Due to the complexity of conducting full-scale experimental tests to characterize the seismic behaviour of buildings (e.g. Bothara et al. 2010), numerical seismic assessment methods are increasingly and widely used within the engineering community (e.g. D’Ayala and Speranza 2003; Lagomarsino et al. 2013; D’Ayala et al. 2015). Reliability and efficiency of numerical analysis have increased in recent years with the advancement of modelling approaches, software tools and computational capabilities. Furthermore, numerical models afford the flexibility of conducting extensive parametric analysis, too expensive and unfeasible for experimental tests. This section presents a discussion on the numerical modelling approach, results of validation and calibration studies for SMM masonry (Sect. 5.1) and the results of non-linear pushover analyses conducted on the PRE-SMM and POST-SMM IBs (Sects. 5.2, 5.3).
Applied element modelling of SMM masonry: validation and calibration
Modelling of masonry is complex because of its heterogenous nature due to the presence of units and mortar having different elastic and non-linear properties. Several methods have been used for studying the structural behaviour of masonry: e.g. limit-analysis based methods (e.g. D’Ayala and Speranza 2003) and numerical methods such as finite element based methods (e.g. Bothara et al. 2018b), discrete element based method (e.g. Lemos and Campos Costa 2017) and applied element based methods (e.g. Karbassi and Nollet 2013). Three approaches can be followed for numerical modelling and analysis of masonry: micro-modelling, simplified micro-modelling and macro-modelling (Lourenço and Rots 1997). In the simplified micro-modelling approach, the mortar layer and the two unit-mortar interfaces are lumped into a zero-thickness joint while the units are slightly expanded on all sides to accommodate the mortar thickness. The joints represent all the deformability and non-linearity of the masonry material while the units are considered as rigid elements. Such modelling strategy is appropriate for the Nepalese SMM construction where the mortar joints have poor stiffness and strength properties compared to the stone units. Furthermore, this approach is a balanced option in terms of the computational cost/time as well as the accuracy of the results compared to the detailed micro-modelling or macro-modelling approaches (Lourenço and Rots 1997). A detailed overview of the modelling strategies and analysis options currently available in the literature for the analysis of masonry structures can be found in D’Altri et al. (2019). Simplified micro-modelling approach using AEM has been chosen for this study as the complete non-linear lateral behaviour of masonry, from the initiation of cracking to the ultimate collapse state, can be studied with this approach (e.g. see Karbassi and Nollet 2013; Malomo et al. 2018).
The version of AEM commercialised in Extreme Loading for Structures (ELS) software (ASI 2018) is used in the present study. In AEM, the masonry is discretized in 3D-elements whose surfaces are connected by a number of deformable springs that simulate the interaction among the blocks with force–deformation laws representing the elastic and post-elastic behaviour of the material as a whole. Two types of springs are used for modelling masonry, ‘unit’ springs connecting the 3D elements of the units, and ‘interface’ springs connecting the individual applied elements to represent the equivalent properties of the mortar and unit-mortar interfaces. A detailed overview of the formulation, constitutive laws, failure criteria etc. for masonry modelling in AEM can be found in ASI (2018) and Malomo et al. (2018).
In this study, to account for the random irregular shape of rubble stone, a triangular 3-D mesh is first created, and then random shaped units are generated by clustering these triangular applied elements (each with 6 degrees of freedom) by means of the ‘unit’ springs. Figure 10 is self-explanatory: different coloured clusters represent stone units which are connected at their faces by the ‘interface’ springs. It should be noted that such modelling technique inherently presents some uncertainty as the wall construction itself presents great variability in terms of shape and size of units and the resulting bond pattern (also refer to Fig. 7).
For the validation and calibration of the proposed numerical modelling strategy, numerically obtained compressive and lateral behaviour of SMM walls are compared against three distinct experimental tests: uniaxial compression behaviour, in-plane shear-compression behaviour and out-of-plane bending behaviour.
Uniaxial compression behaviour
The uniaxial compression test on SMM wallettes conducted by Build Change (2019) in Nepal is numerically reproduced. The test specimens of size 400 mm × 470 mm × 540 mm (thickness × width × height)) were built by local masons using local stone and mud mortar from rural districts and hence are well representative of the constructions considered in this study (Sect. 4). Key material properties from the accompanied mechanical characterization tests are presented in Table 1 and the same set of material properties is used for the numerical analysis in the present study. It is interesting to note that some studies have used an elastic modulus for the Nepalese SMM constructions as high as 850 MPa in their numerical studies (e.g. Bothara et al. 2018b) which is an order of magnitude higher than the recent test results (Table 1).
Figure 11 compares the numerical crack patterns and the stress–strain diagram to the experimental results for the SMM wallete subjected to uniaxial compression. Vertical and inclined cracks are observed which mainly pass through the mud mortar joints, both in the experiment and numerical analysis result. In terms of stress–strain curve, apart from a slight discrepancy in the initial stiffness, the average Young’s modulus as well as peak strength and strain are well predicted by the numerical analysis. The slight difference in the initial stiffness is most probably due to the fact that the test specimen presented some voids and hence showed non-linearity at the beginning of the test.
Lateral shear–compression behaviour
For the validation of the lateral in-plane behaviour of SMM wall, the cyclic shear–compression test conducted by Build Change (2019) is modelled. The wall specimen has a size of 1.2 m (length) × 1.2 m (height) × 0.45 m (thickness) and is built by local masons. Since this test is also from the same test campaign, the material properties listed in Table 1 are applicable. The wall is subjected to a cyclic (loading–unloading) displacement history up to a maximum lateral displacement of 48 mm under a vertical pre-compression of 0.011 MPa.
Figure 12 shows the comparison of ultimate crack pattern under cyclic shear–compression loading. As the cyclic displacement level increases, the cracks become widely distributed all over the wall surface, passing mostly through the mud mortar joint/interfaces. Figure 13 presents the comparison of hysteretic load–deformation behaviour from the AEM analysis and the experiment. Overall, the lateral loading and unloading stiffnesses, peak capacity as well as the ductility are well reproduced by the AEM analysis, although the numerical cycles have a larger hysteresis.
Lateral bending behaviour
In order to validate the bending behaviour of mud mortar masonry walls, the experimental test on a semi-dressed stone masonry pier of dimensions 1400 mm × 500 mm × 350 mm (height × length × thickness) conducted by Pun (2015) is considered. The location of crack at ultimate stage (which was at a height of 925 mm from the bottom in the experimental test, see Fig. 14a) as well as the load–deformation behaviour (Fig. 14b) are well reproduced by AEM analysis.
Thus, the validation and calibration studies presented in Sects. 5.1.1–5.1.3 confirm that AEM can be effectively used for the gravity and lateral load analysis of SMM structures. In all the validation and calibration studies, the AEM numerical models consisted of 3 × 3 pairs of interface springs per shared face and the failure criterion was defined by specifying a separation strain of these interface springs equal to 0.025. These parameters are used in the numerical modelling of the SMM IBs in Sect. 5.2. Once the separation strain limit is exceeded in an interface spring, it has no further tensile capacity, however frictional contact can occur between the adjacent applied elements depending on the loading condition. As the values of the shear modulus and the friction coefficient were not provided by the experimental tests by Build Change (2019), they were obtained as results of the calibration analysis. An average shear modulus of 0.35 times the elastic modulus is found for Nepalese SMM construction, which is in close correlation with codes' recommended relationships (EN 1996-1-1, MSJC 2011). Similarly, a friction coefficient of 0.4 is found from the numerical calibration study. These values of the shear modulus and the friction coefficient are used in the numerical models of the SMM IBs in Sect. 5.2.
For computational efficiency stone units are modelled as rigid elements i.e. the cracks are assumed to develop through the mortar joints only. This assumption is justifiable from the damage results presented in this section, whereby cracks largely develop through the mortar joint and interfaces, due to the substantially lower stiffness and strength properties of the mud mortar compared to that of the stone units.
Numerical modelling and pushover analysis of the SMM IBs
Three dimensional numerical models (Fig. 15) of the PRE-SMM and POST-SMM IBs are created using the modelling strategy discussed in Sect. 5.1 for the SMM walls. Corner stones are created as rectangular elements at all corners and through stones are provided in the wall at an average spacing of 1.2 m in both horizontal and vertical directions, respectively, as per field observations.
For the PRE-SMM IB model, all timber members (floor members, roof members, lintels above the openings, as well as the frames around the openings) are explicitly modelled, as elastic elements, to represent their stiffness contribution to the seismic behaviour of the building. These members are all meshed so that the mesh size is comparable to the size of stone units. The timber-wall interface is represented by the properties of the mud mortar, to represent the actual situation in these constructions.
For the POST-SMM model, the RC seismic bands, confining elements around the openings and the corner reinforcements are all explicitly modelled so that their contribution in improving the global seismic behaviour can be reproduced. The RC elements are meshed such that the size of meshing is comparable to the size of stone units. For this IB model as well, the interfaces between the RC members and the SMM walls are modelled with the properties of mud mortar. The concrete material is assumed to be M15 grade and the reinforcement bars have a yield strength of 415 MPa (NRA 2016c). Although, depending on the level of connection to the load bearing walls, it could add some benefits to the global behaviour of the building, the light timber roof structure is not modelled. Instead, the equivalent mass of the roof structure is applied on the top of masonry walls as lumped mass so as to include its effect on the gravity and lateral loading.
In both the PRE-SMM and POST-SMM IB models, the SMM walls including the cross-wall connections are characterized by the material properties listed in Table 1. Considering that these constructions have stepped strip type foundations with depth greater than 0.5 m, the IB models are assumed to be fixed at the ground level.
Conventional pushover analysis of masonry structures modelled using the element-by-element modelling technique, with discontinuous joint represented by finite-strength springs, is complex as the application of pushover forces or displacements imposed on the structure often cause stress concentration on particular elements or regions thereby causing local failure without affecting the rest of the structure. Thus, the analysis is conducted by simulating the seismic action by a linearly increasing ground acceleration, resulting in an increasing effective earthquake force on the structure which is mass proportional. More details on this pushover loading approach can be found in Adhikari and D’Ayala (2019b).
Although the coefficient of variation (CoV) for different masonry properties in the test results (Build Change 2019) is low because of the small number of samples (three) from one district, the properties of masonry can vary greatly from one building to the next depending on the variation of materials used (stone type, size, shape etc. and the soil used for mortar preparation) as well as the workmanship quality (see also Sect. 4.1). Thus, a sensitivity analysis is conducted by considering the cases of a good material quality (assumed 50% better than the average values reported in Table 1) and a poor material quality (assumed 50% lower than the average material properties) in order to study the effect of variation of material properties in the seismic capacity and fragility/vulnerability functions of the SMM typologies. Such range is chosen based on the CoV reported in literature for rubble stone masonry (see Milosevic et al. 2013).
An Intel Core i7-7600U processor with a speed of 2.80 GHz was used for the modelling and computational analysis and the details of the modelling and the computational effort for the numerical models of the PRE- and POST-SMM IBs are given in Table 2. Although the number of applied elements in the POST-SMM IB model is significantly lower than the same in case of the PRE-SMM IB model, since the same loading steps size and number were used for both models, the analysis time is similar, at about one hour.
Capacity curves and failure mechanisms
Figure 16a compares the capacity curves for the PRE-SMM IB in the longitudinal and transverse directions. The FEMA 356 (FEMA 2000) performance level scale with four distinct performance levels (i.e. operational, immediate occupancy, life safety and collapse prevention) is used in this study. In Fig. 16, the base shear coefficient (also known as the lateral capacity) represents the ratio of the total base shear resisted to the seismic weight of the building. The peak lateral capacity attained by the PRE-SMM IB is very low, i.e. 0.12 g and 0.08 g in the longitudinal and transverse direction, respectively. Due to the poor strength of mortar as well as the random shape of the stone units, the non-linear response starts at a drift as low as 0.05% and the ultimate drift capacity is only about 0.4% in both principal directions. The ultimate drift is selected as the collapse prevention limit when a collapse mechanism is formed, and the cracks' width is greater than 15 mm; triggering a substantial rate of increase of the roof displacement (more than 5 mm), as observed in the capacity curve. Figure 16 also shows the roof drift thresholds representative of the four different performance levels which are used in the seismic performance assessment and fragility analyses in Sect. 6.2.
Figure 16b compares the capacity curves for the PRE-SMM typology for the three different material quality cases, showing substantial change in the ultimate capacity. An analytical study by D’Ayala and Kishali (2012) also found typically low lateral capacity with significant dispersion i.e. 0.07–0.19 g for two-storey Turkish SMM buildings which is comparable to the range for the PRE-SMM typology i.e. 0.02–0.13 g (Fig. 16b) in terms of the dispersion. It is thus important to improve the practice of material preparation as well as workmanship in the wall construction since both of these affect the resulting masonry fabric and hence the material properties. The poor quality case has more severe reduction (about 75%) in the ultimate capacity than the increment (about 60%) in the good quality case. However, in terms of ductility and the drift limits at different performance levels, only slight changes (± 10%) are observed. This is due to the fact that the roof drift is controlled by the crack patterns and crack width which are not affected substantially by the material quality in case of rubble stone masonry with inherently poor interlocking between the units, unless there is an introduction of ductile structural elements.
Figure 17 presents the ultimate collapse mechanisms of the PRE-SMM typology when loaded in the longitudinal and transverse directions. In the case of longitudinal direction loading, the building suffers torsional action due to the irregular distribution of openings and hence the damage starts at the weakest corner triggering the overturning collapse of the corner (Fig. 17a). The long wall with openings also suffers in-plane shear damage which ultimately activates the delamination in the transverse direction. In the case of transverse direction loading, diagonal shear crack originates at the corners of the openings in the short wall and in-plane shear failure including the corner collapse occurs (Fig. 17b). In both cases, the ultimate collapse mechanism involves the collapse of the short walls. These damage patterns are comparable to those observed on these buildings in the 2015 earthquake (see Fig. 1). It is important to notice the crucial function of the longitudinal internal timber structure, which prevents the floor and roof from collapsing, hence reducing the risk of injuries and fatalities, and allowing safe evacuation. The damage levels at the limit of different performance levels are shown and explained in Fig. 18.
Figure 19 shows the actual damage state of the case study IB after the 2015 Gorkha earthquake which was surveyed by the first author during his field trip. Major vertical separation cracks (maximum crack width of about 20 mm) in both direction walls can be seen suggesting that the building was subjected to bi-directional effect of ground motion. Minor diagonal cracks (less than 5 mm) are also present in both long and short walls. Although the building suffered severe vertical separation cracks, it survived collapse.
Both in the numerical analysis as well as in the observed damage (Figs. 18d, 19), excessive out-of-plane overturning of the walls is prevented, and the ultimate collapse mechanisms involves the corner failure and damage of short walls. This also indicates that the timber floor and the roof system promote the global box-type action.
Figure 20a compare the capacity curves of an IB of the POST-SMM typology in the longitudinal and transverse directions. The capacity curves in both longitudinal and transverse direction are similar reaching a peak lateral capacity of 0.6 g. Figure 20 also shows the roof drift thresholds representative of the four different performance levels. These are used in the seismic performance assessment and fragility analyses in Sect. 6.2. Figure 20b compares the capacity curves for the POST-SMM typology for different material quality cases. There is again a considerable dispersion in the lateral capacity (± 30%), ductility (± 20%), and the thresholds life safety and collapse prevention performance levels (± 30%) due to the different quality of materials.
A recent study by Bothara et al. (2018b) also found typically high lateral capacity (about 1.0 g) for single-storeyed SMM school buildings from Nepal (Fig. 21). Slightly higher initial stiffness in the present study is observed, although a one order of magnitude higher elastic modulus is used in the school building analysis. This is justified by the difference in the layout of openings in the two construction typologies, i.e. the school building has several openings on the back wall as well (see Bothara et al. 2018b). In contrast, due to the higher values of tensile strength and shear modulus (and possibly higher value of friction coefficient, although not reported) in the analysis by Bothara et al. (2018b), the yield as well as ultimate strength capacities are considerably higher than the same from the present study.
Figure 22 presents the damage levels at the limits of different performance levels under the pushover analysis in the transverse direction. The seismic bands and the confining elements around the openings successfully contain and prevent the propagation of diagonal shear failure of the wall panels. The final collapse mechanism in POST-SMM buildings is shear sliding failure of masonry portions at the wall-band horizontal interfaces resulting in the compression failure at the corner (see Fig. 22d). Such shear-sliding failure modes are typical when there is low vertical pre-compression and low-strength mortar (Tomazevic 1999).