Vlasis K. Koumousis, Eleni N. Chatzi and Savvas P. Triantafillou

Institute of Structural Analysis and Aseismic Research
National Technical University of Athens
NTUA, Zografou Campus GR-15773, Athens, Greece
E-mail: vkoum@central. ntua. gr


The code “Plastique”, suitable for the inelastic dynamic analysis of R/C structures and its theoretical background are presented. Every structural entity is represented by a single nonlinear element through the implementation of a macro modeling approach. Three different types of 2D-macro elements are formulated namely; beams, columns and shear walls. The structural behavior of each element is evaluated using a flexibility formulation based on both element edge regions that follow a distributed plasticity law. A fiber model is used to define the monotonic strength envelope at each section. The hysteretic behavior of the structural elements is monitored by a smooth hysteretic model of Bouc-Wen type. This model is capable to express the stiffness degradation, strength deterioration and pinching phenomena which are observed in R/C elements under cyclic loading. Plane frames consisting of combinations of plane elements are linked at the levels of floors via diaphragms to assemble the 3D mathematical model of the structure. Solutions are obtained by direct integration of the equations of motion, while an iterative procedure is implemented to satisfy equilibrium at every time step. Finally, a damage analysis is performed using an appropriate damage model. The numerical examples presented herein the reveal the features of the proposed analysis scheme.

Keywords: inelastic analysis, time history, Bouc-Wen, damage index

1 Introduction

A problem of major importance in structural engineering deals with the response of R/C structures subjected to dynamic loading. For load factored linear elastic analysis, suggested by the codes, the results are quite satisfactory, but do not reveal the characteristics of the true behaviour of the structure. However, if inelastic response is taken into account, more refined models are needed as to achieve a realistic behaviour. In recent years, significant research has been carried out in order to overcome the difficulties arising in such an analysis. Difficulties emanate not only from the inherent complexity of R/C structures, but also from the uncertainties related to terms such as dynamical loading, material nonlinearity and hysteresis.

Macro-modeling of structures has been one of the main methods introduced to simulate these complex phenomena. In macro-modeling simulation, the field of knowledge concerning the actual behaviour of reinforced concrete is incorporated in the structure using an element-based approach. In such a way, the well established, from matrix structural analysis, beam element is enriched with a moment curvature envelope describing the behaviour of both end sections, a hysteretic law and a relevant yield penetration rule for the beam. By introducing such an elasto-plastic element, one is able to simulate the gradual shift of the mechanical properties of the element as it passes from the elastic to the inelastic region of its response. The overall behaviour of the structure is assessed using a proper damage index.


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 367-378. © 2006 Springer. Printed in the Netherlands.

An aspect of utmost importance, for a non linear analysis, is the hysteretic rule needed to model the cyclic response of the structure. Over the last twenty years, significant development has occurred in the so-called phenomenological approach of hysteresis. Beginning with Bouc’s original formulation (1967, 1969, 1971) of the single degree degrading hysteresis model with pinching, many modifications have been subsequently introduced, such as the Bouc-Wen model (1976, 1980), the Baber-Noori model (1985, 1986) and the Reinhorn model (1996). These hysteresis models – also known as smooth hysteretic models – are capable of simulating a number of different types of loops using a single smooth hysteretic function affected by a set of user-defined parameters. In doing so, one can easily model the three main phenomena describing the cyclic response of R/C elements namely; stiffness degradation, strength deterioration and pinching behaviour due to bond-slip effects.

Following these rules, many computer programs have been developed, capable to perform a non-linear structural analysis such as DRAIN-2D (Kanaan and Powell,1973), SARCF (Chung et al.,1998; Gomez et al.,1990), IDARC (Park et al., 1978;Kunnath et al., 1992) and ANSR (Oughourlian and Powell,1982). The “Plastique ” code presented herein, although maintains the elastoplastic behaviour within the 2D plane frames, works with a 3D stiffness of the entire structure based on diaphragmatic action.