Estimation of low-velocity impact damage in laminated composite circular plates using nonlinear finite element analysis

Nonlinear finite element analysis is used for the estimation of damage due to low-velocity impact loading of laminated composite circular plates. The impact loading is treated as an equivalent static loading by assuming the impactor to be spherical and the contact to obey Hertzian law. The stresses in the laminate are calculated using a 48 d.o.f. laminated composite sector element. Subsequently, the Tsai-Wu criterion is used to detect the zones of failure and the maximum stress criterion is used to identify the mode of failure. Then the material properties of the laminate are degraded in the failed regions. The stress analysis is performed again using the degraded properties of the plies. The iterative process is repeated until no more failure is detected in the laminate. The problem of a typical T300/N5208 composite [45 °/0 °/ − 45 °/90 °]_s circular plate being impacted by a spherical impactor is solved and the results are compared with experimental and analytical results available in the literature. The method proposed and the computer code developed can handle symmetric, as well as unsymmetric, laminates. It can be easily extended to cover the impact of composite rectangular plates, shell panels and shells.     

A geometrically nonlinear shell finite element for tire vibration analysis

An axisymmetric finite element is developed which includes such features as orthotropic material properties, doubly curved geometry, and both the first and second order nonlinear stiffness terms. This element can be used to predict the equilibrium state of an axisymmetric shell structure with geometrically nonlinear large displacements. Small amplitude vibration analysis can then be performed based on this equilibrium state. The nonlinear path is predicted by using the self-correcting incremental procedure and any point on the path can be checked by using the Newton-Raphson iterative scheme. The present formulation and solution procedure are evaluated by analyzing a series of examples with results compared with alternative known solutions. Examples include: free vibration of an isotropic cylindrical shell, a conical frustum, and an orthotropic cylindrical shell; buckling of a cylindrical shell; large deflection of a clamped disk, a spherical cap, and a steel belted radial tire. The final example is a free vibration analysis of the inflated tire and the natural frequencies obtained compared well with published experimental data.     

Postbuckling of multilayer cylindrical and spherical shell panels reinforced with graphene platelet by isogeometric analysis

The present work fills a gap on the postbuckling behavior of multilayer functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and spherical shell panels resting on elastic foundations subjected to central pinching forces and pressure loadings. Based on a higher-order shear deformation theory and the von Kármán’s nonlinear strain–displacement relations, the governing equations of the FG-GPLRC cylindrical and spherical shell panels are established by the principle of virtual work. The non-uniform rational B-spline (NURBS) based isogeometric analysis (IGA), the modified arc-length method and the Newton’s iteration method are employed synthetically to obtain nonlinear load–deflection curves for the panels numerically. Several comparative examples are performed to test reliability and accuracy of IGA and arc-length method in present formulation and programming implementation. Parametric investigations are carried out to illustrate the effects of dispersion type of the graphene platelet (GPL), weight fraction of the GPL, thickness of the panel, radius of the panel and parameters of elastic foundation on the load–deflection curves of the FG-GPLRC shell panels. Some complex load–deflection curves of the FG-GPLRC cylindrical and spherical shell panels resting on elastic foundations may be useful for future references.

     

A conical shell finite element

In the analysis of rocket and missiles structures one frequently encounters cylindrical and cornica' shells. A simple finite element which fits the above configuration is obviously a conical shell finite element. In this paper stiffness matrix for a conical shell finite element is derived using Novozhilov's strain-displacement relations for a conical shell. Numerical integration is carried out to ge. the stiffness matrix. The element has 28 degrees-of-freedom and is nonconforming. An eigenvalue analysis of the stiffness matrix showed that it contains all the rigid body modes (six in this case) adequately, which is one of the convergence criteria. An advantage of this element is that a cylindrical shell, an annular segment flat plate, a rectangular flat plate elements can easily be obtained as degenerate cases. The effectiveness of this element is shown through a variety of numerical examples pertaining to annular plate, cylindrical shell and conical shell problems. Comparison of the present solution is made with the existing ones wherever possible. The comparison shows that the present element is superior in some respects to the existing elements     

Efficient simulation of low-velocity impacts on composite sandwich panels

For efficiently simulating the damage resistance of sandwich panels subjected to low-velocity impacts, the finite element based damage tolerance tool CODAC has been enhanced. While sandwich structures are very weight efficient and provide integrated acoustic and thermal insulation, impact damage can provoke a significant strength and stability reduction. Therefore, the objective of CODAC is to provide methodologies which reliably simulate impact events and predict impact damage sizes. Since frequent design loops require a quick analysis, efficient deformation and failure models are desired. To achieve a rapid and accurate stress analysis, a recently developed three-layered finite shell element is applied. Failure analysis is based on a progressive damage mechanics approach: Damage initiation is detected by stress-based failure criteria. Material resistance is reduced by appropriate, step-wise linear degradation models. An experimental impact test program on honeycomb sandwich panels is used to validate the impact simulation of the FE-tool CODAC. Comparisons between impact tests and simulations showed that CODAC is capable of accurately and rapidly simulating impact events, which induce barely visible damage. Furthermore, the onset of clearly visible damage is correctly predicted.     

On finite element large displacement and elastic-plastic dynamic analysis of shell structures

Finite element procedures for nonlinear dynamic analysis of shell structures are presented and assessed. Geometric and material nonlinear conditions are considered. Some results are presented that demonstrate current applicabilities of finite element procedures to the nonlinear dynamic analysis of two-dimensional shell problems. The nonlinear response of a shallow cap, an impulsively loaded cylindrical shell and a complete spherical shell is predicted. In the analyses the effects of various finite element modeling characteristics are investigated. Finally, solutions of the static and dynamic large displacement elastic-plastic analysis of a complete spherical shell subjected to external pressure are reported. The effect of initial imperfections on the static and dynamic buckling behavior of this shell is presented and discussed.     

Materially and geometrically nonlinear analysis of laminated anisotropic plates by p-version of FEM

A p-version finite element model based on degenerate shell element is proposed for the analysis of orthotropic laminated plates. In the nonlinear formulation of the model, the total Lagrangian formulation is adopted with moderately large deflections and small rotations being accounted for in the sense of von Karman hypothesis. The material model is based on the Huber-Mises yield criterion and Prandtl-Reuss flow rule in accordance with the theory of strain hardening yield function, which is generalized for anisotropic materials by introducing the parameters of anisotropy. The model is also based on the equivalent-single layer laminate theory. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. Gauss-Lobatto numerical quadrature is used to calculate the stresses at the nodal points instead of Gauss points. The validity of the proposed p-version finite element model is demonstrated through several comparative points of view in terms of ultimate load, convergence characteristics, nonlinear effect, and shape of plastic zone.     

Finite element analysis of laminated shells of revolution

A finite element formulation for the analysis of axisymmetric fibre reinforced laminated shells subjected to axisymmetric load is presented. The formulation includes arbitrary number of bonded layers each of which may have different thicknesses, orientation of elastic axes, and elastic properties. Superparamatric curved elements[17] having four degrees of freedom per node including the normal rotation, are used. Stress-strain relation for an arbitrary layer is obtained from the consideration of three dimensional aspect of the problem. The element stiffness matrix has been obtained by using Gauss quadrature numerical integration, even though the elasticity matrix is different for different layers. The formulation is checked for a cylindrical tube subjected to internal pressure and axial tension, and the results are found to compare very well with the elastic solution [9].     

Nonlinear analysis of an axisymmetric shell using three noded degenerated isoparametric shell elements

An updated Lagrangian formulation of a quadratic degenerated isoparametric shell element is presented for geometrically nonlinear elasto-plastic shell problems. A finite rotation effect is included in the formulation by adopting a co-rotational scheme. The load stiffness matrix has been derived for the treatment of a pressure load. For elasto-plastic behavior, the layered element model is used. The Newton-Raphson iteration method is employed to solve incremental nonlinear equations. For tracking of post-buckling behavior, the work control method is taken into account. Verification of the present technique is obtained by analyzing the available reference problems. Good correlations between the computed results and referenced data can be drawn.     

A high precision triangular laminated anisotropic cylindrical shell finite element

The stiffness matrix for a high precision triangular laminated anisotropic cylindrical shell finite element has been formulated and coded into a composite structural analysis program. The versatility of the element's formulation enables its use in the analysis of multilayered composite plate and cylindrical shell type structures taking into account actual lamination parameters. The example applications presented demonstrated that accurate predictions of stresses as well as displacements are obtained with modest number of elements.     

Large displacement extension of consistent shell element for static and dynamic analysis

The superior performance of the consistent shell element in the small deflection range has encouraged the authors to extend the formulation to large displacement static and dynamic analyses. The nonlinear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear stiffness matrix and the unbalanced load vector for the consistent shell element is presented in this study. Meanwhile, a simplified method for coding the nonlinear formulation is provided by relating the components for the nonlinear B-matrices to those of the linear B-matrix. The consistent mass matrix for the shell element is also derived and then incorporated with the stiffness matrix to perform large displacement dynamic and free vibration analyses of shell structures. Newmark's method is used for time integration and the Newton-Raphson method is employed for iterating within each increment until equilibrium is achieved. Numerical testing of the nonlinear model through static and dynamic analyses of different plate and shell problems indicates excellent performance of the consistent shell element in the nonlinear range.     

Constitutive ply damage modeling, FEM implementation, and analyses of laminated structures

The present work is concerned with the modeling of progressive damage in fiber reinforced polymer laminates and its implementation into a finite element code as constitutive material law. The objective is to predict damage evolution and material degradation due to matrix dominated failure modes (“matrix cracking”). In a previous work, a ply-level continuum damage model based on ply failure mechanisms postulated by Puck has been presented. This model predicts the evolution of the complete tensor of elasticity for the damaged material and is able to capture the effects of stiffness recovery and slanted cracking under transverse compression. In the current work, this damage model is adapted for arbitrary loading paths and implemented within the finite element method in order to analyze complex structures and study their damage behavior including load redistribution due to damage. To demonstrate some key features and the application of the damage model in structural analysis, it is applied in a single element analysis as well as in the simulation of Open Hole Compression tests for which results are compared to experimental data from the literature.     

Linear programming and limit analysis of shell roofs

Limit analysis for cylindrical shell roofs has been formulated as a linear programming problem based on lower bound theorem. The differential equations of equilibrium for a circular cylindrical shell element are transformed into algebraic equations by finite differences. The equilibrium equations and the linearized non-linear yield conditions at various points of the shell are linear functions of the stress resultants. These form the linear constraints of the problem. The load parameter is taken as an objective function and it is maximized using revised simplex method. For a shell of given geometry, stress resultants at various points are obtained to give the optimum collapse load. Thus the versatile technique avoids various trial solutions to achieve best lower bound for complicated shell problems.     

A simple and effective element for analysis of general shell structures

A simple flat three-node triangular shell element for linear and nonlinear analysis is presented. The element stiffness matrix with 6 degrees-of-freedom per node is obtained by superimposing its bending and membrane stiffness matrices. An updated Lagrangian formulation is used for large displacement analysis. The application of the element to the analysis of various linear and nonlinear problems is demonstrated.     

A finite element model for the nonlinear analysis of reinforced and prestressed masonry walls

A materially nonlinear layered finite element model is proposed for the analysis of reinforced and/or prestressed masonry wall panels under monotonie loadings in the plane and/or out of the plane, capable of evaluating both the serviceability load and the ultimate load. An orthotropic incrementally linear relationship and equivalent uniaxial concept are used to represent the behaviour of masonry under biaxial stresses while a uniaxial bilinear elasto-plastic model with hardening is employed for rebar and the so-called ‘power-formula’ is adopted to describe the stress-strain relationship of prestressing steel.

After cracking, the smeared coaxial rotating crack model is adopted and tension stiffening, reduction in compressive strength and stiffness after cracking, and strain softening in compression are accounted for. The modified Newton-Raphson iteration method is employed to ensure convergency of non linear solution.

The proposed finite element model has been tested by a comparison with experimental data available in literature, both for reinforced and prestressed wall panels. The analysis of results shows good agreement between the values obtained by the proposed model and those obtained experimentally.     

Geometrically nonlinear analysis of cylindrical shells using surface-parallel quadratic elements

A moderately thick cylindrical shell isoparametric element that is capable of accurately modeling cylindrically curved geometry, while also incorporating appropriate through-thickness kinematic relations is developed. The analysis accounts for fully nonlinear kinematic relations so that stable equilibrium paths in the advanced nonlinear regime can be accurately predicted. The present nonlinear finite element solution methodology is based on the hypothesis of linear displacement distribution through thickness (LDT) and the total Lagrangian formulation. A curvilinear side 16-node element with eight nodes on each of the top and bottom surfaces of a cylindrical shell has been implemented to model the transverse shear/normal deformation behavior represented by the LDT. The BFGS iterative scheme is used to solve the resulting nonlinear equations. A thin-shallow clamped cylindrical panel is investigated to test the convergence of the present element, and also to compare the special case of the present solution based on the KNSA (von Karman strain approximation) with those computed using the available faceted elements, discrete Kirchhoff constraint theory (DKT) and classical shallow shell finite elements, spanning the entire computed equilibrium path.     

基于Marc的空气弹簧横向特性分析

为实现空气弹簧的横向特性和强度分析,基于有限元非线性接触技术,用Marc对空气弹簧的非线性横向刚度特性进行数值模拟,分析最大横向力作用下空气弹簧各部分的应力分布,横向刚度计算结果与试验结果基本一致,表明空气弹簧的建模与仿真方法的合理性,为空气弹簧优化设计和特性分析提供参考.     

A layered cylindrical quadrilateral shell element for nonlinear analysis of RC plate structures

This paper proposes a simple and accurate 4-node, 24-DOF layered quadrilateral flat plate/shell element, and an efficient nonlinear finite element analysis procedure, for the geometric and material nonlinear analysis of reinforced concrete cylindrical shell and slab structures. The model combines a 4-node quadrilateral membrane element with drilling or rotational degrees of freedom, and a refined nonconforming 4-node 12-DOF quadrilateral plate bending element RPQ4, so that displacement compatibility along the interelement boundary is satisfied in an average sense. The element modelling consists of a layered system of fully bonded concrete and equivalent smeared steel reinforcement layers, and coupled membrane and bending effects are included. The modelling accounts for geometric nonlinearity with large displacements (but moderate rotations) as well as short-term material nonlinearity that incorporates tension, cracking and tension stiffening of the concrete, biaxial compression and compression yielding of the concrete and yielding of the steel. An updated Lagrangian approach is employed to solve the nonlinear finite element stiffness equations. Numerical examples of two reinforced concrete slabs and of a shallow reinforced concrete arch are presented to demonstrate the accuracy and scope of the layered element formulation.     

Large deflection analysis of plates and shells by discrete energy method

The discrete energy method—a special form of finite difference energy approach—is presented as a suitable alternative to the finite element method for the large deflection elastic analysis of plates and shallow shells of constant thickness. Strain displacement relations are derived for the calculation of various linear and nonlinear element stiffness matrices for two types of elements into which the structure is discretized for considering separately energy due to extension and bending and energy due to shear and twisting. Large deflection analyses of plates with various edge and loading conditions and of a shallow cylindrical shell are carried out using the proposed method and the results compared with finite element solutions. The computational efforts required are also indicated.     

Nonlinear finite element computation of the equilibrium,stability and motion of the extensional beam and ring

Discrete Gauss integration of the element total potential energy is applied to the formation of a cubic-cubic C¹ extensible completely nonlinear curved beam finite element. The versatility, accuracy, effectiveness, and robustness of the element, and the Newton-Raphson technique used to solve the nonlinear algebraic stiffness equation set up with it is numerically demonstrated by computations of the nonlinear equilibrium stability and motion of beams and rings.