Analytically derived fragility curves for unreinforced

masonry buildings in urban contexts

D. DAyala University of Bath, United Kingdom

E. Kishali Kocaeli University, Turkey

SUMMARY:

Masonry buildings are the most common form of dwelling worldwide and at the same time one of the most

vulnerable to seismic action. Large numbers of casualties and substantial economic losses are associated with

masonry building partial and total collapses in urban and rural areas. Usually studies of vulnerability of masonry

structures are conducted within an empirical framework, based on past observation and historic damage data.

However empirical approaches have limitation in terms of regional applicability and comparison among different

typological and geographical context. The paper presents an analytical approach, FaMIVE, based on limit state

analysis, which allows defining capacity curves and performance points for masonry structures. The analytical

development of the procedure from derivation of the ultimate capacity to the identification of the damage states

in terms of drift, to the convolution of the capacity and spectral curves to identify performance points for given

level of shaking is presented. Fragility curves are then derived. An application to masonry structures in Turkey

shows the advantages of this approach. This work was carried out within the framework of the WHE-PAGER

project (http://pager.world-housing.net/)

Keywords: Vulnerability functions, Masonry, Performance based assessment,

1. INTRODUCTION

Masonry buildings are vulnerable to seismic actions. A number of analytical procedures exist in

literature for the evaluation of the vulnerability of unreinforced masonry structures. However they

focus mainly on the in-plane behaviours of walls, considering only mechanisms of failure associated

with the shear capacity of piers. For this mechanism to be the effective failure behaviour several

conditions need to be met, among which, small size opening resulting in stiff spandrel and stock piers,

and walls being stabilised by the load of the horizontal structures. However the limits of this approach

has been clearly demonstrated by the analysis of damage patterns and collapses of substantially

different unreinforced masonry building stocks such as the ones recently exposed to the l'Aquila, Italy

and Christchurch, New Zealand earthquake. In both cases the majority of collapses and serious

structural damage are due to out-of-plane failures of walls.

The paper presents a procedure FaMIVE, based on a mechanical approach, which allows to define

capacity curves and performance points for masonry structures of Turkey within the framework of the

N2 method (Fajfar 1999) at the basis of the EC8 assessment guidelines for existing structures. Twelve

different mechanisms are considered and capacity curves are derived to d in terms of lateral capacity

and ultimate displacement. This allows for direct comparison of vulnerability functions and fragility

functions of building stocks in Turkish urban and rural areas, comprised mainly of masonry buildings.

The paper also presents the analytical development of the procedure from derivation of the ultimate

capacity to the identification of the damage states in terms of drift, to the convolution of the capacity

and spectral curves to identify performance points for given level of shaking.

http://pager.world-housing.net/

2. APPLICATION OF FAMIVE METHODOLOGY TO MASONRY TYPOLOGIES

2.1 General Information

For the computation of capacity curves for masonry structures a number of procedures are available in

literature. These are based either on the equivalent frame approach or on the mechanism approach.

Among the first, in the past decade a relatively significant number of procedures aimed at defining

reliable analytical vulnerability function for masonry structures in urban context have been published

(Lang and Bachmann (2004), Erberik (2008), Borzi et al. (2008), Erdik et al. (2003)). Although they

share similar conceptual hypotheses they differ substantially by modelling complexity, numerical

complexity, geographic validity of the model, treatment of uncertainties. Far fewer are the approaches

based on mechanical behaviour, VULNUS (Bernardini et al., 2000) and FaMIVE (DAyala &

Speranza 2003).

2.2. Masonry Typologies

The residential building stock in Turkey is still largely dominated by masonry construction both in

rural and urban areas. The data related to masonry structures in Turkey is provided by Middle East

Technical University (METU) (Erberik 2008, Erberik 2010). For the present study METU provided

data and description of index buildings for the following PAGER structure typologies (Jaiswal et al.

2008) : adobe (A1), rubble stone masonry in mud mortar with earth or metal roof (RS2), massive stone

masonry in lime mortar with timber floors (MS), unreinforced bricks in mud mortar (UFB1),

unreinforced bricks in cement mortar with timber floors (UFB4), unreinforced bricks in cement mortar

with reinforced concrete floors (UFB5) and unreinforced concrete blocks in lime/cement mortar

(UCB). For each typology typical values of a set of geometric parameters were provided as shown in

Table 2.1, together with a set of photos representing typical cases for each class.

Table 2.1. Parameters by typology as provided by Erberik (personal communication).

The pictorial information allowed deriving further parameters, not included in the numerical dataset

but essential to conduct the analysis using FaMIVE, such as number and layout of openings, number

of storey and other more specific construction details, such as the presence of timber bands in rubble

construction. In the following sections the procedure used by FaMIVE to obtain capacity curves is

presented in detail, the rationale used to derive additional input data is discussed, then the results are

presented in terms of capacity curves and finally fragility curves for three limit states are derived for

each typology.

2.3. Methodology for the derivation of the capacity curves

The programme FaMIVE is based on a limit state, mechanical analysis of the external bearing walls

forming a masonry building. The analysis is static equivalent and aims to predict the lateral load

collapse multiplier (expressed in g) which will trigger the onset of a specific failure mechanism. The

procedure is based on a lower bound approach and the detailed analytical developments for a suite of

possible mechanisms are reported in DAyala & Speranza, (2003) for out of plane mechanisms and in

DAyala & Casapulla (2006) for in plane failures. The possibility of occurrence of different

mechanisms is dependent on the geometric configuration of each analysed wall or faade of the

building and its connections to the other structural elements (vertical and horizontal structures).

Among all possible mechanisms computed for each faade, the one that shows the worst combination

between minimum collapse load factor and maximum extent of faade involved in the collapse is

chosen according to a worst product algorithm (DAyala & Speranza, 2002). Application of the

procedure to sites in Turkey and Italy are reported in (DAyala, 2005), (DAyala and Paganoni, 2011).

Although the collapse load factor might be sufficient to generate fragility curves based on lateral

capacity only, in order to obtain performance points, a complete capacity curve, for each building or

faade analysed, is needed. This allows assessing and predicting levels of damage given a specific

demand spectrum, once performance points and damage states are correlated along the capacity curve.

To this end the results obtained with the limit analysis and the mechanism approach need to be recast

in the framework of the capacity spectrum method by associating an elasto-plastic capacity curve to

each mechanism and then further manipulating this to obtain the equivalent SDoF bilinear curve, in

the space Sa-Sd. In the FaMIVE procedure capacity curves are developed by calculating first the

effective stiffness for a wall Keff: this is a function of the type of mechanism attained, the geometry of

the wall and layout of the opening, the constraints to other walls and floors and the portion of other

walls involved in the mechanism:

(2.1)

where is the height of the portion involved in the mechanism, is the estimated elastic modulus

of the masonry as it can be obtained from experimental literature, Ieff and Aeff are the second moment of

area and the cross sectional area, respectively, calculated taking into account extent and position of

openings and variation of thickness over height, k1 and k2 are constants which assume different values

depending on edge constraints and whether shear and/or flexural stiffness are relevant for the specific

mechanism. The effective mass involved in the mechanism is calculated following the same approach:

rfmeffeff V (2.2)

where Veff is the solid volume of the portion of wall involved in the mechanism, is the density of

the masonry are the masses of the horizontal structures involved in the mechanism. Effective

mass and effective stiffness are used to calculate a natural period Teff, which characterise an equivalent

single