Bond-slip behavior of CFRP plate-concrete interface

The paper deals with evaluation of the bond performance between a CFRP plate and concrete with respect to various compressive strengths of concrete and bond lengths of the CFRP plate as parameters. To consider stress conditions in the tensile zone of reinforced concrete (RC) structures, double-lap axial tension tests were conducted for eight specimens with CFRP plates bonded to concrete prisms. In addition, a simple linear bond-slip model for the CFRP plate/concrete joints, developed from the bond tests, was used. To verify the model proposed, a total of seven RC beams were strengthened with CFRP plates and tested in flexure employing various bond lengths, strengthening methods, and numbers of CFRP plates. A nonlinear finite-element analysis, with the bond–slip model incorporated in the DIANA program, was performed for the strengthened RC beams. Also, the results of flexural test and analytical predictions are found to be in close agreement in terms of yield and ultimate loads and ductility.     

Static response and free vibration analysis of FGM plates using higher order shear deformation theory

Free vibration and static analysis of functionally graded material (FGM) plates are studied using higher order shear deformation theory with a special modification in the transverse displacement in conjunction with finite element models. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. The fundamental equations for FGM plates are derived using variational approach by considering traction free boundary conditions on the top and bottom faces of the plate. Results have been obtained by employing a continuous isoparametric Lagrangian finite element with 13 degrees of freedom per node. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present model. Numerical results for different thickness ratios, aspect ratios and volume fraction index with different boundary conditions have been presented. It is observed that the natural frequency parameter increases for plate aspect ratio, lower volume fraction index n and smaller thickness ratios. It is also observed that the effect of thickness ratio on the frequency of a plate is independent of the volume fraction index. For a given thickness ratio non-dimensional deflection increases as the volume fraction index increases. It is concluded that the gradient in the material properties plays a vital role in determining the response of the FGM plates.     

Nonlinear finite element analysis of functionally graded plates integrated with patches of piezoelectric fiber reinforced composite

In this paper, a nonlinear static finite element analysis of simply supported smart functionally graded (FG) plates in the presence/absence of the thermal environment has been presented. The substrate FG plate is integrated with the patches of piezoelectric fiber reinforced composite (PFRC) material which act as the distributed actuators of the plate. The material properties of the FG substrate plate are assumed to be temperature dependent and graded along the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The derivation of this nonlinear thermo-electro-mechanical coupled finite element model is based on the first order shear deformation theory and the Von Karman type geometric nonlinearity. The numerical solutions of the nonlinear equations of the finite element model are obtained by employing the direct iteration method. The numerical illustrations suggest the potential use of the distributed actuator made of the PFRC material for active control of nonlinear deformations of smart FG structures. The effects of volume fraction index of the FG material of the substrate plates and the locations of the PFRC patches on the control authority of the patches are investigated. Emphasis has also been placed on investigating the effect of variation of piezoelectric fiber orientation angle in the PFRC patches on their actuation capability for counteracting the large deflections of FG plates.     

C0 FE model based on HOZT for the analysis of laminated soft core skew sandwich plates: Bending and vibration

Bending and free vibration behaviour of laminated soft core skew sandwich plate with stiff laminate face sheets is investigated using a recently developed C₀ finite element (FE) model based on higher order zigzag theory (HOZT) in this paper. The in-plane displacement fields are assumed as a combination of a linear zigzag function with different slopes at each layer and a cubically varying function over the entire thickness. The out of plane displacement is considered to be quadratic within the core and constant in the face sheets. The plate theory ensures a shear stress-free condition at the top and bottom surfaces of the plate. Thus, the plate theory has all of the features required for accurate modelling of laminated skew sandwich plates. As very few element model based on this plate theory (HOZT) exist and they possess certain disadvantages, an attempt has been made to check the applicability of the refined element model. The nodal field variables are chosen in such a manner that there is no need to impose any penalty stiffness in the formulation. Refined C₀ finite element model has been utilized to study some interesting problems on static and free vibration analysis of laminated skew sandwich plates.     

Optimization of the static and dynamic characteristics of plates with isogrid stiffeners

In this paper, the orientation angles of stiffeners arranged in the form of isogrid configuration over a flat plate are selected to optimize the static and dynamic characteristics of these plates/stiffeners assemblies. The static characteristics are optimized by maximizing the critical buckling loads of the isogrid plate, while the dynamic characteristics are optimized by maximizing multiple natural frequencies of the stiffened plate.

A finite element model is developed to describe the statics and dynamics of Mindlin plates which are stiffened with arbitrarily oriented stiffeners. The model is used as a basis for optimizing separately or simultaneously the critical buckling loads and natural frequencies of the plates per unit volume of the plates/stiffeners assemblies.

Numerical examples are presented to demonstrate the utility of the developed model and optimization procedures. The presented approach can be invaluable in the design of plates with isogrid stiffeners for various vibration and noise control applications.     

An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness

In this study, the bending solution of simply supported transversely isotropic thick rectangular plates with thickness variations is provided using displacement potential functions. To achieve this purpose, governing partial differential equations in terms of displacements are obtained as the quadratic and fourth order. Then, the governing equations are solved using the separation of variables method satisfying exact boundary conditions. The advantage of the purposed method is that there is no limitation on the thickness of the plate or the way the plate thickness is being varied. No simplifying assumption in the analysis process leads to the applicability and reliability of the present method to plates with any arbitrarily chosen thickness. In order to confirm the accuracy of the proposed solution, the obtained results are compared with existing published analytical works for thin variable thickness and thick constant thickness plate. Also, due to the lack of analytical research on thick plates with variable thickness, the obtained results are verified using the finite element method which shows excellent agreement. The results show that the maximum displacement of the plates with variable thickness is moved from the center toward the thinner plate edge. In addition, results exhibit the profound effects of both thickness and aspect ratio on stress distribution along the thickness of the plate. Results also show that varying thickness has not a profound impact on bending and twisting moments in transversely isotropic plates. Five different materials consist of four transversely isotropic and one isotropic, as a special case, are considered in this paper, which it is shown that the material properties have a more considerable impact on higher thickness plate.     

任意四边形杂交Mindlin板单元

本文把基于虚功原理的杂交有限元模型用于板弯曲问题，构造了一个考虑横向剪应变的任意四边形新型板单元，本单元的突出优点是采用了一种比较合理的位移插值函数，使之能较真实地模拟各类板的变形，且用的自由度最少。文中对此单元作了比较广泛的数值试验，计算结果表明它对板厚有相当宽的适用范围，对于各种例题均能在较粗的网格下得到满意的精度．     

Free vibration analysis of piezoelectric coupled annular plates with variable thickness

This paper presents the free vibration analysis of piezoelectric coupled annular plates with variable thickness on the basis of the Mindlin plate theory. No work has yet been done on piezoelectric laminated plates while the thickness is variable. Two piezoelectric layers are embedded on the upper and lower surfaces of the host plate. The host plate thickness is linearly increased in the radial direction while the piezoelectric layers thicknesses remain constant along the radial direction. Different combinations of three types of boundary conditions i.e. clamped, simply supported, and free end conditions are considered at the inner and outer edges of plate. The Maxwell static electricity equation in piezoelectric layers is satisfied using a quadratic distribution of electric potential along the thickness. The natural frequencies are obtained utilizing a Rayleigh–Ritz energy approach and are validated by comparing with those obtained by finite element analysis. A good compliance is observed between numerical solution and finite element analysis. Convergence study is performed in order to verify the numerical stability of the present method. The effects of different geometrical parameters such as the thickness of piezoelectric layers and the angle of host plate on the natural frequencies of the assembly are investigated.     

Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

Modeling of amplitude-dependent damping characteristics of fiber reinforced composite thin plate

A damping model of fiber reinforced composite thin plate with consideration of amplitude-dependent property is established using the Jones–Nelson nonlinear theory in conjunction with the classical laminated plate theory, polynomial fitting method and strain energy method. In this model, the elastic moduli are expressed as the function of strain energy density and the loss factors in the longitudinal, transverse and shear directions are expressed as the functions of excitation amplitude. Moreover, three TC300 carbon/epoxy composite plates are taken as research objects to carry out a case study. One of them is used to determine the amplitude-dependent coefficients of loss factors in fiber reinforced composites by combining the least square method with polynomial fitting method, and the other two plates are used to verify the correctness of the theoretical model. The results of the developed model considering amplitude dependence and experimental test show a good consistency. It is discovered that the viscoelastic effect of epoxy resin materials will contribute to the increased degree of damping. So, if such plate structures exhibit more pronounced viscoelastic characteristics, there will be more significant in the nonlinear degree of amplitude-dependent damping phenomenon.     

Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method

In the present study, a nonlocal continuum model based on the Eringen’s theory is developed for vibration analysis of orthotropic nano-plates with arbitrary variation in thickness. Variational principle and Ritz functions are employed to calculate the size dependent natural frequencies of non-uniform nano-plates on the basis of nonlocal classical plate theory (NCLPT). The Ritz functions eliminate the need for mesh generation and thus large degrees of freedom arising in discretization methods such as finite element (FE). Effect of thickness variation on natural frequencies is examined for different nonlocal parameters, mode numbers, geometries and boundary conditions. It is found that thickness variation accompanying small scale effect has a noticeable effect on natural frequencies of non-uniform plates at nano scale. Also a comparison with finite element solution is performed to show the ability of the Ritz functions in fast converging to the exact results. It is anticipated that presented results can be used as a helpful source in vibration design and frequency optimization of non-uniform small scaled plates.     

A vibration analysis of composite laminated plates

A finite element formulation of the equations governing laminated anisotropic plates using Reddy's higher-order theory is presented. This simple higher-order shear deformable theory takes into account the parabolic distribution of the transverse shear deformation through the thickness of the plate and contains the same unknowns as in the first-order shear deformation theory. Finite element solutions are presented for rectangular plates of different layups, such as cross-ply, antisymmetric angle-ply, and sandwich plates with various material properties, boundaries, and plate aspect ratios. The numerical results are compared with the available closed-form results, the 3-D linear elasticity theory results, and the other available numerical results. A comparison is also made with test data from a laminated cantilever plate.     

Hamilton等参元与智能叠层板的半解析法

根据压电材料修正后的Hellinger-Reissner(H-R)变分原理,建立了各向异性压电材料4节点Hamilton等参元的一般形式．为智能叠层板自由振动问题和带有压电块的叠层悬臂梁的瞬态响应等问题提出了一种新的半解析法．数学模型的基本步骤:将压电层和主体层看成独立的三维体,在平面内离散各层,分别建立各层的方程;根据主体层和压电层在连接界面上广义应力和广义位移的连续条件,联立主体层和压电层的方程得到全结构的控制方程．等参元不限制智能板侧面的几何边界形状、板的厚度和层数,有广泛的应用领域．     

Local radial basis function collocation method for bending analyses of quasicrystal plates

The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.     

A Hybrid Variational Model for the Analysis of Free‐Corner Effects in Layered Plates

A variational model for the assessment of free‐edge and free‐corner effects [1‐3] in thermally loaded rectangular cross‐ply laminate plates is presented. The physical layers of the plate are discretized by an arbitrary number of mathematical layers through the laminate thickness. A layerwise displacement field with unknown interface functions which depend on the inplane coordinates is formulated wherein an a priori assumed layerwise linear thickness interpolation scheme is employed. The application of the principle of minimum elastic potential yields a set of governing Euler‐Lagrange differential equations for the unknown inplane functions which for the special free‐corner problem in rectangular cross‐ply plates can be solved in a closed‐form analytical manner. Boundary conditions of traction free laminate edges are fulfilled in an average sense. Since the approach utilizes a discretization through the plate thickness yet allows for closed‐form solutions within the layer planes for all state variables it is appropriate to speak of a hybrid analysis approach. The presented method allows easy application, can be run on every standard personal computer and is in favourable agreement with comparative finite element calculations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of a higher order finite element on a Winkler foundation

Analyzing thick plates as a construction component has been of interest to structural engineering research for several decades. In particular, thick plates resting on elastic foundations are more specific. Mindlin's plate theory for thick plate analysis and the Winkler theory for elastic foundation analyses have wide applications. The current research considers analysis of isotropic plates on a Winkler foundation according to Mindlin's plate theory. The analysis uses a higher order plate element to avoid shear locking phenomena in the plate. The main features of this element are representation of real displacement functions of the plate perfect and shear locking do not occur at the plates modeled with this element. Derivation of the equations for finite element formulation for thick plate theory uses fourth-order displacement shape functions. A computer program using the finite element method, coded in C++, analyzes the plates resting on an elastic foundation. The analysis involves a 17-noded finite element. The study's graphs and tables assist engineers' designs of thick plates resting on elastic foundations. The study concludes with the computer-coded program, which allows effective use for the shear locking-free analysis of thick Mindlin plates resting on elastic foundations.     

An accurate mathematical study on the free vibration of stepped thickness circular/annular Mindlin functionally graded plates

An analytical solution based on a new exact closed form procedure is presented for free vibration analysis of stepped circular and annular FG plates via first order shear deformation plate theory of Mindlin. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. Based on the domain decomposition technique, five highly coupled governing partial differential equations of motion for freely vibrating FG plates were exactly solved by introducing the new potential functions as well as using the method of separation of variables. Several comparison studies were presented by those reported in the literature and the FEM analysis, for various thickness values and combinations of stepped thickness variations of circular/annular FG plates to demonstrate highly stability and accuracy of present exact procedure. The effect of the geometrical and material plate parameters such as step thickness ratios, step locations and the power law index on the natural frequencies of FG plates is investigated.     

Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements

This paper discusses the vibration and stability analysis of thick orthotropic plate structures using finite elements based on the hybrid-Trefftz formulation. While the formulation can be used for elements of arbitrary geometry, the paper concentrates on the use of a simple and robust triangular element. The key feature of the formulation is to use element interpolations that are consistent for all values of the plate thickness, including the limit when it goes to zero. This eliminates the locking problem automatically and ensures a robust approximation for thick and thin plates. Results for various problems are included to demonstrate the accuracy and efficiency of the element.     

A triangular shell element for vibration analysis of cambered and twisted fan blades

A ten-node triangular shell element of thirty degrees of freedom was successfully developed and applied to plate and shell static analysis, and free vibrations of rectangular plates with different boundary conditions. The element is also suitable for the vibration analysis of pretwisted and cambered fan blades. Natural frequencies of cambered and untwisted fan blades having a rectangular planform with constant thickness are determined for different shallowness ratio and blade thickness. For different blade-tip twist angles, the element can be applied to obtain the frequencies of twisted blades with constant thickness. Frequencies, resulting from the applied element, are compared to those of experimental results which are available in literature, and to those of shallow shell theory and other finite elements. Good correlations and reasonably accurate results are obtained by using a coarse mesh size.     

Optimal control of a variational inequality with applications to structural analysis. II. Local optimization of the stress in a beam. III. Optimal design of an elastic plate

Optimal design with respect to the variable thickness of an elastic beam with unilateral supports under the criterion of minimal value of the maximal stress is presented in Part I. A dual formulation of the state problem (in terms of bending moments) is used and the convergence of some approximations proved.In Part III the variable thickness of an elastic or elasto-plastic plate unilaterally supported on a part of its edge is optimized. For elastic plates with parallel edges a primal finite element model is applied and a convergence result obtained.