Optimal design of symmetric angle-ply laminates for maximum buckling load with scatter in material properties

Abstract

A11 angle-ply laminated plate is optimized with the objective of maximizing critical buckling load. The material properties of the plate are not known precisely but with sorrle scatt,er around their average values. Convex analysis approach is employed to determine the material properties producing the lowest buckling load. T h e laminate is optimized over the ply-angles for the worst case of elastic moduli. The results are given for uniaxial and biaxial loading cases as well as for various aspect ratios. 111 structural design problems it is generally the case tha t the material properties are assurned to be fixed a t some deterministic parameters. However, it is well established tha t t,he elastic and strength properties of structures, and composite struct,urrs in particular, exhibit a scatter, mainly due to manufacturing imperfections. Therefore it is necessary to take the material uncertainty into account in the design of mechanical components to improve the reliability and to avoid premature failure. Consideration of scatter in the material properties becomes even more important in design optimization as the optimal configurations depend heavily on material da ta . Previous studies of the optimal design of composite laminates have been based on the assumption that the propertirs of the composite materials are known in a deterministic Since the scatter in material properties does not,, as a rule, cornply with any predefined distribution function, the rrlethods of convex modeling8 are quite suitable to deal with uncertain material data. This approach was applied to study t,he effect of uncerta i r~t~y i n elastic moduli on the buckling of corrlposite plates and shells.g I11 the present study optimal buckling design of composite laminates is investligated taking into account scatter in the material properties of the c o n posite. Using the methods of convex anx!ysis the elastic properties producirlg the least favorable ( the lowest) critical buckling load are drterrrliiied arid the laminate is optimized over the ply angles for thv worst case of material properties. Buckling a n a l y s i s Consider a simply supported angleply laminated plate of length a , width b and tllickrless 11 i l r thc z, y and z directions, respe~t~ively. T h e plat,r is symrnetrically laminated and balanced with equal number of layers with positive autl negative ply anglcs. It corisists of 11 laycrs so that the tlist,ance fro111 th r r~lidplane, z = 0, to the upper surface of the k;th lamilia is equal to t k = k h l n . 'The ply angle is denoted b j 0 . and tjhe stiffnesscs QiJ are fuuctior~s of 8 , 1, and the vector of elastic moduli E = (El, E2, E3. E4). wherc El = E L , E2 = ET, E3 = VLT and E4 = GLT. The stacking sequence for a symmetric and balanced laminate is of the form ((+@/ B)j),y,r where j = n /4 is the number of +O and -Q combinations. The compression loads per unit length in the z arid y directions are denoted by N, = ,V and IVY = AIL ' , respectively. The equation governing the, hucklilig of the plate under these loading conditions is given b j Copyright c 1994 by the American Institute of Aeronautics and Astronautics, Inc. All right reserved.