The effect of FRP retrofitting on the seismic resilience of low-ductility RC frame structures: a case study

Structure information

A four-story RC frame structure has been designed. For the seismic hazard analysis presented later, the structure is assumed to be located in a site with a seismic fortification intensity of 7 degrees at 0.1g according to the existing codes^[19,20] of China. The seismic design category is Group Ⅰ, the site category is Class Ⅱ, and the ground roughness is classified as C. The floor plan and elevation of the structure are shown in Figure 1. The structure is divided into ground floor and standard floors. The total construction area of the structure is 1, 800 m². The height of the ground floor is 4.8 m, while the height of the standard floors is 4.2 m. The thickness of the floor and roof slabs is 120 mm. The design values for the dead load and live load of the roof are 6.9 and 2.0 kN/m², respectively. For the standard floors, the dead load and live load are 4.3 and 2.0 kN/m², respectively. For the corridors, the dead load and live load are 3.8 and 2.5 kN/m², respectively. The concrete strength used in the design is C30. Due to the low ductility structure usually only considers vertical loads during design, seismic fortification and wind loads are not considered in this structure. The cross-sectional dimensions and reinforcement details of the beams and columns in the structure are shown in Figure 2. The column cross-sectional dimensions are 400 mm × 400 mm, and the beam cross-sectional dimensions are 300 mm × 600 mm. The longitudinal reinforcement of beams and columns is HRB400. All other reinforcement uses HPB235. A finite element model is established for one span of the plane structure. The equivalent load distribution of the structure is illustrated in Figure 3.

Figure 1. Plan and elevation view of structure. (A) Plan view; (B) Elevation view.

Figure 2. Structural reinforcement information. (A) Column; (B) Beam of standard floor; (C) Beam of roof.

Figure 3. Equivalent load distribution of the structure.

Finite element model

Constitutive models of material

(1) Reinforcement steel

The finite element model of the structure is established using the OpenSEES. The reinforcement steel is simulated using the Steel02 material model. This model represents the uniaxial constitutive behavior of the steel and includes isotropic hardening under cyclic loading. This allows it to accurately capture the stress-strain relationship of the steel under repeated loading. The stress-strain constitutive relationship is expressed as:

(1) $$ \begin{equation} \sigma^* = b \varepsilon^* + \frac{(1-b) \varepsilon^*}{(1 + \varepsilon^{*^R})^{1/R}} \end{equation} $$

where $$ \sigma ^* $$ and $$ \varepsilon ^* $$ are the normalized stress and strain, respectively; their calculation expressions are given by

(2) $$ \begin{equation} \sigma^* = \frac{\sigma - \sigma_r}{\sigma_0 - \sigma_r} \end{equation} $$

(3) $$ \begin{equation} \varepsilon^* = \frac{\varepsilon - \varepsilon_r}{\varepsilon_0 - \varepsilon_r} \end{equation} $$

where $$ R $$ is a parameter that influences the shape of the transition curve, and its calculation expression is given by

(4) $$ \begin{equation} R = R_0 - \frac{a_1 \xi}{a_2 + \xi} \end{equation} $$

where $$ b $$ is the strain hardening ratio, which is set to 0.00085 in this study. The values of $$ R_0 $$, $$ a_1 $$ and $$ a_2 $$ are taken as 18, 0.925, and 0.15 according to the recommendations in OpenSEES, respectively.

(2) Concrete

The unconfined concrete regions and the regions confined only by stirrups are simulated using the Concrete01 material model. The peak stress-strain relationship is based on the constitutive model proposed by Scott et al. ^[21]. The confining effect of the stirrups on the concrete is accounted for through a confinement factor. The stress-strain expressions are given by:

(5) $$ \begin{equation} \sigma_c = f_{pc} \left[ \frac{2\varepsilon_c}{\varepsilon_{pc}} - \left( \frac{\varepsilon_c}{\varepsilon_{pc}} \right)^2 \right] \text{,for } 0 \leq \varepsilon_c \leq \varepsilon_{pc} \end{equation} $$

(6) $$ \begin{equation} \sigma_c = \frac{f_{pc} - f_{cu}}{\varepsilon_{pc} - \varepsilon_{cu}} (\varepsilon_c - \varepsilon_{cu}) \text{,for } \varepsilon_{pc} \leq \varepsilon_c \leq \varepsilon _{cu} \end{equation} $$

(7) $$ \begin{equation} \sigma _c=f_{cu} \text{,for } \varepsilon_c \geq \varepsilon_{cu} \end{equation} $$

where $$ \sigma _c $$ and $$ \varepsilon _c $$ represent the compressive stress and the corresponding compressive strain of the concrete, respectively; $$ f_{pc} $$ and $$ \varepsilon _{pc} $$ are the compressive stress and corresponding strain at the peak point, respectively; $$ f_{cu} $$ and $$ \varepsilon _{cu} $$ are the compressive stress and corresponding strain at the ultimate point, respectively.

(3) Stress-strain constitutive model for FRP-reinforced concrete

The model proposed in reference^[3] is used to account for the confinement effect of FRP on RC square columns. FRP-confined concrete components typically exhibit three failure modes: strong, moderate, and weak confinement modes. In the strong confinement mode, the failure of a column is characterized by the crushing of the concrete, immediately followed by the rupture of the FRP material. In the moderate confinement mode, the concrete is crushed first, and then the FRP material ruptures after undergoing some deformation. The weak confinement mode occurs when the FRP material fails prematurely due to either an insufficient number of FRP wrapping layers or adhesive anchorage failure. According to the research in reference ^[3], when the adhesive is intact and the number of FRP reinforcement layers exceeds two, the weak failure mode is unlikely to occur in FRP-confined concrete components. Since the retrofitting scheme discussed later involves at least two layers of FRP, this paper assumes that all components exhibit strong confinement failure modes. The stress-strain relationship of concrete before and after FRP confinement is given in Figure 4.

Figure 4. Stress and strain relationship of concrete before and after FRP confinement.

The ultimate stress and strain for concrete confined by FRP are defined as follows:

(8) $$ \begin{equation} \frac{f_{cu}}{f_c'} = 0.2 + 0.59 \left( \frac{f_{1s}}{f_c'} \right)^{0.20} + 3.47 \left( \frac{f_{1f}}{f_c'} \right)^{0.64} \end{equation} $$

(9) $$ \begin{equation} \frac{\varepsilon_{cu}}{\varepsilon_{c0}} = 2 + 5.06 \left( \frac{f_{1s}}{f_c'} \right)^{0.03} + 73.31 \left( \frac{f_{1f}}{f_c'} \right)^{1.07} \end{equation} $$

where $$ f'_{c} $$ and $$ \varepsilon_{c0} $$ are the peak stress and peak strain of unconfined concrete, respectively; $$ f_{cu} $$ and $$ \varepsilon_{cu} $$ are the ultimate compressive stress and the peak compressive strain of FRP-confined concrete, respectively; $$ f_{ls} $$ and $$ f_{lf} $$ are the effective lateral confining forces provided by stirrups and FRP, respectively.

The peak stress and strain of FRP-confined RC square columns are given by:

(10) $$ \begin{equation} f_{cc} = f'_c \left( 1 + 0.35 \lambda_f + 0.50 \lambda_h + 0.85 \lambda_1 \right) \end{equation} $$

(11) $$ \begin{equation} \varepsilon_{cc} = \varepsilon_{c0} \left( 1 + 2.0 \lambda_f + 2.5 \lambda_h \right) \end{equation} $$

(12) $$ \begin{equation} \lambda_f = \frac{\kappa_a \rho_f f_f}{f_c'} \end{equation} $$

(13) $$ \begin{equation} \lambda_h = \frac{\rho_{st} f_{yt}}{f_c'} \end{equation} $$

(14) $$ \begin{equation} \lambda_1 = \frac{\rho_g f_f}{f_c'} \end{equation} $$

where $$ f_{cc} $$ and $$ \varepsilon_{cc} $$ are the peak compressive stress and strain of FRP-confined concrete, respectively; $$ \kappa_a $$ is the shape factor, defined as the ratio of the effective confined area to the total cross-sectional area; $$ \lambda_f $$ is the characteristic value of the FRP confinement; $$ \lambda_h $$ is the characteristic value of the stirrup confinement; $$ \lambda_l $$ is the characteristic value of the longitudinal reinforcement; $$ \rho_g $$ is the longitudinal reinforcement ratio, excluding the radius of the chamfer; $$ \rho_{st} $$ is the volumetric stirrup reinforcement ratio; $$ \rho_f $$ is the volumetric reinforcement ratio provided by transverse FRP; $$ f_f $$ is the ultimate strength of the FRP, and $$ f_{yt} $$ is the yield strength of the stirrups.

Sections and elements

The finite element of structure is established using the distributed plasticity nonlinear beam-column elements in OpenSEES, as shown in Figure 5. For unreinforced beams and columns, only one element is needed. They are simulated using a fiber section element. For FRP-reinforced RC beams and columns, each component is simulated with three displacement-based nonlinear beam-column elements. These elements represent three regions, including the two ends, which are the plastic hinge reinforced zones and the middle unreinforced zone. Each of these regions is simulated with one nonlinear beam-column element considering distributed plasticity. The number of integration points along the element is 10. Zero-length section elements are used at the roots of beams and columns, and Bond-SP01 material is adopted to consider bond-slip between reinforcement steel and concrete. The element for the unreinforced middle zone is modeled using a fiber section. The number of fibers over a geometric cross-section is 400. The elements for the reinforced zones use the stress-strain constitutive model of FRP-RC. Therefore, the OpenSEES material named FRPConfinedConcrete ^[3] material is used. The hysteresis rule of FRPConfinedConcrete material in this paper could accurately reflect the stress-strain relationship and the deterioration of concrete under cyclic compression. It is determined based on cyclic compression tests of FRP-confined concrete columns. This is a significant improvement when compared to Concrete01 material models, which only modified the backbone curve for monotonic compression, while the hysteresis rule is developed by the test of normal concrete. For a detailed comparison, see the literature ^[3]. It should be noted that the unloading curve is simplified using a two-segment approximation, while the rest of the envelope curve, residual strain, and other aspects are consistent with the relationship described in Section "Constitutive models of material". During the dynamic analysis, the base of the structure is considered rigid, and the convergence tolerance is set to $$ 1.0 \times 10^{-8} $$. In addition, since this paper is concerned with the seismic performance of the structure as a whole, local geometric imperfection is not considered.

Figure 5. Finite element model.

Model validation

The modeling method described above is validated using specimens S1H2C0N2 and S1H1C4N2 from reference ^[22]. Both specimens are column components with an axial compression ratio (ACR) of 0.45. The S1H2C0N2 specimen is a low-ductility specimen, while the S1H1C4N2 has the same design details as the S1H2C0N2. The difference is S1H1C4N2 specimen is reinforced with four layers of CFRP compared to S1H2C0N2. Figure 6 compares the experimental result with the predicted result, showing that the simulation method employed in this study effectively captures the degradation characteristics of low-ductility columns and the seismic performance improvements brought by FRP retrofitting.

Figure 6. Test and predict result for low-ductility column and FRP reinforced low-ductility column. (A) Low-ductility column; (B) Low-ductility column reinforced with CFRP.

Retrofitting scheme

The Pushover analysis of the low-ductility structure model shows that the maximum inter-story displacement occurs in the lower-middle floors, with a noticeable stiffness discontinuity in the middle floors. Therefore, the bottom two floors are identified as the weak stories under seismic loads. According to existing standards^[23,24], the components in weak stories are reinforced with four layers of FRP confinement. For the non-weak story components, two layers of FRP confinement are applied to prevent the stiffness of the bottom two floors from becoming excessively strong compared to the upper floors after FRP retrofitting^[23]. Considering the moment above, the following FRP retrofitting schemes are proposed, as illustrated in Figure 7. In Figure 7, the bottom two floors (1-2 floors) have columns reinforced at their ends by wrapping FRP. The retrofitting height is 1.5 times the column cross-section height. The beam ends are wrapped with four layers of FRP laterally and two layers of FRP longitudinally, with a retrofitting length of two times the column cross-section height. The middle span of the beams is full-length reinforced. The top two floors (3-4 floors) are reinforced in the same positions as the 1 and 2 floors but with half the number of FRP layers. This study assumes the use of a unidirectional CFRP fabric with a weight of 300 g/m² for retrofitting the structure. Each layer has a thickness of 0.167 mm, an elastic modulus of 244 GPa, an ultimate tensile strength of 4, 340 MPa, and an ultimate tensile strain of 0.018. For ease of description later, the unreinforced low-ductility structure is named Case-1, while the FRP-reinforced low-ductility structure is named Case-2.

Figure 7. Retrofitting scheme. (A) Case-1; (B) Case-2.

Assessment framework

The FEMA P-58 methodology, combined with the PACT software, is used to conduct a time-based performance assessment of low-ductility structure (Case-1) and FRP-reinforced low-ductility structure (Case-2). This assessment method evaluates the seismic performance indicators of the building structure under all possible seismic intensities over a specified period (50 years in this study) to determine the annual exceedance frequency and the annual average value of the building performance indicators during its service life.

Building replacement model

Structural occupancy and population density model

According to GB 50189-2015^[30], a model is developed in this study to reflect the variation of office occupancy over time. This model considers China's population density and office habits, as shown in Figure 8.

Figure 8. Population density model for office building in China.

Component and structural fragility functions

In the process of structural resilience analysis, it is essential to determine the fragility functions of both components and structures. The fragility functions for components can be defined by specifying the repair methods and cost information corresponding to different damage states. This allows for assessing the direct economic losses needed for repairing earthquake-damaged buildings and the indirect economic losses resulting from using different retrofitting methods. Meanwhile, the majority of casualties result from partial or complete building collapse. To estimate potential casualties, defining the probability of structural collapse as a function of ground motion intensity is essential. These probabilities are typically represented through collapse fragility functions.

Component fragility functions

(1) Beam-column components

The PACT software provides an extensive database of fragility functions for structural components, including structural components, non-structural components, and internal financial units. However, the fragility functions of the structural components are established from statistical test information on ductile components. Due to the significant difference in the failure characteristics and the deformation capacity corresponding to the performance points of ductile and non-ductile components, using the fragility functions of ductile structural components for low-ductility RC frame structures resilience assessment would underestimate the deformation damage and structural losses caused by earthquakes. According to existing research^[22], insufficient stirrup configuration is the primary cause of low ductility for RC components. Current standards^[19,20] specify that the volumetric stirrup ratio in confined regions of column components must exceed 0.4% to ensure the sufficient ductility of RC components under earthquake actions. Therefore, components with a volumetric stirrup ratio of less than 0.4% are defined as low-ductility components. A database containing 46 low-ductility components and 113 FRP-reinforced low-ductility components has been established to predict structural losses more accurately. Detailed information can be seen in reference ^[31].

According to reference ^[32], the damage states of RC columns are categorized into six levels, including slight damage, minor damage, moderate damage, slight severe damage, severe damage, and collapse. These damage states are labeled as DS1 to DS6, respectively. The damage phenomena and repair methods corresponding to each state are detailed in reference ^[33,34]. The performance points are obtained from the column's backbone curve, as shown in Figure 9. Points B and C represent the nominal yield point ($$ M_y=0.8M_p $$) and peak point of the backbone curve, respectively. Points D and E correspond to a 15% and 25% reduction from the peak moment, respectively. IO (Immediate Occupancy) is defined as the midpoint of the displacement angle between points B and C, LS (Life Safety) is the midpoint between points C and D, and the state beyond point E is defined as collapse.

Figure 9. Definition of performance points.

Since there is significant variability in the fragility function of the components at different ACRs, the fragility functions of the components are established by considering the factor of ACR. With an ACR of 0.3 as the boundary, specimens with ACR less than or equal to 0.3 are classified as low-ACR specimens, while those with ACR greater than 0.3 are classified as high-ACR specimens. Based on the ACR and FRP retrofitting characteristics, all specimens in the database are divided into four groups, including low-ACR low-ductility (LAND) group, high-ACR low-ductility (HAND) group, low-ACR FRP-reinforced low-ductility (LAFND) group, and high-ACR FRP-reinforced low-ductility (HAFND) group.

The characteristic points of the backbone curves for each group of specimens are extracted in Figure 9. After removing outliers using the Peirce criterion, the parameters of the fragility functions are calculated using

(18) $$ \begin{equation} \theta_i = e^{\left(\frac{1}{M} \sum\nolimits_{j=1}^{M} \ln(d_j)\right)} \end{equation} $$

(19) $$ \begin{equation} \beta_i = \sqrt{\beta_{r,i}^2 + \beta_{u,i}^2} \end{equation} $$

where $$ \theta_i $$ and $$ \beta_i $$ are the median value and logarithmic standard deviation of the fragility function.

The parameters of the fragility functions for low-ductility RC columns and FRP-reinforced low-ductility RC columns with different ACRs are shown in Table 2.

Table 2

Fragility functions for low-ductility RC columns before and after FRP retrofitting

Component type	Damage state	Group	$$ \boldsymbol{\theta} $$ (%)	$$ \boldsymbol{\beta} $$	Group	$$ \boldsymbol{\theta} $$ (%)	$$ \boldsymbol{\beta} $$
Low-ductility RC columns B1041.000a(b)	DS1	LAND	0.574	0.533	HAND	0.413	0.778
	DS2		0.971	0.549		0.732	0.628
	DS3		1.353	0.574		1.039	0.597
	DS4		1.634	0.594		1.31	0.486
	DS5		1.923	0.619		1.636	0.45
	DS6		1.937	0.58		1.596	0.381
FRP-reinforced low-ductility RC columns B1041.010a(b)	DS1	LAFND	0.682	0.686	HAFND	0.622	0.838
	DS2		1.431	0.78		0.961	0.787
	DS3		2.193	0.825		1.445	0.811
	DS4		2.897	0.792		2.005	0.758
	DS5		3.954	0.7		2.523	0.744
	DS6		4.498	0.659		3.304	0.815

Table 2 shows that the median value $$ \theta $$ of the fragility functions for low ACR columns is always higher than that for high ACR columns. This indicates that low ACR columns have significantly greater deformation capacity than high ACR columns. Furthermore, comparing the low-ductility columns before and after FRP retrofitting under the same damage states, it is evident that FRP reinforced substantially enhances the deformation capacity of the components. Additionally, it should be noted that for components reinforced with FRP, it is only worthwhile to be repaired if it is in a state of minor damage. The repair cost will exceed the replacement cost if the damage is severe. This study does not consider repair in such cases and will directly proceed with component replacement. Therefore, the low-ductility columns reinforced with FRP are irreparable in damage states DS4, DS5, and DS6. Consequently, the repair costs, repair time, and carbon emissions of these states are accounted for as complete replacements.

(2) Component categories and fragility function numbers

Referring to the fragility categories of components defined in PACT, the number of components in the four-story structure is calculated based on different directions and floors. The results are presented in Table 3.

Table 3

Component categories and fragility function numbers

Direction	Component type	Fragility number	Unit	Numbers of floor
				Floor 1	Floor 2-4	Roof
X direction	Structural components
	Beam-column components (select one)	B1041.000 (low-ductility)	pcs	24	24	—
		B1041.010 (FRP-reinforced)	pcs	24	24	—
	Non-structural components
	Non-structural exterior walls	B1052.001	9.29 m2	15.5	13.56	—
	Non-structural interior walls	C1011.001a	120.77 m2	2.38	2.09	—
	Exterior wall finishes	C3011.002a	83.61 m2	1.72	1.51	—
	Interior wall finishes	C3011.001a	83.61 m2	6.89	6.03	—
Y direction	Structural components
	Beam-column components (select one)	B1041.000 (low-ductility)	pcs	24	24	—
		B1041.010 (FRP-reinforced)	pcs	24	24	—
	Non-structural components
	Non-structural exterior walls	B1052.001	9.29 m2	31	27.13	—
	Non-structural interior walls	C1011.001a	120.77 m2	2.38	2.09	—
	Exterior wall finishes	C3011.002a	83.61 m2	3.44	3.01	—
	Interior wall finishes	C3011.001a	83.61 m2	6.89	6.03	—
Non-directional	Other components
	Ceiling	C3032.003b	55.74 m2	8.07	8.07	—
	HVAC pipes	D3041.011a	30.48 m	6.89	6.89	—
	Firefighting pipes	D4011.021a	30.48 m	6.89	6.89	—
	Roof	B3011.013	9.29 m2	—	—	48.44
	Chiller	D3031.013h	pcs	—	—	1
	Air handling unit	D3052.013k	pcs	—	—	1

Collapse fragility curve and casualties

The seismic fragility analysis of the structure before and after FRP retrofitting is conducted using 22 sets of far-field ground motions recommended by FEMA P695 ^[35]. For each set of ground motions, a random direction is selected as the seismic input. The selected ground motion response spectra are shown in Figure 10. Subsequently, incremental dynamic analysis (IDA) is performed to assess the collapse fragility of the structure. According to the method defined in FEMA 350 ^[36], a structure is considered to have collapsed when the tangent stiffness of the IDA curve drops to 20% of the initial elastic stiffness or when the maximum inter-story drift exceeds the limit of 0.1. According to reference ^[37], the collapse points are fitted to a lognormal distribution using the maximum likelihood estimation method.

Figure 10. Response spectrum and median value.

The collapse fragility curves for Case-1 (unreinforced) and Case-2 (FRP-reinforced) structures are shown in Figure 11. The median and logarithmic standard deviation of the fragility function for Case-1 are 0.744 and 0.602, respectively, while for Case-2, they are 0.828 and 0.541, respectively. This indicates that FRP retrofitting reduces the probability of structural collapse. Assuming that the building collapses with the sidesway failure, the fatality rate and injury rate for the collapse of RC frame structures are assumed to be 15% and 85%, respectively according to reference^[38].

Figure 11. Collapse fragility curves.

Residual drift

Residual drift fragility represents the probability that a structure can be repaired when residual inter-story drift is present after an earthquake. If the building cannot be repaired, the repair costs, repair time, carbon emissions, and energy consumption are assumed to equal those of complete replacement. According to the residual drift fragility function provided in FEMA P-58^[13], the median value is 1%, and the standard deviation is 0.3. The calculation method for structural residual drift is given by:

where $$ \Delta $$ is the inter-story drift obtained through numerical simulation; $$ \Delta_y $$ is the inter-story drift calculated at the yield point of the structure, which corresponds to the maximum inter-story drift at the yield point on the pushover curve; $$ \Delta_r $$ is the overall residual drift of the structure.

Seismic hazard curve

Using the seismic hazard analysis function proposed by Cornell ^[39], the hazard function $$ H(x) $$ can be approximated by an exponential form, as

where $$ H(x) $$ is the annual exceedance probability of a certain intensity earthquake occurring at the design site; $$ k_0 $$ and $$ k $$ are parameters related to the design site and other factors.

According to reference ^[40] and Chinese seismic code ^[19], $$ k_0 $$ and $$ k $$ can be calculated as follows:

where $$ \lambda_D $$ and $$ \lambda_M $$ are the annual exceedance probabilities for the design-based earthquake (DBE) and the MCE, respectively. These probabilities can be calculated based on their return periods, $$ \lambda_D = {1}/{475} = 0.0021 $$ and $$ \lambda_M = {1}/{2475} = 0.0004 $$; $$ IM_D $$ and $$ IM_M $$ are the seismic intensity parameters corresponding to the DBE and MCE in the seismic code, with spectral acceleration $$ S_{\text{a}} $$ being used in this study. Substituting into Equations (22) and (23) yields $$ k = 1.84 $$ and $$ k_0 = 1.35 \times 10^{-5} $$.

According to reference ^[13], eight seismic intensities are selected for structural resilience assessment. The 22 ground motions above are amplitude-modulated to these eight seismic intensities, followed by the structural resilience assessment based on the dynamic time-history analysis data. The selected seismic intensities and the calculated annual exceedance rate are shown in Table 4. The seismic hazard curve for the structure's site is presented in Figure 12.

Figure 12. Seismic hazard curve.

Retrofitting impact factor (RIF)

This study introduces the retrofitting impact factor (RIF) as an indicator to quantify the extent to which FRP retrofitting affects the seismic resilience of low-ductility structures, which is calculated by:

where $$ D_{ND} $$ represents the resilience indicators (replacement costs, replacement time, casualties, $$ etc $$.) for the low-ductility structure; $$ D_{ND-C} $$ indicates the resilience indicators (replacement costs, replacement time, casualties, $$ etc $$.) for the low-ductility structure after FRP retrofitting.

Repair cost

Based on 1000 Monte Carlo simulations, the average repair costs for the case structure under eight different seismic intensities are obtained, as shown in Table 5. The table indicates that FRP retrofitting can reduce the repair costs by up to 9.26% at Intensity 1. However, as the seismic intensity increases, the effectiveness of FRP retrofitting in reducing repair costs gradually decreases. This is because, at very high seismic intensities, although the collapse probability of Case-2 is lower than that of Case-1, a significant number of structural components in Case-2 reach or exceed DS4. As a result, most simulation results indicate a complete replacement of the structure. Additionally, the repair cost calculations for Case-2 include the costs of FRP retrofitting. This results in a negative $$ \delta $$ in some cases, indicating that FRP retrofitting is ineffective.

Based on the classification of component fragility, the structure's repair costs are divided into three categories, including exterior components, interior components, and service systems. The proportion of total repair costs for these categories under eight seismic intensities is shown in Figure 13. It is clear that the proportion of losses attributed to interior components and service systems grows with seismic intensity for both Case-1 and Case-2. This is due to the increased damage to non-structural components such as partition walls, pipes, and ceilings, as well as service systems, resulting in a higher proportion of their repair costs in the total repair costs. This implies that FRP seismic retrofitting does not significantly decrease the damage levels of interior components and service systems.

Figure 13. Repair cost for three categories. (A) Case-1; (B) Case-2.

By combining the site's seismic hazard, the time-based method is further used to assess the seismic resilience of the structure. The annual repair cost probability distribution is shown in Figure 14. It can be observed that the average annual repair costs before and after FRP retrofitting are ＄1, 520.81 and ＄1, 439.84, respectively. This indicates that FRP retrofitting can reduce the average annual repair costs of low-ductility structures by 5.32%.

Figure 14. Annual repair cost probability distribution. (A) Case-1; (B) Case-2.

Repair time

Based on Monte Carlo simulations and considering the total replacement time of the structure, the average repair times for the case structure under eight different seismic intensities are obtained, as shown in Table 6. From Table 6, it is evident that FRP retrofitting consistently reduces the repair time across all intensity levels. Notably, for the Case-2 structure at intensity level 2, the repair time is reduced by up to 8.07%. However, at very high seismic intensities, many structural components in Case-2 reach or exceed DS4, rendering them irreparable and causing the repair time to approach the total replacement time. In such cases, FRP retrofitting becomes ineffective in reducing repair time.

The distribution of repair time across different floors of the case structures under eight seismic intensities is shown in Figure 15. For both case structures, the repair time for the first floor is the highest, accounting for approximately 30%, while the repair times for the other three floors are relatively similar.

Figure 15. Repair time for different floors. (A) Case-1; (B) Case-2.

The time-based evaluation method is further used to assess the seismic resilience of the structure, resulting in the annual repair time probability distribution shown in Figure 16. The calculations reveal that the average annual repair time for Case-1 is 0.2393 days, whereas for Case-2, it is 0.2228 days. This indicates that FRP retrofitting reduces the average annual repair time of low-ductility structures by 6.90%.

Figure 16. Annual repair time probability distribution. (A) Case-1; (B) Case-2.

Casualties

The average numbers of fatalities and injuries under eight different seismic intensities are shown in Tables 7 and 8. Tables 7 and 8 show that as the intensity level rises, the number of fatalities and injuries in the structure gradually increases. Compared to the low-ductility structure, the number of fatalities and injuries in the FRP-reinforced structure is significantly reduced. Comparing the two case structures at different intensities, the Case-1 structure shows potential fatalities and injuries at intensity levels 1 and 2. In contrast, the Case-2 structure exhibits fatalities and injuries only at intensity levels 2 and 6, respectively. This indicates that FRP retrofitting can significantly reduce the casualty rate for occupants. Furthermore, it can be observed that as the intensity level increases, the value of $$ \delta $$ gradually decreases. At intensity level 8, $$ \delta $$ decreases to approximately 50%. This suggests that the benefits of FRP retrofitting in terms of casualty reduction diminish at higher intensity levels. However, it is noteworthy that as the intensity level grows, the reduction in casualties for Case-2 compared to Case-1 progressively increases.

Figures 17 and 18 present the probability distributions of the annual average fatalities and injuries for the example structure. From these figures, it can be seen that the estimated annual average number of fatalities for the Case-1 structure is 0.0002, while after FRP retrofitting, the annual average number of fatalities decreases to nearly zero. Additionally, comparing the annual average number of injuries for the two structures, it is evident that FRP retrofitting significantly reduces the probability of injuries, with the annual average number of injuries decreasing from 0.0033 in the low-ductility structure to 0.0014 in the FRP-reinforced structure. The reduction reaches 57.6%. These results highlight the effectiveness of FRP retrofitting in enhancing the seismic resilience of structures and reducing casualties, particularly in terms of significantly lowering the annual average number of injuries and fatalities.

Figure 17. Annual number of fatalities probability distribution. (A) Case-1; (B) Case-2.

Figure 18. Annual number of injuries probability distribution. (A) Case-1; (B) Case-2.

Carbon emissions

The average carbon emissions for Case-1 and Case-2 under eight different seismic intensities are shown in Table 9. The table shows that Case-2 can reduce structural carbon emissions by 10.43% at intensity level 1 compared to Case-1. Additionally, at intensity levels 2, 3, 5, and 6, FRP retrofitting also reduces the carbon emissions of the structure. At higher seismic intensities, numerous components in the Case-2 structure experience severe damage, leading to repair costs that surpass the replacement cost for much of the structure. Additionally, since the carbon emissions associated with Case-2 include the FRP retrofitting process, this results in a negative reduction efficiency ($$ \delta $$) at certain intensity levels.

The proportion of carbon emissions from the replacement of exterior components, interior components, and service systems for the two cases is shown in Figure 19. For the Case-1 structure, the proportion of carbon emissions from interior components and service systems increases with seismic intensity, but the growth trend is not significant. Interior components account for approximately 52%, while service systems reach up to 4%. Compared to Case 1, the carbon emissions required to repair exterior components in the Case-2 structure are higher, exceeding 60%. Conversely, the carbon emissions for repairing interior components in Case-2 are significantly lower, ranging from 30% to 40%. This is because the carbon emissions from steel used in repairs are much higher than those from concrete and FRP fabric. Meanwhile, a large number of non-structural components and wall finishes in interior components do not require a significant amount of steel. Therefore, the carbon emissions during structural repairs are primarily due to exterior components.

Figure 19. Carbon emissions for three categories. (A) Case-1; (B) Case-2.

Considering seismic hazard, the probability distribution of the annual carbon emissions of the structure is shown in Figure 20. The calculations show that the average annual carbon emissions for the Case-1 structure are 3, 050.17 kg, while for Case-2, it is 2, 831.89 kg. This indicates that FRP retrofitting can reduce the average annual carbon emissions of low-ductility structures by 7.16%.

Figure 20. Annual number of fatalities probability distribution. (A) Case-1; (B) Case-2.

Authors' contributions

Methodology, conceptualization, writing - review & editing, funding acquisition: Yu X

Data curation, visualization, writing - original draft: Li Z

Data curation, visualization, writing - review & editing: Jiang Z

Methodology, investigation, writing - original draft, supervision, funding acquisition: Dai K

Availability of data and materials

Data is available upon request from the authors.

Financial support and sponsorship

The research in this paper has received financial support from the Natural Science Foundation of China (52278492, 52408564), the China Postdoctoral Science Foundation (project number: 2023M733247), the Natural Science Foundation of Henan Province (project number: 232300421322), and the Central Finance of Guangxi Zhuang Autonomous Region Guides Local Science and Technology Development Funds (project number: 2023ZYZX1003).

Conflicts of interest

Yu X is a Youth Editorial Board Member of the journal Disaster Prevention and Resilience, while the other authors have declared that they have no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2024.