Nicola Rizzi

Masonry as structured continuum

A structured continuum model is formulated to describe the behaviour of block masonry modelled as distinct rigid body systems with elastic interfaces. A correspondence between the two motions is obtained by postulating a relationship between the displacement fields of the continuum and the discrete models. The constitutive functions for the dynamic actions of the continuum are derived by equating the power of the two models.SommarioViene presentato un modello di continuo con struttura atto a descrivere il comportamento meccanico di murature a blocchi, pensate come un sistema di corpi rigidi con contatti puntuali elastici. Il moto del sistema discreto e di quello continuo sono messi in relazione postulando una corrispondenzatra i campi di spostamento. Le funzioni costitutive delle azioni dinamiche del modello continuo sono ricavate uguagliando la potenza meccanica spesa in moti corrispondenti.

On the effect of the local overall interaction on the postbuckling of uniformly compressed channels

Abstract On the basis of the general theory of elastic stability due to Koiter, postbuckling analysis of simply supported channels under uniform compression is performed. Attention is essentially focused on local/eularian and local/flexural-torsional simultaneous bucklin modes interaction. The column is treated as a plate assemblage. Linearized expressions for the displacement field are employed while assuming strain-displacements relationships that are linear for the curvatures and up to second order terms for the in-plane strains. The total potential energy is hence written up to third order terms in order to investigate asymmetric buckling phenomena. A discrete model is developed through an automatic procedure of algebraic manipulation, and an extensive parametric analysis is performed. After determining the range of geometric parameters which characterize different types of interaction, it is found that in the postbuckling range the local/eulerian interaction is more dangerous than the local/flexural-torsional one, due to the column higher imperfection sensitivity.

Nonstandard Models for Thin-Walled Beams With a View to Applications

A direct theory of a one-dimensional structured continuum is introduced in order to study the postbuckling behavior of thin-walled beams. A simply supported beam bent by end couples is analyzed showing that, in the case of nonsymmetric cross sections, lateral buckling gives rise to imperfection sensitivity. Then an axially loaded beam is studied taking also into account the interaction between torsional and flexural buckling. The results obtained prove that in this case imperfection sensitivity, though slighter than in the previous case, arises also for symmetric cross sections.

Symbolic manipulation in buckling and postbuckling analysis

Abstract A perturbation procedure for the buckling and postbuckling analysis of elastic structures is shown to be well suited to be implemented as an automatic symbolic manipulation procedure. The postbuckling analysis of a circular arch is considered as an example, and the asymptotic description of the bifurcated equilibrium path is given. The main purposes of the automatic procedure are to generate the representation of the Frechet operator for the strain field and to perform integration by parts. This allows the manipulation of correct expressions of the basic relationships, as the strain-displacement one, without introducing any simplifying assumption or restriction. The perturbation equations are automatically generated and a solution procedure leads to parametric expressions for the coefficients of the asymptotic expansion of the bifurcated path. The symbolic manipulation system used is REDUCE.

A Parametric Postbuckling Analysis of an Asymmetric Two-Bar Frame

ABSTRACT ABSTRACT The modified potential energy approach of flexurally elastic, axially, and shear undeformable beams is employed to investigate the buckling and (asymmetric) postbuckling behavior of the two-bar frame which was experimentally tested by Roorda in 1965. Following a Koiter-type energy approach, a suitable perturbation formulation is, derived from a hybrid functional which is obtained by adding to the potential energy certain extra terms which account for the energy associated with the internal joint forces acting on the beam. In this way the nonlinear constraints expressing continuity of displacement across the joints are accommodated by introducing joint forces acting as Lagrangian multipliers. The inherent simplicity of the model herein proposed allows a thorough parametric analysis to be performed without undue complications up to and including fourth-order terms. The general trend compares favorably with the experimental results. Previous analytical results limited to a third-order analy...

Bifurcation analysis of a circular arch under hydrostatic pressure

SommarioNel presente lavoro si studia la biforcazione di un arco circolare soggetto a pressione idrostatica nel campo elastico facendo uso di un modello di trave cinematicamente esatto. Le corrispondenti equazioni di campo nonlineari vengono risolte utilizzando una tecnica perturbativa. Vengono riportati in diagramma una serie di risultati numerici riguardanti la dipendenza del carico critico e del parametro di carico del secondo ordine dai parametri geometrici e meccanici.SummaryIn this paper the bifurcation analysis of a circular arch under hydrostatic pressure in the elastic postbuckling range is performed by means of a geometrically exact beam model. The relevant nonlinear field equations are solved by utilizing a perturbation technique. A number of numerical results regarding the dependence of the critical load and the second order load parameter on the geometric and mechanical parameters are plotted in diagrams.

On the proper treatment of axial and shear undeformability constraints in post-buckling analysis of beams

Zusammenfassung Das Modell des nichtausdehnbaren und unter Schub nichtverformbaren Tragers wird betrachtet um das Verhalten schlanker Bauwerke nach der Knickung zu untersuchen. Innerhalb des Rahmens einer Koiterschen Energiemethode wird eine best geeignete Perturbationsformulierung von einem gemischten Funktional hergeleitet. Letzteres erhalt man indem zur potentiellen Energie einige besondere Glieder addiert werden, die der von den inneren Kraften an den Tragerenden herruhrenden nichtlinearen Energie Rechnung tragen. Die Losung eines einfachen Tragerproblems wird zur Illustration angegeben.

Symmetric Bifurcation of Plane Frames through a Modified Potential Energy Approach

Abstract The modified potential energy approach to the nonlinear analysis of slender elastic beams is employed to investigate the buckling and postbuckling behavior of two symmetric plane frames, namely a hinged portal frame and a hinged two-bar frame, The beams are assumed to be inextensible and shear undeformable. The “hybrid” functional on which the present analysis is based according to a Koiter-type energy approach is obtained from the total potential energy functional by adding extra terms which account for the nonlinear energy associated with the internal forces applied at the beam ends. Thus the nonlinear constraints on the potential energy (expressed in terms of the cross-section rotation φ only), which are associated with the continuity condition relative to the axial and lateral displacement across the joints, are accounted for by introducing the corresponding joint forces as Lagrangian multipliers. The inherent simplicity of the proposed beam model facilitates the postbuckling analysis of the ...

The Effect of Multiple Buckling Modes on the Postbuckling Behavior of Plane Elastic Frames. Part I. Symmetric Frames

ABSTRACT The postbuckling behavior of a one-bay, two-storey frame with built-in edges and symmetric with respect to midspan is analyzed. Columns are assumed to be inextensible and shear-undeformable, and beams are rigid. Then two buckling modes are possible, that is, sidesway of the lower floor with rigid horizontal displacement of the top floor and sidesway of the top floor with the lower floor undergoing no displacement. Obviously, the two buckling modes occur simultaneously if the ratios EI/h7 (EI being the bending stiffness of a column and h its length) are properly selected. Within the framework of a Koiter-type energy approach a suitable perturbation formulation is derived from a “hybrid” functional which is obtained by adding to the potential energy certain extra terms which account for the nonlinear energy associated with the internal forces applied to the beam at the joints. Results show that the postbuckling behavior of a single buckling mode can be stable or unstable according to the value of t...

Using Symbolic Computation in Buckling Analysis

Asymptotic buckling analysis of elastic structures can be considered a well established procedure (Budiansky, 1974) . It consists in bifurcation analysis of a system of oneparameter differential equations: balance, compatibility and constitutive equations . The aim of this work was to experiment with the use of automatic symbolic computation in the asymptotic bifurcation analysis of elastic beams . To this end an application of the system REDUCE (Hearn, 1971) has been devised in particular for (a) the generation of the formal perturbation equations and (b) the construction of a procedure for solving a specific problem . Further details can be found in Rizzi & Tatone (1985) . A first assessment of the use of symbolic computation systems in structural mechanics can be found in Noor & Andersen (1979), while another application to the solution of perturbation problems is in Noor & Balch (1984) .