Plate Problems Research Articles

Transient Thermoelastic Analysis of Rectangular Plates with Time-Dependent Convection and Radiation Boundaries

Approximate analytical solutions are presented for the transient thermoelastic problem of rectangular plates with time-dependent convection and radiation boundaries. To include the nonlinear radiation boundary, the whole heating process is divided into several time steps, and a linearized approximation is used to simplify the radiation term for each step. The one-dimensional transient temperature along the thickness direction is solved using the technique of the separation of variables. The displacement and stress solutions are obtained by applying the state-space method to the three-dimensional thermoelasticity equations. The accuracy of the present solutions is demonstrated by comparison with the reported results in the open literature and the finite element solutions. In the numerical examples, two kinds of thermal boundaries, namely, time-independent convection boundaries and time-dependent convection and radiation boundaries, are considered to show the availability of the present solutions.

Iterative decoupling procedure for the coupled 1D problems on example of closed cylindrical shell at static loading

The general iterative method of formal decoupling of two coupled 1D problems is suggested. Its essence consists in the alternative consideration of one of two problems as the main one, the aim of which is to produce the eigenfunctions. Then the other problem is considered as an auxiliary one, which is aimed to obtain the particular solution. The latter is presented as a linear combination of already known eigenfunctions, and then is substituted again into the first problem to refine the constants of main governing differential equation and to get the corrected eigenfunctions. The convergent iterative procedure of gradual refinement of the above constants is elaborated. The accuracy and technique of application is considered in details for the closed cylindrical shell problem, which is expanded in Fourier series with respect to circumferential coordinate. This actually leads to coupled 1D problems – one is the plane elasticity problem, and other – the plate problem. The results of calculation show the high accuracy of the method, which allows to get any required number of significant digits in eigenvalues. The comparison with known theoretical and FEM results for radial displacement and axial bending moment distribution at the vicinity of point of concentrated radial force application demonstrates the efficiency and accuracy of the method.

An Analytical Solution on Vibration Reduction and Shear Force Mitigation of Cantilever Mindlin Plates Using Orthogonal Ribs

Vibration of aircraft wings and the dynamic stress concentration at the clamped edge are important research topics due to concerns on the safety of aircrafts. To have a better understanding of the problem, the free and forced vibration response of a ribbed rectangular cantilever plate representing a section of an aircraft wing is investigated in this study. A new analytical solution is developed for the vibration analysis of rib stiffened cantilever plates using Mindlin plate and Timoshenko beam theories alongside the finite integral transform technique. The one- and two-dimensional integral transforms are applied to the governing equations of beams and plates, respectively, where the coupling force components at the interface between the base plate and the beam(s) can be automatically defined during the integral transform. Eventually, the partial differential equations are transformed into a system of linear algebraic equations in which its derivation is rigorous and easily implemented. Good agreements are found between the results of analytical solution, finite element analysis (FEA) and related literature. The solution is then employed to study the vibration suppression of cantilever plates and the shear force at the clamped edge. It is found that the insertion of a pair of orthogonal ribs in the plate can effectively reduce its vibration. An optimum orthogonal ribbing pattern is obtained using multi-objective particle swarm optimization (MOPSO) algorithm, taking into consideration both the vibration suppression of the plate and the maximum induced shear force at the corners of the clamped edge.

A Reissner–Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations

In this work we develop new finite element discretisations of the shear-deformable Reissner–Mindlin plate problem based on the Hellinger–Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence M∈HZ⊂Hsym(Div). The latter results in highly accurate approximations of the bending moments M and in the rotation field being in the Lebesgue space ϕ∈[L2]2, such that the Kirchhoff–Love constraint can be satisfied for t→0. In order to preserve optimal convergence rates across all variables for the case t→0, we present an extension of the formulation using Raviart–Thomas elements for the shear stress q∈RT⊂H(div).We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting.This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries.The robustness of the formulations and the construction of the Hu-Zhang element are tested for shear-locking, curved geometries and an L-shaped domain with a singularity in the bending moments M. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods.

A coordinate transformation based barycentric interpolation collocation method and its application in bending, free vibration and buckling analysis of irregular Kirchhoff plates

AbstractIn this article, a meshless barycentric interpolation collocation method (BICM) with coordinate transformation is proposed to solve the governing partial differential equations (PDEs) for bending, free vibration and buckling problems of irregular plates. First, the coordinate transformation equations for first to fourth derivatives are introduced. Then the PDEs and boundary conditions defined in a physical domain are transformed into a rectangular domain (reference domain) with the derived differential interpolation matrices. The presented BICM can solve the PDEs based on the reference domain and is adaptive to irregular problem domains. Finally, the bending, free vibration and buckling problems of Kirchhoff plates with irregular shapes are studied with the proposed method. To illustrate the effectiveness of the proposed method, the results are compared with those given by theoretical solutions, finite element methods and literature.

Two-dimensional orthotropic plate problems in a thermal environment: Refined crack modelling

Three two-dimensional (2D) refined models are devised for the analysis of orthotropic thin plates with part-through surface and internal cracks under in-plane forces due to temperature differences. The refined models are used to formulate the governing equations for cracked orthotropic thin plates in a thermal environment. Subsequently, the refined models and their ensuing modified forms can be directly applied to all-over and finite-length cracks, respectively. Using an interface-fitted numerical scheme (i.e., matched interface and boundary technique), the resulting equations of motion for the buckling and free vibration of cracked orthotropic thin plates are solved. The effects of various crack types and dimensions on the critical buckling temperatures and vibration frequencies of orthotropic thin plates are investigated. The present models are also validated by comparing their numerical results with the line-spring model and three-dimensional finite-element solutions. Illustrative examples, including boron fiber/epoxy and glass-fibre-reinforced composite materials, are considered to show the superiority of the proposed models over the conventional one, especially in terms of understanding the effect of in-plane forces arising from temperature differences at the cracked location.

The DISCS method for perforation simulation of thin metallic plates by a rigid projectile

The DISCS method had been developed for penetration analysis of an axisymmetric projectile with a slender nose into a semi-infinite medium. Recently, the authors developed an approximate approach to assess perforation of concrete slabs of finite thickness, using the DISCS method. The present paper extends the approximate model to the perforation analysis of ductile metallic thin plates and determines the key parameters of the projectile residual velocity, the target perforation limit, and the ballistic limit. The present investigation focuses on thin metallic plates, the thickness of which is considerably smaller than that of the concrete slabs and smaller in comparison to the projectile nose length. The method is rather simple, easy to implement and provides a fast solution. An idealized target is characterized by a set of discs of small thickness responding in the plate plane, disregarding the local damage around the penetration ductile hole. The in-plane response of the discs during penetration and perforation of an idealized target are calculated. The Residual Velocity upon perforation of the Idealized Target (RVIT) is lower than the residual velocity of the real plate since it disregards the local damage, hence, the RVIT is the lower bound of the true residual velocity of the real perforated thin plate. Therefore, a positive RVIT for the thin plate with a given thickness indicates right away that the real target will be perforated. Using the RVIT parameter, the proposed DISCS method turns into a series of RVIT-perforation problems of plates with thicknesses that are smaller than the real thickness, to determine the decreasing RVIT with increasing thickness, until the full thickness is considered. The analysis is carried out using the modified (hybrid) DISCS method, that aims at analyzing penetration depth into a thick target. The analysis is based on the theoretical field equations and the constitutive equations of the plate material. The proposed method is based on a theoretical model and on engineering insight. Neither empirical constants nor any calibrations are required. Validation of the proposed method is carried out thorough comparisons with numerous test data on ductile aluminum and steel plates of different thicknesses that are struck by rigid projectiles at different impact velocities. All the comparisons demonstrate very good correspondence between the proposed method predictions and the test data.

Homogenisation of Nonlinear Heterogeneous Thin Plate When the Plate Thickness and In-Plane Heterogeneities are of the Same Order of Magnitude

Summary In this work, we propose a new two-scale finite-strain thin plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. For this type of theory, two scales exist, the macroscopic one is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. We consider the case when the plate thickness is comparable to in-plane heterogeneities. We assume that the nonlinear macroscopic part of the model is of Kirchhoff–Love type. We obtain the nonlinear homogenised model by performing simultaneously both the homogenisation and the reduction of the initial three-dimensional plate problem to a two-dimensional one. Since nonlinear equations are difficult to solve, we linearise this homogenised Kirchhoff–Love plate theory. Finally, we discuss the treatment of edge effects in the vicinity of the lateral boundary of the plate.

Research and development of the control system of space multi-dimensional profiling and planing based on PLC

Aiming at the complex processing problem of a rectangular composite plate with warpage in space to remove both sides of the frame and ensure the maximum of the inner core plate of the composite plate after removing both sides of the frame, a space multi-dimensional profiling and planing control system based on PLC is developed. According to the process requirements, the overall design of the space profiling and planing system is carried out, focusing on solving the problem of planing the space warping of the composite board frame. In the aspect of system hardware, the scheme selection, mechanical structure design, pneumatic control scheme design and electrical control scheme design are carried out; In the aspect of system software, based on the TIAPortal software platform, the PLC profiling program is compiled, and the human-computer interaction of the touch screen is designed. The profiling and planing process of composite plate is realized on the prototype, which improves the production efficiency while realizing automation.

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.

APPLICATION OF DISCONTINUOUS SOLUTIONS IN BOUNDARY ELEMENT METHOD FOR THE BENDING PROBLEMS OF KIRCHHOFF PLATES WITH AN ARBITRARY CONTOUR

The discontinuous solutions represent a new direction for the indirect Boundary Element Method (BEM) in the indirect formulation. Discontinuous solutions represent those functions, which when crossing certain lines, the transverse deflection, the slope angle, the bending moment and the generalized shear force may have jumps. These can be used to solve important problems for which the existing methods do not have solutions or a satisfactory accuracy, such as: plates of arbitrary contour, presence of defects, mixed boundary conditions, contact problems, infinite domains etc. In this paper, we describe the methodology of application and the numerical implementation of discontinuous solutions for the bending problems of plates with an arbitrary contour in the classical theory (Kirchhoff). For this purpose, programming codes were developed in the Matlab language, which allowed to calculate the displacements and efforts in the plate. The obtained results were compared with the Finite Element Method (FEM) for different mesh densities.

Bending of Plates with Complex Shape Made from Materials that Differently Resist to Tension and Compression

A new numerical-analytical method for solving physically nonlinear bending problems of thin plates with complex shape made from materials that differently resist to tension and compression is developed. The uninterrupted parameter continuation method is used to formulate and linearize the problem of physically nonlinear bending. For the linearized problem, a functional in the Lagrange form, given on the kinematically possible displacement rates, is constructed. The main unknown problems (displacements, strains, stresses) were found from the solution of the initial problem, which was solved by the Runge-Kutta-Merson method with automatic step selection, by the parameter related to the load. The initial conditions are found from the solution of the problem of linear elastic deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta-Merson scheme are found from the solution of the variational problem for the functional in the Lagrange form. Variational problems are solved using the Ritz method in combination with the R-function method, which allows to submit an approximate solution in the form of a formula – a solution structure that exactly satisfies the boundary conditions and is invariant with respect to the shape of the domain where the approximate solution is sought. The test problem for the nonlinear elastic bending of a square hinged plate is solved. Satisfactory agreement with the three-dimensional solution is obtained. The bending problem of the plate of complex shape with combined fixation conditions is solved. The influence of the geometric shape and fixation conditions on the stress-strain state is studied. It is shown that failure to take into account the different behavior of the material under tensile and compression can lead to significant errors in the calculations of the stress-strain state parameters.

Analytical Study of Stress Distributions around Screws in Flat Mandibular Bone under In-Plane Loading.

A known complication for mechanically loaded bone implants is the instability due to screw loosening, resulting in infection and the non-union of fractures. To investigate and eventually prevent such bone degradation, it is useful to know the stress state in the bone around the screw. Considering only in-plane loadings and simplifying the mandibular bone into an orthotropic laminated plate, the analysis was reduced to a two-dimensional pin-loaded plate problem. An analytic model, based on the complex stress analysis, was introduced to the bone biomechanics field to obtain the stress distributions around the screw hole in the bone. The dimensionless normalized stresses were found to be relatively insensitive to the locations of the screw hole over the mandible. Parametric analyses were carried out regarding the friction coefficient and load direction. It was found that the load direction had a negligible influence. On the contrary, the friction coefficient had a significant effect on the stress distributions. Whether the screw was well bonded or not thus played an important role. The proposed analytic model could potentially be used to study bone failure together with stress-based failure criteria.

Instability and its prediction of an inverted inhomogeneous plate loaded by a subsonic airflow: Theory and experiment

This paper presents a study on the instability of an inverted cantilevered plate with an inhomogeneous cross-section in axial flow from both theory and experiment. The plate is free at the leading edge and clamped at the trailing one. It has a cross-section with discontinuity, bringing difficulty in solving the fluid–structure interaction equation by the traditional methods. We study this problem in state space and transfer its solution into a function approximation problem. We bring no approximation at the first equation level, and the derived instability equation is on the continuum. A numerical scheme is proposed to solve the instability equation. Series expansion and least square method apply to the numerical solutions. The convergence and reliability of the present method are entirely tested. This present modeling method preserves the original information of the problem and facilitates the exploration of the effects of inhomogeneity. Based on the present continuum instability equation, prediction formulas for critical dynamic pressure and instability slope of an inhomogeneous plate are first given and verified. They can apply to the stability prediction of arbitrary cross-sectional plates and have application potential for the plate design. In addition, we report an optimized stiffness distribution of the plate for the maximum critical dynamic pressure. Finally, a wind tunnel test is designed and implemented. A comparative study shows that the present method and prediction formula agree well with the experiment. The present modeling method for a fluid–structure interaction problem in a continuous sense, avoiding approximation at the beginning level, can serve as another new way to address the static instability problem of plates.

Изгиб секторальной пластины: использование систем компьютерной алгебры

Purpose: To investigate the stress-strain state of a thin homogeneous isotropic plate in the form of a sector by numerical-analytical method. To consider the possibility of using computer algebra systems (CAS) to calculate sectoral plates operating under bending due to a transverse load. To demonstrate the effectiveness of applying one of these CASs through the example of the Maple system for calculations using the Ritz method — performing analytical transformations when calculating the integral that determines the total potential energy functional, forming and solving the main resolving system of linear algebraic equations with respect to unknown numerical coefficients in the formula approximating the deflection of the plate, visualization of the obtained solution. Methods: A direct method is used to solve the variational problem of minimizing the functional of the total potential energy of deformation of a thin homogeneous isotropic plate in the form of a sector — the Ritz method. The solution is constructed in the form of a series in terms of basis functions. As basis functions, polynomial functions are chosen that exactly satisfy all boundary conditions. Results: An approximate numerical-analytical solution has been obtained for the problem of bending a sectoral plate in the form of a quarter of a circle, clamped along the contour and loaded with a uniformly distributed load. The effectiveness of using the Maple analytical computing system for solving the problem of bending a sectoral plate by the variational Ritz method is demonstrated. It is shown that the resulting solution quickly converges both for deflection and for bending moments and stresses. Practical significance: The algorithm proposed in the article for solving the problems of bending sectoral plates can be recommended for practical use in determining the stress-strain state of such plates.

Study on the simplified evaluation method of the remaining load-carrying capacity of a corroded steel I-girder end using FEA

In a steel I-bridge, corrosion often occurs severely at the I-girder end rather than at other parts. Therefore, this work mainly discusses the corrosion at the I-girder end as a bearing problem and shear capacity of the end plate. A finite element model (FEM) of a corroded steel I-girder end is developed considering commonly occurring corrosion at first. For the initial imperfection of a steel I-girder end, the initial deformation and residual stress are considered in the model which are simulated by elastic buckling analysis (eigenvalue analysis) and thermal stress, respectively. To investigate the effect of the corrosion pattern on the strength development, the single-area corrosion (flange, stiffener, web) and multi-area corrosion are included in the case study. Finally, the simulation results from the case study using finite element analysis (FEA) are used to formulate the simplified equations for evaluating the remaining load-carrying capacity of a corroded steel I-girder end with the specified corrosion patterns. Furthermore, for the engineering design and preliminary assessment of a corroded steel I-girder end, the simplified equations for evaluating the remaining yield strength ratio are also provided based on the remaining load-carrying capacity.

Unified plate finite elements for the large strain analysis of hyperelastic material structures

This paper proposes a high-order two-dimensional (2D) finite element model for the analysis of isotropic, nearly incompressible hyperelastic material structures based on a decoupled Neo–Hookean strain energy function. The model is based on the Carrera Unified Formulation (CUF) , which allows to automatically implement different kinematics by using an opportune recursive notation. The principle of virtual work and a finite element approximation are exploited to obtain the nonlinear governing equations. Considering the three-dimensional full Green–Lagrange strain components and given the material Jacobian tensor, the explicit forms of tangent stiffness matrices of unified plate elements are presented in terms of the fundamental nuclei, which are independent of the theory approximation order. Several problems of soft material plates under uniform pressure are investigated, including a silicone rubber clamped plate and a simply supported plate made of biological material. The proposed model is compared with literature results including those coming from experiments and numerical solutions. The numerical investigation demonstrated the validity and accuracy of the proposed methodology for the analysis of hyperelastic plates.

Lower bounds for eigenvalues of Laplacian operator and the clamped plate problem

Lower bounds for eigenvalues of Laplacian operator and the clamped plate problem

Nonlinear harmonic vibrations of laminate plates with viscoelastic layers using refined zig-zag theory. Part 1 – Theoretical background

The paper is devoted to the theoretical formulation of the harmonic vibrations problem for laminate plates in the von Kàrmàn geometrically non-linear regime. The plate layers are modelled as viscoelastic using the fractional Zener material with the linear elastic material being a special case. The model of the material is formulated with the separation of the volumetric and deviatoric strain. The laminate plate kinematics is described using the refined zig-zag theory. The vibrations of the plate are formulated using the alternative exponential form with complex-conjugate amplitudes. The harmonic balance method and the time-averaging are applied to derive the amplitude equation of the problem in hand. The efficiency and correctness of the formulation is verified in the continuation paper Part 2 – numerical analysis.

A non-Fourier magneto-thermo-viscoelastic problem of a two-dimensional plate

Viscoelastic materials is a kind of damping materials which has been wildly applied in vibration control of ocean and civil engineering, and viscoelastic materials mostly work in a complex thermoelectric multi-field coupled environment. In this work, a two-dimensional viscoelastic plate is studied with stain relaxation time and viscoelastic relaxation time. The two-dimensional viscoelastic plate is placed in a magnetic field, and the upper surface is subjected to a thermal shock. The problem is investigated by a modified L-S thermo-viscoelastic model with stain, viscoelastic relaxation time, and magnetic field. Normal mode analysis is used to achieve exact analytical expressions of temperature distribution, displacement component and thermal stresses. Numerical computations are conducted for copper-like material and dimensionless results are illustrated graphically. Based on the results, the influence of magnetic field on the distributions of the nondimensional temperature, displacement and stresses are discussed.