José Cirne

Introduction to Finite Element Method

As discussed in Chap. 1, mechanic problems are governed by a set of partial differential equations that are valid in a certain domain and they needed to be solved for evaluating the stress condition of mechanical components. Although analytic methods can be employed to solve linear problems involving partial differential equations, its use to analyze complex structures may be a difficult or, even, an impossible task. Thus, in this chapter, Hamilton’s principle, which one of the most powerful energy principle, is introduced for the FEM formulation of problems of mechanics of solids and structures. The approach adopted in this chapter is to directly work out the dynamic system equations, after which the static dynamic equations can be easily obtained by simply dropping out the dynamic terms

Dynamic behaviour of cork and cork-filled aluminium tubes: Numerical simulation and innovative applications

Abstract Cork is a unique and complex natural cellular material with many industrial applications. The purpose of this paper is to explore a new application field for the use of micro-agglomerate cork as an energy-absorbing medium. A numerical study on the energy absorption capabilities of square and circular cork-filled aluminium tubes with a width or diameter of 80 mm, length of 300 mm and variable thickness was performed with the finite element method code LS-DYNA™. The tubes were impacted uniaxially at 10 and 15 m s-1. The same analysis was carried out on aluminium foam-filled tubes. The results demonstrate that cork filling leads to a considerable increase in the energy absorbed for both section geometries, and that tube thickness plays an important role in the deformation modes and energy absorption. The investigation revealed better results for aluminium foam-filled structures, but demonstrated that micro-agglomerate cork has high potential as an energy-absorbing medium in crash protection applications.

Engineering Computation of Structures: The Finite Element Method

This book presents theories and the main useful techniques of the Finite Element Method (FEM), with an introduction to FEM and many case studies of its use in engineering practice. It supports engineers and students to solve primarily linear problems in mechanical engineering, with a main focus on static and dynamic structural problems. Readers of this text are encouraged to discover the proper relationship between theory and practice, within the finite element method:Practice without theory is blind, but theory without practice is sterile. Beginning with elasticity basic concepts and the classical theories of stressed materials, the work goes on to apply the relationship between forces, displacements, stresses and strains on the process of modeling, simulating and designing engineered technical systems. Chapters discuss the finite element equations for static, eigenvalue analysis, as well as transient analyses. Students and practitioners using commercial FEM software will find this book very helpful. It uses straightforward examples to demonstrate a complete and detailed finite element procedure, emphasizing the differences between exact and numerical procedures.

Stresses from radial loads and external moments in spherical pressure vessels

Abstract In this paper analytical solutions for displacements and stresses in spherical shells over rectangular areas are developed. The analysis is based on spherical shallow shell equations, and solutions are obtained through the use of double Fourier series expressions to represent the displacement and loading terms. Three types of loading were considered: radial load, overturning moment and tangential shear. In order to test the results, an experiment and finite element investigation has been carried out to determine the state of stress in a spherical shell model and results are compared in graphs. The proposed solution is also applied to compute the local stresses around a support leg in a spherical shell, and results are compared with those obtained by the Bijlaard method used in American code ASME VIII, Part 2, and British code PD 5500: 2000.

Finite Element Method for Beams

A beam is a structural member whose geometry is very similar to the geometry of a bar. It is also geometrically a bar of an arbitrary cross-section, by bar it is meant that one of the dimensions is considerably larger than the other two, whose primary function is to support transverse loading. The main difference between the beam and the truss is the type of load that they support. In fact, beams are the most common type of structural component, especially in civil and mechanical engineering. A beam resists to transverse loads mainly through a bending action and, the bending is responsible for compressive longitudinal stresses in one side of the beam and tensile stress on the other beam side. These two regions are separated by the neutral axis in which the stress is zero. The combination of tensile and compressive stresses produces an internal bending moment. Finite element equations for beam-like structures are developed in this chapter.

Advanced FEM Modelling

This chapter presents a discussion on some modelling techniques for the stress analyses of solids and structures. Mesh symmetry, rigid elements and constraint equations, mesh compatibility, modelling of offsets, supports and connections between elements with different mathematical bases are all covered. Advanced modelling of laminated composite materials are also presented.

Finite Element Method for Membranes (2-D Solids)

The development of finite element equations for the stress analysis of two dimensional structures subjected to external loads that are applied within their 2-D geometrical plane will be presented in this chapter. The basic concepts, procedures and formulations can also be found in many existing textbooks [1–4]. The element developed is called membrane or 2D solid element. The finite element solution will solve only the selected mathematical model and that all assumptions in this model will be reflected in the predicted response. Thus, the choice of an appropriate mathematical model is crucial and completely determines the insight into the physical problem that we can obtain by this kind of analysis.

Finite Element Method for Plates/Shells

The development of finite element equations for the stress analysis of two dimensional structures subjected to external loads that are applied transversely to their 2-D geometrical plane will be presented in this chapter. The basic concepts, procedures and formulations can also be found in many existing textbooks [1–3]. The procedure followed in this chapter is to first develop the FE matrices for plate elements, and then the FE matrices for flat shell elements are obtained by superimposing the matrices for plate elements and those for 2D solid plane stress elements developed in Chap. 5. Whereas for general shell finite elements the displacement and the geometry interpolations are obtained by considering also the isoparametric concept.

Mechanics of Solids and Structures

Solid and structural mechanics deal with the elasticity basic concepts and the classical theories of stressed materials. Mechanical components and structures are under a stress condition if they are subjected to external loads or forces. The relationship between stresses and strains, displacements and forces, stresses and forces are of main importance in the process of modeling, simulating and designing engineered technical systems. This chapter describes the important relationships associated with the elasticity basic concepts and the classical mathematical models for solids and structures. Important field variables of solid mechanics are introduced, and the dynamic equations of these variables are derived. Mathematical models for 2D and 3D solids, trusses, Euler-beams, Timoshenko-beams, frames and plates are covered in a concise manner.

Finite Element Method for Trusses

A truss is a structural element that is designed to support only axial forces, therefore it deforms only in its axial direction. The cross-section of the bar can have arbitrary geometry, but its dimensions should be much smaller than the bar length. Finite element developments for truss members will be performed in this chapter. The simplest and most widely used finite element for truss structures is the well-known truss or bar finite element with two nodal points. Such kind of finite elements are applicable for analysis of skeletal type of truss structural systems both in two-dimensional and three-dimensional space. Basic concepts, procedures and formulations can also be found in a great number of existing books [1–3]. In skeletal structures consisting of truss members, the truss elements are linked by pins or hinges without any friction, so there are only forces that transmitted among bars, which means that no moments are transmitted. In the presentation of this concept it will be assumed that truss elements have uniform cross-section. These concepts can be easily extended to treat bars with varying cross-section. Moreover, from the mechanical viewpoint, there is no reason to use bars with a varying cross-section since the force in a bar is uniform.