Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented.

[1]  Mark Walker,et al.  Optimization of symmetric laminates for maximum buckling load including the effects of bending-twisting coupling , 1996 .

[2]  David Bassir,et al.  Multiobjective stacking sequence optimization for laminated composite structures , 2009 .

[3]  F. Bloom,et al.  Handbook of Thin Plate Buckling and Postbuckling , 2000 .

[4]  Paul M. Weaver,et al.  Lay-Up Optimization of Composite Stiffened Panels Using Linear Approximations in Lamination Space , 2008 .

[5]  Joachim L. Grenestedt,et al.  A study on the effect of bending-twisting coupling on buckling strength , 1989 .

[6]  Raphael T. Haftka,et al.  Design of stiffened composite panels by genetic algorithm and response surface approximations , 1995 .

[7]  Akira Todoroki,et al.  Stacking Sequence Optimizations Using Fractal Branch and Bound Method for Laminated Composites , 2001 .

[8]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[9]  Anders Clausen,et al.  Efficient topology optimization in MATLAB using 88 lines of code , 2011 .

[10]  G. Turvey,et al.  Buckling and Postbuckling of Composite Plates , 2012 .

[11]  Antoine Henrot,et al.  Extremum Problems for Eigenvalues of Elliptic Operators , 2006 .

[12]  M. Autio,et al.  Optimization of coupled thermal-structural problems of laminated plates with lamination parameters , 2001 .

[13]  Akram Y Abu-Odeh,et al.  Optimum design of composite plates using response surface method , 1998 .

[14]  M. Friswell,et al.  Optimization of anisotropic composite panels with T-shaped stiffeners including transverse shear effects and out-of-plane loading , 2008 .

[15]  Leonid P. Lebedev,et al.  Functional analysis in mechanics , 2002 .

[16]  E. Brian Davies,et al.  Spectral Theory and Geometry: ICMS Instructional Conference, Edinburgh 1998 , 1999 .

[17]  K. Kiwiel.  Methods of Descent for Nondifferentiable Optimization , 1985 .

[18]  Adrian Lewis,et al.  Foundations of Computational Mathematics, Santander 2005: Eigenvalues and Nonsmooth Optimization , 2006 .

[19]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[20]  Hisao Fukunaga,et al.  Buckling design of symmetrically laminated plates using lamination parameters , 1995 .

[21]  Christos Kassapoglou,et al.  Stacking Sequence Blending of Multiple Composite Laminates Using Genetic Algorithms , 2002 .

[22]  Mark S. Ashbaugh,et al.  Spectral Theory and Geometry: Isoperimetric and universal inequalities for eigenvalues , 1999 .

[23]  Hideki Sekine,et al.  Feasible region in general design space of lamination parameters for laminated composites , 2002 .

[24]  Mitsunori Miki,et al.  Optimum Design of Laminated Composite Plates Using Lamination Parameters , 1991 .

[25]  Barna A. Szabó,et al.  Linear models of buckling and stress-stiffening , 1999 .

[26]  D. Sather,et al.  Branching of solutions of nonlinear equations , 1973 .

[27]  Sergei V Nesterov,et al.  High-Precision Methods in Eigenvalue Problems and Their Applications , 2004 .

[28]  Adrian Lewis,et al.  1 Eigenvalues and nonsmooth optimization , 2005 .

[29]  R. Courant.  Variational methods for the solution of problems of equilibrium and vibrations , 1943 .

[30]  J. Whitney,et al.  The Effect of Transverse Shear Deformation on the Bending of Laminated Plates , 1969 .

[31]  L. W. T. Stafford,et al.  Mathematics for Economists , 1971 .

[32]  Lawrence E. Blume,et al.  Mathematics for Economists , 1994 .

[33]  Bernard Rousselet,et al.  Continuité et différentiabilité d'Éléments propres: Application à l'optimisation de structures , 1990 .

[34]  Stuart S. Antman,et al.  Theodore von Kármán , 2006 .

[35]  Akira Todoroki,et al.  Stacking sequence optimizations using modified global response surface in lamination parameters , 2003 .

[36]  J. Grenestedt,et al.  Composite plate optimization only requires one parameter , 1990 .

[37]  W. T. Koiter,et al.  ELASTIC STABILITY AND POST-BUCKLING BEHAVIOUR , 2001 .

[38]  M. Dauge,et al.  On the asymptotic behaviour of the discrete spectrum in buckling problems for thin plates , 2006 .

[39]  F.-X. Irisarri,et al.  Computational strategy for multiobjective optimization of composite stiffened panels , 2011 .

[40]  Akira Todoroki,et al.  Stacking Sequence Optimizations Using GA with Zoomed Response Surface on Lamination Parameters. , 2000 .

[41]  Trevor Hastie,et al.  The elements of statistical learning. 2001 , 2001 .

[42]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[43]  J. Morlier,et al.  Surrogate modeling approximation using a mixture of experts based on EM joint estimation , 2011 .

[44]  P. Weaver,et al.  On feasible regions of lamination parameters for lay-up optimization of laminated composites , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[45]  R. Haftka,et al.  Elements of Structural Optimization , 1984 .

[46]  J. Grenestedt,et al.  Layup optimization against buckling of shear panels , 1991 .