NUMERICAL AND ANALYTICAL INVESTIGATION OF STABILITY OF THE REINFORCED PLATE

Resume The work is aimed at the construction of an algorithm for studying the equilibrium states of a reinforced plate near critical points, using the first (cubic terms) nonlinear terms of the potential energy expansion. Using geometrically nonlinear analysis of displacement, deformation and stress fields, the Eigenforms of buckling were calculated and bifurcation solutions and solutions for equilibrium curves with limit points were constructed depending on the initial imperfections. Article info Received 19 January 2021 Accepted 5 March 2021 Online 20 August 2021


ISSN 1335-4205 (print version) ISSN 2585-7878 (online version)
of this work [9][10][11], who, using the FEM, numerically, studied the effect of initial imperfections on the bearing capacity of a reinforced plate in the case of multiple and multiple critical loads. In these works, it was found that, indeed, with simultaneous buckling in the total deflection and wave formation in platinum, in the case of coincidence of the critical loads, a double bifurcation is realized as a homeoclinic point of a hyperbolic umbilic. If buckling simultaneously occurs along the general deflection and wave formation in the ribs, then the double semi-symmetric point is an anticlinic bifurcation corresponding to the catastrophe of the elliptical umbilic [11]. Important results, using equations of the first and second geometrically nonlinear approximations (taking into account cubic and quartic terms in the expansion of potential energy), were obtained in the 80s by Manevich [7][8]. He was able to establish that, limiting himself to the first approximation (only cubic terms are taken into account), it is possible to obtain acceptable estimates of the bearing capacity of a compressed reinforced plate if the critical loads of the wave formation in its elements are close to or exceed the critical buckling load according to the general deflection scheme. If the waveform loads are significantly less than the critical load of the total deflection, then taking into account additional terms of the fourth order in decomposition of the potential energy makes it possible to increase the maximum load loss of the bearing capacity of the reinforced plate (on average) by 30-40%. However, taking into account quartic terms

Introduction
Interest in the work of compressed reinforced plates appeared long ago, however, only starting with the work of Koiter and Kuiken [1], Koiter and Pignataro [2], van der Neut [3], Tvergaard [4], Hunt [5] and at a later time Manevich [6][7][8], systems of equilibrium equations were obtained taking into account geometric nonlinearity, which make it possible to analyze the bearing capacity of the mentioned plate either taking into account the total deflection, or taking into account the interaction of this deflection with local waveforms in the ribs or in the plate. Geometrically nonlinear equations for describing the loss of stability of the considered plates, taking into account the interaction of forms, were first presented by Koiter and Pignataro [2], Tvergaard focused his attention on study of plates with double critical loads for the total deflection and for wave formation in the plate [4].
In his work, important and general results were obtained for the calculation of reinforced plates of sufficiently large width and regular arrangement of ribs. For this, he isolated a regular T-shaped fragment, which was investigated in more detail; some of the results obtained by Tvergaard were used by Hunt [5] to construct the bifurcation surface of the homeoclinic bifurcation point corresponding to the catastrophe of the hyperbolic umbilic.
Tvergaard's studies were continued by authors potential energy, taking into account the interaction of these forms, will be written as follows: .
To determine the coefficients a ijk and d 1 , d 2 and d 3 it is necessary to calculate some definite integrals over the rectangular areas of the mentioned T-element of the reinforced plate. Expressions for these integrals: , and a L U , are given in the work of Tvergaard [4] and in a slightly modified form in the work of Manevich [8]. Equations (9) and (10) include the values of the critical parameters 1 m and 2 m , which correspond to the general shape of bending and local wave formation in the plate or ribs. These quantities are found by solving a linear eigenvalue problem. Tvergaard solved a homogeneous boundary value problem for a system of biharmonic equations and eight boundary conditions for conjugation along the line of contact of the rib with the plate: significantly increases the computational complexity due to the need to determine the corrections of the second approximation to eigenforms. In this article, the authors limited themselves to taking into account only the cubic terms in the expansion of the fields of displacements, deformations and stresses, since in design practice such ratios of the geometric parameters of a reinforced plate are used, at which the critical loads of the wave formation in a plate or in the ribs are significantly higher than the critical load of the total deflection.

Semi-analitical solution
Here is considered consider a plate hinged at its ends, supported on one side by a regular set of thin rectangular ribs. The longitudinal edges are free. The plate is compressed by centrally applied forces. The plate material is infinitely elastic. This allows to investigate the stability of the reinforced plate by analyzing the equilibrium of one regular T-shaped fragment (by analogy with Tvergaard and Manevich) equidistant with the rest of the T-shaped fragments. Strains and Hooke's law are expressed as follows: , where: L 1 , H -is a linear operator, L 2 -is a quadratic operator, L 2 (U + V) = L 2 (U) + 2 L 11 (U, V) + L 2 (V), in this expression L 11 -bilinear operator. The total potential energy of the reinforced plate with the retention of terms no higher than the fourth order has the form: . Note that all the three coefficients d 1 , d 2 , and d 3 are negative, which is possible only with a homeoclinic bifurcation point (a variant of the catastrophe of a hyperbolic umbilic). This situation arises during wave formation in a plate. In the case of simple eigenvalues 1 2 ! m m h the bifurcation solutions of the system of nonlinear equations with zero right-hand sides 0 1 2 p p = = turn out to be more complicated: .
If a problem is considered in which the initial imperfection in the total deflection is specified and there is no initial wave formation The most important condition is Equation (21). , , In this case, the equilibrium which corresponds to regular points on the equilibrium curve , , .
These conditions express the relationship between the forces and displacements at the edges of the lefthand and right-hand parts of the plate in relation to the rib [8]. In this case, it is assumed that the total deflection develops along a curve close to a sinusoid corresponding to the buckling of the Eulerian rod. Eight unknown arbitrary constants are found from a system of homogeneous equations, the matrix of coefficients of which is given in the appendix to the work of Tvergaard [4]. Manevich solved the same problem variationally, using the Rayleigh ratio. By setting a different number of half-waves, he found the smallest value of the critical loads of wave formation in the plate or in the ribs.
In Equations (11) and (12), on the right-hand sides, there are values of the initial imperfections 1 p and 2 p . They correspond to buckling shapes in terms of the total deflection and in terms of the shape of local wave formation. If both of these imperfections are nonzero, then the critical point is the limiting point. For a homogeneous bifurcation problem 0 the simplest solutions correspond to the case of multiple loads h turn out to be somewhat more complicated. If the right-hand side of Equation (12) is zero and the right-hand side of Equation (11) has an imperfection 0 1 ! p _ i proportional to the load parameter, then the bifurcation problem is implemented as a search for the critical value of the load at which a plate with a developing initial general deflection loses stability in the form of wave formation in the plate or in the ribs. Let the solutions of the simplest problem for a double semi-symmetric bifurcation point be considered, corresponding to solutions of homogeneous equations of Tvergaard's type. For the general deflection and for the solution describing wave formation in the plate, the mentioned solutions can be written as follows: From the last relations it can be seen that the total deflection of the relative amplitude 1 p is an unconnected deformation. It depends on one coordinate only, while the wave formation with an amplitude 2 p is linearly related to the amplitude of the total deflection. If this amplitude 1 p is equal to zero, then the amplitude 2 p is also equal to zero. For the values of the coefficients of the equations calculated by Tvergaard d 1 = -0.0193, d 2 = 0.6731 and d 3 = 0.1138. The ratio between the bifurcation problem in the presence of an initial imperfection in the form of a general deflection Its solution gives a critical waveform load in the relatively weak ribs of the reinforced plate ( Figure  1). In the graphical form, results for this problem, with various initial imperfections, are presented in the work of Manevich ([8], ch. III, Figure 3.3). The investigated plate had the following dimensionless parameters: For the plate in Figure 1, the values of the coefficients of the Equations, (11) and (12): . , . d d 0 009 1 024 , taken from [8]. Note that this rib plate contains relatively weak elongated cantilever plates. Loss of stability by wave formation should be expected precisely in compressed ribs, and not in the plate. This is indicated by the signs of the coefficients In the problem solved by Tvergaard [4], the critical load is twofold (at the same time there is a general buckling and wave formation in the plate). By specifying a nonzero initial imperfection in the total deflection 0 1 ! p h , it will be possible to plot the sensitivity curve of critical loads at the limiting points depending on the value of the initial deflection 1 p . Calculations according to Equation (23) for k 1 give: . , . , .
These results coincide with the values of ultimate loads with imperfections of the total deflection on the curves constructed by Tvergaard [4]. Consider The condition that the determinant of the Hessian matrix is equal to zero Is added to this equation: p . However, their joint solution is quite difficult. Tvergaard was looking for limit points by directly constructing the equilibrium curve, while Manevich used algorithms for finding the extrema of a function of two variables.
The method of the step-by-step increase in the load, is used in this work, followed by solution of the equilibrium equation to approximately obtain the coordinates of the limiting point and the corresponding maximum load. As an example, the coordinates of the limiting point and the corresponding load are determined for the above stability problem of a reinforced plate with wave formation in the ribs. Substitution of numerical values into Equations (31), after some simplifications, leads to the following equation of the fourth order: With a fixed value of the force parameter . 0 28 1 n = , one finds the minimal modulo negative root of this equation. For that is used a method of the non-singular extensions, proposed by Manuilov in 1971 [12]. If in Equations (11) and (12) the right-hand sides are not equal to zero, then the problem arises of determining the coordinates of the limit point  ; ; In order for a double semi-symmetric bifurcation not to be monoclinic, the condition for the positiveness of the root expression T was established in [13]: , then the wave formation would occur in the edges of the reinforced plate. According to research of Manevich [8], this feature is also valid for multiple critical loads.
Note that signs of coefficients of Equations (11) and (12) significantly affect the interaction of the general deflection with the local wave formation. If this deflection was positive (all the d 0 < i ), then its development provokes the wave formation in the plate, since it would receive additional compression.
On the contrary, the ribs in this case will be somewhat relieved by tensile stresses. If the total deflection is negative , 2 3 h then, on the contrary, the plate will be unloaded and the ribs will be loaded with additional compression. This would cause the rippling in the ribs. The described interaction of the general deflection and the effects of The iterative process of one-sided approximation to the nearest root of a power-law equation is based on local approximation of this equation by a hyperbola or an ellipse, depending on the sign of the curvature at the tangency point. A sufficient condition for one-sided convergence is that all the roots of the polynomial are real. The convergence of iterations to the nearest simple root of the power equation is asymptotically cubic.
If one moves along the equilibrium curve in sufficiently small steps along the load, then jumping over the limit point generates a "hanging" extremum ( Figure 3). This will be indicated by the negative sign of the root expression in the denominator in Equation n is obtained. The corresponding approximate coordinates of the limit point: . , . 0 3135 0 1998 For practical purposes, this is quite sufficient, although this result can be refined by subsequent calculations. The calculated coordinates of the limiting point are close to those shown in Figure 3.3 in the work of Manevich [8].
The relationship between the coefficients of Equations (11), (12) and the types of the wave formation (in the plate or in the ribs) is considered next. Hunt [5], investigating the problem of Tvergaard with a double critical load of the total deflection and wave formation in the plate, established the relationship between the derivatives of potential energy:  with the local waveform of the ribs. This deformation is possible only if the deflection of a negative sign develops. The study of this behavior of the structure is reduced to solving the bifurcation problem of stability of a compressed T-shaped fragment. To take into account the local shape in modeling, the shell-type elements were used, which made it possible to consider this model as a set of plate-strips and take into account not only the membrane deformations of the plate, but the transverse deformations of the rib as well. When solving the linear stability problem, the calculation of critical loads for the first ten eigenforms was obtained. It is important to note that the first eight forms correspond to local forms of the wave formation, and only the ninth -to the form of the plate buckling, like an Eulerian rod E . Mpa, . 196133 002 0 3 n = = h . The first form has seven half-waves, which coincides with the result obtained in [11]. Figure 4 shows the first Eigenform from the linear solution.
Calculating the Euler's critical load, one gets the value P N 2245803 cr = . The result, calculated by the formula, gives an overestimated value of the critical load, since the scheme for calculating the plate as an Euler rack does not take into account the deformation of the plate elements of the bar. Research in a geometrically nonlinear setting without taking into account the initial imperfections 0 1 p = _ i made it possible to obtain a point of unstable bifurcation on the equilibrium curve, corresponding to the shape of the wave formation of the edge. The deformed equilibrium and equilibrium curve are shown in Figures 5 and 6, respectively. The critical load value was: P N . 1288934 01 cr = . Figure 5 shows that additional compressive stresses act in the zones adjacent to the free edge of the rib and cause the wave formation bifurcation at loads lower than those obtained from the linear calculation. This deformed state corresponds to equilibrium at the point of unstable bifurcation. The new branch, after passing the singular point, is falling and unstable.
In cases of the non-zero values of the initial imperfections in the total deflection, to construct the bifurcation curve (Figure 7), one sets the amplitudes of the initial deflections and calculates the critical loads. The results obtained local wave formation is the main feature in behavior of a compressed reinforced plate. When designing reinforced plates, the appropriate geometric parameters should be chosen so as to exclude the nonlinear interactions between the buckling shapes. For this, it is necessary that the critical loads of the wave formation are significantly higher than the critical loads of the general buckling. This problem requires a separate thorough investigation.

Numerical analysis of a T-shaped fragment of a reinforced plate
Algorithms for calculating the critical loads for a reinforced plate, obtaining singular points on equilibrium curves were considered earlier and the relationship between the signs of the coefficients of nonlinear equations and the type of possible wave formation was indicated. By limiting the consideration to one regular T-shaped fragment, one can draw conclusions about the nonlinear behavior of the entire structure containing their totality. A deep understanding of the stability problem, in which there is a question of the interaction of forms, is fully disclosed using the finite element modeling and numerical solutions, taking into account geometric nonlinearity.
In this part of the work, an analysis of the numerical solution using the MSC Software Patran -Nastran software package is presented. Finite elements of a shell type (3131 elements) are selected. The material was considered as absolutely elastic (Е=196133.002MPa, μ = 0.3). Boundary conditions: hinged support along the short sides, along the long sides -floating terminations. The load is applied centrally. Geometric parameters were obtained from the relations of Manevich  The amplitudes of the rib deflections from the vertical plane are much larger in comparison to the similar deviations obtained in the problem without initial imperfection.
The load P cr is maximum in the case when the initial imperfection is set in the total deflection . .  problem of the reinforced plate stability in the case of multiple and non-multiple critical loads. The most dangerous interaction of shapes is the mutual influence of local buckling of rectangular ribs with the shape of the general deflection. A detailed solution to this problem for the reinforced plates is given in [7]. The reduction in bearing capacity reached 60 %. The results of the semi-analytical and numerical solution of stability of the T-shaped fragment of the plate turned out to be quite close. The above equations are suitable for determining the coordinates of singular points on the curves of equilibrium and critical loads.  Figure 10. The equilibrium curve is shown in Figure 11.

Conclusions
The article considers the solution of the nonlinear