Thick rectangular plates—II: The generalized Lévy solution

Lévy's classical solution for the problem of a statically loaded, rectangular plate which is simply supported on two opposite edges and has arbitrary boundary conditions specified on the remaining edges is generalized so as to provide a solution for the problem when transverse shear deformation is taken into account. Such solutions are provided for both Mindlin's plate theory and a new plate theory, recently published, which does not require that normals to the undeformed midsurface of the plate remain straight after deformation of the plate. The results of these analyses are relatively uncomplicated generalizations of the classical results for the transverse displacement, or deflection, of the midsurface of the plate and equally simple expressions for the rotations of normals to the midsurface of the undeformed plate. Several particular examples are presented and numerical results are tabulated for several typical plate geometries in order to compare the predictions of the two theories considered. Pertinent observations and comments concerning the analytical and numerical results are made.     

On the theory of optimal, constant thickness fibre-reinforced plates—III

In the earlier papers in this series optimal fibre layouts have been given for a variety of domain shapes and boundary conditions. The general feature which most of these layouts possess is that the reinforcement fibres are parallel to one another within the various zones. In the present paper attention is focused on cases with non-parallel reinforcement. The most common occurrence of such cases is in slabs with a free edge meeting a supported edge at other than a right angle, and at obtuse corners. A number of results for skew slabs and corners are presented.     

Deflections and free vibrations of laminated plates—Levy-type solutions

This study is concerned with the static deflections and natural frequencies of isotropic, orthotropic/laminated composite plates using a Levy-type solution. Mindlin plate theory is applied in conjunction with the state-space concept to find such solutions. A state-space formulation of such plates is composed of variables having physical meanings, such as moments, shear forces, displacements and rotations. The influences of aspect ratio, ratio, fiber orientation angle, laminate-layer arrangement and ratio of moduli have been investigated. Some numerical results from the present analyses are compared with published results and good agreement is found.     

On the theory of optimal, constant thickness fibre-reinforced plates—II

The investigation of optimal fibre reinforcement layouts begun earlier¹ is continued and extended to another important combination of support conditions. It is also shown that all the deflected shapes for the slabs discussed in the previous paper are everywhere both displacement and slope continuous, and hence are global optima.     

Cross-sectional deformations of rectangular hollow sections in bending: Part I — experiments

This investigation deals with deformations of individual cross-sectional members as flanges and webs in bending of rectangular hollow sections. Part I describes the experimental work, while analytical models developed to predict pre- and post-buckling deformations are presented in a paper to follow (Part II). The experimental program involved rectangular single- and double-chamber aluminium alloy AA6060 extrusions with three different wall thicknesses. The profiles were given two distinct heat treatments to obtain different hardening characteristics. Multiaxial tests were performed to determine the mechanical properties of the materials. The profiles were then bent into a number of different bend radii. Measurements of strains, curvatures, deflections and bending forces were taken. The results show that cross-sectional distortions take place from the very beginning of bending; at first in the form of a uniform sagging-like deformation along the entire length of both sides of the bend until the inner (compressive) flange buckles into several half-waves, superimposing the pre-deformation modes. The instant at which buckling occurs is found to be mainly associated with the width-to-thickness ratio of the flange and the strain hardening characteristic of the material. The magnitude of pre- and post-deformations, however, appears to be more directly related to the actual width of the flange than to its slenderness. The material stress–strain curve is shown to have an increasingly effect on the distortions of members directly sustained to buckling as bending proceeds beyond the onset of buckling, leading to severely concentrated deformations for sections made of low hardening materials. The material has less impact on sagging of the outer (tensile) flange.     

Uniformly loaded, clamped, cross-ply laminated, elliptic plates—An initial failure study

An exact polynomial solution for a uniformly loaded, antisymmetric cross-ply laminated, clamped elliptic plate is combined with the Tsai-Hill failure criterion for an initial flexural failure analysis of GFRP and CFRP plates. Non-dimensionalised initial failure loads and associated plate centre deflections—both of which are significantly dependent on the direction of the applied loading—are presented graphically for the complete range of aspect ratios for plates consisting of varying numbers of laminae. It is also demonstrated that changes in the in-plane boundary conditions produce negligible changes in the initial failure response of the plate.     

Rigid-plastic modelling of blast-loaded stiffened plates—Part II: Partial end fixity, rate effects and two-way stiffened plates

A new rigid-plastic analysis of stiffened plates subjected to uniformly distributed blast loads is developed. In this first part of a two-part paper, a uniform one-way stiffened plate with clamped ends is modelled as a singly symmetric beam, comprised of one stiffener and its tributary plating. Rigid-plastic analysis is then applied to this beam using an idealized piecewise linear bending moment-axial force capacity interaction relation or yield curve. Two solutions to the response are developed. The first solution is in closed form and is based on the solution of the resulting linearized differential equations. The second solution is obtained by approximating the response as a sequence of instantaneous mode responses, where the mode shapes are determined by an extremum principle which maximizes the rate of change of the kinetic energy. This latter solution may be extended to cases involving non-rigid boundaries and two-way stiffening and this is done in the second part of this paper. Here, the two solution methods are applied to several examples of one-way stiffened plates subjected to various blast-type pulses. Good agreement is obtained between the present results and those from elastic-plastic beam finite element and finite strip solutions.     

Effects of fluid inertia forces on the squeeze film characteristics of conical plates—ferromagnetic fluid model

On the basis of the Shliomis ferromagnetic fluid model, this paper is mainly concerned with the influences of convective fluid inertia forces in magnetic fluid‐based conical squeeze film plates in the presence of external magnetic fields. By applying the averaged momentum principle, a lubrication equation governing the film pressure is derived. Some previous contributions can be obtained from special cases of the present studies. Comparing with the non‐inertia non‐magnetic case, better squeeze film performances are predicted for the magnetic fluid‐based conical plates operating with a larger value of the inertial parameter of fluid inertia forces, the volume concentration of ferrite particles and the strength of applied magnetic fields. Some numerical results with specific cone angles are also provided in tables for engineering applications. Copyright © 2012 John Wiley & Sons, Ltd.     

Flywheel energy storage—I: Basic concepts

The basic concepts of flywheel energy storage systems are described in the first part of a two part paper. General equations for the charging and discharging characteristics of flywheel systems are developed and energy density formulas for flywheel rotors are discussed. It is shown that a suspended pierced disk flywheel is competitive with the super-flywheel designs currently being suggested in the literature.In Part II the details of a magnetically levitated spokeless ring flywheel design are provided.     

On plane Prager-structures—I

Prager-structures differ from classical least-weight trusses (or Michell continua) in two respects: the sign of all member forces must be the same and the vertical location of all external loads is to be optimized. A systematic method for constructing plane Prager-structures for any system of vertical loads is outlined and illustrated with examples. It is established that plane Prager-structures (a) always consist of a single funicular and (b) the optimal elevation of the loads also provides an influence line for the minimum weight.     

Hybrid analytical–numerical solution for the shear angle in orthogonal metal cutting — Part I: theoretical foundation

In metal cutting, the shear angle is considered as a fundamental parameter that defines the plastic deformation and the geometry of the process. The present paper presents a further development of the energy method for prediction of the shear angle in case of orthogonal metal cutting. Parallel-sided shear zone model is utilized to describe the geometry of chip formation. The material velocity in the primary shear zone is allowed to change gradually from the bulk material velocity to the chip velocity. The interaction between chip and tool in the secondary shear zone is modeled as sticking to sliding transition. The work material is characterized by an empirical equation, which allows for the influence of temperature, strain, and strain rate as well as their histories. To take into consideration the influence of the temperature on the work material properties, a finite element model (FEM) of heat transfer is employed. The FEM is developed as an adaptive model to reflect the change in the domain geometry. As the work material properties strongly depend on the temperature, an overall iterative calculation procedure including FEM is essential. In Part I, the theoretical basis of the model is described. In Part II the predicted values of the shear angle are compared with data from machining 0.18% C carbon steel over a range of cutting conditions and tool geometry.     

Cross-sectional deformations of rectangular hollow sections in bending: Part II — analytical models

In this part, analytical models to predict the deflection of cross-sectional members such as flanges and webs are developed. The models are based on the deformation theory of plasticity along with the energy method, using appropriate shape functions capable of including the restraining effect of adjacent members. The present method provides explicit solutions of cross-sectional deformations prior to buckling, onset of buckling, as well as post-buckling deformations at different stages of bending. The predictions show that the suck-in of the tensile flange is closely related to geometry parameters, particularly the flange width. Plastic anisotropy appears to be the most significant material parameter. The width-to-thickness ratio tends to be the governing parameter with respect to buckling of the inner (compressive) flange. Also, the strain hardening of the material has a major effect on onset of buckling as well as post buckling deformations. Upon continued bending after buckling, the wavy deformation of the inner flange develops more rapidly than the more uniform deformation of the outer (tensile) flange. For relatively compact sections, however, the deformation mode of the compressive flange resembles that of the tensile flange without any typical buckling waves. There are also obvious interactions between deformations of different members. Comparing the theoretical predictions with the experimental results presented in Part I, a reasonably good agreement was found.     

A unified generalized thermoelasticity; solution for cylinders and spheres

A new unified formulation for the generalized theories of the coupled thermoelasticity based on the Lord–Shulman, Green–Lindsay, and Green–Naghdi models is proposed in this paper. The unified form of the governing equations is presented by introducing the unifier parameters. The formulations are derived and given for the anisotropic heterogeneous materials. The unified equations are reduced for the isotropic and homogeneous materials. Transforming the governing equations into the Laplace domain, they are analytically solved in the space domain for a hollow sphere and cylinder, where a parameter is introduced to consolidate the solution for the sphere and cylinder in a unified form. A thermal shock load is applied to the inner surface of the sphere and cylinder and the results are presented using a numerical inversion technique of the Laplace transform. The results are validated with the known data in the literature.     

Torsion of compound bars—A relaxation solution

This paper deals with a finite-difference solution of the torsion problem of nonhomogeneous and compound prismatic bars. General, governing equations for both problems are developed and the boundary conditions for an interface between parts composed of homogeneous but different materials are stated. The case of multiply connected regions is discussed and integral conditions, analogous to the conditions in multiply connected homogeneous bars, are developed.

Examples illustrating various types of problems are worked out and the accuracy of the method demonstrated by comparison with some known solutions.     

Indenting with pyramids—I. Theory

Using an approximate method, expressions are obtained for the load necessary to indent a rigid-plastic non-hardening solid to a given depth with square-based and triangular-based pyramids. Experimental results will be presented in Part II.     

Modelling of the rolling process—I: Inhomogeneous deformation model

The rolling of material results in the inhomogeneous plastic deformation of the material in the arc of contact between the rolls. To assess the influence of this inhomogeneous deformation, the parameter, ω, developed by Orowan has been incorporated into a theoretical model of the rolling process. The results are compared with recent results published by Alexander in which the assumption of homogeneous deformation or ‘plane sections remaining plane’ was considered.     

On bend-stretch forming of aluminum extruded tubes — I: experiments

Tubular aluminum frame parts for automotive applications are best produced by extrusion. The tubes are then cold formed to the required shape by prestretching, pressurizing and bending them over rigid dies. Tension prevents buckling of the compressed side and significantly reduces the springback on unloading. An unwanted byproduct of the process is distortion of the cross section. It has been found that modest levels of pressure can reduce this distortion. The selection of the level of tension and pressure for optimum forming is presently empirical. The study discussed herein seeks to develop a scientific basis for optimizing forming processes such that buckling is avoided and distortion and springback are minimized. Part I describes a custom bend-stretch-pressure forming facility developed for the study. The facility is operated by one pneumatic and two servohydraulic closed-loop systems. This allows computer control of the process, and affords selectable loading histories. The planar forming process was modeled by approximating the tube as a nonlinear elastic–plastic beam which can undergo large rotations. The model was shown capable of reproducing accurately the loading history experienced by different sections along the length of the part during forming. Representative results from forming experiments involving rectangular aluminum are presented. The results are used to discuss the effect of friction, tension and pressure on the cross-sectional distortion, springback and net elongation of the part. Part II presents a model for establishing the cross-sectional distortion induced during forming. The model is used in conjunction with experimental results to establish ways of optimizing the process.     

Time–frequency analysis of tribological systems—part I: implementation and interpretation

A novel technique adapting the time–frequency analysis has been utilized to characterize stationary and non-stationary signals from tribological interactions. This representation displays time, frequency, and signal magnitude to decipher signals emanating from such interactions. Short-time Fourier transform, Wigner, Coi–Williams, and Zhao–Atlas–Marks distributions are suited to represent stationary and non-stationary signals. Some of the most complex tribological phenomena involve head–disk interactions in magnetic recording systems. Examples drawn from practical head–disk interface tests are analyzed by using the fast Fourier transform algorithm to illustrate the dynamic features of various distributions. Time–frequency representation of output spectrums of laser doppler vibrometer (LDV), strain gage sensor, and acoustic emission (AE) sensor obtained from head–disk experiments giving evidence of stationary and non-stationary behavior are investigated.     

Micro-electrodischarge machining using water as a working fluid—I: micro-hole drilling

Micro-electrodischarge machining (medm) using water as a working fluid is systematically studied to find its characteristics. As a result, the unique advantages of high removal rate, low electrode wear and consequently higher working efficiency, without formation of carbonaceous materials are found under optimum experimental conditions, as compared with the case when kerosene is used. This was achieved by the choice of suitable combinations of electrode and workpiece materials and electrode polarity. Use of a tungsten electrode with straight polarity is exceedingly good with respect to high removal rate and low electrode wear. This makes it possible to obtain a non-tapered straight micro-hole around 0.1 mm in diameter with a certain anticipative working gap. The advantageous properties of this machining method are effectively utilized to drill deep micro-holes with a ratio of depth to diameter of the order of 10–17 as in the case of 2.9 mm depth and 0.17 mm diameter, which is superior to the limits achievable with both electrodischarge drilling (edd) using kerosene and mechanical drilling with a micro-drill.