In the present paper the analysis of the lay -up optimality conditions (derived earlier) for buckling optimization of VAT composite plates is performed. The plates have a symmetric lay -up and loaded along their contour by the in -plane forces. The consideration employs the von Karman approach. The lay -up optimality conditions correspond to the maximization of a single -mode (lowest) buckling eigenvalue. The conditions are comprehensively analyzed. The conclusions of the analysis are pres ented. The key conclusion is that the optimality conditions with and without account of variation of the in -plane forces are the same. Keywords : composite plate, buckling, lay -up, steered fibers, optimality _____________________________________________ ______________________________ © Sergey Selyugin, 2020 2 1. Introduction The buckling optimality of composite plates is studied in the last two -three decades by many authors (see, e.g., the reviews of Ghiasi et al. 2009, 2010 and the book of Falzon and Aliab adi 2008). The works in the field consider the problem mainly from the point of view of numerical solutions. In the paper of Selyugin (2019) the lay -up optimality conditions for buckling optimization of the VAT (steered fiber) composite plates have been derived. The present paper is devoted to the theoretical analysis of the conditions. Section 2 contains some theoretical background. In Section 3 the analysis of the optimality conditions is performed. Section 4 contains the conclusions. 2. Theoretical backgr ound We consider the laminated composite plate with the symmetric lay -up. The lay -up is composed of the orthotropic fiber -reinforced plies of the same thickness plyh and various point - wise orientation angles. The total number of plies i s even and equal to 2K, the generalization to the odd ply number is straightforward and is not considered in the present paper. Fig. 1 illustrate the plate and the notations. 3 Fig. 1 The mid-plane of the flat plate is located in the XY plane . The plate mid-plane is restricted by the piecewise smooth contour C with external no rmal n (having the components m, l) and tangent direction s; the directions n, l 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 1C of the contour C. The remaining part 2C of the contour the plate is not moving in X and Y. The plate is simply supported, i.e., the z-displacement and the normal bendin g moment M acting perpendicular to the contour C are equal to zero. The clamping boundary conditions are not considered in the present paper, the change to the case is straightforward. 4 The X, Y, Z displacements are denoted as u, v, w , respectively. The s igns of the loads are the following. The force flows y xNN, (along X and along Y) are positive in tension, the shear force flow xyN is positive when it decreases the 90 ° angle between the orthogonal lines at the plate surface . The flows are in equilibrium. The Composite Lamination Plate Theory (CLPT) is used for description of plate deflections (see the book of Gibson 1994) . The Classical Lamination Plate Theory (CLPT), based on linear -elastic relat ions for every layer and Bernoulli’s hypotheses, is used for obtaining the equation for the deflections w (see the book of Gibson 1994). The equation is written as follows: 0 24 4 ) 2 (2 2 22 2234 26 34 16 44 22 2 24 66 12 44 11  yxwNywNxwNyxwDyxwDywDyxwD DxwD xy y x (1) where ,6,2,1;6,2,1,  j iDij are the eleme nts 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:               yxwywxw D D DD D DD D D MMM xyyx 22222 66 26 1626 22 1216 12 11 2 (2) 5 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 for the corresponding curvatures: yxwk ywk xwkxy y x  2 22 22 ; ; (3) Then (2) is rewritten in the matrix -vector form as: kD M  (4) With the boundary conditions indicated above, the boundary -value problem for deflections is completed. The distribution of the in -plane stress flows is given by the relations:      x yyx v uvu A N , ,,, (5) where A is the in -plane stiffness matrix, and the colum n-vector T xy y x NNN N ,, (6) the superscript 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. 6 It is supposed that the buckling eigenvalue is a single -modal one. The optimization goal is to design a lay -up maximizing the first (lowest) eigenvalue. As it is shown in Selyugin (2019), the first order necessary local optimality conditions for the ply orientation angle i are written as follows , i,…,K: 0) (61) (2cos) () (4) (2sin3 13 2 2 12 22 1 32   i i i i z z kk k kUU  , i=1,…,K (7) where all angles are measured relatively to the X axis, 1 2 1 ,,,,iizz kk respectively are the largest and the lowest principal curvatures, the angle between the direction 1k and the X axis, the top and the bottom z coordinates of the i-th layer, i=1,…,K . The value of 0z is equal to zero. The relation (7) was obtained in the more simplified case in (Selyugin, 2013) using (Gibson, 1994) background on composites. 3. Analysis of the optimality conditions The first conclus ion coming from (Selyugin, 2019) is that the lay -up optimality conditions in the case of varied in -plane forces of the plate are the same as (7). This means that the in -plane force variation doesn’t influence the optimality conditions. It may be also noti ced from the optimality condition derivation (Selyugin, 2019) that the parameters 2 1, there (they are some Lagrange multipliers) are some generalized in -plane displacements. Further we look at t he typical deflection buckling shape for the 2D uniformly compressed simp ly supported composite plate shown in Figure 2. The Figure will support us in our analysis. 7 Fig. 2. Typical buckling shape of the 2D compressed composite plate. Now we analyze the relation (7) in details. Rewrite (7) wit hout the mutual multiplier as 0) (2cos) () (4) (2sin2 2 12 22 1 32     i i kk k kUU , i=1,…,K (8) The umbilical points (where 2 1kk ) are not discussed below. First we consider the term within the square brackets. When it is equal to zero, we have ) (4) () (2cos 2 1 32 1 2 kkUkkU i (9) We remember that the value of the ratio 32 4UU is about 1.0 (see Selyugin, 2013). When the both principal curvatures 2 1,kk are of the s ame sign (see Fig. 1) and rather close to each other in values (e.g., 5.0 and 4.0) , there is no solution i of (9). But in principle, for some combination of the principal curvature values, there are some solutions of (9). 8 The equatio n 0) (2sin i (10) gives the solutions i or 2i (11) The solutions mean that the ply material direction coincide the first or the second principal curvature direction. It was shown in (Selyugin, 2013) that the sum of (8) for i=1,…,K leads to a relation 0) (2 1..kk Mcurpr xy (12) where ..curpr xyM is the twisting moment measured in the principal curvature axes. The relation (12) means that the moment tensor and the tensor of curvatures are co -axial. In fact, the optimal plate is a locally orthotropic plate. 4. Conc lusions  Possible solutions of the opti mality conditions are discussed;  The variation of the in -plane forces does not influence the optimality conditions obtained without account of such variation;  The plate with optimal lay -up is a locally orthotropic one . Acknowledgement The author is thankful to the colleagues for the fruitful discussions . 9