Non‐linear explicit dynamic analysis of shells using the BST rotation‐free triangle

The paper describes the application of the simple rotation‐free basic shell triangle (BST) to the non‐linear analysis of shell structures using an explicit dynamic formulation. The derivation of the BST element involving translational degrees of freedom only using a combined finite element–finite volume formulation is briefly presented. Details of the treatment of geometrical and material non linearities for the dynamic solution using an updated Lagrangian description and an hypoelastic constitutive law are given. The efficiency of the BST element for the non linear transient analysis of shells using an explicit dynamic integration scheme is shown in a number of examples of application including problems with frictional contact situations

Adaptive finite element strategies based on error assessment

Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error indicators. After that, a new spatial discretization must be defined via h, p or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are clearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics

Computational model for elasto-plastic and damage analysis of plates and shells

Shells and plates are very important for various engineering applications. Analysis and design of these structures is therefore continuously of interest to the scientific and engineering community. Accurate and conservative assessments of the maximum load carried by the structure, as well as the equilibrium path in both elastic and inelastic range are of paramount importance. Elastic behaviour of shells has been very closely investigated, mostly by means of the finite element method. Inelastic analysis on the other hand, especially accounting for damage effects, has received much less attention from the researchers. A computational model for finite element, elasto-plastic and damage analysis of homogenous and isotropic shells is presented here. The formulation of the model proceeds in several stages, described in the following chapters. First, a theory for thick spherical shells is developed, providing a set of shell constitutive equations. These equations incorporate the effects of transverse shear deformation, initial curvature and radial stresses. The proposed shell equations are conveniently used in finite element analysis. A simple C0 quadrilateral, doubly curved shell element is developed. By means of a quasi-conforming technique shear and membrane locking are prevented. The element stiffness matrix is given explicitly which makes this formulation computationally very efficient. The elasto-plastic behavior of thick shells and plates is represented by means of the non-layered model, with an Updated Lagrangian method used to describe a small strain geometric non-linearity. In the treatment of material non-linearities an Iliushin?s yield function expressed in terms of stress resultants is adopted, with isotropic and kinematic hardening rules. Finally, the damage effects modeled through the evolution of porosity are incorporated into the yield function, giving a generalized and convenient yield surface expressed in terms of the stress resultants. Since the elastic stiffness matrix is derived explicitly, and a non-layered model is employed in which integration through the thickness is not necessary, the current stiffness matrix is also given explicitly and numerical integration is not performed at any stage during the analysis. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power

General non-linear finite element analysis of thick plates and shells

AbstractA non-linear finite element analysis is presented, for the elasto-plastic behavior of thick shells and plates including the effect of large rotations. The shell constitutive equations developed previously by the authors [Voyiadjis, G.Z., Woelke, P., 2004. A refined theory for thick spherical shells. Int. J. Solids Struct. 41, 3747–3769] are adopted here as a base for the formulation. A simple C0 quadrilateral, doubly curved shell element developed in the authors’ previous paper [Woelke, P., Voyiadjis, G.Z., submitted for publication. Shell element based on the refined theory for thick spherical shells] is extended here to account for geometric and material non-linearities. The small strain geometric non-linearities are taken into account by means of the updated Lagrangian method. In the treatment of material non-linearities the authors adopt: (i) a non-layered approach and a plastic node method [Ueda, Y., Yao, T., 1982. The plastic node method of plastic analysis. Comput. Methods Appl. Mech. Eng. 34, 1089–1104], (ii) an Iliushin’s yield function expressed in terms of stress resultants and stress couples [Iliushin, A.A., 1956. Plastichnost’. Gostekhizdat, Moscow], modified to investigate the development of plastic deformations across the thickness, as well as the influence of the transverse shear forces on plastic behaviour of plates and shells, (iii) isotropic and kinematic hardening rules with the latter derived on the basis of the Armstrong and Frederick evolution equation of backstress [Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger effect. (CEGB Report RD/B/N/731). Berkeley Laboratories. R&D Department, California.], and reproducing the Bauschinger effect. By means of a quasi-conforming technique, shear and membrane locking are prevented and the tangent stiffness matrix is given explicitly, i.e., no numerical integration is employed. This makes the current formulation not only mathematically consistent and accurate for a variety of applications, but also computationally extremely efficient and attractive

A rotation‐free shell triangle for the analysis of kinked and branching shells

This paper extends the capabilities of previous BST and EBST rotation‐free thin shell elements to the analysis of kinked and branching surfaces. The computation of the curvature tensor is first redefined in terms of the angle change between the normals at the adjacent elements. This allows to deal with arbitrary large angles between adjacent elements and to treat kinked surfaces. A relative stiffness between elements is introduced to consider non‐homogeneous surfaces. This idea is latter generalized to deal with branching shells. Several linear and non‐linear examples are presented showing that the formulation leads to the correct results. Copyright © 2006 John Wiley & Sons, Ltd

A finite difference method for the static limit analysis of masonry domes under seismic loads

The static limit analysis of axially symmetric masonry domes subject to pseudo-static seismic forces is addressed. The stress state in the dome is represented by the shell stress resultants (normal-force tensor, bending-moment tensor, and shear-force vector) on the dome mid-surface. The classical differential equilibrium equations of shells are resorted to for imposing the equilibrium of the dome. Heyman's assumptions of infinite compressive and vanishing tensile strength, alongside with cohesive-frictional shear response, are adopted for imposing the admissibility of the stress state. A finite difference method is proposed for the numerical discretization of the problem, based on the use of two staggered rectangular grids in the parameter space generating the dome mid-surface. The resulting discrete static limit analysis problem results to be a second-order cone programming problem, to be effectively solved by available convex optimization softwares. In addition to a convergence analysis, numerical simulations are presented, dealing with the parametric analysis of the collapse capacity under seismic forces of spherical and ogival domes with parameterized geometry. In particular, the influence that the shear response of masonry material and the distribution of horizontal forces along the height of the dome have on the collapse capacity is explored. The obtained results, that are new in the literature, show the computational merit of the proposed method, and quantitatively shed light on the seismic resistance of masonry domes

East African topography and volcanism explained by a single, migrating plume

Anomalous topographic swells and Cenozoic volcanism in east Africa have been associated with mantle plumes. Several models involving one or more fixed plumes beneath the northeastward migrating African plate have been suggested to explain the space-time distribution of magmatism in east Africa. We devise paleogeographically constrained global models of mantle convection and, based on the evolution of flow in the deepest lower mantle, show that the Afar plume migrated southward throughout its lifetime. The models suggest that the mobile Afar plume provides a dynamically consistent explanation for the spatial extent of the southward propagation of the east African rift system (EARS), which is difficult to explain by the northeastward migration of Africa over one or more fixed plumes alone, over the last ≈45 Myrs. We further show that the age-progression of volcanism associated with the southward propagation of EARS is consistent with the apparent surface hotspot motion that results from southward motion of the modelled Afar plume beneath the northeastward migrating African plate. The models suggest that the Afar plume became weaker as it migrated southwards, consistent with trends observed in the geochemical record