Bending and free vibration analysis of isotropic and multilayered plates and shells by using a new accurate higher-order shear deformation theory

Bending and free vibration analysis of multilayered plates and shells by using a new accurate higher order shear deformation theory (HSDT) is presented. It is one of the most accurate HSDT available in the literature, mainly because new non-polynomial shear strain shape functions (combination of exponential and trigonometric) used in the present theory are richer than polynomial functions, and free surface boundary conditions can be guaranteed a priori. The present HSDT is able to reproduce Touratier’s HSDT as special case. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are then solved via Navier-type, closed form solutions. Bending and dynamic results are presented for cylindrical and spherical shells and plates for simply supported boundary conditions. Panels are subjected to sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. The present results are compared with the exact three-dimensional elasticity theory and with several other well-known HSDT theories. The present HSDT is found to be more precise than other several existing ones for analyzing the bending and free vibration of isotropic and multilayered composite shell and plate structures.     

Bending and free vibration analysis of isotropic and multilayered plates and shells by using a new accurate higher-order shear deformation theory

Bending and free vibration analysis of multilayered plates and shells by using a new accurate higher order shear deformation theory (HSDT) is presented. It is one of the most accurate HSDT available in the literature, mainly because new non-polynomial shear strain shape functions (combination of exponential and trigonometric) used in the present theory are richer than polynomial functions, and free surface boundary conditions can be guaranteed a priori. The present HSDT is able to reproduce Touratier’s HSDT as special case. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are then solved via Navier-type, closed form solutions. Bending and dynamic results are presented for cylindrical and spherical shells and plates for simply supported boundary conditions. Panels are subjected to sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. The present results are compared with the exact three-dimensional elasticity theory and with several other well-known HSDT theories. The present HSDT is found to be more precise than other several existing ones for analyzing the bending and free vibration of isotropic and multilayered composite shell and plate structures.     

Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory

This paper introduces a generalized 5 degrees of freedom (DOF) higher-order shear deformation theory (HSDT) to study the bending and free vibration of plates and shells, which may be used to create other HSDTs. It also introduces a new HSDT for shells that is more accurate than many available HSDTs despite having the same 5DOF, and which is also able to reproduce the well-known Soldatos’ HSDT as special case. The governing equations and boundary conditions of the generalized formulation are derived by employing the principle of virtual work. These equations are solved via Navier-type closed-form solutions. Static and dynamic results are presented for plates and cylindrical and spherical shells with simply supported boundary conditions. Panels are subjected to sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. Results from the new and well-known HSDTs introduced and reproduced based on the present generalized 5DOF HSDT are compared with the exact three-dimensional elasticity solution. The present new HSDT for plates and shells is found to be more accurate than the well-known HSDTs developed by other authors, for analyzing the static and free vibration of isotropic and multilayered composite plates and shells.     

Linearized buckling analysis of isotropic and composite beam-columns by Carrera Unified Formulation and dynamic stiffness method

Abstract

This article introduces a one-dimensional (1D) higher-order exact formulation for linearized buckling analysis of beam-columns. The Carrera Unified Formulation (CUF) is utilized and the displacement field is expressed as a generic N-order expansion of the generalized unknown displacement field. The principle of virtual displacements is invoked along with CUF to derive the governing equations and the associated natural boundary conditions in terms of fundamental nuclei, which can be systematically expanded according to N by exploiting an extensive index notation. After the closed form solution of the N-order beam-column element is sought, an exact dynamic stiffness (DS) matrix is derived by relating the amplitudes of the loads to those of the responses. The global DS matrix is finally processed through the application of the Wittrick-Williams algorithm to extract the buckling loads of the structure. Isotropic solid and thin-walled cross-section beams as well as laminated composite structures are analyzed in this article. The validity of the formulation and its broad range of applicability are demonstrated through comparisons of results from the literature and by using commercial finite element codes.     

非对称Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀Bernoulli-Euler薄壁梁单元自由振动的控制运动微分方程,推导了其精确的动态传递矩阵。采用Bernoulli-Euler弯扭耦合梁理论,假定梁横截面没有任何对称性,考虑了薄壁梁在两个方向的弯曲振动及翘曲刚度的影响。动态传递矩阵可以用于计算非对称薄壁梁及其集合体的精确固有频率和模态形状。针对具体的算例,给出了各种边界条件下固有频率的数值结果并与文献中已有的结果进行了比较,还讨论了翘曲刚度对固有频率和模态形状的影响,结果表明如果忽略翘曲刚度的影响,可能得到毫无意义的结果。     

In‐plane vibrations of shear deformable curved beams

This paper presents the exact dynamic stiffness matrix for a circular beam with a uniform cross‐section. The stiffness matrix is frequency dependent, and the natural frequencies are those that cause the matrix to become singular. Using this matrix the exact natural frequencies of circular beams with various boundary conditions are calculated and compared with available results in the literature. Copyright © 2001 John Wiley & Sons, Ltd.     

Dynamic stiffness analysis of laminated beams using a first order shear deformation theory

In this paper the exact vibration frequencies of generally laminated beams are found using a new method, including the effect of rotary inertia and shear deformations. The effect of shear in laminated beams is more significant than in homogenous beams, due to the fact that the ratio of extensional stiffness to the transverse shear stiffness is high. The exact dynamic stiffness matrix is derived, and then any set of boundary conditions including elastic connections, and assembly of members, can be solved as in the classical direct stiffness method for framed structures. The natural frequencies of vibration of a structure are those values of frequency that cause the dynamic stiffness matrix to become singular, and one can find as many frequencies as needed from the same matrix. In the paper several examples are given, and compared with results from the literature.     

Free vibration analysis of sandwich beams using improved dynamic stiffness method

In this paper, free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness and finite element methods. To determine the governing equations of motion by the present theory, the core density has been taken into consideration. The governing partial differential equations of motion for one element contained three layers are derived using Hamilton’s principle. This formulation leads to two partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are combined to form one ordinary differential equation. Closed form analytical solution for this equation is determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. They are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the known Wittrick–Williams algorithm. After validation of the present model, the effect of various parameters such as density, thickness and shear modulus of the core for various boundary conditions on the first natural frequency is studied.     

中厚椭球壳自由振动动力刚度法分析

介绍精确动力刚度法分析中厚椭球壳自由振动具体实施方法,据环向波数不同将中厚椭球壳自由振动分解为一系列确定环向波数的一维振动;利用控制方程Hamilton形式建立动力刚度关系,用常微分方程求解器COLSYS求解控制方程获得单元动力刚度,用Wittrick-Williams算法求得该环向波数下椭球壳自振频率。数值算例给出中厚圆球壳及椭球壳不同边界条件的自振频率,验证动力刚度法高效、可靠、精确。     

Exact stiffness matrix of mono-symmetric composite I-beams with arbitrary lamination

The exact stiffness matrix, based on the simultaneous solution of the ordinary differential equations, for the static analysis of mono-symmetric arbitrarily laminated composite I-beams is presented herein. For this, a general thin-walled composite beam theory with arbitrary lamination including torsional warping is developed by introducing Vlasov’s assumption. The equilibrium equations and force–deformation relations are derived from energy principles. The explicit expressions for displacement parameters are then derived using the displacement state vector consisting of 14 displacement parameters, and the exact stiffness matrix is determined using the force–deformation relations. In addition, the analytical solutions for symmetrically laminated composite beams with various boundary conditions are derived as a special case. Finally, a finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element results using the Hermitian beam elements and ABAQUS’s shell element.     

Finite element dynamic stability analysis of laminated composite skew plates containing cutouts based on HSDT

The finite element dynamic stability analysis of laminated composite skew structures subjected to in-plane pulsating forces is carried out based on the higher-order shear deformation theory (HSDT). The two boundaries of the instability regions are determined using the method proposed by Bolotin. The numerical results obtained for square and skew plates with or without central cutout are in good agreement with those reported by other investigators. The new results for laminated skew plate structures containing cutout in this study mainly show the effect of the interactions between the skew angle and other various parameters, for example, cutout size, the fiber angle of layer and thickness-to-length ratio. The effect of the magnitude of the periodic in-plane load on the dynamic instability index is also investigated.     

High-order free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method

In this paper, high-order free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness method. The governing partial differential equations of motion for one element are derived using Hamilton’s principle. This formulation leads to seven partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are divided into two ordinary differential equations by considering the symmetrical sandwich beam. Closed form analytical solutions of these equations are determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. The element dynamic stiffness matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by use of numerical techniques and the known Wittrick–Williams algorithm. Finally, some numerical examples are discussed using dynamic stiffness method.     

计及二阶效应的梁杆系统动力响应分析方法研究

基于动力刚度法和有限元理论提出了一种考虑二阶效应计算梁杆动力响应的新方法。通过求解轴向力作用下Bernoulli-Euler 梁横向和轴向挠度自由振动微分方程,利用位移边界条件反解出待定系数,得到了动态精确形函数;使用经典有限元方法推导了考虑截面自身旋转惯量的质量阵和考虑二阶效应的刚度阵,该质量阵和刚度阵各元素均为轴力和圆频率的超越函数;建立了杆系结构瞬态动力学分析的动力平衡方程,给出了稳定和高效的求解方案。对几个典型的算例进行了计算分析,并与通用软件ANSYS 的计算结果进行了比较。计算结果表明:该分析梁杆系统动力响应的新方法具有较高的计算精度和效率,特别是能够准确地计入轴力对于梁杆动力响应的影响。     

Dynamic analysis of partial-interaction composite beams

The state-space method is used to investigate the dynamic behavior of partial-interaction composite beams. The state-space formula is properly established via selecting the appropriate state variables. The characteristic equations of frequency and the corresponding modal shapes of free vibration under generalized boundary conditions are then obtained. Based on the state-space formula and the concept of symplectic inner product, a new route is proposed to prove the orthogonality of vibration modes of composite beams under generalized boundary conditions. By virtue of mode superposition method, we obtain the solution of the inhomogeneous equation for forced vibration and the dynamic response of the partial-interaction composite beams subjected to a uniformly distributed pulse load, a uniformly distributed step load or a moving constant force. Finally, some numerical examples are presented and compared with those available in the literature to validate the present method. The effect of the rigidity of shear connector on dynamic responses of a composite beam under moving constant force is discussed.     

Dynamic analysis of laminated composite plates subjected to a moving oscillator by FEM

This paper presents a finite element model based on the first order shear deformation theory to investigate the dynamic behavior of laminated composite plates traversed by a moving oscillator. The oscillator model is assumed to be consisting of two nodal masses that are connected by means of a spring-damper unit. The governing equations of motion of two sub-systems are separately integrated by applying the Newmark’s time integration procedure. Then, the obtained equations are coupled and the responses of system components are calculated in each time step. The accuracy of algorithm is verified by comparing the numerical results of static, free vibration and simplified moving force problems analysis with the available exact solutions and other numerical results in the literature. Also, the effects of mass ratio, damping ratio of system components, stiffness of suspension system, velocity and eccentricity of moving oscillator on dynamic responses is parametrically studied. This algorithm can be applied to various boundary conditions, lamination schemes and fiber angels.     

Shape control of non-symmetric piezolaminated composite beams

In this paper the exact deflections of generally laminated piezoelectric composite beams are found using a new method, which includes the effect of rotary inertia and shear deformations. The effect of shear in laminated beams is more significant than in homogenous beams due to the fact that the ratio of extensional stiffness to the transverse shear stiffness is high. The exact stiffness matrix is derived, and then any set of boundary conditions, including elastic connections and assembly of members, can be solved as in the classical direct stiffness method for framed structures. In this paper several examples are given, and the posibilities for shape control are investigated.     

Postbuckling and free vibrations of composite beams

An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse deformations of a composite laminated beam accounting for the midplane stretching are derived. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single nonlinear fourth-order partial–integral–differential equation governing the transverse deformations. We find out that the governing equation for the postbuckling of symmetric or asymmetric composite beams has the same form as that of beams made of an isotropic material. Composite beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are considered. A closed-form solution for the postbuckling deformation is obtained as a function of the applied axial load, which is beyond the critical buckling load. To study the vibrations that take place in the vicinity of a buckled equilibrium position, we exactly solved the linear vibration problem around the first buckled configuration. Solving the resulting eigen-value problem results in the natural frequencies and their associated mode shapes. Both the static response represented by the postbuckling analysis and the dynamic response represented by the free vibration analysis in the postbuckling domain strongly depend on the lay-up of the laminate. Variations of the beam’s midspan rise and the fundamental natural frequency of the postbuckling domain vibrations with the applied axial load are presented for a variety of lay-up laminates. The ratio of the axial stiffness to the bending stiffness was found to be a crucial parameter in the analysis. This control parameter, through the selection of the appropriate lay-up, can be manipulated to help design and optimize the static and dynamic behavior of composite beams.     

Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates

This paper presents a generalized hybrid quasi-3D shear deformation theory for the bending analysis of advanced composite plates such as functionally graded plates (FGPs). Many 6DOF hybrid shear deformation theories with stretching effect included, can be derived from the present generalized formulation. All these theories account for an adequate distribution of transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces not requiring thus a shear correction factor. The generalized governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FGP subjected to transverse load for simply supported boundary conditions. Numerical examples of the new quasi-3D HSDTs (non-polynomial, polynomial and hybrid) derived by using the present generalized formulation are compared with 3D exact solutions and with other HSDTs. Results show that some of the new HSDTs are more accurate than, for example, the well-known trigonometric HSDT, having the same 6DOF.     

轴向变速运动弯曲梁的固有频率分析

基于动力刚度矩阵法对轴向变速运动弯曲梁的固有频率进行分析,根据Hamilton原理,推导轴向变速运动弯曲梁的时域控制方程和边界条件,通过傅里叶变换得到频域控制方程和边界条件,求解频域控制方程,并结合位移边界条件和载荷边界条件,建立轴向变速运动弯曲梁的动力刚度矩阵模型;引入Hermite形式的形函数,建立了轴向变速运动弯曲梁的有限元模型。算例中,通过对比现有文献中的结果、有限元模型结果和动力刚度矩阵法模型结果,验证了该文所建立的力学模型,动力刚度矩阵法比有限元法具有更高的精度和效率,分析了轴向变速运动弯曲梁固有频率随着弯曲梁轴向运动速度、加速度、轴向受力、边界条件的变化规律。     

Refined and generalized hybrid type quasi-3D shear deformation theory for the bending analysis of functionally graded shells

The closed-form solution of a generalized hybrid type quasi-3D higher order shear deformation theory (HSDT) for the bending analysis of functionally graded shells is presented. From the generalized quasi-3D HSDT (which involves the shear strain functions “f(ζ)” and “g(ζ)” and therefore their parameters to be selected “m” and “n”, respectively), infinite six unknowns' hybrid shear deformation theories with thickness stretching effect included, can be derived and solved in a closed-from. The generalized governing equations are also “m” and “n” parameter dependent. Navier-type closed-form solution is obtained for functionally graded shells subjected to transverse load for simply supported boundary conditions. Numerical results of new optimized hybrid type quasi-3D HSDTs are compared with the first order shear deformation theory (FSDT), and other quasi-3D HSDTs. The key conclusions that emerge from the present numerical results suggest that: (a) all non-polynomial HSDTs should be optimized in order to improve the accuracy of those theories; (b) the optimization procedure in all the cases is, in general, beneficial in terms of accuracy of the non-polynomial hybrid type quasi-3D HSDT; (c) it is possible to gain accuracy by keeping the unknowns constant; (d) there is not unique quasi-3D HSDT which performs well in any particular example problems, i.e. there exists a problem dependency matter.