Shear Flexible Plate Research Articles

Thermoelastic Stability Behavior of Curvilinear Fiber‐Reinforced Composite Laminates With Different Boundary Conditions

In this article, the thermoelastic stability behavior of curvilinear fiber‐reinforced composite laminates is investigated using a shear flexible plate theory. The laminate is composed of layers that make use of curvilinear fibers to create spatial variation of stiffness due to the continuous change of fiber orientation within the lamina. Such composite laminates, subjected to different thermal loads like uniform and non‐uniform temperature distributions, are considered in this analysis. The governing equations obtained using the principle of minimization of total potential energy, are solved for evaluating the critical buckling temperature. The displacement fields of pre‐buckling of the laminate are evaluated before proceeding for the thermal buckling analysis. The formulation is checked against available solutions in the literature. A detailed parametric study considering important design parameters such as curvilinear fiber angles, lay‐up, thickness ratio, coefficients of thermal expansion, and modular ratio on the thermoelastic stability/buckling behavior of composite laminates is made. The present analysis highlights the substantial change in the critical buckling loads due to the curvilinear fiber angles, lay‐up and boundary conditions. The new results presented here for curvilinear fiber‐reinforced composite laminate with clamped/mixed boundary conditions may be useful in assessing similar studies based on higher‐order theories or other numerical studies. POLYM. COMPOS., 40:2876–2890, 2019. © 2018 Society of Plastics Engineers

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.

Analysis of Piezoelectric Composite Beams and Plates with Multiple Delaminations

In the present work, the effect of delamination or debonding on the static and dynamic characteristics of laminated piezoelectric beams and plates is studied. A four-noded quadrilateral shear flexible plate element is developed to model the damages in composite substrate and in piezoelectric layers. The elastic stiffness and electro-elastic stiffness degradations are introduced at the elemental level through coupled constitutive relations. The undamaged structure is modeled as a single laminate while the delaminated region is modeled as sublaminates whose interface contains the desired delamination or debonding. The continuity of displacement is maintained across the delaminated edge by imposing the strain-based multipoint constraints. Numerical studies are conducted on composite specimens with surface bonded active layers. The results are presented to evaluate the performance of these smart structures in the presence of delamination. For actuators, the performance is measured in terms of the achievable deflection control and in sensors as a measurable output voltage. It is observed that actuator and sensor debonding degrade the capabilities of active materials significantly. Therefore, the study concludes that a damage tolerant approach is essential for the design of smart structural systems to account for damage-induced uncertainty in the functional properties of smart actuators and sensors.

Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method

Here, asymmetric free vibration characteristics and thermoelastic stability of functionally graded circular plates are investigated using finite element procedure. A three-noded shear flexible plate element based on the field-consistency principle is used. Temperature field is assumed to be a uniform distribution over the plate surface and varied in thickness direction only. Material properties are graded in the thickness direction according to simple power law distribution. For the numerical illustrations, aluminum/alumina is considered as functionally graded material. The variation in critical buckling load is highlighted considering gradient index, temperature, radius-to-thickness ratios, circumferential wave number and boundary condition of the plate.

Stability of multi-step flexural-shear plates with varying cross-section

In this paper, multi-story buildings with shear-wall structures and with narrow rectangular plane configuration are modeled as a multi-step flexural-shear plate with varying cross-section for buckling analysis. The governing differential equation of such a plate is established. Using appropriate transformations, the equation is reduced to analytically solvable equations by selecting suitable expressions of the distribution of stiffness. The exact solutions for buckling of such a one-step flexural-shear plate with variable stiffness are derived for several cases. A new exact approach that combines the transfer matrix method and closed from solution of one-step flexural-shear plate with continuously varying stiffness is presented for stability analysis of multi-step non-uniform flexural-shear plate. A numerical example shows that the present methods are easy to implement and efficient.

Stability of multi-step flexural-shear plates with varying cross-section

In this paper, multi-story buildings with shear-wall structures and with narrow rectangular plane configuration are modeled as a multi-step flexural-shear plate with varying cross-section for buckling analysis. The governing differential equation of such a plate is established. Using appropriate transformations, the equation is reduced to analytically solvable equations by selecting suitable expressions of the distribution of stiffness. The exact solutions for buckling of such a one-step flexural-shear plate with variable stiffness are derived for several cases. A new exact approach that combines the transfer matrix method and closed from solution of one-step flexural-shear plate with continuously varying stiffness is presented for stability analysis of multi-step non-uniform flexural-shear plate. A numerical example shows that the present methods are easy to implement and efficient.

Stability of tall buildings with shear-wall structures

An analytical model of a flexural-shear plate is proposed herein for the purpose of buckling analysis. In this paper, tall buildings with shear-wall structures and with narrow rectangular plane configuration are modeled as flexural-shear plates for buckling analysis. The governing differential equation for buckling of a flexural-shear plate with varying cross-section is established. Using appropriate transformations, the equation is reduced to an analytically solvable equation by selecting suitable expressions, such as power functions and exponential functions, for the distribution of stiffness along the height of the plate. The exact solutions for buckling of a flexural-shear plate with variable stiffness are derived. A numerical example shows that the present methods are easy to implement and are efficient.

Erratum for “Buckling of Flexural-Shear Plates” by Q. S. Li

Erratum for “Buckling of Flexural-Shear Plates” by Q. S. Li

Analysis of shear flexible beams, using the meshless local Petrov‐Galerkin method, based on a locking‐free formulation

The problems of shear flexible beams are analyzed by the MLPG method based on a locking‐free weak formulation. In order for the weak formulation to be locking‐free, the numerical characteristics of the variational functional for a shear flexible beam, in the thin beam limit, are discussed. Based on these discussions a locking‐free local symmetric weak form is derived by changing the set of two dependent variables in governing equations from that of transverse displacement and total rotation to the set of transverse displacement and transverse shear strain. For the interpolation of the chosen set of dependent variables (i.e. transverse displacement and transverse shear strain) in the locking‐free local symmetric weak form, the recently proposed generalized moving least squares (GMLS) interpolation scheme is utilized, in order to introduce the derivative of the transverse displacement as an additional nodal degree of freedom, independent of the nodal transverse displacement. Through numerical examples, convergence tests are performed. To identify the locking‐free nature of the proposed method, problems of shear flexible beams in the thick beam limit and in the thin beam limit are analyzed, and the numerical results are compared with analytical solutions. The potential of using the truly meshless local Petrov‐Galerkin (MLPG) method is established as a new paradigm in totally locking‐free computational analyses of shear flexible plates and shells.

Buckling of Flexural-Shear Plates

In this paper, tall buildings with shear-wall structures and with narrow rectangular plane configuration are modeled as flexural-shear plates for buckling analysis. The differential equations that govern the buckling behavior of flexural-shear plates, with and without considering shear deformation of shear-wall structures, are established. The general solutions of uniform flexural-shear plates with various end conditions are derived and used to obtain the eigenvalue equation for multistep flexural-shear plates. The new exact approach that combines the transfer matrix method and the derived closed-form solutions is presented for the buckling analysis of multistep flexural-shear plates. The exact solutions for buckling of nonuniform flexural-shear plates are derived for two types of distributions of variable stiffness. It is proved that a flexural-shear plate with two edges free in the longitudinal direction can be simplified as a flexural bar and that a multistep uniform flexural-shear plate may be treated...

A computational approach for free vibration of non-uniform flexural–shear plates

The concept and an analytical model of a flexural–shear plate are proposed in this paper. The governing differential equation for vibration of such a plate with variably distributed mass and stiffness considering the effects of damping and axial forces is established. Using appropriate transformations, the equation is reduced to a Bessel's equation or an Euler's equation or a differential equation with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the distribution of mass and stiffness along the height of the plate. The exact solutions of one-step non-uniform flexural–shear plates are derived and used to obtain the frequency equations of multi-step non-uniform flexural–shear plates. The new exact approach which combines the transfer matrix method and closed form solutions of one-step flexural–shear plates are proposed for free vibration analysis of multi-step non-uniform flexural–shear plates. The new transfer matrix and the resulting frequency equations are presented. Numerical examples demonstrate that the calculated natural frequencies and mode shapes of tall buildings with narrow rectangular plane configuration which are treated as flexural–shear plates for vibration analysis are in good agreement with the corresponding experimental data.

FREE VIBRATION OF ELASTICALLY RESTRAINED FLEXURAL-SHEAR PLATES WITH VARYING CROSS-SECTION

This paper presents an analytical approach to determining natural frequencies and mode shapes of non-uniform flexural-shear plates with line translational spring and rotational spring supports and line masses under action of axial forces. The governing differential equation for vibration of a non-uniform flexural-shear plate under axial forces is established first. It is shown that it is possible to separate a flexural-shear plate as two beams for free vibration analysis, one is a flexural beam, and the other is a shear beam. The natural frequency of the plate is equal to the square root of the square sum of the two natural frequencies of the two beams, and the mode shape of the plate is the product of the corresponding two mode shapes of the two beams. In this paper, power functions and exponential functions are adopted for describing the distributions of mass and stiffness along the height of the plate as well as the axial forces acting on the plate. The exact solutions for free vibrations of non-uniform flexural-shear plates for several cases that are important in engineering practices are derived. A numerical example shows that the calculated results are in good agreement with the experimental data and it is convenient to apply the proposed method to free vibration analysis of elastically restrained flexural-shear plates with varying cross-section.

Free vibration of flexural-shear plates

In this paper, multi-storey buildings with narrow rectangular plane configuration (narrow buildings) are treated as one-step or multi-step cantilever flexural-shear plates in analysis of free vibration. The governing differential equations for free vibration of flexural-shear plates with variably distributed mass and stiffness are established and reduced to Bessel’s equations by selecting suitable expressions, such as power functions and exponential functions, for the distributions of stiffness and mass along the height of the plates. The general solutions of one-step flexure-shear plates are derived and used to obtain the frequency equation of multi-step cantilever flexural-shear plates. A new exact approach is presented which combines the transfer matrix method and closed form solutions of one step flexural-shear plates. Two numerical examples demonstrate that the calculated natural frequencies and mode shapes of the narrow buildings are in good agreement with the corresponding experimental data. It is shown that it is possible to regard a building with rigid floors as a cantilever flexural bar that is a special case of a cantilever flexural-shear plate. Thus, the proposed methods in this paper are suitable for the calculation of free vibrations of narrow buildings and common shear-wall buildings.

Vibration analysis of flexural-shear plates with varying cross-section

In this paper, multi-storey buildings with narrow rectangular plane configuration (narrow buildings) are treated as cantilever flexural-shear plates in analysis of free vibration. The governing differential equations for free vibration of flexural-shear plates with variably distributed mass and stiffness are established and reduced to Bessel’s equations or Euler’s equation by selecting suitable expressions, such as power functions and exponential functions, for the distributions of stiffness and mass along the height of the plates. The general solutions of flexural-shear plates are derived. Numerical examples demonstrate that the calculated natural frequencies and mode shapes of narrow buildings are in good agreement with the experimentally measured data. It is also shown that it is possible to regard a building with rigid floors as a cantilever flexural bar that is a special case of a cantilever flexural-shear plate. Thus, the methods proposed in this paper are suitable for the calculation of free vibration of narrow buildings and common shear-wall buildings.

Free Vibration Analysis of Flexural-Shear Plates Under the Action of Axial Forces

The governing differential equations for vibration of flexural-shear plates considering the effects of flexural, shear deformation and axial forces are established. The general solutions of a one-step flexural-shear plate are derived and used to obtain the eigenvalue equations of a multi-step flexural-shear plate with various boundary conditions. The new exact approach which combines the transfer matrix method and closed form solutions of a uniform flexural-shear plate with various boundary conditions is presented. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a tall building that is treated as a two-step flexural-shear plate are in good agreement with the experimentally measured data. It is also shown that the effect of shear deformation on the fundamental natural frequency can be ignored, but its effect on the higher natural frequencies should be considered. On the contrary, the effect of axial force (N) on the higher natural frequencies is not significant, and its effect on the fundamental natural frequency is dependent on the ratio of N to the critical buckling force Nc.

A simple non‐layered finite element for the elasto‐plastic analysis of shear flexible plates

AbstractA simple and accurate non‐layered plastic element which is capable of capturing the development of plastic curvatures across the thickness of the plates is presented in this paper. The influence of transverse shear forces on the plastic behaviour of plates is also investigated here. This non‐layered plastic element is based on: (i) the four‐node quadrilateral strain element for shear flexible elastic plates given by the authors; (ii) the plastic hinge formulation outlined by Shi and Atluri; and (iii) the modified Ilyushin's yield function which can account for both the development of plastic deformations across the thickness of plates and the effect of transverse shear forces. The numerical examples given here demonstrate that the present non‐layered plastic element can achieve the results obtained by the layered element model and the transverse shear forces considerably affect the plastic behaviour of plates under certain load conditions.

Accurate force evaluation with a simple, bi-linear, plate bending element

A great deal of attention has been given to the development of simple C ° continuous plate and shell elements based on the shear flexible theories for application to thick plates, sandwich or cellular plates and transversely isotropic or laminated plates. After considerable experimentation using unconventional approaches such as reduced integration, selective integration, mixed methods using discontinuous force fields, etc., it has been possible to develop simple displacement-type elements which can be reliably used. The stress recovery at nodes from such elements is often unreliable as the nodes are usually the points where strains or stresses are least accurate in the element domain. Further, nodal values can reflect severe oscillations at some difficult corner or edge conditions. In this paper, we focus attention on the optimal stress recovery from such an element. This is done after an interpretation of the displacement method as a procedure that obtains strains over the finite-element domain in a least-squares accurate fashion. If a shear flexible element is field-consistent, there are optimal locations at which bending moments and shear forces are accurate in a least-squares sense. These points are identified for the present element and used to study stresses in typical plate problems. Another difficulty faced is the rapid variation of twisting moments at free edges and corners of shear flexible plates and its influence on the shear forces at that edge. A related source of difficulty is the distinction made in Kirchhoff theory between shear forces and the effective shear reactions of that theory. The present study is seen to give accurate enough shear force and twisting moment predictions to allow one to draw the severe conclusion that the use of the Kirchhoff shear reaction at edges in classical plate theory is an ambiguous and unnecessary one and can be avoided. The findings confirm a recent suggestion that it may be more appropriate to have three (as introduced originally by Poisson) instead of the two boundary conditions (as modified by Kirchhoff) usually applied on the edge of a thin plate, especially if that edge is unsupported.

A simple and efficient shear-flexible plate bending element

A shear-flexible triangular element formulation, which utilizes an assumed quadratic displacement potential energy approach and is numerically integrated using Gauss quadrature, is presented. The Reissner/Mindlin hypothesis of constant cross-sectional warping is directly applied to the three-dimensional elasticity theory to obtain a moderately thick-plate theory or constant shear-angle theory (CST), wherein the middle surface is no longer considered to be the reference surface and the two rotations are replaced by the two in-plane displacements as nodal variables. The resulting finite-element possesses 18 degrees of freedom (DOF). Numerical results are obtained for two different numerical integration schemes and a wide range of meshes and span-to-thickness ratios. These, when compared with available exact, series or finite-element solutions, demonstrate accuracy and rapid convergence characteristics of the present element. This is especially true in the case of thin to very thin plates, when the present element, used in conjunction with the reduced integration scheme, outperforms its counterpart, based on discrete Kirchhoff constraint theory (DKT).

Shear-Flexible Orthotropic Plates Loaded in Plane

A new, improved theory is presented for orthotropic plates with thickness-shear flexibility and subjected to in-plane loading. The improvement introduced herein is the modification of the slopes at which the in-plane stress resultants are assumed to act. This is a generalization of the work of Haringx from columns to inplane-loaded plates. Comparison with the classical (Reissner) shear-flexible plate theory shows that the improved theory imposes loading anisotropy on the governing differential equations and tends to predict higher buckling loads than the overly conservative Reissner theory. Comparison with experimental results for uniaxially-compressed, simply supported sandwich plates with glass-fiber-reinforced facings and hexagonal-cell honeycomb cores indicates that the improved-theory results agree slightly better than do the Reissner-theory results.