## Special modelling requirements for nonlinear dynamic analysis

Clause In order to be used in non-linear response-history analysis, member force-displacement

4.3.3.4.3(2) models need only be supplemented with hysteresis rules describing the behaviour in post-elastic unloading-reloading cycles. The only requirement posed by Eurocode 8 for the hysteresis rules is to reflect realistically energy dissipation within the range of displacement amplitudes induced in the member by the seismic action used as input to the analysis. Given that the predictions of non-linear dynamic analysis - especially those for the peak response -are not very sensitive to the exact shape and other details of the hysteresis loops produced by member models, a far more important attribute of the model used for the hysteresis is the numerical robustness under any conceivable circumstance. This is crucial, as it is almost certain that potential numerical weakness of the model will show up in an analysis involving possibly hundreds of non-linear members, thousands of time-steps and, possibly, a few iteration cycles within each step. In some cases, local numerical problems may develop into lack of convergence and global instability of the response. Inertia forces and other stabilizing influences may sometimes prevent local numerical problems from causing global instability; due to the numerical problems, though, local or even global predictions of the response may be in error and - what is worse - it takes a lot of experience and judgement to recognize that predictions are wrong. In general, simple and clear hysteresis models that use just a few rules to describe the response under any cycle of unloading and reloading, small or large, complete or partial, are less likely to lead to numerical problems than elaborate, complex and often obscure models.

Given that within the framework of EN 1998-1 non-linear dynamic analysis is meant to be applied for the evaluation of new buildings designed for a minimum of ductility and dissipation capacity according to this part of Eurocode 8, the non-linear response will be limited to ductile and stable mechanisms of cyclic force transfer and will be prevented in brittle or degrading ones. This facilitates the choice of hysteretic rules, as degradation of stiffness and strength with cycling can be ignored as insignificant. Therefore, the best balance of accuracy, simplicity and reliability is provided by the following types of models for members with a ductile-dominant mechanism of cyclic force transfer:

• For steel or composite (steel-concrete) beams, columns or seismic links in unidirectional cyclic bending and shear with axial force, and for steel or composite (steel-concrete) bracings in tension: an elastic, linearly strain-hardening (bilinear) force-deformation model for monotonic loading and a bilinear cyclic model with kinematic hardening and unloading and reloading branches parallel to those of the monotonic response. 8 For concrete beams, columns or walls in unidirectional cyclic bending with axial force (shear in concrete is a brittle mechanism of force transfer and it is designed for sufficient overstrength with respect to flexure so that it is kept in the elastic range): an elastic, linearly strain-hardening (bilinear) force-deformation model for monotonic loading; linear unloading up to zero-force and linearly reloading thereafter towards the most extreme point reached previously on the monotonic loading curve in the opposite direction. In other words, a model with 'stiffness degradation' but without 'strength degradation' or 'pinching' (e.g. a modified Takeda model,55 according to Otani56).

° For steel or composite (steel-concrete) bracings in alternating tension and compression: an elastic, linearly strain-hardening (bilinear) force-deformation model for monotonic loading in tension, linearly unloading up to the buckling load in compression; shedding load linearly or non-linearly with shortening after buckling; linearly reloading from compression to tension towards the most extreme point reached previously on the monotonic curve in tension.

Non-linear dynamic analysis is considered to excel over its static counterpart (pushover analysis) mainly in its ability to capture the effects of modes of vibration higher than the fundamental mode. For this to be done correctly, member non-linear models should provide a realistic representation of the stiffness of all members up to their yield point. This is far more important than for non-linear static (pushover) analysis because higher modes, when they are important, often involve post-yield excursions in members which stay in the elastic range under the fundamental mode alone. Moreover, in pushover analysis it is primarily (if not only) the determination of the target displacement that is affected by the effective stiffness to yielding. In fact, the target displacement depends only on the global elastic stiffness which is fitted to the capacity curve and is possibly sensitive to the elastic stiffness of certain members which may be crucial for global yielding but are not known before the analysis.

If the response is fully elastic, the peak response predicted through non-linear time-history analysis should be consistent with the elastic response spectrum of the input motion (exactly in the extreme case of a single-degree of freedom system, or in good approximation for a multi-degree of freedom system subjected to modal response spectrum analysis with the CQC modal combination rule). Such conformity is difficult to achieve when using a trilinear monotonic force-deformation relationship for members that takes into account the difference in pre- and post-cracking stiffness of concrete and masonry (e.g. see Takeda55), as allowed by Eurocode 8. Under cyclic loading such models produce hysteretic damping in the pre-yielding stage of the member, which increases with displacement amplitude from zero at cracking to a maximum value at yielding. Similarly to the equivalent viscous damping ratio, in that range of elastic response the elastic stiffness of the trilinear model is not uniquely valued. This ambiguity does not allow direct comparisons with the elastic response spectrum predictions, let alone conformity. For this reason, it is preferable in non-linear dynamic analysis to use member models with a force-deformation relationship which is (practically) bilinear in monotonic loading. After all, it is expected that, at the time it is subjected to a strong ground motion, a concrete or masonry structure will already be extensively cracked due to gravity loads, thermal strains and shrinkage, or even previous shocks. Last but not least, steel (or even composite steel-concrete) members have a (practically) bilinear force-deformation curve under monotonic loading, and it is convenient for computer programs to use the same type of monotonic force-deformation model for all structural materials.

It should be pointed out that in non-linear static (pushover) analysis the effect of using a trilinear monotonic force-deformation relationship for members will be limited to the initial part of the capacity curve and will not give rise to the problems and ambiguities mentioned above in connection with the application of non-linear dynamic analysis.

If non-linear dynamic analysis employs for members a bilinear force-deformation model under monotonic loading, as advocated above, it should also account for the 5% viscous damping ratio considered to characterize the elastic (in this case pre-yield) response. Unless the computer program used for the non-linear dynamic analysis provides the facility of user-specified viscous damping for all modes of practical importance, Rayleigh damping should be used. To ensure a damping ratio not far from 5% for elastic response in all these modes, it may be specified as equal to 5% at:

(1) the natural period of the mode with the highest modal base shear, for analysis under a single component of the seismic action, or at the average of the natural periods of the two modes with the highest modal base shears in two nearly orthogonal horizontal directions, for simultaneous application of the two horizontal components

(2) twice the value of the period in point 1.

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