Thick rectangular plates—I: The generalized Navier solution

Navier's classical solution for the problem of a statically loaded, rectangular plate which is simply-supported on all edges is generalized so as to provide a solution for the problem when transverse shear deformations are taken into account. Such solutions are provided for both Mindlin's plate theory and a new plate theory, recently published, which allows for warping of transverse sections of the plate. The results of the analyses are simple expressions for the Fourier coefficients in the expansions representing the transverse deflection and rotations of normals to the plate midsurface. Although both theories predict the same transverse deflection, the rotations and inplane stresses are different for the two theories. Some observations and comments concerning the effects of transverse shears on the deflections of plates and inplane stresses are made.     

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.     

Accurate representation of interference colours (Michel–Lévy chart): from rendering to image colour correction

Here a work flow towards an accurate representation of interference colours (Michel‐Lévy chart) digitally captured on a polarised light microscope using dry and oil immersion objectives is presented. The work flow includes accurate rendering of interference colours considering the colour temperature of the light source of the microscope and chromatic adaptation to white points of RGB colour spaces as well as the colour correction of the camera using readily available colour targets. The quality of different colour correction profiles was tested independently on an IT8.7/1 target. The best performing profile was using the XYZ cLUT algorithm and it revealed a ΔE₀₀ of 1.9 (6.4 no profile) at 5× and 1.1 (8.4 no profile) at 100× magnification, respectively. The overall performance of the workflow was tested by comparing rendered interference colours with colour‐corrected images of a quartz wedge captured over a retardation range from 80–2500 nm at 5× magnification. Uncorrected images of the quartz wedge in sRGB colour space revealed a mean ΔE₀₀ of 12.3, which could be reduced to a mean of 4.9 by applying a camera correction profile based on an IT8.7/1 target and the Matrix only algorithm (ΔE₀₀ < 1.0 signifies colour differences imperceptible by the human eye). ΔE₀₀ varied significantly over the retardation range of 80–2500 nm of the quartz wedge, but the reasons for this variation is not well understood and the quality of colour correction might be further improved in future by using custom made colour targets specifically designed for the analysis of high‐order interference colours.     

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.     

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.     

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.     

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.     

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.     

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.     

Voltaire's contribution to the spread of Newtonianism—II. Élémens de la philosophie de Neuton: The elements of the philosophy of sir Isaac Newton

Continuing from Part I [Int. J. Mech. Sci.32, 423, 1990]^§, we mainly summarise the contents of, and make comments on, the 25 chapters of Voltaire's Élémen, published in 1738. Mme du Châtelet was the constant companion of Voltaire for 10 years and was very influential during the writing of the latter volume (worthy of being designated a co-author say some), so that space is given over to describing their life and work together during that period, mostly at Cirey, in Montjeu, Lorraine. We include an account of her writing of a translation of Newton's Principia, widely published in France in 1759, and of her early death. Voltaire (and Mme du Châtelet) expended very considerable efforts to spread Newton's philosophy through France.We also outline John Locke's Elements of 1720—hitherto undescribed we believe—Henry Pemberton's A View of Sir Isaac Newton's Philosophy of 1728, and we mention Algarotti's book of 1737. The latter two authors, Voltaire (and Mme du Châtelet) were the major popularisers of Newton's work in Europe though others, s'Gravesande and Keil, for instance, have been held to be more influential in doing so.     

Hybrid analytical–numerical solution for the shear angle in orthogonal metal cutting — Part II: experimental verification

The hybrid analytical–finite element model described in Part I is applied to predict the shear angle for a range of cutting velocity, uncut chip thickness, and two tool orthogonal rake angles. Experimental results and an empirical equation are also presented for the influence of the cutting conditions and cutting tool geometry on the chip–tool contact length. It is shown that there is a linear dependence between the chip–tool contact length/uncut chip thickness ratio and chip thickness/uncut chip thickness ratio over the range of cutting conditions assumed. The increase of the shear angle with the tool orthogonal rake is mostly due to the reduction of the specific shear energy in the primary shear zone and the specific friction energy in the secondary shear zone accompanied by a reduction of the chip–tool contact zone. The uncut chip thickness and cutting velocity influence the shear angle through their effect on the interface temperature and hence on the material flow stress in the secondary shear zone. The change in both parameters does not change significantly the specific shear energy in the primary shear zone. The model results are compared with the experimental results for a work material 0.18% C steel. The agreement between the predicted and experimental results is seen to be exceptionally good.     

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.     

Flywheel energy storage—II: Magnetically suspended superflywheel

This article, the second of a two part paper, describes the general design requirements for a flywheel energy storage system. A new superflywheel energy storage system, using a spokeless, magnetically suspended, composite material pierced disk rotor is proposed. The new system is configured around a permanent magnet (“flux biased”) magnetic suspension system with active control in the radial direction and passive control in the axial direction. The storage ring is used as a moving rotor and electronic commutation of stationary armature coils is proposed. There is no mechanical contact with the rotating ring and long life and low run down losses are projected. A discussion of major components for a 10 kwh system is presented.     

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.     

Machining with applied chip tension—Part II: Experiments

Machining with applied chip tension is the basis of a process, strip peeling, for making small batches of metal special strip products. An approximate slip-line field numerical analysis of the process, related to the matrix method, is presented which shows how pulling stress and its direction affect chip thickness and curvature and tool forces. Pulling parallel to, or within 5° of, the rake face produces straight chips but results in less reduction of chip thickness and tool forces than pulling at more than 5° to the rake face. In the latter case chips are formed curled and are subsequently plastically straightened by the pulling force. Chip failure by plastic straightening and other causes is discussed and it is recommended that the pulling direction should be between 5 and 20° from the rake face. Influences of rake angle and friction stress are also considered.     

基于Lévy飞行微粒群算法的液压系统可靠性优化

针对微粒群算法易于陷入局部最优解、早熟的缺点,将Lévy飞行引入微粒速度迭代公式中,并动态改变微粒群速度迭代公式中Lévy飞行的权重值,提出动态Lévy飞行微粒群算法。根据T-S故障树理论,建立液压支架液压系统的可靠性模型,进而得出可靠性费用目标函数。将提出的动态Lévy飞行微粒群算法应用于液压支架液压系统的可靠性优化中,并通过标准微粒群算法、布谷鸟搜索算法和基于Lévy飞行微粒群算法比较,验证所提出算法的优越性。     

Extended exact least-weight truss layouts—Part II: Unsymmetric cantilevers

In Part I of this study, the optimal bar layout and adjoint displacement fields were derived for cantilever trusses that are symmetric about a horizontal axis. In this paper, the above results are generalized to unsymmetric trusses and a number of further extensions to other support and load conditions are outlined briefly. Most results are verified by comparisons with discretized optimal layouts of trusses and perforated plates.     

Conjectures relating to rigid-plastic plate bending—II

The first paper of this title [P. G. Lowe, Conjectures relating to rigid-plastic plate bending. Int. J. Mech. Sci.30, 365–370 (1988)] restated two conjectured general results relating to plate collapse and went on to apply them to finding accurate lower bound collapse load estimates, as well as the extent of the plastically deforming regions, for a series of clamped plates with regular polygonal shapes.The main geometrical results used there are applicable only to polygonal shapes of plate which have an in-circle. In this paper the geometrical procedures are generalized to apply to any polygonal shape of plate, including non-convex shapes. These more general geometrical results enable additional shapes of plate to be analysed and further evidence is collected which is consistent with the conjectured general results being true.