Introduction

In many nations, earthquakes have considerable effects on human life and assets. The world economy suffers due to the frequent occurrence of these events, forcing structural engineers and researchers to establish numerous methodologies to sustain the economy and human lives. Priestley & Kowalsky (2000) developed the DDBD method to design structures according to their requirements during seismic circumstances [1]. Displacements are the structure’s fundamental response to understanding structural behavior rather than forces. However, displacements have been tested for serviceability limitations after the force-based design procedure. By establishing a maximum displacement limit at the outset and checking for strength at the end, structural engineers may easily predict design needs for significant earthquakes.

Priestley & Kowalsky (2000) proposed the DDBD methodology of performance-based design for buildings to fix the deficiencies and limitations of the conventional method [1]. This technique allows designing a structure to accomplish a specified displacement in the case of a design-level earthquake. The DDBD technique involves multiple phases starting with predicting the seismic deformation of a single degree of freedom system (SDOF) estimated using reports that follow the performance-based design [2]. SDOF system represents the first mode of vibration of the multi-degree of freedom (MDOF) system [3]. Finally, base shear is determined using secant stiffness, rather than assuming initial stiffness, to get more accurate results [4].

In earlier studies, building performance is precisely evaluated by examining the dynamic behavior of different moment-resisting RC frames designed using DDBD [5]. The incremental response spectrum analysis, multi-mode pushover analysis, and modal pushover analysis variant are well-developed methods to understand the nonlinear seismic behavior of structure [6, 7]. Before a decade, the building equipped with various foundation isolation systems acquired some interest in DDBD approach [8]. The accumulated dissipative effect of structural damping may become sufficient to affect the dynamic behaviors of building structures manifestly [9]. Various techniques are invented to preserve structures with optimal response parameters [10, 11]. Changing the influence parameters in this manner increases the structure’s long-term stability, resulting in outstanding numerical behavior. [12, 13].

A comparison of DDBD with force-based design (FBD) has been conducted by considering base shear, maximum displacement, storey drift evaluation, structural failure mechanism, and damage indices calculations to determine the efficiency [14, 15]. In recent years, the DDBD method has shown advancement in irregular buildings, and guidelines have been generated for setback irregularities [16]. The Indian spectra used to know the seismic performance of various RC frames are designed considering LS performance level [17]. The displacement profiles and equivalent viscous damping are efficient parameters that may be directly incorporated during the design of a structure to achieve realistic simulation are investigated [18]. The most relevant parameters from all are chosen for the current study.

The future probability of a structure’s collapse is now the primary concern, which may be determined by generating fragility curves. For a particular seismic event, the representation of the probability of equaling or exceeding a specific damage state of a structure is known as the fragility curve. Kennedy et al. [19] originated fragility curves by considering conditional failure frequency against PGA for a nuclear power plant to evaluate the damage. Recently fragility curves have been widely used for assessing damage states in structures like RC frames [20, 21], elevated water tanks [22], and bridges [23]. Several approaches summarized for fragility analysis are discussed in detail in seismic risk evaluation using various analysis methods [24]. Fragility curves are represented through a standard lognormal distribution function that comprises median and standard deviation [25].

Considering numerous aspects, the functionality of structures under seismic excitation can be predicted after a step-by-step investigation from start to end. In the present study, 4, 8, and 15-storey RC frame structure designs are carried out by applying DDBD, and FBD approaches considering various soil conditions and a maximum considered earthquakes. The displacements in the DDBD approach performed for structures are incorporated for life safety (LS) and collapse prevention (CP) performance limits. The nonlinear static multi-mode pushover study is conducted to account for higher mode effects and calculate the structure’s yield and final displacements. The pushover analysis results determine the response reduction factor (R factor) for these structures, which reveals the structure’s energy dissipation capability under various conditions. The seismic performance of the studied RC frame is estimated by generating fragility curves and determining the probability of exceedance of a particular damage state at the performance point. The seismic damage index provides a quantitative assessment of structural damage, followed by the structural damage state in the future. Finally, the outcomes of this study contribute to implementing the DDBD approach in Indian building regulations, together with general structural design expertise.

Mathematical Modelling

Materials and Geometry Description

4, 8, and 15-storey RC plan symmetric frame structure situated on soft, medium, and hard soil sites is considered to evaluate RC buildings’ performance on various ground motions, as shown in Fig. 1. Each model is categorized as indicated in Table 1 to facilitate identification. The RC frames are designed as special moment resisting fame with code prescribed R factor of 5, fixed-based and ductile detailing is based on IS 13920 (2016) [26] provisions. The building is situated in the most critical zone V (PGA = 0.36 g). Each floor height is 3.2 m, and the bay width is 5.8 m. The structure is subjected to a live load of 3 kN/m2 on ordinary levels and 1.5 kN/m2 on the roof floor. The infill wall’s dead load is transmitted to a beam with a wall thickness of 230 mm, and the infill masonry wall’s density is assumed to be 20 kN/m3. The slab load is calculated at a thickness of 150 mm and a density of 25 kN/m3. M30 concrete grade and Fe500 steel grade are used for steel bars. The building models are designed using two distinct approaches: FBD and DDBD, following IS 456 (2000) [27]. As indicated, the drift constraints selected to implement the DDBD approach are 2% and 4% for LS and CP, respectively [2].

Fig. 1
figure 1

a Plan and elevation of b 4-storey c 8-storey d 15-storey RC frames and section details (in mm) of DMLS models

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Table 1 Model IDs of RC frames at various performance levels and soil types
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FE Modeling

The 3D modeling of RC frame buildings is carried out using the building information modeling solution of MIDAS Gen2021, version 3.1 [29]. The frame elements and their attributes are modeled and assigned by providing material properties and sectional properties to the general beam/tapered beam element type. The vertical loads are converted into masses and are performed as lumped masses, excluding the self-weight of the building. The material’s nonlinear behavior is simulated using the MIDAS GSD tool. The concrete nonlinearity is modeled using the Mander model’s constitutive stress–strain curve [30], and the nonlinear behavior of the steel is determined using the Park strain hardening model [31], as shown in Fig. 2.

Fig. 2
figure 2

Nonlinear material models of a concrete and b steel

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DDBD and FBD Approach

Shabita & Sozen [3] proposed a linear model for evaluating nonlinear response ranges, which serves as the foundation for the DDBD approach. The DDBD approach describes the procedures involved in constructing a structure to accomplish rather than be constrained by a predefined displacement when subjected to a design-level earthquake. Priestley et al. [32] describe the methodology of the DDBD approach adopted in this research as an overall framework that may be used for any structure. The DDBD approach begins with transforming an MDOF system into an SDOF system, which gives a similar response shown in Fig. 3a. In the second stage, the maximum reaction of an SDOF system is represented by a secant stiffness (Ke) at the predetermined design displacement (Δd) (Fig. 3b). The design displacement is established at the commencement of the process, considering performance limits, ductility demands, or material strains. In the third stage, equivalent viscous damping (ξeq) is obtained by calculating the displacement ductility demand at the design displacement (Fig. 3c). Furthermore, the final step determines the effective time period at design displacement using displacement spectra following Indian codal requirements (Fig. 3d).

Fig. 3
figure 3

Phases of the DDBD approach [1]

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Steps involved in the DDBD approach computing lateral load distribution for RC frame building [32]:

  1. 1.

    Design displacement evaluation (∆d)

At the start of the procedure, the design displacement of the structure is estimated by specifying a specific performance limit, as described previously. The displacement of each storey (∆i) is determined considering the mode shapes (δi) given in Eqs. (2–3), which is multiplied by the critical displacement (\(\Delta\) c) at the first mode shape obtained by the design drift limits as specified in FEMA 356 (2000) [2].

$${\Delta }_{\mathrm{i}}={\updelta }_{\mathrm{i}}\left(\frac{{\Delta }_{\mathrm{c}}}{{\updelta }_{\mathrm{c}}}\right)$$
(1)
$${\updelta }_{\mathrm{i}}=\frac{{\mathrm{h}}_{\mathrm{i}}}{{\mathrm{h}}_{\mathrm{n}}} \quad \mathrm{for\, n }\le 4$$
(2)
$${\updelta }_{\mathrm{i}}=\frac{4}{3}\left(\frac{{\mathrm{h}}_{\mathrm{i}}}{{\mathrm{h}}_{\mathrm{n}}}\right)\left(1-\frac{{\mathrm{h}}_{\mathrm{i}}}{{\mathrm{h}}_{\mathrm{n}}}\right)\quad \mathrm{for\, n }> 4$$
(3)

where n denotes the number of stories, hi denotes the height of the ith storey, hn indicates the total height of a building. For buildings having several stories, more than ten, to consider the higher mode effect, Eq. (1) is multiplied by the drift reduction factor ωθ (Eq. (4)).

$${\upomega }_{\uptheta }=1.15-0.0034{\mathrm{H}}_{\mathrm{n}} \le 1.0$$
(4)

Now, the design displacement (Δd) of the substitute structure is calculated from Eq. (5)

$${\Delta }_{\mathrm{d}}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{m}}_{\mathrm{i}}{{\Delta }_{\mathrm{i}}}^{2})}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{m}}_{\mathrm{i}}{\Delta }_{\mathrm{i}})}$$
(5)
  1. 2.

    Determination of the effective mass (meff) and the effective height (heff)

The effective mass (meff) and the effective height (heff) of the RC frame are calculated using equations (6, 7).

$${\mathrm{m}}_{\mathrm{eff}}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{m}}_{\mathrm{i}}{\Delta }_{\mathrm{i}})}{{\Delta }_{\mathrm{d}}}$$
(6)
$${\mathrm{h}}_{\mathrm{eff}}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{m}}_{\mathrm{i}}{\Delta }_{\mathrm{i}}{\mathrm{H}}_{\mathrm{i}})}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{m}}_{\mathrm{i}}{\Delta }_{\mathrm{i}})}$$
(7)

where mi denotes the ith storey mass.

  1. 3.

    Determination of ductility demand (μ)

Ductility demand is evaluated by calculating yield rotation \({(\uptheta }_{\mathrm{y}}\)) and yield displacement (\({\Delta }_{\mathrm{y}}\)) using Eqs. (8, 9)

$${\Delta }_{\mathrm{y}}= {\uptheta }_{\mathrm{y}}*{\mathrm{H}}_{\mathrm{e}}$$
(8)
$${\uptheta }_{\mathrm{y}}= 0.5*{\upvarepsilon }_{\mathrm{y}}*\left(\frac{{\mathrm{L}}_{\mathrm{b}} }{ {\mathrm{H}}_{\mathrm{b}}}\right)$$
(9)

where εy denotes the yield strain of steel, Lb denotes the beam length, and Hb represents the beam depth. Then design ductility demand is found by the ratio between the design and yield displacement (Eq. (10)).

$$\upmu =\frac{{\Delta }_{\mathrm{d}}}{{\Delta }_{\mathrm{y}}}$$
(10)
  1. 4.

    Determination of equivalent viscous damping (ξeq)

Equation (11) gives the equivalent viscous damping \({(\upxi }_{\mathrm{eq}})\) from the ductility demand (μ). The equivalent viscous damping is multiplied with the damping correction factor as per 1893 Part-1 (2002) [33]. The interpolation method is used to find the exact value of the damping modification factor for the specific damping.

$${\xi }_{eq}=5+120\left(\frac{1-{\upmu }^{-0.5}}{\pi }\right) \%$$
(11)
  1. 5.

    Determination of the effective time period (Teff) of the SDOF structure

Figure 4(a) displayed the displacement spectra generated from acceleration spectra for 5% damping. The effective period of the SDOF structure at peak displacement response is determined by entering the design displacement of the substitute SDOF structure Δd and obtaining the effective period Teff from damped response spectra of different models for a critical zone (Fig. 4(b)).

  1. 6.

    Determination of the effective stiffness (Keff) and the design base shear (Vb) of the SDOF structure

Fig. 4
figure 4

a Displacement spectra of various soil types and b Damped spectra of medium soil for zone V

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From the effective time period and the effective mass, the effective stiffness is obtained by Eq. (12), multiplying the effective stiffness (Keff) with design displacement (Δd), and the design base shear (Vb) is evaluated using Eq. (13).

$${\mathrm{K}}_{\mathrm{eff}}=\frac{4{\uppi }^{2}{\mathrm{m}}_{\mathrm{eff}}}{{{\mathrm{T}}_{\mathrm{eff}}}^{2}}$$
(12)
$${\mathrm{V}}_{\mathrm{b}}={\mathrm{K}}_{\mathrm{eff}}*{\Delta }_{\mathrm{ d}}$$
(13)
  1. 7.

    Determination of lateral load distribution (Qi) at each storey

The lateral loads have been distributed at a different storey for RC frames using design base shear given by Eqs. (14, 15),

$${\mathrm{Q}}_{\mathrm{i}}={\mathrm{V}}_{\mathrm{b}}\frac{{\mathrm{m}}_{\mathrm{i}}{\Delta }_{\mathrm{i}}}{\sum_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{m}}_{\mathrm{i}}{\Delta }_{\mathrm{i}}\right)}\quad \mathrm{ for\, n }\le 10$$
(14)
$${\mathrm{Q}}_{\mathrm{i}}={\mathrm{Q}}_{\mathrm{t}}+0.9{\mathrm{V}}_{\mathrm{b}}\frac{{\mathrm{m}}_{\mathrm{i}}{\Delta }_{\mathrm{i}}}{\sum_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{m}}_{\mathrm{i}}{\Delta }_{\mathrm{i}}\right)}\quad \mathrm{for\, n }> 10$$
(15)

where \({\mathrm{Q}}_{\mathrm{t}}=0.1{\mathrm{V}}_{\mathrm{b}}\), Qt is a lateral load at the roof level. Table 2 summarizes an example computation of the above-discussed process using DMLS15 as an illustration model. Table 3 shows the parametric detail evaluated from the DDBD approach of 15-storey models.where, hn represents the height of the roof from the base, W represents the total seismic weight of the RC frame building, δc represents critical normalised mode shape, ωθ represents drift reduction factor, Δd represents design displacement, Δy represents yield displacement, μ represents displacement ductility, meff represents effective mass, Keff represents effective stiffness, Teff represents effective time-period, Vb represents base shear, and Vb/W represents Percentage of lateral force.

Table 2 The sample calculation of lateral load distribution using DDBD for the building Model DMLS15
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Table 3 The parametric details of 15-storey building Models obtained considering the DDBD approach
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The force-based approach is widely applied during structure design, and Indian practice norms are entirely based on this approach. The working stress, the ultimate stress method, and the limit state method come under the category of FBD. The most recent procedure is the limit state method, which encompasses both the limit state of serviceability and strength. When studying any structure utilizing the limit state method, the designer evaluates the linear relationship between stress and strain. The analysis and design are limited to the elastic range of the material property. Specific measures are included in the code of practice to ensure that structures behave in the plastic content of material properties in the event of lateral loads induced by earthquakes. However, this method cannot predict damage to structural components and actual performance in the plastic range. Additionally, regardless of soil conditions, the R factor remains the same for all special moment resisting frames.

In code-based FBD, the design base shear (Vb), as calculated as per IS 1893 Part-1 (2016) [28], shall be distributed along the height of the building using Eq. (16).

$${\mathrm{Q}}_{\mathrm{i}}= {\mathrm{V}}_{\mathrm{b}}* \frac{{\mathrm{W}}_{\mathrm{i}}{{\mathrm{h}}_{\mathrm{i}}}^{2}}{\sum_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{W}}_{\mathrm{i}}{{\mathrm{h}}_{\mathrm{i}}}^{2}}$$
(16)

where Qi = Design lateral force at floor i, Wi = Seismic weight of floor i, hi = height of floor i measured from the base, n = number of stories. The rebar detailing the studied study RC frame building models is shown in Table 4 and Table 5. Only for 15-storey buildings, variation in column dimensions with height is considered for optimal structural design. The time period considered for the RC frame is calculated as per Eq. (17), Where h = height of RC frame building.

Table 4 Reinforcement details of 4-storey and 8-storey RC frame models
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Table 5 Reinforcement details of 15-storey RC frame models
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$$\mathrm{T}= 0.075{\mathrm{h}}^{0.75}$$
(17)

Plastic Hinge Formation

Moment-rotation (M-θ) type plastic hinge model is applied to incorporate nonlinear behavior using the MIDAS Gen2021, v3.1 [29] software as per ASCE 41 (2017) [34]. The skeleton model used to perform the nonlinear analysis is an empirical hysteresis model, characterizing inelastic hysteresis behaviors. The axial component is represented by a central spring, while the two translational components are represented by two springs at either end, with force–displacement relationships. The two flexural components, My and Mz, are represented by springs defined by the connection between the moment and rotation angle. For beam elements, coupled axial force–uniaxial moment behavior is observed by computing the flexural yield strength of a hinge while considering the influence of axial force (P-M). For column elements, coupled axial force-biaxial moment behavior is seen by computing the flexural yield strength of a hinge while taking axial force into account (P-M-M). While locating lumped inelastic hinges, the axial component of a member is affixed to its center. However, both ends are selected for the bending moment components. The modeling criteria and acceptance parameters for beams and columns hinges of RC frame structures are taken as per ASCE 41 (2017) [34].

Multi-mode Pushover Analysis

Earthquake loads are stochastic in nature [35]. This study addresses stochastic loads as lateral design loads, which are incrementally applied until the individual components of a structure yield or buckle. Nonlinear static analysis is a static nonlinear procedure to quantify the nonlinear performance of designed building structures under lateral loading. The influence of higher-mode effects on the structural and nonstructural seismic performance of RC frame structures having a time period greater than one is incorporated by applying multi-mode pushover analysis. A significant number of modes must be considered to capture at least 90% of a structure’s mass involvement to comprehend nonlinear behavior (Table 6). The following are the steps involved in multi-mode pushover analysis [6, 36] from the expansion of the modal pushover analysis [37]:

  • The structure is first analyzed using static pushover analysis in which monolithically increasing lateral loads are applied until the desired displacement is achieved under constant gravity loads. The lateral load pattern on each node is given by the following Eq. (18).

    $$F_{n} \, = \,m\Phi_{n}$$
    (18)
Table 6 Modal analysis results of 15-storey RC frame models
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The peak value of the roof’s movement due to nth mode (urn) when the high-rise building is pushed is taken from the equation (19),

$$u_{rn} \, = \,\Gamma_{n} \Phi_{rn} D_{n}$$
(19)

  • The pushover curve in each mode is then transformed into the capacity curve of the corresponding SDOF system using the modal conversion parameters derived from the same linear (starting) mode shapes [2].

  • Peak inelastic response values of interest, including inelastic displacement, the storey drifts, and plastic hinge rotations, are calculated separately for each mode. The square root of the sum of squares (SSRS) formula is used to estimate the combined peak response values. Figure. 5 represents the drift profiles produced from multi-mode pushover analysis of DSLA15, DMLS15, and DHLS15 and drifts generated after SRSS for all building models. RC frames designed using the DDBD approach are well observed within limits for all assessed performance limit states.

Fig. 5
figure 5

Drift profiles of all participating modes, along with mean drift comparison of various models

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The value of the design base shear and drifts at performance points calculated using the DDBD follow the same pattern (Table 3 & Fig. 5). The base shear values obtained by the FBD approach range between the base shear values generated by the LS and CP performance limitations in the DDBD approach, but the values are slightly nearer to the LS performance criteria revealing that Indian codes design structures are nearer to the 2% drift limit criteria specified in the DDBD approach.

The nonlinear behavior of RC frame buildings is evaluated by generating pushover curves implementing the multi-mode pushover analysis. The pushover curves represent the base shear v/s top displacement graph covering the entire range of structural behavior from linear to yielding, nonlinear, and up to collapse of structures. The inter-storey drift from multi-mode pushover analysis at performance points evaluated for LS models is observed to be less than 2%, i.e., 0.02 × 3.2 = 0.064 m, except DSLS4 and the values for CP models are observed to be less than 4%, i.e., 0.04 × 3.2 = 0.128 m, indicating that the DDBD method is effectively implemented. The RC frame buildings constructed on soft soil observed the maximum inter-storey drift, whereas RC frame buildings constructed on hard soil observed the minimum inter-storey drift.

The plastic hinge development in both design approaches due to seismic loading shows ductile behavior in the nonlinear zone. After the first yielding, plastic hinges began to form in the beam elements in both design approaches due to the adopted strong-column weak-beam philosophy. As yielding continues, plastic hinges in beams increase and column hinges begin forming, indicating a substantial degree of energy dissipation in the elements. A backbone curve arises as the building achieves its load-bearing limit and begins to form collapse hinges. In all the models, sufficient ductility is observed, which will help in preventing column failure caused by flexure and shear.

Figure 6 demonstrates the capacity curve of 15-storey RC frames generated after performing nonlinear static analysis. Compared to FBD, the elastic stiffness (load deflection ratio in an elastic zone) of structures developed using DDBD at the LS performance level is considerably greater. However, models designed using CP performance exhibit less elastic stiffness than FBD, with soft soil having the most significant base shear values and hard soil the lowest. If we consider the medium soil condition, the nonlinear behavior observed for FM15 and DMCP15 reveals a significant difference in ultimate displacement, but the difference between DMLS15 and FM15 is negligible. Based on the abovementioned criteria, models designed using the FBD approach are closer to those designed considering LS performance criteria but have a higher capacity than those designed with CP performance criteria. 4-storey and 8-storey RC frame structures behave identically under lateral loads.

Fig. 6
figure 6

The capacity curves/pushover curves of 15-storey building models

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Seismic Assessment Parameters

Response Reduction Factor (R Factor)

The R factor represents the amount of energy dissipated during an earthquake after yielding. The R factor for a special moment resisting frame is 5, according to the Indian code [28]. In this study, a specific model’s R factor values are derived from a structure’s performance and compared with code-based R factor values. The procedure to determine the R factor based on a building’s performance is described in ATC 19 [38]. The technique is applied to numerous constructions to evaluate the inelastic nature all around the globe [39,40,41]. The R factor evaluation from various factors is presented by Eq. (20).

$${\text{R }} = {\text{ R}}_{{\text{s}}} \times {\text{ R}}_{\mu } \times {\text{ R}}_{\zeta } \times {\text{ R}}_{{\text{R}}}$$
(20)

where Rs is the strength factor calculated from the ultimate shear and the design base shear ratio. Rμ is the ductility factor derived from the performance characteristics of yield displacement and ultimate displacement of the structure,

Rζ and RR are the ductility and redundancy factors, respectively, and both have a value of 1. [22, 38]

Table 7 gives the R factor values of studied models and shows that the performance limit and soil conditions influence the values of R and its components. The strength component governs the R factor in 4-storey RC frames, whereas the ductility factor governs the R factor in 15-storey RC frames. The R values for various soil properties RC frame analyses developed based on DDBD and FBD for seismic zone-V are observed from 4.98 to 10.86. R factors observed for structures constructed using the FBD approach for 4 and 8 stories are incredibly close to or slightly over the code-specified value of 5, indicating that the Indian code provides a safer design [28]. For 15-storey RC frames, more prominent R factors are recorded. The FS15 model has a 12.57% higher R factor value than the FH15 model.

Table 7 R factor values derived considering various factors
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For the design of structures utilizing the DDBD approach, the R factor values increase as the soil changes from soft to hard for each limit state. The values of the R factor for the CP performance state are the greatest compared to other performance criteria, which is not unreasonable given that structures constructed for CP performance are intended to withstand greater displacements. There are no provisions for structural design based on performance limit criteria in the Indian code. Thus, the study recommended that the code-specified unique R not be used for all RC frame structures, regardless of performance restrictions.

Fragility Curve Fitting

The probability of exceeding a particular damage state under specific ground motion is derived by generating fragility curves from pushover curves using Eq. (21) [42]. Building performance should be known at the initial stage in plotting the fragility curve. The capacity spectrum method (CSM) is used to evaluate building performance. The performance point of the RC frame is found for each mode by intersecting the capacity curve with the demand curve as per ATC 40 (1996) and FEMA P58-1 (2018) [43, 44].

$${\text{P}}_{{{\text{k }}\hbox{-}}} \left( {{\text{S}}_{{{\text{di}}}} } \right) \, = {\text{ P }}\left[ {{\text{DS }} \ge {\text{ DS}}_{{\text{k}}} |{\text{ S}}_{{\text{d}}} } \right] \, = \, \Phi \;\left[ {\frac{1}{{{\upbeta }_{{\text{k}}} }}{ }\ln \left( {\frac{{{\text{S}}_{{\text{d}}} }}{{{\text{S}}_{{{\text{d}},{\text{ds}}}} }}} \right)} \right]$$
(21)

where Sd represents the spectral displacement of the capacity curve developed in order to comprehend nonlinear structural behavior, Sd,ds represents the median value of Sd, and Sk represents the threshold limit which is specified in Table 8 [25], βk represents the standard deviation for a given damage state, Φ represents the standard normal cumulative distribution function.

Table 8 More likely damage states, damage state threshold and corresponding mean damage index intervals [19]
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The least-square approach is then utilized to fit the fragility curves [45,46,47]. Nonetheless, some literature [48] has successfully utilized the HAZUS [49] technique to fit buildings’ fragility curves. Table 9 illustrates the estimated uncertainty factor βk, damage state threshold values, and mean spectral displacement at seismic forces equivalent to.

Table 9 Parameter describing the fragility curves and performance point as per FEMA P58-1 (2018) [44]
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MCE for seismic zone-V. In addition, it is assumed that the expected damage state, as indicated by the inelastic spectral displacement Sd, follows the binomial probability distribution as given in Eq. (22).

$${\mathrm{P}}_{\mathrm{k}}=\mathrm{P}\left(\mathrm{DS}={\mathrm{DS}}_{\mathrm{k}}\right)= {\mathrm{P}}_{\mathrm{k}}\left(\mathrm{N},\mathrm{d}\right)=\sum_{k=0}^{n-1}{\left(\genfrac{}{}{0pt}{}{n-1}{k}\right)\mathrm{d}}^{\mathrm{k}}{(1-\mathrm{d})}^{\mathrm{n}-1-\mathrm{k}}; \mathrm{ k }= 0, 1, 2,\dots .\left(\mathrm{N}-1\right)$$
(22)

In this study, the damage grades are designated by N, equal to 5. The value of d indicates the amount of damage, ranging from 0 to 1. d = 0 shows no damage to a building. However, the number d = 1 implies that a frame has sustained total damage. Therefore, Eq. (21) is fitted to the acquired point using Eq. (22) and the least-squares principle. Figure 7 displays fragility curves developed using this approach with a fixed probability of 50%. Using the generated fragility curves, the probability of occurrence (Pk) for each damage state is derived by subtracting the acceptable likelihood of subsequent damage state exceedance according to the following Eq. (23):

Fig. 7
figure 7

Fragility curves fitting for 15-storey models considering damage states

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$${\mathrm{P}}_{\mathrm{k}}={\mathrm{P}}_{\mathrm{k}}\left(\mathrm{k}+1\right)- {\mathrm{P}}_{\mathrm{k}}\left(\mathrm{k}\right)$$
(23)

The fragility evaluation’s reliability depends on the optimal values of the uncertainty factor (β) and the threshold limit values for the various predefined damage states [42]. The uncertainty can be classified as aleatory (random variability) and epistemic (resulting from incomplete knowledge); i.e., manufacturing, construction inaccuracy, and seismic hazard can be reduced using uncertainty factor (β). Equation (24) estimates a fragility function for the optimal uncertainty beta \(\widehat{(\beta })\) by minimizing the sum of squared error (SSE) between normal and binomial distribution.

$$\widehat{\beta }= \genfrac{}{}{0pt}{}{\mathrm{argmin}}{\upbeta }\sum_{\mathrm{j}=1}^{\mathrm{m}}{\left[{\mathrm{P}}_{\mathrm{k}}({\mathrm{S}}_{\mathrm{di}})-\sum_{\mathrm{k}=0}^{\mathrm{n}}{\mathrm{kP}}_{\mathrm{k}}\left(\mathrm{N},\mathrm{d}\right)\right]}^{2}$$
(24)

Sdy represents yield spectral displacement, and Sdu represents an ultimate spectral displacement.

Seismic Vulnerability Calculating Mean Damage Index

The seismic vulnerability is derived in terms of the mean damage index (DSm), which is evaluated from the probability of occurrence of each damage state (Eq. (23)). Using a single metric known as the weighted mean damage index (DSm), the building’s most likely damage condition may be described (Eq. (25)).

$$\mathrm{DSm }=\sum_{\mathrm{i}=1}^{4}\mathrm{k }{\mathrm{p}}_{\mathrm{k}}\left[\mathrm{N},\mathrm{ d}\right]$$
(25)

where k = 0, 1, 2, 3, or 4 depends on the damage state k considered, pk [N, d] = occurrence probabilities of the given damage state. The mean damage index is weighed to examine the structure’s damaged condition. DSm represents a possible damaged state of a building during a given seismic hazard. For instance, DSm = 1.0 indicates the most likely damage state of the facility would be slight.

Table 10 describes the probabilities of surpassing several damage states that can be determined using Fig. 7 represents the probabilities of the predicted damage grade when a chance of 50% is fixed for each damage circumstance. Consider that the performance point of model DSCP15 is 117.26 mm (Table 9) and that the collapse probability of model DSCP15 is calculated to be 7%. (Table 10). The weighted mean damage index (DSm) is then computed as 1.71 by evaluating the expected probability of exceedance for each damage state (Eqs. 23 and 24). However, based on FEMA P695 (2009) [50] guidelines, all RC frames demonstrate adequate seismic performance, with a collapse risk of less than 10%. (Table 10).

Table 10 Probability of exceedance of particular damage state, mean damage index (DSm), and most likely damage state of RC frame models
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The soils and performance constraints imposed by seismic design have a quantifiable impact on the DSm on RC frame-building models (Table 10). Even though their outcomes vary, all models created with varied performance requirements and soil conditions exhibit slight to moderate damage. More surveillance is required for 4-storey RC frames constructed on soft soil sites as complete collapse probabilities are high; hence they are more susceptible to seismic damage. The amount of damage detected for structures built on soft soil is between 1.63 and 2.92, whereas for structures built on hard soil, the range is between 1.04 and 1.71. It is noticed that 8-storey RC frames are relatively safe than 4-storey and 15-storey RC frames.

Compared to DDBD-designed structures, FBD-designed structures are shown to be safe in the event of a total collapse, indicating that the code prioritizes greater safety. In contrast, DDBD-designed structures acquire the entire structural capacity according to design-specified requirements and are also seen to be safe. Hence, DDBD gives a cost-effective design that can be implemented according to the relevance of the construction.

Conclusion

The present research is valuable for determining the anticipated damage to RC frame constructions developed with different parameters. The DDBD-designed models are evaluated according to two performance criteria (i) diverse performance limitations and (ii) distinct soil conditions. The DDBD-designed RC frame models are compared against the FBD technique to determine the efficacy of the DDBD approach. The developed structures are evaluated for nonlinear parameters using multi-mode pushover analysis in MIDAS Gen2021, v3.1 [29]. The seismic assessment of the RC frame models is incorporated by fragility curve generation in a probabilistic manner. The likelihood of exceeding a particular damage state is used to assess the mean damage. Finally, the seismic vulnerability reveals the structure’s future damage state regarding the mean damage index. The following are the key conclusions drawn from the study:

  • The study observed that the R factor values depend on building height, soil conditions, and performance criteria. R factor values cannot be the same for all building categories, and by modifying the R factor value using this study, more cost-effective structures may be constructed. However, due to inaccuracy in rebar placement, RC frame height, poor quality, and poorer building techniques, the actual values of R for real RC frame constructions may be lower than those calculated in this study.

  • While the soil changes from hard to soft, nonlinear analysis reveal that storey drift increases; hence, DDBD has demonstrated effectiveness for various challenges. Moreover, on average, RC frame models on hard soil are slightly damaged, whereas those on medium and soft soil are predicted to experience moderate damage indicating that buildings constructed according to the DDBD method will be economical.

  • DDBD-designed buildings adhere to the structure’s maximum load capacity following design specifications and are rated safe. However, FBD-constructed structures with higher safety standards but the same strength base design are deemed highly safe. All RC frame models produced using DDBD and FBD have a collapse probability of less than 10% when subjected to seismic forces, indicating a lower likelihood of failure [46]. This study showed that structures exhibit remarkable resilience to massive collapse before failure during an earthquake.

The DDBD approach is helpful for structural design as per their importance, as structures can be designed for required performance criteria. The FBD approach also yields good results but has fixed parameters which can affect structure flexibility and result in more expensive. Further, this study can be extended by considering soil-structure interactions on low-rise, mid-rise, and high-rise buildings using the DDBD approach.