Stepped Plates Research Articles

Vibrations of stepped one-way thin rectangular plates subjected to in-plane tensile/compressive force in y-direction on Winkler' s foundation

Differential equations of free/forced vibrations of n- step one-way thin rectangular plates subjected to in-plane tensile/ compressive force in y-direction on Winkler' s foundation are established by using singular functions, their general solutions solved for, expression of vibration mode function and frequency equation on usual supports derived with W operator. Influence functions for various cases deduced here may also be used to solve problems of static buckling or stability for beams and plates in relevant circumstances.

Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions

The free vibration problem of a stepped plate supported on non-homogeneous Winkler elastic foundation with elastically mounted masses is formulated based on Hamilton's principle. The stepped plate is modelled by finite strip method. To overcome the problem of excessive continuity of common beam vibration functions at the location of abrupt change of plate thickness, a set of C 1 continuous functions have been chosen as the longitudinal interpolation functions in the finite strip analysis. The C 1 continuous functions are obtained by augmenting the relevant beam vibration modes with piecewise cubic polynomials. As these displacement functions are built up from beam vibration modes with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as the appropriate order of continuity. The method is further extended to the buckling analysis of rectangular stepped plates. Numerical results also show that the method is versatile, efficient and accurate.

Measurement and prediction of diffuse fields in structures

Measurements and predictions of the response of a variety of plain and coupled plates when excited by a point source have been reported. Tests on plain plates showed that a diffuse field was generated within the first 30 wave transits across the plate. In plates coupled by a thin ligament, around 50 wave transits across one of the substructures was required for a diffuse field to be established in the whole system. This is equivalent to about 25 transits across the whole plate so the time required to set up a diffuse field was not significantly affected, even when the ligament width was only 2.5% of the total plate width. Tests on plates bolted or adhesively bonded together showed that in both cases, although a diffuse field was established in each of the substructures, the damping was too great for a diffuse field to be set up in the whole structure. Tests on stepped plates have shown that the measured surface amplitudes in the thinner sections tend to be much larger than those in the thick sections, but that the energy in the thinner sections is only slightly larger than that in a thick section of similar plan area. The field is not diffuse in the sense that the amplitude or energy density is the same throughout the coupled structure, but the field is diffuse in each substructure.

Vibrations of stepped rectangular thin plates on Winkler's foundation

Differential equations of free/forced vibrations of stepped rectangular thin plates on Winkler's foundation are established by using singular functions, and their general solutions are also solved for expression of vibration mode function and frequency equations on usual supports derived with W operator, as well as forced responses of such plates under different-type loads discussed with Fourier expansion of generalized functions.

VIBRATIONAL OPTIMIZATION OF A MASS-LOADED STEPPED PLATE

In this paper the vibrational characteristics of thin, mass-loaded, stepped plates are investigated. The dimensions of the plates are chosen so that the steps can be thought of as representing the periodically placed stiffeners commonly found in many engineering structures. To achieve this, a classical analytical approach for the analysis of the vibration of a simply supported, stepped plate is first considered. Next, a method for the analysis of such plates carrying concentrated masses is reviewed. The above two analytical methods are then combined to analyse the vibrational behaviour of thin, simply supported and mass-loaded stepped plates. To assess the accuracy of these methods, the resultant frequency responses of the unloaded plate are compared with the Dynamic Stiffness method [1] and those for both the unloaded and mass loaded plates with finite element calculations. For a uniform, mass-loaded plate, there is perfect agreement between the frequency responses obtained from these methods. For the mass-loaded, stepped plate, the agreement is not so complete, the reasons for which are discussed in the paper. The final part of the paper deals with optimisation of the mass positions in order to improve the vibrational behaviour of the plate. In this work, the integral of the frequency response function of the mass-loaded plate over a frequency range containing some 10–15 natural frequencies is regarded as the objective function. The drive and response points are chosen to lie at opposite ends of a plate with high aspect ratio and transverse stiffeners, so that minimizing the frequency response is equivalent to designing vibration isolation characteristics into the plate. The Genetic Algorithm, which is an evolutionary optimization method, is employed to produce the required designs. It is demonstrated that the optimized mass positions significantly improve the vibrational behaviour of the plate.

Effects of nickel and processing variables on the mechanical properties of austempered ductile irons

This study investigates the effects of inoculation techniques, ingate numbers and raw materials on the mechanical properties of ADIs. Five batches of base ductile irons (DI1-DI5) were poured to produce stepped plate castings following designated processing variables. DI1 and DI2 were used to compare the effect of raw materials on the sulphur content of irons and the resulting properties. DI1 and DI4 were produced in order to study the effect of nickel; DI3 and DI5 were poured to evaluate the effect of ingate numbers; DI4 and DI5 were poured to study the effect of inoculation techniques. Effects of these variables on the mechanical properties and microstructures of as-cast and austempered ductile irons (ADIs) were evaluated respectively.Using a given magnesium content (0.033–0.047 wt%), ADIs with a high percentage of sulphur, in the range of 0.004–0.02 wt%, developed a higher percentage elongation. ADIs alloyed with 0.48 wt% Cu, 0.46 wt% Ni and a minimum level of sulphur (0.004 wt%) produced a lower degree of deviation in strength, elongation and impact energy but showed the greatest deviation in the tested hardness of the stepped plate castings. Stream inoculation improved the nodule counts and nodularity of base ductile irons but also introduced dross and/or unreacted inoculant within the stepped plate casting. This was detrimental to the tested elongation of ADIs taken from the stepped plate casting. Two-step austempering treatment could significantly improve the tested elongation of ADIs without significantly affecting the developed strength.

Vibrations of one-way rectangular stepped thin plates on Winkler's foundation

Differential equations of free/forced vibrations of one-way rectangular stepped thin plates on Winkler's foundation are established by using singular functions: their general solutions are solved; expression of vibration mode function and frequency equations on usual supports are derived from W operator: forced responses of such plates under different-type loads are discussed with generalized functions.

VIBRATION ANALYSIS OF STEPPED THICKNESS PLATES

Investigation has been made into various approaches for analyzing the vibration of plates with stepped thicknesses. First, attention has been paid to updating a classical approach for the analysis of such problems, correcting the boundary conditions cited in an earlier paper and dealing with the difficulties that can arise when calculating high order modes. Secondly, contribution has been made to improving the classical finite strip method (FSM) by replacing the “static” shape function of the strip element by a “dynamic” function. This leads to the development of a dynamic finite strip method which improves solution accuracy without compromising model size and which therefore is more efficient than the classical FSM. When compared with the finite element method (FEM), which is also considered here, the advantages of smaller model size and higher accuracy of the dynamic FSM are significant. In order to demonstrate the application of the above approaches, the modes of simply supported plates with uniform and stepped thicknesses have been analyzed. From this numerical study, it is noted that the updated classical approach can be used to obtain a solution for any order mode to any specified accuracy and is the most efficient approach considered in the present study. It is also noted that, compared with the FEM of similar solution accuracy, the dynamic finite strip method normally produces a much smaller model size, so that such calculations are significantly more efficient than for the FEM. The aim of this work is to establish methods for the analysis of stepped plates that might be used in optimization studies where speed of formulation and solution are at a premium. There are, of course, a number of other methods that could be used to tackle such problems, but they lie outside the scope of this work; see for example the papers of Liew and co-authors [1, 2].

Analysis of stress concentration factors for stepped plates based on a crack tip stress intensity approach

The analytical equations given for stress concentration factors are based on the results of finite element analysis of stepped plates subject to uniaxial tension loading. The fillet radii at the stepped transitions were varied over a wide range, leading to elastic stress concentration factors between 1.1 and 8.3 (net stress basis). The parametric equations depend on the previously described concept of a ‘notch configuration factor’. This is similar to the crack configuration factor or compliance function used to modify the basic crack tip stress intensity solutions in the case of finite width or other problems. In the present case of the stepped plate, an energy approach was used to relate the sharp corner stress field to the corresponding sharp crack field, leading to a ‘sharp corner configuration factor’. This factor was then applied to the equation for the stress concentration factor at an elliptical hole in an infinite plate, to give a simple analytical expression for the stepped plate with a radiused fillet. The basic expression was refined further to improve the quality of fit, to an accuracy of 2 per cent with respect to the finite element models.

Free vibration of plates with stepped variations in thickness on non-homogeneous elastic foundations

A finite element model is presented for the analysis of the free vibration of plates with multiple stepped variations in thickness resting on non-homogeneous elastic foundations. Based on Mindlin plate theory, the model includes transverse shear deformation as well as bending-extension coupling in cases of plates with stepped sections eccentrically located with respect to the mid-planes. The section of elastic foundation under a plate element is treated as a separate foundation element. The transverse deformation of these foundation elements is made to be consistent with the deflection of plate elements being supported, resulting in a consistent stiffness matrix for the elastic foundation. Numerical results are in good agreement with the available reported results. The effect of eccentricity of the locations of stepped sections on the natural frequencies is found to be not negligible. The elastic foundations are found to have a significant effect on the fundamental natural frequencies of both uniform and stepped plates. Natural frequencies and mode shapes of rectangular plates and circular plates with multiple eccentrically stepped sections resting on non-homogeneous elastic foundations are presented.

Natural frequencies of rectangular stepped plates using polynomial beam functions with subsectioning

Natural frequencies of thin rectangular plates with single or multiple step variations in thickness and different edge boundary conditions are obtained by employing a novel hybrid method involving a set of beam characteristic polynomials and subsectioning using the Rayleigh-Ritz method. The formulations of the present method involve the division of the plate into strips, the number of strips used depending on the geometric discontinuities. The assumed admissible deflection function of each strip maintains continuity at the interconnecting line of each strip. The natural frequencies obtained by this method are compared with previously published ones and very good agreement is achieved.

An Experimental and Numerical Analysis of Riveted Single Lap Joints

Countersunk and snap riveted single lap joints have been examined both experimentally and numerically. A total of 11 specimens were fatigued to failure with failures occurring in either the plate or the rive***r. The failures have been metallurgically examined to determine the cause of failure.The joints have also been analysed using the finite element method. Initially a single lap joint has been modelled as a ‘stepped plate’ and the results for the stress concentration factor found to be in reasonable agreement with published data. However, the stress concentration for this joint occurred at a point away from the point of failure of a riveted joint.A fuller three-dimensional finite element model has been constructed and the stress patterns around the rivet determined. These stress patterns are discussed in relation to the results from the metallurgical examination.

Shear fracture of unidirectional composites without initial cracks

A number of different types of shear failure in unidirectional composites are considered. It is shown that there is apparently insufficient energy to allow fracture unless large initial cracks are present. However, sufficiently large defects are not normally found in practice. This is explained in terms of the formation of a plastic zone which allows relative deformation, providing energy to drive the fracture without a crack actually being present. Finite element analysis has been undertaken of a unidirectional stepped plate loaded in the fibre direction. Assuming linear elastic behaviour, the strain energy release rate for mode II delamination from the corner of the step increases from zero up to a constant value with increasing crack length. By modelling the plastic zone with non-linear springs it is shown that significant energy is available from the start of crack propagation and therefore that large initial cracks are not required for fracture to occur. Similar effects can be expected in other cases of shear fracture, and may explain why fracture behaviour is apparently consistent with the presence of large initial cracks even though such defects may not in fact exist.

An acoustic transducer system for long-distance ranging applications in air

Transducers for long-distance pulse-echo ranging applications should combine high efficiency and directivity with a relatively wide bandwidth. Since the air has a very low acoustic impedance, the efficient transmission of sound from a solid transducer is one of the major problems. A means to improve the impedance matching between the source and the propagation medium is to increase the radiation impedance by using extensive vibrating plates as radiators. Nevertheless, these extensive radiators will vibrate at a flexural mode resonance, producing a poor directivity due to phase cancellation. In this paper we present a transducer system, based on a stepped-plate radiator, which is able to generate efficiently high-directional radiation in air. To be used for pulse-echo ranging applications, the stepped-plate radiator has been specially adapted in order to increase the bandwidth of the transducer system. In particular, three main features have been introduced: the development of very thin stepped plates; the addition of a matching layer; and the backing of the plate with a damping material. Electroacoustical and pulse-echo characterizations of the transducer system have shown its good performance for use as a level or distance sensor, specially for long-range applications (10–100 m).

Optimal design of rigid-plastic annular plates with piecewise constant thickness

Rigid-plastic stepped annular plates under uniform pressure load are considered. Both plate edges are supported. Four types of boundary conditions are studied. Tresca's yield condition is used. Such plate dimensions are sought for which the plate of constant volume has the maximal load carrying capacity.

Nonlinear thermal and mechanical analysis of step-tapered laminated plate under uniform in-plane load

A method has been developed for the thermal analysis of materially nonlinear carbon fibre reinforced plastic (CFRP) step-tapered laminated plates using a quasi-three-dimensional iso-parametric finite element. The variation with temperature of the coefficient of expansion of CFRP in the transverse direction is included. The nonlinear initial thermal stresses resulting from cooling from the stress-free temperature of 132·2°C (270°F) to 20°C in [+θ/−θ/resin/90°/90°] s and [+θ/−θ/resin/+45°/−45°] s stepped plates were found. All stepped plates were then subjected to a uniform applied stress until first-ply failure was detected inside the laminate according to the maximum strain failure criterion. This nonlinear analysis was based on an initial stress iteration method formulated in a previous paper. In all cases, the resin-rich layer between the laminae that form the step was included. It was found that the initial thermal stresses of some stepped plates are of the same sign as their corresponding cases under an applied tensile stress but without initial thermal stresses included. With initial thermal stresses included, these stepped plates failed at an applied stress lower than for the cases without the initial thermal stresses. In contrast, the initial thermal stress components of other stepped plates are of opposite sign to their corresponding cases under an applied tensile stress only. With the inclusion of the initial thermal stresses, these stepped plates failed at a higher tensile stress than the cases without initial thermal effects. The [+θ/−θ/resin/+45°/−45°] s stepped plates exhibited lower strengths than the [+θ/−θ/resin/90°/90°] s stepped plates. The results suggest that the [+45°/−45°/resin/90°/90°] s laminate is probably better suited to resist tensile loading.

Effect of Transverse Shear and Rotary Inertia on Vibrations of Stepped Thickness Plates

The free vibration characteristics of stepped thickness plates by including the effect of transverse shear deformation and rotary inertia is presented. A method based on the variational procedure in conjunction with the finite difference technique is used to determine the natural frequencies and mode shapes. The convergence characteristics of the present method is studied and the results are compared with existing solutions based on the classical thin plate theory. The effects of the stepped thickness and relative thickness ratio on the vibration characteristics are studied.

Mechanical properties of a femur osteosynthesis using a stepped plate

Mechanical properties of a femur osteosynthesis using a stepped plate

Dispersion curve measurement using Talbot bands

Talbot bands, looking like a channelled spectrum, are produced when a phase plate is partly inserted in the field of a spectroscope. The position and the spacing of the bands in the spectrum of the source directly depends on the optical thickness and therefore the refractive index of the phase plate. In the case of a transmission grating spectroscope, the bands appear only in spectral orders on the same side as the phase plate. The bands can be observed also in case of a two stepped plate, their position being related to the difference of the two refractive indices of the plate, they appear in spectrum orders on the side where the higher index half plate have been inserted in. Using this early known phenomenon in conjunction with modern technology, we propose a method based upon an accurate determination of the positions of these dark bands for computing the dispersion curve of any transparent liquid or solid material, the curve being built up from only one experiment whereas several are required usually. In this paper, after the method have been described, experimental results obtained with known glasses are reported proving the efficiency and the reliability of the technique. Also this method is extended to birefringent materials, it is shown that the extraordinary and ordinary dispersion curve can be obtained simultaneously. Experimental results on liquid crystals materials are reported Proposition d'une nouvelle methode pour obtenir la courbe de dispersion de materiaux transparents. Methode basee sur une determination precise de la position des bandes dans le spectre, associee a une manipulation numerique des donnees conduisant a l'obtention de la courbe de dispersion et ce en une seule experience

IGH axisymmetric modes of vibration of stepped circular plates

Flexurally vibrating stepped plates of extensive area have been shown to be very effective directional ultrasonic radiators [1]. Thus, there is a need to study these vibrating structural elements. A number of papers on this subject have been published in the past few years. Nevertheless, almost all of them have been restricted to the study of plates with one step and/or to the first vibration mode. This paper deals with a study of axisymmetric vibrating stepped plates without any restriction either in the number of steps or in the order of the vibration mode. The only condition, which has been imposed due to the design of the radiators, is that the number of step discontinuities has to be equal to the number of nodal circles. The analytical approach is based on the Rayleigh method. Iterative numerical procedures have been developed to find the solution of the derived set of equations. Experimental tests on plates vibrating with up to seven nodal circles have been performed.