D.J. Dawe

Use of the finite strip method in predicting the behaviour of composite laminated structures

Abstract A description is given of the use of the finite strip method (FSM) in determining the behaviour of composite laminated, prismatic plate and shell structures, with emphasis placed on relatively recent work conducted at The University of Birmingham. Both the semi-analytical and the spline variants of the method are described, and some attention is also paid to “exact” strips. Consideration is given to analyses conducted in the contexts of first-order shear deformation theory and of classical, or thin, theory. The calculation of buckling stresses and natural frequencies of vibration is discussed in detail for single span structures and then, using the spline finite strip approach with variable knot spacings, for multi-span structures and stepped structures. An account is given of the use of the FSM in predicting the post-buckling response of plate structures to progressive end-shortening strain. Brief description is given of the use of the method in predicting the thermal buckling and the transient response to dynamic loading of flat plates. Finally, the calculation of buckling stresses and natural frequencies of sandwich plate structures is considered, based on the adoption of a three-zone plate theory. Numerous examples of the application of the FSM are included in the paper.

Nonlinear transient analysis of rectangular composite laminated plates

The semi-analytical finite strip method is developed for the analysis of the geometrically nonlinear response to dynamic loading of rectangular composite laminated plates. The plates have simply supported ends and their properties are evaluated in the context of first-order shear deformation plate theory. The applied loading acts normal to the plate surface but otherwise may be of a general nature with respect to space and time. Solution to the nonlinear dynamic problem is obtained through use of the Newmark time-stepping scheme in association with Newton–Raphson iteration. Applications are described which relate to isotropic and orthotropic plates, and to laminates. In general a close comparison is demonstrated between the predictions of the developed finite strip approach and those of a finite element method.

Buckling and vibration of finite-length composite prismatic plate structures with diaphragm ends, part I: finite strip formulation

Abstract A finite strip approach is presented for the determination of buckling stresses and natural frequencies of vibration of prismatic plate structures of finite length which have diaphragm ends. The structures are assembled from plate flats which generally are laminates of fibre-reinforced composite material and a very broad specification of material properties is allowed which includes anisotropy and bending/stretching coupling. Modes of buckling or vibration may be of an overall or a local type. To accommodate the presence of applied in-plane shear stress (as well as biaxial direct stress) and material anisotropy the finite strip method is of the multi-term type, i.e., in the strip formulation each displacement-type component is represented by a finite series of products of longitudinal trigonometric functions and crosswise polynomial functions. The out-of-plane properties of plate flats are based in turn on the use of first-order shear deformation plate theory (accounting for the through-thickness shear effects which are often significant) and classical plate theory. Strip properties, as represented by the elastic stiffness matrix, the geometric stiffness matrix and the consistent mass matrix, are derived through potential energy principles and are approximate. However, through the use of a repetitive substructuring scheme, so-called superstrips are created economically with properties which are effectively exact within the confines of the particular background theory. Following a transformation to a global coordinate system, incorporating the effects of eccentric connections as well as accounting for the arbitrary inclination of plate flats, multi-level substructuring procedures are invoked to reduce the number of effective freedoms of the structure. The eigenvalues are determined using the extended Sturm sequence-bisection approach. This paper contains no numerical applications: consideration of such applications is deferred to a companion paper.

Overall and local buckling of sandwich plates with laminated faceplates, part II: Applications

After discussion of related past work, a description is given of a B-spline finite strip method (FSM) for predicting the buckling stresses of rectangular sandwich plates. The core is represented as a three-dimensional solid in which the in-plane displacements vary quadratically through the thickness whilst the out-of-plane displacement varies linearly. The faceplates may in general be composite laminates which are represented, in turn, as either shear-deformable plates or classically-thin plates. The displacement field of a finite strip requires the representation of 12 (for shear-deformable faceplates) or eight (for thin faceplates) fundamental quantities over the strip plan area, with each quantity expressed as a summation of a series of products of longitudinal B-spline functions and crosswise shape functions. Full details are given of the development of elastic stiffness and geometric stiffness matrices. The nature of the solution of the buckling problem is described and this incorporates multi-level sub-structuring, through the use of superstrips. This allows the efficient prediction of buckling stresses for both overall modes and highly-localised, wrinkling-type modes, but description of applications of the developed capability is deferred to the companion Part II paper.

Buckling and vibration of finite-length composite prismatic plate structures with diaphragm ends, Part II: computer programs and buckling application

Abstract In a companion paper details have been given of multi-term finite strip formulations for determining the buckling stresses and natural frequencies of vibration of composite prismatic plate structures of finite length and with diaphragm ends. The formulations are based in turn on the use of shear deformation plate theory and classical plate theory, and include a number of versatile and refined features. Here the implementation of the theoretical work is described. The computer programs BAVAMPAS and BAVAMPAC are introduced and discussed briefly. Thereafter the main body of the paper is concerned with describing numerical applications using these programs. These applications are restricted in this work to problems of buckling under a live applied stress system which might comprise shear stress and direct stress. A wide range of buckling applications is considered, concerning single plates through to complicated plate structures and involving homogeneous, isotropic material through to anisotropic laminated composite material and, in one case, to an unsymmetrically-laminated structure exhibiting bending-stretching material coupling.

Spline finite strip analysis of the buckling and vibration of rectangular composite laminated plates

Abstract A spline finite strip capability is presented for predicting the buckling stresses and natural frequencies of rectangular laminated plates. The plates may have arbitrary lay-ups and general boundary conditions. The spline finite strip method is first developed in the context of first-order shear deformation plate theory and then, by reduction, the method is also developed in the context of classical plate theory. In both approaches the superstrip concept is incorporated into the solution procedure. A considerable range of types of application is described and it is demonstrated that the spline finite strip method is versatile, with good convergence characteristics and accuracy. In these applications, frequent comparison is made with the results of other approaches which comprise a spline Rayleigh-Ritz method, a finite element method, an analytical Rayleigh-Ritz method and a semi-analytical finite strip method.

The vibration and stability of symmetrically-laminated composite rectangular plates subjected to in-plane stresses

Abstract Consideration is given to the twin problems of the elastic buckling of rectangular, symmetrically-laminated composite plates and of the vibration in the presence of applied in-plane stress of such laminates. First-order shear deformation plate theory provides the mathematical model of plate behaviour and the Rayleigh-Ritz and finite strip methods are used to generate numerical results for laminates of thin and moderately thick geometry, with various combinations of standard plate edge conditions. The applied stresses include uniform shear stress as well as direct stresses, and anisotropic material properties can be included. The presented results demonstrate the accuracy of the numerical methods and highlight the very significant influence that transverse shear and related thickness effects can have in the subject problems.

Dynamic instability of composite laminated rectangular plates and prismatic plate structures

Abstract Periodic dynamic loadings may cause dynamic instability of a structure through parametric resonance. In this paper, a B-spline finite strip method (FSM) is presented for the dynamic instability analysis of composite laminated rectangular plates and prismatic plate structures, based on the use of first-order shear deformation plate theory (SDPT). The equations of motion of a structure are established by using Lagranges formulation and they are a set of coupled Mathieu equations. The boundary parametric resonance frequencies of the motion are determined by using the method suggested by Bolotin through a novel development which incorporates the Sturm sequence method and the multi-level substructuring technique to achieve reliability, efficiency and accuracy. Various loading patterns, arbitrary lamination and general boundary conditions are accommodated. A variety of numerical applications is presented to test the developed method and to study the dynamic instability behaviour of single plates and of complicated plate structures under various types of dynamic loading. A dynamic instability index (DII) is devised to measure the degree of instability against certain parameters which include the thickness-to-length ratio, the degree of orthotropy, the fibre orientation, the loading pattern and the boundary conditions.

Buckling and vibration of shear deformable prismatic plate structures by a complex finite strip method

Abstract A finite strip method is presented for the determination of buckling stresses and natural frequencies of vibration of prismatic plate structures assembled from plate flats, which generally are laminates of fibre-reinforced composite material. The finite strip method is of the single-term type, corresponding to the assumption of sinusoidal longitudinal spatial variation of displacement and force quantities. Anisotropic material behaviour and applied in-plane shear stress are accommodated by expressing the strip displacement field in terms of complex quantities. The out-of-plane properties of plate flats are based upon the use of first-order shear deformation plate theory. A family of finite strip models is described and a sub-structuring procedure is utilised to reduce the size of the eigenvalue problem. Presented numerical results reveal the high accuracy and good convergence characteristics of the method, as well as indicating the influence of through-thickness shear effects in a range of circumstances.

Finite strip post-local-buckling analysis of composite prismatic plate structures

Abstract A finite strip method is described for the analysis of the geometrically non-linear elastic response of composite laminated, orthotropic prismatic plate structures subjected to progressive uniform end shortening. Attention is restricted to local buckling/post-buckling behaviour so that certain simplifying assumptions related to the insignificance of movements of plate junctions can be invoked. Analyses are based on the use of both classical plate theory and first-order shear deformation plate theory and a range of finite strip models is available for use in the contexts of each of these plate theories. A description is given of a number of applications involving the post-local-buckling behaviour of box sections and top-hat-stiffened and blade-stiffened panels. In one application considering a laminated box section, results are generated using a commercial finite element package and these are seen to compare closely with the predictions of the presented finite strip method.