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 approa ch. 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 ov er the plate are taken into account. The physical meaning of the optimality conditions is discussed. Keywords : composite plate, buckling, lay-up, steered fibers, optimality ___________________________________________________________________________ © Sergey Selyugin, 2019 2 1. INTRODUCTION Buckling analysis of composite plates attracts attention of the numerous researche rs since 80-ies years of the last century. The account of the effect allows obtaining lighter design solutions and us ing the capacit y of the material more efficiently. The book of Turvey and Marshall (1995) seems to be the first one summarizing the state - of-the-art for buckling and postbuckling of the composite plates. The more modern review s may be found in the papers of Ghiasi et al . (2009, 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 plate s 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 numer ical analysis that there is an influence of the 2D str ess distribution on the buckling eigenvalue. The present paper considers the influence theoretically. The theoretical results for the VAT pl ate buckling optimization are absent now. The present paper de als with the derivation and analysis of the lay -up optimality condition s for the buckling level maximization of the steered fiber composite plates. Section 1 presents the short Introduction. 3 Section 2 describe s the main assumptions and the theoretical bac kground . 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. 2. THEORETICAL BACKGROU ND In the present paper we consider the laminated composite plate wi th the symmetric lay - up. The plate thickness is h. 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 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. Fig. 1. 4 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 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 in Z direction. 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 the description of the plate deflections (see the book of Gibson 1994) . The Classical Lamination Plate Theory (CLPT), based on linear -elastic relations for every layer and Bernoulli’s hypotheses, is used for obtaining the equation for the deflections w (see the books of Gibson 1994, Lekhnitzky 1956, Reddy 2004). 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 elements of the bending stiffness matrix D, coupling the bending/twisting moments and various second derivatives of the deflecti on w with respect to x and y. The coupling is written as follows: 5               yxwywxw D D DD D DD D D MMM xyyx 22222 66 26 1626 22 1216 12 11 2 (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 notation s yxwk ywk xwkxy y x  2 22 22 ; ; (3) Then (2 ) is rewritten in the matrix -vector form as: kD M  (4) 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 1967)    xy xy y y x x k M kM kMdS A    2 (5) where is the variation symbol. Then the increase of the total strain potential energy ~ is A~ (6) The quantity of the total strain potential energy and its variation are written as follows: ) 2 2 221 21(~ 22 662 22 262 22 162 22 22 22 22 122 22 11                 yxwDyxw ywDyxw xwDywDyw xwDxwD dS (7) ) 2 2 221 21(~ 22 662 22 262 22 162 22 22 22 22 122 22 11                 yxwDyxw ywDyxw xwDywDyw xwDxwD dS (8) 6 The work, performed by the external forces, is             yw xwNywNxwNdS Wxy y x 2212 2 (9) The total energy of the syst em W U~ (10) Because of energy balance a t the buckling state w the energy variation δU is equal to zero: 0~ W U (11) 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 stationar ity conditions of ( 10) with respect to w, under the proper boundary conditions, lead to the buckling equation ( 1). The st atement 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 stationar ity form of the following ratio (under the same boundary conditi ons as before) : 0~ )0(    W (12) where             yw xwNywNxwN dS Wxy y x)0(2 )0(2 )0( )0( 21 21 (13) 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 Ra yleigh relation: )0(~ W (14) 7 The distribution of the in -plane s tress flows are given by the relations :      x yyx v uvu A N , ,,, (15) where A is the in -plane stiffness matrix . The rhs column -vector is T xy y x NNN N ,, (16) 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 s atisfy the corresponding boundary conditions. We will also use below some relations for the laminate d plate presented in Selyugin (2019 a, 2019b ). For the orthotropic lamina, we follow the approach and the notations of (Gibson 1994, Section 2.6). Fig. 2 shows the orthotropic lamina. The stress tensor components within the lamina in the local coordinates ( x, y) are:              xyyx xyyx Q Q QQ Q QQ Q Q   266 26 1626 22 1216 12 11 (17) where Q is the lamina stiffness matrix, and xy y x ,, are the lamina infinitesimal strain tensor components. 8 Fig. 2. Orthotropic lamina with material orientation (1, 2). The components of the lamina stiffness matrix are (see Gibson 1994, Section 2.6):      4cos ) (214sin 2sin214sin 2sin214cos 2cos4cos4cos 2cos 3 4 1 663 2 263 2 163 2 1 223 4 123 2 1 11 U U U QU U QU U QU U U QU U QU U U Q      (18) where 4 3 2 1 ,,, UUUU are the elastic constants of the orthotropic layer. The stress tensor components within the lamina in the coordinates ( x, y) are:              xyyx xyyx Q Q QQ Q QQ Q Q   266 26 1626 22 1216 12 11 (19) where Q is the lamina stiffness matrix, and xy y x ,, are the lamina infinitesimal strain tensor components. The components of the lamina stiffness matrix are the known algebraic 9 functions of the four layer elastic constants 4 3 2 1 ,,, UUUU and of the trigonometric functions of the angle ϴ. The angle is the angle between the fiber direction and the x-axis (measured in the direction of the counter -clockwise rotation from x to y). According to (Gibson 1994), the derivatives of the lamina stiffness matrix ijQ w.r.t. the orientation angle ϴ are:      4sin44cos4 2cos4cos4 2cos) (2 4sin44 4sin4 2sin24 4sin4 2sin2 3663 2263 21626 16 31226 3 22216 3 211 UdQdU UdQdU UdQdQ Q UdQdQ U UdQdQ U UdQd         (20) The d etermination of ijA and ijD via the lamina parameters is performed according to the general formulas (Gibson 1994), ij=11, 12, 22, 16, 26, 66 :    K kk k kij ij z z Q A 11 (21)    K kk k kij ij z z Q D 13 13 31 (22) where 1,k kzz are the top and the bottom z-coordinates of the l ayer ( z=0 corresponds to the mid-plane). The derivatives of the stiffness matrices A, D w.r.t. the lamina angles k are, ij=11, 12, 22, 16, 26, 66 ; k=1,…, K: 10   kkij k k kijkkij k k kij dQd zzddDdQd zzddA    ) (31) ( 3 131   (23) where k is the layer orientation angle. 3. FIRST VARIATION OF T HE BUCKLING EIGENVAL UE 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 (Gibso n, 1994) 00 )0( ,)0( ,)0( ,)0( ,  yy xxyyxy xx N NN N (24) - the loading conditions by the X and Y loads )0( )0(,yn xnN N at 1C )0( )0( )0()0( )0( )0( yn y xyxn xy x NmNlNNmNlN  (25) - the kinematic in -plane conditions at 2C 00  vvuu (26) The subscript after comma above and below means the differentiation wrt to the subscript letter. 11 The Lagrangian L is written as follows:                              x yyx TC Cyn y xy xn xy xyy xxy yxy xx vuvu NA dv u dC NmNlN NmNlN dCN N N N dWL , ,,, 12 1)0( )0( )0( 1)0( )0( )0( 1)0( ,)0( , 2)0( ,)0( , 1 )0( 2 1~     (27) where 3 2 1 2 1 2 1 2 1 ,,,,,,,,  are the Lagrange multipliers ; the column -vector  is composed of 3 2 1,, . The Lagrangian variation (to be equal to z ero for the optimal design ) is written in the form:                                     K i ST ST x y y xT vu NC Cyn y xy xn xy xyy xxy yxy xx N D w NA dS vuvu NA dSv u dC NmNlN NmNlN dCN N N N dW W WL i 1)0( 1 , , , ,)0( 1 ,,2 1)0( )0( )0( 1)0( )0( )0( 1)0( ,)0( , 2)0( ,)0( , 1 )0( )0( )0( ;;~ ~ ~ )0(2 1)0(              (28) 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 varia tion 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 f ifth 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: 12        Sxy y x y y x x NNww Nw NwdSW W , ,2 ,2 , )0( )0(221~ )0( (29)           ST ST x y y xT NNA dS vuvuNA dS    1 , , , ,1;; )0( (30)                      Sx y y xST x y y xT vu v udS y u vx v u dl vuvuNA dS ,3 ,2 ,3 ,1 3 23 1 , , , ,1 , , cos, cos ;;   (31) As ,u are the variations of the inde pendent quantities, then we obtain from variation of L wrt u, v (with account of (31)) : 0,3 ,1y x ; 0,3 ,2x y in Γ (32) 0), cos( ), cos(3 1   ny nx  ; 0), cos( ), cos(2 3   ny nx  at 1C (33) 02 1 at 2C (34) But (32)-(33) leads to 03 2 1  (35) Hence, (30) is equal to zero. Finally, omitting some simple transformations, we obtain:                              2 1 1 221 )0( 1 2 2 12 21 1 )0()0( )0(, , ,2 ,1)0(2 , ,2)0(2 , ,1 )0(~ 222 CT CTD y x x yy yx x T m lml NdC m l l mm ml l NdCW WwwWwWw N d L          (36) where N  contains the independent variations. It follows from (36) that 02 1 at 2C (37) 13 and 1 1 ; 2 2 at 1C (38) Also 02)0(2 , ,1 Wwx x ; 02)0(2 , ,2 Wwy y ; 02)0(, , ,2 ,1 Wwwyx x y in Γ (39) The relations (37) and (39) lead to the determination of 2 1, . The only item in (36) with the variation of the ply angles is           K ii i i i ply i D zz kk kkUUhUW 13 13 2 2 12 22 1 32 3 )0() (61) (2cos) () (4) (2sin 4~     (40) where 1 2 1 ,,,,iizz kk 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 relation ( 40) was obtained in more simplified case in (Selyugin, 2013) using the Section 2 background on composites. The relation ( 40) means that the lay -up optimality condition s are 0) (61) (2cos) () (4) (2sin3 13 2 2 12 22 1 32   i i i i zz kk kkUU  , i=1,…,K (41) The conditions are identical with the conditions obtained without account of the in-plane force variation (Selyugin, 2013). 4. DISCUSSION The above -obtained results clearly demonstrate that there is a n 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. 14 5. 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 optimalit y conditions may be used for obtain ing the proper lay -up sensitivities ACKNOWLEDGEMENT The author is thankful to the colleagues for the fruitful discussions .