Introduction

Masonry building is a widespread construction method due to its cost-effectiveness and ease of construction. The number of unreinforced buildings, including residential houses, educational facilities, government offices, and historic structures, is significant, especially in southern Asia. Most of these structures are built using traditional methods without any seismic provisions. The lateral load capacity of a masonry building is low due to its inadequate tensile strength and low ductility (Feng et al., 2011; Murtyl et al., 2004). These buildings are seismically vulnerable due to poor mortar connections, wall openings, improper wall sections and dimensions, and lack of vertical confinement. The lateral load capacity of existing or new masonry constructions can be increased by utilizing seismic protection techniques (Choudhury et al., 2015; Milani et al., 2017, 2018; Murtyl et al., 2004). For this purpose, strengthening-based retrofitting methods are commonly utilized. Seismic base isolation, which is utilized to reduce seismic demand by lengthening the natural period, is also an alternative strategy. Despite being acknowledged as a promising technology, base isolation is rarely used in conventional and brick structures due to high cost, weight, and unavailability, particularly in developing countries (Kelly, 2002; May, 2002; Pan et al., 2005). As a result, cost savings and the simple design concept of the base isolator are important considerations.

For low-to-medium-rise buildings, several low-cost base-isolations such as friction interfaces or sliding interfaces composed of marble, graphite powder, screened gravel, rubber-sand, and paraffin wax between structure and foundation have been proposed (Li, 1984; Nanda et al., 2015; Qamaruddin et al., 1986). The scrap tire pad (STP) isolator, which is made by stacking STP one above another without adhesive, was proposed by Turer and Özden (2008). But the application of unbonded layers of STP is impractical. A discarded Kart tire filled with recycled elastomeric materials and aggregates was proposed by Hadad et al. (2017). It has the potential to improve seismic performance. Further research is needed on fire resistance, the effects of ambient conditions, tire dimensions, and the size and amount of filling grains on dynamic response. A recycled tire bearing (RTB) was studied by Morales et al. (2018). It consists of tires that have been cut through their diameter and inserted between the structural slab of a designated room and a service floor surface. The vertical vibration, rocking motion, and pre-compression force are described as significant challenges for this system.

The scrap tire rubber pad (STRP) isolator is an attractive and low-cost alternative to SREI or FREI applicable for ordinary structures, as proposed by Mishra (2012). It is made by bonding several tire pads stacked one above another. A STRP isolator can be used without the thick endplates that are necessary to fasten the isolator to the structural base in the case of SREI. As a result, the cost of the isolator is significantly reduced. An isolator which is utilized without any bonding is known as an unbonded STRP (U-STRP). This type of isolator transfers the inertia force through friction. Since there are no difficulties relating to mechanical fastening, this type of isolator can be used in existing structures as well. In the case of an existing structure, the lifting of the superstructure method can be used to install the U-STRP isolator without interfering with serviceability. The feasibility of an unboded STRP isolator has been studied by Mishra (2012) and Zisan and Igarashi (2021). It shows that the stiffness of the U-STRP isolator degrades with lateral displacement due to roll-over deformation, which reduces the seismic demands of supporting structures. Again, the U-STRP isolator exhibits stiffness hardening at large deformation when the vertical edges of the isolator touch the structural elements during roll-over deformation as shown in Fig. 1. This isolator is stable until the roll-over deformation reaches 250% shear displacement (Zisan and Igarashi, 2021). This feature is unavailable in the conventional SREI or bonded isolator. This feature is advantageous in limiting the shear displacement during a maximum earthquake event (Van Engelen et al., 2016). Some of the tire-derived base-isolators have already been tested for seismic isolation of frame structures. A recycled rubber-fiber reinforced bearing (RR-FRB) made with recycled tire chips, carbon, and polyester fibers exhibits more advantages than that of SREI or FREI in terms of dissipation capacity, manufacturing cost, and weight (Spizzuoco et al., 2014). Shaking table tests (Calabrese et al., 2015; Maddaloni et al., 2017; Losanno et al., 2019) reveal significant improvement in isolated structures and an excellent re-centering capability of RR-FRBs. The drawback is the low shear strain and tensile capacity.

Fig. 1
figure 1

Lateral deformation and force–displacement relationship of bonded and unbonded STRP isolators

Full size image

Mishra et al. (2012), Mishra and Igarashi (2013) and Mishra et al. (2013) examined the stiffness and damping of U-STRP isolators as well as the hyperelastic properties of scrap tires. It was claimed that the U-STRP isolator has damping ratios of 10–22% and vertical-to-horizontal stiffness ratios of 450–600, above 150, which are critical for isolation material (Eurocode 8, 2004). A pseudo-dynamic test of the U-STRP isolator (Mishra et al., 2014) and a numerical analysis of a three-story STRP base-isolated building indicate that the STRP isolator is feasible for low-to-medium-rise buildings. The numerical analysis ignored the friction-based unbonded boundary condition of the U-STRP isolator and also the softening and hardening characteristics. Later, Zisan and Igarashi () investigated the lateral performance and deformation capacity of unbonded square and strip-shaped U-STRP isolators. These studies covered the influence of the length-to-width ratio and bearing height on the horizontal stiffness, damping, and seismic demand. It also provides a theoretical formulation for the lateral stiffness of U-STRP isolators. To judge the effectiveness of the U-STRP isolator in a real building, it is necessary to assess the performance of a U-STRP base-isolated building in real-time conditions. Therefore, the objective of this paper is to analyze the seismic vulnerability of masonry buildings resting on a three-dimensional model of U-STRP isolators. Both square-shaped and strip-shaped isolators having the same stiffness properties are taken into consideration. Pushover and fragility analyses are performed in order to assess the performance and vulnerability of the base-isolated building.

FE modelling of masonry building

Figure 2 shows a two-story masonry building with a plan dimension of 15 m × 10 m, typically found in the rural areas of Bangladesh. The overall height of the building above the grade beam is 6260 mm, and the story height is 3130 mm. All brick columns are equally spaced by 5000 mm in the X (strong axis) and Y (weak axis) directions. The principal components of the building are the reinforced concrete floor, grade beam, and load-bearing brick wall. The brick-made grade beams are 390 × 600 mm in size, and U-STRP isolators are placed below the grade beams. All the brick columns are 390 × 390 mm in size, and the brick walls and the concrete slab are 130 mm. Openings for doors and windows are 1100 × 2200 mm and 1400 × 1400 mm, respectively. To check the premature damage of walls, a reinforced concrete lintel of size 100 × 200 mm is inserted on top of wall openings. The unit weight of concrete and that of the brickworks is assumed to be 2400 kg/m3 and 1950 kg/m3, respectively.

Fig. 2
figure 2

FE model of masonry building considered in this study

Full size image

The material parameters for the masonry units are taken from Choudhury et al. (2015) and those for the reinforced concrete are found in Jain and Kennedy (1974) and Crisfield (1982). These parameters are listed in Table 1. Isotropic nonlinear material that exhibits cracking in both tension and compression is used for masonry units through a simplified stress–strain relationship, which is shown in Fig. 3 (Choudhury et al., 2015; Habieb et al., 2017). In the figure, fy and fu are the yield and ultimate compressive strength of a masonry wall, respectively. The symbol ft is used to indicate the tensile strength. The stress–strain in tension is linearly elastic until cracking occurs, followed by a softening branch, which indicates micro-crack propagation in the material. The behavior of the masonry unit in compression is linearly elastic until 1.8 MPa, followed by a 2.4 MPa crushing stress and a linear softening branch. The reinforced concrete elements are assumed to behave elastically. This assumption is quite reasonable since the strength of concrete is greater than that of the mortar. The yield strength for concrete is 31.6 MPa. The cracking behavior of mortar primarily determines the overall damage of a masonry structure since its tension capacity is deficient. The peak tensile strength of concrete and that of mortar is 2.0 MPa and 0.15 MPa. These values comply with the experimental data available in the literature and codes of practice. The tension softening modulus is 3.62 GPa and 0.123 GPa for concrete and mortar, respectively, around one-tenth of the elastic modulus. Both concrete and masonry units are modeled using an eight-node, isoparametric, arbitrary hexahedral element.

Table 1 Properties of concrete and masonry units utilized in the FE model
Full size table
Fig. 3
figure 3

Simplified nonlinear behavior of masonry units in compression and tension

Full size image

The maximum strength in tension and compression is calculated using the Mohr–Coulomb yield criterion under plane stress conditions. The yield surface is based on the generalized Mohr–Coulomb model, which was developed by Drucker and Prager. It can be mathematically expressed by (MSC Marc, 2020):

$$f{\text{ }} = \alpha {\text{J}}_{1} {\text{ + J}}_{2} ^{{1/2}} - \frac{{\mathop \sigma \limits^{{\text{ - }}} }}{{\sqrt 3 }},$$
(1)

where \(\mathop \sigma \limits^{{\text{ - }}}\) is the deviatoric stress and α is the material constant. J1 and J2 are the 1st and 2nd invirants of the deviatoric stress tensors given bellow.

$${\text{J}}_{1} {\text{ = }}\sigma _{{{\text{ij}}}}$$
(2)
$${\text{J}}_{2} {\text{ = }}\frac{1}{2}\sigma _{{{\text{ij}}}}^{{\text{d}}} \sigma _{{{\text{ij}}}}^{{\text{d}}}$$
(3)

The isotropic hardening rule is assumed using the Drucker and Prager model, described as follows:

$${\text{c}} = \frac{\sigma }{{\left[ {3\left( {{\text{1 - 12}}\alpha ^{2} } \right)} \right]^{{1/2}} }}{\text{andsin}}\phi {\text{ = }}\frac{{3\alpha }}{{\left( {{\text{1 - 12}}\alpha ^{2} } \right)^{{1/2}} }},$$
(4)

where c is the cohesion and ϕ is the angle of friction. In the current study, the cohesion and the angle of friction are assumed to be 0.15 MPa and 30°, respectively (Choudhury et al., 2015; Habieb et al., 2017).

Figures 4 and 5 show the natural period and mode shapes of the fixed-base and base-isolated masonry buildings. The first mode period of the fixed-base building is 0.15 s, which is comparable to the period of a low-rise masonry building estimated by the flowing equation (Calvi, 1999).

Fig. 4
figure 4

Period and mode shapes of fixed-base masonry buildings

Full size image
Fig. 5
figure 5

Period and mode shapes of DBI masonry buildings

Full size image
$$T = 0.04 {h}^{3/4}$$
(5)

Here, h is the total height of the building in meters, and period T is in seconds. In comparison with the fixed-base building, Table 2 indicates that the period of the STRP base-isolated building is substantially higher than that of the fixed-base building. The first and second modes of the building are translational modes. The third mode is rotation, whereas the fourth one is the combined mode. In base-isolated buildings, the first mode is the largest model in which the STRP isolator is subjected to large displacement.

Table 2 Period of fixed base, DBI and SBI buildings
Full size table

FE modelling of U-STRP isolators

The total mass of the masonry building is 3011 kN. The mass of the isolator is negligible and disregarded. The size of the isolator is determined based on the pressure on the isolator and the aspect ratio of the isolator. Though AASHTO-LRFD recommends a minimum aspect ratio of 3.0 for bonded isolators, Zisan and Igarashi (2021) recommend an aspect ratio of 3.0 for U-STRP. A minimum aspect ratio is assumed to be 2.50 to keep the pressure on the isolator greater than 2.0 MPa. Figure 6 illustrates the hysteresis loops of U-STRP of size 180 mm × 180 mm × 72 mm under 1.5 MPa and 2.20 MPa compressions, respectively. The horizontal axis represents the elastomer strain (u/tr), in which u is the displacement of the isolator and tr is the total rubber thickness. It shows that the U-STRP isolator experiences no sliding until 200% shear displacement. Since there is no rigid foundation in masonry buildings, large strip-shaped and uniformly distributed square-shaped U-STRP isolators are employed. The strip-shaped isolation is termed as SBI, whereas the uniformly distributed square-shaped isolator is named as DBI system. Figure 7 demonstrates the layout of isolators in the SBI and DBI systems, including the size of isolators. In the SBI layout, corner isolators are 450 × 180 mm, isolators at the periphery and middle position are 675 × 180 mm, and two inside isolators are 900 × 180 mm. In the DBI layout, 36 isolators equally spaced in both X and Y directions have a dimension of 180 × 180 mm. The minimum aspect ratio of these isolators is 2.50, and the height is 72 mm. The average pressure on isolators in SBI and DBI layouts is 2.18 MPa and 2.02 MPa, respectively. These pressures are substantially below 5.0 MPa, acceptable for 200% shear displacement without exceeding tension 2G ~ 3G. Here, G is the shear modulus of tire material. The shear modulus of U-STRP isolators subjected to 3.3–10 MPa compression and 100% shear strain is 0.48–0.75 MPa (Mishra et al., 2014), and the effective damping ratio is 12–16%. The shear modulus at 100% shear and 3.00 MPa working stress is 0.60 MPa. The total area of SBI and DBI isolators is 1377 × 103 mm2 and 1361 × 103 mm2, respectively. Utilizing a 72 mm thick isolator, the stiffness and period of the masonry building with SBI and DBI systems are about 13.6 kN/mm and 0.95 s, respectively. The lateral stiffness of 180 × 180 × 72 mm size isolator calculated from Fig. 6b is 646, 544, 422, 360, and 600 N/mm at 25%, 50%, 100%, 150%, and 200%, respectively. The stiffness and period of the SBI and DBI systems calculated from the minimum stiffness of 180 × 180 × 72 mm isolator are 15.12 kN/mm and 0.90 s. This period is substantially longer than the period of a fixed base masonry building and close to the finite element analysis results.

Fig. 6
figure 6

Force–displacement relationship of 180 × 180 × 72 mm U-STRP isolator

Full size image
Fig. 7
figure 7

Different layout of U-STRP isolator for masonry building (mm)

Full size image

Contact modeling

The base-isolated building is rested on a rigid plane called the bottom plane. It can move horizontally in the direction of applied displacement or be fixed in position in a static condition. This rigid plane does not undergo any deformation nor carry any stress and is modeled as a load control element. The control node is defined at the center of the plane, subjected to ground acceleration in dynamic analysis or fixed boundary conditions in static analysis. Touch contact is assumed among isolators, grade beams, and the rigid plane. The friction coefficient, contact algorithm, and contact procedure are the same as described in Zisan and Igarashi (2021). The node-to-segment contact algorithm is utilized in the computation of contact force. The so-called mixed-method based on Herrmann’s formulation is used for the large-strain analysis of incompressible rubber. The geometric nonlinearities are included using the updated Lagrangian formulation. The single-step Houbolt operator, a default implicit method used by Marc-Mentat, is used for integration. The Single-Step Houbolt operator γ1 = 1.5 and γ = − 0.5 is assumed. The Newton–Raphson iteration with a relative force and displacement tolerance equal to 0.1 N and 0.1 mm, respectively, is used.

FE model verification

The STRP-4/1 isolator, which has four layers of STRP and a length-to-width ratio of 1.0, has been used in the experiment and FE model verification. The details of the STRP-4/1 isolator and experiment conditions can be found in Mishra (2012). Table 3 shows the comparison of stiffness values of the STRP-4/1 isolator obtained from the past loading test, FE analysis, and an analytical solution (Zisan and Igarashi, ). The vertical stiffness, KV, and the compression modulus Ec are found from the pressure solution approach (Tsai & Kelly, 2002) as follows:

Table 3 Horizontal and vertical stiffness: experimental, FE analysis, and analytical solution
Full size table
$${\text{K}}_{\text{v}}= \text{ } \frac{{\text{E}}_{\text{c}}\text{ A}}{{\text{t}}_{\text{r}}}$$
(6)
$${\text{E}}_{{\text{c}}} = {\text{ }}\frac{{12{\text{G}}_{{\text{e}}} {\text{S}}^{2} }}{{{\text{(aa)}}^{2} }}\left( {1 - \frac{{{\text{tan}}h(\alpha a)}}{{\alpha a}}} \right){\text{and}}\alpha {\text{ = }}\sqrt {\frac{{12{\text{G}}_{{\text{e}}} {\text{(1 - }}\nu _{{\text{f}}}^{2} {\text{)}}}}{{{\text{E}}_{{\text{f}}} {\text{t}}_{{\text{f}}} {\text{t}}_{{{\text{ri}}}} }},}$$
(7)

where α denotes the degree of reinforcement flexibility. Ef and tf are the elastic modulus and equivalent thickness of steel cord, respectively, and tri is the thickness of the elastomer layer. Figure 8a demonstrates that the lateral force–displacement relationship derived from the FE analysis and that from the experiment are in good agreement. Table 3 shows that the stiffness at various displacement levels obtained by FE analysis has good agreement with the experimental and analytical results. Figure 8b displays the vertical load–displacement relationship in which the vertical stiffness is defined by the slope of the cyclic loading component. The vertical stiffness of the STRP-4/1 isolator matched well with the analytical solution but was significantly overestimated as compared to the test findings. The vertical stiffness in real specimens is reduced since the FE model overlooked the sliding and twisting effects of steel cables (Costello, 1997). The vertical stiffness of the STRP specimen is also affected by the curvature caused by the roundness of tires. The vertical stiffness of the STRP specimen can be improved by flattening it by surface treatment and pre-compression during the bonding process. Table 4 demonstrates that the FE analysis-derived effective damping of STRP-4/1 is 16.9% greater than the experimental measurement.

Fig. 8
figure 8

Force–displacement relationship of STRP-4/1 a lateral load, b vertical load

Full size image
Table 4 Effective damping (β %): experiment and FE analysis
Full size table

Pushover analysis

The effectiveness of U-STRP base isolation is assessed using the damage state of the masonry building. The damage states for fixed-base buildings and those for U-STRP base-isolated buildings are obtained from pushover analysis. For this purpose, the pushover curve of the analyzed buildings is obtained by plotting base shear with respect to the roof displacement. The lateral displacement of the building base is assumed in the global X, Y, and 45°-direction. The stiffness of the floors is substantially higher than that of masonry units, so the lateral displacement of the floor nodes is assumed to be the same. The first natural vibration mode is considered in pushover analysis since it provides maximum displacement in the base-isolation in a low-rise base-isolated structure. The inter-story drift obtained from the pushover analysis defines the damage state of the masonry building. Table 5 shows the threshold value of inter-story drift and corresponding damage states introduced by Calvi (1999) and used in this study.

Table 5 Threshold value for damage state and damage criteria in masonry building (Calvi, 1999)
Full size table

According to Table 5, the states of damage in a masonry building can be classified into no damage, slight damage, moderate damage, extensive damage, and collapse damage. Figure 9 shows the performance level on the pushover curve at different damage levels and inter-story drifts as described in Table 5. Masonry structures do not display a clear elastic limit since the cracking load in masonry buildings is significantly smaller than that of a reinforced concrete structure. Besides, masonry structures follow progressive collapse. Based on these criteria, Calvi (1999) merged the first two damage states into a single one called “no damage” and “slight damage.”

Fig. 9
figure 9

Performance level on the pushover curve

Full size image

The damage states of U-STRP base-isolated buildings are determined using the same criteria used for fixed-base buildings, as described in Table 5 and Fig. 8. The same methodology for fixed-base building is used by Thuyet et al. (2018). Thuyet et al. (2018) considered a damaged state, DS5, for damage or instability of the isolator, which is defined by the isolator displacement of 170% shear, the instant of commencement of hardening in an unbonded isolator. A U-STRP isolator exhibits progressive roll-over deformation and no instability until 250% shear strain. However, tension within a U-STRP isolator exceeds 2G ~ 3G for displacement exceeding 200% shear, which could be defined as the damage state of the isolator, DS5. The tension failure of the isolator is ignored in the current study.

Figures 10 and 11 show the contour for the principal value of cracking strain at different damage levels of non-isolated and base-isolated masonry buildings. These damage states are calculated based on the drift ratio within the masonry unit only. It does not consider the strain within the isolator. That is why the contours between fixed-base and base-isolated structures are comparable. The strains that fall outside of the range [0.003 − 0.003] indicate collapse cracking. It shows that the strain is diagonally propagated both in non-isolated and base-isolated buildings. At moderate damage state DS2, strain in the fixed-base building and that of the base-isolated building is within the range [0.003 − 0.003], as shown in Figs. 10a, d, g, and 11a, d, g. In the extensive and collapsed damage states, cracks in the masonry walls that are parallel to the loading directions are significant. These cracks propagate in the diagonal direction and originate from the wall openings. At the collapsed level, horizontal cracks were also generated on the upper floor, especially within the wall perpendicular to the loading direction. At the collapsed state DS4, almost the entire wall of the upper floor experiences a cracking strain larger than 0.003.

Fig. 10
figure 10

Cracking contour in different damage states of masonry buildings for displacement in the strong axis

Full size image
Fig. 11
figure 11

Cracking contour in different damage states of masonry buildings for displacement in the weak axis

Full size image

Figure 12 shows the pushover curve for fixed-base and base-isolated buildings due to the top floor displacement. The building is subjected to displacement at the top floor in the strong axis (X), weak axis (Y), and 45°-direction. Floor displacement is applied until failure occurs either within the masonry components or the isolators. It is observed that failure occurs in the masonry elements only due to strain exceeding the limit value. The moderate, extensive, and collapsed damage states are indicated in the pushover curves by the marker. The damage states and corresponding displacement capacities are estimated based on the inter-story drift criteria, as presented in Fig. 9. The inter-story drift and damage in the upper story are more significant than that in the lower story. Figure 13 shows the displacement capacity at different damage states. The horizontal axis of the plot indicates the displacement component for loading. It was found that the displacement capacity of the U-STRP base-isolated building is significantly higher than that of the non-isolated building. The capacity of non-isolated buildings for different loadings is comparable. The displacement capacity in a U-STRP base-isolated building is substantially higher in the strong axis than that of the weak axis. The DBI building has a higher capacity than that of the SBI building in the strong axis, and the capacities of both buildings are comparable in the weak axis. In non-isolated buildings, moderate, extensive, and collapsed damage occur at the top floor displacement of about 5 mm, 15 mm, and 20.0 mm, respectively. Base isolation significantly delays these damage states with respect to the floor displacement, and the DBI system shows superior performance to the SBI system. At the moderate damage state, the displacement capacity of the base-isolated building is 55–75 mm for X directional loading, whereas 40–49 mm for Y directional and 45° loadings. In an extensive damage state, the displacement capacity in the X direction is 132–148 mm and 78–98 for load in the Y and 45° directions. At the collapse level, the displacement capacity in the strong axis is 215–240 mm, and it is 114–140 in the weak direction and 45° directions.

Fig. 12
figure 12

Pushover curve of base-isolated and non-isolated masonry buildings

Full size image
Fig. 13
figure 13

Displacement capacity of different masonry buildings at different damage states

Full size image

Therefore, a U-STRP base isolation is efficient in mitigating the seismic vulnerability of masonry buildings in terms of damage state delaying. The shear strain in the isolator is denoted γ in the pushover curves, which indicates that the shear strain in the DBI system is higher than that of the SBI. The maximum shear strain in the strong and weaker axes is 215% and 114%, respectively, in the SBI system. The same is 240% and 114% in the DBI building. Due to the 45° orientation of the loading, the maximum shear strain in the X and Y directions is 129% and 91% in the SBI building. In the DBI building, it is 140% and 125%. It is found that the U-STRP base isolated building reaches the damage state DS5 only when the load is acting in the X-direction because of the low aspect ratio of the isolator.

Figure 14 shows the base shear capacity of masonry buildings at different damage states. The base shear capacity in a U-STRP base-isolated building is lower than that of a fixed-base building. Until damage state DS2, the base-shears in DBI and SBI buildings are comparable. Then, at damage states DS3 and DS4, base shear in the SBI building is slightly higher than that of the DBI building.

Fig. 14
figure 14

Base shear in different masonry buildings at different damage states

Full size image

Fragility analysis of masonry buildings

The fragility curve generated from the pushover analysis is a popular method used in the seismic vulnerability assessment of buildings (El-Maissi et al., 2022; Kassem et al., 2019; Nazri et al., 2018). The fragility curve is a cumulative lognormal probability distribution function that graphically represents the probability of reaching or exceeding a given damage state for a given intensity measure (spectral displacement or inter-story drift). At a given spectral displacement Sd, the conditional probability of being in or exceeding a particular damage state, DS, as per HAZUS (FEMA, 2003; Nazri et al., 2018; El-Maissi et al., 2022), is as follows:

$$P[{\text{DS}}|{\text{S}}_{{\text{d}}} ]{\text{ = }}\Phi \left[ {\frac{{\text{1}}}{{\beta _{{{\text{DS}}}} }}{\text{ln}}\left( {\frac{{{\text{S}}_{{\text{d}}} }}{{\mathop {\text{S}}\limits_{{{\text{d,DS}}}}^{{{\text{ - }}}} }}} \right)} \right],$$
(8)

where \({\stackrel{\mathrm{-}}{\text{S}}}_{\text{d,DS}}\) is the median value of the lognormal distribution of the spectral displacement that produces the damage state threshold. The value of \({\stackrel{\mathrm{-}}{\text{S}}}_{\text{d,DS}}\) at different damage states found from the pushover curves are listed in Table 6. The median of spectral displacement  \({\stackrel{\mathrm{-}}{\text{S}}}_{\text{d,DS}}\) for a damaged state is the displacement corresponding to 50% of the type of building that would reach or exceed the damage state. Φ is the standard normal cumulative distribution function. βDS is the standard deviation of the natural logarithm of spectral displacement of the damage state, DS. The dispersion βDS depends on the uncertainties in the seismic demand, such as the derivation of the hazard curve, the variability of the seismic intensity, the uncertain definition of the limit state threshold, and the variability of the capacity of buildings. A value of βDS (ATC-58, 2012; FEMA, 2003; Thuyet et al., 2018) is 0.65 for the low-rise building that is used for the current study. These codes suggest a value of 0.75, 0.70, and 0.65 for buildings designed to old, moderate, and modern codes, respectively. The spectral displacement Sd is the displacement capacity as given in the pushover curve. The capacity spectrum in acceleration-displacement (AD) format is obtained by dividing the shear force in the pushover diagram by the seismic weight.

$${\text{S}}_{\text{a}}\text{ = }\frac{\text{V}}{{\text{W}}},$$
(9)

where Sa is the spectral acceleration, V is the base shear, and W is the seismic weight. The seismic demand of masonry buildings in AD format is derived from the traditional elastic acceleration spectrum, Sae, in which spectral acceleration is expressed in terms of the period using the following equation.

Table 6 Median of spectral displacement \({\stackrel{\mathrm{-}}{\text{S}}}_{\text{d,DS}}\)(mm) at different damage states
Full size table
$${\text{S}}_{{{\text{de}}}} {\text{ }} = \frac{{{\text{T}}^{2} }}{{4\pi ^{2} }}{\text{S}}_{{{\text{ae}}}} .$$
(10)

here Sde is the elastic displacement spectrum corresponding to period T and fixed viscous damping. Figure 15 shows the capacity spectrum and demand spectrum of masonry buildings in AD format. The demand spectrum is the 5% damped elastic response spectrum at the seismic site class of D, such as SS = 0.25 and S1 = 0.10, as specified in ASCE/SEI 7-10. Here, SS and S1 mean mapped spectral response acceleration parameters at short and 1 s periods, respectively. According to Fig. 15, the seismic demand of the masonry building is around 30 mm for the assumed site.

Fig. 15
figure 15

Acceleration–displacement response spectrum (ADRS) of fixed base and base-isolated building

Full size image

Figure 16 shows the comparison of the fragility curves for the fixed-base and base-isolated masonry buildings. The comparison shows that the seismic vulnerability of the U-STRP base-isolated building for a spectral displacement is substantially lower than that of the fixed base building. Again, for a fixed spectral displacement, the seismic vulnerability along the strong axis, as shown in Fig. 16a, is considerably lower than that of the weak axis, as shown in Fig. 16b. The lower stiffness of masonry buildings yields high lateral drift, and masonry buildings fail due to exceedance of the strain limit. Besides, isolators in SBI systems are aligned parallel to the weak axis of the building to increase the rigidity. Therefore, the lower aspect ratio of the strip-shaped isolator and the higher stiffness of the building in the strong axis reduce seismic vulnerability. The fragility curves also show that the 45° aligned displacement is more vulnerable than that of the displacement in the principal axis.

Fig. 16
figure 16

Comparison of fragility curves of fixed base and base-isolated building

Full size image

The seismic vulnerability of masonry buildings is figured out in site class D such that SS = 0.25 and S1 = 0.10 to understand the efficiency of U-STRP isolation. The damage probability of different building systems at 30 mm spectral displacement is marked in the figures. The probability of exceedance for collapse damage, DS4 at 30 mm, is about 100% in the fixed base building, which indicates complete collapse in site class D such that SS = 0.25 and S1 = 0.10. In the same seismicity, the probability of exceedance for moderate damage DS2 is 48% and 73% in the strong and weak axes of the SBI building, respectively. The probability of exceedance at the same damage state is 65% and 77% in the DBI building. For 45° aligned loading, the seismic vulnerability in SBI and DBI systems at moderate damage DS2 is comparable and falls in the range of 70–75% and 66–77%, respectively. The probability of exceedance for extensive and collapsed damage states is about zero in the base-isolated buildings. Therefore, U-STRP base-isolation is efficient in reducing the seismic vulnerability of masonry buildings.

Conclusion

This paper focused on the performance and seismic vulnerability of masonry buildings supported by strip-shaped and uniformly distributed square-shaped U-STRP isolators. A two-story masonry building conventionally designed with material properties is modeled and analyzed. The structure is analyzed for loads acting along the principal axes and in the inclined direction (at an angle of 45° with the principal axis). The damage states of masonry buildings and their seismic vulnerability are determined based on inter-story drift and fragility analysis. From the analysis results, the following conclusions can be drawn:

  • The U-STRP base-isolation mitigates the seismic vulnerability by delaying the damage occurrence. The displacement capacity of the uniformly distributed square-shaped U-STRP base-isolated building is higher than that of the strip-shaped base-isolated building. The displacement capacity of the masonry building decreases when the load is applied at 45°.

  • Lateral strain in the uniformly distributed square-shaped U-STRP isolator system is higher than that of the strip-shaped isolator system. The maximum strain exceeds 200% but remains below 250% even after the collapsed damage. The lateral strain for a 45° aligned load is substantially low. In the extensive and collapsed damage, the base shear in the strip-shaped isolator system is slightly higher than that of the uniformly distributed square-shaped U-STRP isolator system.

  • Fragility curves of masonry buildings corresponding to different damage states indicate a significant reduction in the seismic vulnerability of the base isolated building as compared to that of the fixed base building.