Evaluation of Existing Reinforced Concrete Buildings for Seismic Retrofit through External Stiffening: Limit Displacement Method

Abstract

This study introduces an alternative approach to the assessment of the earthquake resistance of existing buildings and the evaluation of the requirements for seismic retrofit through external stiffening. Instead of assessing individual structural components, we evaluate the entire building using a nonlinear static pushover analysis. Earthquake resistance of the building is defined as a ratio between the total energy that the existing structure can absorb and the energy at its ideal (upgraded) state estimated using the capacity curves obtained from the pushover analysis. The required retrofitting can then be assessed as the stiffness needed to be added to achieve the desired resistance. The study establishes the definition for the ideal capacity of an existing structure, safe displacement limits, and a procedure for deriving the target capacity curves and earthquake-resistance factors. The proposed procedure is applied to a benchmark example, demonstrating that upgrading sub-standard RC structures can be achieved by adding external stiffening members without strengthening individual components. The study also shows that traditional assessment procedures have limited capability to evaluate earthquake resistance in existing buildings with low ductility. The developed procedure provides an essential tool for comparative assessments of retrofitting scenarios and profitability evaluations.

1. Introduction

The importance of this research stems from the need to reevaluate the level of protection of existing structures against seismic actions and to determine the seismic requirement for the retrofitting process. The current approach to the assessment of existing buildings is based on an evaluation of the strength of structural elements and performance ratios. First, the critical elements are identified by an expert, based on engineering standards, and then, they are checked against the prescribed performance criteria. Finally, these critical elements are treated in the proposed seismic upgrade. As these local measures can only marginally affect the capacity and stiffness of the whole building [1], this approach often leads to disproportionate levels of retrofit demand that greatly exceed the requirements of the current seismic provisions. In general, the assessment of individual components is more time-intensive and it does not provide the quantified assessment of retrofit needs.

In this paper, we propose a displacement-based model that treats the structure as a whole object without evaluating its components. This approach allows for the quantitative assessments of the earthquake resistance of existing buildings and a level of the required seismic retrofit.

Nowadays, the legal provisions in many countries require a reassessment of the earthquake resistance of existing structures that were designed for gravity loads only or to substandard design codes. The deficiency of seismic resistance in these buildings is due to the lack of ductility rather than the lack of strength, which is a result of poor reinforcement detailing and inadequate design methods [2]. Moreover, these sub-standard buildings suffer from the corrosion of reinforcement, lack of concrete cover, joint problems [3], and poor design of weak and flexible stories [4]. The identification of sufficient levels of seismic upgrade for these buildings is complicated by high uncertainties in the evaluation of real existing conditions, which in turn limits the possibility of the quantitative assessment of required retrofit levels.

The procedures for the assessment of the earthquake resistance of existing buildings can be broadly classified into two categories: force-based and displacement-based methods. The force-based assessment approaches start from the assessment of member and joint-strength capacities, which define the elastic period and a class of the building frame. The base shear demands are then calculated and used to evaluate the members’ strength demands [2]. The displacement-based approaches are the performance-based methods defining the ultimate response of a structure in terms of the displacement parameters, such as roof displacements, inter-story drift ratios, rotations at joints, and ductility ratios. These parameters are assessed against the ones associated with the specific levels of the seismic hazard. The main advantage of the displacement methods is that it accounts for the ductility of the system [2,5,6,7]. The displacement-based methods assume definitions of the target displacements for each assessment case (building) independently. The most common method that defines the target displacement is the Capacity Spectrum Method (CSM), which predicts the maximum response of the structures based on the corresponding earthquake design spectrum [6,7,8,9,10]. Another common approach is the Displacement Coefficient Method (DCM), which estimates the peak displacement based on the spectral response of an elastic system and defines it using a series of coefficients depending on the effective fundamental period of the structure [10]. Lately, a simplified Direct Displacement-Based Assessment (DDBA) has been proposed for the seismic design of concrete buildings [2]. This method was originally developed by Priestley and colleagues [2,5,11]. It was demonstrated that the Nonlinear Static Pushover Analysis (NSPA) can be applied to the seismic assessment of RC buildings and evaluations of the limit displacements under various structural conditions. It was also shown that DDBA can be used to evaluate the effectiveness of various retrofit scenarios. The assessment of integrated solutions for combined energy efficiency improvement and the seismic strengthening of existing buildings was also discussed in [12]. The rapid seismic damage assessment of RC frames via a performance-based approach was presented in [13].

Few studies are dedicated to the development of various specific assessment models. Binda and colleagues suggested applying the NSPA to assess the capacity of masonry structures [14]. Zamppieri et al. used the nonlinear time–history analysis to examine the safety ratios for arch bridges subjected to different earthquake scenarios [15]. Mendes et al. reviewed assessment approaches to the rehabilitation of historic buildings, which focus on damage evaluation in the components of the main stability system [16]. The displacement-based criteria for possible seismic retrofitting scenarios were discussed in [17,18]. The dissipation energy-based method for the seismic retrofit of existing frames was presented in [19,20]. A seismic retrofit approach based on general stiffening of the building and displacement-based evaluation was proposed in [21,22,23]. Another approach to reducing the cumulative damage in existing sub-standard buildings by introducing viscous dampers was presented in [24].

The current provisions of various international standards for assessing the seismic performance of existing structures are diverse. The New Zealand Standard [25] utilizes classical methods for evaluating the resistance of existing buildings. It includes displacement-based methods and performance models and recommends the use of a time–history analysis. Japanese provisions [26,27] focus on changing the properties of the structure to increase ductility and stiffness. It emphasizes the use of base isolation devices for low-energy absorption and an improved seismic response. The US standards are based on the performance theory. ASCE provisions [28] require examining the structure as a collection of components rather than a whole object. It provides guidelines for evaluating the seismic performance of existing structures. FEMA-27 [29] emphasizes improving ductility by restoring capacity in various connectors of the structure. It also considers changes in damping levels as the structure moves between different positions.

In general, the developed methodologies for the earthquake-resistance assessment of the existing buildings are summarized in

Table 1. Assessment methods for the earthquake resistance of existing buildings.

The Underlying Principle of the Assessment Method	References
Strength assessment of critical joints 1	[1]
Identification of damage/survival levels using performance points 2.	[30]
A statistical approach is based on the risk ratio 2	[30,31,32,33]
Based on the performance ratio (CSM) 3	[31,34]
Strength assessment of critical joints 1	[31,33,35,36]
Assessment based on the ductility ratio 3	[37,38]
Based on the analog or approximation modeling like the cantilever structure 3. Elastic–plastic behavior 3.	[1]
Assessment based on lateral displacement limitation 4.	[23,39,40]
Elastic–plastic assessment based on the energy balance 3.	[41,42,43]
A combination of approaches.	[44,45]

¹ Force-based method, ² damage and risk assessment of the structure, ³ elastic–plastic analysis, ⁴ displacement-based method.

. One can see a variety of methods involving different underlying principles that have been proposed in the literature. However, most of these methods do not provide a quantitative measure for the lack of resistance and cannot be used for the evaluation of the efficiency of retrofit procedures.

The retrofit procedures for upgrading RC buildings that have been developed in recent decades can be classified into two main groups. The first group includes the methods that upgrade the structure at local levels of individual elements, while the approaches from the second group deal with the structure as a whole object by introducing so-called global measures [46]. The most popular local treatment is the jacketing of existing RC members, which can be performed using concrete, steel, or composite materials [10,23,47,48]. This local reinforcement improves the load-bearing capacity and ductility of critical structural elements. The global measures, on the other hand, aim at enhancing the lateral resistance of the building or decreasing the seismic demand. Bracing systems, cables, steel exoskeletons, and shear and infill walls are among the structural systems used to improve lateral resistance [23]. The seismic demand can be decreased by using base isolation [43] and various energy dissipation systems [41].

In this study, we assess the global earthquake resistance of buildings through the displacement-based approach. The proposed procedure will allow for an evaluation of the seismic demand of the tested building. Here, we translate this demand into the amount of lateral stiffening required to provide the sub-standard building with sufficient earthquake resistance. In the verification case study, this additional stiffening is supplied using the external shear walls, which are proportioned based on the estimated demand. A similar retrofitting solution for RC sub-standard buildings was discussed in [23,38,49,50]. It was shown that the shear walls were very efficient in controlling lateral drifts and reducing damage to frame members. The main drawback of this method is that the existing foundation system needs to be strengthened to resist the increased overturning moment and the larger weight of the structure. The advantages of applying this external upgrade included lower installation costs when compared to local repairs, no need to shut down the building, and the use of conventional construction materials. The efficient application of retrofitting through external stiffening requires reliable estimates of the amount of shear wall that is needed to achieve the desired level of resistance. To the best of the author’s knowledge, none of the existing assessment methods can provide this evaluation. The current study aims to address this research gap. It is expected that the same procedure can also be applied when additional stiffness is provided externally through exoskeletons or external braces. In the case of internal braces or infill walls, the necessary stiffening can be similarly estimated using standard methods of structural analysis.

The method we propose here capitalizes on several ideas explored in the previous studies. Firstly, we utilize the idea of the displacement limit by introducing a safety target displacement for the existing building. Then, we employ the fact that the yield displacements vary little with respect to the structural stiffness. Since the yield displacement is essentially defined by the geometry of the structure and material it is built of, it can be assumed that it is independent of the stiffness of the structure [1,23,38,51]. As a result, we can reasonably assume that the yield displacement of the existing building is not significantly affected by the stiffness added to the structure during the retrofitting process. Finally, we use the energy balance to define the resistance factor for the building as reviewed in [42,52]. The main challenge of the proposed procedure is the definition of the safety target displacement for existing buildings that is not presently available in the literature. The ability to evaluate the displacement limits will allow us to determine the seismic demand for a specific structure and thus assess the retrofitting efficiency.

To determine the displacement limits, the building structure is described as a whole object, while the overall resistance is assessed using equilibrium force-displacement curves. The ability of the structure to resist seismic loads is examined using the Nonlinear Static Pushover Analysis (NSPA). The seismic loads are defined based on the requirements and recommendations of the relevant standards. The additional seismic demand is determined in terms of the additional stiffness required to maintain the displacement limits.

The paper is structured as follows. Section 2 deals with the characterization of typical sub-standard buildings that require seismic upgrades. Section 3 discusses the applicability of available procedures to provide the safety displacement limits for typical structures. Section 4 presents the derivations of the proposed method and discusses the assessment of the seismic demands for sub-standard existing buildings. Section 5 is devoted to the definition of the resistance factor, which is based on the safety displacement limits and the energy balance. Section 6 presents a case study that demonstrates the ability of the developed method to evaluate the earthquake capacity of existing buildings and predict the efficiency of retrofitting procedures.

2. Characterization of Typical Sub-Standard Structures

To define a benchmark structure for the evaluation of the applicability limits of various assessment procedures for the earthquake resistance of existing buildings, we start with a review of typical sub-standard reinforced concrete (RC) buildings. We surveyed 18 seismically sub-standard buildings with similar structural systems built in Israel between 1960 and 1980. The survey included the geometry, inspection of the structural system, and a series of non-destructive tests to measure the material properties of concrete and reinforcement steel and the reinforcement ratios of all structural components. The seismic deficiency of these buildings was broadly recognized, but the actual level of seismic resistance had never been assessed [53]. All of these buildings are conventional RC structures with three to four stories and an open floor. The structural arrangements consist of two to four RC load-bearing frames and non-bearing infill walls. The RC frames are made of drop edge beams and small section columns that were originally designed for gravity loads only. The connectors between beams and columns exhibit limited resistance to bending moments and interstory drifts. The structural features of the surveyed RC frames are summarized as follows:

• Cross-sections of the columns and beams range from 20/30 to 20/50 cm;
• The concrete strength is 18 MPa;
• Concrete sections are not confined and have very few links;
• The reinforcement ratio (RR) in the cross-sections of the beams and columns ranges from 0.3% to 1.25%;
• The frames are on pinned supports due to the very limited resistance to bending. Connecting beams between the columns are provided at the ground level, but these beams do not contribute to the connection between the columns and the foundation;
• The strength of the reinforcement steel is 200 MPa;
• The infill walls add weight, but do not serve as a part of the stability system.

The detailed characteristics of the surveyed buildings can be found in [53].

Based on this survey, we define three typical benchmark frames that will be used in the following sections for the evaluation of various assessment models. These frames are two-, three-, and four-bay RC frames that represent a four-story building; see Figure 1. The structural parameters of these frames only differ in the number of spans, which leads to different levels of lateral stiffness. The ductility in these frames depends upon the available level of reinforcement. In the following numerical simulations, three values of the reinforcement ratios in the beams and the columns are considered as 0.3, 0.8, and 1.25%. Note that for the sake of simplicity, the infill walls are not included in the structural analysis, and the connections between the beams and columns are assumed to be rigid.

As a first step toward the development of the proposed procedure, three benchmark structures are analyzed using the mode-superposition method using the commercial computational package ATIR [54]. Note that this procedure is recommended by the Israeli standard [55]. The compliance of the structures with the resistance criteria prescribed by [56] is checked by examining the story drifts and stability factors. The interstory drifts are assessed based on the maximum obtained value (Δ) and maximum Interstory Drift Ratio (IDR_max). These indicators are commonly used for seismic performance assessments (e.g. [57,58]). The detailed building specifications and seismic loads are presented in Appendix A, while the results of this analysis are given in Appendix B and are summarized below as follows.

The interstory drifts are greater than what is allowed by Israeli standards [56]. The first floor is particularly vulnerable, as indicated by its high stability coefficient (see

Table A3. Mode-superposition analysis results for Frame 1.

Story	Story Height [m]	Level [m]	Interstory Drift, Δmax [mm]	IDRmax, %	Allowable Drift [60, mm]	θ	Comments
1	2.5	2.50	44.50	1.78%	12.5	0.1533	Δ and θ exceeded [56] limits
2	3.2	5.70	30.60	0.95%	16	0.0721	Δ exceeded [56] limits
3	3.2	8.90	22.50	0.70%	16	0.0430	Δ exceeded [56] limits]
4	3.2	12.10	14.20	0.44%	16	0.0190	Within limits [56]

,

Table A4. Mode-superposition analysis results for Frame 2.

Story	Story Height [m]	Level [m]	Interstory Drift, Δmax [mm]	IDRmax, %	Allowable Drift [60, mm]	θ	Comments
1	2.5	2.50	50.3	2.01%	12.5	0.1790	Δ and θ exceeded [56] limits
2	3.2	5.70	32.9	1.03%	16	0.0808	Δ exceeded [56] limits
3	3.2	8.90	23.9	0.75%	16	0.0490	Δ exceeded [56] limits]
4	3.2	12.10	14.9	0.47%	16	0.0240	Within limits [56]

and

Table A5. Mode-superposition analysis results for Frame 3.

Story	Story Height [m]	Level [m]	Interstory Drift, Δmax [mm]	IDRmax, %	Allowable Drift [60, mm]	θ	Comments
1	2.5	2.50	43	1.72%	12.5	0.192	Δ and θ exceeded [56] limits
2	3.2	5.70	27.4	0.86%	16	0.085	Δ exceeded [56] limits]
3	3.2	8.90	19.9	0.62%	16	0.052	Δ exceeded [56] limits]
4	3.2	12.10	12.3	0.38%	16	0.025	Within limits [56]

). When compared to FEMA-356 guidelines [59], the IDR_max of the 1st floor exceeded the acceptable limit for buildings to remain occupied immediately after an earthquake (Immediate Occupancy Performance Level [59]). Due to the uniform rigidity of all stories and the absence of infill walls, which leads to equal story shear strength, no weak or soft stories appear in these cases. This outcome is not surprising as it is commonly accepted that these buildings would not withstand any strong ground motion. Yet, this evaluation does not provide a quantitative measure for the lack of their earthquake resistance and the degree of vulnerability of these buildings. In the next section, we discuss the ways for such assessments.

3. Assessment of the Earthquake Resistance

We start the discussion of the quantitative assessment of earthquake resistance by investigating of applicability of the existing models, such as the CSM and the DCM. We apply both methods to the benchmark frames 1, 2, and 3 and determine the ranges where these models can be applied. We evaluate the capacity of the frames using the Nonlinear Static Pushover Analysis (NSPA) implemented in the commercially available computation package SAP2000 [60] (see Appendix C for details).

3.1. Capacity Spectrum Method (CSM)

The CSM is based on the interaction between two spectrum curves: a demand curve (DC) and a capacity spectrum curve (CSC). The intersection of these curves defines a performance point (PP), the point that designates the peak capacity of the examined structure. This PP provides a base for the assessment of critical components of the structure and their required upgrade. Here, we apply the CSM to Frames 1, 2, and 3 using the seismic parameters used in the previous example; see

Table A2. Seismic load specification—applied for all examples.

Parameter	Description	Value	Units
T	Fundamental period	2.07	s
Z	Expected horizontal ground acceleration coefficient	0.23	m/s × 2
Sa	Horizontal spectral acceleration factor	0.13	m/s × 2
I	Importance factor	1	-
Cd	Design response spectrum factor	0.08	-
K	Load reduction coefficient	1.5	-
S	Site infrastructure coefficient	D	-
Ss	Spectral acceleration at short time periods	0.58	m/s × 2
S1	Spectral acceleration in periods of 1 s	0.11	m/s × 2
Fv	Site coefficient at long time periods	2.38	-
Fa	Site coefficient at short time periods	1.38	-
SD1	Spectral design acceleration in short time periods	0.2618	-
SDS	Spectral design acceleration in periods of 1 s	0.80	-
T0	Is the lower limit of the period of the constant spectral acceleration branch	0.065	s
Ts	Is the upper limit of the period of the constant spectral acceleration branch	0.327	s
TL	Is the value defining the beginning of the constant displacement response range of the spectrum	4	s

. CSC is derived by converting the load-displacement curve calculated using NSPA into S_a (Spectral Acceleration) vs. S_d (Spectral displacement), (e.g., see Ref [6]). The results obtained for Frame 3 are presented in Figure 2 for various reinforcement ratios. It is seen that a low RR of 0.3% produces a high ductility ratio of 3.11 and no apparent PP; see Figure 2a. The apparent PP appears when the RR reaches 1.25%, whereas the ductility ratio is 1.25; see Figure 2c. The results obtained for Frames 1 and 2 are summarized in

Table 2. Summary of the analysis results for the benchmark frames obtained via NSPA.

	Frame 1			Frame 2			Frame 3
RR, %	1.25	0.8	0.3	1.25	0.8	0.3	1.25	0.8	0.3
μ	2.67	2.09	1.86	1.27	1.65	2.41	1.31	1.50	3.11
ue, [m]	0.1422	0.0970	0.0333	0.1380	0.0850	0.0309	0.1360	0.0840	0.0282
umax [m]	0.3800	0.2029	0.0618	0.1765	0.1400	0.0744	0.1783	0.1261	0.0877
δt, [m]	0.1620	0.1570	0.1740	0.1530	0.1500	0.1570	0.1460	0.1430	0.1510
μcr	1.16	1.69	4.93	1.11	1.80	4.96	1.08	1.75	5.20
VC, kN	47	30.9	10.9	77.6	53.4	18.3	130.55	70.705	20.556
Κ, kN/m	330.5	318.6	330.3	562.3	628.2	592.2	959.9	841.7	728.9

μ—the ductility ratio, u_e—the elastic limit, u^max—the maximal displacement, δ_t—the DCM target displacement, μ_cr—the critical ductility ratio, V_C is the shear force at the elastic limit, and K is the stiffness to the elastic limit computed as the slope at the linear section of CC.

. It can be observed that in Frame 2, the ductility decreases with the increase in the reinforcement ratio, while the degree of this decrease is less than in Frame 3. Furthermore, in the case of a more flexible Frame 1, this trend is reversing and a slight increase in the ductility is observed for higher reinforcement ratios. The influence of the frame lateral stiffness on the ductility levels also depends on the reinforcement ratios. In the case of the low RR of 0.3%, the ductility ratio is increasing from 1.86 for Frame 1 to 3.11 for Frame 3. While for higher reinforcement ratios the ductility ratios decreased from 2.09 to 1.5 and from 2.67 to 1.31 for RR = 0.8% and 1.25%, respectively. This complex behavior can be ascribed to the low strength of concrete and relatively small cross-sections of all flexural members. In a multi-degree-of-freedom frame structure, ductility is achieved through the gradual formation of plastic hinges at the connection between the columns and beams. In the case of a low RR = 0.3%, the plastic hinge development is slower for stiffer frames, which results in increasing ductility. For RR of 0.8% and 1.25%, the low strength of concrete of only 18 MPa (Section 2) in conjunction with the higher RR increases the concrete stress block and reduces the ductility levels.

It can be observed for all tested frames that the CSM is valid in a certain range of ductility ratios and that the capacity of more brittle structures cannot be assessed directly using this model. If the CSM assessment of a structure does not provide a PP, it signifies a deficiency in earthquake resistance. The degree to which the structure fails to reach the PP gives valuable information about the specific improvements needed, such as increased strength, deformation capacity (ductility), or damping, to enable the structure to withstand the imposed demands.

3.2. Displacement Coefficient Method (DCM)

The DCM is a displacement-based method, which provides the target displacement (TD) for the examined structure calculated using an empirical formula [59]. In this method, the load-displacement capacity curve (CC) does not need to be transformed into the spectral domain [6]. Figure 3 shows the curves obtained via NSPA for Frame 3 with various reinforcement ratios together with the TD calculated using the DCM. It is seen that in the cases with the low reinforcement ratios of 0.3% and 0.8%, the failure occurs before it reaches the target displacement limit; see Figure 3a,b. The increase in the RR decreases the ductility and provides the target displacement within the plastic range of the CC; see Figure 3c. Similar results are also obtained for Frames 1 and 2 in which the DCM can only be applied in the cases with the highest RR of 1.25%; see Table 2. To further investigate the applicability of the DCM, we define the critical ductility ratio, μ_cr—the minimal ratio ensuring that TD is within the range of the corresponding capacity curves. The critical ductility presented in Table 2 also defines the minimum ductility value that ensures the formation of a performance point. These values are obtained via interpolation, and they demonstrate that at the low reinforcement levels, the DCM can only be applicable at very high ductility levels. It can be observed that the critical ductility increases with a decrease in the reinforcement ratios, and it varies in each case.

Considering that the stiffness of a structure dictates the elastic displacement limit, the DCM can offer a meaningful displacement target at high ductility levels and provide insights into the extent to which the structure deviates from meeting the target displacement for less ductile ones.

Another important observation for the development of the proposed procedure can be found in Table 2. It can be observed that for a given reinforcement ratio, the displacement at the elastic limit varies little with respect to the stiffness of the frame. For example, for RR = 1.25%, u_e changes from 0.1422 m for K = 330.5 kN/m to 0.138 m for K = 562.3 kN/m and 0.136 m for K = 959.9 kN/m. This result validates the assumption that the yield displacement can be considered independent of the structural stiffness of the frame.

3.3. Safety Target Displacement

The safety target displacement is a concept used in this study, which defines the maximum displacement that the structure can attain during an earthquake without reaching the point of failure. As shown in Section 3.2, the DCM has limited ability to provide the safety displacement limits and thus cannot be employed for the assessment of the frames with low ductility. Alternatively, we propose to use the energy balance [41] to define the safety target displacement (STD). This displacement limit will be unique for each specific existing structure and will lead to a quantifiable measure of its earthquake resistance. We define earthquake resistance as a ratio between the energy the structure can absorb at its current state and the energy it will be able to absorb at the ideal upgraded state. These energies can be estimated using the capacity curves that are calculated through the NSPA. The procedure establishing the seismic demand at the ideal state is developed in the following section.

It should be noted that in the case of high-ductility structures, the DCM can be used to define the peak target displacement. This, however, will still require the definition of the ideal capacity curve.

4. Seismic Demand in Existing Buildings

To define the seismic demand for an existing structure without descending to an examination of individual members and joints, there is a need to determine the target ductility curve (TDC) for an existing structure. The TDC represents the ideal capacity curve for the assessed structure (test structure) that ensures sufficient resistance and defines the target displacement (TD). To the best of our knowledge, no current approach for the assessment of existing buildings can provide this evaluation; thus, this section is dedicated to the development of such a method. The seismic retrofit then will constitute an addition of an appropriate amount of stiffness to the structure without any treatment of individual components. The main feature of this approach is that the building is retrofitted through global stiffening in the two main directions rather than strengthening of specific joints and members. Consequently, the seismic demand is defined as the additional stiffness to be provided to ensure the TD limits.

4.1. Target Ductility Curve (TDC)

The target ductility curve (TDC) describes the seismic demand for perfect elastic–plastic behavior, which is defined in terms of the base shear force and top story displacement; see Figure 4. The location of TDC is bound between two limit states: from below by the capacity curve (CC) obtained for the assessed structure and from above via an elastic equilibrium curve describing the non-ductile structure (NDS), as shown in Figure 4a. To define the TDC for the test structure, we need to establish two points: the yield point (Point A) and the safety target displacement (Point B). Point A defines the elastic limit, which corresponds to the development of the first plastic hinge and designates the peak base shear force. The displacement at this point does not change after the retrofitting through global stiffening since the additional stiffness does not alter the existing joints [1,51], which is illustrated in Figure 4a and reads as

$$U_e^{max}=U_e^{cap}=U_e^{demand}$$

(1)

where $U_e^{max}$ is the maximum displacement for the non-ductile structure (NDS curve), $U_e^{cap}$ is the elastic displacement obtained via NSPA for the tested structure (CC), and $U_e^{demand}$ is the elastic displacement in the TDC.

Point B defines the ductility demand for the tested structure and its maximum TD. The positions of Points A and B can be estimated using the energy balance [42], which is illustrated in Figure 4b and is derived as follows.

The elastic energy of the corresponding non-ductile structure (NDS), E_e, is defined as [42]

$$E_e={\int }_0^{U_e^{\mathrm{m}\mathrm{a}\mathrm{x}}}{C}_e\left(U\right)dU$$

(2)

where $C_e\left(U\right)$ is the linear function of the displacement. The energy at the NDS state is then defined using Figure 4b as follows:

$$E_e={\displaystyle \frac{U_e^{\mathrm{m}\mathrm{a}\mathrm{x}}·V_e^{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}$$

(3)

where $V_e^{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum base shear force for the NDS, which is given as follows:

$$V_e^{max}=\sum W_i\ast S_a$$

(4)

where $S_a$ is the design spectral acceleration, W_i is the weight of the i^th story, and $\sum W_i$ is the total weight. The total energy to be dissipated by the retrofitted structure, $E_t$, is defined as follows:

$$E_t={\int }_0^{U_{STD}^{demand}}{C}_{e-p}^{demand}\left(U\right)dU$$

(5)

where $C_{e-p}^{demand}\left(U\right)$ is the function describing the TDC.

Using the definitions in Figure 4b we obtain the following:

$$E_t={\displaystyle \frac{U_e^{demand}·V_e^{demand}}{2}}+\left(U_{TD}^{demand}-U_e^{demand}\right)·V_e^{demand}$$

(6)

where $V_e^{demand}$ is the maximum shear force defined by the TDC. According to the energy balance principle [42], the amount of dissipated energy in the NDS equals the total dissipated energy with the ductile behavior. This balance is expressed using Equation (3) and (6) as follows:

$${\displaystyle \frac{U_e^{max}·V_e^{max}}{2}}={\displaystyle \frac{U_e^{demand}·V_e^{demand}}{2}}+\left(U_{TD}^{demand}-U_e^{demand}\right)·V_e^{demand}$$

(7)

There are two unknown parameters, $V_e^{demand}$ and $U_{TD}^{demand}$, in Equation (7). As discussed in Section 3.2, the target displacements for existing structures with ductility levels higher than the critical ductility can be estimated using the DCM (see Table 2). In general, we have three different scenarios that are illustrated in Figure 5. In the case shown in Figure 5a, the ductility of the tested structure is greater than the critical ductility, $\mu >{\mu }_{cr}$, which allows the use of the DCM to define $U_{TD}^{demand}$. Figure 5b presents a case of a non-ductile structure, $\mu ~0$. In this case, $U_{TD}^{demand}$ cannot be greater than the elastic capacity and should be taken as $U_{TD}^{demand}=U_p^{cap}.$ Once $U_{TD}^{demand}$ is determined in both these cases, $V_e^{demand}$ can be found using the energy balcance, Equation (7).The third case shown in Figure 5c, represents the structure with ductility levels less than the critical one, $\mu <{\mu }_{cr}.$ The DCM cannot be used in this case as its target displacement is greater than the total displacement, and $U_{TD}^{demand}$ will be determined as follows.

Using Figure 4a, we define the following ratios:

$${\displaystyle \frac{V_e^{demand}}{U_e^{demand}}}={\displaystyle \frac{V_C}{U_1}}$$

(8)

We assume the safety margin as $U_p^{cap}-U_p^{demand}$ and obtain

$$U_e^{demand}-U_1=U_p^{cap}-U_p^{demand}$$

(9)

Then, following Equations (1), (3) and (6)–(9), we determine the ratio (8) as follows:

$${\displaystyle \frac{V_e^{demand}}{U_e^{demand}}}={\displaystyle \frac{V_c}{U_e^{demand}+U_{TD}^{demand}-U_p^{cap}}}$$

(10)

where V_c is the maximum shear load obtained from the capacity curve. By solving Equations (7) and (10) with the equality (1), $V_e^{demand}$ and $U_{TD}^{demand}$ can be calculated for the specific test structure. Finally, using (4) we define Points A and B and build the TDC curve; see Figure 4a.

As a final step, we need to check that the obtained $U_{TD}^{demand}$ is smaller than $U_p^{cap}$ and $U_e^{cap}-U_1\ge U_p^{cap}-U_{TD}^{demand}$. The later condition is required to ensure the safety margin. If the obtained $U_{TD}^{demand}$ is greater than $U_p^{cap}$, $U_{TD}^{demand}$ should be offset by $U_e^{cap}-U_1$ obtained from the previous step and the calculations for $V_e^{demand}$ repeated using Equation (7) alone.

The key steps of this procedure are illustrated in Figure 4a and can be summarized as follows:

Step 1: Calculate the capacity curve (CC) of the existing structure based on available structural data using the NSPA.

Step 2: Create the non-ductile capacity curve (NDS) based on Equation (4) using design spectral accelerations at the given location.

Step 3: Develop the target ductility curve (TDC) by computing the coordinates of Points A and B (see Figure 4a) using Equations (7) and (10).

4.2. Seismic Demand—Conversion to Stiffness

The TDC curve defined in Section 4.1 is used to determine the seismic demand for the test structure. Following the approach proposed here, we treat the structure as a whole object and need to define the amount of stiffness required to satisfy the seismic demand. In the displacement-based design, the performance of existing components is ensured by limiting the horizontal displacements to the TD. To constrain the displacements within the TDC range, the additional stiffness should be provided to the existing structure, as generally illustrated in Figure 6. This additional stiffness can be calculated based on the difference between the existing capacity curve (CC) and the seismic demand (TDC); see Figure 4a.

Following the definitions of Figure 4a, the additional stiffness, $∆K$, can be calculated as follows:

$$\mathsf{\Delta }K={\displaystyle \frac{V_e^{demand}-V_c}{U_e^{demand}}}$$

(11)

4.3. Seismic Demand—Conversion to Stability Members

Following the idea of retrofitting through global stiffening, the additional stiffness defined in Equation (11) is added to the test structure in the form of an external additional member. The retrofit objective is to limit the top floor displacement to the TD value without dealing with individual structural components. For example, an option for the stiffening retrofit applied to Frame 3 is presented in Figure 7. Although Figure 7 shows a shear member attached to the exterior of the building, it is important to note that it can be placed at various locations and realized using different shear elements. To estimate the additional stiffness, we assume that the stiffening components at each story level have a similar design and define the seismic load using the 1st mode distribution as provisioned by the equivalent static analysis [38]. The additional stiffness is defined as the base shear due to a unit top story displacement that can be estimated using common methods of structural mechanics. For example, taking the heights and masses of each story as equal and using structural dynamic textbook formulas [52], the top floor displacement can be expressed as

$$D^{top}={\displaystyle \frac{\left(64·H_{q4}+40.5·H_{q3}+20·H_{q2}+5.5·H_{q1}\right)·h^3}{3·E·I}}$$

(12)

or more generally as

$$D^{top}={\displaystyle \frac{h^3}{3EI}}\sum _{i=1}^N{C}_i{·H}_{qi}$$

(13)

where $C_i$ is the contribution of a seismic force $H_{qi}$ to the top floor displacement, N is the number of stories, E is Young’s modulus of the material used for the stiffening, and I is the moment of inertia of the stiffening member. Thus, the additional stiffness demand is determined as

$$∆K={\displaystyle \frac{3·E·I}{h^3}}·{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{H}_{qi}}{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{C_i·H}_{qi}}}$$

(14)

Assuming that a shear member is to be used for stiffening, the required additional cross-section can be calculated as follows:

$$I={\displaystyle \frac{∆\mathrm{K}·h^3}{3·E}}·{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{C_{i}H}_{qi}}{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{H}_{qi}}}$$

(15)

Note that Equation (15) is valid for different materials and structural arrangements that can be used for the seismic retrofit.

5. Resistance Factor for the Assessment of Existing Buildings

Assessment of the earthquake resistance of existing structures in different geophysical locations requires a quantitative factor that can help a decision-maker to determine retrofitting and demolition policies and recommendations that will improve the safety status of the buildings in the given area. As discussed in Section 3, the CSM and DCM models that are widely accepted as the methods of choice in the literature are unable to describe the earthquake resistance of low ductility structures, nor are they able to derive the resistance factor from the true properties of the structure. Several attempts have been made to define the resistance factor as a ratio between the original capacity of a structure and the required capacity [7,61]. However, the definition of a target structure and the assessment of the capacity demands were not given in these studies. On the other hand, the methodology proposed in the current study allows for the qualitative assessment of the resistance through a comparison of the capacity curves of an existing structure with a corresponding ideal TDC of the same structure. This comparison can be expressed as a ratio between the energy the existing structure is capable of dissipating to the amount of energy the ideal target structure would be able to dissipate, whereas both energies can be calculated using the curves established in Section 3 and Section 4 as follows. The amount of energy the ideal target structure should be able to dissipate, $E^{demand}$, is defined as the integral of TDC, which is graphically represented in Figure 8 as the shaded area below the curve. The amount of energy the actual structure dissipates under the earthquake loading is obtained using its capacity curve (CC); see Figure 4a. However, in practice, we suggest limiting this value by using a safety target displacement (STD) defined at the right integration boundary, as shown in Figure 8, where this reduced energy amount is illustrated by a cross-hatched area. This energy reduction will provide a margin of safety and account for joints’ detailing, which is not considered in the analysis. Thus, the resistance factor, RF, will be defined as the ratio between the reduced energy, E^red and E^demand, as

$$RF={\displaystyle \frac{E^{red}}{E^{demand}}}$$

(16)

It can be seen from Equation (16) that the RF varies as 0 < RF < 1.0. RF→0 defines an engineering structure with a very low level of earthquake resistance. This type of structure with no actual stability system was discussed in [39]. At the upper limit of RF = 1.0, we see the upgraded structure that is retrofitted using the methodology developed in the current study. This resistance factor can be used to compare the seismic performance of similar structures located in different seismic regions, to assess the retrofit costs and to define the amount of retrofitting required. It can also be an inductive factor for decision-making, such as upgrading or demolition when it is implemented along with cost estimates.

6. Case Study

The main feature of the proposed assessment procedure is that it capitalizes on the structural properties of the tested structure and the geo-seismic conditions of its location. The proposed model is not limited to specific types of structures or seismic conditions. However, the assessment outcome is test dependent. For example, for a structure located in a seismic active area, one can expect it to have a lower resistance ratio than an identical structure located in an area with a lower probability of an earthquake. The lower resistance factor indicates higher seismic requirements and higher retrofitting costs. To demonstrate the implementation of the proposed methodology and verify its ability to evaluate the earthquake resistance of an existing building, we apply it to an existing building of a Frame 3 type, calculate the resistance factor, and evaluate the potential retrofit scenario. We also assess the applicability of the CSM and DCM models, as models of choice that are accepted in the scientific and engineering literature.

6.1. Description of the Tested Structure

The tested structure represents an existing four-story building, which is located in the City of Beit She’an, North Israeli district. The structural arrangement of this building is similar to the benchmark Frame 3. The building was built in 1973, and it represents a typical concrete frame structure built in Israel before 1980. The stability system consists of concrete frames that were designed for gravity loads without the consideration of the horizontal (seismic) loads. The beams are continuous beams with a negative reinforcement over the supports and an overlap for the column reinforcement at the joints. This system of beams and columns creates joints with a limited capacity to transfer bending moments and can only provide a partial stability system. There are four frames in every direction with spans of 5 m. The total plan size is 15 m × 15 m, and the total height is 12.1 m; see Figure 9. The material properties and seismic loads are given in Appendix A.

First, we analyze the tested structure using a commonly accepted procedure and evaluate the ability of the stability system to resist earthquake loads. To this end, the Israeli standard [56] for earthquake-resistant structures was adopted. SI-413 is based on elastic performance with ductility-reduction factors, which follows the philosophy of Eurocode [62] According to this procedure, the tested structure was examined using the mode superposition method using ATIR software, STRAP 2022 [54]. The seismic performance attained by the building is summarized in

Table 3. Seismic performance check for the non-retrofitted structure.

	Story 1	Story 2	Story 3	Story 4	Comments
Vi (kN)	39.60	47.80	62.60	114.70
Δel,i (mm)	47.5	30.3	22.1	13.9	1st, 2nd, and 3rd stories exceeded [56] drift limits
θ	0.3596	0.1582	0.0975	0.0475	1st and 2nd storeys are not stable [56]
IDRmax	1.90%	0.947%	0.656%	0.434%	1st storey exceeded FEMA IO level [59]

. It observes that (1) the maximum story drifts at the first three levels exceeded the allowable limits, (2) the vertical elements at the 1st and 2nd stories are not stable, and (3) the 1st story exceeded FEMA-356 requirements for the Immediate Occupancy (IO) performance level.

The obtained results reveal that the structure cannot withstand the earthquake loads prescribed by the standard. It is evident that the structure is deficient in seismic resistance and requires a retrofit. However, these results do not allow for the quantitative assessment of this deficiency, nor do they provide an estimate for the required seismic upgrade. The limit displacement method (LDM) developed in this study is applied next.

6.2. Assessment of the Tested Structure Using the Limit Displacement Method

The procedure developed in Section 3, Section 4 and Section 5 is applied next for an evaluation of the capacity curve (CC) under the existing structural conditions and calculations of the target displacement curve (TDC) defining the ideal conditions for the structure at hand. Then, these parameters are used to calculate the resistance factor (RF) and to assess the additional stiffness (ΔK) required for the seismic retrofit.

6.2.1. Capacity Curve (CC)

The CC of the structure is defined as the base shear–top floor displacement curve and it is calculated through the NSPA using the commercial software SAP2000, Version 10.07 [60]. The model of the structure included the control point and the joint properties, which we used in 2D modal analysis, along with all the relevant features of the existing structure. The obtained CC is presented in Figure 10. The curve describes the elastic–plastic behavior of the structure subjected to the horizontal seismic load. It shows the maximum elastic displacement, $U_e=0.0766m$, the total displacement, and $U_p=0.1517m$, the peak shear force capacity, $V_c=280.0kN$. The detailed results of the analysis are also presented in Table A3, Appendix D. The obtained curve indicates a low-capacity structure with a relatively low ductility.

In order to compare the developed limit displacement method with the existing procedures, we examine the applicability of the DCM and CSM procedures. The application of the DCM procedure yields a limit displacement of 0.153 m. As seen in the NSPA results (Figure 10), the frame fails at the total displacement of 0.1517 m, which is below the DCM limits. Thus, the DCM approach cannot be used for the assessment of this structure.

According to the CSM procedure, the CC obtained via NSPA (Figure 10) needs to be transformed into the capacity spectrum curve (CSC) as described in Section 3.3. In this case, the interaction between the CSC and DC shows no performance point, as illustrated in Figure 2a, which also indicates that the tested structure is not earthquake resistant. It is seen that none of these methods provides a reliable estimate for the target performance point to which the structure should be retrofitted. The detailed results of both assessments are given in Appendix D.

6.2.2. Target Ductility Curve (TDC)

The TDC curve for the tested structure is built using the procedure described in Section 4.2, through the following steps:

• The maximum base shear force is calculated using Equation (4) as V_emax = 1187.0 [kN], where the total weight of the tested structure, W, and the spectral ground acceleration, S_a, are as given in Appendix D.
• The NDS curve is derived using the elastic displacement of 0.07720 [m] and the base shear V_emax = 1187.0 [kN]; see Figure 11.
• The elastic energy is calculated using Equation (3) as E_e = 45.46 [kN·m]. × m].
• Substituting these values in Equations (6) and (7), we obtain the following system:

$$45.46={\displaystyle \frac{0.0766·V_e^{demand}}{2}}+\left(U_{STD}^{demand}-0.0766\right)·V_e^{demand}$$

(17a)

$${\displaystyle \frac{V_e^{demand}}{0.0766}}={\displaystyle \frac{280.0}{0.0766+U_{STD}^{demand}-0.1517}}$$

(17b)

By solving system (15), V_e^demand and U_STD^demand are obtained as 651.91 [kN] and 0.108 [m], respectively. These values define Points A and B required for the TDC, which are plotted in Figure 11 as well.

6.2.3. Ideal Performance Point (IPP)

The created ideal TDC is mapped into the spectrum acceleration–displacement curve and plotted in Figure 12 together with the CSC of the CSM and the demand curve (DC). The intersection of the TDC and the spectrum demand (DC) indicates the ideal performance point (IPP). It is seen that the IPP (A = 793.50; D = 76.22) is within the elastic region and indicates safer performance of the existing elements without the development of plastic joints.

The change in the location of the performance point proves the advantage of improving the resistance by adding global stiffness to the whole structure. The new position of the performance point ensures that no failure will occur to the structure’s elements during an earthquake. The additional stiffness limits the displacements in general and provides an increase in the loading capacity. The existing structural elements reach the plastic regime at the higher load levels, while the ductility does not change.

6.2.4. Resistance Factor

As described in Section 5, the resistance factor is the ratio between the total energy dissipated by the tested structure and the energy that the ideal structure would be able to dissipate. These values can be calculated using the CC and TDC plotted in Figure 11 as follows:

$$RF={\displaystyle \frac{E^{red}}{E^{demand}}}={\displaystyle \frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\cdot 0.0766\cdot 280.0+\left(0.1517-0.0766\right)\cdot 280.0}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\cdot 0.0766\cdot 651.91+\left(0.108-0.0766\right)\cdot 651.91}}={\displaystyle \frac{31.75}{45.44}}=0.698$$

(18)

The obtained value 0.698 indicates that the tested structure requires an additional 20% of the energy absorption capacity.

6.3. Stiffening and Seismic Demand

The seismic demand of the tested structure is expressed in terms of the additional stiffness that should be added to the structure externally. Following the procedure described in Section 4, this additional stiffness is calculated using Equation (14) as

$$∆K={\displaystyle \frac{651.91-280.0}{0.108}}=3443.61\left[\mathrm{k}\mathrm{N}/\mathrm{m}\right]$$

(19)

If the stiffening is provided by external shear members, the additional cross-section can be calculated using Equation (15) as

$$I={\displaystyle \frac{L^3\cdot ∆K}{3\cdot E\ast \left(P\right)}}=0.043\left[m^4\right]$$

(20)

where I is the moment of inertia of a section that should be added, and E is Young’s modulus of concrete suggested for the stiffening of the building. The additional stiffness can be supplied by providing two external members in each direction, as shown in Figure 13, which can be dimensioned using Equation (20) as follows. The thickness of the member is defined as b = 0.3 [m], and then, the length in the action direction, a, is calculated as

$$a={\left({\displaystyle \frac{1}{4}}·I\cdot {\displaystyle \frac{12}{b}}\right)}^{1/3}=0.75\left[m\right]$$

(21)

6.4. The Stability Check after the Proposed Retrofit

To check the efficiency of the proposed retrofit, the stability coefficients are calculated following [56] as

$${\theta }_i={\displaystyle \frac{W^{total}.{∆}_{el,i}.k}{V^i.h^i}}\le 0.10$$

(22)

where θ_i is the stability factor of the ith story, W^total is the total weight of the tested structure, ${∆}_{el,i}$ is the drifts of ith story, k is the reduction factor (k = 1.5), $V^i$ is the shear load at ith story, and hⁱ is the story heights.

The results of the seismic performance check are presented in

Table 4. Stability check results for the retrofitted structure.

	Story 1	Story 2	Story 3	Story 4	Comments
Vi (kN)	55.60	53.30	41.60	124.30
Δel,i (mm)	0.32	0.363	3.85	1.89	Not exceeded
θ	0.0023	0.0027	0.0104	0.0262	P-Δ effects are insignificant [56]
IDRmax, %	0.0128%	0.0113%	0.120%	0.059%	Not exceeded [56]

. It is seen that all stability coefficients are within the allowed range $\le$ 0.1, the interstory drifts do not exceed the values provisioned by the standard [56], and IDR_max values are all under the 1% required by FEMA-365 [59]. This check verifies the proposed retrofit scenario.

6.5. Verification of TDC Procedure

To validate the basic assumptions used to develop the proposed model, the capacity curve is calculated for the retrofitted structure (RCC) and presented in Figure 14. It can be observed that the RCC yield displacement and CC yield displacement are within a reasonable margin and differ by 3%. It is also seen that the shear capacity at RCC (566.33 kN) is greater than the demand shear capacity of (651.91 kN, Figure 11) by 13.0%, which indicates that the retrofitted structure has sufficient seismic resistance. The performance level of the retrofitted structure is examined in Figure 15 by comparing the spectrum acceleration–displacement curves. The CSM performance point indicates that the retrofitted structure is at a reliable performance level.

These results demonstrate that the proposed model provides a reliable estimate for the amount of additional stiffening required to achieve sufficient seismic capacity in a retrofitted structure.

6.6. Case Study Summary

The tested structure is located in a seismically active area and cannot satisfy the current earthquake resistance provisions. It has been shown that the DCM and CSM models are not able to provide an adequate assessment of the structure. The proposed limit displacement model has provided a resistance ratio of 0.797, which indicates a severe lack of seismic resistance. Using the energy balance, the target ductility demand curve has been established, which is used to calculate the seismic demand. The proposed retrofit scenario included four shearing members of 0.30/0.75 m sections in each direction. It has been shown that the performance point defined through the CSM for the upgraded structure is significantly improved.

7. Conclusions

In a range of the methods used for the assessment of existing buildings, the displacement-based methods are among the most reliable ones. However, the commonly used CSM and DCM models for the evaluation of earthquake resistance have limited ability to provide a quantitative measure for the lack of such resistance. Moreover, it has been shown that these methods are not suitable for the assessment of low-ductility structures. In such cases, these methods might only be able to provide the seismic demand estimate for higher damping levels.

The outcomes of the current study can be summarized as follows:

• An alternative approach to the identification of the seismic demands and corresponding resistance factors has been developed. The building is considered as a whole object and the structural capacity is evaluated in terms of the base shear force–top floor displacement equilibrium curve.
• The seismic retrofit is achieved through global stiffening of the existing structure without the treatment of individual components.
• The ideal target structure is obtained by defining the limit target displacement, which is determined through the energy balance.
• The seismic resistance factor is expressed as a ratio between the amounts of energy the existing structure is able to dissipate to those an ideal (upgraded) structure would be able to dissipate.
• The amount of stiffening required to achieve the ideal energy-absorbing capacity defines the seismic demand. Both seismic demand and resistance factors depend on the structural properties of the existing structure and the seismic conditions of the location.
• The minimum ductility in the existing structures that ensures the formation of a performance point depends on the available level of reinforcement. Underenforced structures or those with heavily corroded steel reinforcements will have high levels of critical ductility and thus cannot be assessed using the CSM or DCM methods, as the performance point cannot be determined.
• The ability of the developed procedure to assess an existing building is demonstrated through a case study by applying the method to a real RC building.
• The case study has been carried out for a low-ductility structure, which cannot be analyzed using the commonly accepted DCM and CSM models. First, it is shown, using the mode-superposition method, that the structure lacks adequate earthquake resistance. Then, the non-linear pushover analysis is used to determine the resistance factor and seismic demand for the target structure. The retrofit scenario has been proposed based on the obtained results. The upgraded structure has been checked and deemed to satisfy the seismic requirements.

It has been demonstrated that the developed limit displacement model provides a reliable approach for the quantitative assessment of existing buildings. This method can be used for evaluations of a wide range of existing buildings and the comparative assessment of sub-standard structures. The resistance factor derived here provides a quantitative measure of the amount of required retrofit for any structure regardless of the geophysical location or specific site conditions. Since this factor is independent of a particular retrofit scenario, it allows for the grading of existing buildings for the profitability of potential upgrades. This grading system would assist policymakers in creating tiered incentive programs where buildings with high profitability scores receive large grants or tax breaks for retrofit. This would also encourage building owners to undertake retrofits when they might otherwise hesitate due to uncertainties in upfront costs.

The case study presented in this paper validates that the seismic upgrade of subs-standard low-rise RC buildings can be performed by upgrading the global stiffness of the building rather than improving the capacity and ductility of individual structural members. To allow for the use of the method by design engineers, it is recommended to include, in future studies, the effects of the infill walls and bracing systems, to investigate the effects of different types of foundations, and to verify the method for medium-rise buildings.