Rectangular Plates Research Articles

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

The buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formidable obstacle to the attainment of analytical solutions for these problems. Despite the availability of various approximate/numerical methods for addressing these challenges, the literature lacks accurate analytic solutions. In this study, we employ the symplectic superposition method, a recently developed method, to effectively analyze the buckling problem of rectangular thin plates analytically. These plates have four supported corners and four free edges. To achieve this, the problem is divided into two sub-problems and solve them separately using variable separation and symplectic eigen expansion, leading to analytical solutions. Finally, we obtain the resolution to the initial issue by superposing the sub-problems. The current solution method can be regarded as a logical, analytical, and rational approach as it begins with the basic governing equation and is systematically derived without assuming the forms of the solutions. To examine various aspect ratios and in-plane load ratios of rectangular thin plates, which are supported at their four corner points with four edges free, we provide numerical examples that demonstrate the buckling loads and typical buckling mode shapes.

Tension buckling and postbuckling of nanocomposite laminated plates with in-plane negative Poisson’s ratio

Abstract Mechanical metamaterials with negative Poisson’s ratio (NPR) have emerged as a novel class of engineering material, and have attracted increasing attention in various engineering sectors. Most studies available on the buckling problem of laminated plates with positive or NPR are those under uniaxial compression. Here, we report that the buckling phenomenon may occur for auxetic nanocomposite laminated plates under uniaxial tension when the unloaded edges of the plates are immovable. Two types of nanocomposites are considered, including graphene/Cu and carbon nanotube/Cu composites. Governing equations of the auxetic nanocomposite laminated plates are formulated based on the framework of Reddy’s higher-order shear deformation theory. In modeling, the von Kármán nonlinear strain–displacement relationship, temperature-dependent material properties, thermal effects, and the plate–substrate interaction are considered. The explicit analytical solutions for postbuckling of auxetic nanocomposite laminated plates subjected to uniaxial tension are obtained for the first time by employing a two-step perturbation approach. Numerical investigations are performed for tension buckling and postbuckling behaviors of auxetic nanocomposite laminated rectangular plates with in-plane NPR rested on an elastic substrate under temperature environments.

Analysis of free in-plane vibrations of a rectangular plate with various boundary conditions canonical form using the modified Riley-Ritz method

This paper investigates the free in-plane vibrations of rectangular sheets under various boundary conditions using an improved Riley-Ritz method. A novel approach involving graph theory and canonical forms is introduced, marking a first in the study of symmetric structures. The research presents an advanced version of Riley’s theory for accurately computing the natural frequencies of structures, showing significant improvements in efficiency and precision over traditional methods like Monte Carlo simulations. Key findings include the ability of the upgraded Riley theory and graphs to conduct comprehensive structural reliability and sensitivity analyses, particularly in evaluating changes in failure risk and aiding in precise design. Numerical analysis demonstrates the method’s rapid convergence and accuracy, proving its effectiveness in structural analysis. The study also explores the impact of geometric parameter variations on the free vibrations of rectangular sheets, offering crucial insights for various engineering applications. These findings have broad implications in mechanical, marine, aerospace, and civil engineering, particularly in the design and analysis of structural components such as fuselages, airplanes, missiles, and tank bottoms.

Damping loss factor identification based on energy finite element method

A new method for the identification of damping loss factor (DLF) is proposed in this paper, combining energy finite element method (EFEM), energy density testing theory and particle swarm optimization (PSO) algorithms. An experimental system is designed to identify the DLF of a rectangular plate made in steel based on the theoretical formulation. The identification results are compared with the power injection method (PIM) results. Besides, the identification method is modified by considering the range of test deviation. This research shows that the new method is less dependent on the location of the selected exciting points and the location of the measurement points in far-field conditions, and it is more consistent with the PIM method and has a higher accuracy within the correction range. In addition, the method can be used to obtain an accurate identification results with fewer measurement points compared with PIM, which is of high practicality.

Design and Optimization of Heatsink for an Active CPU Cooler using Numerical Simulations

Heat generation in CPUs with high computing performance has been a very serious issue that deteriorates their performance. To ensure that CPUs operate at their maximum potential, it is essential to maintain their temperature below 80 °C. Forced convection coolers comprising of a heatsink and fan are considered the most effective means of satisfying operating temperature requirements for a CPU in order to ensure its maximum performance. A heatsink design paired with a fan with airflow speed of 80 cubic feet per minute (CFM) for a CPU Cooler targeted at CPUs with maximum heat generation of 380 watts operating in ambient temperature of 25 °C, is developed using numerical methods of Computational Fluid Dynamics (CFD) and topology optimization using ANSYS Mechanical and ANSYS Fluent. Various fin profiles, fin arrangements, fin numbers and heatsink materials are comparatively analyzed. The best results from the comparative analyses are combined to present a base design capable of keeping the CPU temperature below 80 °C, which is the requirement for ensuring maximum computing performance. A 30 fin heatsink with covered rectangular plate fins in an arc-shaped placement configuration is determined to provide the maximum cooling performance. In materials, Silicon Carbide resulted in the lowest CPU temperature of 78°C, followed by copper at 84 °C. Silicon Carbide heatsink successfully managed to satisfy the requirements for maximum CPU performance. Copper heatsink is unlikely to cause CPU failure, but it fails to meet conditions for maximum CPU performance. In addition, this base design is then optimized for material cost reduction using topology optimization which resulted in 13% reduction in material cost with a negligible cooling performance reduction of only 0.32%. The overall design of the cooler can be improved by incorporating fan design and various CPU load conditions into the design parameters in future research.

Three-dimensional analysis of thermoelastic damping in couple stress-based rectangular plates with nonlocal dual-phase-lag heat conduction

Since thermoelastic damping (TED) plays a decisive role in the functioning of micro/nanoresonators, accurate assessment of its magnitude in these tiny systems is imperative. Successfully simulating the thermomechanical behavior of various mechanical elements requires consideration of heat transfer in all three principal directions. Furthermore, there is no denying the impact of size on the mechanical and thermal domains. In light of these reasons, the current study strives to create a three-dimensional (3D) size-dependent TED model for rectangular micro/nanoplates via the use of the modified couple stress theory (MCST) and nonlocal dual-phase-lag (NDPL) heat transfer model concurrently for the first time. To accomplish this, by applying the Galerkin method to the 3D NDPL-based heat equation, the relation of temperature field is derived. Moreover, the coupled thermoelastic constitutive relations are extracted by including the couple stress effect. Lastly, by exploiting the obtained relations in the energy dissipation approach, an analytical TED formula in the form of infinite trigonometric series is attained. To appraise the rightness of derived formulation, the method of model reduction is applied. Moreover, an elaborate convergence analysis is implemented to realize the number of sufficient terms of the provided solution that result in well-grounded outcomes. A rigorous simulation study is also accomplished on the role of 3D heat conduction, scale parameters of MCST and NDPL model, boundary constraints, geometry and material in the variations of TED. Findings imply the definite impact of heat conduction dimension on TED value in rectangular plates with a large ratio of thickness to length and width.

Flutter investigation and deep learning prediction of FG composite wing reinforced with carbon nanotube

Abstract The flutter of a composite wing reinforced with functionally graded carbon nanotubes (CNTs) has been investigated. A rectangular plate models a supersonic wing with cantilever boundary conditions. To determine displacement fields of a moderately thick plate, shear deformation theory is used. Using the Hamilton principle, a first-order piston theory was used to simulate supersonic airflow. This study examines four types of CNT thickness. Also, four different CNT distribution patterns are investigated. In a two-layer asymmetric composite, the effects of patch mass, mass distribution, fiber orientation angle, and distribution of CNTs were examined. Moreover, the results are compared and verified with other studies. A greater mass ratio led to a smaller flutter boundary, while a longer added mass increased the flutter boundary. A variation in the distribution pattern in CNT fiber orientation results in a distinct behavior of the flutter boundary for asymmetric composites with increasing orientation angles. The artificial neural network is utilized to predict the damping ratio, and the results showed great accuracy compared to the study results. Hyperparameter tuning is employed for better optimizing the predictive models.

Hamiltonian system-based analytic thermal buckling solutions of orthotropic rectangular plates

New analytic solutions for the thermal buckling of orthotropic rectangular plates are presented using the symplectic superposition method (SSM). The SSM deals with higher-order partial differential equations within the Hamiltonian framework, distinguishing it from conventional methods in the Lagrangian framework. The solution procedure involves dividing the original problem into two subproblems, utilizing the techniques in the symplectic space, like separating variables and symplectic eigen expansion, to achieve analytic solutions, and using superposition to obtain the final solutions. One notable advantage of the SSM is the absence of predetermined assumptions, which is a departure from conventional semi-inverse methods. The new analytic solutions under challenging non-Lévy-type boundary conditions (BCs) were rarely addressed in prior research due to their complex mathematical characteristics, which reveals the main novelty. Comprehensive thermal buckling results are provided for isotropic/orthotropic plates under various BCs, validated by those from the finite element method with a maximum difference being 2.7374% and literature with a maximum difference being 5.9201%. Parameter analyses quantify the effects of BCs, aspect ratio, and thickness-to-length ratio on critical buckling temperatures and mode shapes.

The Problem of the Reliability of Bending Models for Composite Plates of Medium Thickness

Most refined bending models of medium-thick plates, which consider transverse shear and partial compression deformations, differ little. However, despite a significant increase in the order of the governing differential equations, the results obtained from their equations give mainly a small increase in accuracy compared to the existing theories. On the other hand, such an increase in the order of the constructed systems of differential equations requires a significant increase in the effort required to solve them, complicates their physical interpretation, and narrows the range of people who can use them, primarily engineers and designers. Therefore, developing a plate-bending model that incorporates all the above factors and is on par with previously applied theories regarding the complexity of the calculation equations remains relevant. For example, most of the applied theories that do not consider transverse compression cannot be used to solve problems of contact interaction with rigid and elastic dies and bases because it is impossible to satisfy the conditions at the contact boundary of the outer surface of the plate, as well as the boundary conditions at the edges of the plate. Therefore, to provide guaranteed accuracy of the results, some researchers of these problems have introduced such a concept as “energy consistency” between the functions of representation of the displacement vector components, their number, the order of equations, and the number of boundary conditions. The authors, based on the developed version of the model of orthotropic plates of medium thickness, investigate the problem of taking into account the so-called “energy consistency” effect of the bending model, depending on the order of the design equations and the number of boundary conditions, as well as its usefulness and disadvantages in practical calculations. The equations of equilibrium in displacements and expressions for stresses in terms of force and moment forces are recorded. For rectangular and circular plates of medium thickness, test problems are solved, and the numerical data are compared with those obtained using spatial problems of elasticity theory, as well as the refined Timoshenko and Reissner theories. An analysis of the obtained results is provided.

Static Convergence Behaviour of Discrete Kirchhoff-Mindlin Triangular Element on Rectangular Functionally Graded Material Plates

Rapid infrastructure development is consistent with the increase in innovation in structural elements. One of the main parameters in infrastructure development is materials. An advanced composite material called Functionally Graded Material (FGM) has been widely used. FGM is composed of the combination of two or more materials. FGM increases the bond strength between the layers compared to conventional composite materials, eliminating stress at the interface layers, and reducing cracks. However, further studies are required to understand the behaviour of the FGM plate. Finite element analysis (FEA) was considered to evaluate the convergence behaviour of the FGM plates. The Discrete Kirchhoff-Mindlin Triangular (DKMT) element is employed in the analysis. Studies regarding the FGM plates with the DKMT element were limited to square shapes and skew shapes. Thus, this study aims to study the convergence behaviour of the rectangular FGM plates composed of ceramic and metal. The FEA was carried out in different types of meshing, ratio a/h, ratio a/b, the power-law index, and boundary conditions. The analysis results indicate that the application of the DKMT element in analyzing the FGM plates gives good asymptotic and convergence behaviour. Thus, this method has proved reliable and sustainable.

Experimental analysis of biocementation technique for sealing cracks in concrete water storage tanks

Water supply systems are vital infrastructures that provide an indispensable public service to society. An important percentage of water losses is caused by the development of cracks in water storage tanks, so efficient crack sealing repair works must be investigated. Biocementation has been used with good results for sealing cracks in many construction concrete infrastructures (e.g., buildings), as an alternative to standard materials, such as polymeric resins and cement mortars. This research aims at the experimental evaluation of biocimentation effectiveness in terms of watertightness for sealing cracks structures in contact with pressurized water, to be further applied to repair cracks in water storage tanks. Biocementation treatment is applied in cracks artificially created in small rectangular concrete plates 4 cm thick, with three crack widths (i.e., 0.1, 1 and 10 mm) to cover different real cases in which the repair with this technique is viable. The 10 mm width cracks were filled with sand before the treatment. Different rounds of bacteria Sporosarcina pasteurii are applied, being the efficiency of the treatment investigated by performing watertightness tests through variable water head tests starting at 10 kPa (1 m of water). Dissolution is discarded through measurements of ultrasonic pulse velocities before and after the watertightness tests and the results are explained by images of thermographic camera. The presence of biocement is confirmed by mineralogical analysis of the precipitate extracted from the cracks after breaking the plates through bending tests. Obtained results are very promising since the biocement precipitated can completely stop water flow in the 0.1 mm cracks and, for the cracks with 1 mm and 10 mm widths, the treatment can reduce the initial flow rate, on average, by 95% and 98%, respectively. The performance of the 10 mm width cracks previously filled with sand was similar to that of the 1 mm cracks, where no sand was used. The material strength recovered after the treatment is not as good as desirable: although a maximum recovery of 95.5% of the initial strength (average values) is found for the smallest crack width and it was almost null for the other two crack widths. These results demonstrate that the technique is effective when watertightness is required, but not when material strength must also be recovered. Further research is required to investigate the maximum water pressures that can be applied to each case, eventual effects of biocimentation on water quality and the durability of the treatment.

Experimental and numerical analysis of uni-axial buckling of single-phase functionally graded porous polymeric sandwich plates

The porosity gradient functionally graded material (PFGM) is one of the most popular types of FGM, in which the porosity in the material is made to change in the specified direction. This study looks into the buckling problems of rectangular sandwich plates made of single-phase porous functionally graded materials (PFGMs), commonly used in aircraft structures and biomedical applications. A compression test was performed on the 3D-printed polymeric FG specimens bonded with two thin solid face sheets on the upper and lower surfaces. The critical stress of well-designed and fabricated 3D printed FGM plate samples with various metal core types is determined using a PC installed on universal testing equipment (UTM). The effect of different essential parameters (such as porous ratio, gradient exponent, and aspect ratio) on buckling load and total deformation were explored.The finite element method (FEM) was used to run a numerical simulation on elastic buckling using ANSYS 2021 R1 software to validate the experimental results. The load-displacement relationships and deformed morphologies were investigated using experiments and numerical analysis. The topology arrangement and relative density of the polymer core were examined using the SEM micro-tomography test based on porosity distribution to check the resistance of the sandwich to buckling load. PETG/Al sandwich plates have been found to have critical buckling loads that are 2.52 % higher than PLA/Al sandwich plates, while TPU/Al sandwich plates show increased essential loads of buckling of 5.139 %. The FEM and experiment results show that the existence of porosity in the PLA core in the PFGM plate can reduce the buckling strength tremendously, about 10.52% and 6.8 %, respectively. It was evident that the numerical results show a good agreement with the experimental findings, with a maximum discrepancy of no more than 12 % occurring at the (TPU/Al) sandwich plate with a porosity of 30%.

Termostressed state of a three-layer rectangular plate under non-stationary convective heating conditions

The study considers a rectangular isotropic plate with a layered irregular structure. It is convectively non-stationarily heated by an external environment. The initial relationships of the non-stationary heat conduction and thermoelasticity problem are formulated using a five-mode mathematical model based on the shear deformation theory of thermoelasticity. Using the methods of Fourier and Laplace integral transforms, general solutions have been obtained for the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a hinge-supported plate along its edges. A numerical analysis of the temperature field, radial deflections, normal forces, bending moments, and normal stresses, depending on geometric parameters and the Bi criterion, has been performed for a three-layer plate. The materials of its layers are made of ceramics and metal. The temperature and mechanical parameters have been analyzed for the layering configuration of the plate: metal-ceramic-metal.

The compelled fluctuations of the rectangular two-layer a piecewise homogeneous plate of the constant thickness

In the given work it is solved private applied problems of fluctuation of rectangular and infinite two-layer plates on the basis of the approached equation of the sixth order received in article. Frequencies of own fluctuations, influences of parameters of making plates on these frequencies, viscosity influences etc. are investigated The received theoretical results for the decision of dynamic problems of cross-section fluctuation a piecewise homogeneous two-layer plates of a constant thickness when a material of the top layer piecewise ‒ a homogeneous two-layer plate elastic, and another satisfies models of Marksville, that is viscoelastic, allow to count more precisely the is intense-deformed condition of plates at non-stationary external loadings.

Mathematical modeling of bending of a thin orthotropic plate clamped along the contour

Within the framework of Kirchhoff’s theory, a new approach to constructing a solution to the problem of modeling the bending of a thin rectangular orthotropic plate clamped along the contour, which is under the influence of a load normally distributed over its surface, is proposed. the solution to the inhomogeneous biharmonic equation for an orthotropic plate is obtained in the form of a partial sum of a double series in Chebyshev polynomials of the first kind. To find the coefficients in this expansion, the boundary value problem is reduced by the collocation method to a system of linear algebraic equations in matrix form using the properties of these polynomials. Based on matrix and differential transformations, expressions for bending moments and shearing forces are obtained. the results of calculations of the bending of the middle surface of the plate under different loads on the plate are presented, which demonstrate the effectiveness of the proposed approach.

Model Equations of the Theory of Elasticity in Strains: Classical and New Formulations

The article is devoted to the construction of model equations of the theory of elasticity with respect to deformations. Classical and new versions of boundary value problems of the theory of elasticity in strains are considered. In the classical version, model equations in strains are constructed within the framework of the Beltrami-Michell equations. A new version of model equations in strains is based on a new formulation of boundary value problems of the theory of elasticity in stresses. Discrete equations are constructed using the finite-difference method for two-dimensional problems. The well-known problem of tension a rectangular plate with a parabolic load applied on opposite sides has been solved. By comparing the numerical results of boundary value problems in classical and new formulations, as well as the Timoshenko-Goodier solution, the validity of the formulated model equations in strains and the reliability of the obtained numerical results are ensured.

Two-dimensional double horizon peridynamics for membranes

<abstract> <p>In this study, a two-dimensional "double-horizon peridynamics" formulation was presented for membranes. According to double-horizon peridynamics, each material point has two horizons: inner and outer horizons. This new formulation can reduce the computational time by using larger horizons and smaller inner horizons. To demonstrate the capability of the proposed formulation, various different analytical and numerical solutions were presented for a rectangular plate under different boundary conditions for static and dynamic problems. A comparison of peridynamic and classical solutions was given for different inner and outer horizon size values.</p> </abstract>

ПРИМЕНЕНИЕ СИЛОВЫХ КОМПОЗИТНЫХ КОНСТРУКЦИЙ ДЛЯ МАШИН АПК

The equipment involved in the agro-industrial complex is subject to active wear and corrosion. The modern solution is to replace the body steel elements with composite ones, but they have not become widespread due to some conservatism of the industry. (Research Purpose) The research purpose is substantiating the effectiveness of the use of polymer composite materials for power body elements of agricultural transporting machines. (Materials and methods) It was noted that the replacement of traditional steel products with composite ones makes it possible to achieve a significant reduction in the mass of products while increasing their reliability, increasing the resource due to the absence of corrosion destruction and minimizing the cost of conservation and storage. (Results and discussion) A comparative analysis was performed using the example of a trailer of an agricultural vehicle, all-metal and composite versions of the side design were considered. The presented options were substantiated on the basis of a qualimetric assessment by an expert method according to the criteria: cost, prospects, material consumption, environmental friendliness, maintainability, corrosion and wear resistance. Analysis of the calculation result showed that the experts preferred the composite version compared to the traditionally used steel sides. An accumulation histogram was constructed according to the data obtained, showing the significance of the final assessment for each criterion (Conclusions) Proved the feasibility of introducing power body elements based on polymer composite materials for machines of the agro-industrial complex. A design based on polymer composite materials of the side of an agricultural trailer in the form of a five-layer rectangular plate of sandwich-structured type capable of withstanding loads arising during operation has been proposed. The use of this design will significantly reduce the mass while increasing corrosion resistance and reliability, which will reduce the cost of repair, maintenance and consumption of fuel and lubricants.

Термические напряжения в изоляции тяговых электродвигателей локомотивов

Thermomechanical processes and cyclic temperature changes in the structure of insulation of traction electric motors lead to thermal gradient appearance. Increased temperature widens the elements of winding, damages insulation material (thermal fatigue, extreme deformation, creeping, etc.). Methods for the numerical study of stress-strain state of the material of the insulation of traction electric motors of diesel locomotives on different stages of static and cyclic loading have been developed. It is determined, that the main factor defining the intensiveness of insulation aging is the impact of alternating cyclic thermo-mechanical stresses, but not thermomechanical destruction of impregnating composition. Analytical expressions for the evaluation of stress-strain state of insulation of traction motor diesel locomotives are received and they are distinguished by the fact that they require a relatively small amount of basic data and let conduct accurate calculations taking into account the peculiarities of the electric drive system. An approximate predictive model for the evaluation of interphase thermal stresses in the assembly of the materials of winding slots of the traction electric motor of a diesel locomotive is developed. The insulation material is considered as linear elastic when the deformation level is less than the yield strength. The joining components can be considered from the view point of structural analyses as elongated rectangular plates suffering from linear elastic strains. The analyses results can be used for evaluation of thermomechanical stresses in the impregnating materials of traction electric motors of diesel locomotives and their analogues.

МОДЕЛИРОВАНИЕ ПРОЦЕССОВ ВЛАГОПОГЛОЩЕНИЯ ДРЕВЕСИНЫ ПРИ ИЗМЕНЕНИИ ДАВЛЕНИИ СРЕДЫ

On the basis of differential equations of material balance in a hermetic chamber and heat and mass transfer in the object under study under given boundary conditions, a mathematical model describing technological processes occurring with changes in medium pressure has been developed. The methodology of experimental studies of the process of material saturation in a unit operating in a periodic mode at varying medium pressure is presented. Birch veneer is taken as a model material for experimental studies, the heat and mass transfer characteristics of which are widely known. The preparation of the experiment consisted in the production of samples of the saturated material, in the form of rectangular plates with dimensions of 30 * 90 mm and a thickness of 4 mm. The results of physical and mathematical modeling are reflected in the form of profiles of changes in moisture content and temperature along the cross section of the material during saturation. It is established that the active process of saturation of the veneer with moisture under changing medium pressure is observed during the first 15 minutes, and by this time the temperature in the center of the material exceeds 120 ° C. The results of mathematical modeling were obtained using the Visual Basic for Applications software environment. The discrepancy between the calculated and experimental data does not exceed 15%. The developed mathematical model of the process makes it possible, when the capacities of sources (steam generator) and drains (condenser and vacuum pump) vary, to recommend the modes of conducting the technological process or to select the capacities of the steam generator and pump in accordance with the required law of pressure change in the technological process. For physical modeling of the investigated process in the processing of wood raw materials, an experimental setup was created, on which experimental data confirming the adequacy of the developed model were obtained.