LAY-UP OPTIMALITY CONDITIONS FOR BUCKLING LEVEL MAXIMIZATION OF VAT (STEERED FIBER) COMPOSITE PLATES

In the present paper the flat composite plates in buckling are studied. The plates have a symmetric lay-up and loaded along their contour by the in-plane forces. The consideration employs the Kirchhoff hypothesis and the von Karman approach. The lay-up optimality conditions for a single-mode (lowest) buckling eigenvalue are derived using the proper variational principles and the variation calculus. Both the bending terms and the terms following from the redistribution of the 2D stresses over the plate are taken into account. The physical meaning of the optimality conditions is discussed.


INTRODUCTION
Buckling analysis of composite plates attracts attention of the numerous researchers since 80-ies years of the last century. The account of the effect allows obtaining lighter design solutions and using the capacity of the material more efficiently.
The book of Turvey and Marshall (1995) seems to be the first one summarizing the stateof-the-art for buckling and postbuckling of the composite plates. The more modern reviews may be found in the papers of Ghiasi et al. (2009Ghiasi et al. ( , 2010 and in the book of Falzon and Aliabadi (2008).
We also mention some monographs important to the considered matter. Ohsaki and Ikeda (2007) consider mainly the numerical aspects of the design optimization under the eigenvalue constraints.
In the book of Kassapoglou (2010) the modern approaches to the buckling analysis of the composite plates are described. The approaches are used at the Universities to educate the engineers in structural mechanics.
The buckling analysis of the composite plates is performed, as a rule, numerically, using the Rayleigh-Ritz or similar kinematic approaches based on the kinematic variational principles (indicated, e.g., in Washizu 1982).
In several papers (e.g., Ijsselmuiden et al. 2010) it was noticed in the numerical analysis that there is an influence of the 2D stress distribution on the buckling eigenvalue.
The present paper considers the influence theoretically. The theoretical results for the VAT plate buckling optimization are absent now.
The present paper deals with the derivation and analysis of the lay-up optimality conditions for the buckling level maximization of the steered fiber composite plates.
Section 1 presents the short Introduction.
Section 2 describes the main assumptions and the theoretical background.
In Section 3 we derive the first variation of the buckling eigenvalue.
Section 4 is devoted to the result discussion.
Section 5 presents the conclusions.

THEORETICAL BACKGROUND
In the present paper we consider the laminated composite plate with the symmetric layup. The plate thickness is h. The lay-up is composed of the orthotropic fiber-reinforced plies of the same thickness ply h and various point-wise orientation angles. The total number of plies is even and equal to 2K, the generalization to the odd ply number is straightforward and is not considered in the present paper. Fig. 1 illustrates some notations used in the paper. The mid-plane Γ of the flat plate is located in XY plane. The mid-surface is restricted by the piecewise smooth contour C with the external normal n (the components of the normal vector are l, m) and the tangent direction s; the directions n, s and Z create a right-hand triplet.
The coordinate system XYZ is a Cartesian one.
The plate is loaded by the in-plane forces, acting at the part 1 C of the contour C. The remaining part 2 C of the contour the plate is not moving in X and Y. The plate is simply supported in Z direction.
The X, Y, Z displacements are denoted as u, v, w, respectively.
The signs of the loads are the following. The force flows are the elements of the bending stiffness matrix D, coupling the bending/twisting moments and various second derivatives of the deflection w with respect to x and y. The coupling is written as follows: (2) We denote the left-hand side of (2) as a vector-column M  , and as k  the vector-column at the right-hand side (which is multiplied to D matrix). Also we use the notations (3) Then (2) is rewritten in the matrix-vector form as: Now we discuss the energy balance for the plate element dxdy. For a feasible deflection w we have the variation of the structural work of the internal forces (see the book of Volmir where is the variation symbol. Then the increase of the total strain potential energy 

is
The quantity of the total strain potential energy and its variation are written as follows: The work, performed by the external forces, is The total energy of the system Because of energy balance at the buckling state w the energy variation δU is equal to zero: Making formal variations in (8)-(10), integrating by parts and taking into account the boundary conditions of the considered types and the in-plane equilibrium, we obtain the equilibrium equation (1). Hence, the stationarity conditions of (10) with respect to w, under the proper boundary conditions, lead to the buckling equation (1). The statement is a content of the kinematic variational principle for the plate buckling.
Following the book of Washizu (1982), the kinematic variational principle may be rewritten as the stationarity form of the following ratio (under the same boundary conditions as before): and the superscript (0) corresponds to some constant initial (pre-buckling) force flows. The flows should be multiplied by the eigenvalue λ for obtaining the buckling force flows.
For found buckling mode the eigenvalue satisfies the Rayleigh relation: The distribution of the in-plane stress flows are given by the relations: where A is the in-plane stiffness matrix. The rhs column-vector is T means transposing, and index after comma means the differentiation w.r.t. the corresponding coordinate.
The components of the N  must satisfy the equilibrium equations, and the in-plane displacements must satisfy the corresponding boundary conditions.
We will also use below some relations for the laminated plate presented in Selyugin (2019a, 2019b).
For the orthotropic lamina, we follow the approach and the notations of (Gibson 1994, Section 2.6).   According to (Gibson 1994) The determination of ij A and ij D via the lamina parameters is performed according to the general formulas (Gibson 1994), ij=11, 12, 22, 16, 26, 66: where k  is the layer orientation angle.

FIRST VARIATION OF THE BUCKLING EIGENVALUE
Now we proceed with the determination of the variation of the eigenvalue. It is supposed that the eigenvalue is a single-modal one. Our goal is to design a lay-up maximizing the first (lowest) eigenvalue.
For making the variation we use the Lagrange function L uniting the goal function and the constraint of the problem.
The considered constraints of the problem are: -the equilibrium equations (Gibson, 1994) -the kinematic in-plane conditions at The subscript after comma above and below means the differentiation wrt to the subscript letter.
The Lagrangian L is written as follows: The Lagrangian variation (to be equal to zero for the optimal design) is written in the form: where the first item is equal to zero due to the above-indicated variational principle describing the plate stability, the second item is due to the bending stiffness variation and is derived in Selyugin (2013) (the influence of the in-plane force variation was considered as small and negligible there), the third item is due to the in-plane force variations influencing the eigenvalue, the fourth item corresponds to the in-plane equilibrium, the fifth item corresponds to the loading, the sixth item corresponds to the in-plane boundary conditions, the seventh item corresponds to (15), the eighth item corresponds to the ply orientation angle i  , i=1,…,K, variation influencing (15).
Performing the usual transformations based on the Gauss divergence theorem, we obtain:  u are the variations of the independent quantities, then we obtain from variation of L wrt u, v (with account of (31)): But (32)-(33) leads to Hence, (30) is equal to zero.
Finally, omitting some simple transformations, we obtain: The relations (37) and (39)  The only item in (36) with the variation of the ply angles is The conditions are identical with the conditions obtained without account of the in-plane force variation (Selyugin, 2013).

DISCUSSION
The above-obtained results clearly demonstrate that there is an influence of the 2D stress distribution on the buckling eigenvalue. The influence is due to the quantity ) 0 ( W in the denominator of (14). The quantity contains the in-plane forces.

CONCLUSIONS
 The first order necessary local lay-up optimality conditions for the plate buckling level maximization are derived;  The influence of the in-plane forces to the optimal buckling eigenvalue comes through the denominator quantity ) 0 ( W containing the forces;  The derived optimality conditions may be used for obtaining the proper lay-up sensitivities

ACKNOWLEDGEMENT
The author is thankful to the colleagues for the fruitful discussions.