Seismic failure mode interaction for the equivalent frame modeling of unreinforced masonry structures Simon Petrovc ˇic ˇ, Vojko Kilar ⇑ University of Ljubljana, Faculty of Architecture, Zoisova 12, SI-1000 Ljubljana, Slovenia article info Article history: Received 13 September 2012 Revised 26 March 2013 Accepted 28 March 2013 Available online 7 May 2013 Keywords: Unreinforced masonry Equivalent frame Failure mode interaction Plastic hinges SAP2000 Pushover analysis IN2 method IDA abstract The paper presents a relatively simple and computationally less demanding technique for the modeling and analysis of regular unreinforced masonry (URM) structures. This technique is based on the equivalent frame approach, and incorporates linear beam elements and the plastic hinge concept. The complex seis- mic failure mechanism of masonry piers is expressed by a single failure mode interaction surface (an ‘‘FMI surface’’), taking into account the influence of variation in the pier’s vertical loading, and its bending moment distribution. The effect of the governing mechanical and geometrical parameters which deter- mine the shape of the FMI surface is presented and discussed. For modeling purposes, the ultimate lateral strength of a masonry element is expressed as a section which cuts through the FMI surface. A single fail- ure mode interaction plastic hinge (an ‘‘FMI hinge’’) for each masonry frame element is introduced by combining specific failure modes, taking into account their minimum envelope. Calculations were carried out using the commercially available computer program SAP2000 Ultimate, and the validity of the pro- posed modeling procedure was confirmed by means of a comparative analysis of an URM wall assem- blage which has already been studied by other researchers, using different modeling techniques and analysis software. The effect of the vertical loading acting on piers was studied, as well as the formation of typical failure mechanisms throughout the structure. The final part of the paper broadens the research to a fictitious 3D URM structure, where the out-of-plane behavior of piers has been considered alongside the standard in-plane failure mechanisms. Results obtained using the incremental N2 method were com- pared, for a range of ground-motion intensities, with selected results obtained using incremental nonlin- ear dynamic analysis. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction In the seismically active zones of southern and central Europe unreinforced masonry (URM) structures are one of the most com- mon types of buildings. Some of them possess a high historical va- lue, and could therefore be classified as part of the architectural heritage, requiring special attention with regard to their preserva- tion and retrofitting measures. In practice, however, the accurate prediction of the seismic response of such structures often proves to be difficult, not only due to the complex geometrical features of their individual architectural parts, but also due to the composite and non-homogeneous nature of URM. Due to the great variety of structural configurations and materials used, rational ap- proaches for the assessment of the seismic safety levels of such buildings are needed. The development of simplified non-linear analysis procedures was first recognized about four decades ago (e.g. [1–4]). Even nowadays, one of the most widely used modeling approaches for the idealization of regular URM buildings remains the ‘‘equivalent frame modeling approach’’ [5–10]. In the latter loads (vertical or lateral) are only applied to specific nodes and the mathematical model of the structure comprises of one-dimen- sional beam elements consisting of (i) vertical structural compo- nents (piers), (ii) coupling horizontal structural components (spandrels), and (iii) rigid elements between the pier-spandrel intersections, which account for connection zones that are nomi- nally much stiffer than their connecting elements. URM buildings are usually built of load-bearing masonry walls, arranged in different orthogonal planes and in most cases con- nected together by means of flexible diaphragms (timber floors). In a seismic event, the damage to such structures usually includes in- and out-of-plane failures of structural elements, which can be taken into consideration only if a comprehensive 3D model of the whole building is used [9,13]. Nevertheless, in practice 2D models which account only for in-plane resistant mechanisms are usually applied to the analysis of URM structures [10]. The fea- sibility of the simplified and cost-effective 2D analytical models as a tool for practical design and/or assessment of typical masonry structures should therefore be used with caution [11]. In general, 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.03.050 ⇑ Corresponding author. Tel.: +386 1 2000720; fax: +386 1 4257 414. E-mail addresses: simon.petrovcic@fa.uni-lj.si (S. Petrovc ˇic ˇ), vojko.kilar@fa. uni-lj.si (V. Kilar). Engineering Structures 54 (2013) 9–22 Contents lists available at SciVerse ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct