Solutions For Plates Research Articles

Buckling analysis of orthotropic plates with various boundary conditions using a new hyperbolic shear displacement model: an analytical approach

PurposeThis paper introduces a closed-form solution for analyzing the buckling behavior of orthotropic plates using a refined plate theory with four variable parameters, leveraging a new hyperbolic shear displacement model.Design/methodology/approachThe proposed theory incorporates a quadratic variation of transverse shear strains across the plate’s thickness and satisfies zero traction boundary conditions on both the upper and lower surfaces without employing shear correction factors. The governing equations are derived from the principle of minimum total potential energy. Closed-form solutions for rectangular plates, with two opposite edges simply supported and the remaining two edges subjected to arbitrary boundary conditions, are obtained using the state space approach to the Levy-type solution. Comparative studies are conducted to validate the accuracy of the obtained results.FindingsThe paper successfully examines and discusses in detail the effects of boundary conditions, loading conditions, variations in modulus ratio and thickness ratio on the critical buckling load of orthotropic plates.Originality/valueThis study presents a novel and precise method for evaluating the buckling behavior of orthotropic plates. The refined plate theory, without the need for shear correction factors, offers significant insights and improvements in understanding the critical buckling load under various conditions, contributing valuable knowledge to the field of structural analysis.

Analytical free vibration solution of orthotropic thin plates with three edges rotationally restrained and one edge free

Abstract Orthotropic thin plates are essential in aerospace applications due to their high strength-to-weight ratio, which is critical for efficient structural performance. The mechanical properties of these structures are influenced by numerous factors, with free vibration characteristics being particularly significant. However, analytical solutions for free vibration are often limited to plates with simple boundary conditions (BCs). Few exact analytical solutions exist for plates with complex non-Lévy-type or rotationally restrained BCs. This study aims to analyze the free vibration behavior of orthotropic thin plates featuring three rotationally restrained edges and one free edge, employing the finite integral transform (FIT) method. A key advantage of this method is its simplicity and versatility, as it does not require the predefinition of a deflection function. By setting different values of the rotating fixed coefficient, the study offers analytical free vibration solutions for plates with various combinations of rotationally restrained, simply supported, clamped, and free BCs without additional derivations. The solutions derived using the FIT method are validated by comparison with numerical results obtained from ABAQUS software and available analytical solutions, confirming the method’s accuracy and reliability. Parameter analyses are conducted to examine the effects of aspect ratio, BCs, material properties, and rotating fixed coefficient on non-dimensional frequency parameters and corresponding vibration modes. The results reveal that these parameters significantly influence the frequency characteristics of orthotropic thin plates. The results can be used as a benchmark for validating other analytical and numerical methods.

Layerwise generalized formulation solved via a boundary discontinuous method for multilayered structures. Part 1: Plates

Abstract This article presents for the very first time a layerwise bending analytical solution for cross-ply laminated composite plates with clamped boundary conditions at all edges. The displacement field is implemented within the framework of the Carrera unified formulation at a layer level by employing Legendre polynomials. The governing equations are obtained by using the principle of virtual displacements statement. This work utilizes the boundary-discontinuous double Fourier series to provide analytical solutions. The high accuracy of the proposed solution is demonstrated by comparing the results with three-dimensional finite-element model (FEM) solutions of multilayered and sandwich plates for various side-to-thickness ratios. In conclusion, the highly accurate proposed solution might be used as a benchmark problem for new analytical or FEMs.

The Factorial Fundamental Frequency Explicit Formulas for Homogenous Rectangular Thin Plates with Four Free Edges

PurposeThis study aims to investigate fundamental modal solutions for rectangular thin plates with four free edges: (1) to provide a novel criterion for fundamental mode identification, and (2) to derive high accurate explicit formulas for calculating the fundamental frequency.MethodTo achieve the study objectives, an investigation into the fundamental mode space was conducted, varying structural and material parameters including the plate’s dimensions (a, b), thickness (h), and Poisson’s ratio (ν). A novel, explicit criterion for fundamental mode identification was formulated, incorporating a six sigma tolerance tailored for specific aspect ratios. Ordinary least squares (OLS) multiple linear regression methodology was integrated within a dedicated process to identify optimal factor combinations for factorial frequencies based on 56 features. This methodology allowed for the derivation of factorial frequency formulas as a function of a, b, h, and ν, providing explicit expressions for calculating the fundamental frequency.ConclusionThe study concludes that the global fundamental mode shapes of rectangular thin plates with four free edges are primarily governed by the aspect ratio a/b, Poisson’s ratio ν, and the thickness-induced shearing effect h on both dimensions. The derived explicit formulas for calculating the fundamental frequency exhibit exceptional accuracy, with a maximum error of −0.217% compared to extensive finite element analysis results, surpassing current state-of-the-art accuracy. This work presents physically meaningful, explicit formulations and introduces a novel perspective in plate and shell research, particularly relevant in scenarios with adjacent free edges and similar global modes.

Accurate Nonlinear Bending Analytical Solutions Setting New Benchmarks for Rectangular Thin Plates with Four Clamped Edges

The von Kármán plate equations have been challenging to solve accurately, especially for two-dimensional problems. This paper presents more accurate nonlinear analytical solutions for rectangular thin plates with four clamped edges, providing a new benchmark beyond Lévy’s solutions. The main features of the present method are that the deflection is expanded by the double series of the classical beam eigenfunctions, the Airy stress function satisfying the geometric deformation compatibility equation corresponds to the nonlinear coupling relationships between the plate deflection and the in-plane force and/or displacement boundary conditions. These solutions can verify existing nonlinear numerical and approximate analytical solutions, offering a robust basis for engineering design. The central deflection error is less than 0.01%, and the central stress error is below 0.6%, exceeding the accuracy of the existing literature.

Isogeometric Analysis of Functionally Graded Plates using High-Order Theories

Functionally Graded Materials are composites, usually of ceramic and metal, in which the volume fraction of their constituents varies smoothly along an interest direction. These materials were initially proposed as a solution for thermal barriers development, but are currently used in different applications. The composition variation results in a gradual change in their properties, enhancing the performance of structures subjected to thermal and mechanical loads, but making structural analysis more complex. Plates, on the other hand, are three-dimensional flat structures in which the thickness is much smaller than their other two dimensions. Different theories have been proposed for structural analysis of plates. These theories can be grouped based on the considerations adopted for the behavior of transverse shear deformations. In this regard, there are the Kirchoff-Love Theory (also known as the Classical Plate Theory - CPT), which disregards the effect of these deformations, the Reissner-Mindlin Theory (also known as the First-Order Shear Deformation Theory - FSDT), which considers shear deformations constant throughout the plate thickness, and Higher-Order Shear Deformation Theories (HSDT’s), which consider a nonlinear variation of shear deformations through different approaches. Among these theories, the most robust and accurate ones are evidently the HSDT’s. However, it is important to clarify that they require the displacement field to have a C1 continuity, which is complex for isoparametric finite elements. Thus, Isogeometric Analysis emerges as a more viable alternative by employing high continuity elements based on Non-Uniform Rational B-Splines (NURBS) functions. Therefore, this paper presents a NURBS-based isogeometric formulation for buckling and free vibration analyses of functionally graded plates using a HSDTs. A series of tests were conducted to assess the accuracy of this formulation considering examples available in the literature with different geometries, boundary conditions, and materials. The obtained results present excellent agreement with the reference solutions for thin and thick plates.

Experimental research on heat dissipation by microchannel phase transition

Abstract Given the characteristics of some missile-borne electronic equipment, such as narrow space, high heat flux, limited power supply, and high reliability in a short time, the idea of phase change heat dissipation is proposed, absorbing a large amount of heat from electronic components through the vaporization phase change of working fluid, to ensure its working performance. The heat dissipation device is a cold plate for micro-channel phase change heat dissipation. Because it is difficult to study phase-change heat dissipation by simulation calculation, the experimental scheme is designed. A set of experimental systems that can not only simulate the working mode of the cold plate in high altitude and low-pressure environments but also carry out multi-parameter exploration is built with conventional and simple devices and instruments. Through extensive and repeated experiments, it has been concluded that: for the same mass of phase change working medium, the smaller the pressure difference between the working medium in the cold plate and the cold plate outlet is, the longer the temperature control time of the cold plate is. When the surface heat load of the cold plate is 400 W, the mass of the phase change working medium is 270 g, and the cold plate temperature control time is up to 10 min. When the working medium of phase transformation is 35% ethanol solution, the cooling and temperature equalization effect of a cold plate are the best, and the maximum temperature of the cold plate surface is stable between 70±1°C for a long time.

Free vibration analysis of rectangular plates resting on an elastic foundation

Abstract Rectangular plate on elastic foundations is a common structural form in aerospace fields, while there still lacks a general analytical method for free vibration analysis of this kind of structure. Thus, the extended separation-of-variable method is used to obtain the analytical free vibration solutions for plates with all kinds of homogeneous boundary conditions. The mode functions are assumed to be in separation-of-variable form, and the natural frequencies of two-direction mode functions are mathematically independent. Then, the free vibration solutions are obtained according to the two-direction boundary conditions. The correctness of the present results is verified via numerical comparisons. Parameter study shows that the elastic foundation significantly affects the flexural vibration behaviors of rectangular plates, and the influence is dependent on boundary conditions and mode orders of rectangular plates.

Vibration analysis using the theory of exponential shear deformation for laminated plates

In this research, the free vibration response of laminated composite plates is studied using a refined exponential shear strain theory. The most interesting feature of this theory is that it allows exponential distributions of transverse shear strains, and verifies zero-shear boundary conditions on the plate surfaces without the use of shear correction factors. Some stiffness constants are written as a series. The number of independent unknowns in the present theory is four, compared with five in other shear deformation theories. The equations of motion are obtained from Hamilton's principle and Navier's method is used to determine the exact solution for antisymmetric cross-laminated plates. The numerical results found in the present analysis for free vibration are presented and compared with those available in the literature. The proposed theory is not only accurate, but also effective in predicting the fundamental frequencies of laminated composite plates.

Dynamic response of double-layer rectangular sandwich plates with graded foam cores under blast loading

In this paper, dynamic response of a fully clamped double-layer (DL) rectangular sandwich plate with graded foam cores (GFC) under blast loading is studied theoretically and numerically. Based on the yield criterion, an analytical model for dynamic response of a DL rectangular sandwich plate with GFC is established under blast loading. With the use of the inscribing and circumscribing squares of the exact yield locus, the bound of the analytical solution of the dynamic response of a double-layer rectangular sandwich plate with GFC is obtained. By neglecting the effect of bending moments, the membrane mode solution of a DL rectangular sandwich plate with GFC in large deflection is obtained. Finite element analysis (FEA) is performed using ABAQUS/Explicit software. The effects of interlayer factor, average yield strength of graded foams, and gradient properties of graded foams on the dynamic response of DL rectangular sandwich plate with GFC are considered. The analytical predictions are in excellent accord with the numerical ones. It is demonstrated that the proposed analytical model is effective to predict the blast response of a DL rectangular sandwich plate with GFC.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

Experimental study on rotary inter-module connections with corrosion-resistant metal materials: Under axial tension

Modular construction is an innovative construction technology that has attracted considerable attention for its pronounced advantages in reducing reliance on on-site labor, speeding up construction, enhancing productivity, and ensuring superior quality. To extend the applicability of modular construction to structures in corrosive environments, such as ocean floating structures for exploring renewable energy, a rotary inter-module connection employing corrosion-resistant metal materials was proposed herein. Six full-scale inter-module connection specimens made of grade S30408 austenitic stainless steel, grade S220503 duplex stainless steel, and grade A96061-T6 aluminum alloy were tested under axial tension. The load-displacement relationships of each specimen were recorded to enable a thorough analysis of the strength, stiffness, and ductility of the tested specimens. As compared to the conventional carbon steel counterparts, the stainless steel specimens showed superior load-bearing capacity with improved ductility, while the aluminum alloy specimens experienced brittle fracture at the bolt shank-bottom plate junction with reduced load-bearing capacity. Three typical failure modes were identified in the experimental study, namely, plastification of the connector and lower corner fitting, plastification of the connector and both corner fittings, and fracture of the bolt. The load-strain relationships of each specimen were also discussed to analyze the load transfer mechanism and structural behavior of the connection. In addition, the applicability and accuracy of the simplified calculation methods based on beam bending theory and Navier solution for rectangular thin plates, respectively, were discussed based on the test results.

Integrated analytical, experimental, and numerical study of the effect of hydro-static loading on the vibrational response of clamped circular plates

This paper presents a comprehensive investigation, comprised of analytical, experimental and numerical approaches, into the interaction between a water cavity of varying fluid cavity height and the vibration of a thin circular plate subjected to its hydro-static pressure. We extend the application of the strong modal coupling method to derive a solution for this unexplored physical problem by utilising classic plate theory, Fourier-Bessel series formulation and the uncoupled solution of a thin clamped circular plate with uniform radial tension. Using a set of geometric and physical properties for the system, the resonance frequencies, response functions and non-dimensionalised added virtual mass incremental (NAVMI) factors are calculated and investigated as a function of hydro-static pressure and fluid cavity height, providing novel fundamental insights into the physical system. We construct an experimental rig to embody the conditions of the analytical investigation for the purpose of validation and to uncover experimental insights into the physical problem. Finite element analysis (FEA), employing modal analysis and transient dynamic analysis, is used to further validate and extend the analytical insights while more accurately mirroring the experimental conditions. The response functions, resonance frequencies and NAVMI factors and their dependence on the cavity pressure were experimentally measured and numerically simulated, with direct comparisons made with the analytical model. A high degree of accuracy for the analytical model is validated, along with its ability to describe the underlying physical phenomena. The validated analytical model is then leveraged to perform fundamental explorations into the modal compositions of the coupled system modes as a function of cavity height and hydro-pressure, the deformation of the coupled system mode shapes and a parametric sensitivity analysis on the effects of plate radii and plate thickness on the coupled system resonance frequencies and NAVMI factors. In totality, this study provides detailed modelling and prediction of the frequency response, resonance frequencies, added mass factors, modal contributions, and deformation of the coupled mode shapes, offering comprehensive insights with wide applicability.

A physics-informed machine learning model for global-local stress prediction of open holes with finite-width effects in composite structures

Fast and accurate methods are required to predict stresses in the vicinity of open and closed holes in composite structures, especially in a global-local modelling context as applied during the design of airframe structures. Fast analytical solutions for infinite-width anisotropic plates with open holes do not consider finite-width effects. Heuristic methods and semi-analytical solutions can be used to towards addressing such effects. To improve the accuracy and speed of these respective methods, we use machine learning (ML) methods trained on high-fidelity finite element analyses to make finite-width corrections. However, such methods require large amounts of training data to reduce errors to satisfactory levels. Therefore, in this study, the fusion of analytical solutions with machine learning is performed. We develop an analytical solution-informed ML model that is as fast as an analytical solution and superior in accuracy to analytical solutions with heuristic finite-width scaling. Our informed ML model offers accuracies equal to analytical solutions for the infinite-width case, and it is capable for use in a global-local modelling context, under uniaxial and biaxial loading. Our informed ML model outperforms prediction accuracy across all cases compared to uninformed ML models and requires a significantly lower size training dataset size.

Passing-through I-plates-to-SHS moment resisting joints subjected to symmetric bending moments

When using hollow structural section (HSS) members in multi-storey buildings, beam-to-column moment resisting connections’ design raises critical questions to tackle. With conventional welded and bolted joints’ response being generally very flexible, the design resistance becomes governed by a deformation criterion rather than a strength criterion. This underscores the necessity for smart reinforcement techniques. A widespread solution is the diaphragm approach, in which the stiffeners usually protrude over the tubular column. The solution of plates passing through the HSS column is another stiffening option which was studied recently for CHS, yet, SHS columns have not been subject to comparable scrutiny. The objective of this paper is to study experimentally, numerically, and analytically the mechanical behaviour of I-beam-to-SHS column moment resisting joints using passing through I-plates under symmetric bending moments. The testing of two full-scale specimens corroborates previous findings with CHS columns in the elastic regime. However, significant deviations were observed in the post-peak response, revealing new insights into the behaviour of SHS columns. Failure was due to progressive buckling of passing through plates and yielding of tube-wall in transverse compression. A finite element model using solid and contact elements is developed and validated against experimental data. This model underscores how loads are redistributed between the tube wall and passing plates. A simplified version of the FE model is used to perform a thorough parametric numerical analysis on 101 configurations expanding upon the experimental test database by varying geometrical parameters such as passing plate thickness, tube, and beam dimensions. Finally, an analytical model integrating the different components of the joint is proposed to evaluate the initial rotational stiffness and the bending resistance.

Analysis of composite honeycomb sandwich panels under blast loads

ABSTRACT This paper presents a novel isogeometric model to analyze the static bending, fre, and forced vibration characteristics of a sandwich of two-layer plates made of auxetic honeycomb core and laminated three-phase polymer/GNP/fiber skin subjected to a double blast load. The two layers are restricted from vertical movement by elastic pins but they can still move in the horizontal plane. The unique feature of the refined first-order shear plate theory with a distribution function g(z) is used to replace the shear correction factor, thereby allowing the shear stress at the top and bottom surfaces to be accurately calculated. The results are derived by the exact Navier solutions for rectangular plates and the IGA method with arbitrary boundary conditions for elliptical and rectangular plate structures. A comprehensive investigation was conducted to evaluate the influence of some parameters on the oscillation of sandwich two-layer plates subjected to double blast load.

Thermal entrance solution for parallel plates with asymmetric arbitrary wall heat flux

Thermal entrance solution for parallel plates with asymmetric arbitrary wall heat flux

Dynamic analysis of cracked thick composite shells by the Boundary Element Method

This article presents a numerical formulation based on the Boundary Element Method for the transient dynamic analysis of cracked thick symmetrical composite shells. The integral formulation uses the static fundamental solutions for thick orthotropic symmetric plates and the anisotropic plain elasticity fundamental solution. Domain integrals associated to distributed loads, curvature and inertial terms are evaluated employing the Radial Integration Method. The crack was modeled using the sub-region method. The developed formulation is implemented computationally and validated through the analysis of several proposed examples. The obtained results demonstrate the validity and robustness of the developed formulation.

Nonlinear Poisson-Boltzmann solutions for charged parallel plates: When opposite charges repel.

I present an exact solution of the Poisson-Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, Lc,1 < Lc,2. For separations less than Lc,1, the force between plates is repulsive. It switches to attractive at Lc,1, but with the electric potential having the same signon both plates. For L > Lc,2, the force remains attractive, and the potential at the plates has the same signas the charge on each plate. I also describe charge regulation, determined by pKa, and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. The implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.

Implementation of a high-order spatial discretization into a finite volume solver: Applications to turbomachinery test cases using an eddy-viscosity turbulence closure

In this study, the implementation of a high-order spatial discretization method into a Finite Volume solver is presented. Specific emphasis is put on the analysis of the performance over selected turbomachinery test cases. High-order numerical discretization is achieved by adopting the cell-centered Least-Square reconstruction, which is implemented in the in-house solver HybFlow. The validation of the adopted methodology is performed by assessing the solution of a turbulent flat plate with zero pressure gradient, using a eddy-viscosity transitional model. The test case also evidences the effect of the discretization of gradient-based source terms when a high-order reconstruction methodology is used. In the second part of the paper, the solver is used for the solution of relevant two-dimensional turbomachinery test cases, assessing the impact of 2nd and 3rd order reconstruction on the prediction of the aerodynamics and the heat transfer for respectively a low-pressure blade and a high-pressure turbine vane. It is shown how a high-order reconstruction allows for obtaining a better prediction of turbomachinery aerodynamics, with lower number of elements. The benefits over heat transfer predictions in high Reynolds number conditions are instead limited to the reduction of heat transfer coefficient spikes in under-resolved regions of the blade. Eventually, the methodology is validated for a three-dimensional low-pressure turbine cascade with realistic boundary layer inflow conditions.