Earthquake Resistant Engineering Structures V 375 Simplified non-linear time-history analysis based on the Theory of Plasticity J. L. Domingues Costa1, R. Bento2, V. Levtchitch3 & M. P. Nielsen1 1Department of Civil Engineering, The Technical University of Denmark 2ICIST – DECivil, IST, The Technical University of Lisbon, Portugal 3Frederick Institute of Technology, Nicosia, Cyprus Abstract This paper aims at giving a contribution to the problem of developing simplified non-linear time-history (NLTH) analysis of structures for which dynamical response is mainly governed by plastic deformations so as to be able to provide designers with sufficiently accurate results. The method to be presented is based on the Theory of Plasticity. Firstly, the formulation and the computational procedure to perform time-history analysis of a rigid-plastic single degree of freedom (SDOF) system are presented. The necessary conditions for the method to incorporate pinching as well as strength degradation are outlined. The procedure is applied to a typical SDOF system and results are compared with NLTH analysis commonly used for design purposes. Secondly, by means of the Virtual Work Principle, the definition of the equation of motion of a desired collapse mechanism of a multi degree of freedom (MDOF) system is presented. This equation is of the same type as in the SDOF case, and therefore the procedure presented in the first part of the paper may be used. The method is applied to a 4-story reinforced concrete frame structure. Results are compared to those derived by a conventional NLTH analysis and found to be encouraging. Keywords: non-linear time-history analysis, theory of plasticity, rigid-plastic material, collapse mechanism, virtual work principle. 1 Introduction It is well known that the most powerful and rational technique to assess the dynamical response of a structure subjected to an earthquake is a NLTH analysis. WIT Transactions on The Built Environment, Vol 81, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) 376 Earthquake Resistant Engineering Structures V However, structural engineers are faced with difficulties when applying the different versions available nowadays of this tool. Even for simple structures designed to have localized inelastic behaviour, the analysis requires significant computational effort and there are difficulties in choosing the appropriate hysteretic behaviour at the plastic hinges regions. Therefore, simplification is needed. Plasticity Theory provides rational and efficient solutions concerning the analysis and design of structures that deform in the inelastic range, which is the case under severe earthquake motion. In this paper, Plasticity Theory is used to design and to assess the dynamical response of structures with the simplification of rigid-plastic behaviour for the plastic hinge regions and rigid behaviour for the remaining part of the structure. Special emphasis is put on reinforced concrete structures. Two main justifications for this may be exposed: The use of the extremum principles of the theory allows the choice of a suitable collapse mechanism where ductility demands may be realistically assigned to the plastic hinges regions. If properly designed, R/C structures may have significant levels of ductility, which means that it is reasonable to neglect the contribution of elastic deformations. 2 The rigid-plastic single-degree-of-freedom system 2.1 Formulation of the equation of motion The equation of motion of a SDOF system solely subjected to base motion may be easily formulated expressing Newton’s second law in a coordinate system moving with the ground: ma⋅−=−⋅rg() t Rt () ma () t (1) or, in a more convenient form, Rt() at()=− at () (2) rgm Here t is the time, m is the mass of the SDOF system, ar(t) is the acceleration of the mass in the moving coordinate system, R(t) is the total force applied to the SDOF system and ag(t) is the ground acceleration. The values of ag(t) are derived from the accelerogram record. Here we shall assume that the only contribution to R(t) is the resultant force on the mass from internal forces. Any type of damping is disregarded as the only source of energy dissipation considered is due to plastic deformations. For illustrative purposes consider a typical SDOF system namely a fixed, vertical weightless column with length L, supporting a given mass, m, that can only move in the horizontal direction. The non-linear behaviour of the SDOF system is specified by the flexural behaviour of the plastic hinge in the cross- section located at the base, cross-section A, which is of the rigid-plastic type. For simplicity, the positive yield moment has the same absolute magnitude, MP, as the negative yield moment. The term θ is the relative rotation in the hinge. WIT Transactions on The Built Environment, Vol 81, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) Earthquake Resistant Engineering Structures V 377 Bending, shear and axial stiffnesses are infinite at any cross section of the column. M A m M P L θ A -M P A Figure 1: Typical SDOF system with rigid-plastic behaviour. Fig.2a) shows the sign convention adopted. The ground displacement and the displacement of the mass are positive to the right-hand side. The bending moment at the fixed end, MA(t), is positive when giving rise to tension in the right-hand side of the column. To express dynamical equilibrium, d’Alembert’s principle is used. In Fig. 2b) the free-body-diagram of the mass in the coordinate system moving with the ground is shown. a) b) m m ag(t) m m ar(t) M A (t) L L M A(t) + ag Figure 2: a) SDOF system subjected to ground acceleration and b) corresponding free-body-diagram for the mass. The dynamical equilibrium condition yields: Mt() A −⋅ma() t =⋅ ma () t (3) L gr M ()t Dividing eqn (3) by m, eqn 2 with Rt()= A is reached. L Finally, one has to associate the behaviour of cross-section A with the dynamical state of the system. To carry out this task, one has to identify the transition conditions between rigid and plastic behaviour. The system changes from rigid to plastic behaviour when the bending capacity is exhausted. From the sign convention expressed above and from the moment vs. rotation relationship shown in Fig. 1, one may easily conclude that exhaustion of the positive bending capacity at cross-section A implies positive rate of change of the rotations of the cross-section meaning that the mass is WIT Transactions on The Built Environment, Vol 81, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) 378 Earthquake Resistant Engineering Structures V moving in the negative direction. Conversely, the exhaustion of the negative bending capacity of cross-section A, implies the mass to move in the positive direction. This is the same to say that the velocity of the mass in the moving coordinate system, vr(t), is positive when “negative plastic behaviour” at cross- section A takes place and is negative if “positive plastic behaviour” occurs. The system may change from plastic to rigid behaviour whenever the relative velocity of the mass is zero. The bending moment at the moment of transition corresponds to the bending capacity associated with the sign of the previous plastic deformations. When the system has rigid behaviour, MA(t) is within the extreme values MP or –MP, and therefore the rotation at cross-section A is constant and equal to the rotation at the transition moment. Obviously, the same applies to the relative displacement, dr(t). The complete formulation of the equation of motion for the SDOF system depicted in Fig. 1 may now be written. vrr (t)== a (t) 0 If MAPP() t∈⇒ ][-M ,M d (t)= Constant r (4) Mt() If M() t=≠ {}-M ,M and v (t) 0 ⇒ a (t)= A −a() t APPr rmL⋅ g It appears that the rigid-plastic approach to model the non-linear behaviour at the plastic hinge, yields an extremely simple formulation of the equation of motion as all the information necessary to compute the dynamical response is in the form of the set of expressions in (4). Additionally, it should be noted that when plastic behaviour takes place at the plastic hinge regions the effect of the base motion, given by ar(t), is determined by the absolute acceleration of the mass M given by the ratio ± P ,for the present case. This parameter may be of great mL⋅ usefulness to estimate the dynamical response of the system as the amount of plastic deformations has a direct correlation with the magnitude of this parameter. Moreover, as it will be shown in the next paragraph, it may also be associated with the transition conditions between different types of behaviour. Finally, it should be noted that the formulation for any other type of SDOF system with rigid-plastic behaviour, would yield similar expressions as (4) regardless the type of the internal force mechanism. 2.2 The computational procedure to perform a time-history analysis of a rigid plastic SDOF system In the following, a step-by-step integration procedure for the computation of the dynamical response of a SDOF system to any type of base motion is presented. The response is evaluated at each successive time interval i with length ∆t sufficiently small to justify that ag(t) is assumed constant and equal to the average, ag,i, between the value at the beginning and at the end of that time interval, ag(ti) and ag(ti+∆t), respectively. Thus: WIT Transactions on The Built Environment, Vol 81, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) Earthquake Resistant Engineering Structures V 379 at+atgi() gi()+∆ t a() t== a for t∈ t; t +∆