Bending Of Thin Plates Research Articles

Finite element analysis of plates by an accurate strain-based element

Plates analysis has been a very important research area for a long time. In this perspective, in this paper, finite element analysis of thin plates in bending is carried out using a new four node strain-based plate element. The strain-based approach and Kirchhoff theory for thin plates have been used to create the strain field of this element. This approach consists in having a displacement field from a proposed strain field. This developed model contains essential degrees of freedom at each corner node which are three, one displacement and two rotations. The displacement field is based on the assumed functions for various deformations which satisfy the compatibility equations, and the displacement functions are obtained by integrating these deformation functions. The numerical validation of this developed element was held through several examples from the literature. The obtained results from these numerical tests prove the good accuracy and performance of this element.

Chien-physics-informed neural networks for solving singularly perturbed boundary-layer problems

A physics-informed neural network (PINN) is a powerful tool for solving differential equations in solid and fluid mechanics. However, it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives. In this paper, we introduce Chien’s composite expansion method into PINNs, and propose a novel architecture for the PINNs, namely, the Chien-PINN (C-PINN) method. This novel PINN method is validated by singularly perturbed differential equations, and successfully solves the well-known thin plate bending problems. In particular, no cumbersome matching conditions are needed for the C-PINN method, compared with the previous studies based on matched asymptotic expansions.

Sensitivity Analysis of the Factors Affecting the Ground Heave Caused by Jet Grouting

Jet grouted piles are widely used to reduce post-construction settlement of soft clay roadbeds. Nevertheless, it is easy to cause ground heave due to the jet grouted pile. According to the analytical method and numerical method, a sensitivity analysis of the factors affecting ground heave caused by a single jet grouted pile was performed. It is found that the influence of each parameter on ground heave is in the following order: grout pump pressure > embankment load > soil type (including the cohesion, friction angle, and Young’s modulus) > pile diameter > pile length. Considering the effect of the pump pressure on the ground heave is more significant, based on the analytical method of ground heave caused by a single jet grouted pile combined with the solution of small-deflection bending of a circular thin plate, the calculation method for the suggested limit grout pressure for construction under different embankment heights was established. Suggested values of theoretical grout pump pressure were given to prevent ground heave from harming the pavement of operating highways. This study provides some theoretical basis for the subsequent research on the jet grouted pile.

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.

Ritz Variational Method for the Analysis of Thin Rectangular Plate Bending Problems with Adjacent Edges Clamped and Simply Supported Using the Super position of Trigonometric Series and Polynomial Basis Functions

Ritz Variational Method for the Analysis of Thin Rectangular Plate Bending Problems with Adjacent Edges Clamped and Simply Supported Using the Super position of Trigonometric Series and Polynomial Basis Functions

Comparative Analysis and Unified Derivation of Reissner’s Equations For 2D Bending of Thick Plates and Timoshenko’s Equations for Bending of Beams

Currently, calculations of flexural deformations of lithospheric plates are carried out on the basis of Kirchhoff’s theory of bending of thin plates formulated about 170 years ago. The paper examines the possibility of refining these calculations based on the theory of bending of thick plates by Timoshenko and Reissner. A new unified derivation is presented of the Timoshenko equations for 2D bending of beams and the Reissner equations for bending of slabs by direct transformation of the general elasticity equations with a simple approximate replacement of power cubic functions with effective linear ones. This derivation offers a simpler and more detailed understanding of the difference between the equations and the meaning of the simplifications made in these theories. By comparing the analytical solutions of the Timoshenko and Reissner equations with the existing test analytical solutions of the exact elasticity equations, quantitative estimates of the accuracy of these theories are presented.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

Stress-Relief Engineering in a N-Doped C-Modified Hierarchical Nanoporous Si Anode with a Microcurved Pore Wall Structure for Enhanced Lithium Storage.

The commercialization of alloy-type anodes has been hindered by rapid capacity degradation due to volume fluctuations. To address this issue, stress-relief engineering is proposed for Si anodes that combines hierarchical nanoporous structures and modified layers, inspired by the phenomenon in which structures with continuous changes in curvature can reduce stress concentration. The N-doped C-modified hierarchical nanoporous Si anode with a microcurved pore wall (N-C@m-HNP Si) is prepared from inexpensive Mg-55Si alloys using a simple chemical etching and heat treatment process. When used as the anode for lithium-ion batteries, the N-C@m-HNP Si anode exhibits initial charge/discharge specific capacities of 1092.93 and 2636.32 mAh g-1 at 0.1 C (1 C = 3579 mA g-1), respectively, and a stable reversible specific capacity of 1071.84 mAh g-1 after 200 cycles. The synergy of the hierarchical porous structure with a microcurved pore wall and the N-doped C-modified layer effectively improves the electrochemical performance of N-C@m-HNP Si, and the effectiveness of stress-relief engineering is quantitatively analyzed through the theory of elastic bending of thin plates. Moreover, the formation process of Li15Si4 crystals, which causes substantial mechanical stress, is investigated using first-principles molecular dynamic simulations to reveal their tendency to occur at different scales. The results demonstrate that the hierarchical nanoporous structure helps to inhibit the transformation of amorphous LixSi into metastable Li15Si4 crystals during lithiation.

Boundary Value Problems in a Theory of Bending of Thin Micropolar Plates with Surface Elasticity

Boundary Value Problems in a Theory of Bending of Thin Micropolar Plates with Surface Elasticity

Multi-Scale-Matching neural networks for thin plate bending problem

Physics-informed neural networks are a useful machine learning method for solving differential equations, but encounter challenges in effectively learning thin boundary layers within singular perturbation problems. To resolve this issue, multi-scale-matching neural networks are proposed to solve the singular perturbation problems. Inspired by matched asymptotic expansions, the solution is decomposed into inner solutions for small scales and outer solutions for large scales, corresponding to boundary layers and outer regions, respectively. Moreover, to conform neural networks, we introduce exponential stretched variables in the boundary layers to avoid semi-infinite region problems. Numerical results for the thin plate problem validate the proposed method.

The Effect of TIG Welding Heat Input on the Deformation of a Thin Bending Plate and Its Weld Zone

Heat input is a crucial parameter in the process of welding thin plates. It has a direct impact on the quality of the weld and the degree of deformation caused during welding. This study investigates the impact of heat input on the deformation of a thin bending plate and its weld zone using the thermoelastic–plastic finite element method. The accuracy of the model is ascertained using the non-contact inspection method utilizing digital image correlation technology. The welding deformation patterns of thin bending plates with a radius of 500 mm were analyzed at various welding heat inputs ranging from 173 J/mm to 435 J/mm. The results indicate that the finite element prediction model proposed in this paper is highly accurate. It has been observed that, under this range of thermal input, the thin bent plates undergo saddle deformation. By examining the correlation between heat input and the maximum deformation outside of the plane, it has been determined that a heat input of at least 50 J/mm is required for the thin bending plate to experience out-of-plane deformation. Additionally, as the level of heat input increases, so too does the out-of-plane deformation of the thin bending plate. After the completion of the welding cooling process, the transverse shrinkage at the weld seam of the thin bend plate is twice that of the longitudinal shrinkage. However, the transverse deformation of the bend plate is not significantly different from the longitudinal deformation.

A four-node rectangular plate finite element using Airy functions with transverse shear

PurposeAmong different types of engineering structures, plates play a significant role. Their analysis necessitates numerical modeling with finite elements, such as triangular, quadrangular or sector plate elements, owing to the intricate geometrical shapes and applied loads. The scope of this study is the development of a new rectangular finite element for thin plate bending based on the strain approach using Airy's function. It is called a rectangular plate finite element using Airy function (RPFEUAF) and has four nodes. Each node had three degrees of freedom: one transverse displacement (w) and two normal rotations (x, y).Design/methodology/approachEquilibrium conditions are used to generate the interpolation functions for the fields of strain, displacements and stresses. The evolution of the Airy function solutions yielded the selection of these polynomial bi-harmonic functions. The variational principle and the analytical integration approach are used to evaluate the basic stiffness matrix.FindingsThe numerical findings for thin plates quickly approach the Kirchhoff solution. The results obtained are compared to the analytical solution based on Kirchhoff theory.Originality/valueThe efficiency of the strain based approach using Airy's function is confirmed, and the robustness of the presented element RPFEUAF is demonstrated. Because of this, the current element is more reliable, better suited for computations and especially intriguing for modeling this kind of structure.

Corrections to the Theory of Elastic Bending of Thin Plates for 2D Models in Reissner’s Approximation

Corrections to the Theory of Elastic Bending of Thin Plates for 2D Models in Reissner’s Approximation

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

Corrections to the Theory of Elastic Bending of Thin Plates for 2D Models in the Reissner Approximation

Elastic bending stresses and strains in the lithosphere are typically calculated based on the Kirchhoff–Love (thin) plate theory. The criterion for its applicability is the smallness of plate thickness to length ratio. In oceanic plates, due to the buoyant force of the mantle, the main deformations are not uniformly distributed along the plate but are concentrated in the vicinity of the subduction zone. Therefore, the effective length of the bending part of the plate is a few fractions of the actual length, and the criterion of a thin plate is partially violated. In this paper, we analyze the possibility of applying thick plate bending equations. The existing variational theories of 3D bending of thick plates are much more complicated than the Kirchhoff–Love theory, as they involve solving three differential equations instead of one, and have limited application due to their complexity. Since geophysical applications frequently use 2D models, in this paper we analyze in detail the potential and accuracy of the thick plate bending theory for 2D models. After the conversion to the 2D plane strain and plane stress approximation, the original 3D Reissner thick plate bending equations are written out in the form similar to the Kirchhoff equations with additive corrections and are supplemented with the explicit formulas for longitudinal displacement. The comparison of the analytical solutions of the 2D Reissner equations with the exact solutions shows that the 2D approximation only provides a correction for the plate bending function. However, this correction refines the Kirchhoff–Love theory by almost an order of magnitude. At the same time, the solution of the equations in this case turns out to be almost as simple as the solution of the thin plate equations.

Numerical and experimental modeling of the behavior of flexible shell elements of structures

This article is devoted to computer and experimental modeling of the behavior of flexible elements of shell structures, and the development of effective algorithms for solving emerging nonlinear boundary value problems of their calculation and optimization of parameters. At the same time, the probability of the results obtained using different approaches to the construction of a nonlinear theory is established. Their comparative analysis, error estimation, which in this case is given by calculation according to linear and corresponding nonlinear theories, is carried out. The results of calculated data and experimental studies of the behavior of real structural elements are compared.
 The results of a comparative analysis of the application of two mathematical models of deformation of flexible shell elements, obtained by direct integration of boundary value problems of shell mechanics, by the finite element method and experimental research, are presented.
 The problems of axisymmetric nonlinear bending of thin ring plates, corrugated membranes of a sinusoidal profile and bellows as a shell of rotation with a complex meridian shape are considered.
 Using the necessary optimality conditions of the principle of maximum L. S. Pontryagin obtained the results of the optimal design of a flexible corrugated membrane with a sinusoidal profile of the highest sensitivity. The results are presented in the form of tables, photos and graphs.

Bending of a clamped thin isotropic plate by the Kantorovich method using special polynomials

The problem of bending a thin isotropic rectangular plate clamped on all four sides under the action of a normal load uniformly distributed over its surface is considered. An analytical solution of the boundary value problem for the resolving differential equation with respect to the normal deflection of the plate is obtained by the method of L. V. Kantorovich using special-type polynomials satisfying homogeneous boundary conditions. A feature of these polynomials is the so-called “quasi-orthogonality” property of the first and second derivatives, which leads to the splitting of the system of ordinary differential equations of the L. V. Kantorovich method into separate ordinary differential equations that are easily solved analytically. However, this property of polynomials is only approximately fulfilled. Two solutions are compared: an analytical one under the assumption of “quasi-orthogonality” of the first and second derivatives of polynomials and a numerical-analytical one without this assumption. The stress-strain state in the neighborhoods of corner points has been studied. It is shown that the moments and shear forces tend to zero when approaching the corners of the plate, as well as a double change in the sign of the shear force on the edge of the plate in the neighborhoods of the corner points.

Application of the Ritz method for the bending and stress analysis of thin rectilinear orthotropic composite sector plates

This paper aims to investigate the linear elastic behavior of composite sector plates with rectilinear orthotropy in terms of displacement and stress fields determination. This variety of plates presents layers reinforced with unidirectional fibers organized along straight directions that make a variable angle with the generic radial direction. The fundamental equations for rectilinear orthotropic composite sector plates are obtained according to the Classical Lamination Theory. The mid-surface deflection of the sector plates is derived on the basis of an original analytical approach that combines the principle of virtual works with the Ritz method. Different boundary conditions in terms of displacement constraints are taken into account and, accordingly, the relative discretized forms of the displacement field are written as a linear combination of expressly obtained approximation functions. Commonly employed symmetrical stacking sequences are considered to demonstrate the accuracy and reliability of the proposed method in the final test cases.

Analysis of Size-Dependent Linear Static Bending, Buckling, and Free Vibration Based on a Modified Couple Stress Theory

The purposes of this paper are to study bending, buckling, and vibration by considering micro-scale effects using the Kirchhoff thin-plate theory and to consider small deflections, neglecting higher-order nonlinear terms. The governing equations for the bending, buckling, and vibration of the system are obtained using the equilibrium method coupled with the Kirchhoff thin-plate theory and a modified couple stress theory (MCST). The concept of the equivalent bending stiffness (EBS) of micro-thin plates is proposed to describe the scale effect. The Navier method is used to obtain analytical solutions for the bending, buckling, and free vibration of thin plates under simply supported boundary conditions with scale effects. The numerical results are presented to investigate the influence of scale effects on deflection, critical buckling load, buckling topography, and thin-plate natural frequency. The results show that the scale effect increases the equivalent stiffness of the thin plate, which leads to a decrease in deflection, a larger critical buckling load, and an increase in natural frequency, but does not affect the buckling topography. The MSCT is invalid when the thickness is greater than 10 times the scale effect parameter, thus defining the scope of application of the scale effect. This research study may contribute to the design of micro-scale devices such as MEMSs/NEMSs.

A homotopy-based wavelet method for extreme large bending analysis of heterogeneous anisotropic plate with variable thickness on orthotropic foundation

A novel homotopy-based decoupled wavelet numerical strategy is developed to investigate the large-deflection bending of variable-thickness anisotropic thin plate constituted by heterogeneous material resting on orthotropic foundation. The anisotropic plates are described undergoing sinusoidally initial imperfections with inhomogeneous thickness and exponentially varying elastic modules, while the subgrade reaction involves the nonlinear support effects by Winkler-type foundation along with Pasternak-type orthotropy. Geometric nonlinearity by uneven thickness and large deformation, material nonlinearity by heterogeneity and contact nonlinearity of orthotropic foundation are simultaneously considered with the highly coupled and strongly nonlinear variable-coefficient Von Kármán equations derived. The accuracy and convergence of the present formulation are validated in excellent agreement with published results, while the new results of extreme bending solutions are provided, which are few given by other methods and further verify the effectiveness and superiority of proposed wavelet procedures. Parametric investigations of influences of material orthotropy and inhomogeneity, unevenness of thickness, amplitude of initial imperfection along with orthotropy of nonlinear Winkler–Pasternak foundation have been studied, which are of great significance on the extremely large bending properties.