Deflection Function Research Articles

Buckling behavior of orthotropic thin plates using analytical and machine learning methods

Designing plate structures poses challenges due to potential instabilities and the complexities involved in obtaining accurate analytical solutions, especially for non-Lévy-type boundary conditions. This study employs the finite integral transform method to analyze the buckling behavior of orthotropic thin plates with complex boundary conditions. In solution procedure, the governing high-order partial differential equations are transformed into a system of linear algebraic equations, yielding exact and rapidly converging analytical solution. The method is simple and general and does not need to predetermine the deflection function. The correctness of the method is validated by comparison with numerical simulations performed using the ABAQUS software. Furthermore, Gene Expression Programming (GEP) is utilized to develop empirical models that predict buckling coefficients of isotropic and orthotropic plates under classical and non-classical boundary conditions. Material properties, aspect ratio, rotating fixed coefficient, and boundary conditions are used as input parameters, with simplified mathematical formulations to predict the buckling coefficient. Model performance is assessed using parametric analysis and statistical tests to ensure accuracy and generalization. Further investigation shows aspect ratio and rotating fixed coefficient are significant variables influencing buckling coefficients under classical and non-classical boundary conditions, respectively, followed by boundary conditions and material property. The performance of the GEP model is further assessed by comparing with linear and non-linear regression models. The results show that GEP outperforms the regression models, showing its higher prediction accuracy. This study not only addresses the challenges of designing thin plates with complex boundary conditions, but it also presents an effective machine-learning method for predicting buckling behavior. The analytical solution can be used as a benchmark for validating new analytical and numerical methods. The equations developed using GEP to predict the buckling coefficient of plates provide a useful tool for extrapolating results beyond the scope of this study.

Deflection function of a novel bolt-connected precast concrete floor

Owing to the post-cast procedure and relatively complex joint configuration, composite precast floors limit the full display of advantages of precast concrete systems. The development of fully assembled precast floors is of necessity; therefore, a novel bolt-connected precast concrete floor was proposed in this study to meet the increasing demand for quick assembly, lightweight, and good insulation in flooring systems. The proposed floors consist of precast sandwich unit slabs that are fully assembled to an integral floor with dry-type bolted connectors. Field static loading tests were conducted on four full-scale precast concrete floors. The test results indicate that the floors exhibited a two-way flexural ability due to the involvement of the bolted connectors. The restraining mechanism at anchorage and load-transferring mechanism at joints were investigated through a theoretical analysis. Based on the theory of plates and shells, a simplified theoretical method of solving the deflection function of the floors was established and validated to provide a reference for designing such fully assembled floors. The comparison between the theoretical and experimental results indicates that the arrangement of joint connectors could affect the theoretical calculation accuracy due to its influence on the performed continuity simplification. The application scope of the proposed theoretical method was evaluated through a finite element analysis.

Thermomechanical Nonlinear Vibration of Axially Loaded Functionally Graded Material Cylindrical Panels Including Porosity

For the first time, the combined influences of axial compressive load, porosity, geometric imperfection, elastic edge restraint, and elevated temperature on the nonlinear free vibration of functionally graded material (FGM) cylindrical panels are investigated in this paper. Unlike previous studies, the structural model considered in this paper includes practical situations and conditions arising in the design process and application of FGM cylindrical panels. The properties of material constituents are assumed to be temperature-dependent, and the effective properties of porous FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on first-order shear deformation theory, taking into account von Kármán–Donnell nonlinearity and initial geometric imperfection. Analytical solutions are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to derive a time differential equation containing both quadratic and cubic nonlinear terms. This differential equation is numerically integrated employing the fourth-order Runge–Kutta scheme to determine the frequencies of nonlinear free vibration. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies of porous FGM cylindrical panels. The study reveals that natural frequencies are strongly decreased and that frequency nonlinearity is more significant due to axial compressive loads.

Free vibration and transient response of initially compressed CNT-reinforced composite plates

This paper investigates the linear free vibration and nonlinear transient response of carbon nanotube (CNT)-reinforced composite plates subjected to pre-existent compressive load in thermal environments. CNTs are reinforced into matrix according to uniform and functionally graded distributions. The properties of constitutive materials are assumed to be temperature-dependent while the effective properties of nanocomposite are determined using an extended rule of mixture. Governing equations are established within the framework of classical plate theory incorporating von Kármán nonlinearity and initial geometrical imperfection. Analytical solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to obtain a time differential equation including both quadratic and cubic nonlinear terms. Fourth-order Runge–Kutta numerical integration scheme is employed to determine dynamical deflection-time response of nanocomposite plates. Numerical analyses are presented to consider the influences of CNT volume fraction, CNT distribution, initial compressive load, geometrical imperfection, elevated temperature and plate geometry on the natural frequencies and nonlinear dynamical response of nanocomposite plates. The study reveals that the natural frequency and dynamical deflection are strongly decreased and increased when initial compressive load is increased, respectively. Numerical results also find that the natural frequencies are enhanced and dynamical deflection is dropped as a result of increase in volume percentage of CNTs, respectively.

Determination of flexural stiffness of composite timber beams using DIC and wavelet transform of deflection lines

The article presents an experiment of four-point bending of timber I-beams, the deflections of which were measured using digital image correlation (DIC). The deflection functions were subjected to a wavelet transform operation, which used the second derivative of the Gaussian distribution to determine the curvature of the beams’ axis. On this basis, the distribution of flexural stiffness of the elements was estimated, showing the potential usefulness of the DIC technique in production control and laboratory testing of prefabricated timber elements. The limitations of the proposed research method related to the need to perform numerical integration operations on discrete experimental data subject to measurement errors are also presented

ЖҰҚА ИЗОТРОПТЫ ПЛАСТИНАЛАРДЫҢ ИІЛУІН ТАЛДАУ

In the field of modern construction, thin isotropic plates are widely used. The development of calculation methods and mathematical models for thin isotropic plates of rectangular shape is an important issue. The article examines the bending of a rectangular thin isotropic plate. The method of separation of variables is employed to derive the plate calculation theory. New formulas for the function of deflection and internal forces of the plate are determined. A simplified version of the classical theory is presented, defining the solution to the plate bending problem using simple polynomials. The values of the deflection function and internal forces of a rectangular thin isotropic plate are presented in the form of tables and diagrams. The obtained calculation results are compared with Navier's method and the finite element method. The results obtained completely coincide with the results obtained by other methods and thereby confirm the correctness of the proposed version of the theory.

Solution of nonlinear vibration problem of shear deformable multilayer nonhomogeneous orthotropic plates using Poincare-Lindstedt method

In this study, one of the first attempts is made to solve the nonlinear (NL) vibration problem of shear-deformable multilayer plates consisting of nonhomogeneous orthotropic layers (NHOLs) using the Poincaré-Lindstedt method. First, the shear deformation theory (SDT) for homogeneous plates is extended to multilayer plates composed of NHOLs. In the framework of von-Karman type nonlinear theory, the basic relations of the plates in question are established and then NL equations of motion based on four functions such as rotation angles, Airy stress and deflection functions are derived. Then, NL-partial differential equations (NL-PDEs) are reduced to NL-ordinary differential equations (NL-ODE) and the solution of NL-ODE is performed for the first time by the modified Poincaré −Lindstedt method, yielding new amplitude dependent expressions for NL frequency, and for the ratio of NL frequency to linear (NL/L) frequency for multilayer plates consisting of NHOLs. Finally, detailed parametric studies are carried out to gain insight into the effects of various factors such as shear strains, non-homogeneity, number and array of layers on the NL frequencies under different rectangular plate characteristics.

Glory interference spectroscopy in Sr atom

Slow (meV) photoelectron imaging spectroscopy is employed in the experimental study of near-threshold photoionization of strontium atoms in the presence of an external static electric field. Specifically, the study is devoted to the glory effect, that is, the appearance of an intense peak at the center of the recorded photoelectron images, when dealing with m= 0 final ionized Stark states (m denoting the magnetic quantum number). This critical effect is formally identical to that encountered in classical scattering theory, where, for a nonzero value of the impact parameter, the zero-crossing of the deflection function leads to a divergent classical differential cross section. By recording the magnitude variation of this glory peak as a function of electron excitation energy, we observe that, besides the traces of classical origin, it also exhibits intense quantum interference and beating phenomena, above and below the zero-static-field ionization threshold. We study both, single- and two-photon ionization of Sr, thus enabling a comparison not only between the different excitation schemes, but also with an earlier work devoted to two-photon ionization of Mg atom by Kalaitzis et al (2020 Phys. Rev. A 102 033101). Our recordings are analyzed within the framework of the Harmin–Fano frame transformation Stark effect theory that is applied to both the hydrogen atom and a non-hydrogenic one simulating Sr. We discuss the various aspects of the recorded and calculated glory interference and beating structures and their ‘short time Fourier transforms’ and classify them as either atom-specific or atom independent. In particular, we verify the ‘universal’ connection between the glory oscillations above the zero-field threshold and the differences between the origin-to-detector times of flight corresponding to pairs of classical electron trajectories that end up to the image center.

Polarization and frequency multiplexed multi-channel focusing terahertz vortex with a birefringent dielectric metasurface

We propose an approach to generate polarization and frequency multiplexed focusing vortex beams using a birefringent dielectric metasurface in terahertz (THz) range. The designed metadevice consists of Si nanopillars with different sizes. The transmission efficiency of each unit cell under orthogonal polarized illumination can exceed 70% at both operating frequencies 1 THz and 1.2 THz. Combining the functions of beam deflection, focusing, and vortex beam generation in the interference holography design strategy, polarization and frequency multiplexed multi-channel focusing THz vortex beams with different topological charges and generation positions can be achieved, which can greatly improve the transmission capacity of THz communications. Combining its compact and efficient features with multiplexing methods, our designed metadevice has enormous potential for application in THz vortex generation and information processing.

Some Comments Concerning the Preparation of and Fatigue Testing of the Aircraft’s Cable-Control System

Abstract The currently accepted rules that are applied to the aircraft cable-control systems’ operational use are based on the reactive maintenance idea and the comparative tests, inspections, and diagnostics performed at the mandatory intervals. Fatigue tests of the aviation cables are commonly conducted by bending in the range of ± 90° with constant load. Aircraft cable-control systems are subject to a number of random loads and deformations. Additionally, forces and their values are modified by the wear and tear of cable-pulley raceways, elastic deformations, and changes caused by temperature. The actual values of tension of aviation cable-control systems are relatively low, and bending usually does not exceed the maximum of ± 35°. Moreover, the forces characteristic of the control cables are nonlinear functions of the control surface deflection. This means that the typical fatigue tests we employ help with only comparative estimations and acceptance tests. It is not possible to estimate the operational durability of the systems and forecast inspections and diagnoses intervals based on the mentioned results. The present article utilizes the operational profiles of selected aircraft categories to determine the stochastic load-related deflection spectra for the preparation of cable fatigue-testing programs. Operation profiles are built considering a group of aircraft belonging to the same category, performing similar missions, for example, training missions, photogrammetric missions, aircraft towing, e.q., and having a similar share in the total resource. The special stands for the selected cable fatigue tests have been proposed. The cable test stand ensures the real stochastic loads for the cable use and other actual conditions of load. The proposed stand enables the simultaneous testing of more than one cable at different deformation parameters, for example, wrap angles. The results of the proposed method and tests can be used to estimate the operational durability of aviation-control systems as well as for inspection and diagnosis intervals as well.

Собственные колебания упругой прямоугольной CFCF-пластинки при одноосном растяжении-сжатии ее плоскости

The influence of a uniaxial distributed load in the plane of a thin rectangular plate on the value of the first natural frequency of its oscillations has been studied. We considered a plate, two opposite faces of which are clamped, and the other two are free (CFCF-plate, C-clamped edge, F-free edge). The load is applied to the clamped edges. The main differential equation for the coordinate component of the deflection function and all the boundary conditions of the problem are fulfilled exactly using two hyperbolic-trigonometric series in two coordinates and an additional function depending on one variable x. The problem was reduced to an infinite system of linear algebraic equations with respect to one sequence of uncertain coefficients, containing as parameters the magnitude of the load and the frequency of oscillations. For a number of load values, the natural frequencies of oscillations were found by enumerating frequency values in combination with the method of successive approximations when solving a reduced system of linear algebraic equations. To ensure acceptable accuracy of calculations, the number of terms in the series (the size of the reduced system), the number of iterations and the number of significant digits in the mantissa when calculating non-trivial coefficients of the system were changed. A square plate was considered as an example. Based on the calculation results, a graph was constructed of the dependence of the first natural frequency of oscillations on the magnitude of tension-compression forces, which is a curve close to a parabola. Under Eulerian compressive load, the oscillations stop. The shapes of natural vibrations changed slightly and were similar to the shape of the curved surface of a plate under the action of a uniform transverse load. The purpose of this study is to create an effective algorithm for calculating the natural frequencies of oscillations of a CFCF-plate when the uniaxial tensile-compression load of its clamped faces changes.

The Reissner–Ritz Method for Solving the Deflection Function of the Crown Pillar in the Stope and Its Application in the Crown Pillar Failure Analysis

The Reissner–Ritz Method for Solving the Deflection Function of the Crown Pillar in the Stope and Its Application in the Crown Pillar Failure Analysis

Effects of Pipe Deflection and Arching on Stress Distribution and Lateral Earth Pressure Coefficient in Buried Flexible Pipes

In this study, full-scale laboratory tests were conducted on a 315 mm diameter HDPE pipe under shallow buried and localised surface loading conditions to investigate the effects of pipe deflection and arching on stress distribution and the lateral earth pressure coefficient. The tests were validated using 2D finite element software, and further analyses were carried out through parametric studies. These studies considered variations in pipe stiffness, burial depth, backfill properties and pavement stiffness to increase the reliability of the test results. For a shallowly buried HDPE pipe, a comprehensive explanation is provided regarding the evolution of the lateral earth pressure coefficient within the central soil prism. Initially set at Ko conditions, this coefficient tends to shift towards Kp with increasing arching and transitions to Ka with weakening arching. The findings suggest that stress predictions in the crown region of shallow buried flexible pipes are achievable through the application of Terzaghi’s arching theory, contingent upon an accurate estimation of the lateral earth pressure coefficient for the central soil prism. Furthermore, the horizontal deflection of the pipe at the springline results in compressive behaviour and passive effects in the surrounding backfill in this specific region. This situation demonstrates that the horizontal stresses at the springline and the lateral earth pressure coefficient can be reliably estimated by considering them as functions of the horizontal deflection of the pipe.

Quantum State-Resolved Nonadiabatic Dynamics of the H+NaF → Na+HF Reaction

The H + NaF reaction is investigated at the quantum state-resolved level using the time-dependent wave-packet method based on a set of accurate diabatic potential energy surfaces. Oscillatory structures in the total reaction probability indicate the presence of the short-lived intermediate complex, attributed to a shallow potential well and exothermicity. Ro-vibrational state-resolved integral cross sections reveal the inverted population distributions of the product. The HF product favors an angular distribution in the forward hemisphere of 30°–60° within the collision energy range from the threshold to 0.50 eV, which is related to the nonlinear approach of the H atom to the NaF molecule. Quantum generalized deflection functions show that the low-J partial waves contribute primarily to the backward scattering, while the high-J partial waves govern the forward scattering. The correlation between the partial wave J and the scattering angle ϑ proves that the reaction follows a predominant direct reaction mechanism.

Composite phase modulated beam steering controllable reflective metasurface

Terahertz metasurface functional devices as an effective method to control terahertz waves have attracted extensive attention from researchers. In order to enhance the functionality and flexibility of the metasurface and adapt to diverse application scenarios and demands, a beam-steering controllable reflective metasurface is designed by combining the Pancharatnam-Berry phase principle and the phase change material vanadium dioxide in this work. The metasurface unit consists of five layers, they being the top layer that is a metal patterned layer, the third layer that is made of vanadium dioxide and located between the dielectric layers with different thickness, the dielectric layer that is made of polytetrafluoroethylene (PTFE), and the bottom layer that serves as a metal reflective layer. The metasurface units are rotated based on the Pancharatnam-Berry phase principle to obtain four metasurface units with fixed phase differences in between, after which the metasurface units are arranged in two dimensions based on the generalized Snell reflection law to obtain the desired phase-gradient deflected reflection beam. The insulating state-metallic state transition of the vanadium dioxide layer on the metasurface can change the phase gradient of the preset metasurface, thereby realizing the on/off function of deflection. The simulation results show that when the vanadium dioxide is in the insulating state, the phase gradient of the designed metasurface appears, and the metasurface can deflect the vertically incident circularly polarized wave with specific angle anomalies in a operating band of 1.1–2.0 THz; when the vanadium dioxide is in the metallic state, for the same operating band of the same metasurface, the phase gradient of the metasurface disappears, and the metasurface mirror reflects the vertically incident circularly polarized waves, thereby realizing the function switching. This design provides new possibilities for modulating the terahertz reflected beam, which will have potential applications in terahertz wireless communication and radar systems.

EQUILIBRIUM OF A NORMALLY LOADED ELASTIC SHALLOW CYLINDRICAL SHELL WITH DISLOCATIONS AND DISCLINATIONS

We consider the problem of the equilibrium of a normally loaded elastic shallow rectangular circular cylindrical shell, which contains sources of internal stress in the form of fields of continuously distributed edge dislocations and wedge disclinations or other sources, for example, thermal ones. Based on the modified system of Karman equations for the equilibrium of an elastic plate with dislocations and disclinations, a system of nonlinear equilibrium equations for the shell was constructed. The resulting system of equations differs from the Karman system of equilibrium equations for an elastic shallow cylindrical shell under the action of a normal load by the presence on the right side of the continuity equation of a nonzero function, called the incompatibility function, which is expressed through the densities of edge dislocations and wedge disclinations. The edges of the shell are freely clamped or hinged. In the absence of internal stress sources, this system transforms into a system of equilibrium equations for a shallow shell, and in the case of an infinitely large radius of curvature (and the presence of internal stress sources) into a modified system of Karman equilibrium equations for a flexible elastic plate with dislocations and disclinations. To solve the system of nonlinear shell equilibrium equations, the Newton – Kantorovich method in combination with the difference method is proposed. For the case of small values of the normal load and small values of the incompatibility function, linearization is carried out with respect to the deflection and stress function. As a result, a linear boundary value problem was obtained in the form of a system of differential equations for the deflection and the stress function, which is solved by the difference method. The solution of the linearized system is used as an initial approximation in the implementation of the Newton – Kantorovich method. Examples of numerical calculations of deflection and stress function for given values of normal load and incompatibility function are given, and corresponding graphs are constructed.

RITZ METHOD IN THE DISCRETE APPROXIMATION OF DISPLACEMENTS FOR SLAB CALCULATION

Introduction: The FEM reduces the problem of structural analysis for various building structures to the formation and solution of a system of linear algebraic equations. For this purpose, there are techniques available for obtaining FE stiffness and flexibility matrices where the main structural deformation characteristics are taken into account. However, the FEM can also be considered as a special case of the Ritz method in the discrete approximation of the required functions. In the functional of full potential deformation energy with regard to the considered structure, all adopted stress-strain state characteristics are taken into account. Since it is difficult or impossible to find continuous approximation functions both in the classic version of the Ritz method and in the Bubnov–Galerkin method for some types of edge restraint in such building structures as beams, slabs, or shells, it is possible to use the Ritz method in the discrete approximation of the required functions (by analogy with the FEM). This paper presents a method of such calculations using slab calculations as an example. It is shown that, due to introducing some notations (operators), the process of finding the coefficients of the system of linear algebraic equations creates no difficulties and is easily programmable. The proposed method is not an alternative to the FEM, which is the most effective numerical method for the calculation of complex three-dimensional building structures. Purpose of the study: We aimed to create a method for calculating slabs by the Ritz method in the discrete approximation of the deflection function for edge restraint cases when it is difficult or impossible to find continuous approximation functions in the classic version of the Ritz method and the Bubnov–Galerkin method. Methods: Based on the application of the Ritz variational method in the discrete approximation of displacements for slab calculation, all the basic relations for rectangular finite elements with 12 degrees of freedom are obtained, and an algorithm for forming the coefficients of the system of linear algebraic equations is developed. Results: For the first time, the solution by the Ritz method in the discrete approximation of slab displacements is obtained for the case when two edges of the slab are rigidly restrained and other two edges are free. In this case, the correct solution of the above problem is possible only with the use of the proposed method and FEM. For the test problem, we performed a comparison of the results of the calculation using the proposed method with the results using the classic Ritz method, which showed their very close agreement. The accuracy of the obtained results was assessed.

Investigation of the vibration problem of sandwich heterogeneous orthotropic plates

In this study, the solution of the free vibration problem of sandwich plates consisting of heterogeneous orthotropic layers is presented. First, the Kirchhoff-Love theory (KLT) for homogeneous plates is extended to sandwich plates composed of heterogeneous orthotropic layers. Within the framework of Donnell theory, after establishing the basic relations of multilayer plates, governing equations are derived based on Airy stress and deflection functions. The solution of the governing equations is carried out by the Galerkin method and the analytical expression is found for the linear frequency of sandwich plates consisting of heterogeneous orthotropic layers. Finally, the effects of various factors such as heterogeneity, number of layers and their arrangement on the free vibration frequency of sandwich plates are examined.

Variational fractional-order modeling of viscoelastic axially moving plates and vibration simulation

Based on the thin plate theory and D’Alembert’s principle to establish the equilibrium equation for viscoelastic axially moving plates, this paper establishes the ternary governing equation of variable fractional order for viscoelastic axially moving plates by using a variable fractional order constitutive relationship for viscoelastic materials. Shifted Chebyshev wavelets are introduced for approximating the deflection function, and numerical solutions for the governing equations are given. The effectiveness and accuracy of the algorithm in this paper is illustrated by convergence analysis, error correction and numerical examples. Finally, the algorithm is used to simulate the vibration of axially moving plates with different moving speeds and different boundary conditions. Meanwhile, Gaussian white noise was introduced to investigate the vibration of the viscoelastic axially moving plate under pure noise environment, simple harmonic load-same direction noise environment and simple harmonic load-opposite direction noise environment, respectively. And the vibration comparison of PP material plate and LLDPE material plate is also carried out. The above research conclusions are consistent with the existing literature, indicating that algorithm proposed by this paper is applicable to numerical simulation and research on viscoelastic axially moving plates.

Nonlinear thermomechanical vibration of initially stressed functionally graded plates with porosities

AbstractThe large‐amplitude vibration of initially stressed functionally graded material (FGM) rectangular plates with porosities is investigated in this paper. All edges of the plate are simply supported and uncompressed edges are tangentially restrained. Initial stresses are caused by in‐plane compressive load acting on freely movable edges or thermally induced forces at restrained edges. Pores present in FGM through even and uneven distributions. Material properties are assumed to be temperature‐dependent and, due to the presence of the pores, the effective properties of FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on the classical plate theory taking into account von Kármán nonlinearity and initial geometric imperfection. These equations are solved using approximate analytical solutions along with the Galerkin method to obtain a time differential equation containing quadratic and cubic nonlinear terms. Nonlinear frequencies are determined by means of numerical integration employing the fourth‐order Runge–Kutta scheme. Parametric studies are carried out to analyze the effects of material and geometry properties, porosity volume fraction and distribution, pre‐existent compressive and thermal loads, imperfection and degree of tangential constraints of unloaded edges on the natural frequencies and nonlinear to linear frequency ratio of FGM plates.