Thin Rectangular Plates Research Articles

Blow up result for a viscoelatic plate equation with nonlinear source

We consider a viscoelastic plate equation with nonlinear source and partially hinged boundary conditions. Our goal is to show analytically that the solution blows up in finite time. The background of the problem comes from the modeling of the downward displacement of suspension bridge using a thin rectangular plate. This result shows that in the present of a nonlinear source such as the earthquake shocks, the bridge will collapse in finite time

Static Analysis of thin Rectangular Plate Resting on Elastic Foundation Using Modified Vlasov Model

A four-nodded rectangular plate bending element based on Kirchhoff theory resting on elastic foundation using modified Vlasov model. For evaluate the two soil parameter for Vlasov foundation the modulus of elasticity and Poisson ratio of the soil is assumed constant from top surface to the top of bottom rigid base. All the deformation stiffness matrix of plate and subsoil are evaluated using finite element method. A Matlab code is developed and convergence study is carried out and then some realistic cases of plate under static load on elastic foundations are solved to determine the static response. The results, thus obtained, are compared, with the available results obtained by other researchers. It behaves extremely well for thin plates and convergence rate is high.

Analytical free vibration solutions of rectangular edge-cracked plates by the finite integral transform method

Analytical free vibration solutions of rectangular edge-cracked thin plates are obtained by the finite integral transform (FIT) method. While the previous studies focus on the mechanical behaviors of intact plates, this study establishes an FIT-based new scheme for analyzing the free vibration characteristics of rectangular thin plates with an edge crack. The plates may have any combined conditions of simply supported, clamped, and free boundaries. In specific solution scheme, an edge-cracked plate is decomposed into four subdomains, and a double cosine FIT is imposed on the fourth-order partial differential governing equations of each subdomain, which yields the relationship between the transformed modal deflections and naturally defined unknowns. By substituting the exterior boundary conditions of the plate, continuity conditions across the subdomains, and free conditions at the crack into the inverse transforms, the natural frequencies and unknowns are successively determined via the established homogeneous linear algebraic equations. The modal deflections are provided by further substituting the obtained unknowns into the inverse transforms. Comprehensive benchmark results for the edge-cracked plates under representative boundary conditions are shown. The results are well verified by other solution methods. A parametric study quantitatively reveals the effects of boundary conditions and crack length on the natural frequencies of edge-cracked plates. Due to the rigorous but easy-to-implement mathematical derivations, this study presents a novel solid way for exploring analytical solutions of vibration problems.

A large-deflection deformation reconstruction method for semi-flexible plate nozzle based on strain-moment relationship

To enhance the health detection capability of the semi-flexible plate nozzle in a supersonic wind tunnel, a large-deflection deformation reconstruction algorithm for a semi-flexible plate nozzle based on the strain-moment relationship is developed. The multi-jack semi-flexible plate nozzle in the supersonic wind tunnel is studied; its mechanical model of contour assembly is established. The differential equation of flexible plate contour curve is deduced, and the elliptic integral expression of the flexible plate deformation is given. Based on the deformation theory of a rectangular thin plate, the deformation reconstruction algorithm is modified, and its correction effect is simulated and analysed with the finite element method. The maximum deviation after correction is only 10.78% compared with that before correction. The 0.3 m semi-flexible plate nozzle is used as the experimental platform to carry out the experimental study. The experimental results show that under the 6 Mach number, the large-deflection deformation reconstruction results based on strain-moment relationship are in good agreement with the experimental results. The coordinate deviation of the nozzle throat position is about 6.12% compared with the structural deformation, thus verifying that the deformation reconstruction algorithm in this paper is effective and accurate.

An Accurate Computational Method for Buckling of Orthotropic Composite Plate with Non-Classical Boundary Restraints

New accurate buckling analysis for rectangular orthotropic thin plates with complicated non-classical boundary restraints are conducted through adopting the finite Fourier integral transform approach. Non-classical boundaries such as rotationally restrained edges increase the mathematical difficulty in processing problems of plates, which leads to rare analytical results for benchmark use. The proposed approach is implemented in the framework of integral transform theory, in which trial function for the deflection is not necessary, and offers uniform solution procedures for problems of plates with various boundaries via adopting different integral kernels. The main merits of the approach employed is to enable one to change the complicated title problem into dealing with linear algebraic equations easily solved. Via altering the rotational spring factors introduced, buckling behaviors of plates with Levy-type boundaries and non-Levy-type boundaries can also be studied. Finally, all the given results including critical load factor and mode shape match the FEM analysis exactly, which illustrate the accuracy and validity of the method.

An analytical approach for modelling the vibroacoustic behaviour of a heavy fluid-loaded plate near a free surface

Predictions of the vibroacoustic response of a point-force excited baffled thin rectangular plate immersed in a heavy fluid and near a free surface are presented using an analytical model. The equations of motion are solved by Fourier analysis, where the eigenfunctions of plate vibration form the basis of spatial expansion for fluid loading. Vibroacoustic indicators, including the plate velocity, acoustic pressure, and acoustic power, are predicted using the analytical approach and verification is performed by comparison with finite element simulations. The results have shown that variations in the height of the free surface can have a significant effect on these indicators. From the vibration response, added mass effect due to heavy fluid loading is altered and further investigated with the explicit evaluation of an added mass ratio for different free surface heights for the first five plate modes. For a given height of a free surface, standing waves can form between the free surface and baffled plate at specific excitation frequencies and slightly alters the acoustic pressure spectra. This condition also presents an effect on the acoustic power, where the first standing wave frequency dictates the efficient sound radiation to the far field.

Thermal buckling analysis of concrete rectangular thin plates considering material damage

Thermal buckling analysis of concrete rectangular thin plates considering material damage

Study on the Mechanism of Elastic Instability Caused by Natural Growth in Orthotropic Material.

Compared to synthetic materials, naturally grown biological materials have more specific behavioral patterns and life connotations in their morphological evolution over millions of years of environmental evolution on Earth. In this paper, we investigate the physical mechanisms and manifestations of out-of-plane deformation instability. Firstly, the origin of the instability phenomenon caused by the growth of the leaf is introduced. Leaf instability problems are modeled using rectangular thin plates. Secondly, the variation in the critical intrinsic strain with the principal shear modulus is obtained by numerical solution. The post-buckling behavior of the growth instability is further analyzed by general static analysis, and we obtain the phase diagram of morphogenesis of thin plant organs as functions of the principal shear modulus and off-axis angle. The research results enhance the understanding of the mechanism of elastic instability caused by natural growth in orthotropic materials.

Development of Polynomial Based Program for Nonlinear Isotropic Rectangular Thin Plate

The work is aim at the development of a computer program for the nonlinear analysis of rectangular thin isotropic plate on Ritz method. Twelve boundary conditions were analyzed which include: SSSS, CCCC, CSCS, CSSS, CCSS, CCCS, CCFC, SSFS, CCFS, SCFC, CSFS, and SCFS. General expressions for displacement and stress functions for large deflection of isotropic thin rectangular plate under uniformly distributed transverse loading were obtained by direct integration of Von karman’s non-linear governing differential compatibility and equilibrium equations. Polynomial function as shape function was on the decoupled Von Karman’s equations to obtain particular stress and displacement functions respectively. Non-linear total potential Energy was formulated using Von Karman equilibrium equation and Ritz method was deployed in this formulation. A computer based program was developed using Matlab programming language to circumvent the challenges involved in solving the governing differential equations of thin rectangular plates. The developed program is capable of determining deflection and stresses at any point of the plate against the usual method of evaluating deflection at the center. The results obtained were compared with those of previous researchers The comparison made are only for SSSS, CCCC and CCCS plates. It was so because the remaining boundary conditions considered in this work have not been researched upon by previous researchers. From results obtained, the average percentage differences recorded for SSSS, CCCC, and CCCS plates for the present and previous studies are 4.01978%, 3.7646%, and 5.02% respectively. The percentage differences for the three plates compared are within acceptable limit of 0.05 or 5% level of significance in statistics. From the comparison made, it was obvious that an excellent agreement was observed in all cases thus indicating applicability and validity of the polynomial function and computer program for solving exact plate bending problems.

Selected Problems of Random Free Vibrations of Rectangular Thin Plates with Viscoelastic Dampers.

The main motivation of this work was to present a semi-analytical extension of the correspondence principle in stochastic dynamics. It is demonstrated for the stochastic structural free vibrations of Kirchhoff–Love elastic, isotropic and rectangular plates supported by viscoelastic generalized Maxwell dampers. The ambient temperature of the plate affects the dampers only and is included in a mathematical model using the frequency–temperature correspondence principle. The free vibration problem of the plate–viscoelastic damper system is solved using the continuation method and also the Finite Element Method (FEM). The stochastic approach begins with an initial deterministic sensitivity analysis to detect the most influential parameters and numerical FEM recovery of the polynomial representation for lower eigenfrequencies versus these parameters. A final symbolic integration leads to the first four basic probabilistic characteristics, all delivered as functions of the input uncertainties.

Polynomial Based Nonlinear Analysis of CCCS Thin Isotropic Rectangular Plate

This work is aimed at formulating a polynomial function for the nonlinear analysis of CCCS isotropic rectangular thin plate. The previous researchers used trigonometry function as their shape function on the decoupled Von Karman’s equations to obtain particular stress and displacement function respectively. Trigonometry function can only be used effectively for SSSS and CCCC plates; apart from these boundaries conditions its efficiency reduces. This present work hence used a polynomial function to formulate the approximate shape function for the CCCS plate. Direct variational calculus was used applied on Von Karman’s equations to obtain the general form of minimized total potential energy which serves as a platform for the determination of coefficient factor( Amplitude or coefficient of deflection). The numerical values of CCCS plate under unit load were obtained using Amplitude equation formulated. These values were obtained for various aspect ratio (ranging from 1 to 1.5 with an increment of 0.1). This work was compared with the previous work [1] and the percentage difference in the results are within the acceptable limit. This results indicate that the approach adopted by the present work is adequate, reliable and satisfactory for the analysis of CCCS rectangular plate.

Nonlinear Analysis of SSSS Thin Isotropic Rectangular Plate Using Polynomial as Shape Function

This paper presents non¬linear analysis of isotropic rectangular plate with all edges simply supported (SSSS) under a uniform distributed load using polynomial series as shape function, and under the application of Ritz method. There are several analytical tools available for analyzing isotropic rectangular plate. Among all the available analytical methods, polynomial offers a better approach. Earlier researchers on SSSS plate were, however, centered on the use trigonometric series and approximate method in form of numerical and energy method. General expressions for displacement and stress functions for large deflection of isotropic thin rectangular plate under uniformly distributed transverse loading were obtained by direct integration of Von Karman’s non- linear governing differential compatibility and equilibrium equations. Total potential energy functional was formulated based on the derived displacement and stress functions. Subsequently, the formulated total potential energy functional was minimized and resulted to a general amplitude equation of the form K1∆3+K2∆+K3. Where K1, K2 and K3 are coefficients of amplitude equation and ∆ is the deflection coefficient (factor). Newton-Raphson method was used to evaluate the deflection coefficient. Values of ∆ from Timoshenko and that from present study were compared with an aspect ratios ranging from 1.0 to 1.5 with an increment of 0.1. From results obtained, the average percentage difference for the pervious and present studies is 4.0198%. The percentage difference for the plate was within acceptable limit of 0.05 or 5% level of significance in statistics. From the comparison, an excellent agreement exist between the present and previous works. Thus, this indicate applicability and validity of the polynomial function for solving exact plate bending problem.

Large Deflection Analysis of Clamped Thin Rectangular Isotropic Plate under the Action of a Uniform Distributed Load

This present paper deals with large deflection of clamped thin rectangular isotropic plate under uniformly distributed load. The analysis of clamped plates is of great important and its structural behavior has not been completely understood. Previous studies have made a great in road in the analysis of clamped plates through varied approaches mostly assuming trigonometric series as displacement field. A variational method based on the principle of minimization of total potential energy has been employed through assumed displacement field. Therefore, the von Karman differential equation was decoupled through direct integration retaining its non-linear character. The aim of this present studies is to provide solution to analysis structural behavior of clamped (CCCC) thin rectangular isotropic plate under uniform distributed. The analysis was accomplished through a theoretical formulation of shape function based on polynomial series and subsequently, the formulated shape function was used on Ritz energy method. The resulting functional was minimized to obtain a general amplitude equation of the form K1∆3+K2∆+K3. Where K1, K2 and K3 are coefficients of amplitude equation and ∆ is the deflection coefficient (factor). Newton-Raphson method was used to evaluate the deflection coefficient. Values of ∆ from Timoshenko and that from present study were compared with an aspect ratios ranging from 1.0 to 1.5 with an increment of 0.1. From results obtained, the average percentage difference for the pervious and present studies is 3.7646%, The percentage difference for the plate was within acceptable limit of 0.05 or 5% level of significance in statistics. It is shown from the comparison that good agreement exists between the present and previous works and that this approach provides simply, direct, straightforward and highly accurate solutions for this family of problems.

Nonlinear Analysis of SSSS Thin Isotropic Rectangular Plate Using Polynomial as Shape Function

This paper presents non¬linear analysis of isotropic rectangular plate with all edges simply supported (SSSS) under a uniform distributed load using polynomial series as shape function, and under the application of Ritz method. There are several analytical tools available for analyzing isotropic rectangular plate. Among all the available analytical methods, polynomial offers a better approach. Earlier researchers on SSSS plate were, however, centered on the use trigonometric series and approximate method in form of numerical and energy method. General expressions for displacement and stress functions for large deflection of isotropic thin rectangular plate under uniformly distributed transverse loading were obtained by direct integration of Von Karman’s non- linear governing differential compatibility and equilibrium equations. Total potential energy functional was formulated based on the derived displacement and stress functions. Subsequently, the formulated total potential energy functional was minimized and resulted to a general amplitude equation of the form K1∆3+K2∆+K3. Where K1, K2 and K3 are coefficients of amplitude equation and ∆ is the deflection coefficient (factor). Newton-Raphson method was used to evaluate the deflection coefficient. Values of ∆ from Timoshenko and that from present study were compared with an aspect ratios ranging from 1.0 to 1.5 with an increment of 0.1. From results obtained, the average percentage difference for the pervious and present studies is 4.0198%. The percentage difference for the plate was within acceptable limit of 0.05 or 5% level of significance in statistics. From the comparison, an excellent agreement exist between the present and previous works. Thus, this indicate applicability and validity of the polynomial function for solving exact plate bending problem.

Nonlinear Analysis of Isotropic Rectangular Thin Plates Using Ritz Method

This research work presents Nonlinear Analysis of Isotropic Thin Rectangular Plates using Energy Principle (Ritz method). Isotropic thin Rectangular plate having different twelve boundary conditions were analyzed and these boundary conditions were formed by combination of three major supports – Clamp, C; Simply supported, S; and Free, F.; General expressions for displacement and stress functions for large deflection of isotropic thin rectangular plate under uniformly distributed transverse loading were obtained by direct integration of Von karman’s non-linear governing differential compatibility and equilibrium equations. Polynomial function instead of trigonometry function as was with previous researchers was used on the decoupled Von Karman’s equations to obtain particular stress and displacement functions respectively. Non-linear total potential Energy was formulated using Von Karman equilibrium equation and Ritz method was deployed in this formulation. This equation was fully converted to potential energy by multiplying all the terms in it with displacement, w and the formed total potential energy, π consists of potential energy of internal forces and potential energy of external forces. This formulated total potential energy π, could give an accurate approximation of displacement field if the parameters were properly chosen. However, we assumed deflection, w to be ∆H1, and stress function, ɸ to be ∆2H2 and substituted into the formulated potential energy. H1 and H2 are profiles of the deflection and stress function respectively, and ∆ is deflection coefficient factor of the plate. Potential energy formulated contains deflection coefficient factor to the power of four. This potential energy was minimized by differentiating it partially with respect to coefficient factor reducing to cubic form.

A Semi-analytical Solution for the Dynamic Analysis of a Rectangular Viscoelastic Plate Subjected to a Moving Inertial Load

A semi-analytical method is developed to determine the response of a thin rectangular plate made of a general viscoelastic material to the excitation of a moving inertial load. The governing equation of the general problem is derived in the Laplace domain, which, for any particular viscoelastic model, is transformable into a system of differential equations in the time domain. Any standard procedure can then be readily employed to solve this system of equations. Using this method, sample response spectra are presented, through which the effect of viscosity, mass and velocity is scrutinized. The results show that, when the viscosity is not large enough, inertial terms cannot be ignored, especially when a heavy load is travelling at a high velocity.

Free vibration analysis of rectangular thin plates with corner and inner cutouts using C1 Chebyshev spectral element method

This paper presents a novel higher-order C1 continuous Chebyshev spectral element method for the free vibration analysis of rectangular thin plates with corner and inner cutouts. To achieve C1 continuity at the interfaces of adjacent elements, the Hermite interpolation functions through nonuniformly distributed Chebyshev nodes are employed to construct the element displacement field. A Gauss–Lobatto type quadrature is used to efficiently evaluate the element matrices. Comparison with other well-established methods show that the proposed element has good convergence and accuracy. The influence of cutout size, location and eccentric ratio on the natural vibration characteristics of plates are investigated.

Free Vibration Analysis of Rectangular Thin Plates Resting on Nonhomogeneous Elastic Foundations by Using Matched Interface and Boundary Algorithms and their Multi-Domain Formulations

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations.

New Analytical Free Vibration Solutions of Thin Plates Using the Fourier Series Method

This article aims at analytically solving the free vibration problem of rectangular thin plates with one corner free and its opposite two adjacent edges rotationally-restrained, which is difficult to handle by conventional semi-inverse approaches such as the Levy solution and Naiver solution, etc. Based on the classical Fourier series theory, this work presents a first endeavor to treat the two-dimensional half-sinusoidal Fourier series, which is quite similar to the Navier’s form solution, as the solution form of plate deflection. By utilizing the orthogonality of the present trial function and the Stoke’s transformation technique, the present solution procedure converts the complicated plate problem into solving sets of linear algebra equations, which heavily decreases the difficulties. Therefore, the present approach enables one to solve the title problem in a unified, simple and straightforward way, which is very easily implemented by researchers. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through utilizing different types of Fourier series and it can be extended for further dynamic/static analysis of plates under different shear deformation theories. Moreover, without any extra derivation processes, new, precise analytical free vibration solutions for plates under three non-Levy-type boundary conditions are also obtained by choosing different rotating fixed coefficients. Consequently, we present more than 400 comprehensive free vibration results for plates with classical/non-classical boundaries, all the present results are confirmed by FEM/analytical solutions and can be used as benchmark data for further research.

Ti:Sa Crystal Geometry Variation vs. Final Amplifiers of CPA Laser Systems Parameters

In this paper, the different Ti:Sapphire crystal configurations of the final amplifiers, depending on the Chirped Pulse Amplification laser system parameters, such as the repetition rates and pulse energy, are discussed. Restrictions placed on the final Ti:Sa amplifiers with a high repetition rate are discussed. The repetition rate of these systems is limited due to the crystal overheating, which leads to stress fracturing or significant beam distortion. The heating density threshold leading to stress fracturing was calculated and taken as the limit of the upper level of the possible pump average power. On the basis of these calculations, the highest repetition rates and corresponding thermolens focal distances were estimated for conventional crystal geometry of the most suitable thicknesses. It was demonstrated that conventional crystal shapes, such as a thin disc, can be used for systems with repetition rates below a few hundred Hz if several Joules of the output pulse energy are required. The rectangular thin crystal plate geometry was more suitable for Ti:Sa amplifiers with repetition rates above 1 kHz. Finally, the parameters of rectangular thin crystal plate Ti:Sa laser amplifier with an output energy above 3 J per pulse for a laser system with more than 100 TW pulse power and 1 kHz repetition rates are presented.