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.     

Linear constraint equations for continuous support conditions in finite element analysis

Discretised structural models such as by finite elements imply discretised support conditions. In some cases such as plates on elastic foundation or slabs on large interacting columns an improved formulation of the continuous support conditions is desirable. This can be achieved by means of linear constraint equations. The numerical treatment of linear constraints is discussed for the method of elimination of variables as well as for the method of Lagrange multipliers. Then specific constraint equations for different accuracy requirements are derived, which can be used to constrain rectangular flat shell elements of arbitrary shape functions. These constraints introduce six generalized displacements according to the rigid body motions of the element and transmit the corresponding generalized reactions on the nodal degrees of freedom in a way consistent with distributed reactions. The effect on the strain energy of a square shell element is shown for the different constraint equations. As an application, the linear constraints are used to represent the continuous interaction of columns with the plate in a flat slab structure. Comparison of the finite element solutions with analytical results shows that the derived constraint equations allow a considerably improved formulation of continuous support conditions.     

Stability of cylindrical shell panels subjected to follower forces based on a mixed finite element formulation

A mixed finite element formulation is developed from a weak variational priniciple. This formulation is applied to stability analysis of cylindrical shell structures subjected to follower loading. Bilinear trial functions are used for all field variables. The rectangular curved elements presented here satisfy the continuity requirements for the field variables at the element interface. Two examples of a cantilevered cylindrical shell panel under different kinds of loading are solved.     

Thin shell and surface crack finite elements for simulation of combined failure modes

In this study we present a new approach to analyse cracked shell structures subjected to large geometric changes. It is based on a combination of a rectangular assumed natural deviatoric strain thin shell finite element and an improved linespring finite element. Plasticity is accounted for using stress resultants. A power law hardening model is used for shell and linespring material. A co-rotational formulation is employed to represent nonlinear geometry effects. With this, one can carry out nonlinear fracture mechanics assessments in structures that show instabilities due buckling (local/global), ovalisation and large rigid body motion. By numerical examples it is shown how geometric instabilities and fracture compete as governing failure mode.     

A finite element formulation for shells of negative Gaussian curvature

An expression for the strain energy of a shell of negative Gaussian curvature, including thickness shear deformations and without neglecting z/R in comparison with unity, is derived. Then a curved trapezoidal finite element formulation based on the principle of minimum potential energy is obtained. The shell element has eight nodes with 40 degrees of freedom and at each node there are three displacements and two rotations. The formulation is applicable for both thin and moderately thick shell analysis. The performance of this finite element is verified by applying it to some problems existing in the literature.     

A finite element formulation for large amplitude flexural vibrations of thin rectangular plates

Large amplitude flexural vibrations of rectangular plates are studied in this paper using a direct finite element formulation. The formulation is based on an appropriate linearisation of strain displacement relations and uses an iterative method of solution. Results are presented for rectangular plates with various boundary conditions using a conforming rectangular element. Whenever possible the present solutions are compared with those of earlier work. This comparison brings out the superiority of the proposed formulation over the earlier finite element formulation.     

The simulation of sheet metal forming processes via integrating solid-shell element with explicit finite element method

Solid-shell elements can be seen as a class of typical double-surfaced shell elements with no rational degrees of freedom, which are more suitable for analyzing double-sided contact problems than conventional shell elements. In this study, a solid-shell finite element model is implemented into the explicit finite element software ABAQUS/Explicit as a user-defined element, through which the sheet metal forming processes are simulated. The main feature of this finite element model is that the solid-shell element formulation is embedded into an explicit finite element procedure, compared to the previous studies on the solid-shell elements under the implicit finite element framework. To obtain a straightforward element, a complete integration scheme is adopted. No loss of generality, a twelve-parameter enhance assumed strain method is employed to improve the element’s behavior. Two benchmarks from the NUMISHEET conference and a U-channel roll-forming process are simulated with this explicit solid-shell finite element model. The calculated results are comparable with experimental and numerical results presented in the literatures.     

Hybrid strain based three node flat triangular shell elements—II. Numerical investigation of nonlinear problems

In a companion paper [M. L. Liu and C. W. S. To, Comput. Struct. 54, 1031–1056 (1995)] theories and incremental formulation of nonlinear shell structures discretized by the finite element method are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Based on the theory and incremental formulation explicit element stiffness and mass matrices of three node flat triangular shell finite elements are derived. In the present paper the derived element matrices are applied to nine examples. The latter include static and dynamic response analysis of shell structures with geometrical, material, and geometrical and material nonlinearities. The formulation adopted and element matrices derived are found to be accurate, flexible and applicable to various types of shell structures with geometrical and material nonlinearities.     

Non-linear strain-displacement equations exactly representing large rigid-body motions. Part III. Analysis of TM shells with constraints

The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko-Mindlin-type (TM) shell elements has been considered in Part I of the present work. In Part II it has been developed an enhanced mixed finite element formulation that allows using load increments that are much larger than possible with existing geometrically exact displacement-based shell element formulations. In this paper the developed formulation is employed to solve frictionless contact problems for TM shells undergoing finite deformations and interacting with rigid bodies. The contact conditions are incorporated into the assumed stress-strain TM shell formulation by applying a perturbed Lagrangian procedure with the fundamental unknowns consisting of 6 displacements and 11 strains of the bottom and top surfaces of the shell, 11 conjugate stress resultants and the Lagrange multiplier, associated with a nodal contact force, through using the non-conventional technique. The efficiency and accuracy of the proposed finite element formulation are demonstrated by means of several numerical examples.     

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.     

Fitting Polynomial Volumes to Surface Meshes with Voronoï Squared Distance Minimization

We propose a method for mapping polynomial volumes. Given a closed surface and an initial template volume grid, our method deforms the template grid by fitting its boundary to the input surface while minimizing a volume distortion criterion. The result is a point‐to‐point map distorting linear cells into curved ones. Our method is based on several extensions of Voronoi Squared Distance Minimization (VSDM) combined with a higher‐order finite element formulation of the deformation energy. This allows us to globally optimize the mapping without prior parameterization. The anisotropic VSDM formulation allows for sharp and semi‐sharp features to be implicitly preserved without tagging. We use a hierarchical finite element function basis that selectively adapts to the geometric details. This makes both the method more efficient and the representation more compact. We apply our method to geometric modeling applications in computer‐aided design and computer graphics, including mixed‐element meshing, mesh optimization, subdivision volume fitting, and shell meshing.     

Isoparametric axisymmetric shell elements with temperature gradients for heat conduction

This paper presents a finite element formulation for axisymmetric shell heat conduction where temperature gradients through the shell thickness are retained as primary nodal variables. The element geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the middle surface nodal point normals. The element temperature field is approximated in terms of element approximation functions, the nodal temperature, and the nodal temperature gradients. The weak formulation of the two-dimensional Fourier heat conduction equation in cylindrical coordinate system is constructed. The finite element properties of the shell element are then derived using the weak formulation and the element temperature field approximation. The formulation permits linear temperature gradients through the shell thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the element. Furthermore, the element can also have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented and a comparison is made with the theoretical results.     

Stability and phase speed for various finite element formulations of the advection equation

This paper analyzes the stability and accuracy of various finite element approximations to the linearized two-dimensional advection equation. Four triangular elements with linear basis functions are included along with a rectangular element with bilinear basis functions. In addition, second-and fourth-order finite difference schemes are examined for comparison. Time is discretized with the leapfrog method. The criss-cross triangle formulation is found to be unstable. The best schemes are the isosceles triangles with linear functions and the rectangles with bilinear basis functions.     

Sensitivity of structural joint stiffnesses with respect to beam properties: A hybrid approach

In the optimization of frame structures the sizes of the beam members change, as do the stiffnesses of the joints where such members meet. In this paper a method for calculating the design sensitivities of structural joints to changes in the size of the members, using a finite element formulation, is presented. The method uses the initial joint stiffness, predicted from a more detailed shell finite element model or experimental data, to calculate the design sensitivies for any number of joint members. The method is developed into a computer program that does not require a finite element model of the joint. The formulation of the method and a test case are presented.     

A stress analysis technique for plate and shell built-up structures with junctions and its application to minimum-weight design of stiffened structures

A finite element formulation using the penalty function method to analyse exactly the junctions of plate and shell built-up structures is suggested for an isoparametric shell element. The connectivity condition at the junction is added to the potential energy functional by the penalty parameter and the interpolating function of displacements. This formulation yields an integral-type stiffness matrix of the special junction elements, which can directly evaluate the surface tractions at the junction. For applying the technique suggested here to the optimum design of structures with junction parts, a design sensitivity analysis formulation for the adhesive special element is also developed. The technique is applied to the minimum-weight design problems of isotropic and composite laminated plates with a stiffener subjected to stress constraints.     

考虑刚-柔-热耦合的板结构多体系统的动力学建模

对热载荷作用下中心刚体与大变形薄板多体系统的动力学建模问题进行研究.基于Kirchhoff假设,从格林应变和曲率与绝对位移的非线性关系式出发,推导了非线性广义弹性力阵,用绝对节点坐标法建立了大变形矩形薄板的有限元离散的动力学变分方程.为了考虑刚体姿态运动、弹性变形和温度变化的相互耦合作用,推导了热流密度与绝对节点坐标之间的关系式.引入系统的运动学约束方程,建立了中心刚体-矩形板多体系统的考虑刚-柔-热耦合的热传导方程和带拉格朗日乘子的第一类拉格朗日动力学方程.为了有效地提高计算效率,将改进的中心差分法和广义-α法相结合,求解热传导方程和动力学方程,差分后的方程通过牛顿迭代法耦合求解.对刚-柔耦合和刚-柔-热三者耦合两种模型的仿真结果进行比较表明,刚体运动对温度梯度和热变形的影响显著.此外,本文建模方法考虑了几何非线性项,因此也考虑了热膨胀引起的轴向变形对横向变形的影响.     

Analysis of laminated shells with laminated stiffeners using rectangular shell finite elements

A finite element analysis of laminated shells reinforced with laminated stiffeners is described in this paper. A rectangular laminated anisotropic shallow thin shell finite element of 48 d.o.f. is used in conjunction with a laminated anisotropic curved beam and shell stiffening finite element having 16 d.o.f. Compatibility between the shell and the stiffener is maintained all along their junction line. Some problems of symmetrically stiffened isotropic plates and shells have been solved to evaluate the performance of the present method. Behaviour of an eccentrically stiffened laminated cantilever cylindrical shell has been predicted to show the ability of the present program. General shells amenable to rectangular meshes can also be solved in a similar manner.     

Thin element applications of a new formulation for C structural elements

A new formulation was proposed recently for the removal of the shear and membrane locking mechanisms from the finite element equations of the structural C⁰ shell, plate and beam elements. The use of full integration with the proposed formulation does not allow development of the zero energy modes or the softening effects, usually associated with the use of the technique of reduced integration in C⁰ plate and shell element applications. In the present paper a beneficial side effect of the new formulation is presented with regard to the development of the purely machine dependent locking. Questions concerning the introduction of softening effects by the new formulation in some flat C⁰ plate/shell element applications are addressed.     

Dynamic response of plate systems by combining finite differences, finite elements and laplace transform

A numerical method for the determination of the dynamic response of large rectangular plates or plate systems to lateral loads is proposed. The method is a combination of the finite difference method, the finite element method and the Laplace transform with respect to time. The plate system is considered as an assemblage of a small number of big rectangular superelements whose stiffness matrices are derived with the aid of the finite difference method in the Laplace transform domain. These superelements are then used to formulate and solve the problem by the finite element method in the transformed domain. The dynamic response is finally obtained by a numerical inversion of the transformed solution. External viscous or internal viscoelastic damping as well as the elastic foundation interaction effect can easily be taken into account. The method is illustrated and its merits demonstrated by means of numerical examples.     

On the finite element formulation of bifurcation mode of creep buckling of axisymmetric shells

In this paper, a finite element formulation is given in detail for the creep buckling of an axisymmetric shell. A special emphasis is placed on the bifurcation mode of creep buckling. A bifurcation point is determined by examining the shape of the potential energy in the vicinity of an axisymmetric equilibrium state obtained from a creep deformation analysis in the prebuckling stage. To illustrate the capability of the finite element formulation, a numerical example is presented for the creep buckling of a shallow spherical shell subjected to a uniform external pressure. In this analysis, not only the axisymmetric snap-through type but also the asymmetric bifurcation one are considered as buckling modes.