio. Here, the value of the index i=1 corresponds to the radial direction, i=2 -corresponds to the circumferential direction. Load acting on the shell can be represented as the sum of the static and dynamic component: 3)

Original Scientifi c Paper
We represent the functions and as a sum of components defi ning the initial state of the shell ( and ) and components describing deviation of the system in the process of oscillation ( and ): Write the function the infl uence of inertial forces in form: Equations (5) linearize the small deviations of the system relation to the initial equilibrium state [02, 06, 07]. The system of equations is separated into system, which defi nes the initial deformed state the shell and the system describing the oscillations of a shell. The fi rst system of equations is similar to the one in [01, 02]. Second, after linearization written as: The last of the equations in (5) is obtained by double differentiation on and replacement according to (2). Support contour is modeled as the elastic fi xing, which is a uniform representation of the different types of supports. The boundary conditions of the problem are taken in the following form: For the elastic fi xing with elasticity coeffi cients of the support contour m and n: On the top closed shell as: On the free upper edge of the shell, which is not closed at the top as: where is the static load applied to the upper edge of the contour in form evenly distributed force and evenly distributed bending moment, respectively. We assume that the shell is affected by a harmonic disturbance load: where is a frequency of disturbing force, is a function describing the distribution of the load amplitude values along the generatrix of shell. Here is the amplitude values of the uniformly distributed load acting on the element, is the value of the load integral, calculated for all elements, starting from the element closest to the center of the shell, to the current element (i-th element): The values of are calculated for all elements before the procedure for assembling the generalized matrices and vectors of the problem. The resulting fi nite-element relations have a simple structure, and the calculation of the matrices and vectors of the system (13) does not require the use of numerical integration, which positively affects the accuracy of the results obtained. The boundary conditions, consisting in equating to zero the values of one or several functions, are implemented by the usual technique for the fi nite element method. The boundary conditions (7)  Assembling generalized matrices and vectors of the problem is usually carried out for the fi nite element method. The system of equations obtained after assembling and taking into account the boundary conditions is solved by the Gauss method or by another suitable method for solving systems of linear algebraic equations.

CHECKING THE RESULTS
To establish the convergence of fi nite-element method plotted Nr force change on the partition of the shell with a different number of elements in radial direction. When the number of elements is more than 15-20, the difference between the obtained values is less than 0.1%. To cheking of the results obtained using the developed methodology, a test calculation of the isotropic shell of a spherical shape was carried out according to the undeformed scheme. These results were compared with the solution obtained by the fi nite element method in displacements formulation using the SCAD program. The difference in values was not more than 5%. The correctness of the results obtained is confi rmed by the amplitude-frequency characteristic of the shell (see Figure 1). The frequencies at which resonance is observed are very close to the values of the eigenfrequency of the oscillations of the shell obtained by the methodology [02]: 13477, 24261, and 51948 Hz.

INVESTIGATION OF DYNAMIC RESPONSE OF SHELL
The infl uence of the static component of the load on the shell , which determines the initial stressstrain state of the shell, on the amplitude values of forces and displacements is investigated. We considered a top closed shallow shell of revolution of spherical shape with rigid fi xed support contour made of isotropic material. Geometrical parameters of the shell: .
Here is the rising height of the shell, and a is the radius of the base. The amplitude value of the dynamic component of the load was assumed to be unity. Based on the results of the performed calculations graphs are constructed for the changing of the function and the defl ection w of shell in radial direction (see Figures 2-4). For convenience of comparison, the values of the functions are given to a range of values from 0 to 1. The dashed lines show graphs calculated from the linear theory. The graphs in Figures 2-4 show that with a constant dynamic load, with an increase in the static component of the load, the amplitude values of functions and displacements increase during the process of oscillation of the shell. This effect is explained by the nonlinearity of the shell deformation model. In Figure 5 shows the amplitude-frequency response of the shell for various values of the static component of the load, which determines the initial stress-strain state. As can be seen from the fi gure, when the static load changes, the graphs of the amplitude-frequency response of the shell change with a shift in the resonance frequencies toward a decrease.

CONCLUSIONS
The developed fi nite element makes it possible to realize the effective methodology of dynamic analysis of orthotropic fl exible shells of revolution with allowance for the initial deformed state. The carried out researches of the infl uence of the shell's initial stress-strain state on the amplitude values of the forces and displacements that appear in the process of forced vibrations demonstrate the need to take this into account when performing a dynamic analysis of non-linear shallow shells.