Geometrically nonlinear flexural analysis of hygro-thermo-elastic laminated composite doubly curved shell panel

In this article, nonlinear flexural behaviour of laminated composite doubly curved shell panel is investigated under hygro-thermo-mechanical loading by considering the degraded composite material properties through a micromechanical model. The laminated panel is modelled using higher order shear deformation mid-plane kinematics and Green–Lagrange geometric nonlinear strain displacement relations. In the present case, all the nonlinear higher order terms are included in the mathematical model to obtain the exact flexure of the structural panel. The nonlinear system governing equations are derived using variational method and discretised using the nonlinear finite element steps. Numerical results are computed through direct iterative method and validated by comparing with those published results available in open literature. Finally, wide variety of numerical examples are computed using the proposed model to address the effect of hygrothermal conditions, geometrical and material parameters and support conditions on the flexural behaviour of laminated composite doubly curved shell panel.     

Transverse vibration of symmetrically laminated rectangular composite plates

A general numerical method for estimation of the vibrational response of symmetrically laminated rectangular composite plates is presented. This is based on the Rayleigh-Ritz method using the admissible 2-D orthogonal polynomials to derive the governing eigenvalue equation. The natural frequencies and mode shapes for the laminated plates are obtained by solving this governing eigenvalue equation. Several test problems are solved to demonstrate the accuracy and flexibility of the proposed method. The present results, where possible, are verified with those available values from the literature. The effects of the material, number of layers and fibre orientation upon the frequencies and mode shapes are discussed. This study may provide valuable information for researchers and engineers in design applications.     

Delamination modeling in doubly curved laminated shells for free vibration analysis using zigzag theory-based facet shell element and hybrid continuity method

We present a finite element (FE) formulation for the free vibration analysis of doubly curved laminated composite and sandwich shells having multiple delaminations, employing a facet shell element based on the efficient third-order zigzag theory and the region approach of modeling delaminations. The methodology, hitherto not attempted, is general for delaminations occurring at multiple interfacial and spatial locations. A recently developed hybrid method is used for satisfying the continuity of the nonlinear layerwise displacement field at the delamination fronts. The formulation is shown to yield very accurate results with reference to full-field three-dimensional FE solutions, for the natural frequencies and mode shapes of delaminated shallow and deep, composite and highly inhomogeneous soft-core sandwich shells of different geometries and boundary conditions, with a significant computational advantage. The accuracy is sensitive to the continuity method used at the delamination fronts, the usual point continuity method yielding rather poor accuracy, and the proposed hybrid method giving the best accuracy. Such efficient modeling of laminated shells with delamination damage will be of immense use for model-based techniques for structural health monitoring of laminated shell-type structures.     

Asymptotic finite strip analysis of doubly curved laminated shells

On the basis of the Hellinger–Reissner (H–R) principle, an asymptotic finite strip method (FSM) for the analysis of doubly curved laminated shells is presented by means of perturbation. In the formulation the displacements and transverse stresses are taken as the functions subject to variation. Imposition of the stationary condition of the H–R functional, the weak formulation associated with the Euler–Lagrange equations of three-dimensional (3D) elasticity is obtained. Upon introducing a set of appropriate dimensionless scaling and bringing the transverse shear deformations to the stage at the leading-order level, the weak formulation is asymptotically expanded as a series of weak-form equations for various orders. An asymptotic FSM according to the present formulation is then developed where the field variables are interpolated as a finite series of products of trigonometric functions and crosswise polynomial functions independently. Through successive integration, the present formulation turns out that three mid-surface displacement degrees-of-freedom (DOF) and two rotation DOF for each node in a strip element are taken as the independent unknowns in the system equations for various orders. The solution procedure for the leading-order level can be repeatedly applied level-by-level in a consistent and hierarchic way. Application of the asymptotic FSM to a benchmark problem is demonstrated.     

Nonlinear hygro-thermo-elastic vibration analysis of doubly curved composite shell panel using finite element micromechanical model

Nonlinear free vibration behavior of laminated composite curved panel under hygrothermal environment is investigated in this article. The mathematical model of the laminated panel is developed using Green–Lagrange-type geometrical nonlinearity in the framework of higher-order mid-plane kinematics. The corrugated composite properties are evaluated through the micromechanical model and all the nonlinear higher-order terms are included in the present model for the sake of generality. The equation of vibrated panel is obtained using Hamilton's principle and discretized with the help of the finite element steps. The solutions are computed numerically using the direct iterative method. The effect of parameters on the nonlinear vibration responses is examined thoroughly by solving the wide variety of numerical examples.     

Nonlinear free vibration analysis of shape memory alloy embedded laminated composite shell panel

High specific strength and stiffness are characteristics desired for aircraft and launch vehicle domains to enhance the payload gain and performance. The mechanical properties of the composites can be further tailored by embedding structural components, such as shape memory alloys, into the passive composite structure. The present study is primarily focused on the nonlinear free vibration analysis of spherical and cylindrical composite shell panels embedded with shape memory alloy fibers. The nonlinear finite element governing equations based on the higher-order shear deformation plate theory and principle of virtual work with nonlinear von-Karman strain displacement relations are employed for the analysis. The temperature-dependent material properties of shape memory alloy are considered in the formulation. A nine-noded isoperimetric element is accounted for synthesizing the element for the finite formulation. The Young's modulus and the recovery stress vary with temperature and higher nonlinearity. The incremental method is used to generate the inputs for the temperature-dependent nonlinear properties of materials. The temperature change is divided by many small temperature increments. The temperature-dependent material properties are assumed constant during the small increment. The mechanics of shape memory alloy in substrate are presented and the governing equation of laminated composite with shape memory alloy is obtained and implemented in the MATLAB 7.8 program.     

Nonlinear vibration of symmetrically laminated spherical caps with flexible supports

Summary Based on a nonlinear theory for laminated spherical caps, the nonlinear vibration of symmetrically laminated thin spherical caps with flexible supports is investigated by a multi-mode approach. A Fourier-Bessel series solution is formulated and satisfies the associated boundary conditions. The governing equations expressed in terms of transverse displacement and stress function are fulfilled by the Galerkin procedure and the method of harmonic balance. Numerical results in the nonlinear vibration are presented for various boundary conditions, cap rises, numbers of layers and material properties. Present results are compared with available data.     

Stochastic free vibration analyses of composite shallow doubly curved shells – A Kriging model approach

This paper presents the Kriging model approach for stochastic free vibration analysis of composite shallow doubly curved shells. The finite element formulation is carried out considering rotary inertia and transverse shear deformation based on Mindlin’s theory. The stochastic natural frequencies are expressed in terms of Kriging surrogate models. The influence of random variation of different input parameters on the output natural frequencies is addressed. The sampling size and computational cost is reduced by employing the present method compared to direct Monte Carlo simulation. The convergence studies and error analysis are carried out to ensure the accuracy of present approach. The stochastic mode shapes and frequency response function are also depicted for a typical laminate configuration. Statistical analysis is presented to illustrate the results using Kriging model and its performance.     

Effects of geometric imperfections on vibration of compressed shear deformable laminated composite curved panels

Summary The vibrational behavior of geometrically imperfect single and multilayered composite double-curved shallow panels subjected to a system of tangential compressive/tensile edge loads in the pre- and postbuckling ranges is investigated. The effects of transverse shear deformations, lamination, the character of in-plane boundary conditions, and of transverse normal stress are incorporated and their influence is emphasized.Numerical illustrations enabling one to compare the obtained results based on higher order and first order shell theories with their classical counterparts, based on the Love-Kirchhoff model are presented and conclusions related to their range of applicability are outlined.     

Natural frequencies of symmetrically laminated rectangular plates with free edges

This paper is concerned with the estimation of natural frequencies of undamaged laminated rectangular plates having completely free edges. A possible application of the result is use as a reference in the determination of location and extent of defects in laminated plates from measurements of natural frequencies of damaged plates. The Rayleigh-Ritz energy approach is employed here to obtain the approximate natural frequencies of symmetrically laminated plates. The plates are considered to be thin and to undergo small deflections, satisfying Kirchhoff's hypothesis. A complete power series is used as admissible functions for in-plane and out-of-plane displacements. In the numerical examples the accuracy of the present results is compared with experimental and finite element analysis results. The effects of aspect ratio and fiber orientation on the natural frequencies of symmetrically laminated plates are documented. For each frequency, the associated mode shape is identified and tabulated in this paper.     

A postprocess method for laminated shells with a doubly curved nine-noded finite element

A numerical method that predicts through-the-thickness stresses accurately by using in-plane displacement of Efficient Higher Order Shell Theory (EHOST) as a postprocessor is implemented in nine-noded doubly curved shell element. In the present study, an efficient postprocess method is developed in the framework of shell finite element without losing the accuracy of solutions. This method consists of two steps. First is to obtain the relationship between shear angles of First Order Shear Deformation Theory (FSDT) and EHOST. Second is to construct accurate displacement and stress fields from the FSDT solution by using EHOST displacement fields as a postprocessor. To obtain accurate transverse shear stresses, integration of equilibrium equation approach is used. In the course of calculating transverse shear stresses, the computation of third derivatives of transverse deflection is required. Simply supported curved panels and finite cylinder problems demonstrate economical and accurate solution of laminate composite shells provided by the present method. The present postprocess method should work as an efficient tool in the stress analysis of multilayered thick shells.     

Free vibration of symmetrically laminated plates using a higher-order theory with finite element technique

A C⁰ finite element formulation of the higher-order theory is used to determine the natural frequencies of isotropic, orthotropic and layered anisotropic composite and sandwich plates. The material properties that are typical of high modulus fibre reinforced composites are used to show the parametric effects of plate aspect ratio, length-to-thickness ratio, degree of orthotropy, number of layers and lamination angle/scheme. The present theory is based on a higher-order displacement model and the three-dimensional Hooke's laws for plate material. The theory represents a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strains through the plate thickness. A special mass matrix diagonalization scheme is adopted which conserves the total mass of the element and includes the effects due to rotary inertia terms. The results presented should be useful in obtaining better correlation between theory and experiment, and to numerical analysts in verifying their results.     

对称层合折板结构自由振动分析的移动最小二乘无网格法

提出了一种求解对称层合折板结构自由振动问题的移动最小二乘无网格法。以作者提出的折板无网格模型为基础,将对称层合折板结构视为由不同平面上对称层合板组成的复合结构。先基于一阶剪切变形理论,由移动最小二乘近似推导出各对称层合板的刚度和质量矩阵,再利用板与板间的位移协调条件,将各板的刚度和质量矩阵叠加得到整个结构的刚度和质量矩阵,推导出描述层合折板结构自由振动行为的控制方程。文末算例表明由本文方法得到的解与采用壳单元得到的ANSYS有限元解吻合良好,证明了本文方法的准确性。     

In-plane free vibrations of single-layer and symmetrically laminated rectangular composite plates

This work presents accurate upper-bound solutions for free in-plane vibrations of single-layer and symmetrically laminated rectangular composite plates with an arbitrary combination of clamped and free boundary conditions. In-plane natural frequencies and modes shapes are calculated by the Ritz method with a simple, stable and computationally efficient set of trigonometric functions. Reliability of the method is assessed by comparison with known accurate solutions for isotropic plates and specially orthotropic single laminates. The extensive results presented here for the first time can be considered benchmark data against which other methods may be compared and validated in the future. Influence of fiber orientation, stacking sequence, degree of orthotropy, aspect ratio and boundary conditions upon the in-plane vibration behavior are also discussed.     

Active damping of geometrically nonlinear vibrations of doubly curved laminated composite shells

This paper deals with the analysis of active constrained layer damping (ACLD) of geometrically nonlinear transient vibrations of doubly curved laminated composite shells. Vertically/obliquely reinforced 1–3 piezoelectric composite (PZC) and active fiber composite (AFC) materials are used as the materials of the constraining layer of theACLD treatment. The Golla–Hughes–McTavish (GHM) method has been implemented to model the constrained viscoelastic layer of the ACLD treatment in time domain. The first-order shear deformation theory (FSDT) and the Von Kármán type non-linear strain displacement relations are used for analyzing this coupled electro-elastic problem. A three dimensional finite element (FE) model of doubly curved laminated smart composite shells integrated with ACLD patches has been developed to investigate the performance of these patches for controlling the geometrically nonlinear transient vibrations of the shells. The numerical results indicate that the ACLD patches significantly improve the damping characteristics of the doubly curved laminated cross-ply and angle-ply shells for suppressing their geometrically nonlinear transient vibrations. It is found that the performance of the ACLD patch with its constraining layer being made of the AFC is significantly higher than that of the ACLD patch with vertically/obliquely reinforced 1–3 PZC constraining layer. The effects of variation of piezoelectric fiber orientation in both the obliquely reinforced 1–3 PZC and the AFC constraining layers on the control authority of the ACLD patches have also been investigated.     

平面曲梁面内自由振动有限元分析的p型超收敛算法

该文采用状态空间法研究了圆弧曲梁面内动力学特性。通过选择合适的状态变量,建立了相应的状态空间列式,并给出了固有频率和振动模态的求解过程。进一步通过引入辛内积的概念,建立了三种工程常见边界条件(简支、固支和自由)下圆弧曲梁面内振动模态关于质量和旋转惯量的正交关系式。在此基础上,运用模态叠加法给出了非齐次状态方程的解析解,并得到了圆弧曲梁在竖向移动集中荷载作用下的瞬态响应。数值算例结果表明该文方法是十分精确和可靠的。     

Non-linear free vibrations of symmetrically laminated, slightly compressible cylindrical shell panels

A geometrically non-linear dynamic shell theory presented by the authors in an earlier work is used to study the non-linear free vibrations of symmetrically laminated cylindrical shell panels. The theory accounts for arbitrary lamination constructions, anisotropy, and slight compression across the thickness. In this paper, this theory is used to derive the equation of motion of the panel with quadratic and cubic non-linearities and symmetric lamination schemes. The symbolic manipulator Mathematica™ is used to perform the Rayleigh-Ritz procedure and derive a single-mode approximation to the vibration of the panel. The Lindstedt-Poincare perturbation technique is used to analyze the resulting non-linear differential equation of motion and study the effects of non-linearities on the dynamics of free vibrations of the panel. A numerical example of a symmetrically laminated graphite/epoxy shell panel is used to demonstrate the procedure. The numerical example shows that non-linearities are of the hardening type and are more pronounced for smaller opening angles. Moreover, it shows that the larger-amplitude motions are dominated by the lower modes.     

Analysis of laminated bimodulus composite thin shells of revolution using a doubly curved quadrilateral finite element

This paper presents a finite element analysis of laminated bimodulus composite thin shells of revolution using a 48 d.o.f. doubly curved quadrilateral finite element. All the three displacements of the shell element reference surface are expressed as products of one-dimensional first-order Hermite interpolation polynomials. The constitutive relationship for a bimodulus composite is assumed to depend on the fibre-direction strain experienced by each orthotropic layer. Consequently the true state of strain and the corresponding constitutive relationship in a bimodulus composite structure are to be determined iteratively. The true state of stress/strain is obtained by specifying a maximum error in the locations of the two neutral surfaces (one along each of the orthogonal reference axes) in the shell. The use of the quadrilateral shell finite element is validated by solving the problem of (i) a freely supported single layer (0°) bimodulus composite square plate and (ii) a freely supported single layer (0°) cylindrical panel, which are subjected to sinusoidal transverse loading and for which analytical solutions are available. Next, the problems of a single layer (90°) pinched cylindrical shell and a single layer (0°) open crown hemispherical shell are solved to show the ability of the present program.     

Nonlinear bending and vibration of symmetrically laminated orthotropic elliptical plate with simply supported edge

Summary This paper deals with the static large deflection and the large amplitude free flexural vibration of a symmetrically laminated orthotropic elliptical plate with a simply supported immovable edge. A unified approximate solution of the dynamic von Karman-type equations of the plate in terms of the deflection and two inplane displacements is presented by use of a modified Galerkin procedure. Two inplane equations of equilibrium and the associated inplane boundary conditions are satisfied exactly. Numerical computations are performed for static and dynamic responses of homogeneous and symmetric cross-ply elliptical plates with different axes ratios, numbers of layers and high-modulus material properties. In the static and dynamic linear cases, present results are compared with available data.Notations A nondimensional amplitude (=w ₀/h) - A _i, B_i constants in Eq. 912) - A _ij, D_ij plate stiffnesses defined in Eqs. (4) - a, b, h semi-major axis, semi-minor axis and thickness of elliptical plate - C _ij ^(k) reduced stiffnesses of thek-th layer - d _ij, e_ij plate constants defined in Eqs. (4) - E _L, E_T, G_LT principal and shear moduli of orthotropic material - F nondimensional time function - G ₁, G₂ constants in Eq. (11) - J _i() differential operator defined in Eq. (4) - l, m direction cosines - M _x, M_y, M_xy bending and twisting moments per unit length - M ,M ,M nondimensional moments defined in Eqs. (8) - N _x, N_y, N_xy membrane forces per unit length - N ,N ,N nondimensional membrane forces defined in Eqs. (9) - Q, q nondimensional and dimensional transverse load per unit area - T, T ₀ nondimensional nonlinear and linear periods respectively in Eq. (22) - t time - U, V, W nondimensional displacements defined in Eqs. (5) - u ⁰, v⁰, w midsurface displacement components alongx, y andz directions, respectively - w ₀ maximum deflection - x, y, z rectangular cartesian coordinates - _i constant coefficients in Eqs. (17) and (18) - coefficient of nonlinear term in Eq. (21) - , nondimensional coordinates defined in Eqs. (5) - normalization factor in Eq. (19) - axis ratio,a/b - v, v _LT Poisson's ratios of isotropic and orthotropic materials - , ₀ mass per unit area and volume, respectively - nondimensional time defined in Eqs. (5) - ₀ nondimensional fundamental linear frequency in Eq. (21) With 5 Figures     

Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation

The static, dynamic, and free vibration analysis of a functionally graded material (FGM) doubly curved panel are investigated analytically in the present paper. The FGM Panel is originated from a rectangular planform and its principle curvatures are considered to be constant. All mechanical properties of the FGM panel are assumed to vary continuously through the thickness according to a power law formulation except Poisson’s ratio, which is kept constant. A Pasternak-type elastic foundation containing damping effects is considered to be in contact with the panel during deformation. The elastic foundation reacts in both compression and tension. Equations of motion are established based on the first order shear deformation and the modified Sanders shell theories. Following the Navier type solution, the established equations are reduced to time-dependent ordinary differential equations. Using the Laplace transform, the time-dependency of the problem is eliminated. The solutions are obtained analytically in the Laplace domain and then are inverted to the time domain following an analytical procedure. Finally, the analytical results are verified with those reported in the literature.