Design of z-shaped reinforced panels of composite materials under compression and shear

. When designing thin-walled aircraft structures, as a rule, the main limitations are associated with ensuring stability. The objects of research in this work are composite z-shaped reinforced panels of the wing box of an aircraft. For the early stages of the design of load-bearing panels, it is necessary to evaluate the weight of the design decisions made, taking into account possible defects. The paper proposes an analytical technique for designing reinforced panels, taking into account the use of the condition of uniform stability and the presence of possible regulated defects in the skin that may occur in further operation. The task of determining the parameters of panels of minimum weight is reduced to minimizing the function of one variable, which is the ratio of the height to the reinforcement step. To take into account the indicated regulated defects, parametric studies were carried out using the finite element method, on the basis of which refinement coefficients were added to the analytical ratios of the stiffness parameters of the reinforced panels. The paper presents the results of studies of orthotropic panels with defects in the form of through holes under loading with compressive and tangential forces. The results are presented in the form of graphs that show possible changes in the critical stresses of thin skins with defects.


Introduction
Let us consider the design features of the upper reinforced panels made of composite materials (CM) of the aircraft wing box. When designing thin-walled structures, the main ones, as a rule, are the constraints associated with ensuring stability. Consider z-shaped reinforced panels loaded with compressive and tangential forces. Note that at the early stages of designing modern CM structures, it is necessary to take into account defects of various categories [1]. In this regard, we will further consider the refinement of the analytical methodology [2] for the design of reinforced composite panels from CM, based on the condition of uniform stability, taking into account the introduction of additional coefficients that take into account possible defects resulting from impacts [3]. It should be noted that in practical strength calculations, these defects from impacts are considered in reserve in the form of through holes. Thus, further we will assume that the critical stresses of local and general buckling are equal and, in this case, the skin can have regulated damage in the form of a through hole. The effects of defects in the skin between stringers will be determined by parametric studies of smooth orthotropic panels with a hole in compression and shear. Note that studies of the location of holes in the panel from the point of view of changing the critical buckling stresses will be of particular interest [4]. Now we note that already at the early stages of designing modern composite structures, the aircraft developer, as a rule, creates special design conditions, where he describes, among other things, possible regulated defects from impact actions and coordinates this document with the aviation authorities. These data are the basis for the subsequent determination of lists of elementary and structurally similar samples for subsequent tests. We also note that these lists, as a rule, include samples that simulate possible manufacturing and operational defects in order to assess their impact on the strength and fatigue characteristics of load-bearing panels [5].
Let's consider some publications devoted to the design of reinforced composite panels under stability and load-bearing capacity constraints. Note that a large review of publications since 2000. until 2015 made in the work of Ni X., Prusty G., Hellier A.. Optimal design problems are considered in Faggiani A and Falzon B G . Separately, we single out the monograph Falzon B G, Aliabadi M H. , which presents interesting results of experimental, numerical and analytical studies related to the problems of both stability and supercritical behavior of composite panels [6,7]. The use of parametric studies for the design of composite aircraft structures was proposed in the works of Baranovski S., Mikhailovskiy K.. A modern multilevel approach to the design of CM structures is presented in the review work by Pogosyan M. We also note the publications of Gavva L. devoted to the calculations and design of structurally anisotropic panels [8,9].
The methodology for designing load-bearing composite panels while providing supercritical strength is presented . The article proposes a method for calculating and designing composite reinforced panels, taking into account the supercritical behavior of the skin and the elastic behavior of stringers. An applied method for determining the bearing capacity in compression of thin-walled composite structures with impact damage, based on the use of the finite element method, was proposed . The works present interesting results of numerical studies of the stability of rectangular panels, taking into account various cutouts [10]. The conducted studies of the panels showed that the critical buckling stresses in the presence of various cutouts can have different values of critical stresses depending on the design parameters of the considered panels and cutouts. Thus, to determine potentiallycritical points of location of possible defects, it is advisable to use parametric numerical studies.

Model and method
The purpose of this study is to develop an analytical methodology for designing reinforced CM panels, taking into account the effect of through defects. In this case, it is necessary to solve the following problems: to obtain analytical relationships to determine the optimal parameters of equally stable panels in the presence of defects and to conduct numerical studies to take into account the effect of through defects on critical stresses. These numerical studies of the panels will be carried out for smooth panels under uniform loading with compressive and tangential forces. Separately, we note that the further development of the proposed methodology for designing reinforced panels can be associated with taking into account the influence of uneven loading and refinement of the corresponding coefficients based on, for example, interesting results given in monographs [11].

Basic ratios for calculating the stability of reinforced composite panels
Now let's consider a method for determining the parameters of reinforced composite panels of minimum weight in the presence of possible damage that is associated with the consequences of impacts on the skin. We will take into account the conditions of equal stability and then write down the main relations for the case under consideration [12]. The critical force of the total buckling of the panel in compression along the stringers is determined for the hinged post by the formula Where D11reduced bending stiffness of the panel in the longitudinal direction, L (t)length (pitch) of the stringers.
The critical stresses of local buckling of the skin and the wall of an orthotropic stringer under the condition of hinged support, taking into account longitudinal compression, are calculated using the equation [17] Here  -skin thickness, b -characteristic size (stringer pitch (Fig. 2)).
For a stringer flange, taking into account the hinged support of the panel with one free long edge, the critical stresses are found by the formula Where K (i) -stability factors E -some modulus of elasticity for reduction. From the compatibility conditions for deformations, we have Where Next, to take into account the defect from impact actions in the skin, we introduce the coefficient , which will be determined below in a numerical study of compressive critical stresses, taking into account the location of the defect: (1) = (1) . We also introduce the coefficient (1) = ( (1) − ) (1) ⁄ , where d is the diameter of the damage in the skin. Flexural stiffness of a z-shaped reinforced panel, taking into account the coefficient of influence of a skin defect (1) write in the form Where , 2 = 1.5 2 (3) (1) , 3 = (2) 12 , b1= (1) (a1+a3), b2=a2a3, and  -design parameter selected from design relationships [0.2, 0.5] and conditions b (3) =b (2) . The reduced thickness of the reinforced panel is determined by the formula = (1) + (2) (2) .
Further, by making some transformations of the equations written above, we can reduce the optimal design problem to the problem of minimizing the function of one variable k = h/t, which is the ratio of the height to the reinforcement step. From design considerations, it is clear that the parameter can vary within the segment k [0,1]. To calculate the optimal parameters of z-shaped reinforced panels (Fig. 2), we write an expression for the reduced thickness of the reinforced panel and a system of equations for determining all geometric parameters of the reinforced panel Indexes in parentheses indicate the elements of the panel (Fig. 2): (1), (2), (3) -indicate the parameters related to the skin, web and stringer shelf, respectively; ( ) -function of one variable to be minimized [13].

Determination of parameters of z-shaped reinforced composite panels of minimum shear weight
Applying the relations written earlier for the optimal design of z-shaped reinforced panels, one can obtain a system of equations similar to that written above (10)- (11). In the case of shear flows we have, taking into account through defects ,

Numerical studies of orthotropic rectangular panels in the presence of through defects in compression
Let us further consider numerical studies of thin carbon fiber panels in the presence of defects in the form of circular cutouts. It should be noted that the panels were loaded with longitudinal compressive and shear forces and had all-round hinge support [14,15]. The task of the study in this case was to determine the potentially critical geometric location of the defect in the plane of the panel in terms of changes in critical stresses. For example, a panel with geometric parameters a * b = 300 * 100 mm (Fig. 3) and with symmetrical laying was considered ℎ = 0.1.

Compression of an orthotropic panel with a through defect
Figures 4 and 5 show the dependences of the change in relative critical compressive forces with a change in the location of the defect in the direction of the X and Y axes. Figure 6 shows the dependence of the relative critical stresses with a change in the hole diameter [16,17]. The obtained calculated values of critical stresses are presented in a dimensionless form with respect to the critical stresses of an undamaged panel. So the relative stresses are relative = critical critical , Where relative -critical panel stresses without defect, critical defect -critical stresses of a panel with a defect.    Figure 8 and Figure 9 show similar dependences of the change in relative critical shear stresses with a change in the location of the defect in the direction of the X and Y axes, as well as with a change in the diameter of the defect [18]. In this case, the relative shear stresses there is -critical shear stresses of the panel without defect, critical defect -critical stresses of the panel with a defect.

Discussion
When designing modern composite panels, taking into account various defects is a prerequisite [19]. The parameters of various defects for thin-walled structures affect the allowable stresses both in terms of strength and stability. Thus, modification of the analytical methodology for designing panels with the introduction of through-thickness coefficients is a necessary measure for further expert assessments of changes in the weight characteristics of panels. Note that the results of a parametric study of the stability of panels with through defects in compression and shear have characteristic dependences showing a decrease in critical stresses compared to undamaged smooth panels. The plot of the relative normal compressive stresses with a change in the defect diameter (Fig. 6) deserves special attention, which shows that a large hole does not correspond to large critical stresses [20]. This means that when developing test programs for thin composite panels, it is necessary to carry out parametric studies to select potentially critical impact points and it is possible to indicate the impact energy limit to obtain damage of a critical size in terms of stability.

Conclusion
In this paper, a modified analytical method is proposed for determining the optimal parameters of reinforced weight panels, taking into account possible through defects. The technique can be used at the early stages of design to assess the change in the reduced thickness of the panel when varying the values of the calculated flows and the size of the allowed defects. The example parametric studies of critical buckling stresses of orthotropic rectangular panels with through defects may be of particular interest to developers of composite structures and can be continued to obtain analytical approximating dependences.