A Solid Element Tailored for 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.

In the previous paper the rigid flat plate element was derived for the discrete limit analysis of thin shell structures of arbitrary shape, and numerical results under the assumption of small deformations were shown on the standard elasto-plastic problems. In general the effect of geometrical nonlinearity such as large deformations and buckling behaviors is one of the main factors which determine the ultimate strength of thin-walled structures, however, the method for the general treatment of geometrical nonlinearity is not necessarily given in the conventional plastic analysis.In the present report the general computational algorithm for the discrete limit analysis of thin-walled plate and shell structures is presented, by using the flat rigid plate element of triangular shape. The present algorithm is characterized by the following statements :(1) The discrete analysis on the limit strength is possible for arbitrary thin-walled structural members, in which the effects of geometrical as well as material nonlinearities are taken into account.(2) In the application to the structural stability problems a family of simulation models with low degrees of freedom can be derived, including Shanley model in the plastic buckling problem of columns and Yoshimura buckle pattern observed in the non-axisymmetric buckling behavior of circular cylindrical shells, which can be effectively used in qualitative investigations on the instability phenomena of thin-walled structural members.(3) The formulation expressed in terms of stresses and strains is adopted instead of resultant forces, so that all kinds of constitutive equations can be easily introduced and the application to general material-nonlinear problems such as visco-elasto-plastic problems is possible.

Shape sensitivity analysis of thin-shell structures

A finite element model to the analysis of thin shell structures with cutouts/inclusions is developed based on shear deformation theory. An eight-noded isoparametric shell element is considered with six degrees of freedom at each node. A convergence study is under taken to arrive at a suitable finite element mesh. The loading conditions include uniform tension, uniform pressure. The finite element results presented in this work are shown to be in good agreement with available analytical and experimental data and the direction for future work is also given.

Further development of three dimensional tri-linear elements based on a modified enhanced strain methodology is presented. The new formulation employs Taylor expansions of the derivatives of the isoparametric and enhanced shape functions in local coordinates. With this approach, only nine enhanced modes are needed for developing a dilatational-locking free element. Furthermore, the formulation permits a symbolic integration of the element tangent matrix, and more efficient static condensation procedure due to uncoupling of the enhanced modes. Good results in the analysis of thin shell structures, using only one 3D element in the thickness direction of the shell, are also presented.

This thesis presents new numerical models for analysis of thin beam and arch structures and thin plate and shell structures based on the combined finite-discrete element method (FDEM). The new models perform quasistatic and dynamic analysis. The thesis also presents an overview of the established numerical tools for the analysis of arch and shell structures, together with their strengths and disadvantages. The bases of the combined finite-element method and its strengths are also presented. The main motivation was the development of precise, robust and computationally efficient numerical models. The model for the analysis of thin beam and arch structures uses straight two-noded rotation free finite elements, and takes into account linear-elastic material behaviour, finite displacement, finite rotations and small strains. The thesis describes axial and bending carrying mechanisms of a beam finite element in detail. The presented model is implemented into the open source FDEM package ‘Yfdem’. Performance of the new numerical model is demonstrated on simple benchmark tests by a comparison with known analytical and numerical results. Its application for the seismic analysis of arch structures is presented. This thesis also presents a new numerical model for analysis of thin plate and shell structures. The model uses three-noded triangular finite elements, and takes into account linear-elastic material behaviour, finite rotations, finite displacements and small strains. The thesis describes membrane and bending carrying mechanisms for triangular finite element in detail. The model is implemented into the open source FDEM package ‘Yfdem’. Performance of the new numerical model was demonstrated on simple benchmark tests by comparing it with the known analytical and numerical results. Its application in typical problems of stability of plate and shell structures is presented. Directions for further development and upgrade of the model in performing contact interaction and material nonlinearity are given.

In the unpredictable world of various natural phenomena under which the shell structures are designed to function, their intended strength and dynamic stability may become negotiable when subjected to unknown parameters. This dilemma for the structural engineer is a common feature of most marine shell constructions. The general governing equations of equilibrium and dynamic stability for thin shell structures subjected to follower loads and undergoing large deformations have already been developed for the general shell defined in a monoclinically convected coordinate system.In a previous report, the complex peninsular shaped instability regions of a shell at various disturbed equilibrium positions have been numerically illustrated. This paper expands on the same direction and reaches at a demonstration of the disturbance threshold curves for the dynamic stability of a shell. The numerical results for subsequent equilibrium positions show that the shell stability slips suddenly over a threshold point of dynamic equilibrium from the heteronomous state to an autonomous state of self-sustained motions. Beyond this point the dynamic behaviour becomes highly irregular until the static critical point is reached, at which deep shells generally fail and the shallow shells enter the next phase of post-collapse stability.

The article is devoted to a further analysis of the natural vibrations of inhomogeneous shells under the action of static loads. The method of investigation is based on a unified methodology that combines the problems of static stability and the vibrations of elastic shells. The problems of natural vibrations take into account the presence of a prestressed state of the shell structure from the action of static loads. The presence of a static load significantly affects the spectrum of the natural frequencies of the shell. This approach allows us to determine the critical load by the dynamic criterion.The method of investigating of inhomogeneous shells is based on the uniform methodological positions of the 3-d geometrically nonlinear theory of thermoelasticity and the finite-element method in the form of the moment finite-element scheme. So, a thin shell is considered by this method as a three-dimensional body which is modeled throughout the thickness by one isoparametric solid finite element with multilinear shape functions.Two nonclassical hypotheses are used to describe the stress–strain state of a thin inhomogeneous shell. The kinematic hypothesis of deformed straight line in the thickness direction: though stretched or shortened during deformation, a straight segment along the thickness remains straight. This segment is not necessarily normal to the mid-surface of the shell. The displacements are assumed distributed linearly along the thickness, which is conventional in the theory of thin shells. The static hypothesis compressive assumes that the stresses in the fibers are constant throughout the thickness of the shell.Modal analysis of a shallow ribbed panel demonstrates the effectiveness of the developed method. The natural frequencies and mode shapes are determined at each increment of static loading.

An unsymmetric 8-node hexahedral solid-shell element based on ANS and incompatible concepts for thin shell analysis

Piezocomposites are attracting widespread interest since they can offer greater flexibility and better performances in specific applications with respect to traditional piezoelectric wafers. Design of piezocomposites requires accurate homogenisation models for the prediction of the equivalent electro-mechanical properties. In macro-scale models of structures with piezocomposite transducers, these properties are adopted in order to avoid the complexity of the piezocomposite microstructure. In the case of smart structures the accurate modelling of the actuation and the response of the structure is of primary importance. If classical structural finite elements are not sufficiently accurate, higher order or solid elements should be adopted. Thanks to adaptation or mixed-dimensional methods, it is possible to adopt computationally expensive higher order or solid elements only in some sub-domains of the structure. In this work, the modelling of smart structures equipped with thin piezoelectric transducers is considered in a multi-scale framework. Micromechanical homogenisation models are developed and employed for the prediction of the equivalent properties of piezocomposites. A micromechanical model based on the concept of inclusion is proposed to investigate the influence of the shape of the inclusions, of the constituent materials and of the polarisation on the equivalent properties. It has been found that fibre-shaped inclusions should be considered in order to obtain piezocomposites with strong piezoelectric effect and to have at the same time high direction-dependence. The equivalent properties of Macro Fiber Composites are determined via the Asymptotic Homogenisation Method (AHM) with an analytical solution and with a numerical solution via FEM which takes into account the effect of the electrodes. AHM analytical solution is adopted to investigate the effect of the material properties of the matrix on the overall piezocomposite. Results indicate that low values of the Young's modulus and of the Poisson's ratio yield high directional dependence in the piezoelectric properties. A laminated design with anisotropic layers and a piezocomposite layer is investigated via UFM. A configuration with maximum directional dependence in terms of equivalent piezoelectric strain constants is proposed, whereas maximum directional dependence in terms of piezoelectric stress constants is proved to be not achievable with such a design. Hierarchical finite elements for structural analyses based on a Unified Formulation (UF) by Carrera are developed and coupled via the Arlequin method proposed by Ben Dhia. Solid, plate and beam finite elements for mechanical and for piezoelectric problems are presented. Via UF, higher order and piezoelectric elements can be formulated straightforwardly. These elements are combined in variable kinematic solutions in the Arlequin framework. Higher-order elements are adopted locally where the stress field is three-dimensional, whereas the remaining parts of the structure are modelled with computationally cheap lower-order elements. Two electro-mechanical coupling operator for the Arlequin method in the context of piezoelectric analyses are proposed. Results are validated towards monomodel solutions and three-dimensional analytical and numerical reference solutions. Accurate solutions are obtained reducing the computational costs

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.

Donnell’s thin shell theory and basic equations based on the wave propagation method discussed in detail here, is used to investigate the natural frequencies of thin finite length circular cylindrical shells under various boundary conditions. Mode shapes are drawn to explain the circumferential mode numbern and axial mode numberm, and the natural frequencies are calculated numerically and compared with those of FEM (finite element method) to confirm the reliability of the analytical solution. The effects of relevant parameters on natural frequencies are discussed thoroughly. It is shown that for long thin shells the method is simple, accurate and effective.

A curved finite element for general thin shell structures

The paper introduces a numerical shape-finding procedure for very thin continuous shells (thickness less than 10 cm): an assigned loading distribution is applied on a point-wise suspended membrane, which, after its deformation, is inverted to give the geometry of a reinforced concrete continuous shell. The procedure is an extension to the continuum case of previous ones introduced for gridshells. In the past it has been employed in an empirical way by several designers. The geometrically non-linear analysis of the membrane is carried out taking account of large displacements and finite rotations. Finally, the structural analysis of the shell, including stability aspects, is performed, for example, to check and control the membranal behaviour of the structure.

A novel approximate theoretical solution is presented in this paper for the boundary disturbance of thin elastic shells of revolution with a nearly spherical middle surface and constant thickness. The indicated solution is a procedure of successive corrections, which was originally used by S. D. Poisson in handling non‐linear differential equations and introduced into thin shell analysis by E. F. Burmistrov.