A simplified non-linear modelling strategy to generate fragility curves for old masonry buildings

Highlights

• Modelling the global nonlinear response of masonry structures.

• An efficient numerical method to compute fragility curves.

• Fragility curves derived for a large dataset of structures in a typology.

• A global structural damage indicator based on the eigenfrequency drop off.

Abstract

In the context of historical seismology, studying the behaviour of historic masonry buildings is of great importance, as they are witnesses of past events. While interesting methods can be found in the literature to assess the seismic vulnerability of masonry structures subject to strong earthquakes, the topic of moderate seismicity, as encountered in many European countries, is still to be investigated. The present work proposes a global methodology to build fragility curves for existing masonry buildings. An efficient computational method to address the non-linear response of masonry structures is presented. The method is based on the modal decomposition of the structural response. An equivalent non-linear single degree of freedom oscillator is identified for the main modes. The modelling strategy enables to carry out numerous computations with a low computational effort, allowing for a probabilistic approach. To build fragility curves, a damage indicator based on the frequency shift is chosen. To validate the proposed approach, the simplified model of a real masonry building is compared to a full non-linear time history analysis. The method is eventually illustrated by the derivation of fragility curves for industrial masonry buildings.

Introduction

For design purposes, a level of seismic intensity based on the known past earthquakes is defined. In moderate seismic regions, the instrumental seismology is not old enough to rely on measured earthquakes only. The definition of a macro-seismic intensity for historical earthquakes is needed, often based on archival testimony. However, the data available to quantify these historical earthquakes may not be sufficient. In particular, if no considerable damage is reported in the identified historical documents, the uncertainties related to the macro-seismic intensity and to the epicentre location of the earthquake may be high.

Recent methodologies [1] propose to introduce structural analyses in the process of defining the macroseismic intensity of historical earthquakes. The main idea is to update the distribution of intensities or magnitudes of the considered earthquake by means of a Bayesian approach, combining historical data, fragility curves and in-situ observed damages. This approach is very promising in the context of low to moderate seismic activity, such as encountered in France, associated to a large amount of old structures as well as cultural heritage. The method can be used as a way to reduce the uncertainties associated to the process of assigning magnitudes to historical earthquakes. For this purpose, the building stock is classified in typologies for which fragility curves are computed [2], [3]. This paper focuses on the derivation of such fragility curves for old masonry buildings subject to low to moderate earthquakes.

The structural analysis of historical masonry buildings can be handled with different numerical strategies as presented in [4], [5]. A global methodology to derive fragility functions for masonry structures can be found in [6], along with a set of fragility curves for specific building typologies. Macro-element models are commonly used for the generation of fragility curves [5], [7]. An exemple of numerical limit analysis based on macro-elements can also be found in [8]. Nevertheless, finite element (FE) based modelling has several advantages, as explained below, if an adequate amount of computational resources [9] is available. FE models are used to address the seismic vulnerability of historical structures exhibiting several local modes [10], [11], [12], or also within the framework of limit analysis [13], [14]. The macro-element approach is very interesting for a relatively high seismic level, when a global state of damage with large macro-cracks is reached for the structure. In the context of low seismic activity however, the seismic load generally leads to low damages. The cracks are rather small and distributed in the structures. In this case, a FE computation is more relevant for the structural analysis, and several masonry models have been developed based on a continuous approach [15], [16], [17], [18]. To build fragility curves based on this approach, a large number of computations must be carried out for a structure typology. Indeed, uncertainties on the structure as well as on the material must be taken into account. Thus, an efficient numerical method must be adopted.

In this paper, a global modelling strategy is presented, based on the modal decomposition of the structure and the identification of non-linear single degree of freedom (NLSDOF) oscillators. Despite its simplicity and the use of certain hypotheses, non-linear simple oscillators allow to obtain an accurate global response and information regarding the damage state of the structure considering global indicators. The method is suitable for structures for which local modes are not preponderant or do not lead to strong damage. After a detailed presentation of the developed modelling strategy, the method is applied to the case of a real historical masonry structure. The computed response is first compared to a full 3D non linear time history analysis (NLTHA) of the building. The probabilistic framework is then addressed. From the computed structural responses, the method used to derive fragility curves is presented. The definition of a structural damage criterion is discussed, and fragility curves are derived for the considered case-study.