Buckling Behavior of Long Symmetrically Laminated Plates Subjected to Shear and Linearly Varying Axial Edge Loads

A parametric study of the buckling behavior of infinitely long symmetrically laminated anisotropic plates that are subjected to linearly varying edge loads, uniform shear loads, or combinations of these loads is presented. The study focuses on the effects of the shape of linearly varying edge load distribution, plate orthotropy, and plate flexural anisotropy on plate buckling behavior. In addition, the study examines the interaction of linearly varying edge loads and uniform shear loads with plate flexural anisotropy and orthotropy. Results obtained by using a special purpose nondimensional analysis that is well suited for parametric studies of clamped and simply supported plates are presented for [±θ]s thin graphite-epoxy laminates that are representative of spacecraft structural components. Also, numerous generic bucklingdesign charts are presented for a wide range of nondimensional parameters that are applicable to a broad class of laminate constructions. These charts show explicitly the effects of flexural orthotropy and flexural anisotropy on plate buckling behavior for linearly varying edge loads, uniform shear loads, or combinations of these loads. The most important finding of the present study is that specially orthotropic and flexurally anisotropic plates that are subjected to an axial edge load distribution that is tension dominated can support shear loads that are larger in magnitude than the shear buckling load. Introduction Buckling behavior of laminated plates that are subjected to combined loads is an important consideration in the preliminary design of aircraft and launch vehicles. The sizing of many structural subcomponents of these vehicles is often determined by stability constraints. One subcomponent that is of practical importance in structural design is the long rectangular plate. These plates commonly appear as subcomponents of stiffened panels used for wing structures and as semimonocoque shell segments used for fuselage and launch vehicle structures. Buckling results for infinitely long plates are important because they often provide a useful conservative estimate of the behavior of finite-length rectangular plates, and they provide information that is useful in explaining the behavior of these finite-length plates. Moreover, knowledge of the behavior of infinitely long plates can provide insight into the buckling behavior of more complex structures such as stiffened panels. An important type of long plate that appears as a subcomponent of advanced composite structures is the symmetrically laminated plate. In the present paper, the term “symmetrically laminated” refers to plates in which every lamina above the plate midplane has a corresponding lamina located at the same distance below the plate midplane, with the same thickness, material properties, and fiber orientation. Symmetrically laminated plates remain flat during the manufacturing process and exhibit flat prebuckling deformation states. These characteristics and the amenability of these plates to structural tailoring provide symmetrically laminated plates with a significant potential for reducing structural weight of aircraft and launch vehicles. Thus, understanding the buckling behavior of symmetrically laminated plates is an important part of the search for ways to exploit plate orthotropy and anisotropy to reduce structural weight. In many practical cases, symmetrically laminated plates exhibit specially orthotropic behavior. However, in some cases, such as [±45]s laminates, these plates exhibit anisotropy in the form of material-induced coupling between pure bending and twisting deformations. This coupling is referred to herein as flexural anisotropy, and it generally yields buckling modes that are skewed in appearance. The effects of flexural orthotropy and flexural anisotropy on the buckling behavior of long rectangular plates that are subjected to single and combined loading conditions are becoming better understood. For example, recent in-depth parametric studies that show the effects of anisotropy on the buckling behavior of long plates that are subjected to compression, shear, pure inplane bending, and various combinations of these loads have been presented in references 1 through 5. The results presented in these references indicate that the importance of flexural anisotropy on the buckling resistance of long plates varies with the magnitude and type of the combined loading condition. However, none of these studies supply results for plates loaded by uniform shear and a general linear distribution of axial load across the plate width. Both the uniform axial compression and the pure in-plane bending loads are special cases of the general linear distribution of axial edge loads. Results for this class of loadings are useful in the design of aircraft spar webs and panels that are located off the neutral axis of a fuselage or launch vehicle that is subjected to overall bending and torsion loads. Moreover, the importance of neglecting flexural anisotropy in a buckling analysis is practically unknown for this class of loadings. 2 One objective of the present paper is to present buckling results for specially orthotropic plates that are subjected to uniform shear, a general linear distribution of axial load across the plate width, and combinations of these loads in terms of useful nondimensional design parameters. Other objectives are to identify the effects of flexural anisotropy on the buckling behavior of long symmetrically laminated plates that are subjected to the same loading conditions and to present some previously unknown results that show some unusual behavior. Results are presented for plates with the two long edges clamped or simply supported and that are free to move in their plane. Several generic buckling-design curves that are applicable to a wide range of laminate constructions are also presented in terms of the nondimensional parameters described in references 1, 2, 5, and 6. Symbols Am , Bm displacement amplitudes (see eq. (22)), in. b plate width (see fig. 1), in. D11, D12, D22, D66 orthotropic plate-bending stiffnesses, in-lb D16, D26 anisotropic plate-bending stiffnesses, in-lb E1, E2, G12 lamina moduli, psi nondimensional buckling coefficient associated with critical value of an eccentric in-plane bending load (see eq. (21) and fig. 1(a)) nondimensional buckling coefficient associated with critical value of a uniform shear load (see eq. (20) and fig. 1(a)) shear and in-plane bending buckling coefficients, defined by equations (20) and (21), respectively, in which anisotropy is neglected in the analysis nondimensional buckling coefficient associated with critical value of a uniform axial compression load (see eq. (18) and fig. 1(a)) nondimensional buckling coefficient associated with critical value of a uniform transverse compression load (see eq. (19) and fig. 1(a)) L1, L2, L3, L4 nondimensional load factors defined by equations (14) through (17), respectively nondimensional membrane stress resultants of system of destabilizing loads defined by equations (10) through (13), respectively nondimensional membrane stress resultants of system of subcritical loads defined by equations (10) through (13), respectively N number of terms in series representation of out-of-plane displacement field at buckling (see eq. (22)) Nb intensity of eccentric in-plane bending load distribution defined by equation (5), lb/in. Nxc intensity of constant-valued tension or compression load distribution defined by equation (5), lb/in. Nx, Ny, Nxy longitudinal, transverse, and shear membrane stress resultants, respectively (see eqs. (5), (7), and (8)), lb/in. membrane stress resultants of system of destabilizing loads (see eqs. (6) through (9)), lb/in. membrane stress resultants of system of subcritical loads (see eqs. (6) through (9)), lb/in. nondimensional loading parameter (see eqs. (14) through (17)) and corresponding value at buckling (see eqs. (18) through (21)), respectively out-of-plane displacement field at buckling defined by equation (22), in. x, y plate rectangular coordinate system (see fig. 1), in. nondimensional parameters defined by equations (1), (2), (3), and (4), respectively in-plane bending load distribution parameters (see fig. 1 and eq. (5)) Kb nb1 ( )cr ≡