Exact solution of bending problem of clamped orthotropic rectangular thin plates

The generalized integral transform technique (GITT) is employed to obtain an exact solution for the bending problem of fully clamped orthotropic rectangular thin plates. The use of the GITT approach in the analysis of the transverse deflection equation leads to a coupled system of fourth order differential equations (ODEs) in the dimensionless longitudinal spatial variable. The resulting transformed ODE system is then numerically solved by making use of the subroutine DBVPFD from IMSL Library. Numerical results with automatic global accuracy control are obtained for different values of aspect ratio. Critical comparisons with previously reported numerical results are performed with excellent agreement. A set of reference results for clamped orthotropic rectangular plates is also provided for future covalidation purposes, which are verified by those obtained from the general-purpose finite element software package ABAQUS.Indisponível

New exact series solutions for transverse vibration of rotationally-restrained orthotropic plates

The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.apm.2018.08.033 © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/The exact series solutions of plates with general boundary conditions have been derived by using various methods such as Fourier series expansion, improved Fourier series method, improved superposition method and finite integral transform method. Although the procedures of the methods are different, they are all Fourier-series based analytical methods. In present study, the foregoing analytical methods are reviewed first. Then, an exact series solution of vibration of orthotropic thin plate with rotationally restrained edges is obtained by applying the method of finite integral transform. Although the method of finite integral transform has been applied for vibration analysis of orthotropic plates, the existing formulation requires of solving a highly non-linear equation and the accuracy of the corresponding numerical results can be questionable. For that reason, an alternative formulation was proposed to resolve the issue. The accuracy and convergence of the proposed method were studied by comparing the results with other exact solutions as well as approximate solutions. Discussions were made for the application of the method of finite integral transform for vibration analysis of orthotropic thin plates.Natural Sciences and Engineering Research Council of Canada [Discovery Grant, Grant No. 203154

Vibration Serviceability of Cold-Formed Steel Floor Systems

Excessive vibration in response to human activities has been a significant problem associated with lightweight steel floor systems, especially cold-formed steel (CFS) floors. Methods for accurately predicting these vibrations and evaluating floor systems are not readily available to the design community. The limited amount and complexity of research on the vibration serviceability of lightweight steel floor systems have shown an urgent need for further investigation. The objective of this research is to evaluate how human walking affects the performance of lightweight steel floor systems. Four important aspects that influence floor vibration performance are investigated: rotationally restrained floor joist ends, structural properties of CFS floors, human-structure interactions, and the applicable design guidelines. The investigation was carried out using an analytical approach in which CFS floor systems are modelled by equivalent orthotropic plates, and the equivalent structural properties are determined by using the Rayleigh method. The method of finite integral transform is extended to obtain the exact series solutions of the bending and vibration of orthotropic plates with rotationally restrained edges. The analytical/numerical results are compared to the results obtained in previous methods and experimental investigations. Then, the significant effects of human occupants on the dynamic properties and responses of lightweight steel floors are examined through the proposed damped plate-oscillator model, which determines frequencies and damping ratios through analytical analysis of coupled floor-occupant systems. The predicted results are compared with previous test results. Three loading models--moving force, moving damped-oscillator, and moving and stationary damped-oscillators are subsequently proposed to obtain the dynamic responses of floor systems to human walking. The analytical results from the three models are compared with the previous test results. After that, parametric studies are conducted on the effects of step frequency, damping ratio, human-to-structure mass ratio, and walking path. The foregoing investigations provide a comprehensive understanding of the dynamic performance of lightweight steel floors affected by human walking. Finally, design guidelines are developed for lightweight CFS steel floors in residential constructions. The floors are classified into three categories based on their fundamental frequencies, i.e. low-, mid-, and high-frequency floors. For each category, the corresponding design criterion and method are proposed. It is the author's desire that the contributions made in this thesis research help engineering practitioners better understand the dynamic responses and vibrational characteristics of lightweight CFS floor systems, particularly on human-structure interactions and ultimately lead to the efficient design of lightweight CFS floor systems that resisting the vibration induced by human walking

Analytical approaches to vibration analysis of circular, annular and sectorial plates subjected to classical and arbitrary boundary conditions – a literature survey

Plates are one of the most important structural components used in many industries like aerospace, marine and various other engineering fields and thus motivate designers and engineers to study the vibration characteristics of these structures. A lot of research work and studies have been done to study its vibration characteristics. This paper is a review of existing literature on vibration analysis of circular, annular and sector plates. The aim of this paper is to compile prominent studies related to circular, annular and sector plates subjected to classical and arbitrary boundary conditions under different supports and loadings. This review also identifies the analytical methods and approaches used to study the vibration characteristics of circular, annular and sector plates based on classical plate theories, Mindlin plate theory and higher order shear deformation theories. Few important citations related to functionally graded circular, annular and sector plates have also been included. Apart from helping researchers and engineers to identify relevant literature quickly and easily, this review will also help them to apply some of these analytical methods to study the vibration characteristics of other 2D and 3D built up and coupled structures

Stability analysis of three-dimensional thick rectangular plate using direct variational energy method

This study investigated the elastic static stability analysis of homogeneous and isotropic thick rectangular plates with twelve boundary conditions and carrying uniformly distributed uniaxial compressive load using the direct variational method. In the analysis, a thick plate energy expression was developed from the three-dimensional (3-D) constitutive relations and kinematic deformation; thereafter the compatibility equations used to resolve the rotations and deflection relationship were obtained. Likewise, the governing equations were derived by minimizing the equation for the potential energy with respect to deflection. The governing equation is solved to obtain an exact deflection function which is produced by the trigonometric and polynomial displacement shape function. The degree of rotation was obtained from the equation of compatibility which when equated to the deflection function and put into the potential energy equation formulas for the analysis were obtained after differentiating the outcome with respect to the deflection coefficients. The result obtained shows that the non-dimensional values of critical buckling load decrease as the length-width ratio increases (square plate being the highest value), this continues until failure occurs. This implies that an increase in plate width increases the probability of failure in a plate. Hence, it can be deduced that as the in-plane load on the plate increase and approaches the critical buckling, the failure in a plate structure is abound to occur. Meanwhile, the values of critical buckling load increase as the span-thickness ratio increases for all aspect ratios. This means that, as the span-thickness ratio increases an increase in the thickness increases the safety in the plate. It also indicates that the capacity of the plate to resist buckling decreases as the span-depth ratio increases. To establish the credibility of the present study, classical plate theory (CPT), refined plate theory (RPT) and exact solution models from different studies were employed to validate the results. The present works critical buckling load varied with those of CPT and RPT with 7.70% signifying the coarseness of the classical and refined plate theories. This amount of difference cannot be overlooked. The average total percentage differences between the exact 3-D study (Moslemi et al., 2016), and the present model using polynomial and trigonometric displacement functions is less than 1.0%. These differences being so small and negligible indicates that the present model using trigonometric and polynomial produces an exact solution. Thus, confirming the efficacy and reliability of the model for the 3-D stability analysis of rectangular plates

Vibration analysis of cracked aluminium plates

This research is concerned with analytical modelling of the effects of cracks in structural plates and panels within aerospace systems such as aeroplane fuselage, wing, and tail-plane structures, and, as such, is part of a larger body of research into damage detection methodologies in such systems. This study is based on generating a so-called reduced order analytical model of the behaviour of the plate panel, within which a crack with some arbitrary characteristics is present, and which is subjected to a force that causes it to vibrate. In practice such a scenario is potentially extremely dangerous as it can lead to failure, with obvious consequences. The equation that is obtained is in the form of the classical Duffing equation, in this case, the coefficients within the equation contain information about the geometrical and mass properties of the plate, the loading and boundary conditions, and the geometry, location, and potentially the orientation of the crack. This equation has been known for just over a century and has in the last few decades received very considerable attention from both the analytical dynamics community and also from the dynamical systems researchers, in particular the work of Ueda, Thompson, in the 1970s and 1980s, and Thomsen in the 1990s and beyond. An approximate analytical solution is obtained by means of the perturbation method of multiple scales. This powerful method was popularized in the 1970s by Ali H.Nayfeh, and discussed in his famous books, ‘Perturbation Methods’ (1974) and ‘Nonlinear Oscillations’ (1979, with D.T.Mook), and also by J.Murdock (1990), and M.P.Cartmell et al. (2003) and has been shown to be immensely useful for a wide range of nonlinear vibration problems. In this work it is shown that different boundary conditions can be admitted for the plate and that the modal natural frequencies are sensitive to the crack geometry. Bifurcatory behaviour of the cracked plate has then been examined numerically, for a range of parameters. The model has been tested against experimental work and against a Finite Element model, with good corroboration from both. In all events, this is a significant new result in the field and one that if implemented within a larger damage detection strategy, could be of considerable practical use

Vibration analysis of a plate with an arbitrarily orientated surface crack

This research presents a vibration analysis for a thin isotropic plate containing an arbitrarily orientated surface crack. The work has been motivated by the well known applicability of various vibrational techniques for structural damage detection in which the detection and localisation of damage to thin plate structures at the earliest stage of development can optimise subsystem performance and assure a safer life, and is intended to be an enhancement to previous work on cracked plates for which the orientation of the crack angle was not included. The novelty of this research activity has been in the assimilation of a significantly enhanced crack model within the analytical model of the plate, in modal space, and taking the form of a specialised Duffing equation. The governing equation of motion of the plate model with enhanced crack modelling is proposed to represent the vibrational response of the plate and is based on classical plate theory into which a developed crack model has been assimilated. The formulation of the angled crack is based on a simplified line-spring model, and the cracked plate is subjected to transverse harmonic excitation with arbitrarily chosen boundary conditions. In addition, the nonlinear behaviour of the cracked plate model is investigated analytically from the amplitude-frequency equation by use of the multiple scales perturbation method. For both cracked square and rectangular plate models, the influence of the boundary conditions, the crack orientation angle, crack length, and location of the point load is demonstrated. It is found that the vibration characteristics and nonlinear characteristics of the cracked plate structure can be greatly affected by the orientation of the crack in the plate. The dynamics and stability of the cracked plate model are also examined numerically using dynamical systems tools for representing the behaviour of this system for a range of parameters. Finally the validity of the developed model is shown through comparison of the results with experimental work and finite element analysis in order to corroborate the effect of crack length and crack orientation angle on the modal parameters, as predicted by the analysis. The results show excellent predictive agreement and it can be seen that the new analytical model could constitute a useful tool for subsequent investigation into the development of damage detection methodologies for generalised plate structures

Analytical approaches to vibration analysis of circular, annular and sectorial plates subjected to classical and arbitrary boundary conditions – a literature survey

Plates are one of the most important structural components used in many industries like aerospace, marine and various other engineering fields and thus motivate designers and engineers to study the vibration characteristics of these structures. A lot of research work and studies have been done to study its vibration characteristics. This paper is a review of existing literature on vibration analysis of circular, annular and sector plates. The aim of this paper is to compile prominent studies related to circular, annular and sector plates subjected to classical and arbitrary boundary conditions under different supports and loadings. This review also identifies the analytical methods and approaches used to study the vibration characteristics of circular, annular and sector plates based on classical plate theories, Mindlin plate theory and higher order shear deformation theories. Few important citations related to functionally graded circular, annular and sector plates have also been included. Apart from helping researchers and engineers to identify relevant literature quickly and easily, this review will also help them to apply some of these analytical methods to study the vibration characteristics of other 2D and 3D built up and coupled structures

Analytical approaches to vibration analysis of circular, annular and sectorial plates subjected to classical and arbitrary boundary conditions – a literature survey

Plates are one of the most important structural components used in many industries like aerospace, marine and various other engineering fields and thus motivate designers and engineers to study the vibration characteristics of these structures. A lot of research work and studies have been done to study its vibration characteristics. This paper is a review of existing literature on vibration analysis of circular, annular and sector plates. The aim of this paper is to compile prominent studies related to circular, annular and sector plates subjected to classical and arbitrary boundary conditions under different supports and loadings. This review also identifies the analytical methods and approaches used to study the vibration characteristics of circular, annular and sector plates based on classical plate theories, Mindlin plate theory and higher order shear deformation theories. Few important citations related to functionally graded circular, annular and sector plates have also been included. Apart from helping researchers and engineers to identify relevant literature quickly and easily, this review will also help them to apply some of these analytical methods to study the vibration characteristics of other 2D and 3D built up and coupled structures