Vibration analysis of variable stiffness laminated composite sandwich plates

Abstract

This paper presents the free vibration analysis of a variable stiffness laminated composite sandwich plates. The fiber orientation angle of the face sheets (Skin) is assumed to vary linearly with the x-axis. The problem formulation is based on the higher-order shear deformation plate theory HDST C⁰ coupled with p-version of finite element method. The elements of the stiffness and mass matrices are calculated analytically. The sandwich plate is presented with a uniform mesh of four p-elements and the convergence properties are achieved by increasing the degree p of the hierarchical shape functions. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as plate thickness, core to face sheets thickness ratio, orientation angle of curvilinear fibers and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New comparison study of vibration response of laminated sandwich plate between the straight and curvilinear fibers is presented.     

Effect of fiber orientation on fracture toughness of laminated composite plates [0°/θ°]s

Fracture toughness of single edge notched fiber reinforced composite plates is investigated experimentally. Load–displacement curves for unidirectional carbon fiber/epoxy resin reinforced composite plates are obtained experimentally under tensile load. Fracture toughness is obtained by determining failure loads. For numerical study, ANSYS is used. Material properties of laminates are calculated with classical laminated plate theory and applied to the finite element model by using plane element. Stiffness matrix of laminates is determined and shell element is chosen for numerical solution. Critical stress intensity factors are calculated with Displacement Correlation Method under experimental failure load conditions.     

Load–displacement behavior of glass fiber/epoxy composite plates with circular cut-outs subjected to compressive load

An experimental study of the behavior of woven glass fiber/epoxy composite laminated panels under compression is presented. Compression tests were performed on to 16 fiber-glass laminated plates with and without circular cut-outs using the compressed machine. The maximum load of failure for each of the glass-fiber/epoxy laminated plates under compression has been determined experimentally. A parametric study was performed as well to investigate the effects of varying the centrally located circular cut-out sizes and fiber angle-ply orientations on to the ultimate load. The experimental work revealed that as the cut-out size increases, the maximum load of the composite plate decreases. Also, it has been observed that cross-ply laminates possess the greatest ultimate load as compared to other types of ply stacking sequences and orientations.     

Free vibration analysis of bimodulus laminated plates using higher‐order plate element

Abstract

A C° isoparametric higher‐order plate element is developed to analyze the free vibration of bimodulus laminated plates. The equations of motion for the higher‐order plate theory are also derived variationally. The natural frequencies and neutral surface locations are determined for benchmark problems. The numerical results are compared to available analytical solutions, and excellent agreement is observed. Obviously, the present formulation is more accurate than the first‐order theory.     

An experimentally validated numerical method for investigating the air blast response of basalt composite plates

ABSTRACT

Nonlinear air blast response of basalt composite plates is analysed by using a generalized differential quadrature (GDQ) method, which requires less solution time and decreases the complexity compared to finite element method. A test environment that contains a shock tube is designed and set to experiment on the transient response of blast loaded laminated plates. Experimental and numerical results show a good agreement in terms of displacement, strain, and acceleration versus time. The responses of glass/epoxy, Kevlar/epoxy, and carbon/epoxy composite plates are also investigated by using GDQ method and the results are compared with the basalt/epoxy composite plate and discussed.     

Stochastic critical stress intensity factor response of single edge notched laminated composite plate using displacement correlation method

Abstract

The second-order statistics of critical stress intensity factor (SIF) of single edge notched fiber reinforced composite plates with random system properties and subjected to uniaxial tensile loadings is investigated. This paper is an extension of reference (Lal and Kapania, 2013) by the present authors by considering more number of input random system parameters for higher accuracy. A C⁰ finite element method based on a higher-order shear deformation plate theory using displacement correlation method via isoparametric quarter point element is proposed for basic formulation. A stochastic finite element method using first-order perturbation technique and Monte Carlo simulation (MCS) is employed to examine the mean, coefficient of variance, and probability density faction of critical first mode SIF. The effect of different fiber orientations, crack length, plate thickness, a number of layers, and the lamination schemes with random system properties on the statistics of SIF of single edge crack laminated composite plate is evaluated. The tensile failure load is predicted using Hashin’s failure criteria. The present approach is validated with results available in literature and by employing independent MCS.     

采用分层理论计算层合板的固有频率和振型

采用有限元方法并结合分层理论对复合材料层合板的固有频率和振型进行理论计算,再用实验验证。通过分层有限元模型求解层合板的位移模式,对层合板固有频率进行计算。分析有限元网格数,铺设角度、铺设层数等对固有频率的影响,获得层合板自由振动的前九阶振型,采用实验进行了验证。结果表明,提出的方法可较精确计算层合板的固有频率和振型。     

Free vibration analysis of continuous grading fiber reinforced plates on elastic foundation

The free vibration characteristics of rectangular continuous grading fiber reinforced (CGFR) plates resting on elastic foundations have been studied, based on the three-dimensional, linear and small strain elasticity theory. The foundation is described by the Pasternak or two-parameter model. The CGFR plate is simply supported at the edges and is assumed to have an arbitrary variation of fiber volume fraction in the thickness direction. Suitable displacement functions that identically satisfy the simply supported boundary conditions are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method (DQM) to obtain natural frequencies. Convergence studies have been performed on CGFR plates on the elastic foundations. It is shown that the present method has a rapid convergent rate, stable numerical operation and very high accuracy. Besides results for CGFR plate with arbitrary variation of fiber volume fraction in the plate’s thickness are compared with discrete laminated composite plate. The main contribution of this work is to present useful results for continuous grading of fiber reinforcement in the thickness direction of a plate on elastic foundation and comparison with similar discrete laminate composite plate. Results indicate the advantages of using CGFR plate with graded fiber volume fractions over traditional discretely laminated plates.     

A partial hybrid stress model for the refined C1 higher‐order plate theory

Abstract

The partial hybrid stress model is applied to the refined C ¹ higher‐order plate theory in this paper. The displacement model is adopted in the flexural part and the hybrid stress model in the transverse shear part. The plate concept is introduced and the governing equations of plate are derived variationally from the modified Hellinger‐Reissner principle. This new plate element is demonstrated to be more accurate than displacement formulation in the analysis of orthotropic thick laminated plates. Moreover, the through thickness distribution of transverse shear stress is precisely predicted.     

Geometrically nonlinear NURBS isogeometric finite element analysis of laminated composite plates

This research present the development of geometrically nonlinear NURBS isogeometric finite element analysis of laminated composite plates. First-order, shear-deformable laminate composite plate theory is utilized in deriving the governing equations using a variational formulation. Geometric nonlinearity is accounted for in Von-Karman sense. A family of NURBS elements are constructed from refinement processes and validated using various examples. k-refined NURBS elements are developed to study thin plates. Isotropic, orthotropic and laminated composite plates are studied for various boundary conditions, length to thickness ratios and ply-angles. Computed center deflection is found to be in an excellent agreement with the literature. For thin plate analysis, linear and k-refined quadratic NURBS element is found to remedy the shear locking problem. k-refined quadratic NURBS element provide stabilized response to distorted, coarse meshes without increasing the order of the polynomial, owing to the increased smoothness of solution space.     

An n-sided polygonal finite element for nonlocal nonlinear analysis of plates and laminates

In this study, a locking-free n-sided C¹ polygonal finite element is presented for nonlinear analysis of laminated plates. The plate kinematics is based on Reddy's third-order shear deformation theory (TSDT). The in-plane displacements are approximated using barycentric form of Lagrange shape functions. The weak-form Galerkin formulation based on the kinematics of TSDT requires the C¹ approximation of the transverse displacement over the polygonal element. This is achieved by embedding the C⁰ Lagrange interpolants over a cubic Bernstein-Bezier patch defined over the n-sided polygonal element. Such an approach ensures the continuity of the derivative field at the inter-element edges. In addition, Eringen's stress-gradient nonlocal constitutive equations are used in the present formulation to account for nonlocality. The effect of geometric nonlinearity is taken by considering the von Kármán geometric nonlinearity. Examples are presented to show the effect of nonlocality, geometric nonlinearity, and the lamination scheme on the bending behavior of laminated composite plates. The results are compared with analytical solutions, conventional FEM results, and with those available in the literature. Shear locking is addressed considering reduced integration and consistent interpolation techniques. The patch test is used to check the convergence of the element developed.     

Large reference displacement analysis of composite plates. Part II: Computer implementation

This investigation concerns itself with the computer implementation of the dynamic formulation of thin laminated composite plates consisting of layers of orthotropic laminae that undergo large arbitrary rigid body displacements and small elastic deformations. A finite element preprocessor computer program is developed to automatically generate the invariants of the laminae, which may have arbitrary orientations. The laminae invariants are then used to obtain the invariants of the elements and the composite laminated plate. The consistent and lumped mass formulations of the invariants of motion of composite plates are compared and it is concluded that the two methods are comparable, if a fine enough finite element mesh is used. The structure of the dynamic equations of motion, based on the formulation presented in Part I of this paper, is examined. Non-linear centrifugal and Coriolis forces arising as the result of the finite rotations of the laminae are defined, and the solution schemes of the resulting non-linear differential equations of motion are discussed. Numerical examples illustrating the differences between homogeneous isotropic and laminated composite plates are presented. An RSSR (Revolute-Spherical-Spherical-Revolute) mechanism is used in the numerical examples, with the coupler modelled as a laminated plate flexible body. It is found that the inertia of the plate contributed greatly to the transverse deformation. The effects of laminae orientation is also investigated.     

Elastic stability of skew laminated composite plates subjected to biaxial follower forces

The present study investigates the elastic stability of skew laminated composite plates subjected to biaxial inplane follower forces by the finite element method. The plate is assumed to follow first-order shear deformation plate theory (FSDPT). The kinetic and strain energies of skew laminated composite plate and the work done by the biaxial inplane follower forces are derived by using tensor theory. Then, by Hamilton's principle, the dynamic mathematical model to describe the free vibration of this problem is formed. The finite element method and the isoparametric element are utilized to discretize the continuous system and to obtain the characteristic equations of the present problem. Finally, natural vibration frequencies, buckling loads (also the instability types) and their corresponding mode shapes are found by solving the characteristic equations. Numerical results are presented to demonstrate the effects of those parameters, such as various inplane force combinations, skew angle and lamination scheme, on the elastic stability of skew laminated composite plates subjected to biaxial inplane follower forces.     

Transient analysis and control of delaminated composite plates in hygrothermal environment using active fiber composite actuator

Abstract

In the present article, the transient analysis and control of delaminated composite plates under hazardous environmental conditions using active fiber composite (AFC) is discussed. Top and bottom layers of the laminated composite plate are embedded AFC layers. The present investigation utilizes AFC as an actuator and sensor. A finite element model for centrally located delamination is developed and coded in Matlab. The proportional controller is used to control the undesirable response in real time. The transient response of the smart delaminated plate is studied for different temperatures and moisture conditions. The feedback control of the dynamic response is performed with the help of velocity and displacement feedback gain to the AFC actuator. The key observations from the numerical studies are; the dynamic response and the frequency response of composite plate increase due to delamination and also with the increase of the temperature and moisture concentrations. The response reduces when the feedback control loop is activated. So, the overall performance of the delaminated plate structure in hygrothermal environment may be enhanced.     

层合板频率响应与声功率双目标优化设计

利用遗传算法对复合材料层合板结构的固有频率间隔和辐射声功率进行双目标优化设计。利用分层理论结合有限元模型求解层合板的固有频率和振速分布。通过声辐射模态理论,计算层合板结构辐射声功率。以铺设角度作为设计变量,第一阶与第二阶固有频率间隔和辐射声功率作为双目标优化目标函数,以某4层的层合板结构为例,采用目标加权法优化目标函数。研究了不同权重系数、不同频率时固有频率间隔最大化和声功率最小化对应的优化铺设角度。由数值分析结果可知:不同的权重系数比下获得的Pareto最优解不同;在同一权重系数下,两个优化目标所起的作用不尽相同;随着频率的增加,选择相对较大的权重系数可使Pareto最优解较好地兼顾两个优化目标。     

A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells

 The formulation of a nonlinear composite shell element is presented for the solution of stability problems of composite plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thin and thick laminated plates and shells by incorporating bending moment and transverse shear resultant forces. The composite element is free of both membrane and shear locking behaviour by using the assumed natural strain method such that the element performs very well as thin shells. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The proposed formulation is computationally efficient and the test results showed good agreement. In addition the effect of the viscoelastic material is investigated on the postbuckling behaviour of laminated composite shells. Received: 6 February 2002 / Accecpted: 6 January 2003 ID=" Present address: School of Civil Engineering, Asian Institute of Technology     

Finite element analysis of laminated composite stiffened plates using FSDT and HSDT: A comparative perspective

Significance of using higher-order shear deformation theory (HSDT) over the first-order shear deformation theory (FSDT) for analyzing laminated composite stiffened plates is brought out using the finite element method (FEM). For this purpose, a C⁰ HSDT, is extended for application to stiffened configurations, for linearly elastic static and natural vibration analysis. The spatial displacement fields of both the plate and the stiffener are derived as functions of reference plane variables using Taylor series expansion. The developed computational tool is employed for analyzing systems having varying configurations using the FSDT and two different HSDTs, and their comparative effects are systematically studied, demonstrating the need for using HSDT instead of FSDT, for obtaining accurate structural response of such stiffened configurations.     

The Displacement Perspective During Ultimate Failure of Composite Laminates

This paper deals with the studies on the state of displacement of symmetric and anti-symmetric angle-ply and cross-ply laminated composite plates during its ultimate failure, subjected to transverse static load. First-order shear deformation theory (FSDT) is employed in conjunction with the finite element approach using eight-noded quadratic isoparametric element. The free vibration analyses of isotropic and laminated composite plates are carried out to ensure the overall validity of the present finite element formulation. The mid surface of the laminate is considered as the reference plane. The principal material directions in different laminae are oriented to produce a laminated structural element capable of resisting loads in several directions. The stiffness of a composite laminate is obtained from the properties of the constituent laminae. The affected stiffness of the failed lamina is discarded completely after the failure of weakest ply. The rigidity matrix of the laminate with remaining laminae is re-established. The re-evaluation process continues until the laminate fails completely. To investigate the displacement behaviour of laminates during the ultimate failure, parametric studies are carried out for different cases by varying the stacking sequences, fiber orientations, layer thicknesses, aspect ratios and the number of layers in the laminate. The comparison of results in terms of non-dimensional natural frequencies and ply-by-ply failure analyses obtained from the present investigation are made with those available in the reported literature.     

含纤维波纹缺陷复合材料层合板的损伤分析

纤维波纹是复合材料层合板制备过程中的一种常见缺陷,会导致其刚度和强度显著下降,有效地预测含波纹缺陷复合材料层合板的失效强度具有显著的意义。基于此,本文采用解析的方式分别构造了纤维波纹呈正弦起伏与余弦起伏状的复合材料层合板模型。利用该模型,以Tsai-Wu准则作为失效判据,研究了一种含纤维波纹的碳纤维/环氧树脂复合材料层合板在受压情况下的损伤演化过程,得到了碳纤维/环氧树脂复合材料层合板的初始损伤强度。与有限元方法计算得到的损伤位置和损伤强度非常吻合,验证了本文算法的正确性。另外,相比于有限元方法,本文所述计算方法具有模型构造简单、计算效率高等优点,便于快速分析和确定含纤维波纹缺陷复合材料层合板的损伤位置与损伤强度。      

Multiscale analysis of laminated plates with integrated piezoelectric fiber composite actuators

This study is concerned with the detailed analysis of fiber-reinforced composite plates with integrated piezoceramic fiber composite actuators. A multiscale framework based on the asymptotic expansion homogenization method is used to couple the microscale and macroscale field variables. The microscale fluctuations in the mechanical displacement and electric potential are related to the macroscale deformation and electric fields through 36 distinct characteristic functions. The local mechanical and charge equilibrium equations yield a system of partial differential equations for the characteristic functions that are solved using the finite element method. The homogenized electroelastic properties of a representative material element are computed using the characteristic functions and the material properties of the fiber and matrix. The three-dimensional macroscopic equilibrium equations for a laminated piezoelectric plate are solved analytically using the Eshelby-Stroh formalism. The formulation admits different boundary conditions at the edges and is applicable to thick and thin laminated plates. The microscale stresses and electric displacement in the fibers and matrix are computed from the macroscale fields through interscale transfer operators. The multiscale analysis procedure is illustrated using two model problems. In the first model problem, a simply-supported sandwich plate consisting of a piezoceramic fiber composite shear actuator embedded between two graphite/polymer layers is studied. The second model problem concerns a cantilever graphite/polymer substrate with segmented piezoceramic fiber composite extension actuators attached to its top and bottom surfaces. Results are presented for the homogenized material properties, macroscale deformation, macroscale average stresses and microscale stress distributions.