## Final Report Summary - NOUS (Probabilistic Inverse Models for Assessing the Predictive Accuracy of Inelastic Seismic Numerical Analyses)

Research project NOUS deals with developing probabilistic inverse models for assessing the predictive accuracy of seismic inelastic time history analyses. The reliability of the engineering demand parameters (EDPs) obtained as outputs of such numerical simulations is challenged by the presence of uncertainties in the model parameters as well as in the model itself: modeling errors, poor comprehension of the numerous energy dissipation mechanisms, and so on. Bringing together experts in probabilistic computational mechanics, earthquake engineering, nonlinear material science and statistics, the research project aims at providing innovative useful numerical tools to researchers, designers and analysts for decision making regarding the seismic risk and structural safety of designing and existing structures. More specifically, research is oriented toward developing probabilistic inverse numerical models to quantify model intrinsic uncertainty in numerical inelastic time history analysis of structures in seismic loading.

NOUS project relies on the assumption that the damping forces that have to be introduced in seismic inelastic time history analyses so that simulation accurately predicts actual experimental response can reflect the intrinsic uncertainty in the inelastic structural model. These damping forces, as commonly computed with Rayleigh damping assumption in the structural Earthquake Engineering community, have first been thoroughly investigated. A comprehensive study of Rayleigh damping has shown that controlling the damping ratio actually resulting from Rayleigh damping throughout inelastic time history seismic analysis is challenging. Consequently, Rayleigh damping introduces uncertainty in the modeling. An analytical method for optimal control of the damping ratios has been proposed and used to perform a probabilistic sensitivity analysis to quantify the impact of uncertain damping ratios on EDPs of interest for performance-based design. A probabilistic nonlinear structural model of a reinforced concrete frame structure tested on a shaking table has been developed, and Monte-Carlo simulations with efficient sampling procedure have been run in parallel to propagate the initial damping uncertainty in a tractable procedure. For some seismic ground motions, large scatter in the EDPs is observed, which proves that Rayleigh damping forces can significantly challenge the predictive accuracy of inelastic seismic numerical analyses.

At this point, it is clear that both inelastic structural model and damping model introduce uncertainty in inelastic seismic numerical analyses. The main issue then is: How to quantify this uncertainty? or in other words: How to assess the predictive accuracy of these analyses? To answer this question, a probabilistic inverse approach has been developed.

A Bayesian inverse approach that provides a measure of model intrinsic uncertainty while remaining computationally tractable has been developed. The approach relies on a probabilistic structural finite element model and on an experimental dataset with recordings of displacement and acceleration time histories at the degrees of freedom of the finite element model of the tested structure. It consists in identifying – in a Bayesian framework – the set of model parameters for which the model numerical response best fits the experimental observations, and then to use the residual error as a measure of model intrinsic uncertainty. This strategy raises two main issues: (i) How to run the tens or hundreds of thousands of numerical inelastic time-history seismic analyses needed to sample the space of the random parameters in Bayesian updating procedure?, and (ii) How to characterize the residual error?

Those two questions can find answers by appealing to the original concept of discrepancy forces that has been developed within NOUS project. The discrepancy forces are the forces needed to satisfy the equilibrium equation when the experimental data are imposed to the structural finite element model. Computing the discrepancy forces is much less time consuming than solving classical time history inelastic seismic analyses. This can been exploited to run tractable Markov chain Monte Carlo (MCMC) analyses in the purpose of identifying the set of model parameters that leads to the smallest discrepancy forces. Therefore, the errors to be minimized in the Bayesian updating process are not defined as outputs of the numerical analysis but as the discrepancy forces. The residual discrepancy forces computed with the identified set of optimal parameters can then characterize model intrinsic uncertainty. Indeed, discrepancy forces gather all the error sources in the numerical model because they are the forces needed to satisfy system equilibrium.

Finally, the issue of reducing model intrinsic uncertainty has also been addressed in NOUS project. To that purpose, the work has been focused on reinforced concrete frame structures.

Part of the seismic energy imparted to the system in ground motion is absorbed through numerous inelastic mechanisms at the material level. In concrete, hysteresis loops are experimentally observed in unloading-reloading cycles, which is a source of energy dissipation that contributes to the overall structural damping. Explicitly accounting for such material energy dissipative behavior is expected to increase the predictive accuracy of the models. In this perspective, a stochastic multi-scale approach has been developed. Two scales are considered: the macroscale where an equivalent homogenous concrete model capable of representing key features of 1D concrete response is retrieved, and a mesoscale where heterogeneous local nonlinear response coupling damage and plasticity is assumed. Local response at mesoscale is modeled in the framework of thermodynamics with internal variables and is seen as the homogenized response of mechanisms that occur at the micro- or nano- underlying scales. Spatial variability at mesoscale is introduced using stochastic vector fields generated by spectral representation. Homogenized macroscopic response is recovered using standard averaging method from micromechanics. It is demonstrated that, for suitable parameterization of the mesoscale (structure of the random fields), the homogenized response at macroscale is independent of the realization of the random field that represent heterogeneities at mesoscale: a representative macroscopic response is obtained. Besides, the resulting macroscopic response represents salient features of concrete compressive 1D response in cyclic loading. This material model has been implemented in a fiber beam structural element and the capability of the latter for generating damping has been demonstrated.

The work developed within NOUS project should allow for assessing the predictive accuracy of numerical models used for simulating the time history response of nonlinear structures in seismic regions. It should also open new perspectives toward the rational modeling of damping effects in such simulations. This work consequently participates to the continuous effort in the earthquake engineering community toward providing reliable engineering demand parameters that can be used for improving seismic risk management methods. This ultimately impacts the design or retrofitting actions of buildings in seismic hazard, the way insurance companies and stakeholder deals with seismic risk, as well as the designing of emergency plans to save lives in case of an earthquake.

Publications issued from the project as well as contact details can be found on the researcher webpage: http://www.mssmat.ecp.fr/cms/lang/fr/mssmat/jehel_pierre

NOUS project relies on the assumption that the damping forces that have to be introduced in seismic inelastic time history analyses so that simulation accurately predicts actual experimental response can reflect the intrinsic uncertainty in the inelastic structural model. These damping forces, as commonly computed with Rayleigh damping assumption in the structural Earthquake Engineering community, have first been thoroughly investigated. A comprehensive study of Rayleigh damping has shown that controlling the damping ratio actually resulting from Rayleigh damping throughout inelastic time history seismic analysis is challenging. Consequently, Rayleigh damping introduces uncertainty in the modeling. An analytical method for optimal control of the damping ratios has been proposed and used to perform a probabilistic sensitivity analysis to quantify the impact of uncertain damping ratios on EDPs of interest for performance-based design. A probabilistic nonlinear structural model of a reinforced concrete frame structure tested on a shaking table has been developed, and Monte-Carlo simulations with efficient sampling procedure have been run in parallel to propagate the initial damping uncertainty in a tractable procedure. For some seismic ground motions, large scatter in the EDPs is observed, which proves that Rayleigh damping forces can significantly challenge the predictive accuracy of inelastic seismic numerical analyses.

At this point, it is clear that both inelastic structural model and damping model introduce uncertainty in inelastic seismic numerical analyses. The main issue then is: How to quantify this uncertainty? or in other words: How to assess the predictive accuracy of these analyses? To answer this question, a probabilistic inverse approach has been developed.

A Bayesian inverse approach that provides a measure of model intrinsic uncertainty while remaining computationally tractable has been developed. The approach relies on a probabilistic structural finite element model and on an experimental dataset with recordings of displacement and acceleration time histories at the degrees of freedom of the finite element model of the tested structure. It consists in identifying – in a Bayesian framework – the set of model parameters for which the model numerical response best fits the experimental observations, and then to use the residual error as a measure of model intrinsic uncertainty. This strategy raises two main issues: (i) How to run the tens or hundreds of thousands of numerical inelastic time-history seismic analyses needed to sample the space of the random parameters in Bayesian updating procedure?, and (ii) How to characterize the residual error?

Those two questions can find answers by appealing to the original concept of discrepancy forces that has been developed within NOUS project. The discrepancy forces are the forces needed to satisfy the equilibrium equation when the experimental data are imposed to the structural finite element model. Computing the discrepancy forces is much less time consuming than solving classical time history inelastic seismic analyses. This can been exploited to run tractable Markov chain Monte Carlo (MCMC) analyses in the purpose of identifying the set of model parameters that leads to the smallest discrepancy forces. Therefore, the errors to be minimized in the Bayesian updating process are not defined as outputs of the numerical analysis but as the discrepancy forces. The residual discrepancy forces computed with the identified set of optimal parameters can then characterize model intrinsic uncertainty. Indeed, discrepancy forces gather all the error sources in the numerical model because they are the forces needed to satisfy system equilibrium.

Finally, the issue of reducing model intrinsic uncertainty has also been addressed in NOUS project. To that purpose, the work has been focused on reinforced concrete frame structures.

Part of the seismic energy imparted to the system in ground motion is absorbed through numerous inelastic mechanisms at the material level. In concrete, hysteresis loops are experimentally observed in unloading-reloading cycles, which is a source of energy dissipation that contributes to the overall structural damping. Explicitly accounting for such material energy dissipative behavior is expected to increase the predictive accuracy of the models. In this perspective, a stochastic multi-scale approach has been developed. Two scales are considered: the macroscale where an equivalent homogenous concrete model capable of representing key features of 1D concrete response is retrieved, and a mesoscale where heterogeneous local nonlinear response coupling damage and plasticity is assumed. Local response at mesoscale is modeled in the framework of thermodynamics with internal variables and is seen as the homogenized response of mechanisms that occur at the micro- or nano- underlying scales. Spatial variability at mesoscale is introduced using stochastic vector fields generated by spectral representation. Homogenized macroscopic response is recovered using standard averaging method from micromechanics. It is demonstrated that, for suitable parameterization of the mesoscale (structure of the random fields), the homogenized response at macroscale is independent of the realization of the random field that represent heterogeneities at mesoscale: a representative macroscopic response is obtained. Besides, the resulting macroscopic response represents salient features of concrete compressive 1D response in cyclic loading. This material model has been implemented in a fiber beam structural element and the capability of the latter for generating damping has been demonstrated.

The work developed within NOUS project should allow for assessing the predictive accuracy of numerical models used for simulating the time history response of nonlinear structures in seismic regions. It should also open new perspectives toward the rational modeling of damping effects in such simulations. This work consequently participates to the continuous effort in the earthquake engineering community toward providing reliable engineering demand parameters that can be used for improving seismic risk management methods. This ultimately impacts the design or retrofitting actions of buildings in seismic hazard, the way insurance companies and stakeholder deals with seismic risk, as well as the designing of emergency plans to save lives in case of an earthquake.

Publications issued from the project as well as contact details can be found on the researcher webpage: http://www.mssmat.ecp.fr/cms/lang/fr/mssmat/jehel_pierre