Linear and Geometrically nonlinear analysis of plates and shells by a new refined non-conforming triangular plate/shell element

A refined non-conforming triangular plate/shell element for linear and geometrically nonlinear analysis of plates and shells is developed in this paper based on the refined non-conforming element method (RNEM). A conforming triangle membrane element with drilling degrees of freedom in Cartesian coordinates and the refined non-conforming triangular plate-bending element RT9, in which Kirchhoff kinematic assumption was adopted, are used to construct the present element. The displacement continuity condition along the interelement boundary is satisfied in an average sense for plate analysis, and the coupled displacement continuity requirement at the interelement is satisfied in an average sense, thereby improving the performance of the element for shell analysis. Selectively reduced integration with stabilization scheme is employed in this paper to avoid membrane locking. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plate bending problems and plane problems or for the geometrically nonlinear analysis of thin plates and shells with large displacement, moderate rotation but small strain.     

A new finite element scheme for instability analysis of thin shells

This paper describes a new finite element scheme for the analysis of instability phenomena of arbitrary thin shells. A computationally efficient procedure is proposed for calculating the non-linear stiffness and tangential stiffness matrices for a doubly-curved quadrilateral element defined by co-ordinate lines. The essential feature is the explicit addition of the non-linear terms into the rigid-body motion of the element. Thus the non-linear and tangential element stiffness matrices can easily be generated by transforming the generalized element stiffness matrix for linear analysis, and the non-linear terms of these matrices are separated into a number of component terms multiplied by the rigid-body rotations. These component terms can be stored permanently and used to calculate efficiently the non-linear and tangential stiffness matrices at each iteration. Illustrative examples are presented which confirm the validity of the present approach in the analysis of instability phenomena of thin plates and shells.     

Refined discrete quadrilateral degenerated shell element by using Timoshenko's beam function

A refined discrete degenerated 20‐DOF quadrilateral shell element RQS20 is proposed. The exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary. The re‐constitute method for shear strain matrix is adopted. The proposed element can be used for the analysis of both moderately thick and thin plates/shells, and the convergence for the very thin case can be ensured theoretically. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements. Most important of all, it is free from the membrane and shear locking phenomena for extremely thin plates/shells, on the one hand, and it can also avoid the phenomenon of oscillatory solutions for thick plates/shells case on the other. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element large deflection analysis of shallow shells

A finite element formulation is presented to study the non-linear buckling of arbitrary shallow elastic thin shells with general boundary conditions and subjected to conservative pressure loading. Pre and post buckling behaviour of a large number of shallow and semi deep doubly curved shells is studied in detail. Unsymmetrical bifurcation paths of a shallow spherical shell subjected to uniform inward pressure are also investigated.     

A rectangular finite element for analysing composite multilayered shallow shells in statics,vibration and buckling

This paper presents a new 32-degree-of-freedom finite element of multilayered composite, moderately thick, shallow shells. The element is a four-node C¹ rectangular element and is built from standard interpolations but with a new kind of kinematics which allows us to exactly ensure the continuity conditions for displacements and stresses at the interfaces between the layers of a laminated structure and the boundary conditions at the upper and lower surfaces of the shell. The transverse shear deformation which is represented by cosine functions is of a higher order and allows us to avoid shear correction factors. The element is evaluated on standard problems and in comparison with exact three-dimensional and analytical solutions for multilayered plates and shells in statics, vibrations and stability.     

板壳概率变分原理

本文建立了薄板、薄扁壳及考虑剪切变形板壳的概率变分原理及概率广义变分原理,它们是建立概率有限元法、概率有限条法及各种概率样条函数方法的理论基础。     

A shell element for the prediction of residual load-carrying capacities due to delamination

This paper deals with layered plates and shells subjected to static loading. The kinematic assumptions are extended by a jump function in dependence of a damage parameter. Additionally, an intermediate layer is arranged at any position of the laminate. This allows numerical simulation of onset and growth of delaminations. The equations of the boundary value problem include besides the equilibrium in terms of stress resultants, the local equilibrium in terms of stresses, the geometric field equations, the constitutive equations, and a constraint which enforces the correct shape of a superposed displacement field through the thickness as well as boundary conditions. The weak form of the boundary value problem and the associated finite element formulation for quadrilaterals is derived. The developed shell element possesses the usual 5 or 6 degrees of freedom at the nodes. This is an essential feature since standard geometrical boundary conditions can be applied and the elements are applicable to shell intersection problems. With the developed model, residual load-carrying capacities of layered shells due to delamination failure are computed.     

Transient analysis of FGM and laminated composite structures using a refined 8-node ANS shell element

In this paper, we investigate the vibration analysis of functionally graded material (FGM) and laminated composite structures, using a refined 8-node shell element that allows for the effects of transverse shear deformation and rotary inertia. The properties of FGM vary continuously through the thickness direction according to the volume fraction of constituents defined by sigmoid function, but in this method, their Poisson’s ratios of the FGM plates and shells are assumed to be constant. The finite element, based on a first-order shear deformation theory, is further improved by the combined use of assumed natural strains and different sets of collocation points for interpolation the different strain components. We analyze the influence of the shell element with the various location and number of enhanced membrane and shear interpolation. Using the assumed natural strain method with proper interpolation functions the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. The natural frequencies of plates and shells are presented, and the forced vibration analysis of FGM and laminated composite plates and shells subjected to arbitrary loading is carried out. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To validate and compare the finite element numerical solutions, the reference solutions of plates based on the Navier’s method, the series solutions of sigmoid FGM (S-FGM) plates are obtained. Results of the present theory show good agreement with the reference solutions. In addition the effect of damping is investigated on the forced vibration analysis of FGM plates and shells.     

样条函数配点法分析薄板的压曲及自由振动与薄壳的稳定问题

本文采用单五次B一样条函数配点分析了薄板的压曲以及自由振动,用九次样条函数配点分析了几种薄壳的稳定问题。计算了不同的例题并与解析解进行比较,证明样条函数配点法是分析板、壳稳定问题的简便、有较的方法。     

A refined non‐linear non‐conforming triangular plate/shell element

A refined non‐conforming triangular plate/shell element for geometric non‐linear analysis of plates/shells using the total Lagrangian/updated Lagrangian approach is constructed in this paper based on the refined non‐conforming element method for geometric non‐linear analysis. The Allman's triangular plane element with vertex degrees of freedom and the refined triangular plate‐bending element RT9 are used to construct the present element. Numerical examples demonstrate that the accuracy of the new element is quite high in the geometric non‐linear analysis of plates/shells. Copyright © 2003 John Wiley & Sons, Ltd.     

Refined 15‐DOF triangular discrete degenerated shell element with high performances

A refined discrete degenerated 15‐DOF triangular shell element RDTS15 with high performances is proposed. For constructing the element displacement function, the exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary, and the re‐constitute method for shear strain matrix is adopted. The proposed element can be used in the analysis of both moderate thick and thin plates/shells. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements, and also passed the inf–sup test for free cylindrical shell problems and satisfied both the bending‐ and membrane‐dominated test. Copyright © 2004 John Wiley Sons, Ltd.     

复合材料板壳分析的样条有限点法

本文提出一个分析复合材料板壳的样条有限点法，建立了静力分析、热效应分析、稳定性分析及动力分析的新计算格式。这个方法是以B样条函数、高阶剪切变形理论及变分原理为基础而建立的，由于采用样条离散化，不存在剖分协调问题。利用这个方法分析复合材料板壳，不仅计算简便，而且精度也高。     

Non-linear coupled multi-field mechanics and finite element for active multi-stable thermal piezoelectric shells

In this paper, a coupled multi-field mechanics framework is presented for analyzing the non-linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations in thermal environments. The mechanics incorporate coupling between mechanical, electric and thermal fields and encompass geometric non-linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and are combined with the kinematic assumptions of a mixed-field shear-layerwise shell laminate theory. A finite element methodology and an eight-node coupled non-linear shell element are developed. The discrete coupled non-linear equations of motion are linearized and solved, using an extended cylindrical arc-length method together with a Newton–Raphson technique, to enable robust numerical predictions of non-linear active shells transitioning between multiple stable equilibrium paths. Validation and evaluation cases on laminated cylindrical strips and cylindrical panels demonstrate the accuracy of the method and its robust capability to predict non-linear response under thermal and piezoelectric actuator loads. Moreover, the results illustrate the capability of the method to model piezoelectric shells undergoing large shape changes by actively jumping between stable equilibrium states and quantify the strong relationship between shell curvature, applied electric potential, applied temperature differential and induced shape change. Copyright © 2008 John Wiley & Sons, Ltd.     

Hybrid stress finite element model for non-linear shell problems

The present paper describes a hybrid stress finite element formulation for geometrically non-linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger-Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi-field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non-linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton-Raphson iteratioin. Numerical examples presented in the paper include complete snap-through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non-linear shell problems of high complexity.     

An eighteen-node solid element for thin shell analysis

An eighteen-node, three-dimensional, solid element with 54 degrees of freedom is presented for the finite element analysis of thin plates and shells. The element is based on the Hellinger-Reissner principle with independent strain. The assumed independent strain is divided into higher and lower terms. The stiffness matrix associated with the higher order independent strain plays the role of stabilization matrix. A modified stress-strain relation decoupling inplane and normal strain is used to incorporate thin shell behaviour. Numerical results demonstrate that, with a properly chosen set of assumed strain, this element is effectively free of locking even for very thin plates and shells.     

A hybrid equilibrium element for folded plate and shell structures

This paper extends hybrid equilibrium formulation concepts, previously used with success for planar problems, to the analysis of folded plates and curved shells. A 2D hybrid equilibrium flat shell quadrilateral element is formulated for linear analysis, where detailed consideration is given to the implication of slope discontinuity when the element is used for non‐planar domains. Benchmark plate bending, folded plate and curved shell problems are modelled using equilibrium and conforming elements for comparison. In models of the latter two problems, torsional moments may be released along lines of slope discontinuity, and the effects of this assumption for the folded plate are studied by analysing a third type of model composed of 3D solid brick elements. The comparisons demonstrate an excellent performance from the new hybrid equilibrium analysis method for folded plates and curved shells. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary element analysis of reinforced shear deformable shells

In this paper, stiffened shear‐deformable shells are analysed using the boundary element method. Coupled boundary integral equations are presented for describing curved shells under general loading conditions. The equations are based on boundary integral equations for plane stress and plate bending, with coupling terms arising from the curvature of the shell. Domain integrals are transformed into boundary integrals using the dual reciprocity technique. Stiffeners are modelled as curved beams, continuously attached to the shell. Numerical solutions calculated using the present method are compared with finite element results in two examples. Copyright © 2002 John Wiley & Sons, Ltd.     

Axisymmetric nonlinear analysis of shallow spherical shells by a combined boundary element and finite difference method

This paper is concerned with the development of the mixed boundary element method and finite element method for the analysis of spherical annular shells under axisymmetric loads. The boundary element techniques are used to solve the equilibrium equation of shells and the central difference operator is adopted to deal with the compatibility equations. Iterative techniques are used throughout the analysis procedure. A number of numerical examples are given in the paper to illustrate the validity of the present approach.     

Finite element method for non-linear forced vibrations of circular plates

Geometric non-linearities for large amplitude free and forced vibrations of circular plates are investigated. In-plane displacement and in-plane inertia are included in the formulation. The finite element method is used. An harmonic force matrix for non-linear forced vibration analysis is introduced and derived. Various out-of-plane and in-plane boundary conditions are considered. The relations of amplitude and frequency ratio for different boundary conditions and various load conditions are presented.