Rayleigh-Ritz Energy Method Research Articles

Free in-plane vibration analysis of rectangular plates by the method of superposition

The superposition method is introduced as a means for obtaining analytical-type solutions for free in-plane vibration of rectangular plates. The governing differential equations and boundary conditions are expressed in dimensionless form. The problem of free in-plane vibration of the completely free rectangular plate is resolved for illustrative purposes. Convergence is found to be rapid and excellent agreement between computed results and those obtained by previous authors utilizing the Rayleigh–Ritz energy method is obtained. It is pointed out that following procedures analogous to those utilized in resolving lateral plate vibration problems, in-plane free vibration problems related to point supported plates, plates with in-plane elastic boundary support, etc., are now amenable to solution by this method.

The behaviour of damage tolerant hat-stiffened composite panels loaded in uniaxial compression

Damage tolerant hat-stiffened thin-skinned composite panels with and without a centrally located circular cutout, under uniaxial compression loading, were investigated experimentally and analytically. These panels incorporated a highly postbuckling design characterised by two integral stiffeners separated by a large skin bay with a high width to skin-thickness ratio. In both configurations, the skin initially buckled into three half-wavelengths and underwent two mode-shape changes; the first a gradual mode change characterised by a central deformation with double curvature and the second a dynamic snap to five half-wavelengths. The use of standard path-following non-linear finite element analysis did not consistently capture the dynamic mode change and an approximate solution for the prediction of mode-changes using a Marguerre-type Rayleigh-Ritz energy method is presented. Shortcomings with both methods of analysis are discussed and improvements suggested. The panels failed catastrophically and their strength was limited by the local buckling strength of the hat stiffeners.

Delamination buckling of stitched laminates

Delamination buckling of stitched laminates can cause high interlaminar stresses which, in turn, lead to delamination growth. This study presents a three-dimensional model for the analysis of local buckling of stitched composite laminates with an embedded elliptical delamination near the surface. Effects of stitching, delamination sizes, delamination orientation, and stacking sequence on buckling strains are investigated by the Rayleigh–Ritz energy method. The results show that stitching has a significant effect on buckling strains and buckling mode. Buckling strains obviously increase with increase in the equivalent spring constant, k, of the stitching thread, but decrease gradually in amplitude as the delamination sizes increase. In addition, the delamination orientation and stacking sequence simultaneously affect the buckling strain.

An analysis of impact force in plain-weave glass/epoxy composite plates subjected to transverse impact

In the present study, a new analytical method is developed for predicting the impact force from the dynamic strain of composite plates subjected to transverse impact. For this, two assumptions are introduced in this study. Firstly, the impact force and dynamic strain can be separated into frequency and amplitude. Secondly, the amplitude of the impact force corresponds to that of the dynamic strain at any frequency. By applying the Rayleigh–Ritz energy method and Lagrange’s principle to a rectangular plate, the governing equation is derived, and the equation is solved by an optimal design technique. To verify this analytical method, impact tests are performed on plain-weave glass/epoxy composite plates having various thicknesses. The impact forces obtained by this analytical method agree well with the experimental results for all plate thicknesses studied. Additionally, the dynamic response, such as the impact force, is greatly influenced by the thickness. This behavior is attributed to the different energy absorbing mechanisms: the absorbed energy is governed by the plate deflection in a thin plate, whereas the energy is governed by the local indention in a thick plate.

A Simplified Approach to Analysis of Large Deflections for Rectangular Plates with a Cross-Stiffener.

In this study, the nonlinear Berger equation for large deflection problems of isotropic plates and the finite-difference method are used to obtain solutions for deformation in rectangular elastic plates with a cross-stiffener. Numerical solutions are presented in order to illustrate the influence of the stiffener on the center deflection in uniformly loaded rectangular plates, which are supported by two beams orthogonally intersecting at the center of the plate. The numerical procedure is simpler than the procedures based on the von Karman theory, and takes into account the continuity conditions related to particular loads more rigorously than tha variational methods, such as the Rayleigh-Ritz energy method and finite element method.

Buckling mode transition in hat-stiffened composite panels loaded in uniaxial compression

A combined experimental and analytical study of a hat-stiffened carbon-fibre composite panel loaded in uniaxial compression was investigated. A buckling mode transition was observed in the panel's skin bay which was not captured using non-linear finite-element analysis. Good correlation between experimental and numerical strain and displacement results was achieved in the prebuckling and initial postbuckling region of the loading history. A Marguerre-type Rayleigh-Ritz energy method was applied to the skin bay using representative displacement functions of permissible mode shapes to explain the mode transition phenomenon. The central criterion of this method was based on the assumption that a change in mode shape occurred such that the total potential energy of the structure was maintained at a minimum. The ultimate strength of the panel was limited by the column buckling strength of the hat-stiffeners.

A Nonlinear Multi-Domain Stress Analysis Method for Surface-Mount Solder Joints

Solder joint fatigue failures are a potential reliability hazard in surface-mount electronic packages under cyclic thermal loading environment. Proper design and reliability assessment are thus crucial to ensure the fatigue endurance of the electronic packages. Accurate modeling of the stress and strain fields within the solder joint under cyclic thermal loading condition is of extreme importance since ultimately, a reasonable fatigue life estimation depends not only on a appropriate fatigue model, but more fundamentally, on accurately predicted stress and strain fields. Modeling stress and strain fields in solder joint in surface-mount electronic packages have never been an easy task since solder undergoes elastic, plastic and time dependent creep during each loading and unloading cycle. Some of the existing closed-form stress analysis models tend to oversimplify this complicated viscoplastic stress state, thus failing to give a reasonable prediction of the solder joint fatigue endurance. Extensive finite element analyses require prohibitive investment in terms of the analysis time and analyst expertise, especially when full scale elastic, plastic and creep analyses are performed. A generalized multi-domain approach proposed earlier by the authors is further developed in this paper to obtain the stress and strain fields in J-leaded surface-mount solder joint undergoing elastic-plastic deformation, under cyclic thermal environment (Ling et al., 1995). The Rayleigh-Ritz energy method based on a multi-field displacement assumption is used. In a previous paper (Ling et al., 1995), the results for analysis within elastic region had been demonstrated and were proved to be in agreement with finite element analysis. In this paper we further develop the methodology into plastic deformation region. Hysteresis loops for both the global and the local CTE mismatch problem can finally be generated. Results for two-dimensional elastic-plastic analysis are presented in the current paper. Creep deformation can be further modeled with this scheme by using time-stepping incremental techniques, and will be presented in a future paper. The final goal of this research is to predict the stress, strain and energy density distributions in the solder joint with reasonable accuracy. The fatigue assessment of the solder joint can then be performed by combining results from this stress analysis model with an appropriate damage model, for example, the energy-partitioning fatigue model (Dasgupta et al., 1992).

A rayleigh-ritz approach to modeling bending and shear deflections of gear teeth

The Rayleigh-Ritz energy method was used to study the shear effect of an involute gear tooth. The gear tooth was simulated by a tapered plate model subjected to a concentrated load. The plate deflections, including shear deformation, were determined and compared with the theoretical values and experimental data. The comparisons indicate that the deflections of the shear plate model are higher than those computed from the thin plate models which neglects the shear effects. On the other hand, the experimental results are higher than those of the shear plate model due to the base flexibilities of the experimental models. The shear model deflections are also shown to be in excellent agreement with finite element results. The shear plate model could replace the finite element model since it is more computationally efficient and its results are accurate enough for most engineering purposes.

Integrated aeroelastic control optimization of laminated composite lifting surfaces

An integrated design approach for aeroelastic control of laminated composite lifting surfaces is formulated and demonstrated. The design is posed as an optimization problem to determine the optimum ply orientation angles and the control law which maximizes the aeroelastic instability speed of an actively controlled laminated composite lifting surface subject to limited control authority and structural frequency constraints. The lifting surface is simulated by a rectangular symmetric cantilevered composite plate. The formulation incorporates the Rayleigh-Ritz energy method, two-dimensional incompressible unsteady aerodynamic theory, optimum control design, and parameter optimization techniques. The particular control configuration consists of two point-force inputs applied at the leading and trailing edges of the lifting surface. In conjunction with the laminated compositeplate structural model, the control configuration effectively simulates an active flexible lifting surface. The effectiveness of the integrated approach is evaluated by comparing the optimum designs obtained by the method with designs obtained by a design procedure using aeroelastic tailoring and optimum control design as well but in a nonintegrated fashion. The illustrative studies resulted in a design that exhibited 102% higher aeroelastic instability speed with 294.4% lower control cost than designs obtained by a nonintegrated design procedure.

Buckling and post-buckling behaviour of rectangular plates under linearly varying compression and shear: part 1 — Theoretical analysis

The buckling and post-buckling behaviour of thin, rectangular, simply supported flat plates subjected to end compression which varies linearly in the direction of loading and simultaneously loaded in shear is the subject of this paper. Part 1 presents a theoretical analysis of the problem. In this analysis the Rayleigh-Ritz energy method is utilised, and numerical minimisation of the total potential energy is applied to obtain the magnitude of the parameters of the assumed deflection functions. These are then used to define and describe other post-buckling characteristics of the plate. A simplified imperfection analysis is used to examine certain cases. The wide range of results presented covers many aspects of plate post-buckling behaviour.

Buckling of antisymmetric angle- and cross-ply rectangular plates under shear and compression

The paper deals with buckling of antisymmetric angle- and cross-ply, simply supported rectangular flat plates under shear and compressive loads. The Rayleigh-Ritz energy method is used to find the buckling load, by assuming a double series form of trial functions for the buckled displacement field. Both the symmetric and antisymmetric buckling modes are considered. The effect of the number of layers, the aspect ratio and the lamination angle on the buckling load is studied. Results are presented for plates subjected to (i) shear loads only and (ii) both shear and uniaxial compressive loads; for the latter case, the results are presented in the form of interaction curves.

Buckling and Post-Buckling Behavior of Elliptical Plates: Part I—Analysis

A polynomial series expansion for displacements is used in conjunction with the Rayleigh-Ritz energy method to produce buckling and post-buckling stress solutions for an elliptically-shaped surface layer that has been delaminated from the main load-bearing body. Plate deformations are induced by a combined in-plane displacement field applied to the plate boundary and normal pressure. Convergence of the plate solution is assessed by systematically increasing the number of displacement terms in the series expansion. The convergence of membrane and bending stresses at the plate boundary was generally slow and nonuniform. The degrees-of-freedom necessary for a satisfactory solution typically increase with increasing complexity or magnitude of the plate deformations. By employing as many as 77 displacement terms, practically exact stress solutions are obtained for a wide variety of basic delamination plate problems. The proposed solution procedure is highly efficient and economical, and it may be easily extended to other plate geometries or loading conditions.

Transient response of graphite/epoxy and Kevlar/epoxy laminates subjected to impact

The influence of different parameters on the impact behavior of laminated composite plates is considered analytically. A Rayleigh-Ritz energy method was used to spatially discretize the time-varying boundary value problem and a set of coupled, ordinary differential equations in time were obtained based on the discretized system Lagrangian. The effects of shearing deformation, bending-twisting coupling, and nonlinear contact behavior were included in the model. The resulting equations were integrated using the implicit Newmark beta method without the effects of rotary inertia. The results indicate that the effective mass of the plate is often an important effect in the response to impact events. In general, the influence of the constitutive behavior dominates for very low velocity impact, whereas the target mass properties become more important as the impactor velocity increases. This importance of mass clearly shows that impactor kinetic energy is not sufficient to characterize the impactor as the impactor mass is shown to have a large influence on the resulting dynamic behavior. In addition to these parameters, the effects of preload and material properties are considered and discussed.

Effect of Finite Width on Deflection and Energy Release Rate of an Orthotropic Double Cantilever Specimen

The problem of an orthotropic cantilevered plate subjected to a uniformly distributed end load is solved by the Rayleigh-Ritz energy method. The result is applied to laminated composite, double cantilevered specimens to estimate the effect of crack tip constraint on the transverse curvature, deflection and energy release rate. The solution is also utilized to determine finite width correction factors for fracture energy characterization tests in which neither plane stress nor plane strain conditions apply.

The Buckling of a Delaminated Sublaminate on an Elastic Foundation

A one-dimensional model of a delamination in a laminated structure is considered in order to evaluate the effective edge fixity coefficient and the load at which the delaminated sublaminate buckles. A Rayleigh-Ritz energy method is applied to the model and the ef fective boundary fixity coefficient, c' , and the relative delamination tip deflection, wd/wc, are determined. Appreciable reductions in the buckling load, as compared to classical clamped boundary conditions, and nonzero deflections at the delamination front are pre dicted. The model also provides a rule of thumb as to the area around a delaminated sub laminate which must be modeled as an elastic foundation in order to properly predict the buckling behavior of the sublaminate.

A micromechanical model for non-linear viscoelastic behavior of particle-reinforced rubber with distributed damage

A model based on micromechanics for predicting effective viscoelastic stress-strain equations and microcrack growth in particle-reinforced rubber (or other relatively soft viscoelastic matrix) is described. Geometric idealization of the microstructure follows that of the composite spheres assemblage and generalized self-consistent scheme originally used for linear elastic composites without damage. The approach combines a perturbation analysis of the matrix, which becomes more accurate as the particle volume fraction is increased, with the Rayleigh-Ritz energy method for predicting mechanical response of the composite. Results for linear elastic behavior with crack growth are first obtained, and then extensions to linear and non-linear viscoelastic behavior are discussed. It is shown that the elasticity theory may be easily extended to predict mechanical response of a viscoelastic composite, and that an approximate equation governing microcrack growth is analogous to one for an aging elastic material. Finally, a limited assessment of the theory is made through comparison with some existing theoretical effective modulus results and experimental data on a particle-filled rubber.

Buckling and vibration of composite folded-plate structures of finite length in combined shear and compression

An accurate method is developed for analysing the critical buckling loads of folded-plate structures, notably longitudinally stiffened panels, of finite length, with diaphragm supported ends, when the component plates are loaded in combined biaxial compression and shear and the mode is of an overall type. The plates may be either isotropic or orthotropic; the latter are very nearly representative of most laminated fibre-reinforced composites. A Rayleigh-Ritz energy method is used, with the mode represented by a series of exact buckling modes of the structure subjected to the biaxial compressive loads alone. This results in a standard algebraic eigen-value problem, of order equal to the number of compression modes included in the series. A simple way of ranking the compression modes in order of relative importance is derived. The analysis has been incorporated into a general purpose computer program called casiopeia, which is capable of generating automatically any specified number of compression modes, and thence computing points on the interaction curve relating critical values of the compression and shear for any prismatic folded-plate structure of this type. Its efficiency is enhanced by providing a limited capability for assembling the structure from substructures. The program was applied to six stiffened panels that have previously been analysed by N. A. S. A. using an extremely expensive finite element method, and which provide the only accurate solutions available for comparison. The results were in close agreement. Finally it is explained how small modifications to the analysis enable natural frequencies of vibration to be obtained for folded-plate structures when the component plates carry combined compression and shear.

Compression Buckling Tests of Laminated Graphite-Epoxy Curved Panels

Ten different orientation arrangements (in various panel thicknesses) were used in fabricating 72 panels. Eighty-four tests were performed, and the range of R/t studied was 100 to 400. All specimens had a 12-in. outside radius and were 13-in. long by 9-in. wide. The results included development of the moire grid-shadow technique and the South well plot method for nondestructive testing as well as observations of the effects of orientation and R/t on the buckling behavior and imperfection sensitivity of such shells. Satisfactory correlation was obtained with a laminated anisotropic curved panel analysis using the Rayleigh-Ritz energy method, which included a crude assumption that the thickness variations in the panels could be used to measure imperfection amplitude.

Flexural Vibrations and Timoshenko's Beam Theory

This paper is a study of flexural elastic vibrations of Timoshenko beams with due allowance for the effects of rotary inertia and shear. Two independent formulations are developed, one based on the concepts proposed by Timoshenko and the other on the extended Rayleigh-Ritz energy method. The results obtained from the two formulations are matched in order to determine the correcting coefficients in the simplified formulae which are proposed for the frequency of the first two flexural branches. The proposed formulae are shown to achieve greater accuracy in describing the flexural motions. New accurate solutions are offered for several cross-sectional geometries which enable the accuracy of the available methods to be assessed. Correction coefficients are evaluated for several sections and are compared with previous work.

Analysis and Behavior of Microsound Surface Waveguides

An analysis is presented for the evaluation of frequency spectra and mode configurations of orthotropic microsound waveguides of arbitrary cross section. The analysis is based on the extended Rayleigh-Ritz energy method together with the concept of discretization of the guide cross section. The formulation satisfies all the bound- ary conditions and extracts with high accuracy a predeter- mined number of lowest frequencies and their modes. The method is employed to study the behavior of several different types of microsound surface waveguides of general interest. The results and their discussions reveal the influences of cross- sectional geometry and material properties on guide's perfor- mance. The solutions offered are general and provide a useful basis for the design of microsound surface waveguides.