Classical Plate Theory Research Articles

Revisiting Kirchhoff–Love plate theories for thin laminated configurations and the role of transverse loads

The present article aims to re-derive a (de-)homogenization model for particularly investigating the behavior of thin laminated plates withstanding transverse loads. Instead of starting with Kirchhoff-type of assumptions, we directly apply perturbation analysis, in terms of the small parameter introduced by the thinness of composite plates, to the original three-dimensional governing elastostatic equations. The present article sees its intriguing points in the following three aspects. First, it is shown that transverse loads applied on a thin laminated plate induce an in-plane stress response, which essentially differs from the case of single-layered homogenous plates. A scaling law estimating the magnitude of the in-plane stresses due to transverse loads is then given, and a size effect in such induced in-plane stresses arises. Second, the stress state at any position of interest in the original three-dimensional configuration can be asymptotically estimated following a (de-)homogenization scheme, and the (de-homogenization) accuracy is shown, both theoretically and numerically, to be at a same order of magnitude as the thickness-to-size ratio. Third, the asymptotic analysis here identifies the right order of magnitude for the transverse normal strain, which is often set to vanish, leading to the so-called Poisson’s locking problem in classical thin plate theories.

Non-contact strain measurement to eliminate strain gages in vibration-based high cycle fatigue testing

Digital Image Correlation (DIC) is a non-contacting, camera-based technique that calculates full-field displacements and strains by comparing digital images taken before and after an object is deformed. During a vibration-based fatigue test, DIC has an advantage over strain gages in that it is non-contacting and does not accumulate damage during the test. In this work, DIC was implemented to build strain-velocity calibration curves as an alternative to strain gages. First, a curve fit was applied to DIC displacements and strains along the free edge of the plate using an approximate solution for the mode shape of a cantilevered plate. In total, the curve fits were applied to three sets of DIC data: (i) the raw strains calculated with DIC; (ii) the in-plane U-displacements from which the raw DIC strains were computed; and (iii) the out-of-plane W-displacements observed in the direction of motion. Second, classical plate theory was used to calculate strains by taking derivatives of each of the applied curve fits. Third, the peak strains from each curve fit were used to build the strain-velocity calibration curves. Further, a Monte Carlo Method uncertainty analysis was performed to estimate the uncertainty of the curve fitted DIC and strain gage measurements. Of the three curve-fits, the DIC strains derived from the out-of-plane displacements provided the most precise measurements relative to a strain gage at all excitation levels used to build the calibration curves.

Nonlinear Dynamics Suppression of Bending Deflection in a Composite Laminated Plate Using a Beam Stiffener Due to Critical Buckling Load and Shear Force

In this paper, the nonlinear dynamics behavior of the bending deflection of a stiffened composite laminated plate is suppressed using beam stiffeners at different fiber volume fractions and different aspect ratios. The non-periodic motion and chaos in a stiffened composite laminated plate is detected using the largest Lyapunov exponent parameter and power density function of a fast Fourier transform (FFT). The critical buckling load is calculated at different thickness ratios, numbers of stiffeners, lamination angles and stiffener–depth ratios based on different boundary conditions. The nonlinear response of the bending deflection is analyzed analytically, numerically and experimentally. The analytic solution has been derived using Levy and Navier solutions of classical laminate plate theory at different boundary conditions (CLPT). The numerical simulation was conducted using the ANSYS program while the experiment test was carried out using a strain gauge through a strain meter device. Experimentally, a Southwell plot is used to investigate the value of the critical buckling load. The combined loading are the in-plane compression mechanical load and shear force. All the values of the largest Lyapunov exponent are positive, which gives indication to non-periodic motion and chaos. The nonlinear dynamics behavior of the bending deflection is decreased with the increasing of number of stiffeners in which the value of largest Lyapunov exponent has been decreased. The nonlinear dynamics behavior is increased with the increasing of aspect ratios and fiber volume fractions. The system with an aspect ratio (2.5) and fiber volume fraction (υf = 80%) for an un-stiffened plate is more chaotic than the other systems.

Buckling analysis of isotropic and orthotropic square/rectangular plate using CLPT and different HSDT models

Buckling analysis of simply supported isotropic and orthotropic rectangular plate has been performed using Classical plate theory (CLPT) and various higher-order shear deformation theories (HSDT) developed by Reddy, Karama, Shi, and Touratier. The considered HSDT models utilize parabolic, trigonometric, and exponential distribution functions for shear strain along the transverse direction. The total minimum potential energy approach has been applied to find the governing equations using a generalized higher-order displacement field. The non-dimensional stiffness parameters are calculated for all the considered HSDT models employing governing differential equations. Subsequently, the governing differential equations have been solved using the Navier method. The non-dimensional critical buckling load (Ncr) as per different HSDT models has been discussed for various plate thicknesses. The Ncr of the rectangular plate considering HSDT and CLPT models for various plate thicknesses has been plotted to analyze the numerical result. Finally, a detailed discussion has been done to compare the Ncr values obtained using different shear deformation theories for isotropic and orthotropic plates. From the above critical analysis, the authors wanted to investigate the best suitable plate thickness range for CLPT as well as HSDT models for better accuracy in bucking load calculation.

Nonlinear Free Vibration Analysis of In-plane Bi-directional Functionally Graded Plate with Porosities Resting on Elastic Foundations

This paper deals with the nonlinear free vibration analysis of in-plane bi-directional functionally graded (IBFG) rectangular plate with porosities which are resting on Winkler–Pasternak elastic foundations. The material properties of the IBFG plate are assumed to be graded along the length and width of the plate according to the power-law distribution, as well as, even and uneven types are taken into account for porosity distributions. Equations of motion are developed by means of Hamilton’s principle and von Karman nonlinearity strain–displacement relations based on classical plate theory (CPT). Afterward, the time-dependent nonlinear equations are derived by applying the Galerkin procedure. The nonlinear frequency is determined by using modified Poincare–Lindstedt method (MPLM). Numerical results are obtained in tabular and graphical form to examine the effects of some system key parameters such as porosity coefficients, distribution patterns, gradient indices, elastic foundation coefficients, aspect ratio and vibration amplitude on the nonlinear frequency of the porous IBFG plate. To validate the analysis, the results of this paper have been compared to the published data and good agreements have been found.

ELECTROMAGNETIC RESPONSE OF LAYERED MAGNETO-ELECTRO-ELASTIC THIN RECTANGULAR PLATE UNDER MODERATELY LARGE DEFLECTION

The analytical solution of a layered thin magneto-electro-elastic rectangular plate is presented. The governing equation is based on classical laminate plate theory and von Karman's stress function; capturing the effects of moderately large deflection. Electromagnetic fields are determined in terms of mechanical unknowns by solving Maxwell's equations of electrostatics and magnetostatics. The condensation of electric and magnetic state into plate kinematics, coupled with stress function definition, derives the governing non-linear partial differential equation of motion. Solution for both simply supported and clamped transverse boundary condition is obtained using Galerkin method, which reduces the system into an ordinary differential equation of cubic and quadratic nonlinearity. Numerical results of laminated plate with constituent piezoelectric barium titanite and piezomagnetic cobalt ferrite is produced utilizing electromagnetic boundary and continuity conditions. The effect of transverse elastic boundary condition on through the thickness variation of electric and magnetic potential under linear and moderately large deflection is produced.

Effect of large deflection on laminated composite plates

In this study, the effect of large deflection on the natural frequency of pre-stressed laminated composite plates with different lamination orientations is investigated. An external distributed vertical load is applied at the free edge of the plate when the other edge is clamped, and then the loading edge of the deflected plate is fixed without removing the load to model a pre-stressed curved plate case. The non - linear deflection curve of the large deflected laminated plate is obtained from the large deflection equation. The large deflection is limited to the deflection length corresponding to 25% of the plate length. The thin curved plate is modeled by employing the classical plate theory with a finite element analysis approach. Four different lamination orientations are used which are [00 00 00 00], [900 00 00 900], [00 450 -450 00] and [00 900 00 900], which are used for the examination of large deflection. Besides, the natural frequency parameter of the present model, which is performed in MATLAB, is compared with ANSYS to verify the reliability and validity of the present model. The load parameter that forms the curved plate causes the different mode shapes for each lamination configuration

Simply Supported Polygon Plate Frequencies using Classical and Shear Deformation Theories

In this paper, the frequency equation relationship between classical plate theory and a third order single variable plate theory is derived. The third order single variable plate theory used herein is considerably simpler in the class of higher order plate theories. The theory uses a single variable to formulate the plates. The theory is governed by only one differential equation. As the formulation involves a single unknown function, plate analysis using this theory goes almost in the similar lines of classical plate theory. Hence, the complexity associated with plate analysis is considerably less compared to other higher order plate theories. This paper focuses on obtaining the frequency equation relationship between the third order single variable plate theory used herein and the classical plate theory. The relationships between the third order plate theory and classical plate theory frequency equations are derived. Using the derived equations, the frequencies have been calculated for the case of regular polygon plates with simply supported edge conditions. Frequencies calculated herein are compared with other shear deformation plate theory frequencies. The obtained plate frequency results are in good agreement and are accurate.

Free oscillation analysis of rectangular plate of porous FGM material placed on a Winkler elastic base by analytical method

In this paper, classical plate theory is used to analyse free oscillation of rectangular plates made of porous FGM material, edge joints on Winkler elastic base. Three different types of pore distribution: uniform distribution, symmetric irregular distribution and asymmetrical irregular distribution are investigated. The reliability of the analytical solution as well as the calculation program written on Matlab are verified with some published results and with the results calculated by the SAP2000 structural calculation software. The influence of the material parameters, the geometrical dimensions of the plate, as well as the background coefficient on the free oscillation frequency of the plate are evaluated.

Wave Mode Control of Cantilever Slab Structure of T-Beam Bridge with Large Aspect Ratio

The cantilever plate structure in a T-beam bridge with a large aspect ratio will cause vibration under the influence of environmental disturbance and self-stress, resulting in fatigue damage of the plate structure. Wave control based on elastic wave theory is an effective method to suppress the vibration of the cantilever plate structure in a beam bridge. Based on the classical thin plate theory and the wave control method, the active vibration control of the T-shaped cantilever plate with a large aspect ratio in the beam bridge is studied in this paper. The wave mode control strategy of structural vibration is analyzed and studied, the controller is designed, the vibration mode function of the cantilever plate is established, and the control force/sensor feedback wave control is implemented for the structure. The dynamic response of the cantilever plate before and after applying wave control force is analyzed through numerical examples. The results show that the response of the structure is intense before control, but after wave control, the structure increases damping, absorbs the energy carried by the elastic wave in the structure, weakens the sharp response, and changes the natural frequency of the structure to a certain extent.

Flexural Analysis of Thick Isotropic Rectangular Plates Using Orthogonal Polynomial Displacement Functions

This paper analysed the flexural behaviour of SSSS thick isotropic rectangular plates under transverse load using the Ritz method. It is assumed that the line that is normal to the mid-surface of the plate before bending does not remain the same after bending and consequently a shear deformation function f (z) is introduced. A polynomial shear deformation function f (z) was derived for this research. The total potential energy which was established by combining the strain energy and external work was subjected to direct variation to determine the governing equations for the in – plane and out-plane displacement coefficients. Numerical results for the present study were obtained for the thick isotropic SSSS rectangular plates and comparison of the results of this research and previous work done in literature showed good convergence. However, It was also observed that the result obtained in this present study are significantly upper bound as compared with the results of other researchers who employed the higher order shear deformation theory (HSDT), first order shear deformation theory (FSDT) and classical plate theory (CPT) theories for the in – plane and out of plane displacements at span – depth ratio of 4. Also, at a span - depth ratio of and above, there was approximately no difference in the values obtained for the out of plane displacements and in-plane displacements between the CPT and the theory used in this study.

Flexural Analysis of Thick Isotropic Rectangular Plates Using Orthogonal Polynomial Displacement Functions

This paper analysed the flexural behaviour of SSSS thick isotropic rectangular plates under transverse load using the Ritz method. It is assumed that the line that is normal to the mid-surface of the plate before bending does not remain the same after bending and consequently a shear deformation function f (z) is introduced. A polynomial shear deformation function f (z) was derived for this research. The total potential energy which was established by combining the strain energy and external work was subjected to direct variation to determine the governing equations for the in – plane and out-plane displacement coefficients. Numerical results for the present study were obtained for the thick isotropic SSSS rectangular plates and comparison of the results of this research and previous work done in literature showed good convergence. However, It was also observed that the result obtained in this present study are significantly upper bound as compared with the results of other researchers who employed the higher order shear deformation theory (HSDT), first order shear deformation theory (FSDT) and classical plate theory (CPT) theories for the in – plane and out of plane displacements at span – depth ratio of 4. Also, at a span - depth ratio of and above, there was approximately no difference in the values obtained for the out of plane displacements and in-plane displacements between the CPT and the theory used in this study.

Nonlinear forced vibration of rectangular plates by modified multiple scale method

In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined.

Role of residual thermal stress on the electrochemical performance of a solid-state half-cell

Synthesizing a solid electrolyte layer with a positive electrode layer requires a high-temperature sintering process to improve the interface contact between the two layers. This may generate substantial residual thermal stress between the components during cooling. In this study, we develop a theoretical model to investigate the influence of residual thermal stress on the electrochemical performance of a solid-state half-cell (solid electrolyte–positive electrode). The model accounts for the stress–diffusion interaction and electrochemical reaction and is based on the classical plate theory. The numerical results indicate that, although the residual thermal stress could effectively improve the half-cell capacity, the structural mechanical reliability is reduced. The improvement in the battery capacity is found to be highly dependent on the ratio of the thermal expansion coefficients of the solid electrolyte and the positive electrode. The thermal expansion coefficients determine whether the operating temperature needs to be raised or reduced. Additionally, under the influence of residual thermal stress, reasonable control of the thickness of a solid electrolyte can significantly improve battery capacity. Finally, by examining the stress drop at the interface, we found that introducing residual thermal stress is detrimental to the structural mechanical reliability of the half-cell, meaning that the conflicting demands for improved capacity and mechanical reliability seem irreconcilable.

Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.

Natural element hierarchic models for laminated composite plates

This paper presents the hierarchic models for laminated composite plates in the framework of 2-D natural element method (NEM), aiming at the accurate and effective numerical analysis of stress distributions and buckling loads. The current study was inspired by the fact that the classical composite plate theories are restricted to thin structures and most of hierarchic models have been implemented by FEM. In the hierarchic models, the displacement field is expressed in the product of the 2-D in-plane unknown functions and the 1-D out-of-plane assumed monomials. The in-plane unknown functions, which are supported on the mid-surface of composite plate, are approximated by the 2-D locking-free NEM. The approximated strains and stresses are smoothened by the CPU time-effective strain recovery technique, and furthermore the transverse shear stresses are enhanced by the equilibrated post-processing technique. The proposed hierarchic models are applied to two types of laminated composites, the conventional orthotropic laminated plate and the laminated plate with functionally graded CNTRC layers. Through the numerical experiments, the fundamental features of hierarchic models are investigated in terms of the model level, the structure thickness and the grid density. And the deflections, stresses and buckling load parameters obtained by the present models were compared with 3-D exact solutions and the advanced plate theories. From the comparison, the reliability and excellence of the proposed hierarchic models have been justified.

Analytical Study of Bending Characteristics of an Elastic Rectangular Plate using Direct Variational Energy Approach with Trigonometric Function

In this paper, an analytical three-dimensional (3D) bending characteristic of an isotropic rectangular thick plate with all edges simply supported (SSSS) and carrying uniformly distributed transverse load using the energy technique is presented. The three-dimensional constitutive relations which involves six stress components were used in the established, refined shear deformation theory to obtain a total potential energy functional. This theory obviates application of the shear correction factors for the solution to the problem. The governing equation of a thick plate was obtained by minimizing the total potential energy functional with respect to the out of plane displacement. The deflection functions which are in form of trigonometric were obtained as the solution of the governing equation. These deflection functions which are the product of the coefficient of deflection and shape function of the plate were substituted back into the energy functional, thereafter a realistic formula for calculating the deflection and stresses were obtained through minimizations with respect to the rotations and deflection coefficients. The values of the deflections and stresses obtained herein were tabulated and compared with those of previous 3D plate theory, refined plate theories and, classical plate theory (CPT) accordingly. It was observed that the result obtained herein varied more with those of CPT and RPT by 25.39% and 21.09% for all span-to-thickness ratios respectively. Meanwhile, the recorded percentage differences are as close as 7.17% for all span-to-thickness ratios, when compared with three dimensional plate analysis. This showed that exact 3D plate theory is more reliable than the shear deformation theory which are quite coarse for thick plate analysis. Doi: 10.28991/esj-2021-01320 Full Text: PDF

Semi‐analytical method for the stress analysis of transverse cracks in composite structures

AbstractIn this paper a semi‐analytical approach is developed which enables the computation of the stress field in the vicinity of a transverse crack in symmetric cross‐ply laminated plates subjected to an in‐plane tensile load. Due to the assumption of a plane state of strain, the analytical model can be reduced to a two‐dimensional plate strip. After a discretization of the considered laminate into mathematical layers with respect to the thickness direction, the displacements for the undamaged part are derived by making use of the classical laminate plate theory. To incorporate transverse cracks into the formulation, the closed‐form analytical solution is upgraded by a so‐called ‘local solution’ that incorporates unknown displacement functions, specified in the interfaces of the numerical layers, as well Lagrangian interpolation functions of different order. The governing equations are determined by means of the principle of minimum elastic potential and eventually yield a quadratic eigenvalue problem which requires numerical evaluation. The obtained formulation of the previously defined interface displacements includes unknown ‘weight factors' which have to be characterized by adequate boundary and continuity conditions. Consequently, by solving the resulting linear system of equations, a full‐scale representation of the displacement‐, strain‐ and stress field within the damaged laminate is obtained. The semi‐analytical method is verified by comparison with numerical results of detailed finite element simulations. Using only a fraction of the computational effort, the results of the semi‐analytical approach show comparable accuracy to the finite element method.

Axisymmetric nonlinear behavior of functionally graded saturated poroelastic circular plates under thermo-mechanical loading

In functionally graded saturated poroelastic circular plates with immovable simply supported and clamped rims, the axisymmetric nonlinear bending under transverse thermo-mechanical loading has been parametrically studied and compared with the axisymmetric postbuckling and nonlinear bending under thermal loading. Based on the classical plate theory, Love–Kirchhoff hypotheses and Sander’s assumptions, the general coupled nonlinear radial and transverse equilibrium equations, central continuity, symmetry and boundary conditions has been derived in ordinary and state-spatial forms. The corresponding difference equations have been achieved by using the generalized differential quadrature method. The equations have been assembled and numerically solved by using the Newton–Raphson iterative algorithm. The effects of the mechanical and thermal loads, pore distribution type, porosity parameter, Skempton’s coefficient, and thickness and boundary condition type on the behavior of the deflection, whether caused by thermo-mechanical bending, thermal postbuckling, or thermal bending, have been investigated in detail. From the parametric study, a novel quantity determining bending behavior has been found. The axisymmetric themo-mechanical nonlinear bending deflection is inversely and nonlinearly proportional to thermal load when the quantity is greater than a critical value and is nonlinearly proportional to thermal load when the quantity is less than a critical value. It was verified that the plate behavior complies with the general rules known for FG saturated poroelastic circular plates and with those known for metal–ceramic functionally graded circular plates whose governing equations are mathematically analogous to those of the current research.

Geometrical Dimensional Effect on Natural Frequency of Single Layer Graphene in Armchair Configuration

Graphene has remarkable strength, such as yield strength and elasticconstant. The dynamic behaviour of graphene sheet is affected bygeometrical variation in atomic arrangement. This paper introducedgraphene with armchair atomic structure for estimating fundamental naturalfrequencies. The presented analysis can be useful for the possible highfrequency nanomechanical resonator systems. The analytical formulation,based on classical plate theory and continuum solid modelling based finiteelement method have been performed for estimation of fundamental naturalfrequencies of single layer graphene sheet (SGLS) with different boundaryconditions. The free edge and clamped edge boundary conditions have beenconsidered. For simplifying analytical formulations, Blevins approach fordynamic solution has been adopted and for validating analytical results.The finite element analysis of SLGS has been performed using ANSYSsoftware. The effect of variation in geometrical parameters in terms ofwidth and length of SLGS has been analysed for realization of ultra-highfrequency based nanomechanical resonator systems