Nonlinear finite element analysis of composite shell under impact

Large deflection dynamic responses of laminated composite cylindrical shells under impact are analyzed by the geometrically nonlinear finite element method based on a generalized Sander’s shell theory with the first order transverse shear deformation and the von-Karman large deflection assumption. A modified indentation law with inelastic indentation is employed for the contact force. The nonlinear finite element equations of motion of shell and an impactor along with the contact laws are solved numerically using Newmark’s time marching integration scheme in conjunction with Akay type successive iteration in each step. The ply failure region of the laminated shell is estimated using the Tsai-Wu quadratic interaction criteria. Numerical results, including the contact force histories, deflections and strains are presented and compared with the ones by linear analysis. The effect of the radius of curvature on the composite shell behaviors is investigated and discussed.     

Dynamic response and damage of composite shell under impact

The dynamic response and damage of laminated composite cylindrical shell subjected to impact load is numerically investigated using the finite element method. A nine-node isoparametric quadrilateral element based on Sander's shell theory is developed, in which the transverse shear deformation is considered. A semi-empirical contact law that accounts for the permanent indentation is incorporated into the finite element program to evaluate the contact force. The Newmark time ingegration algorithm is used for solving the time dependent equations of the shell and the impactor. The Tsai-Wu failure criterion is used to estimate the failure of the laminated shell. Numerical results, including the contact force history, interlaminate damage zone, and failure indices in the shell are presented. Effects of curvature, impact velocity and mass of impactor on the composite shell behaviors are discussed.     

Optimally reinforced cutouts in laminated circular cylindrical shells

The problem of designing a cutout in a load bearing structural member in the form of a shell, such that the cut structure maintains its stress state with a minimal departure from the stress state of the uncut structure is addressed herein. Symmetrically laminated composite circular cylindrical shells under hydrostatic compression and axial pressure are considered. Shallow thin shell (Donnell shell theory) lamination theory is utilized. The original (uncut) stiffness of the shell structures is recovered considerably by appropriately designing an edge reinforcement around the cutout. The buckling load of the designed shells are analyzed via the finite element method. An experimental investigation has been carried out to verify some of the results obtained from the finite element analysis. In the work presented, the reinforcement is modeled as a one-dimensional rod/beam type structural element.     

Viscoelastic and geometrically nonlinear behavior of laminated cylindrical shells

A viscoelastic and geometrically nonlinear finite element analysis is performed to investigate the stress relaxation and deflection of a laminated cylindrical shell under thermal loading. Incremental viscoelastic constitutive equations are derived to predict the stress relaxation. The finite element program is developed using a 3-D degenerated shell element, the first order shear deformation theory and the updated Lagrangian formulation. The viscoelastic and geometrically nonlinear analysis is executed for laminated shells with cross-ply and angle-ply stacking sequences, and its results are compared with those obtained from geometrically linear and viscoelastic analyses. The numerical results show that viscoelasticity and geometrical nonlinearity affect on the deflections and stresses of laminated cylindrical shells.     

Nonlinear bending response of laminated composite spherical shell panel with system randomness subjected to hygro-thermo-mechanical loading

In the present paper, the effect of random system properties on transverse nonlinear central deflection of laminated composite spherical shell panel subjected to hygro-thermo-mechanical loading is investigated. System properties such as material properties, thermal expansion coefficients, hygro-contraction coefficients, load intensity and lamina plate thickness are taken as independent random variables. The higher order shear deformation theory and von-Karman nonlinear kinematics are used for basic mathematical formulation. The elastic and hygrothermal properties of the composite material, which are considered to be dependent on temperature and moisture concentration, have been obtained based on micromechanical modeling. A direct iterative based C⁰ nonlinear finite element method combined with mean centered first-order perturbation technique (FOPT) proposed by present authors for the plate is extended for the spherical shell panel subjected to hygro-thermo-mechanical loading. The influences of random system properties with plate geometry, stacking sequences, support conditions, fiber volume fraction and temperature, and moisture distributions on the response of laminated spherical shell panel are examined in detail. The performance of the proposed approach is validated through comparison with those available in the literature and independent Monte Carlo simulation (MCS).     

Linear and nonlinear vibration analysis of sandwich cylindrical shell with constrained viscoelastic core layer

Damping characteristics of three-layered sandwich cylindrical shell for thin and thick core viscoelastic layers are studied using semi-analytical finite element method. The finite element method is developed based on the linear and nonlinear variations of the displacement distribution through the thickness of the core layer. Transient vibration has been conducted using the developed linear and nonlinear models and shown that the nonlinear formulation exhibits more damping property than the linear model. The effect of geometric nonlinearity due to the large deformation of the shell has also been considered assuming small strain and moderate rotation. Different assumptions based on the continuity and discontinuity in transverse shear stresses and slope of in-plane displacements are considered in the finite element formulation and their effects have been investigated. Considering nonlinearity of eigenvalue problem due to the frequency dependent property of viscoelastic material, an efficient algorithm has been developed to find the natural frequencies and loss factors of the viscoelastic cylindrical shell considering large deformation. The effect of imperfect bonding between the layers has also been investigated in the modeling and it is shown that slippage between layers at the interfaces leads to reduction in loss factor at the majority of modes.     

Postbuckling of shear deformable cross-ply laminated cylindrical shells under combined external pressure and axial compression

A postbuckling analysis is presented for a shear deformable cross-ply laminated cylindrical shell of finite length subjected to combined loading of external pressure and axial compression. The governing equations are based on Reddy's higher order shear deformation shell theory with von Kármán–Donnell type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of shear deformable laminated cylindrical shells under combined loading cases. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, unstiffened or stiffened, moderately thick, antisymmetric and symmetric cross-ply laminated cylindrical shells for different values of load-proportional parameters.     

Curing induced residual stresses in laminated cylindrical shells

A viscoelastic finite element analysis is presented to investigate residual stresses occurred in a laminated cylindrical shell during cure. An incremental viscoelastic constitutive equation that can describe stress relaxation during the cure is derived as a recursive formula which can be used conveniently for a numerical analysis. The finite element analysis program is developed on the basis of a 3-D degenerated shell element and the first order shear deformation theory, and is verified by comparing with an one dimensional exact solution. Viscoelastic effect on the residual stresses in the laminated shell during the cure is investigated by performing both the viscoelastic and linear elastic analyses considering thermal deformation and chemical shrinkage simultaneously. The results show that there is big difference between viscoelastic stresses and linear elastic stresses. The effect of cooling rates and cooling paths on the residual stresses is also examined.     

Optimum design of composite laminated plates via a multi-objective function

This paper deals with the optimum design of composite laminated plates under stiffness and gauge constraints. A multi-objective function which combines the plate weight and the strain energy stored in the plate by weighting parameters is introduced. This objective function is minimized while satisfying constraints such as the structural deformation and the limits on design variables. Both ply orientation angles and ply thicknesses of the composite plate are used as the design variables. The stiffness analysis is performed by the finite element method in which a triangular element is used that is suitable for the analysis of thin to thick plates and includes the transverse shear effects. Analyses of the derivatives of the objective function and the constraint functions with respect to the design variables is performed analytically. The mathematical programming method called the constrained variable metric is used to solve this optimum problem. An example is provided for the optimal design of a rectangular laminated plate.     

Free vibration of a laminated composite shell of revolution: Effects of shear non-linearity

Effects of shear non-linearity on free vibration of a laminated composite shell of revolution are investigated using a semi-analytical method based on the Reissner–Mindlin shell theory. The coupling between symmetric and anti-symmetric vibration modes of the shell is considered in the shear deformable shell element employed in this study. The Hahn–Tsai non-linearly elastic shear stress–shear strain relation is adopted. Numerical examples are given for laminated composite circular cylindrical and conical shells with various boundary conditions. The numerical results indicate that shear non-linearity may reduce significantly the fundamental frequencies of cross-ply composite shells of revolution.     

Dynamic finite element method for generally laminated composite beams

A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.     

Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness

The free vibration analysis of laminated conical shells with variable stiffness is presented using the method of differential quadrature (DQ). The stiffness coefficients are assumed to be functions of the circumferential coordinate that may be more close to the realistic applications. The first-order shear deformation shell theory is used to account for the effects of transverse shear deformations. In the DQ method, the governing equations and the corresponding boundary conditions are replaced by a system of simultaneously algebraic equations in terms of the function values of all the sampling points in the whole domain. These equations constitute a well-posed eigenvalue problem where the total number of equations is identical to that of unknowns and they can be solved readily. By vanishing the semivertex angle (α) of the conical shell, we can reduce the formulation of laminated conical shells to that of laminated cylindrical shells of which stiffness coefficients are the constants. Besides, the present formulation is also applicable to the analysis of annular plates by letting α=π/2. Illustrative examples are given to demonstrate the performance of the present DQ method for the analysis of various structures (annular plates, cylindrical shells and conical shells). The discrepancies between the analyses of laminated conical shells considering the constant stiffness and the variable stiffness are mainly concerned.     

Assessment of shell theories for hybrid piezoelectric cylindrical shell under electromechanical load

The assessment of classical lamination shell theory and first-order shear deformation theory is presented for simply supported finite circular cylindrical hybrid shell with cross-ply composite laminate as elastic substrate under electromechanical static load. Navier-type solutions are obtained and used in threedimensional equilibrium equations and transverse strain—displacement relation to obtain transverse stress components and improved value of deflection. These solutions are assessed by comparison with the threedimensional solution. The error in the two-dimensional shell theories increases as the shell becomes thicker and it is more for the patch loads in comparison to the uniformly distributed and sinusoidal loads.     

Non-linear free vibration analysis of laminated orthotropic cylindrical shells

A method to predict the influence of geometric non-linearities on the natural frequencies of an empty laminated orthotropic cylindrical shell is presented in this paper. It is a hybrid of finite element and classical thin shell theories. Sanders—Koiter non-linear and strain-displacement relations are used. Displacement functions are evaluated using linearized equations of motion. Modal coefficients are then obtained for these displacement functions. Expressions for the mass, linear and non-linear stiffness matrices are derived through the finite element method (in terms of the elements of the elasticity matrix). The uncoupled equations are solved with the help of elliptic functions. The frequency variations are first determined as a function of shell amplitudes and then compared with the results in the literature.     

Thermally induced vibration control of cylindrical shell using piezoelectric sensor and actuator

Shell type components and structures are very common in many mechanical and structural systems. Modeling and analysis of adaptive piezothermoelastic shell laminates represent a high level of sophistication and complexity. In this paper a finite element model is developed for the active control of thermally induced vibration of laminated composite shells with piezoelectric sensors and actuators. The present model takes into account the mass, stiffness and thermal expansion of the piezoelectric patches. A C_o continuous nine-node degenerated shell element is implemented to model the structure. The piezoelectric sensing layer senses the structural vibration and a suitable voltage applied in the piezoelectric actuator layer suppresses the oscillation. Actuator and sensor are coupled together with a control algorithm so as to actively control the dynamic response of the structure in a close loop. Numerical results are generated for a cylindrical shell and it is observed that thermally induced vibration of a laminated cylindrical shell can be suppressed through the application of piezoelectric sensor and actuator. Effects of variation in control gain and piezoelectric layer area coverage (PAC) have been studied. Higher control gain is more effective in damping out the vibration. Although the damping is enhanced by increase in PAC, increase beyond a certain level may not be useful in view of smaller efficacy and increased weight.     

Postbuckling of cross-ply laminated cylindrical shells with piezoelectric actuators under complex loading conditions

A postbuckling analysis is presented for a cross-ply laminated cylindrical shell with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness and the electric field is assumed to be the transverse component E_z only. The material properties are assumed to be independent of the temperature and the electric field. The governing equations are based on the classical shell theory with a von Kármán–Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of hybrid laminated cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical thin shells with fully covered or embedded piezoelectric actuators subjected to combined mechanical loading of external pressure and axial compression, and under different sets of thermal and electric loading conditions. The effects played by temperature rise, applied voltage, shell geometric parameter, stacking sequence, as well as initial geometric imperfections are studied.     

Effect of transverse shear on nonlinear vibration and postbuckling of anti-symmetric cross-ply imperfect cylindrical shells

Based on the Timoshenko-Mindlin kinematic hypothesis, the Donnell-type shell theory is extended to include transverse shear and rotary inertia for the nonlinear analysis of an anti-symmetrically laminated cross-ply circular cylindrical shell. The resulting governing equations are expressed in terms of a stress function, two rotations and transverse displacement. The shell is assumed to be all-simply-supported and all-clamped. A solution is formulated by a multi-mode approach and the method of harmonic balance for nonlinear vibrations. The corresponding postbuckling problem is treated as a special case. The results are compared with available data.     

A new hybrid-mixed composite laminated curved beam element

In this study, we piesent a new efficient hybrid-mixed composite laminated curved beam element The present element, which is based on the Helhnger-Reissnet vatiational principle and the first-order shear deformation lamination theory, employs consistent stress parameters coriespondmg to cubic displacement polynomials with additional nodeless degrees in order to lesolve the numerical difficulties due to the spurious constraints The stress parameters are eliminated and the nodeless degrees are condensed out to obtain the (6X6) element stiffness matrix The present study also incorporates the straightforwaid prediction of interlaminar stresses from equilibrium equations Seveial numencal examples confirm the superioi behavior of the present composite laminated curved beam element     

含孔复合材料层合板的疲劳寿命研究

建立了三维有限元模型,分析了复合材料层合板的应力场。使用修正Hashin失效准则判定复合材料的失效模式,并突降失效单元的材料性能。疲劳载荷引起复合材料刚度降和强度降依靠缓降模型实现。笔者将突降模型和缓降模型植入有限元模型中,模拟了复合材料层合板在拉伸和压缩疲劳载荷下的渐进损伤过程,并计算了层合板的纵向刚度损伤和疲劳寿命。层合板的纵向刚度损伤具有三阶段特点,与试验观察是一致的。层合板疲劳寿命预测值与试验值吻合地很好。     

Vibration of completely free composite plates and cylindrical shell panels by a higher-order theory

This paper deals with the free vibration of open, laminated composite, circular cylindrical panels having a rectangular plan-form and all their edges free of external tractions. The material arrangement of the shell panels considered may vary from this of the single isotropic (or special orthotropic) layer to that of a general angle-ply lay-up. The analysis is based on the application of the Ritz approach on the energy functional of the Love-type version of a unified shear deformable shell theory. A through-thickness parabolic distribution of the transverse shear deformation is mainly assumed but, for comparison purposes, numerical results that are based on the assumptions of the classical Love-type shell theory are also presented. The Ritz method is a powerful analytical technique since, provided that a complete set of trial functions is employed, it can provide the exact solution of the problem considered in infinite series forms. The mathematical formulation is therefore presented in a general form, appropriate for any set of basis functions. The variational approach is, however, finally applied in conjunction with a complete functional basis made of the appropriate admissible orthonormal polynomials.