E.K. Koltsakis

On the numerical treatment of nonconvex energy problems of mechanics

The present paper presents three numerical methods devised for the solution of hemivariational inequality problems. The theory of hemivariational inequalities appeared as a development of variational inequalities, namely an extension foregoing the assumption of convexity that is essentially connected to the latter. The methods that follow partly constitute extensions of methods applied for the numerical solution of variational inequalities. All three of them actually use the solution of a central convex subproblem as their kernel. The use of well established techniques for the solution of the convex subproblems makes up an effective, reliable and versatile family of numerical algorithms for large scale problems. The first one is based on the decomposition of the contigent cone of the (super)-potential of the problem into convex components. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. The third one is based on the fact that nonconvexity in mechanics is closely related to irreversible effects that affect the Hessian matrix of the respective (super)-potential. All three methods are applied to solve the same problem and the obtained results are compared.

The B.E.M. in plane elastic bodies with cracks and/or boundaries of fractal geometry

The scope of the present paper is to present a method which examines the influence of the fractal geometry on the stress and strain fields in cracked plane elastic bodies through a B.E. scheme combined with an iterative approximation procedure. The method proposed here is based on the description of the fractal as the attractor of a deterministic or a random iterated function system. It is an iterative method which approximates the fractal boundary by classical C1-curves in order to avoid additional singularities. The method proposed may be seen as an extension of the classical B.I.E.M. to the case of bodies having cracks and/or boundaries of fractal geometry.

Interfacial debonding in composites via mathematical programming methods; the material inclusion problem for lubricated and non-lubricated interfaces

Abstract Debonding at the interfaces between the matrix and the material inclusions in composites is strongly nonlinear and is the main cause of the nonlinear behaviour of composite materials. The main difficulty in solution of debonding problems is that one does not know a priori the contact and noncontact regions and this inherent high nonlinearity prevents the effective use of a classical structural analysis approach. These problems are usually formulated as variational inequality problems which in the case of elastic matrix-elastic (or rigid) inclusion problems are equivalent to certain inequality constrained quadratic programming problems. Clear distinction is made between lubricated and non-lubricated interfaces, i.e. free and zero tangential sliding, respectively, and algorithms for both cases are given. These problems are solved by means of an appropriately modified optimization algorithm. The influence of some material parameters on debonding is shown.

Softening behaviour of continuous thin-walled steel beams. Two numerical methods

Abstract The problem of the computation of arbitrary multi-span continuous thin-walled beams with a softening moment resistance is addressed in the present work. Two numerical methods are exposed and comparison with the procedure that the Eurocode provides to this end is made. The application of the methods was stimulated by experimental evidence found in the literature, which establishes the existence of a softening branch in the bending of a thin-walled section before failure. The ability to perform exact numerical treatment of the softening behaviour allows full exploitation of the potential of the cross-section and, most importantly, the avoidance of pitfalls in the estimation of the ultimate strength.

Connections of CHS concrete-filled diagonals of X-bracings

Abstract The scope of the present paper is the numerical investigation of the behaviour of concrete-filled CHS bars used as X-bracings diagonals. The weak point for an X-brace is the point where the two diagonals intersect where, in order that the two bars remain co-planar, one of them is usually interrupted. In this case, a usual practice is to connect the interrupted bar to the continuous one by means of two gusset plates each one welded along a generatrix of the continuous bar. The result is, that the axial load is transferred as an inside-the-cross-section loading situation of the continuous bar thus causing a prohibitively excessive ovalisation of the cylindrical shell of the CHS. The in-plane transfer mechanism of the forces causes an increase of the axial flexibility of the interrupted bar by a factor of 50–300 thus making this type of connection too flexible and therefore unsuitable for use in X-braces. Filling the tubular braces with concrete is already known as a promising technique against local buckling: the subject of the present work is the assessment the effect of this technique to the axial flexibility of the interrupted bar. From the numerical simulation point-of-view, the problem is formulated as a unilateral contact problem between the concrete in-fill and the steel shell and a parametric analysis is made to elucidate flexibility and strength aspects of this connection for the most frequently used CHS profiles.

Three-dimensional adhesive contact laws with debonding: a nonconvex energy bundle method

Abstract In the present paper we develop a new numerical approach to the problem of structures having two or more parts of them adhesively bonded together like e.g. in sandwich structures. The adhesive material is idealized by a nonmonotone, possibly multivalued stress–strain law, which is three-dimensional (3D) and introduces a nonconvex nonsmooth energy function in the problem. The problem is formulated as a hemivariational inequality, whose solution(s) must render the potential energy substationary. We apply here the proximal bundle method and more specifically the optimization programme NSOLIB, based on first order polyhedral approximations of the locally Lipschitz continuous objective function. This algorithm permits the determination of at least one substasionarity point, e.g. of an equilibrium problem. An example of a 3D finite element model, illustrates the effectiveness of the proposed mehod.

Teaching Physics: Utilization of Scratchboard in Laboratories’ Activities

In this work we present two teaching modules, designed and realized using Scratchboard and Scratch. The two modules concern the measurement of the speed of sound in the air, by wave interference, and a consideration of oscillation’s phenomena. Both of the modules have been implemented in the school laboratory, providing reliable measurements and getting the students engaged in a higher level than the usual one.

Validation of a CHS connection used in X-bracing diagonals

Abstract The present paper investigates the behaviour of a connection widely used at the co-planar intersection of circular hollow sections (CHS), a case very common in X-bracings. The performance of this type of connection is validated by means of finite element modelling. Our connection is made up of two separate gusset plates, welded externally along the generatrices of the cylindrical surface of a CHS (in an ear-like manner), so as to accommodate the terminal gusset plates of the interrupted CHS: in this way the two bars intersect and remain on the same plane. Shell elements are used to model the CHS bar connection along several diameters on either side of the gusset plates (ears). The stiffness of the cylindrical shell that is called upon to undertake the axial force of the interrupted bar, drew our attention as it affects the structural behaviour of the X-brace well before local strength phenomena assume any importance. The actual result of this paper is that the stiffness of the interrupted length of the CHS bar is significantly less than that of the continuous bar: the interrupted bar of the X-bracing cannot develop any stresses whenever this type of connection is used, as any strain applied to its end will be cumulated in the connection zone. This effectively causes our X-brace to function with only one of its diagonals, i.e. the continuous one. Ending-up with a single-diagonal bracing can be dangerous even in the absence of material non-linearities or large displacement effects, not to mention an earthquake situation. These results are also compared with a formula derived by plane stress considerations and are found in good agreement. A total of 117 variants of this connection are examined for the purpose of this analysis and the stiffness reduction results are presented in a schematic form.