Fundamental solutions for thick sandwich plates

A method to evaluate the fundamental solutions of thick sandwich plates by the use of the Fourier transform technique is developed. The fundamental solution is presented in the Fourier integral form and the calculation of this solution is discussed in detail. A special care is taken for representation and calculation of the hypersingular integrals entering in this presentation.     

Buckling of thick skew plates

The paper considers the elastic buckling of thick skew plates. Up to now, very little work has been done on this subject and the paper aspires to fill this gap. Based on the principle of stationary total potential energy and the recently developed pb-2 Rayleigh-Ritz method as a solution procedure, buckling solutions for skew plates of various aspect ratios, skew angles and boundary conditions are determined. The tabulation of these results, not available thus far, should be useful to designers and researchers who may use them as benchmark values to test the validity and convergence of their numerical techniques and softwares for thick plate analysis.     

基于拉伸振动精确化理论求解含孔厚板弹性波散射问题

针对采用弹性力学平面问题求解波动/振动时常产生较大误差的问题,基于厚板拉伸振动精确化方程,采用复变函数方法对含孔平板中弹性波散射与动应力集中问题进行了研究。利用正交函数展开的方法将待解的问题归结为对一组无穷代数方程组的求解。给出了含椭圆孔厚板拉压弹性波散射与动应力集中的数值结果。研究结果表明:动应力集中系数与分布取决于入射波数、平板厚度、椭圆偏心率等无量纲化参数。     

Triangular finite element for analysis of thick laminated plates

The development of a general triangular C⁰ element, based on an assumed quadratic displacement potential energy approach, is presented for the analysis of arbitrarily laminated thick plates. The element formulation assumes transverse inextensibility and layerwise constant shear-angle. Convergence of transverse displacement, moments and stresses, the effects of two different Gauss quadrature schemes and comparison of the present solutions with the available analytical/finite-element results also form a part of the investigation. Furthermore, numerical results indicate close agreement between the LCST (layerwise constant shear-angle theory) and the three-dimensional elasticity theory with the length (or width) to thickness ratio as low as 4. Detailed comparison of the LCST-based finite-element solutions with those based on the CST (constant shear-angle theory) and the CLT (classical lamination theory) clearly demonstrates the superiority of the former over the latter two, especially in the prediction of the distribution of the in-plane displacements and stresses through the laminate thickness. This paper also introduces a new non-dimensionalized parameter, Δθ*, which is shown to be a very useful measure for classification of the laminated plates and the suitability of different plate theories over various ranges of length-to-thickness ratio.     

钢结构厚板的冲击韧性研究

随着钢材在我国大跨度桥梁工程中广泛应用,钢材板厚也日益增加。对于厚板钢材,厚度增大使钢板的应力应变状态发生变化,中心偏析现象严重,更容易降低钢板厚度方向的韧性,且在低温环境下,这种现象表现尤为突出。而现行相关规范缺乏对厚钢板冲击试验规定。根据厚板低温冲击韧性试验数据,对钢板厚度变化对冲击韧性影响进行了探讨,并且利用Boltzmann函数对试验结果进行拟合,为钢结构厚板工程中冲击韧性取值提供参考作用。     

A mixed 3D finite element for modelling thick plates

Based on the Hellinger-Reissner variational principle, we formulate a mixed 3-d finite element for plate bending. This element is used to model thick plates and alleviates the problem of shear-locking in plates with large length/thickness ratios. The computer code which was used here, is available.     

Hierarchic finite elements for moderately thick to very thin plates

We consider the numerical solution of Reissner-Mindlin plates. The model is widely used for thin to moderately thick plates. Generally low order finite elements (with degree one or two) are used for the approximation. Unfortunately the solution degenerates very rapidly for small thickness (locking phenomenon). Non standard formulations of the problem are usually combined with low order finite elements to weaken or possibly eliminate the locking of the numerical solution. We introduce a family of hierarchic finite elements and we present a set of numerical results for the plate problem in its plain formulation. We show that reliable solutions are achieved for all thicknesses of practical interest by using high order finite elements. Moreover, the hierarchic structure allows a low cost assessment of the quality of the solution.     

On two mixed variational principles for thick laminated composite plates

The basic behavior of the shear deformation plate theories derived from the recently proposed Reissner's and Jing-Liao's functionals is studied in this paper. The difference between these two functionals is on the energy form of transverse normal stress. In Reissner's functional, the transverse normal stress has to be assumed as well as displacements and transverse shear. On the other hand, the independent variables in Jing-Liao's functional include only displacements and transverse shear. The comparison of these functionals is made through two examples, one with assumed transverse normal stress and the other without. It is found that for the cases studied in this paper Jing-Liao's functional provides a simpler, sometimes even better, alternative approach than Reissner's functional in formulating shear deformation theory of thick plates.     

Axisymmetric postbuckling of orthotropic thick annular plates

Summary This study deals with the geometrically nonlinear postbuckling response of cylindrically orthotropic thick annular plates subjected to inplane radial compressive load at the outer edge. Clamped and simply supported plates with a free hole and a plugged hole have been considered. Governing differential equations are expressed in terms of transverse displacementw, shear rotation and stress function . Polynomial expansions are employed for these three field variables and the orthogonal point collocation method has been used to obtain the discretised equations. Buckling loads have been determined by solving a linear eigen-value problem using the method of inverse iterations. Postbuckling loads for different central deflections have been obtained by solving the nonlinear differential equations. The effect of thickness ratio, orthotropic parameter, annular ratio and the edge conditions has been investigated. The effect of shear deformation on the buckling loads increases with the orthotropic parameter and the thickness-to-radius ratio.Notations a, b, h Outer and inner radii and thickness of the plate - E ,E ,v ,v Young's moduli, Poisson's ratios - Orthotropic parameter:E /E _r - G _rz /E _r - u ^*,u ₀ ^* Radial displacement, radial displacement at mid plane - r, , z Cylindrical polar coordinates - , ^* Strains and stresses - w ^*,w ,w Transverse displacement, shear rotation and stress function - w, , Dimensionless deflection, shear rotation and stress function - N _r ,N ,Q _r Inplane stress resultants and transverse shear - M _r ,M Moment resultants - k ² Shear correction factor - , (r–b)/(a–b), b/(a–b) - a/h - _r ^b , ^m Radial bending stress, circumferential membrane stress - N, _i Number and radii of collocation points - Eigenvalue for buckling With 6 Figures     

Vibrations of cracked rectangular FGM thick plates

Accurate first-of-its-kind solutions of the free vibration characteristics of side-cracked rectangular functionally graded material (FGM) thick plates are reported. From a brief review summary of available shear deformable plate theories, the well-established Reddy third-order plate theory apropos to cracked FGM thick plates is utilized. A novel Ritz procedure is developed incorporating special admissible functions – appropriately named in this study as crack functions – that properly account for the stress singularity behaviors in the neighborhood of a crack tip, and that properly account for the discontinuities of displacements and slops across a crack. Material properties of the FGM plates are assumed to vary continuously in the thickness direction according to the Mori–Tanaka scheme or a simple power law. The proposed special admissible functions accelerate the convergence of the extensive non-dimensional frequency solutions summarized. The first known non-dimensional frequencies of simply-supported and cantilevered cracked aluminum (Al) and ceramic (zirconia (ZrO₂)) or alumina (Al₂O₃) FGM thick plates of moderate thickness ratio (side-length to plate thickness, b/h = 10) are accurately determined. The effects of the volume fraction in the modeling of material distribution in the thickness direction and of cracks with different lengths, locations and orientations on the non-dimensional frequencies are investigated.     

Large deflection behaviour of thick composite plates

A rectangular quadratic finite element which includes large displacements, strains and rotations is developed in this paper. The effect of extra quadratic terms in the strain—displacement relationships on the large deflection of angle-ply plates is examined first. It is further shown that the fibre orientation and the boundary condition have a predominant effect on the nonlinear deflection of a unidirectional laminate. Basic work relating to the large deflection behaviour of thin and think antisymmetric and unsymmetric layered plates is also reported. Results are presented for simply supported and clamped boundary conditions.     

Crack propagation in PCBA-PMMA sandwich plates

The crack propagation in PCBA (Lexan)-PMMA (Plexiglas) sandwich plates has been studied by means of the high-speed photography together with a dynamic caustics method. Various phenomena were observed in these experiments: the time lag between the two cracks propagating into the two phases of the sandwich, the time coincidence of the two propagating cracks and phenomena of acceleration, deceleration and bifurcation of the propagating cracks. More precisely, the initial crack begins to propagate first into the brittle (PMMA) phase while a second crack begins to propagate later into the ductile (PCBA) phase of the sandwich plate. The time lag and the time coincidence depend on the nature and the degree of compatibility of the two phases of the sandwich plate.     

A hybrid stiffness matrix for orthotropic sandwich plates with thick faces

A stiffness matrix is derived for a rectangular orthotropic sandwich plate element with thick faces by the assumed stress hybrid approach. It will be demonstrated that an independent polynomial can be used for the internal forces of the sandwich plate by the membrane theory (assumption of thin faces) and also for the additional moments and shear forces of the faces. The boundary displacements are approached solutions of the homogeneous differential equation of a sandwich beam.     

The boundary element method for thick plates on a Winkler foundation

In this paper the application of the boundary element method to thick plates resting on a Winkler foundation is presented. The Reissner plate bending theory is used to model the plate behaviour. The Winkler foundation model is represented by continuous springs which are directly incorporated into the governing differential equation. The fundamental solutions are constructed using operator decoupling technique. These fundamental solutions represent three different cases depending on the problem constants. The explicit forms of the boundary and internal point kernels are given in all cases. Quadratic isoparametric boundary elements are used to model the plate boundary. Several examples are presented to demonstrate the accuracy of the present formulation. © 1998 John Wiley & Sons, Ltd.     

A Serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates

A hybrid-stress element is developed for the analysis of thin and moderately thick plates. The independent transverse displacement and rotations are interpolated by the 12-node cubic Serendipity shape functions. All components of stress are included and 36β stress assumption is used. The element stiffness possesses correct rank and numerical results indicate that the element does not lock in the thin-plate limit. Results obtained using the present element are compared with those obtained using a 12-node assumed-displacement based Mindlin plate element with reduced integration; the present hybrid-stress element is shown to yield superior accuracy for all cases considered. In addition, the accuracy of the present element is compared against that of analogous 4-node and 8-node hybrid-stress Mindlin plate elements.     

Crack width evaluation in FRP reinforced concrete members

This study proposes a model for calculating the average width of cracks in reinforced concrete (RC) elements using non metallic reinforcement bars (FRP). This model has already been applied to the case of steel reinforcing bars successfully. It considers the influence of the percentage of reinforcement and the concrete strength. It emphasizes how the mechanical features of FRP (fiber reinforced polymer) bars, and particularly their modulus of elasticity, can affect the crack width. The model is validated with experimental results available in the literature. An example of its application for the calculation of the crack width is shown.

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Résumé  Cet article présente l'application d'un modèle de prévision de l'amplitude des fissures en présence d'armatures non métalliques et de classes de résistance du béton allant de 30 à 80 MPa. L'analyse théorique est utilisée pour interpréter les résultats d'essais expérimentaux sur des traverses en béton armé contenant des polymères renforcés de fibres (FRP) soumises à des forces de traction. Le but de cette recherche était d'étudier et de prévoir la largeur des fissures, en utilisant comme paramètres de base la contrainte dans l'acier et la distance moyenne entre les fissures. La recherche décrit aussi l'amplitude des fissures, pour la même traverse, en présence d'armatures en acier et en polymères renforcés de fibres (FRP).

     

Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

Lower bound static approach for the yield design of thick plates

The present work addresses the lower bound limit analysis (or yield design) of thick plates under shear‐bending interaction. Equilibrium finite elements are used to discretize the bending moment and the shear force fields. Different strength criteria, formulated in the five‐dimensional space of bending moment and shear force, are considered, one of them taking into account the interaction between bending and shear resistances. The criteria are chosen to be sufficiently simple so that the resulting optimization problem can be formulated as a second‐order cone programming problem (SOCP), which is solved by the dedicated solver MOSEK . The efficiency of the proposed finite element is illustrated by means of numerical examples on different plate geometries, for which the thin plate solutions as well as the pure shear solutions are accurately obtained as two different limit cases of the plate slenderness ratio. In particular, the proposed element exhibits a good behavior in the thin plate limit. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of thick plates by radial basis functions

In this paper, we use a higher-order shear deformation theory and a radial basis function collocation technique for predicting the static deformations and free vibration behavior of thick plates. Through numerical experiments, the capability and efficiency of this collocation technique for static and vibration problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.     

The vibration characteristics of thick rectangular multilayered plates

We develop the analysis and numerical calculations for the exact free vibration characteristics of simply-supported, rectangular, thick, multilayered composite plates. We assume that each layer of the composite plate is of arbitrary thickness, is perfectly bonded to adjacent layers, possesses up to orthotropic material symmetry and that its material crystallographic axes are oriented either parallel or perpendicular to the plate's boundaries. Exact formal solutions are obtained for the individual layers which are, in turn, used to relate the field variables at the upper and lower layer surfaces. The solution is carried through by the successive application of appropriate interfacial continuity conditions between adjacent lamina. We facilitate our analysis and its numerical implementation by making use of the matrix transfer technique, thereby rendering the order of the resulting eigenproblem independent of the number of lamina. Numerical results are given in the form of scaled natural frequency versus nondimensionalized mode numbers. The effect of the plate's microstructure on its vibrational characteristics is investigated by examining changes in natural frequency and modal response for selected laminate lay-ups and material combinations.