Analysis of stability and bearing capacity of reinforced panels made of composite materials under shear

When performing stress verification analysis of wing spar supported composite walls it is essential to estimate local stability of smooth wall and general stability of orthotropic panel under shear forces. Almost square fragment of spar wall supported with ribs (between two ribs) is examined in this paper. In order to estimate general stability and load-bearing capacity of orthotropic panel geometrically nonlinear problem is examined. When examining linear problem compact expression for critical shear flow definition is explicitly written in order to estimate general stability with ultimate loading. When examining nonlinear problem a system of two equations in deflection amplitude is obtained. Besides nonlinear problem of smooth wall (skin) local stability and postbuckling state estimation with possible geometrically nonlinear behavior is examined. Analytical solution of orthotropic rectangular panel with shear geometrically nonlinear problem is obtained. Based on method of design by postbuckling state with critical shear stress provided with buckling a method of skin thickness definition is suggested. In order to apply obtained solutions to design activities method of orthotropic panels design by stability and postbuckling state with corresponding load level is shown.


Introduction
Let us examine the composite reinforced wall of the wing spar of light aircraft. It is assumed that the overall dimensions of the wall fragment (figure 1) between the ribs are commensurable with one another (L≈B). The design of a spar wall structure generally requires analysis of local and overall shear stability under limit loads, as well as an assessment of the load-bearing capacity with ultimate loading.
It should be noted that the actual task is to determine the stability and load-carrying capacity of the square structural-orthotropic panel with shear taking the peculiarities of the buckling shape into account provided the commensurability of the sides L≈B (figure 1).
In addition, for thin wall structures of small aircraft, the local form of buckling is acceptable for load-bearing panels (including thin walls of spars) with loads close to ultimate level. In this case, geometrically nonlinear analysis of orthotropic rectangular panels with L>b (figure 1) in the refined formulation is necessary. Considering the fragment of a reinforced wall (figure 2), it should be noted that when assessing the postbuckling behavior and local buckling of this case (figure 2), it is correct to assume rigid support conditions along the long sides. In this case, it is also necessary to use analysis methods for stability and evaluation of possible postbuckling behavior taking into account the selection of an appropriate function of deflection due to shear. In addition, the methodology for design of orthotropic rectangular panels based on postbuckling state is presented further on. In this case the panel thickness is determined provided critical stress values with buckling.  It should be noted that the considered problems of stability and postbuckling behavior are essential, and weight characteristics of aircraft structures depend on the efficiency of their solution. Contemporary theories for composite structures design and analysis taking strength, stability and postbuckling behavior problems into account are shown in fundamental studies [1][2][3]. Reviews [4][5] as well as numerical and analytical solutions of composite panels stability problems with shear [6][7] are of certain interest. The results of studies of the behavior of composite panels are given in [8][9][10][11][12]. Composite panels design method based on postbuckling state using analytical solutions of geometrically nonlinear problems is shown in papers [13][14].

Problem statement and basic equations
To solve the stability and load-bearing capacity problems, we use geometrically nonlinear basic equations. Let us write down the equation of strain compatibility for the orthotropic panel in accordance with the theory of composite structures in the following form [3]     Hereinafter the generalized stiffness of orthotropic panels according to [3] is determined by the relations  we have   2  22  11  12  12  11 22  12 , , , , The second nonlinear equation of Karman type for orthotropic panels of following form For a structurally orthotropic panel the bending stiffness D11 is determined with consideration of reinforcing elements by relations (2) taking the "smearing" hypothesis of stringers into account, and the remaining bending stiffnesses D22, D12 and D33 refer to the skin and are calculated by relations (5).
To solve the geometrically nonlinear problem using the Bubnov-Galerkin method we will use following equations a b with Wk being deflection function. Let us also write down the equations determining stress function [3] 19th International Conference "Aviation and Cosmonautics" (AviaSpace-2020) Journal So, next we consider two geometrically nonlinear problems. First one for a structurally orthotropic (stiffened) square panel, and second one for a smooth orthotropic rectangular panel.
3. Applied methods for calculating stability and postbuckling state Let us first examine the geometrically nonlinear problem of the structural-orthotropic panel taking into account the geometric parameters corresponding in the general case to a square shape with a ≈ b, where a and b are the length and width of the panel.
Under the boundary conditions corresponding to the hinge support with tangential forces qxy acting, let us represent the deflection as 1 2 x After substituting expression (8) the solution of which is a stress function of the following form To assess the stability of the square panel, next we need to examine the linear system of equations It should be noted that the obtained nonlinear system (12)-(13) at a given load numerically allows us to determine the deflection amplitudes f1 and f2, and to find the stress state of the structuralorthotropic panel based on the definition of the stress function (7). Now let us examine the second problem concerning the stability and load-bearing capacity of a smooth orthotropic rectangular panel with rigid support along the long sides. It should be noted that the solution of the verification analysis for this case is shown in paper [14]. Here we also present the problem of design based on postbuckling state provided the critical stress ху  is reached with loads close to ultimate level.
In this case, the deflection of the orthotropic skin can be written as follows Based on the solution of the geometrically nonlinear problem we obtain the stress function of Airy [14] 4E , The tangential stress is obtained from the above mentioned function F. Under the action of shear