In the present paper the flat composite plates in buckling are studied. The plates have a symmetric lay -up and loaded along their piecewise -smooth contour by the in -plane forces. The consideration employs the Classical Lamination Plate Theory . A reciprocity theorem for the buckling states of the plate is derived and analyzed . The theorem may be used as a benchmark for the structural analysis and the optimization software. Keywords : composite plate, buckling, lay -up, steered fibers, reciprocity _______ ____________________________________________________________________ © Sergey Selyugin, 2019 2 1. INTRODUCTION Buckling of the anisotropic composite plates is thoroughly examined in many papers (see the books of Lekhnitzky 1956, Turvey and Marshall 1995, Falzo n and Aliabadi 2008, Kassapoglou 2010). The theory of the phenomenon is well -developed. Nevertheless , in the theory there are several aspects not explored until now . A study of such an aspect is performed in the present paper. The paper is devoted to the derivation and analysis of a reciprocity theorem for the buckling states of the anisotropic composite plates. In Section 2 the theoretical background is presented. Section 3 deals with the theorem. The conclusions are presented in Section 4. 2. THEORETICAL BACKGROUND Below we consider a flat thin composite plate restricted by a piecewise -smooth contour. The mid-plane of the plate coincides the X-Y plane of the Cartesian coordinate system XYZ. The plate has a symmetric lay -up, is restricted by a piecewise -smooth contour and is loaded by the in -plane loads along the contour. The C lassical Lamination Plate Theory (CLPT) is used for the description of the plate deflections (see the book of Gibson 1994). The boundary conditions considered are the simple support and/or the clamping (note that the zero deflection boundary condition is a part of the indicated boundary conditions) . 3 Following the book of Gibson (1994), we consider the buckling equilibrium of the infinitely small element of the plate. The equation, de scribing the force balance at buckling, is written as follows (see also the book of Volmir 1967): 0 22 22 22          yQ xQ yxM yM xM y x xy y x (1) where y x xy y x QQ MMM ,, ,, respectively are the x and y bending moments, the twisting moment, the buckling -induced forces acting normally to x and y edges of the element, with the moments and forces being calculated for the unit length edge. The Classical Lamination Plate Theory (CLPT), based on linear elasticity relations for every layer and Berno ulli’s hypotheses, is used for transition from (1) to an equation for 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 (2) where ,6,2,1;6,2,1,  j iDij are t he 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:               yxwywxw D D DD D DD D D MMM xyyx 22222 66 26 1626 22 1216 12 11 2 (3) 4 The quantities y xNN, , xyN in (2) are the corresponding stress flows. Below we will denote the left-hand side of (3) as a vector M , and the vector at the right -hand side (which is multiplie d to D matrix) as k . Also we use the notation yxwk ywk xwkxy y x  2 22 22 ; ; (4) Then (3) is rewritten in the matrix -vector form as: kD M (5) For y xQQ, as the functions of the external loads the following relation is valid (see the books of Mikhlin 1964, Gibson 1994, Volmir 1967). xy xy yy xxy xkN kN kN yQ xQ2    (6) When we insert (6) and (3) into (1), we obtain (2) resulting from the substitution. We introduce the quantities y x, and the vector  , composed of them: yw xw y x  ; (7)     yx   (8) Obviously (Gibson 1994) F Q (9) where the matrix F depends on x, y and is written as:     y xyxy x N NN N F (10) 5 The total strain potential energy ~ and the work of the internal forces W are written respectively as follows:       kDkdS kDkkD kkD kD kkD kD dS k T xyxyy xyx y yx x 21) 22 221 21( )(~ 2 6626 162 22 122 11 (11)          F dS N N NdS WT yx xy yy xx212 21)(2 2 (12) where the superscript T means mat rix transposing. 3. A RECIPROCITY THEORE M We consider further a reciprocity theorem for two different buckling states. We will denote the values related to the first buckling state by the superscript and the values related to athe second buckling state by the superscript . The two buckling states correspond to, generally speaking, different loadings. We consider the following term and implement integr ation by parts, keeping in mind the simple support or the clamped boundary conditions:            T y xxy y x T QdSyQ xQwdSyxM yM xMwdS kMdS2 22 22 2 (13) On the other hand   k MdS kDkdS kMdST T T (14) This leads to the relation 6   T TQdS QdS (15) which means, that the work performed by the force factors of the state on the deflections of the state , is equal to the work performed by the force factors of the state on the deflections of the state . Substituting Q via  from (9), we obtain:     F dS F dST T (16) The obtained theorem is an analogue of the known Betti theorem of the structural mechanics in application to the buckling states of the composite plate. It should be noted that the buckling eigenvalues are located within the F matrice s as the multipliers. It should be also noted that the theorem is obviously val id for the isotropic plates . 4. CONCLUSIONS  A reciprocity theorem for two buckling states is derived and discussed;  The theorem may be used as a benchmark for the buckling analy sis of the composite plates . ACKNOWLEDGEMENT The author is thankful to the colleagues for the fruitful discussions. 7