Deflection Of Thin Plate Research Articles

Enhancing the Mechanical Performance of Concrete Slabs through the Incorporation of Nano-sized Iron Oxide Particles (Fe2O3)

The utilization of recycled iron in durable concrete production has gained attention for enhancing sustainability and resource efficiency. Simultaneously, incorporating nanoparticles as supplementary cementitious materials (SCMs) offers significant benefits. Introducing nano-sized iron particles (Fe2O3) into the cement paste results in a compact microstructure, improving strength, and durability. In this study, we investigate the bending behavior of concrete slabs reinforced with Fe2O3 nanoparticles using the non-local quasi-3D shear deformation theory based on Eringen's non-local differential constitutive relations. To characterize the elastic material properties of the nanocomposite, we employ Eshelby's homogenization model. In order to extend the applicability of our findings, we assume that the concrete plate rests on Kerr's foundation, which includes a shear layer connected to upper and lower springs. By deriving the equations of motion using the principle of virtual work, we establish a comprehensive framework for analyzing the bending of the concrete plate. To solve the equilibrium equations for a simply supported concrete plate, we present Navier's analytical solutions. Our investigation considers various influential parameters, such as the concentration of Fe2O3 nanoparticles in the concrete matrix, the elastic constants of the soil medium, different types of bending loads, and size-dependent nonlocal parameters. One of the most captivating findings of this study is that the incorporation of 30 wt% of iron nanoparticles in concrete leads to a remarkable improvement of 60% in the elastic properties of the material. Additionally, this same amount of iron nanoparticles has shown the potential to reduce the deflection of thin plates by over 60%.

Determining residual stress profile induced by end milling from measured thin plate deformation

Lots of residual stress data is usually required to support the machining process optimization of the thin-walled component. In this work, an inverse method to determine the residual stress profile for end milling titanium alloy Ti-6Al-4V is developed based on the measured thin plate deformation. Firstly, the hyperbolic tangent function is introduced to parameterize the residual stress profile of end milling. Then the mapping relationship between the residual stress profile and the bending deflection of thin plate is established. Furthermore, a solution algorithm to determine the characteristic coefficients of the residual stress profile from the measured deflection is proposed. And the residual stress profiles under different cutting conditions are inversely solved. Finally, six groups of end milling verification experiments are conducted and a high average prediction accuracy of 90.19% is obtained. The proposed reverse method can quickly obtain the residual stress profile by measuring the machining deformation of thin plate.

Uticaj temperaturskog gradijenta na savijanje tankih ploča

Within the theory of thermo-elasticity, the temperature field of thin plates is commonly defined via two parameters: temperature in the mid-plane and linear temperature gradient normal to the mid-plane. First, the paper analytically proves the justification of that assumption in machine structures. Then, in an analytical closed form, applying the integral transformation method, the thin plate deflection caused by a constant temperature gradient is defined. It is shown that, in that case, the plate deflection does not depend on its thickness but only on the plate dimensions in the mid-plane. Analytically defined values are compared to corresponding values obtained by applying the thin plate finite element, where the temperature field is described using the two mentioned parameters. This finite element is defined and programmed within the Komips program package. The influence of the temperature gradient on the behavior of constructions mostly depends on the type of material. That is why the behavior of some structural elements made of brass, steel, and concrete is analyzed in this paper.

АППРОКСИМАЦИЯ ПРОГИБОВ ПЛАСТИНОК, ЛЕЖАЩИХ НА ВИНКЛЕРОВОМ ОСНОВАНИИ

The purpose of this research is to develop the method of shape factor interpolation for calculating the maximum deflection of thin plates on an elastic Winkler base, which are widely used in modeling the operation of elements of building constructions of buildings and structures. The above calculation method allows to obtain solutions based on direct analytical dependences, the argument of which is an integral characteristic of a flat convex one-connected area - the shape factor. This characteristic has applications in a number of problems of mathematical physics and is known from the works of scientists G. Polia and G. Szegö. The shape factor was first applied to the calculation of plates by Professor V.I. Korobko. The method of interpolation by shape factor was developed by Professor A.V. Korobko. When determining the maximum deflection of thin plates on an elastic base, some parameters of the problem are considered as functions of the shape factor of the plate in consideration and are determined by the type of boundary conditions on its contour. The present study is devoted to the construction of approximating functions for continuous sets of plates of characteristic outlines and boundary conditions. The paper presents functions for calculating the maximum deflection of elastic plates in the form of isosceles triangles, rhombuses and rectangles. The plates with various combinations of hinged support and rigid pinch along their individual sides, loaded with a continuous uniformly distributed load, are considered. The established functional dependences are intended for direct use in the calculation of plates of the specified outlines, as well as for obtaining reference solutions during interpolation of the values of maximum deflections of plates of more complex outlines.

Flexural and sinking stiffnesses investigation of metal, glass fiber reinforced plastic and hybrid laminates through a semi‐experimental method

AbstractThe hybrid metal/GFRP laminate is a kind of typical hybrid structure consisting of thin metal sheets and glass fiber reinforced plastic laminates, which combines the benefits of low cost, high strength, excellent impact resistance, and good corrosion resistance. In this work, a semi‐experimental method based on a nondestructive test and the analytic solution of the thin plate's deflection for the flexural and sinking stiffnesses identification of hybrid metal/GFRP laminates was proposed and validated by corresponding numerical simulations. Then, in total 17 different specimens (including metal, GFRP and hybrid metal/GFRP) were prepared and tested by the proposed semi‐experimental method to investigate their flexural and sinking stiffness performances in multiple perspectives (including stiffness per unit mass/volume/price/surface density). The analytic solutions show that the flexural and sinking stiffnesses are proportional to macro equivalent moduli of laminate specimens and proportional to the third power of thicknesses of specimens. The test results show that the stiffnesses per unit mass/density of the 7‐ply GFRP laminate can be comparable to those of the variable‐thickness rolled blank (VRB) made of steels, but the stiffnesses per volume/price of the 7‐ply GFRP laminate are only one fifth to one quarter of those of VRB. It is also found that though the hybrid metal/GFRP laminate with the combination sequence of aluminum plate/5‐ply GFRP/aluminum plate has the lowest GFRP layers' volume fraction, but has the best performances of stiffnesses per unit mass/density. Besides, the VRB part can significantly improve the stiffnesses per unit volume of hybrid metal/GFRP laminates.

A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates

In this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.

Existence of polynomial solutions of degree 4 of the Monge-Ampère equation. Large deflections of thin plates

We provide necessary and sufficient conditions for the solvability of a simplest Monge-Ampère equation, assuming that both the right-hand side and the solution are polynomials of degree 4. We give a constructive method of solution of the basic system of algebraic equations corresponding to the Monge-Ampère operator under the above conditions on the prescribed polynomial. Applications to large deflections of thin plates are presented. Bibliography: 9 titles.

Research on the Extraction Technology of Gas-Integrated Working-Face in Close Coal Seam Mining in Permafrost Area

In order to understand the mechanical deformation characteristics of the permafrost layer and the stress-fracture-gas seepage law of the superimposed mining of the proximity coal seam group, a coupled gas-solid model of the proximity coal seam in the permafrost area was constructed based on the theoretical model of small deflection of thin plate and the deformation control equation, gas transport equation, porosity evolution equation and permeability evolution equation of the coal rock body, and the analysis of the dependent project. The analysis was carried out at the NMT coal mine. The results show that the multiyear permafrost seam in the NMT coal mine has a tendency to move to the left and will produce extensive pull damage near the right end. The initial and periodic collapse steps are 192.87 m and 43.24 m, respectively; the pore and fissure structural characteristics of the NMT coal seam are not conducive to gas transport, and the stress and fissure fields have a nonlinear increasing effect. The study also proposed the “protected seam mining and bottom extraction lane directional drilling group comining technology” to increase the daily production of working face by 25.8%. The results of the study can be used as a reference for similar projects.

Optimization of a Radial Point Interpolation Meshless strategy for strain gradient nanoplates

A parameter optimization of the Radial Point Interpolation Meshless Method (RPIM) is presented in this work for solving the static bending analysis of Kirchhoff nanoplates which include the first order strain gradient theory. Optimization in meshless strategies is often required since shape parameters, of the Radial Basis Functions (RBFs) used, strongly influence the numerical results when different geometries, boundary conditions or various mechanical aspects are considered. Due to the introduction of a first order strain gradient theory within Kirchhoff plate framework, to approximate the bending degrees of freedom, a Hermite RPIM is used. The static deflection of thin plates with different aspect ratios and boundary conditions is analysed. The results of the static analysis are compared with the solutions available in the literature and good agreement among all presented results as well as convergence behaviour is shown.

Time history response of an elastic thin plate on a transversely isotropic multilayered medium due to vertical loadings

An efficient theoretical solution is proposed to investigate the time history response of an elastic circular thin plate bonded on multilayered transversely isotropic soils. Based on the analytic layer-element method, dynamic differential equation governing time dependent deflection of thin plate is solved with the global stiffness matrix equation of layered soils. In this way the compatible deformation at the plate-soil interface can be rigorously satisfied. After adopting the numerical inversion of Laplace-Hankel integral transform, the displacement and stress responses of the plate-soil system are eventually obtained in the time domain. The numerical results are validated by comparing with the finite element method (FEM). Parametric studies are also carried out to elucidate the influence of the external load form, plate rigidity, material anisotropy and soft stratum on the time history behavior of plate-soil system. It’s indicated that significantly influence has been bought by the shallowed buried weak layer, and as the buried depth decreases the elastic plate becomes more flexible.

Nonlinear free vibration analysis of piezoelectric laminated plate with random actuation electric potential difference and thermal loading

Solutions to nonlinear free vibration analysis of an arbitrarily laminated thin rectangular plate subjected to thermo-piezoelectric load is investigated. The governing equation of motion is based on von Kármán’s deflection of thin plates to capture the effects of moderate geometric nonlinearity. Induced strain actuation assumption is confined to a linear model for applied low electric fields and varying temperature distribution along the thickness direction. The solution for lateral displacement field for simply supported and clamped condition edges are determined by displacement-stress; mixed formulation. The applied Galerkin’s projection reduces the governing nonlinear partial differential equation to a time dependent ordinary differential equation. Solutions are obtained using a predictor-corrector numerical integration scheme. The effects of randomly applied actuation potential and thermal loading on nonlinear frequency ratio are examined. The marginal values provided using various probabilistic methods are found to be in good agreement with each other.

Characteristics of a novel thermal-induced epoxy shape memory polymer for smart device applications

With the increasing demand for the application of electronic technology, high performance, multi-function and intelligent control have been the main developing trend of smart material. In this paper, a novel thermal-induced epoxy shape memory polymer (ESMP) is developed and utilized as substrate material for smart flexible electronic device. Owing to the advantage of thermal-induced ESMP, circuit substrate can be rigid-flexible controllable at different working temperature. Firstly, the ESMP samples are prepared and then the mechanical and shape memory characteristics are studied respectively. Secondly, the mechanical behaviors are analysed based on thin plate deflection theory and finite element simulation software on the engineering application. Meanwhile, corresponding experimental tests are designed and performed to corroborate the theory and simulation results. At last, ESMP based test circuit board is designed and manufactured. The electrical signals under large deformation are also measured and compared with typical circuit board. The results show that the prepared ESMP, as a smart material, has good reliability, flexibility and great application prospects in the field of intelligent devices.

The Improved Complex Variable Element-Free Galerkin Method for Bending Problem of Thin Plate on Elastic Foundations

In this paper, the improved complex variable element-free Galerkin (ICVEFG) method is proposed for solving the bending problem of thin plate on elastic foundations. In the ICVEFG method, the approximation function regarding the deflection of thin plate is formed with the improved complex moving least-squares (ICVMLS) approximation, the discrete equation is obtained from Galerkin weak form of bending problem of thin plate on different elastic foundations, and essential boundary conditions are considered based on penalty method. As the ICVMLS approximation is based on the complex variable theory, it can obtain the shape function quickly with high precision. Three sample problems are used to discuss the advantages of the ICVEFG method, and the numerical results show that the ICVEFG method presented in this paper has a fast convergence speed and great computational accuracy.

Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials

• Two-dimensional linear and nonlinear Berger equation investigated. • Meshless multiple-scale Pascal polynomials method applied to fourth order equations. • Accurate results on complexly shaped domains with low number of collocation points . • Efficiency of proposed method demonstrated based on six problems. In this study, we employ Pascal polynomial basis in the two-dimensional Berger equation, which is a fourth order partial differential equation with applications to thin elastic plates. The polynomial approximation method based on Pascal polynomial basis can be readily adapted to obtain the numerical solutions of partial differential equations. However, a drawback with the polynomial basis is that the resulting coefficient matrix for the problem considered may be ill-conditioned. Due to this ill-conditioned behavior, we use a multiple-scale Pascal polynomial method for the Berger equation. The ill-conditioned numbers can be mitigated using this approach. Multiple scales are established automatically by selecting the collocation points in the multiple-scale Pascal polynomial method. This method is also a meshless method because there is no requirement to establish complex grids or for numerical integration. We present the solutions of six linear and nonlinear benchmark problems obtained with the proposed method on complexly shaped domains. The results obtained demonstrate the accuracy and effectiveness of the proposed method, as well showing its stability against large noise effects.

Large Deflections of Elastic Rectangular Plates

It is known that elastic large deflections of thin plates are governed by von Karman nonlinear equations. The analytical solution of these equations in the general case is unfeasible. Samuel Levy, in 1942, showed that large deflections of the rectangular plate can be expressed as a double series of sine-shaped harmonics (deflection harmonics). However, this method gave no way of creating the computer algorithm of solving the problem. The stress function expression taken in the Levy's method must be revised to find the approach that takes into account of all possible products of deflection coefficients. The algorithm of solving the problem for the rectangular plate with an arbitrary aspect ratio under the action of the lateral distributed load is reported in this paper. The approximation of the plate deflection is taken in the form of double series proposed by Samuel Levy. However, the expression for the stress function is presented in the form that incorporates products of deflection coefficients in the explicit form in distinction to the Levy's expression. The number of harmonics in the deflection expression may be arbitrary. The algorithm provides composing the system of governing cubic equations, which includes the deflection coefficients in the explicit form. Solving the equation system is based on using the principle of minimum potential energy. A method of the gradient descent is applied to find the equilibrium state of the plate as the minimum point of the potential energy. A computer program is developed on the basis of the present algorithm. Numerical examples carried out for the plate model with 16 deflection harmonics illustrate the potentialities of the program. The results of solving the examples are presented in the graphical form for the plates with a different aspect ratio and may be used under designing thin-walled elements of airplane and ship structures.

Modelling and analysis for a pilot relief valve using CFD method and deformation theory of thin plates

The current work is concerned with modelling and analysis for a pilot relief valve, thus successfully bringing a systematic method for designing and analyzing similar valves. The essence of the work is to solve two important problems, one for positions of the pilot valve influenced by flow force and the other is for the opening of the relief valve governed by a thin annular plate. The computational fluid dynamics (CFD) method is used to present the flow force. Using a series of experiments, the flow rate versus pressure drop shows the rationality of the CFD results. In order to obtain the opening of relief valve with higher accuracy, the large deflection theory of thin plates is adopted. An equivalent method for replacing the concentrated force is innovatively proposed so that all of the loads of the plates can be given by a unified expression, which reduces the number of the governing equations and intermediate boundary conditions. For presenting a very simple and reliable method for solving the governing equation, an unconstrained nonlinear optimization is innovatively introduced to solve the deflection of the thin annular plate. Being verified by finite-element method (FEM) of the relief valve, the equivalent method and optimization can solve deflection of thin plates rapidly and accurately. Reflected through a complete model for the pilot relief valve, the theoretical flow rate of the pilot relief valve is consistent with experimental conclusion. Once again, the comparisons bring us insight into the accuracy of the method adopted in the current work.

Nonlinear Vibration Analysis of Stochastic Thin Plates Based on Lindstedt-Poincar\'{e} Perturbation Method

Nonlinear vibration computational problem of stochastic thin plates in large amplitude was investigated here. Based on the nonlinear theory of large deflection of thin plates, we derived the governing equations of nonlinear free vibration of isotropic thin plates. And solved the governing equations by Lindstedt- Poincare Perturbation Method Presented herein are asymptotic analytical solutions for the frequency function of the thin plates with fixed parameters. Based on the theory of random field, after mathematical treatments of the probability density function of random vector expressions, such as determining the interval of integrating, substituting variable and transforming integral order, then the probability density function of nonlinear vibration frequency of stochastic thin plates can be obtained directly. And combined with Monte Carlo Simulation analysis and our previous work (1), the result of the method in this paper is validated, which testifies that the method in this paper is effective and of preferable precision.

Analytical modeling of residual stress and the induced deflection of a milled thin plate

Predictions of residual stress-induced deformations on thin aerospace parts are still among the most challenging and relevant problems in engineering applications. In this paper, a new methodology based on analytical modeling of machining mechanics is introduced in order to predict thin plate deflection induced by residual stresses given material attributes, process configurations, and parameters. The model uses an elasto-plastic constitutive behavior algorithm to predict the plastic stress, strain, and thus residual stresses. A relaxation procedure is applied upon residual stresses in order to quantify the elastic deflection when the plate is unloaded from cutting forces. Experimental validations are presented on multi-pass milling of aerospace grade aluminum alloy Al7050-T7451. The newly developed analytical model shows promise to predict the residual stresses and the thin plate deflection profile under various cutting conditions.

切向气流对激光加热金属板非熔化穿孔效应的影响

On the basis of elastic bending theory,the bending deflection expression by both quadrate and circinal facula irradiations is deduced for burn-through effect with no-melting of metal heated by a laser caused by tangential airflows.A rupture criterion for burn-through with no-melting is given out by the Mises theory.The study indicates that the material softened by laser heating is the primary mechanism for burn-through effect with no-melting.The maximal deflection of thin plate in the spot area is related to the airflow speed,spot diameter,plate thickness and the elasticity modulus(expression is U2a4/Eh3),the damage temperature is related to the airflow speed,spot diameter and the plate thickness(expression is(Ua/h)2),and the damage threshold under the circinal facula irradiated is higher than that under the quadrate facula.The numerical result indicates that for aluminum alloy shell,the damage threshold of laser energy can decrease much up to 40%-50% under 0.8 Ma tangential airflows,but it can decrease less(about 20%) for a stainless steel shell.The damage threshold decrease under airflows needs more attention.

Free vibration analysis of thin plates by using a NURBS-based isogeometric approach

An isogeometric finite element method is presented for natural frequencies analysis of thin plate problems of various geometries. Non-Uniform Rational B-Splines (NURBS) basis function is applied for approximation of the thin plate deflection field, as for description of the geometry. The governing and discretized equation for the free vibration of the Kirchhoff thin plates is obtained using the standard Galerkin method. Several numerical examples are illustrated to demonstrate the effectiveness, robustness and accuracy of proposed method and compared with the theoretical solutions and other numerical methods.