Seismic fragility models for typical non-engineered URM residential buildings in Malawi

Malawi is an earthquake-prone country that lies within the East African Rift. A large proportion of its population lives in non-engineered single-storey constructions made of clay bricks and low-strength mortar. Walls are typically single-skin and often lack adequate wall-to-wall connections, leaving them vulnerable to seismic actions. This work reports a comprehensive study on the seismic fragility of unreinforced masonry buildings of the Malawi housing stock. The probability of exceeding different levels of in-plane/out-of-plane damage is estimated by considering the aleatory and epistemic uncertainties of the problem. Inter-building and intra-building variability are accounted for by adopting material test results and building survey data collected in Malawi. The in-plane capacity of building walls is calculated through a finite element model that considers the orthotropic properties of masonry. The out-of-plane capacity is computed using an analytical solution, developed for walls in one way bending. In addition, record-to-record variability is considered. The new country-specific fragility models result more conservative that global estimates, which reflects the high vulnerability of Malawian masonry buildings. These fragilities can be integrated into catastrophe modelling platforms for earthquake risk assessment in Malawi and in the wider East African region.


Introduction
Malawi has one of the highest rates of population growth in the world.According to the United Nations [1], the total population will increase from 18.6 million in 2019 to 38.1 million by 2050.This trend will inevitably result in increased exposure to natural disasters, such as earthquakes and floods [2][3][4].Additionally, owing to the unregulated and inadequately planned nature of urban growth in Malawi, informal settlements, typical of rural areas, will be expanded at the outskirts of major cities, such as Lilongwe and Blantyre [5].Residential buildings are characterised by poor engineering and construction quality, and are therefore highly vulnerable to natural hazards [6].The location of Malawi, spanning over the southern branch of the active East Africa Rift, makes the country susceptible to M w 7.0 (or even greater) earthquakes [7].Moderate seismic hazard, high vulnerability, and increasing exposure are attracting the attention of international agencies towards the implementation of Disaster Risk Reduction (DRR) actions in Malawi [8].
The urgency of developing risk-informed DRR policies comes with the need for tailored risk assessment models for the region.Only a few studies are currently available on the topic.In 2016, Goda et al. [7] presented a first-generation probabilistic seismic risk model for Malawi.The model was developed as a combination of recent findings in seismic hazard [9] and global models for exposure/vulnerability [10,11].Regarding structural vulnerability, the study emphasised that the adoption of global empirical models [10,11] may lead to potential bias and significant variation in the prediction of collapse rates.Moreover, the classification of the Malawian building stock through global building classes can be rather subjective and consequently uncertain and questionable [12].More recently, the Global Earthquake Model (GEM) Foundation has released a new set of models for seismic risk assessment, including a global fragility/vulnerability database [13], a global hazard model [14], and an exposure database for different geographical regions, including East Africa [15].According to GEM estimates, the average annual loss of the residential building stock of Malawi is USD 10.3 million, roughly 0.025% of the total asset replacement cost.This estimate can be further improved by reducing the uncertainties around the current hazard data and by incorporating country-specific vulnerability models.
Consulting with the technical-scientific literature, it is clear that Malawi, when compared with other seismic-prone nations, lacks countryspecific vulnerability/fragility models for earthquake risk assessment.Only recently, Novelli et al. [16] have proposed a first set of analytical fragilities for non-engineered constructions in Malawi.The work (i) presents the geometry of 646 façades of 327 houses, classified by structural features and potential failure modes and (ii) derives fragility curves by adopting the FaMIVE (Failure Mechanism Identification and Vulnerability Evaluation) mechanical approach [17].Building on this previous research, the objective of this work is to enrich the current portfolio of fragility curves for the country focusing on informal, non-engineered, unreinforced masonry (URM) residential buildings (Fig. 1).In particular, by referring to the classification of Kloukinas et al. [12], URMs with the following characteristics are considered ("UFB*" in [12]): (i) fired bricks in cement mortar assemblage, (ii) absence of timber/concrete lintels, (iii) light metal roof (no diaphragm action), (iv) rectangular plan shape, and (v) single storey.
In order to derive fragility curves representative of actual URM residential buildings in the country, the analysis is carried out by considering the database reported by Novelli et al. [16], that includes information of houses located in different areas of the country, as shown in Fig. 2a.The dataset contains the geometric data of the surveyed houses that defines the inter-building variability of the construction class.Fig. 2b-c shows histograms of the façades' heights and thicknesses, indicating that the geometrical characteristics of the façades are highly variable.Additionally, a set of experimental tests on local masonry [19,20] is used to characterise the mechanical properties in a statistical manner.
The methodology adopted to derive seismic fragilities consists of two parts: (i) an In-Plane (IP) damage assessment is carried out by analysing each façade with a nonlinear orthotropic Finite Element (FE) model that accounts for the regularity (stretcher bond) of the masonry assemblage; (ii) an Out-Of-Plane (OOP) damage assessment is executed on each façade through an analytical closed-form procedure developed by Giordano et al. [21].Both assessments start with the estimation of the nonlinear static capacity curve (pushover) of the façade.Subsequently, the amplitude of the seismic Intensity Measure (IM) that triggers a specific Damage State (DS) is calculated by adopting two procedures from the literature: SPO2IDA (Static Pushover to Incremental Dynamic Analysis) by Vamvatsikos and Cornell [22] and a CSM-based procedure (Capacity Spectrum Method [23]) proposed by Lagomarsino [24] and adapted by Giordano et al. [25].The final fragility curves are calculated by combining the assessments from the two methods on each façade in a consistent probabilistic manner.Section 2 presents the fragility assessment methodology of this study.Section 3 discusses the experimental response of typical Malawian masonry and presents a probabilistic model to be used for fragility derivation.Lastly, Section 4 discusses the fragility results with reference to existing studies from the literature and underlines the relevance of this work towards the implementation of country-specific seismic risk assessment analyses for Malawi.

Fragility assessment methodology
Malawian non-engineered URM buildings are characterised by several structural weaknesses that affect their seismic behaviour [16].Consequently, the fragility assessment procedure should reflect these deficiencies by introducing suitable considerations in the analyses [25].Given the lack of adequate wall-to-wall/wall-to-roof connections and the absence of boxbehaviour due to flexibility of the roof [8], this study considers two main assumptions: (i) the building's fragility is governed by the fragility of its constituting façades (e.g.[25][26][27]), and (ii) the IP and OOP damage assessment of URM façades can be decoupled (e.g.[28,29]).The fragility methodology is schematically depicted in Fig. 3 and described in the following list.a) Stochastic generation of typological buildings (Fig. 3a) The first step of the methodology consists of generating an ensemble of buildings that reasonably represents the UFB* housing stock in Malawi.646 façade geometries are extracted from the 327 buildings of the database in Fig. 2 (normally one façade for the short direction of the building and one façade for the perpendicular long direction) [16].This represents the geometrical aleatory variability.For each façade, a set of URM material properties is assigned in a probabilistic manner.In particular, façades that belong to the same building are characterised by different mechanical parameters to take into account the variability of construction quality within the same construction [12].The probabilistic distributions from which the random properties are extracted are based on experimental tests of local URM materials [19], as discussed in Section 3.This reflects the material aleatory variability.b) IP and OOP capacity of façades (Fig. 3b) The damage assessment of each façade is carried out by considering the IP and OOP nonlinear static (pushover) responses.The IP capacity is estimated with an FE analysis of the façade carried out in OpenSees [30] through STKO (Scientific ToolKit for OpenSees) [31].To account for the non-symmetric response of irregular façades, the IP pushover is estimated for both positive and negative directions of the seismic forces.The OOP capacity is quantified with an analytical mechanics-based closed-form procedure developed by Giordano et al. [21].The numerical-IP and analytical- OOP capacity curves are then represented by bilinear/multi-linear models according to De Luca et al. [32].Details of the IP/OOP capacity estimations are reported in Sections 2.1 and 2.2.c) Fragility curves of façades (Fig. 3c) For each façade's pushover diagram, a set of DS fragilities is generated by adopting two methodologies: SPO2IDA [22] (described in Section 2.3.1) and CSM [23][24][25] (reported in Section 2.3.2).According to these techniques, the DSs are defined in terms of specific displacement thresholds on the pushover curve [13,33] (details in Section 2.3.3).The exceedance probability of a generic DS# is a function of a seismic IM.The IM selected for this study is the Peak Ground Acceleration (PGA).The adoption of the two assessment techniques (SPO2IDA/CSM) for each damage mode (IP/OOP) is The fragility sets of the façades are subsequently combined to obtain DS fragilities of each building.For a given damage mode (IP/OOP), DS#, and assessment technique (SPO2IDA/CSM), a building's DS# fragility corresponds to the maximum envelope of the DS# fragilities of its Fig. 3. Seismic fragility assessment methodology.
N. Giordano et al. constituting façades (i.e.maximum probability of damage).Subsequently, the DS# fragility of the building class is calculated by averaging the fragility curves of all the buildings for each PGA.With the four sets of fragility curves listed in point (c) above, two final additional sets (IP and OOP) can be derived to account for the epistemic uncertainty of the assessment technique.For a given damage mode and a given PGA, the fragility curves of buildings derived with SPO2IDA and CSM are averaged to obtain a final fragility estimate (mean curve).

In-plane capacity
The IP capacity of a URM building can be estimated with a variety of methods that range from simplified analytical procedures to computationally expensive numerical techniques.Among the numerical approaches, many studies model buildings as FE equivalent frames, where the URM walls are discretised into a system of nonlinear piers, nonlinear spandrels, and rigid nodes [34,35].The equivalent frame represents a reasonable simplification to assess the IP response of URMs and, owing to its low computational cost, can be used to perform probabilistic analyses (e.g.[36][37][38]).However, several studies have pointed out that the assumptions related to the frame discretisation are questionable [39], which increases the uncertainty of the assessment results [40].An alternative to the equivalent frame is the Nonlinear Continuum FE (NC-FE) approach.It is usually implemented in two ways: by modelling masonry units and mortar joints with different constitutive models (micro-modelling), or by representing masonry as a single material with homogenised mechanical properties (macro-modelling) [41].With this latter method, the different failure modes of masonry, such as cracking/ crushing of units/joints and sliding of units are smeared out in a continuum [42,43].Macro-modelling is generally preferred to the former since it requires less computational effort.There are various past studies on the experimental validation of NC-FE models (e.g.[42,[44][45][46]), including results that are applied to historical URM buildings (e.g.[47][48][49][50]).In contrast, fragility assessment studies on URM building typologies based on NC-FE analyses are lacking in the literature.Two main reasons are the need for comprehensive information on URM mechanical parameters of the investigated structural typology (which are not usually available) and the computational cost of NC-FE models in performing a large number of seismic fragility analyses.
With the aims of executing NC-FE analyses and reducing the simplifications associated with IP modelling, the above-mentioned shortcomings are addressed in this work.The computational cost of NC-FE is reduced by using an implicit (at structure level) -explicit (at material level) nonlinear solver available in STKO [31,51] and implemented in OpenSees [30].Additionally, taking advantage of available experimental results of Malawian masonry (Section 3), the URM orthotropic constitutive model is characterised in a probabilistic way.Specifically, each façade is discretised with quadrilateral plane-stress elements.Orthotropic material properties are assigned to the tension-compression plastic-damage model developed by Pelà et al. [43] and implemented in STKO [31,51].Once the FEM model is built, nonlinear static pushover analysis is carried out by considering an inverse-triangular force distribution, as suggested by international codes [52].

Example of IP capacity of URM façade
An example of the procedure to identify the IP capacity is provided in Fig. 4 perforated wall (right side of the door) and a slender pier (left side of the door).The two portions are linked through a spandrel.The dimensions of the door and the window are 0.8 m × 1.8 m and 0.4 m × 0.7 m, respectively.The masonry mechanical properties are the median values of the distributions reported in Section 3.3.The mesh size of the quadrilateral elements is 100 mm × 100 mm.Fig. 4a-b-c show the damage patterns of the façade computed at different displacement levels for loading to the left, while Fig. 4d reports the force-displacement response of the façade.Fig. 4a corresponds to the maximum horizontal capacity (11.3 kN): cracks appear at the corners of the window and the bottomright of the spandrel.Fig. 4b corresponds to a post-peak horizontal capacity equal to 61% of the maximum.A rocking mechanism of the topleft side of the perforated wall is activated.Subsequently, Fig. 4c shows an additional rocking mechanism of the left pier and consequent rotation of the spandrel.The corresponding horizontal capacity has decreased to 39% of the maximum capacity.These results highlight the complexity of the IP response of URM façades, especially in the absence of floor/roof constraints that prevent relative displacements of the top nodes.With such constraints, experimental quasi-static tests on URM façades have shown a very stable force-displacement response (e.g.[53,54]).

Out-of-plane capacity
The OOP capacity of the façades is quantified by adopting the analytical closed-form mechanical model developed by Giordano et al. [21] for the fragility assessment of traditional URM buildings in Nepal [25].The method adopts three main assumptions: (i) as validated experimentally [55][56][57][58][59], the OOP capacity of a vertically spanning URM wall is governed by rocking; (ii) for the case of the cantilever configuration (i.e. with no constraint at the top of the wall), the wall is modelled as a rigid body connected to the ground with a nonlinear hinge; and (iii) the nonlinear response of the hinge is defined through the moment-curvature diagram of the base (critical) cross-section [60].
Utilizing the experimental data on OOP tests of Malawian URM that provide the tensile capacity of bed-joint interfaces, the cantilever model developed in [21] is improved by taking into account the tensile contribution of mortar joints in the initial-elastic phase.Fig. 5 depicts an example of an OOP pushover curve with reference to the façade of Section 2.1.The structure behaves in a linear elastic manner until the maximum tensile strength of the base bed-joint is achieved (cracking).After the exceedance of the tensile strength, the façade behaves according to the nonlinear OOP response described by Giordano et al. [21].

Estimation of façade's fragility for a given DS#
As illustrated in Fig. 3c, for a given façade, the estimation of DS fragilities is carried out with SPO2IDA [22] and CSM [25].In Sections 2.3.1.and 2.3.2,these approaches are described.As previously mentioned, the IM selected in this work is the PGA, since it is a reasonable measure for low-rise masonry buildings (e.g.[25,61,62]) and this information is available from national seismic hazard maps (e.g.[7,9]).In line with previous studies (e.g.[25,37,38]), DSs are defined as displacement thresholds on the pushover curve of the building.Section 2.3.3 reports the definition of DS limits with reference to IP and OOP responses.

SPO2IDA assessment
The first step of the SPO2IDA assessment is to convert the force--displacement pushover curve into a spectral-acceleration (S a ) versus spectral-displacement (S d ) capacity curve through the single degree of freedom (SDOF) equivalence (e.g.Eurocode 8 [52]).Subsequently, a simplified multi-linear model of the capacity curve is calculated according to De Luca et al. [32].The multi-linear curve is then converted into a set of equivalent Incremental Dynamic Analysis (IDA) curves by adopting SPO2IDA [22].Three curves, representing the 16th, 50th, and 84th percentiles of equivalent IDAs, are quantified to reflect the recordto-record variability.Given a DS# displacement threshold on the multilinear capacity curve, the corresponding damage fragility is estimated by considering a lognormal distribution model.The median value S a,50% corresponds to the 50th percentile estimate of the IDA.The dispersion is defined as β = 0.5 × ln (S a,84% /S a,16% ).The resulting fragility is expressed in spectral-acceleration for the equivalent vibration period of the SDOF, T eq .This period is directly calculated from the multilinear model.For instance, it has an IP average value of 0.065 s (CoV = 0.36) over the database of façades.To increase the practical usability of the fragility curves, the spectral-acceleration values are converted to PGA through a scaling relationship based on the Ground Motion Prediction Equation (GMPE) by Boore et al. [63].This GMPE is suitable for shallow Fig. 5. Simulated OOP response of a representative façade.The star represents the cracking limit for tensile failure of the base bed-joint.
N. Giordano et al. crustal earthquakes in active tectonic regions and it is based on global data.The conversion from S a (T eq ) to PGA is carried out with Equation (1): where PGA GMPE and Ŝa ( T eq ) GMPE are the average PGA and S a (T eq ), respectively, from the GMPE, considering the case of normal faulting, top 30 m average shear wave velocity of 300 m/s [64], distance interval of 1 km to 30 km and magnitude interval of 5 Mw to 8 Mw [7,9].Fig. 6a shows the described procedure for a generic DS#.

CSM assessment
The CSM assessment is carried out in reference to Giordano et al. [25].In analogy with SPO2IDA, the force-displacement pushover curve is firstly mapped into the S a -S d plane.Subsequently, for the sole IP capacity curve, a simplified bilinear model is calculated using the procedure presented by De Luca et al. [32].Given a DS# displacement on the capacity curve, the following analysis steps are carried out to derive the corresponding fragility curve: -934 smoothed S a -S d spectral shapes [65] from the SIMBAD (Selected Input Motions for displacement-Based Assessment and Design) database [66] are scaled to intersect the pushover curve at the DS# displacement, by using the CSM [21,23,24].This results in a set of 934 PGAs.-The 934 PGAs are sorted by their absolute scale factor, ASF = max {SF; 1/SF}, where the scale factor, SF, is the ratio between the scaled PGA (estimated with the CSM) and the PGA of the recorded ground motion [25].Subsequently, the first (lowest ASF) 100 PGAs are selected.As discussed by Giordano et al. [25], this selection procedure preserves record-to-record variability while removing PGA values with excessively high scale factors [67].-Lastly, the cumulative distribution of the 100 PGAs is fitted with a lognormal model to derive the DS# fragility.
Fig. 6b depicts the CSM assessment for a given façade and DS#.

Definition of DSs
DSs are defined as displacement thresholds on the capacity curve.For the IP curve, the indications by Martins and Silva [13] are adopted: -S d,DS1 (slight damage) corresponds to 75% of the yielding displacement S d,y of the bilinear/multi-linear model; -S d,DS2 (moderate damage) is defined as 0.5 × S d,y + 0.33 × S d,u , where S d,u is the ultimate displacement of the multi-linear/bilinear model.S d,u is defined on the numerical capacity curve as the displacement at 80% post-peak capacity; -S d,DS3 (severe damage) corresponds to 0.25 × S d,y + 0.67 × S d,u ; -S d,DS4 (near collapse) is equal to S d,u .
OOP DSs are defined according to previous studies [13,25,33]: -S d,DS1 (slight damage) corresponds to the displacement at cracking (S d, y ), which is highlighted with a star in Fig. 5.

Experimental data of local materials
An extensive experimental programme was carried out in Malawi to characterise the mechanical properties of local masonry materials [19,20].In this work, four sets of experimental tests are used to define the probabilistic material model of URM: • Vertical compression tests on masonry prisms were carried out according to EN 1052-1 [68].The prisms were each composed of five bricks having a nominal size of 200 mm × 90 mm × 50 mm.A total of twenty-four tests are considered in this study: (i) six tests characterised by a cement-to-sand ratio of 1:4 (the recommended value according to MS791-1 [69]) and realised in favourable (F) construction conditions (with clean and dampened bricks); (ii) six tests with a cement-to-sand ratio of 1:4 and realised in unfavourable (U) conditions (bricks laid in a dry state and in the presence of dust); (iii) six tests characterised by a cement-to-sand ratio of 1:6 (minimum value indicated by the Safer House Construction Guidelines [70]) and realised in favourable conditions; and (iv) six tests with a cement-to-sand ratio of 1:6 and constructed in unfavourable conditions.The mechanical properties that can be obtained from the vertical compression tests are: the elastic modulus in the vertical direction (i.e.perpendicular to the bed-joints) (E v ), the compressive strength at the elastic limit in the vertical direction (f c0 ), the ultimate compressive strength in the vertical direction (f cp ), and the corresponding strain in the vertical direction (ε p ). Fig. 7a-b shows the stress-strain results of the 1:4 and 1:6 tests, respectively.It is observed that the material behaviour varies significantly in strength, stiffness, and strain capacity, which can only be reflected by using an advanced numerical modelling technique (Section 2.1).In addition, the construction conditions (F/U) do not consistently affect the stress-strain results for either 1:4 or 1:6 cement-to-sand ratios.
• Diagonal compression tests on masonry panels were carried out by using the in-situ testing procedure presented by Brignola et al. [71].The dimension of the masonry panels was approximately 1060 mm × 1060 mm with a thickness of 90 mm (one-brick thick).A total of twelve experimental results are considered: (i) six texts characterised by a cement-to-sand ratio of 1:4 and favourable construction conditions; and (ii) six tests characterised by a cement-to-sand ratio of 1:6 and unfavourable construction conditions.The masonry properties extracted from the diagonal compression tests are the shear modulus (G) and the equivalent shear strength (τ 0 ).Fig. 8a-b shows the stress-strain results of the 1:4 and 1:6 tests, respectively.As expected, the 1:4 panels (Fig. 8a) are considerably stronger than the 1:6 panels, but, at the same time, more brittle.• Compression tests on bricks were carried out according to the Malawian standard MS6 [72] to quantify their strength (f b ).In total, 54 test results are considered.The sample exhibits a quite low median value (4.32 MPa) and high dispersion (coefficient of variation ≈ 0.40) with respect to typical international benchmark values (e.g.N. Giordano et al. [73]), which is a consequence of the non-industrial production process of Malawian bricks.• OOP four point bending tests on masonry panels (bending parallel to the bed-joints) were carried out according to EN 1052-2 [74].The dimension of the URM specimens was 760 mm × 420 mm with thickness of 90 mm.A total of nine tests are considered where four of them are characterised by a 1:4 cement-to-sand ratio and favourable construction conditions, while five of them had a 1:6 cement-to-sand ratio and unfavourable construction conditions.In terms of mechanical parameters, the OOP bending tests provide the tensile strength of the mortar bed-joints (f t,bj ).

Numerical modelling of experimental tests
The experimental results reported in Section 3.1 are used as a benchmark for the calibration of the IP numerical analyses (Section 2.1).As previously discussed, given the periodic bond (stretcher bond) of typical Malawian brick masonry, the orthotropic material model by Pelà et al. [43] (OrthotropicMaterial in STKO [31]) is adopted.In addition to the mechanical parameters obtained from the tests (Section 3.1), additional parameters have to be defined for the orthotropic model.Therefore, the following assumptions are considered: -The elastic modulus in the horizontal direction (parallel to the bedjoints), E h , is quantified according to the formula that Mojsilovic [75] calibrated with 130 experimental compression tests on URM walls: were G can be assumed as 35% of E v as in Wilding et al. [76].Equation ( 2) is also consistent with the results of recent experimental investigations on URM panels tested in two orthogonal directions [77]; -The Poisson's ratio ν is assumed equal to 0.15 according to global data on masonry mechanical properties [78]; -The compressive strength of masonry in the horizontal direction is conservatively assumed as half of the value in the vertical direction to account for the weakness of head joints in low-quality masonry [8].
Fig. 9a shows a comparison between the results of a vertical compression test and a numerical simulation (test 1:4(F)-1).It is observed that the numerical model can accurately reproduce the experimental response of the prism.The average absolute error between experimental and numerical curves is 2.9%.The modelling of the full set of the tests provides an average absolute error between 2.4% and 9.7% (mean value: 5.6%).Fig. 9b shows a comparison between the results of a diagonal compression test and a numerical simulation (test 1:6(U)-6).In this case, the numerical model is less accurate in reproducing the experimental behaviour of the panel.This is the consequence of larger uncertainties in the modelling of diagonal compression tests with respect to vertical compression tests [71,79].The experimental versus numerical comparison on test 1:6(U)-6 is characterised by an average absolute error of 9.0%.Additionally, looking at the full set of the tests, the average absolute error is between 4.6% and 17.3% (mean value: 10.3%).

Probabilistic models for masonry material properties
The mechanical properties extracted from the tests are used to define the URM material model in a probabilistic way.Each material property is represented by a lognormal distribution whose parameters (median and lognormal standard deviation) are reported in Table 1.Additional parameters required for conducting the IP numerical analyses are taken from the literature.Particularly, the fracture energy in compression (G c ) and in tension (G t ) are described with uniform probability distributions.The upper and lower limits of these distributions are taken from da Porto et al. [79]: G c = 5.0 N/mm to 20.0 N/mm and G t = 0.018 N/mm to 0.05 N/mm.It is worth mentioning that these intervals are consistent with the fracture energy values adopted for the numerical vs. experimental comparisons described in Section 3.2.Fig. 10a shows the probabilistic distributions of the main strength parameters, f c0 , f cp , τ 0 , and f t,bj , while Fig. 10b shows the distributions of the stiffness parameters E v , E h , and G.

Discussion of the seismic fragility results
Fig. 11 shows an example of the derivation of DS# fragility for the UFB* building class.In particular, it refers to the IP damage mode, SPO2IDA assessment technique, and DS4.The grey lines represent the fragility curves of the portfolio of buildings where each curve is the maximum envelope of the façades' fragilities of a building.The solid red line represents the mean fragility at conditional PGA.The black lines (solid and dash-dotted) represent the mean +/-standard deviation fragilities to illustrate the dispersion around the mean result.Lastly, the red dashdotted line indicates the lognormal approximation of the mean fragility.For a given probability of exceedance, the dispersion around the mean estimate can be quantified with the formula reported in Section 2.3.1.,β = 0.5 × ln (PGA 84% /PGA 16% ).For instance, at P(D ≥ DS4|PGA) = 0.5 (median value of the fragility curve), the dispersion β is equal to 0.55.This value indicates that the sources of inter-/intra-building variability introduced in the model adequately reflect the typical confidence interval of fragility curves of portfolio of buildings (e.g.[80]).
The full set of mean fragility estimates is shown in Fig. 12, while the corresponding lognormal parameters (median η and lognormal standard deviation β) are summarised in Table 2. Fig. 12a-b shows the IP damage fragilities assessed with SPO2IDA and CSM, respectively.Fig. 12c-d includes the corresponding OOP fragility curves.Lastly, Fig. 12e-f depicts the average SPO2IDA-CSM fragility estimates.In addition, fragility curves for brick masonry buildings developed by GEM [13] and relevant to Malawi are plotted along with the IP estimates.
The key findings of the study are summarised as follow: The IP fragility estimates (Fig. 12a-b-e) are characterised by a close similarity between the DS3 and DS4 curves.This can be explained by looking at the IP force-displacement response of the façades (example in Fig. 4d).In most cases, the post-peak response presents a rapid decay in horizontal capacity.Consequently, the DS4 displacement (defined at 80% post-peak capacity) is close to the displacement at peak horizontal capacity.In other words, the similarity between DS3 and DS4 is the result of the brittle IP response of the façades, as discussed in Section 2.1.1.
The OOP fragility curves (Fig. 12c-d-f) present lower median PGA values with respect to IP estimates.This is an expected result given the well-known vulnerability of non-engineered URMs against OOP loads [8,25].Therefore, UFB* buildings are more likely to experience partial OOP failures of walls rather than widespread IP damage.
The comparison between CSM and SPO2IDA provides interesting insights on the epistemic uncertainty of seismic assessment methods.Looking at the IP damage mode, the CSM results are more conservative than SPO2IDA, in agreement with previous research findings [81].In particular, median values of DS fragilities are approximately 20% lower for CSM with respect to SPO2IDA.Both techniques provide reasonable fragility results, therefore, to account for the epistemic uncertainty with regard to the seismic performance assessment, the mean IP fragilities (Fig. 12e) should be considered.In contrast, when looking at the OOP damage mode, the CSM η values are consistently higher than the corresponding SPO2IDA estimates  (about two times higher).More generally, the OOP-SPO2IDA fragilities seem too conservative with respect to experimental evidence [56] and previous research works on OOP fragility of URMs [25,28,29].Meanwhile the CSM fragilities are in line with the literature and should be selected to characterise the OOP vulnerability of the UFB* category.Lastly, the comparison between the IP fragilities of this study and the ones of GEM (Martins and Silva [13]) displays a satisfactory match on the median values and a very good fitting of the dispersions β.In particular, the GEM fragilities provide larger median estimates with respect to the results of this work since: (i) they have been derived with nonlinear time history analysis on equivalent SDOFs (a less conservative technique with respect to nonlinear static analysis/simplified IDA), and (ii) the nonlinear response of the equivalent SDOF is quantified by assuming a global response of the URM building.

Conclusions
The fragility model presented in this paper refers to typical URM brick-in-cement houses in Malawi.The work merges research findings on structural materials and in-field data on building geometries.This allows the estimation of new country-specific fragility curves, currently not available in the technical-scientific literature.The procedure adopted to derive fragility curves provides estimates for in-plane and out-ofplane damage potential.The epistemic uncertainty around seismic assessment methods is also addressed by using two established approaches, the CSM and SPO2IDA.The key finding of the study is that non-engineered Malawian houses are highly vulnerable to out-of-plane damage, something that could be prevented by improving construction details such as wall-to-wall and wall-to-roof connections.If out-ofplane action is controlled, masonry walls are more likely to sustain inplane damage first.In general, the fragility of Malawian URMs appears larger than the estimate provided by global models.In risk analyses, it is suggested to adopt IP-mean fragility sets (Fig. 12e) for buildings with good construction details and OOP-CSM fragilities (Fig. 12d) for low-quality buildings.In conclusion, the new fragility curves represent a consistent advancement towards the implementation of earthquake risk modelling studies in the context of Malawi and in the wider East African region.

Data access statement
The underlying data for this study were drawn from the referenced literature.All the data produced in this work are provided in full within this paper.

Fig. 4 .
Fig. 4. IP numerical response of a representative façade where the contour plots show the material tensile damage (d + ).(a) damage pattern at maximum horizontal capacity, (b) damage pattern at 61% post-peak capacity, (c) damage pattern at 39% post-peak capacity, (d) force-displacement diagram.

Table 1
Probabilistic parameters of material properties extracted from the experimental tests.

Table 2
Lognormal parameters of seismic fragility curves.