Analytical solutions for single- and multi-span functionally graded plates in cylindrical bending

A recently developed plate theory using the concept of shape function of the transverse coordinate parameter is extended to determine the stress distribution in an orthotropic functionally graded plate subjected to cylindrical bending. The transfer matrix method is presented to derive the shape function. The equations governing the plate deformation are then solved analytically using the transfer matrix method for arbitrary boundary conditions. For a simply supported functionally graded plate, a comparison of the present solution with the exact elasticity solution, the first- and third-order shear deformation plate theories is presented and discussed. It is demonstrated that the present method yields more accurate stresses than the first- and third-order shear deformation theories. The effect of boundary conditions and inhomogeneity of material on the displacements and stresses in functionally graded plates are investigated. A multi-span functionally graded plate with arbitrary boundary conditions is further considered to demonstrate the efficiency of the present method.     

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.     

Representation of the solution of refined bending theory for isotropic plates

A solution of the bending problem for isotropic plates in a refined statement based on the system of six-order differential equations is proposed. A procedure for determining the general solutions of the corresponding biharmonic and metaharmonic equations is suggested. A method for satisfying the boundary conditions is given. The results of numerical studies of the stress state of an infinite plate with an elliptic cavity are given.     

功能梯度板弯曲和自由振动分析的简单精化板理论

论文提出了一种可用于分析功能梯度板弯曲和自由振动行为的简单精化板理论.该理论分析功能梯度板的弯曲时只需三个未知量,而分析功能梯度板的自由振动时只需一个未知量.与包含三个未知量的经典板理论相比,论文提出的简单精化板理论考虑了横向剪切效应,提高了计算准确度.与一阶剪切变形板理论不同,该简单精化板理论引入了多项式型剪切应变函数,满足板上下表面剪切应力为零的边界条件,因此不需要剪切修正.通过与已有文献的比较,验证了该简单精化板理论的准确性和便捷性,并基于该简单精化板理论研究了功能梯度板的弯曲和自由振动力学行为.     

A quadrilateral element based on refined global-local higher-order theory for coupling bending and extension thermo-elastic multilayered plates

In this paper a refined higher-order global-local theory is presented to analyze the laminated plates coupled bending and extension under thermo-mechanical loading. The in-plane displacement fields are composed of a third-order polynomial of global coordinate z in the thickness direction and 1,2–3 order power series of local coordinate ζ_k in the thickness direction of each layer, which is identical to the 1,2–3 global-local higher-order theory by Li and Liu [Li, X.Y., Liu, D., 1997. Generalized laminate theories based on double superposition hypothesis. Int. J. Numer. Methods Eng. 40, 1197–1212] Moreover, a second-order polynomial of global coordinate z in the thickness direction is chosen as transverse displacement field. The transverse shear stresses can satisfy continuity at interfaces, and the number of unknowns does not depend on the layer numbers of the laminate.Based on this theory, a quadrilateral laminated plate element satisfying the requirement of C¹ continuity is presented. By solving both bending and thermal expansion problems of laminates, it can be found that the present refined theory is very accurate and obviously superior to the existing 1,2–3 global-local higher-order theory. The most attractive feature of this theory is that the transverse shear stresses can be accurately predicted from direct use of constitutive equations without any post-processing method. It is also shown that the present quadrilateral element possesses higher accuracy.     

A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates

In the present paper, a new sinusoidal higher-order plate theory is developed for bending of exponential graded plates. The effects due to transverse shear and normal deformations are both included. The number of unknown functions involved in the present theory is only five as against six or more in case of other shear and normal deformation theories. The theory accounts for sinusoidal distribution of the transverse shear strains, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. Based on the sinusoidal shear and normal deformation theory, the position of neutral surface is determined and the governing equilibrium equations based on neutral surface are derived. There is no stretching–bending coupling effect in the neutral surface-based formulation, and consequently, the governing equations of functionally graded plates based on neutral surface have the simple forms as those of isotropic plates. Numerical results of present theory are compared with three-dimensional elasticity solutions and other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and efficient in predicting the bending response of exponential graded plates.     

BENDING RESPONSES OF AN EXPONENTIALLY GRADED SIMPLY-SUPPORTED ELASTIC/VISCOELASTIC/ELASTIC SANDWICH PLATE

The bending response for exponentially graded composite (EGC) sandwich plates is investigated.The three-layer elastic/viscoelastic/elastic sandwich plate is studied by using the sinusoidal shear deformation plate theory as well as other familiar theories.Four types of sandwich plates are considered taking into account the symmetry of the plate and the thickness of each layer.The effective moduli and Illyushin’s approximation methods are used to solve the equations governing the bending of simply-supported EGC fiber-reinforced viscoelastic sandwich plates.Then numerical results for deflections and stresses are presented and the effects due to time parameter,aspect ratio,side-to-thickness ratio and constitutive parameter are investigated.     

Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions

In this paper, an exact solution to the governing equations of the bending of a variable-thickness inhomogeneous rectangular plate is presented. The procedure is applicable to variable-thickness inhomogeneous rectangular plates with two opposite edges simply supported. The remaining ones subjected to a combination of clamped, simply supported, and free boundary conditions and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for variable-thickness inhomogeneous orthotropic plates as for uniform-thickness homogeneous isotropic plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of the inhomogeneity, aspect ratio, thickness parameter and degree of non-uniformity on the deflections and stresses are investigated. This paper is partially supported by the Deanship of Scientific Research at King AbdulAziz University (Grant no. 172/427).     

An efficient and simple refined theory for nonlinear bending analysis of functionally graded sandwich plates

In this paper, an efficient and simple refined theory is presented for nonlinear bending analysis of functionally graded sandwich plates. The theory presented is variationally consistent, does not require the shear correction factor, and gives rise to transverse shear stress variations such that the transverse shear stresses vary parabolically across the plate thickness, satisfying shear-stress-free surface conditions. The energy concept along with the present theory and the first- and third-order shear deformation theories is used to predict the large deflection and the stress distribution across the thickness of functionally graded sandwich plates.     

Elasticity solutions for functionally graded plates in cylindrical bending

The plate theory of functionally graded materials suggested by Mian and Spencer is extended to analyze the cylindrical bending problem of a functionally graded rectangular plate subject to uniform load. The expansion formula for displacements is adopted. While keeping the assumption that the material parameters can vary along the thickness direction in an arbitrary fashion, this paper considers orthotropic materials rather than isotropic materials. In addition, the traction-free condition on the top surface is replaced with the condition of uniform load applied on the top surface. The plate theory for the particular case Of cylindrical bending is presented by considering an infinite extent in the y-direction. Effects of boundary conditions and material inhomogeneity on the static response of functionally graded plates are investigated through a numerical example.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

Winkler地基上变厚度圆板的轴对称弯曲

本文提出了Winkler地基上变厚度圆板轴对称弯曲的传递矩阵算法。首先,根据贝塞尔函数理论获得了等厚度圆板和环板单元在任意荷载作用下轴对称弯曲的解析解,这些解均由通解和特解两部分组成。基于这些解析解,导出了等厚度圆板和环板单元的传递矩阵。然后沿径向将变厚度圆板划分成一个等厚度圆板单元和一系列等厚度环板单元,应用传递矩阵算法原理获得了变厚度圆板的整体传递矩阵。引入圆板的边界条件,给出了该板每条节线上的挠度、径向转角、径向弯矩和径向剪力。最后,讨论了受均布荷载作用的简支线性变厚度圆板的弯曲,将本文数值解与解析解进行比较,证实了本文方法的有效性,并简要地讨论了地基参数对板挠度和径向弯矩的影响。     

A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses

A two-dimensional solution is presented for bending analysis of simply supported functionally graded ceramic–metal sandwich plates. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity and Poisson’s ratio of the faces are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. We derive field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Numerical results of the sinusoidal, third-order, first-order and classical theories are presented to show the effect of material distribution on the deflections and stresses.     

混合边界各向异性矩形板的静力分析

采用一般解析解和配点法相结合的方法,求解混合边界各向异性矩形板的弯曲问题.先由弯曲挠度的微分方程求出各种类型的齐次解和特解,然后组成一般解析解,再将板的每个边等分 为很多微小的段,仅对每一微段的中点建立应满足的边界条件,由全部边界条件方程式即可求得全部积分常数.以每边一半边界为平夹、另一半边界为简支或自由的方板为例进行了计算,并与四边均为简支的方板进行了对比,表明理论简单,结果实用.     

Stroh-like formalism for the coupled stretching–bending analysis of composite laminates

Due to the two-dimensional nature of thin plates, the lamination theory considering the composite laminates with in-plane and plate bending problems coupling each other is treated in this paper by using complex variable formulation. By following the steps of Stroh formalism for two-dimensional linear anisotropic elasticity, a displacement complex variable formalism developed by the other researchers was introduced and re-derived in a different but more Stroh-like way. In addition, a brand-new mixed formalism (mixed use of displacements and stresses as basic functions) is established to compensate the displacement formalism. In order to transfer all the related formulae and mathematical techniques of the Stroh formalism to these two formalisms, the general solutions for the basic equations of lamination theory and their associated eigenrelations have been purposely arranged in the form of Stroh formalism. Moreover, by using the presently developed mixed formalism, the explicit expressions for the fundamental matrix and eigenvectors are obtained first time for the most general composite laminates. Furthermore, letting the coupling stiffness vanish, the formalism has been reduced to the case of symmetric laminates and checked by a recently developed Stroh-like formalism for the plate bending problems. The comparison between Stroh formalism for two-dimensional problem, Stroh-like formalism for plate bending problem, displacement formalism and mixed formalism is then made at the end of this paper, and through their connection some useful relations are obtained.     

Efficient higher-order shear deformation theories for bending and free vibration analyses of functionally graded plates

In this paper, various efficient higher-order shear deformation theories are presented for bending and free vibration analyses of functionally graded plates. The displacement fields of the present theories are chosen based on cubic, sinusoidal, hyperbolic, and exponential variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theories is reduced and hence makes them simple to use. Equations of motion are derived from Hamilton’s principle. Analytical solutions for deflections, stresses, and frequencies are obtained for simply supported rectangular plates. The accuracy of the present theories is verified by comparing the obtained results with the exact three-dimensional (3D) and quasi-3D solutions and those predicted by higher-order shear deformation theories. Numerical results show that all present theories can archive accuracy comparable to the existing higher-order shear deformation theories that contain more number of unknowns.     

环扇形薄板弯曲问题的环向辛对偶求解方法

根据平面弹性与薄板弯曲问题的相似性原理,极坐标系板弯曲的弯矩函数被引入作为原变量,并通过恰当的辛内积定义建立了环扇形薄板弯曲问题的一个辛几何空间. 然后应用类Hellinger-Reissner变分原理,导出了辛几何空间的对偶方程,从而将环扇形薄板弯曲问题导入到辛对偶求解体系. 于是,分离变量和本征展开的有效数学物理方法得以实施,给出环扇形薄板弯曲问题的一个分析求解方法. 具体讨论了两弧边简支和两弧边一边固支一边自由薄板的本征问题,分别导出它们对应的本征值超越方程和本征向量,并给出原问题本征展开形式的通解. 最后,给出了两个算例的分析解并与已有文献或数值方法的解进行了对比,结果表明该方法有很好的收敛性和精度.      

On the simple and mixed first-order theories for functionally graded plates resting on elastic foundations

In this article, the bending response of a functionally graded plate resting on elastic foundations and subjected to a transverse mechanical load is investigated. An accurate solution for the functionally graded plate with simply supported edges resting on elastic foundations is presented. The interaction between the plate and the elastic foundations is considered and included in the equilibrium equations. Pasternak’s model is used to describe the two-parameter elastic foundations, and get a special case of Winkler’s model by considering one-parameter of elastic foundation. A relationship between the simple and mixed first-order transverse shear deformation theories is presented. Numerical results for deflections and stresses of functionally graded plates are investigated. Comparisons between the results of the simple and mixed first-order theories are made, and appropriate conclusion is formulated. Additional boundary conditions at the edges of the present plates are investigated.     

On the Bending, Vibration and Stability of Laminated Rectangular Plates with Transversely Isotropic Layers

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Exact solutions for magneto-electro-elastic laminates in cylindrical bending

Analytical solutions are derived for the cylindrical bending of multilayered, linear, and anisotropic magneto-electro-elastic plates under simple-supported edge conditions. We construct the general solution in terms of a simple formalism for any homogeneous layer, from which any physical quantities can be solved for the given boundary conditions. For multilayered plates, we derive the solution in terms of the propagator matrices. A special feature of cylindrical bending, which distinguishes itself from the three-dimensional plate problem, is that the associated eigenvalues for any homogeneous layer are independent of the sinusoidal mode, and thus need to be solved only once. Typical numerical examples are also presented for a piezomagnetic plate, a two-layered piezoelectric/piezomagnetic plate, and a four layered piezoelectric/piezomagnetic plate, with different span-to-thickness ratios. In particular, the piezoelectric and piezomagnetic fields show certain interesting features, which give guidance on the development of piezoelectric/piezomagnetic thin-plate theories. Furthermore, it is shown that the variations of the elastic, electric, and magnetic quantities with thickness depend strongly upon the material property and layering, which could be useful in the analysis and design of smart composite structures with sensors/actuators.