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This paper presents the generalized theory of the most important energy principles in structural analysis. All derive from two basic complementary theorems denoted as the principles of virtual displacements and virtual forces. Both exact and approximate methods are discussed and particular attention is paid to the derivation of upper and lower limits. The theory is not restricted to linearly elastic bodies but includes ab initio the effect of non‐linear stress‐strain laws and thermal strains. Finally the basic principles are illustrated on a number of simple examples in preparation for the more complex ones to appear in Parts II and III.

Purpose − The purpose of this paper is to propose a novel and simple strategy for construction of hybrid‐“stress function” plane element. Design/methodology/approach − First, a complementary energy functional, in which the Airy stress function is taken as the functional variable, is established within an element for analysis of plane problems. Second, 15 basic analytical solutions (in global Cartesian coordinates) of the stress function are taken as the trial functions for an 8‐node element, and meanwhile, 15 unknown constants are then introduced. Third, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of the displacements along element edges, which are interpolated by element nodal displacements. Finally, the whole system can be rewritten in terms of element nodal displacement vector. Findings − A new hybrid element stiffness matrix is obtained. The resulting 8‐node plane element, denoted as analytical trial function (ATF‐Q8), possesses excellent performance in numerical examples. Furthermore, some numerical defects, such as direction dependence and interpolation failure, are not found in present model. Originality/value − This paper presents a new strategy for developing finite element models exhibits advantages of both analytical and discrete method.

A discussion of teaching points for the graphical construction, and the analytical energy methods

Proposes two tentative projects for ordering the scientific concepts in relation to a systemic theory of knowledge, based on cognitive sciences. It tries to develop some basic ideas of Wiener in the framework of modern physics. The concept of complementarity is extended to ideologies. Cybernetics, it is assumed, could have an important role in the transformation of conflicts into synergetic trends.

The purpose of this paper is to analyse the accuracy of the thrust force of a linear actuator computed with different finite elements models.

A series of 2D and 3D models corresponding to different levels of approximation of the original problem are considered. A reliable error estimator based on dual magnetostatic formulations is used.

A 3D model does not necessarily ensure more accurate results than a 2D model. Because of limitations on the number of mesh elements, the discretisation error in 3D can be of the same order of magnitude as the error introduced by the 2D approximation.

The results emphasise the need to consider errors arising from different simplifications with respect to one another, in order to avoid improvements of the model increasing the complexity but not improving the accuracy of the results.

This teaching case intends to be a tool for academic purposes as a way to show the different assessments an investor should make and the many problems he/she may face, when evaluating a megaproject. It reviews the experience of two large corporations in Chile, intending to build a major hydroelectric generation project in Chile while facing major opposition from environmental NGOs and other stakeholders. Although in the view of many industry experts and consultants Hidroaysén was a good and necessary project, the environmental implications and some of the project’s stakeholders created a deadlock.

This teaching case was written with the idea of being used as a tool for classes in order to discuss the implications of environmental issues in big projects. The research was based on particular information of the project, financial data of the companies involved, and other public sources (news, interviews, etc.).

The conclusion of this case is that private initiatives, without the right alignment of political actors and civil society, could face the risk of being blocked and not being executed.

COP21 guidelines for responsive investment could be a guideline to follow, aligning private interest with development for countries in the third world.

We offer a way to analyze external impacts on a project of this kind, that using a common framework (COP21 guidelines) could avoid risks taking all considerations into the project.

Error indicators are introduced as part of a simple computational procedure for improving the accuracy of the finite element solutions for plate and shell problems. The procedure is based on using an initial (coarse) grid and a refined (enriched) grid, and approximating the solution for the refined grid by a linear combination of a few global approximation vectors (or modes) which are generated by solving two uncoupled sets of equations in the coarse grid unknowns and the additional degrees of freedom of the refined grid. The global approximation vectors serve as error indicators since they provide quantitative pointwise information about the sensitivity of the different response quantities to the approximation used. The three key elements of the computational procedure are: (a) use of mixed finite element models with discontinuous stress resultants at the element interfaces; (b) operator splitting, or additive decomposition of the finite element arrays for the refined grid into the sum of the coarse grid arrays and correction terms (representing the refined grid contributions); and (c) application of a reduction method through successive use of the finite element method and the classical Bubnov—Galerkin technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Bubnov—Galerkin technique. The similarities between the proposed computational procedure and a preconditioned conjugate gradient (PCG) technique are identified and are exploited to generate from the PCG technique pointwise error indicators. The effectiveness of the proposed procedure is demonstrated by means of two numerical examples of an isotropic toroidal shell and a laminated anisotropic cylindrical panel.

A computational procedure is presented for the efficient non‐linear dynamic analysis of quasi‐symmetric structures. The procedure is based on approximating the unsymmetric response vectors, at each time step, by a linear combination of symmetric and antisymmetric vectors, each obtained using approximately half the degrees of freedom of the original model. A mixed formulation is used with the fundamental unknowns consisting of the internal forces (stress resultants), generalized displacements and velocity components. The spatial discretization is done by using the finite element method, and the governing semi‐discrete finite element equations are cast in the form of first‐order non‐linear ordinary differential equations. The temporal integration is performed by using implicit multistep integration operators. The resulting non‐linear algebraic equations, at each time step, are solved by using iterative techniques. The three key elements of the proposed procedure are: (a) use of mixed finite element models with independent shape functions for the stress resultants, generalized displacements, and velocity components and with the stress resultants allowed to be discontinuous at interelement boundaries; (b) operator splitting, or restructuring of the governing discrete equations of the structure to delineate the contributions to the symmetric and antisymmetric vectors constituting the response; and (c) use of a two‐level iterative process (with nested iteration loops) to generate the symmetric and antisymmetric components of the response vectors at each time step. The top‐ and bottom‐level iterations (outer and inner iterative loops) are performed by using the Newton—Raphson and the preconditioned conjugate gradient (PCG) techniques, respectively. The effectiveness of the proposed strategy is demonstrated by means of a numerical example and the potential of the strategy for solving more complex non‐linear problems is discussed.

The expressions of the material derivative of differential forms in the language of vector analysis are introduced. These formulae allow us to describe naturally the electromechanical coupling, and the coupling term appears to be a volume integral. A general approach to compute forces is then proposed, which takes that fact into consideration. The method is applicable in 2D and 3D with dual formulations. Numerical evidences of its efficiency are given.

This unusual book gives a combined account of three different branches of dynamics, dealing with particles, fluids and solid bodies. The preface explains the history of the course at the Massachusetts Institute of Technology on which the book is based. When instruction in aeronautical engineering was started at M.I.T., some forty years ago, the branch of the subject that was most fully developed from a rational viewpoint was the theory of dynamic stability. The students, with only an elementary training in mathematics and mechanics, found this theory of stability too difficult to grasp, and a course was therefore started to bridge the gap between the mathematics and mechanics that the students already knew and the complex problems of aircraft stability.