This paper discusses the fact that the cured shape of thin unsymmetric laminates do not conform to the predictions of classical lamination theory. Rather than being saddle shaped, as predicted by the classical theory, the paper shows that thin unsymmetric laminates cure into a shape of a right circular cylinder. This anomalous behavior has been observed by many but the paper serves to quantify the effect and to inspire investigators to begin thinking about using the phenomenon to advantage. The paper indicates that the anomalous behavior is repeatable and that thicker laminates con form to the predictions of the classical theory. Laminates of the [0/902 /θ ]T, [02 /θ2 ] T and [04 /θ4 ] T families are investigated for this be havior and it is shown the principal curvature directions of the cylindrical shapes are predictable.

Some Observations on the Cured Shape of Thin Unsymmetric Laminates

Aeroelastic tailoring technology is reviewed with reference to the historical background, the underlying theory, current trends, and specific applications. The specific application discussed include the Transonic Aircraft Technology program, an Advanced Design Composite Aircraft, the Wing/Inlet Advanced Development program, and the forward-swept wing. Finally, the future of aeroelastic tailoring and the development of an aeroelastic tailoring analysis and design tool under the Automated Strength-Aeroelastic Design program are examined.

Aeroelastic tailoring - Theory, practice, and promise

A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates

A summary of the recent developments in the analysis of laminated beams and plates with an emphasis on vibrations and wave propagations in presented. First, a review of the recent studies on the free-vibration analysis of symmetrically laminated plates is given. These studies have been conducted for various geometric shapes and edge conditions. Both analytical (closed-form, Galerkin, Rayleigh-Ritz) and numerical methods have been used. Because of the importance of unsymmetrically laminated structural components in many applications, a detailed review of the various developments in the analysis of unsymmetrical ly laminated beams and plates also is given. A survey of the nonlinear vibrations of the perfect and geometrically laminated plates is presented next. It is seen that due to the bending-stretching coupling, the nonlinear behavior of the unsymmetrically laminated perfect and imperfect plates, depending upon the boundary conditions, may be hardening or softening type. Similar behavior also is observed for imperfect isotropic and laminated plates. Lastly, the developments in studying the wave propagation in laminated materials are reviewed. It is seen that a significant effort has been spent on developing appropriate continuum theories for modeling the composite materials. Some recent studies on the linear and nonlinear transient response of laminated materials also are described.

Recent Advances in Analysis of Laminated Beams and Plates, Part II: Vibrations and Wave Propagation

A new shear deformation theory for sandwich and composite plates is developed. The proposed displacement field, which is “m” parameter dependent, is assessed by performing several computations of the plate governing equations. Therefore, the present theory, which gives accurate results, is relatively close to 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature.

A new higher order shear deformation theory for sandwich and composite laminated plates

The energy formulation presented earlier is extended to include the analysis of plane anisotropic rectangular plates with non-uniform plate properties and non-uniform in-plane and distributed normal loadings. The formulation includes linear theory stability analysis, the calculation of natural frequencies and mode shapes, and analysis of displacement due to lateral loads. The extension presented in the present paper is formulated in a manner such that the non-uniform properties and loadings can be handled with no appreciable increase in difficulty over the formulations previously presented except for the evaluation of certain more general integrals. An easily automated method of evaluating these more general integrals is presented.

Analysis of Anisotropic Plates II

In a current article, Whitney [1] has presented exact solutions for certain problems of the bending of unsymmetrically laminated plates. This note considers an approximate method of solution for such plates which, although approximate, yields solutions that compare favorably with those presented by Whitney. Furthermore, since the approximate method reduces the problem to an equivalent anisotropic bending problem, the techniques that have been developed for the analysis of orthotropic and anisotropic plates can be directly utilized. It should be noted that the possibility of using the approximate analysis discussed herein was first suggested in [2] on the basis of experimental evidence. The present note presents an explanation of why this approximation does not introduce large errors, and presents numerical comparisons with Whitney’s exact solutions.

Approximate Solutions for Unsymmetrically Laminated Plates

The requirements on the boundary conditions of the assumed series in a Ritz solution for anisotropic plates is discussed for classical and elastically restrained boundary conditions. A convenient method is indicated for handling elastically restrained boundary conditions which satisfies the natural boundary conditions for orthotropic or iso tropic plates and which utilizes the integration formulas given in a previous paper. Extensive numerical results are presented to illus trate the solution method for series satisfying only the geometrical boundary conditions and for series satisfying both natural and geo metrical boundary conditions.

Anisotropic Plate Analysis-Boundary Conditions

The off-axis tension test is frequently used to characterize the strength behavior of fibrous composite materials. This test is, ideally, uniform tension applied along a single axis. Because this axis does not coincide with one of the principal axes of the material, i.e., since the fibers are neither parallel nor perpendicular to this loading axis, the state of stress or strain referred to the principal axes of the material includes biaxial tension and shear. Such combined stress information is needed to assess the interpolation formulas or failure theories that are based upon 0-degree and 90-degree test specimens. Recently Pagano and Halpin [1] have pointed out that the shear coupling compliance S16 causes a non-uniform state of stress and strain in such tensile specimens when the ends are clamped, and furthermore this non-uniformity cannot be eliminated by simply allowing the ends of the specimen to rotate. An analytical solution was presented in [1] which indicated the aforementioned effect. This solution was obtained by integrating the equations for the plane stress condition from the linear theory of elasticity, but in order to solve the problem only the centerline at the specimen ends was rigidly clamped. Rizzo [2] has presented numerical solutions utilizing the finite element method to indicate the effects of full clamping and free rigid rotation of the ends of the specimens. These results are consistent with the closed form solutions of reference [1] ] and also with the experimental data presented by Wu and Thomas [3]. In the present note, further numerical results are presented which indicate the accuracy of the closed form solution presented in [1], and these results are further used to design a reasonable off-axis tensile coupon.

Off-Axis Tensile Coupon Testing

An experimental study of the uniaxial compressive stability of rectangular boron-epoxy laminated plates is presented. The plates are clamped on the loaded edges and either clamped or simply- supported on the sides. The buckling loads are determined by means of Southwell plots and are compared to analytical results obtained with a previously published method. Experimental results are presented for twenty composite plates and four metal plates. Good agreement between the experimental and analytical results is shown for both symmetrical and unsymmetrical plates, and for plates strongly aniso tropic as well as orthotropic and isotropic.

J.E. Ashton

Papers

Analysis of Anisotropic Plates II

Approximate Solutions for Unsymmetrically Laminated Plates

Anisotropic Plate Analysis-Boundary Conditions

Off-Axis Tensile Coupon Testing

Experimental Study of the Stability of Composite Plates