Peculiarities of calculation and design of smooth orthotropic panels with considering geometric nonlinear behavior

. When calculating the stability and bearing capacity of the panels of the caisson of the wing of an aerospace aircraft, it is necessary to take into account the presence of a heat-shielding coating, which also acts as an elastic base (EB) for the skin. The subject of this work is thin smooth orthotropic rectangular panels with an elastic base that does not perceive compressive and shear flows acting on the panel. The aim of the work is to develop methods for evaluating and designing composite panels associated with an elastic foundation, taking into account the possible geometrically nonlinear behavior under loads close to the design level. Note that the indicated level of loading can be realized only in full-scale static tests in the experimental justification of the strength of the wing structure. The paper presents analytical solutions of a geometrically nonlinear problem by the Bubnov-Galerkin method for rectangular orthotropic panels, taking into account the ER under the action of compressive and tangential forces. Based on the solutions obtained, methods for determining the thicknesses of orthotropic panels are proposed, taking into account two possible criteria: either reaching the ultimate strength stresses in a possible supercritical state, or reaching the limiting values of the deflection amplitude with geometrically nonlinear behavior. The last specified condition is a consequence of the requirements for the strength of the adhesive bond between the orthotropic skin and the heat-shielding layer. The paper considers smooth rectangular orthotropic panels with rigid support along the long sides and hinged support along the short sides.


Introduction
When designing the load-bearing panels of the wing box of an aerospace aircraft, it is necessary to take into account the presence of tiles of a heat-shielding coating, which performs the function of an elastic base (ES) for the skin. The subject of this work is thin orthotropic rectangular panels with regard to the elastic base, which should be made in the form of tiles and should not absorb compressive and tangential flows acting on the panel from bending and torsion of the wing. The aim of the work is to develop analytical methods for calculating and designing composite panels associated with an elastic foundation, taking into account the supercritical behavior at loads close to the design level. Note that in this case it is necessary to solve the following problems [1]. First, it is necessary to solve analytically geometrically non-linear problems for orthotropic panels associated with the CR. Note that further we will consider analytical solutions of geometrically nonlinear problems obtained by the Bubnov-Galerkin method. Secondly, it is necessary to obtain analytical relationships to determine the optimal thicknesses of orthotropic skins for two possible options for limiting strength or deflection amplitude, taking into account the supercritical behavior of the skin at the calculated loading level. In this case, the optimal thicknesses of orthotropic panels are determined by choosing the maximum values from the two specified criteria options. Note that the specified design level of loading must be implemented during full-scale static tests to experimentally substantiate the static strength of the wing structure. In this paper, the following variant of the boundary conditions will be considered: rigid support along the long sides and hinged support along the short sides. Consider separately two variants of loading in compression and in shear, taking into account different expressions for the deflection [2,3].
Let's consider some works that are devoted to the design of composite panels, taking into account the limitations in strength, stability and load-bearing capacity [4]. The review by Ni X., Prusty G., Hellier A. provides an analysis of publications since 2000. to 2012, devoted to the calculations and design of composite panels. Further, the work of Baranovski S., Mikhailovskiy K. which is included in a series of thematic publications devoted to versatile parametric studies of composite structures, should be noted [5]. The review article by Pogosyan M. proposes a comprehensive program for modeling composite structures using finite models of various levels. A "computational and experimental research pyramid" is also proposed and its connection with the complex modeling program for modern developments of composite structures is shown [6]. The article Gavva L., Firsanov V. presents an analytical review of calculation methods and experimental approaches to the analysis of the stress-strain state of structurally anisotropic composite panels of aircraft structures [7]. In another work, Gavva L. proposed a new model and an analytical review of approaches to studying the problem of buckling of structurally anisotropic aircraft panels. The article by Bokhoeva L. presents the results of the development of the optimal design of the multilayer wing console of an unmanned aerial vehicle, taking into account experimental justification [8]. Separately, we note the work of Bokhoeva L. devoted to assessing the crack resistance characteristics of a compressed composite panel with initial delamination [9][10][11]. An interesting monograph by Falzon B G, Aliabadi M H. is devoted to the study of the supercritical behavior and stability of composite panels, which contains the results of analytical, numerical and experimental studies related to the design of reinforced composite panels, taking into account the features of ensuring stability and analysis of the supercritical state [12]. Next, we note the monograph by Mitrofanov O., which presents the methodology for designing load-bearing composite panels according to the supercritical state, which will be used below in this work. We also note that the consideration of geometrically nonlinear problems for composite panels associated with an elastic foundation can make appropriate changes to the traditional form of the solution without taking into account the elastic foundation [5,6]. Next, we will use the approach proposed , where a method for designing orthotropic panels according to the supercritical state in compression is proposed, taking into account the limitations on strength or deflection with all-round hinged support. Taking into account the technical formulation of the problems considered below (Fig. 1), in this case, it will be necessary to consider the panels with rigid support along the long sides and hinged support along the short ones, which somewhat complicates obtaining analytical solutions to geometrically nonlinear problems [13][14][15].

Model and method
Let us write the compatibility equation for deformations of an orthotropic panel, taking into account geometrically nonlinear relationships, in the form [16] where Fstress function Erie, Lmoperators , Ex, Ey, Gxy, xyaverage characteristics of the composite package [9]. We write the second nonlinear Karman-type equation in the form , where Here Dmnbending stiffness of orthotropic panel [12]; qzreaction from the elastic base (EB), which can be represented as where Сielastic foundation coefficients.
For the analytical solution of a geometrically nonlinear problem for an orthotropic panel, taking into account the elastic foundation by the Bubnov-Galerkin method, we will use the expression , where Wkdeflection function.
Note that the membrane stresses in the middle surface of an orthotropic plate with supercritical behavior are determined from the definition of the Airy function F .

Panels in compression
Consider the analytical solution of the geometrically nonlinear problem of panels under compression by a longitudinal flow . Assuming that the panel has an all-round rigid support along the long sides, we represent the deflection in the form where а , and further for rectangular panels we have n=1. Substituting equality (5) into equation (1), we obtain the Airy function   (7) can be written in an abbreviated form without making a special notation for the notation We also write the expression for the stresses in the middle surface, we write it based on the definition of the stress function Where Е .
Next, we rewrite expression (9) with respect to the deflection amplitude, taking into account the condition that the longitudinal compressive stresses reach the limit values in absolute value х and further, taking into account the sign, we have Here -stress surface function for determining potential-critical points (PCP), in which stresses reach the maximum stress modulus. Note that in simple cases with hinged support, the expression for the function turns out to be quite simple and allows you to determine the PCT without additional calculations. In this case, to determine the coordinates of the PCT, it is necessary to use calculations based on the system of equations: , Е у [17]. Turns out to be quite simple and allows you to determine the PCT without additional calculations. In this case, to determine the coordinates of the PCT, it is necessary to use calculations based on the system of equations: . Further, for a given laying of an orthotropic panel, it is possible to numerically determine the critical parameters of wave formation, which depend on the ratios of geometric parameters and stiffness ratios. Secondly, it is necessary to determine the PCT numerically by examining the function Е х у . Thirdly, when using the criterion of strength at substitute value f 2 from Equation (10) to Equation (8), write down the equation for the panel thickness and perform the calculations in PCT. Fourth, when considering the stiffness criterion for f=fmax using formula (8), it is possible to determine the coordinates of points with the maximum deflection amplitude, then calculate the thickness using equation (5) at the indicated points. Fifth, compare the values obtained at the two previous stages of thicknesses and choose the maximum thickness [18].

Panels when sheared
Now consider an orthotropic panel loaded with tangential flows, taking into account the influence of an elastic foundation. In this case, we represent the deflection in the form , where -the tangent of the slope of the waves, s is the distance between the nodal lines.
After substituting the deflection (12) into the strain compatibility equation (1), we can obtain an equality whose solution is the stress function с с (13) When considering particular solutions of the homogeneous equation, one can obtain coefficients that take into account the orthotropy of the structure , and for shear stresses, we can write Where Е с с .
After substituting expressions (12) and (13) into equation (4) and some cumbersome transformations, one can obtain a non-linear relationship with respect to the deflection amplitude (f 0) This equation can be rewritten in an abbreviated form without special notation At pxyδ=qxy in the case of small deflections, from the last equality we have the expression + + (17) and considering further the system of equations (18) It is possible to determine the critical parameters of wave formation and solve the problem of stability for an orthotropic panel under the action of tangential flows qxy and then solve the problem of geometrically nonlinear behavior of the delamination of an orthotropic structure under shear, taking into account the chosen boundary conditions, taking into account the elastic foundation [19]. Note that the procedures of the design technique for a panel loaded with tangential flows can be written in the following form. First, for a given laying of an orthotropic panel, from system (18) it is possible to numerically determine the critical parameters of wave formation, which depend on the ratios of geometric parameters and stiffness ratios, which depend on the laying. Secondly, it is necessary to determine the PCT numerically by examining the function . Thirdly, when using the criterion of strength at substitute value f 2 from Equation (14) to Equation (16), write down the equation for the panel thickness and carry out the calculations in PCT. Fourth, for the option of considering the criterion for rigidity at f=fmax using formula (12), determine the coordinates of points with the maximum deflection amplitude, then calculate the thickness using equation (16) at the indicated points. Fifth, compare the thicknesses obtained at the previous two stages and choose the maximum thickness value [20].

Conclusion
It should be noted that the proposed methods can be used at the early stages of design when evaluating the design decisions made for load-bearing panels. This paper presents analytical solutions to a geometrically nonlinear problem for orthotropic panels connected to an elastic foundation and loaded with compressive or tangential forces. We also note that in the case of combined loading, a geometrically nonlinear problem can also be solved taking into account the use of the deflection type indicated in expression (12). In addition, in the general case, for designing according to the methodology, a more complex strength criterion can be used.
On the basis of analytical solutions of geometrically nonlinear problems for orthotropic panels associated with CR with rigid support along the long sides, in this paper, methods are proposed for determining the optimal thicknesses obtained from the condition of achieving ultimate values in terms of static strength during supercritical behavior or achieving limiting values of the deflection amplitude.