Time-Dependent Fragility Functions and Post-Earthquake Residual Seismic Performance for Existing Steel Frame Columns in Offshore Atmospheric Environment

Abstract

This paper evaluates the time-dependent fragility and post-earthquake residual seismic performance of existing steel frame columns in offshore atmospheric environments. Based on experimental research, the seismic failure mechanism and deterioration laws of the seismic behavior of corroded steel frame columns were revealed. A finite element analysis (FEA) method for steel frame columns, which considers corrosion damage and ductile metal damage criteria, is developed and validated. A parametric analysis in terms of service age and design parameters is conducted. Considering the impact of environmental erosion and aging, a classification criterion for damage states for existing steel frame columns is proposed, and the theoretical characterization of each damage state is provided based on the moment-rotation skeleton curves. Based on the test and numerical analysis results, probability distributions of the fragility function parameters (median and logarithmic standard deviation) are constructed. The evolution laws of the fragility parameters with increasing service age under each damage state are determined, and a time-dependent fragility model for existing steel frame columns in offshore atmospheric environments is presented through regression analysis. At a drift ratio of 4%, the probability of complete damage to columns with 40, 50, 60, and 70-year service ages increased by 18.1%, 45.3%, 79.2%, and 124.5%, respectively, compared with columns within a 30-year service age. Based on the developed FEA models and the damage class of existing columns, the influence of characteristic variables (service age, design parameters, and damage level) on the residual seismic capacity of earthquake-damaged columns, namely the seismic resistance that can be maintained even after suffering earthquake damage, is revealed. Using the particle swarm optimization back-propagation neural network (PSO-BPNN) model, nonlinear mapping relationships between the characteristic variables and residual seismic capacity are constructed, thereby proposing a residual seismic performance evaluation model for existing multi-aged steel frame columns in an offshore atmospheric environment. Combined with the damage probability matrix of the time-dependent fragility, the expected values of the residual seismic capacity of existing multi-aged steel frame columns at a given drift ratio are obtained directly in a probabilistic sense. The results of this study lay the foundation for resistance to sequential earthquakes and post-earthquake functional recovery and reconstruction, and provide theoretical support for the full life-cycle seismic resilience assessment of existing steel structures in earthquake-prone areas.

1. Introduction

Strong earthquakes have caused serious building damage and casualties, resulting in major economic losses that have seriously affected normal social development. In recent years, the probability of structural collapse and casualties in earthquakes has significantly decreased owing to the continuous development of the Performance-Based Earthquake Engineering (PBEE) concept. However, most non-collapsed structures have suffered serious damage, which is either difficult to repair or has no repair value, creating “standing ruins” [1]. In particular, with the demand for green and sustainable development in urban and rural areas, challenges remain, including imperfect emergency rescue systems and difficulties in post-earthquake reconstruction and restoration of normal daily life. Therefore, researchers have proposed the concept of “seismic resilience,” which characterizes the ability of structures or cities to resist seismic disturbance and quickly recover after earthquakes. It is also considered the core of next-generation PBEE [2,3].

The component fragility model is a key part of the structural seismic resilience assessment, which represents the conditional probability of components reaching or exceeding a specific damage state (DS) under a given engineering demand parameter (EDP), thus quantifying the seismic performance [4]. Simultaneously, by defining the repair method and cost information associated with different damage states, the subsequent reinforcement scheme and repair cost for earthquake-damaged structures can be predicted, thereby providing support for the seismic resilience evaluation of the structures. FEMA-P58 [5] systematically established the fragility functions of key components of structural systems, such as concrete, steel, masonry, and wooden structures. Goksu [6] proposed fragility functions for reinforced concrete columns incorporating recycled aggregates by compiling test data from the literature. Mohammadgholibeyki [7] developed fragility functions for steel-plate concrete composite shear walls using available test data. Wang [8] determined damage indices and performed fragility assessments for coupled low-yield-point steel plate shear walls. Haghpanah [9] re-evaluated the seismic fragility functions of cold-formed steel framed shear walls using the FEMA P-58 methodology.

However, most previous studies neglected the impact of the construction history, service environment, and age on the fragility of existing structural components without the scientific quantification of the actual seismic resistance. Existing structural components affected by environmental corrosion and aging undergo significant deterioration in their seismic performance [10,11]. Therefore, Mirzaeefard [12] developed time-dependent fragility curves for corroded pile-supported wharves by updating the limit states using incremental dynamic analysis. Zhao [13] developed fragility functions for corroded welded joints and proposed reduction factors for seismic capacity under the combined effects of corrosion and earthquake damage. With the rapid development of machine learning techniques, Zhang [14] evaluated the life-cycle seismic performance of offshore small-to-medium-span bridges based on a Long Short-Term Memory (LSTM) neural network and the Shapley interpretation method. Xu [15] proposed a machine learning-aided rapid estimation method for failure mode identification and multilevel drift ratio capacity of corroded circular RC columns. Fu [16] collected 158 shear tests of CRC beams and established a predictive time-dependent shear strength model for corroded reinforced concrete beams based on a gradient boosting regression tree. However, research on the machine learning-based time-dependent performance of steel components remains absent. In addition, owing to the differences between the Chinese and international codes, the component fragility functions established by methodologies such as FEMA P-58 are not suitable for direct use in the assessment of the seismic losses and resilience of structures in China [17].

The prediction of the residual seismic behavior of earthquake-damaged components is also an important part of seismic resilience assessment and an important basis for the functional restoration and reconstruction of post-earthquake structures, which is of great significance for improving disaster prevention and mitigation capabilities and the scientific decision-making level [18]. The JBPDA Guideline [19] and FEMA 306 [20] adopted a reduction factor η to determine the residual seismic capacity of the structural components. Alwashali [21] studied the influence of pre-damage levels on the residual seismic performance of RC shear walls using cyclic loading tests. Harrington [22] performed a seismic performance evaluation of RC frames with retrofitted columns designed to meet the ASCE 41damage levels (with CP, LS, and IO states). Khanmohammadi [23] studied the residual capacity of mainshock-damaged prestressed bridges subjected to vertical earthquake ground motions. Chiu [24] experimentally quantified the post-earthquake residual seismic capacity of RC columns with flexural, flexural-shear, and shear failure modes. Zhang [25] developed a pattern recognition approach to quantitatively assess the residual structural capacity of earthquake-damaged tall buildings by using sequential nonlinear response history analyses. Cavaleri [26] investigated the residual out-of-plane stiffness and strength of infills damaged by in-plane cyclic loads using an extended numerical experiment. Using a machine learning method, Asgarkhani [27] proposed a reliable prediction model for the residual seismic behavior of steel moment-resisting frames, considering the soil-structure interaction.

However, previous evaluations of residual seismic behavior are mainly based on theory, omitting the association between damage characteristics and residual seismic behavior, the uncertainty of the structure itself (such as geometric features, material properties, and service age), and loading conditions. This restricts the effective implementation of post-earthquake performance evaluation and safety assessments.

Table 1. Comparisons of previous vs. current studies.

Related Studies	Novelty of the Studies
	Component Seismic Fragility (Yes or No)	Post-Earthquake Residual Seismic Behavior (Yes or No)	Environmental Corrosion and Aging Impact (Yes or No)	Machine Learning Techniques (Yes or No)	Relationships Between the Damage Characteristics and Residual Seismic Capacity (Yes or No)	Structural Variable Parameters (Yes or No)
Literature [5]	Yes	No	No	No	No	Yes
Literature [6,7,8,9]	Yes	No	No	No	No	Yes
Literature [12,13]	Yes	No	Yes	No	No	Yes
Literature [14,15,16]	Yes	No	Yes	Yes	No	Yes
Literature [19,20]	No	Yes	No	No	Yes	Yes
Literature [21,22,23,24,26]	No	Yes	No	No	Yes	No
Literature [25,27]	No	Yes	No	No	Yes	Yes
This study	Yes	Yes	Yes	Yes	Yes	Yes

provides a brief summary of the comparisons between the previous and current research.

This study begins with a review of experimental studies on the seismic behavior of corroded steel frame columns. Chloride-accelerated corrosion and low-cycle loading tests were conducted on six steel frame columns, and the seismic failure mechanism and deterioration laws of the seismic behavior are revealed. A finite element analysis (FEA) method for corroded steel frame columns is then developed. Subsequently, a classification criterion and theoretical characterization of the damage states is proposed, thus establishing a time-dependent fragility model for existing steel frame columns in offshore atmospheric environments. Furthermore, using the swarm optimization back-propagation neural network (PSO-BPNN) model, a quantitative correlation between the characteristic variables (service age, design parameters, and damage level) and residual seismic capacity is constructed. Finally, by combining the time-dependent fragility and residual seismic behavior evaluation modes, the expected value of the residual seismic capacity for existing multi-aged steel frame columns at a given drift ratio is obtained in a probabilistic sense. The research design is illustrated in Figure 1.

2. Low-Cycle Loading Tests

To investigate the seismic failure mechanism and deterioration laws of the seismic performance of corroded steel frame columns subjected to chloride erosion, six steel frame column specimens were designed based on relevant design codes [28,29] and engineering examples. All specimens were fabricated using Q235B hot-rolled H-shaped steel with a cross-sectional specification of HW 250 × 250 × 9 × 14 mm, which was produced by Shaanxi Iron and Steel Group Co., Ltd. in China. Figure 2 shows the geometries and details of the specimens. Referring to the ISO 9227 standard [30], accelerated corrosion in a chloride environment was performed on all the column specimens using the ZHT/W2300 environment simulation system, as shown in Figure 3. Subsequently, low-cycle loading tests were conducted on the corroded column specimens. The test setup and loading protocols are shown in Figure 4. The test parameters include the corrosion level (quantified by the mass loss rate η) and axial compression ratio n (see

Table 2. Test parameters of steel frame column specimens.

Specimen No.	Section Specification (mm)	Axial Compression Ratio n	Accelerated Corrosion Time t (h)	Mass Loss Rate η (%)
SFC-1	HW 250 × 250 × 9 × 14	0.3	0	0.00
SFC-2	HW 250 × 250 × 9 × 14	0.3	960	3.06
SFC-3	HW 250 × 250 × 9 × 14	0.3	1920	5.33
SFC-4	HW 250 × 250 × 9 × 14	0.3	2880	8.02
SFC-5	HW 250 × 250 × 9 × 14	0.2	1920	5.33
SFC-6	HW 250 × 250 × 9 × 14	0.4	1920	5.33

Notes: n = N/Af_y, where N is the design axial load, A is the section area, f_y is the yield strength. η = (m₀−m₁)/m₀, where m₀ and m₁ are the specimen mass before and after corrosion respectively.

). Further details of the tests can be found in the literature. [31].

The damage processes of test columns SFC-1 to SFC-6 were similar. At the initial loading, all the columns were in the elastic stage. When loaded to a drift ratio level of 1.5–2%, the columns yielded and entered the elastic-plastic stage, owing to the strain hardening effect, in which visible rotational deformation occurred at the column end, accompanied by the surface rust layer peeling off. However, the deformation recovered after unloading owing to the low level of plastic development. At the 2~3% drift ratio level, slight local buckling was observed at the column end flanges, as well as residual deformation after unloading, owing to the relatively high level of plastic development. At the 3–4% drift ratio level, the horizontal load reached its maximum and then began to decrease, indicating that the columns entered the elastic-plastic softening stage. At this stage, local buckling of the flange was evident, and the web plate warped, indicating the formation of a plastic hinge at the column’s end. When the drift ratio reached 5%, the plastic hinge at the column end was fully developed, and the bearing capacity dropped below 30% of the maximum, indicating column failure.

The difference lies in the fact that with an increase in the corrosion level or axial compression ratio, the drift ratio levels associated with steel column yielding, plate local buckling (i.e., plate plastic deformation), and maximum bearing capacity decrease.

Table 3. Main damage phenomena of specimens.

Specimen No.	Drift Ratio Level of Flange Local Buckling	Drift Ratio Level of Web Local Buckling	Drift Ratio Level of Maximum Load	Drift Ratio Level of Failure	Plastic Hinge Length (mm)	Failure Mode of Cross Section	Damage State
							DS1	DS2	DS3	DS4
SFC-1	3% (2)	4% (2)	4% (1)	5% (2)	340		1.60%	2.62%	3.13%	4.79%
SFC-2	3% (2)	4% (1)	3% (2)	5% (1)	325		1.54%	2.51%	3.00%	4.49%
SFC-3	3% (1)	4% (1)	3% (2)	5% (1)	315		1.47%	2.48%	2.99%	3.96%
SFC-4	2% (2)	3% (1)	3% (1)	4% (2)	330		1.37%	2.36%	2.86%	3.67%
SFC-5	3% (2)	4% (2)	4% (1)	5% (2)	330		1.64%	3.21%	4.00%	4.99%
SFC-6	2% (2)	3% (2)	3% (1)	4% (2)	280		1.42%	2.43%	2.94%	3.82%

summarizes the main damage phenomena for each specimen, and Figure 5 shows the final failure mode.

Figure 6 shows the bending moment-rotation (M − θ) hysteresis curves of column specimens, where M and θ can be expressed as Equations (1) and (2), respectively.

$$M=P\cdot L+N\cdot \Delta$$

(1)

$$\theta =\Delta /L$$

(2)

where P is the horizontal load; N is the axial pressure; Δ is the horizontal displacement; L is the effective height of the column.

As shown in Figure 6a–d, as the corrosion level increased, the bearing capacity gradually decreased, the degradation of strength and stiffness intensified, and the ductility and energy dissipation declined, indicating that the seismic performance deteriorated. This is primarily because corrosion damage weakens the column’s cross-sectional size and causes degradation of the steel’s mechanical properties. In addition, as shown in Figure 6c,e,f, steel frame columns with high axial compression had a relatively low bearing capacity, small hysteresis loop area, significant degradation in strength and stiffness, and poor ductility and energy dissipation. This is mainly due to the prominent second-order (P − Δ) effect, which accelerates local plate buckling. This indicates that in the seismic design of steel frame columns, the axial compression ratio should be strictly controlled, and full life-cycle anti-corrosion measures should be implemented.

3. Finite Element Analysis

Using ABAQUS 6.13 software, a three-dimensional finite element model considering the corrosion damage and ductile metal damage criteria was established to numerically simulate the test steel frame columns.

Corrosion in steel is primarily divided into uniform and pitting corrosion. In practical situations, owing to differences in the external corrosion conditions and the material itself, the corrosion morphology of steel has high randomness, without a strict distinction between uniform and pitting corrosion, and is mostly a superposition of the two forms, as shown in Figure 7. Uniform corrosion only causes a reduction in plate thickness but does not lead to material property degradation, and the mechanical performance of the components can be assessed by the residual thickness. However, pitting corrosion can cause stress concentration, resulting in the degradation of the material strength and ductility, making it difficult to predict the performance of components through the thickness reduction. Therefore, this study divided corrosion damage into two parts: uniform corrosion and pitting corrosion, to comprehensively consider the different impact mechanisms on steel column behavior.

(1)

Uniform corrosion was characterized by a reduction in the cross-sectional size of the column.

(2)

Pitting corrosion was characterized by the degeneration laws of the mechanical properties of steel with an increasing mass loss rate η, as shown in Equation (3), as discussed in the author’s previous research [31]. The maximum residual thickness was used to determine the mechanical properties.

$$\left\{\begin{array}{l}{f}_y/f_{y0}=1-0.473\eta \hspace{1em}{R}^2=0.962\\ f_u/f_{u0}=1-0.546\eta \hspace{1em}{R}^2=0.973\\ \delta /{\delta }_0=1-1.589\eta \hspace{1em}{R}^2=0.940\\ E_s/E_{s,0}=1-0.472\eta \hspace{1em}{R}^2=0.845\end{array}\right.$$

(3)

where f_y, f_u, δ, and E_s are the yield strength, tensile strength, elongation and elastic modulus of corroded steel, respectively; f_y0, f_u0, δ₀, and E_s,₀ are the original mechanical properties; η is the mass loss rate (<14%), and R² is the coefficient of determination.

Figure 8 shows the FEA model of the corroded steel frame columns in which the reduced section size and deteriorated mechanical properties, as well as the Kinematic Hardening constitutive model considering the Bauschinger effect [31], were adopted. Moreover, a ductile metal damage criterion was applied in the modelling, and the damage initiation was represented as [32]:

$${\epsilon }_{\mathrm{f}}^{\mathrm{pl}}=C/\left(3{\eta }_{\mathrm{ave}}\right),\hspace{1em}1/3\le {\eta }_{\mathrm{ave}}$$

(4)

$${\eta }_{\mathrm{ave}}=\left({\displaystyle {\int }_0^{{\epsilon }_{\mathrm{f}}^{\mathrm{pl}}}{\sigma }_{\mathrm{m}}/{\sigma }_{\mathrm{e}} \mathrm{d}{\epsilon }^{\mathrm{pl}}}\right)/{\epsilon }_{\mathrm{f}}^{\mathrm{pl}}$$

(5)

$${\epsilon }^{\mathrm{pl}}={\displaystyle \frac{\sqrt{2}}{3}}{\left[{\left({\epsilon }_1-{\epsilon }_2\right)}^2+{\left({\epsilon }_2-{\epsilon }_3\right)}^2+{\left({\epsilon }_3-{\epsilon }_1\right)}^2\right]}^{1/2}$$

(6)

where ${\epsilon }_{\mathrm{f}}^{\mathrm{pl}}$ is the fracture initiation strain; ${\epsilon }^{\mathrm{pl}}$ is the equivalent plastic strain; ${\eta }_{\mathrm{ave}}$ is the stress triaxiality; ${\sigma }_{\mathrm{m}}$ is the hydrostatic stress; ${\sigma }_{\mathrm{e}}$ is the Mises equivalent stress; ${\epsilon }_1$, ${\epsilon }_2$ and ${\epsilon }_3$ are the three principal strains. Based on existing literature [31,32,33] and repeated tests, a fracture initiation strain of 0.1–0.3, a linear damage evolution path, and fracture displacement of 1.5–2.5 mm were suggested.

All parts of the model employed C3D8R solid elements and were divided using an adaptive meshing technique with a maximum element size of 15 mm. Local mesh refinement was performed at the column ends, with a minimum element size of 3 mm. Binding constraints were used to simulate welding actions among the parts. The mode boundary and loading conditions were aligned with the tests, with a fixed-end constraint at the bottom of the model and out-of-plane constraints at the sides of the model. Lateral cyclic loads and axial pressure were applied at the column top using “coupling command” and “pressure form,” respectively. Furthermore, an initial imperfection based on the first-order buckling mode of the eigenvalue analysis was introduced to efficiently capture the plate local buckling at the column end, and the maximum defect value was taken as h/5 00 [29]. An implicit Newton-Raphson algorithm was applied. The initial increment size was 0.01, the minimum increment size was 1 × 10⁻⁶, and the maximum number of increments was 1 × 10⁶.

Figure 5 and Figure 6 present a comparison between the FEA and test results. The failure modes of the FEA agreed well with those of the tests, with both forming plastic hinges at the column-end. The stress distribution law generally agrees with practice. The hysteresis curves calculated by FEA were in good agreement with the test hysteresis curves, with coordinated shapes and development rules. The maximum differences in terms of the maximum bearing capacity M_max, ultimate rotation θ_u, and cumulative energy dissipation E_t (the total area of the hysteresis loops) were less than 5%, 12%, and 17%, respectively. These results reflect the stress and deformation performance of the corroded steel frame columns under low cyclic loading, verifying the accuracy and reliability of the FEA method.

4. Time-Dependent Fragility Analysis

4.1. Fragility Function

The component fragility function characterizes the conditional probability of components reaching or exceeding a specific damage state (DS) for a given engineering demand parameter (EDP) [5], as follows: It quantitatively depicts the seismic performance of components from a probabilistic perspective by considering uncertainties such as geometric features, material properties, construction and modelling, load conditions, and history. FEMA P-58 [5] and previous research [6,7,8,9] used a lognormal distribution to develop component fragility functions, which are expressed as follows:

$$P\left(\mathrm{DS}\ge d{s}_i\left|\mathrm{EDP}=x\right.\right)=\mathsf{\Phi }\left[{\displaystyle \frac{\ln \left(x/{\overline{\theta }}_i\right)}{{\beta }_i}}\right]$$

(7)

where Φ[·] is the standard normal cumulative distribution function; ${\overline{\theta }}_i$ and β_i are the median and logarithmic standard deviation, respectively, which can be calculated by

$${\overline{\theta }}_i=e^{{\displaystyle \frac{{\displaystyle \sum _{j=1}^n\ln (x_j)}}{n}}}$$

(8)

$${\beta }_i=\sqrt{{\beta }_{\mathrm{r},i}^2+{\beta }_{\mathrm{u},i}^2}$$

(9)

$${\beta }_{\mathrm{r},i}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{j=1}^n{\left[\ln \left(x_j/{\overline{\theta }}_i\right)\right]}^2}}{n-1}}}$$

(10)

where n is the sample quantity, x_j is the jth EDP value, and β_r,i and β_u,i are the logarithmic standard deviations considering essential and subjective uncertainties, respectively. FEMA-P58 [5] suggests that β_u,i = 0.1 when n > 5, and β_u,i ≥ 0.25 when n ≤ 5.

The main stages of establishing component fragility curves are as follows: (1) Select the EDP index that can characterize the seismic behavior of the components. (2) Divide the damage levels of the components and identify the corresponding quantitative limits. (3) A dataset is constructed to calculate the probability distribution of the fragility function parameters. (4) Eliminate outliers based on the Peirce criterion and optimize the dataset. (5) Confirm the median and logarithmic standard deviation at each damage level using the maximum likelihood estimation method and construct the fragility curves. (6) Perform goodness-of-fit tests on the fragility curves based on the Lilliefors method (represented as Equations (11)–(13)), verifying whether they meet the assumed lognormal distribution and examining the correlation between the data samples and fitting values.

$$D=\max \left|P\left(x_j\right)-S_n\left(x_j\right)\right|$$

(11)

$$S_n\left(x_j\right)=\left(j-0.5\right)/n$$

(12)

$$D_{\mathrm{crit}}=0.895/\left(n^{0.5}-0.01+0.85n^{-0.5}\right)$$

(13)

where P(x) and S_n(x) are the fitted fragility and sample cumulative distribution function values, respectively; D is the maximum absolute difference between the two functions; D_crit is the upper limit of D at the 5% significance level, which can be calculated using Equation (13) and the Monte Carlo simulation method. By comparing D and D_crit, significant differences in the functional statistics were identified. When D ≤ D_crit, the assumption of a lognormal distribution of the fragility function is accepted, and the goodness-of-fit test passes. Otherwise, the hypothesis is rejected and the test fails. Figure 9 shows the implementation process of the component fragility curves.

4.2. Classification Criteria of Damage States

Currently, there are many descriptions of the damage states of steel structures and components in performance-based seismic design and damage assessment studies. FEMA P-58 [5], ASCE 41 [34], FEMA 356 [35], EC8-3 [36], HAZUS [37], GB 50011-2010 [28], and GB/T 38591-2020 [38] have proposed classification criteria and theoretical characterizations of the damage state for steel components; however, they all ignore the effects of environmental factors and service age on the performance index of existing steel components.

Combining previous studies [5,34,35,36,37,38] and the typical failure modes in the test steel columns, the damage states of the steel frame columns were divided into five levels: insignificant, slight, moderate, severe, and complete damage, numbered DS0-DS4, respectively.

Table 4. Definition of damage states.

Damage States	Description of Damage Phenomena	Repair Methods	Visual Illustrations
Insignificant (DS0)	Basically intact, with no deformation at the column end In the elastic stage	Repair or replacement of the exterior hangings
Slight (DS1)	Visible rotational deformation at the column end Flange yielding and web yielding in the rotational deformation area. Surface rust layer or coating peeling off	Cosmetic repair coating application low repair cost
Moderate (DS2)	Slight flange local buckling in the rotational deformation area Residual deformation existing after unloading Relatively high plastic development Approaching the maximum bearing capacity	Heat straightening of the buckled flanges Relatively high repair cost
Severe (DS3)	Obvious flange local buckling Web local buckling Plastic hinge forming at the column end Strength attenuation and stiffness degradation.	Heat straightening of the buckled flanges and web replacement or retrofitting in case of too large buckling deformation High repair cost
Complete (DS4)	Plastic hinge is fully developed Ductile tearing Vertical bearing capacity is lost	Replacement or retrofitting High repair cost

lists the descriptions of the damage phenomena and illustrations for each damage state, as well as the corresponding repair methods.

Previous studies [5,34,38] have shown that the drift ratio, as an index of the seismic behavior of steel frame components, can better represent the relationship between the damage level and EDP and have provided the theoretical characterization of each damage state based on the M − θ skeleton curves. On this basis, the equivalent yield point B of the skeleton curve was defined as reaching the slight damage state DS1; point D at 2/3 BC segment was defined as reaching the moderate damage state DS2; peak point C was defined as reaching the severe damage state DS3; and point E, at which the bending bearing capacity decreased by 25%, was defined as reaching the complete damage state DS4, as shown in Figure 10. Table 2 presents the quantitative drift ratio limits for the different damage states of the tested corroded steel columns (SFC-1~6).

4.3. Dataset for Fragility Function Parameters

Given the distortion of component fragility evaluation owing to the limitations and heterogeneity of test data, this study developed a fragility function parameter dataset based on the verified FEA method of corroded steel columns and established the fragility curves of existing multi-aged steel frame columns in an offshore atmospheric environment.

For H-shaped steel components, the plate width-thickness ratio and its composition, axial compression ratio, and corrosion damage are the main factors affecting the seismic performance. Five design parameters were considered: axial compression ratio n, flange width-thickness ratio b_f/t_f, web depth-thickness ratio h_w/t_w, slenderness ratio L/h, and section depth-width ratio h/b, as well as the difference in service age T in the offshore atmospheric environment. A total of 1790 FEA model samples of existing steel frame columns were established. Figure 11 shows the analysis parameters used.

The offshore atmospheric environment refers to the coastal atmospheric area affected by marine chloride ions, and the research objects were existing steel frame columns in offshore atmospheric environment urban areas (within 5 km of the coastline). The rusting time of a steel structure is primarily determined by the failure time of its anti-corrosion coating. However, the factors and mechanisms that affect the failure of anti-corrosion coatings are complex, making it difficult to accurately evaluate the initial rusting time of the steel. Therefore, on-site durability inspections were conducted on in-service steel components in offshore cities (such as Qingdao and Quanzhou in China) (see Figure 12) to investigate the initial rusting time and corrosion depth of steel structures at different service ages in an offshore atmospheric environment.

Based on the survey results, relevant specifications for steel structure corrosion prevention [39,40], and existing research findings [41,42], the initial rusting time of existing steel structures in an offshore atmospheric environment was recommended as 30 years. Meanwhile, with reference to the C3 corrosion environment defined in ISO 12944-2 (low salinity coastal areas, steel corrosion rate K = 25–50 μm/a) [39], the average corrosion rate of steel r_corr was recommended as 0.04 mm/a. The modelling procedure for corrosion damage of steel columns caused by the service age is shown in Figure 13.

To obtain the relationship between the steel mass loss rate η and the corrosion depth d, the following assumptions were made: (1) there was no change in steel density before and after corrosion, that is, ρ₀ = ρ_T; (2) because the component length was much larger than the section width and height, corrosion only occurred in the section dimensions of the components, that is, L₀ = L_T. Thus, the mass-loss rate η can be represented as

$$\eta ={\displaystyle \frac{m_0-m_T}{m_0}}={\displaystyle \frac{{\rho }_0{A}_0{L}_0-{\rho }_T{A}_T{L}_T}{{\rho }_0{A}_0{L}_0}}={\displaystyle \frac{A_0-A_T}{A_0}}={\displaystyle \frac{A_{\mathrm{corr}}}{A_0}}$$

(14)

where m₀, ρ₀, A₀ and L₀ are the initial mass, density, cross-section area, and length of components, respectively; m_T, ρ_T, A_T, and L_T are the corresponding physical quantities at T-year service age; A_corr is the section corrosion area.

For a H-shaped steel component with a section specification h × b × t_w × t_f (see Figure 14), initial cross-section area $A_0=2b{t}_{\mathrm{f}}+t_{\mathrm{w}}\left(h-2t_{\mathrm{f}}\right)$, and cross-section corrosion area at T-year service age $A_{\mathrm{corr}}=2bd+d\left(h-2t_{\mathrm{f}}\right)$, $d=r_{\mathrm{corr}}\cdot T$, the mass loss rate η can be converted to:

$$\eta ={\displaystyle \frac{2bd+d\left(h-2t_{\mathrm{f}}\right)}{2b{t}_{\mathrm{f}}+t_{\mathrm{w}}\left(h-2t_{\mathrm{f}}\right)}}={\displaystyle \frac{2b\cdot r_{corr}\cdot T+r_{corr}\cdot T\cdot \left(h-2t_{\mathrm{f}}\right)}{2b{t}_{\mathrm{f}}+t_{\mathrm{w}}\left(h-2t_{\mathrm{f}}\right)}}$$

(15)

Combining Equation (15) with Equation (3), the mechanical properties of the corroded steel at T-year service life can be obtained as follows:

$$\left[\begin{array}{l}{f}_y/f_{y0}\\ f_u/f_{u0}\\ \delta /{\delta }_0\\ E_s/E_{s,0}\end{array}\right]=\left[\begin{array}{l}1\\ 1\\ 1\\ 1\end{array}\right]-\left[\begin{array}{l}0.473\\ 0.546\\ 1.589\\ 0.472\end{array}\right]\cdot {\displaystyle \frac{2b\cdot r_{corr}\cdot T+r_{corr}\cdot T\cdot \left(h-2t_{\mathrm{f}}\right)}{2b{t}_{\mathrm{f}}+t_{\mathrm{w}}(h-2t_{\mathrm{f}})}}$$

(16)

In addition, to accurately reflect the seismic performance of existing steel frame columns, the average value of the mechanical properties of steel should be considered in numerical modelling. Combined with engineering information and test data [31,43], the strength and elastic modulus of steel followed a normal distribution, and the ratios of the mean to standard values for the yield strength and elastic modulus were 1.21 and 1.08, respectively, with a variation coefficient of 0.15.

The GEM report [44] states that for a class of structures or components in urban buildings, a unified performance index limit should be employed, regardless of the differences in their inherent properties (such as geometric size, material properties, and boundary and load conditions). However, this study and previous experimental studies [43,45] have shown that under environmental erosion, the mechanical and seismic performance of steel components deteriorates significantly with an increase in the Therefore, in the statistical analysis of the drift ratio limit at different damage states for all column models, only the influence of service age T on column behavior was considered, while ignoring the influence of factors such as the section depth-width ratio h/b, plate width-thickness ratio (b_f/t_f and h_w/t_w), axial compression ratio n, and slenderness ratio L/h.

4.4. Fragility Analysis

The skeleton curves of 1790 numerical models of existing steel frame columns were extracted, and the drift ratios corresponding to each damage state were obtained. By integrating the test and simulation data and removing the outliers based on the Pierce criterion, four common types of distribution functions, namely Lognormal, Gamma, Weibull, and Lorentz distributions, were selected to estimate the probability distribution of the drift ratio for each damage state. Figure 15 shows an example of the drift ratio probability distribution and frequency histogram for 50-year-old columns with different damage states. It can be seen that the lognormal distribution has the largest correlation coefficient R² and good significance level (Prob > F value being close to zero), indicating an optimal fitting, which is consistent with the recommendation of FEMA P-58 [5] and previous research [6,7,8,9].

By integrating the test and simulation data, and removing the outliers based on the Pierce criterion, the fragility function parameters (median $\overline{\theta }$ and logarithmic standard deviation β) of existing steel frame columns at different service ages can be calculated using Equations (8)–(10), as presented in

Table 5. Fragility function parameters of existing steel frame columns at different service ages.

Service Age	Damage ID	Fragility Function Parameters		Lilliefors Test
		$\overline{\mathit{\theta }}$	β	D	Dcrit	Result
30 years	Slight (DS1)	0.0150	0.166	0.035	0.046	Pass
	Moderate (DS2)	0.0290	0.257	0.038		Pass
	Severe (DS3)	0.0360	0.305	0.049		Fail
	Complete (DS4)	0.0500	0.302	0.044		Pass
40 years	Slight (DS1)	0.0146	0.154	0.040		Pass
	Moderate (DS2)	0.0276	0.247	0.028		Pass
	Severe (DS3)	0.0340	0.293	0.052		Fail
	Complete (DS4)	0.0486	0.284	0.043		Pass
50 years	Slight (DS1)	0.0141	0.139	0.029		Pass
	Moderate (DS2)	0.0262	0.233	0.038		Pass
	Severe (DS3)	0.0320	0.274	0.066		Fail
	Complete (DS4)	0.0458	0.263	0.047		Fail
60 years	Slight (DS1)	0.0135	0.133	0.032		Pass
	Moderate (DS2)	0.0243	0.215	0.04		Pass
	Severe (DS3)	0.0294	0.250	0.058		Fail
	Complete (DS4)	0.0390	0.238	0.054		Fail
70 years	Slight (DS1)	0.0128	0.129	0.034		Pass
	Moderate (DS2)	0.0228	0.204	0.036		Pass
	Severe (DS3)	0.0274	0.236	0.044		Pass
	Complete (DS4)	0.0350	0.228	0.050		Fail

and Figure 16. With an increase in service age, the corrosion damage of steel frame columns increased continuously, the seismic performance gradually deteriorated, and the median drift ratios $\overline{\theta }$ of slight, moderate, severe, and complete damage states gradually decreased. When the service age increased to 50 years, the $\overline{\theta }$ corresponding slight, moderate, severe, and complete damage states decreased by 6.0%, 9.7%, 11.1%, and 8.4%, respectively. At 70 years of age, the values decreased by 14.7%, 21.4%, 23.9%, and 30.0%, respectively. Simultaneously, the logarithmic standard deviation β of each damage state gradually decreased, indicating that the uncertainty of the drift ratio limit evaluation for steel columns reduced when subjected to environmental erosion. Overall, environmental erosion negatively affected the fragility of the steel frame columns.

Table 6. Typical classification of damage states for steel structures.

Current Specifications	Category	Damage States-Median Demand ($\overline{\mathit{\theta }}$)
This study (30 years)	Steel frame column	DS1	DS2	DS3	DS4
		0.015	0.029	0.036	0.050
FEMA P-58 [5]	Column Base Plates	DS1	DS2	DS3
		0.04 (strong axis)	0.07(strong axis)	0.10(strong axis)
	non-RBS Beam-Column Moment Connections	DS1	DS2	DS3
		0.03	0.04	0.05
HAZUS [37]	Steel moment frame	Slight	Moderate	Extensive	Complete
		0.003	0.006	0.015	0.040
FEMA 356 [35]	Steel structures	IO	LS	CP
		0.007	0.025	0.050
GB 50011-2010 [28]	Steel structure vertical component	Basically intact	Slight	Moderate	Extensive
		0.003	0.005	0.01	0.018
GB/T 38591-2022 [38]	Steel column (0.3 < n < 0.5)	IO	K	LS	CP
		1.25 θy	(8.125–11.7 n) θy	(15–23.3 n) θy	(18–28.3 n) θy

lists the typical classification of the damage states for steel structures. The damage levels of steel columns provided in FEMA P-58 [5] are aimed at column base plates, without reflecting the typical failure modes of steel frame columns. Table 6 provides a summary of this. The definition of damage in FEMA 356 [35] and HAZUS [37] is based on the overall structure, while that in this study is based on steel columns from the component level, and there are design differences between the Chinese and American codes. Compared with GB 50011-2010 [28] and GB/T 38591-2020 [38], the damage state limits of steel frame columns within 30 years in this study were larger because the design codes tend to be conservative, with safety redundancy. The proposed definition and theoretical characterization of the damage states are complementary to the current specifications.

Overall, considering the impact of environmental erosion and aging, a classification criterion and theoretical characterization of damage states, as well as the corresponding repair methods for existing steel frame columns, were proposed, improving the classification standards for the damage levels of existing steel structures in the current specifications.

By substituting the fragility parameters in Table 5 into Equation (7), the fragility curves of the existing steel frame columns under different service ages in offshore atmospheric environments can be established, as shown in Figure 17. It can be observed that (1) as the degree of damage increased (from DS0 to DS4), the fragility curve gradually shifted to the right. (2) With an increase in service age, the exceedance probability of each damage state at the same drift level for the steel columns increased continuously. This is because corrosion damage reduces the steel’s strength and weakens the cross-sectional size, thereby degrading the seismic performance of the steel columns.

Furthermore, based on the aforementioned fragility curves, the damage probability matrices at different service ages were derived (see Equation (17)), as shown in Figure 18. It is clear from Figure 17 and Figure 18 that at a drift ratio level of 0.5%, all multi-aged steel frame columns were in an insignificant damage state. At a drift ratio of 1.0%, all multi-aged columns suffered slight damage. At a drift ratio level of 1.5%, moderate damage was observed in 30, 40, and 50-year-old columns, while severe damage was observed in 60 and 70-year-old columns. At a drift ratio of 2%, complete damage was observed in all multi-aged columns. With an increase in the drift ratio, the damage accumulation intensified, and the damage level of the steel frame column increased. For example, at a drift ratio level of 2%, the probability matrices of 50-year-old steel frame columns reaching insignificant, slight, moderate, severe, and complete damage were 0.6%, 87.0%, 8.2%, 4.2%, and 0.1%, respectively. At a drift ratio level of 4%, the probability matrices were 0.0%, 3.6%, 17.2%, 48.9%, and 30.3%, respectively.

$$P_{\mathrm{matrix}}\left[{\mathrm{DS}}_i\left|\theta \right.\right]=\left\{\begin{array}{ll}1-P\left[{\mathrm{DS}}_{i+1}\left|\theta \right.\right]& i=0\\ P\left[{\mathrm{DS}}_i\left|\theta \right.\right]-P\left[{\mathrm{DS}}_{i+1}\left|\theta \right.\right]& 0<i<4\\ P\left[{\mathrm{DS}}_i\left|\theta \right.\right]& i=4\end{array}\right.$$

(17)

After the initial yielding of the steel columns (approximately 1.5% drift ratio), the damage class at the same drift ratio level increased significantly with an increase in service age. For example, at the elasto-plastic drift ratio limit of 2% for steel structures in GB 50011-2010 [28], the probability matrices of severe damage to columns with 40, 50, 60, and 70-year service ages were 3.4%, 4.2%, 5.9%, and 8.4%, respectively, with increases of 30.8%, 61.5%, 126.9%, and 223.1%, respectively, compared to columns within a 30-year service age (probability of 2.6%). At the drift ratio limit of 4% for SMFs in AISC 341-10 [46], the probability matrices of complete damage to columns with 40, 50, 60, and 70-year service ages were 24.6%, 30.3%, 55.1%, and 72.1%, respectively, with increases of 6.96%, 31.7%, 139.6%, and 213.5%, respectively, compared to columns within 30-year service age (probability of 23.0%).

The Lilliefors test was performed on the established fragility curves, and the results are presented in Table 4. Figure 19 shows an example of the comparison between the theoretical and empirical fragility of 50-year-old steel-frame columns. At a significance level of 0.05, for the severe damage (DS3) of steel columns with 30, 40, 50, and 60-year service ages, and the complete damage (DS4) of steel columns with 50, 60, and 70-year service ages, the goodness-of-fit tests of the fragility curves failed. However, the difference between the D and D_crit values was not significant, indicating that the fragility function established using the lognormal distribution assumption did not cause obvious errors in the results. The goodness-of-fit tests of the fragility curves for the other damage states were qualified.

4.5. Time-Dependent Fragility Model

To continuously evaluate and predict the full life-cycle fragility of steel frame columns, a time-dependent fragility model for existing multi-aged steel frame columns is proposed. The fragility model was established by calibrating the fragility parameters, namely the median $\overline{\theta }$ and logarithmic standard deviation β. Therefore, based on Table 5, the relationship between the fragility parameters and service age was constructed using regression analysis, presenting a time-dependent fragility model for existing steel frame columns (see Equations (18)–(20)). Figure 20 shows an example of the variation in the fragility parameters with service age in the severe damage state.

Table 7. Regression coefficients for the median and dispersion values at different damage states.

Damage ID	Regression Coefficients
	$\overline{\mathit{\theta }}$		Logarithmic Standard Deviation β
	${\mathit{a}}_{\mathit{i}\_\overline{\mathit{\theta }}}$	${\mathit{b}}_{\mathit{i}\_\overline{\mathit{\theta }}}$	${\mathit{a}}_{\mathit{i}\_\mathit{\beta }}$	${\mathit{b}}_{\mathit{i}\_\mathit{\beta }}$
Slight (DS1)	0.017	−5.50 × 10−5	0.192	−9.50 × 10−4
Moderate (DS2)	0.034	−1.57 × 10−4	0.300	−1.38 × 10−3
Severe (DS3)	0.043	−2.18 × 10−4	0.362	−1.81 × 10−3
Complete (DS4)	0.063	−3.96 × 10−4	0.360	−1.94 × 10−3

lists the regression coefficients for the median and dispersion values of the different damage states.

$${\overline{\theta }}_i(T)=a_{i\_\overline{\theta }}+b_{i\_\overline{\theta }}\cdot T$$

(18)

$${\beta }_i(T)=a_{i\_\beta }+b_{i\_\beta }\cdot T$$

(19)

$$P\left[\mathrm{DS}\ge d{s}_i\left|\mathrm{EDP}=x\right.\right]\left(T\right)=\mathsf{\Phi }\left[{\displaystyle \frac{\ln \left(x\right)-\ln (a_{i\_\overline{\theta }}+b_{i\_\overline{\theta }}\cdot T)}{a_{i\_\beta }+b_{i\_\beta }\cdot T}}\right]$$

(20)

Considering the uncertainty of the design parameters and durability damage, a time-dependent fragility model of existing steel frame columns was established, achieving continuous evaluation and prediction of the full life-cycle seismic fragility of steel frame columns.

5. Residual Seismic Behavior Evaluation

With the continuous advancement of performance-based seismic design concepts, the seismic capacity of building structures has significantly improved, and the probability of structural collapse has been greatly reduced, even when subjected to major earthquakes. However, although the number of collapsed structures during major earthquakes has decreased, the number of damaged structures that require detailed evaluation to determine whether to repair or demolish has increased. Scientific and efficient damage assessment and residual seismic capacity prediction of earthquake-damaged buildings are important for resistance to sequential earthquakes and post-disaster functional recovery and reconstruction of building structures, which are of great significance for promoting earthquake prevention and disaster mitigation capabilities and scientific decision-making levels.

5.1. Loading Scheme

The residual seismic behavior of the earthquake-damaged steel frame columns was evaluated based on the established 1790 numerical models of the existing steel frame columns and the determined damage classes (Table 3 and Table 4). First, each model was preloaded to four different damage levels (DS0–DS3), and then main loading was applied to the damaged model to examine the residual seismic behavior [21]. The loading protocol is illustrated in Figure 21 and listed in

Table 8. Loading schedule for residual seismic behavior.

Damage ID	Pre-Loading
	Drift Ratio θ and Cycle Number
	±0.375%	±0.5%	±0.75%	±1%	±1.5%	±2%	±3%	±4%
DS0	2	2	5
DS1	2	2	2	2	2	5
DS2	2	2	2	2	2	2	5
DS3	2	2	2	2	2	2	2	5
ID	Main-loading
	Incremental drift ratio θIc and cycle number
	±0.375%	±0.5%	±0.75%	±1%	±1.5%	±2%	±3%	±4%	±5%	…
DS0–DS3	2	2	2	2	2	2	2	2	2	2

. Referring to AISC 341-10 [46], two loading cycles were imposed at 0.375%, 0.5%, 0.75%, 1%, 1.5%, 2%, 3%, 4%, and drift ratio levels. In the pre-loading stage, five loading cycles were performed for the target drift ratio level with the set damage state, followed by unloading. Subsequently, the main loading was implemented based on the residual deformation, that is, adopting the incremental drift ratio θ_Ic. In ABAQUS, the damage caused by preloading was introduced into the column models using a “predefined field” command.

5.2. Residual Seismic Capacity Analysis

Figure 22 shows an example of the moment-rotation hysteretic curves for 50-year-old steel frame columns with different earthquake damage levels, with the design parameters n = 0.3, b_f/t_f = 8.61, h_w/t_w = 24.67, h/b = 1, and L/h = 6. The red curves represent the M − θ_Ic hysteretic curves of the damaged columns during the main loading stage, while the grey curves represent the M − θ hysteretic curves of the undamaged columns. The blue curves represent the hysteretic curves of preloading, and the green curves represent the residual deformation caused by preloading (“predefined field” in ABAQUS). It can be observed that before reaching the target drift ratio with the set damage class, compared with the intact steel frame column, the hysteresis loop areas of the earthquake-damaged column were reduced, and the stiffness decreased significantly. After reaching it, for the minor damage states (DS0, DS1), there was no significant difference in the strength and stiffness between the intact and damaged columns; however, for the greater damage states (DS2, DS3), strength attenuation and stiffness degradation occurred due to reaching or exceeding the maximum bearing capacity.

To study the influence of the damage level, design parameters, and service age on the residual seismic performance of earthquake-damaged steel frame columns, the residual stiffness K_res, bearing capacity M_res, and energy dissipation capacity E_res of the damaged column (i.e., the initial stiffness, maximum moment, and cumulative energy dissipation on the hysteretic curve of the main loading stage) were extracted and regularized with the values (K, M_max, E) of the intact column, where the initial stiffness could be calculated based on an initial loading of 0.375% drift ratio.

Figure 23 shows the typical variation relationships between K_res/K, M_res/M_max, E_res/E, and the column parameters at different earthquake damage levels. It can be observed that (1) in the insignificant damage state (DS0), the steel columns suffered less damage, and the initial stiffness, bearing capacity, and energy dissipation were reduced to some extent, but within 8%, as the parameters changed. (2) For damage states DS1–DS3, under the research range of parameters and same conditions, with an increase in the service age T, K_res/K exhibited a linear decline, M_res/M_max decreased with a negative exponential trend, and E_res/E decreased linearly. (3) With an increase in the axial compression ratio n, K_res/K, M_res/M_max, and E_res/E all decreased with a negative exponential trend. (4) With an increase in the flange width-to-thickness ratio b_f/t_f and web depth-to-thickness ratio h/t_w, K_res/K and M_res/M_max decreased exponentially, and E_res/E decreased linearly. (5) The effect of the slenderness ratio L/h on K_res/K and E_res/E was irregular, but M_res/M_max decreased gradually as L/h increased. (6) With an increase in the section depth-width ratio h/b, K_res/K, M_res/M_max, and E_res/E decreased gradually.

The JBDPA Guideline [19] proposed a reduction factor η to evaluate the residual seismic capacity of the components. However, this reduction factor is a manifestation of comprehensive capacity reduction, which integrates stiffness, ductility, and energy reduction, and cannot be simply multiplied by the original capacity to achieve residual capacity. Research has shown that the reduction values are conservative [21], while FEMA306 [20] separated the reduction factor into component stiffness, strength, and deformation capacity. A comparison with the analysis results in this study shows that the reduction values recommended in FEMA306 overestimate the stiffness degradation for moderate and severe damage (factors of 0.5 and 0.2), and the strength for severe damage (factor of 0.3), as they simultaneously adopt the universal value without considering the impact of service age and design parameters.

5.3. Implementation of PSO-BPNN Model

Ref. [31] indicated that web shear instability failure of H-shaped steel members should be avoided in project design because it restricts plastic development and degrades the bearing performance. Therefore, by excluding the parameter analysis results that caused web shear instability and high dispersion, the datasets of K_res/K, M_res/M_max, and E_res/E for existing steel frame columns under different design parameters, service ages, and damage levels were obtained. Subsequently, using the PSO-BPNN model in MATLAB R2021a [47,48], the nonlinear mapping relationships between the characteristic variables (design parameters, service age, and damage level) and residual seismic capabilities (K_res/K, M_res/M_max, and E_res/E) of existing steel frame columns were established (see Equation (21)). The operational flow of the PSO-BPNN model is shown in Figure 24.

$$\left\{\begin{array}{l}{\displaystyle \frac{K_{\mathrm{res}}}{K}}=g (T, n, b_f/t_f, h_w/t_w, L/h, h/b,D{S}_i)\\ {\displaystyle \frac{M_{\mathrm{res}}}{M_{\max }}}=u (T, n, b_f/t_f, h_w/t_w, L/h, h/b,D{S}_i)\\ {\displaystyle \frac{E_{\mathrm{res}}}{E}}=z (T, n, b_f/t_f, h_w/t_w, L/h, h/b,D{S}_i)\end{array}\right.$$

(21)

(1) PSO-BPNN mode architecture

In the PSO-BPNN mode, the information transmission relationships are as follows:

Hidden layer:

$$h_j=f({\displaystyle \sum _{i=1}^n{x}_i{w}_{ij}}+b_j)$$

(22)

Output layer:

$$y_k={\displaystyle \sum _{j=1}^m(h_j\cdot w_{jk})}+b_k$$

(23)

Activation function (e.g., Sigmoid function):

$$f(x)={\displaystyle \frac{1}{(1+e^{-x})}}$$

(24)

where x_i is the input variable; w_ij and b_j are the weight and bias corresponding to the input variable, respectively; h_j is the output variable for the hidden layer; w_kj and b_k are the weight and bias corresponding to the output variable, respectively; y_k is the output variable for the output layer; f (x) is the activation function.

The updated equations for velocity v_i and position x_i of the particles are as follows:

$$v_i^{t+1}=\omega v_i^t+c_1{\epsilon }_1\left(P_i^t-x_i^t\right)+c_2{\epsilon }_2\left(P_g^t-x_i^t\right)$$

(25)

$$x_i^{t+1}=x_i^t+v_i^{t+1}$$

(26)

where t is the number of iteration steps; ω is the inertia-weight factor; c₁ and c₂ are the learning factors; ε₁ and ε₂ are the combined random values between [0, 1]; P_l and P_g are the local best position and global best position, respectively, for each iteration. The particle velocity and position were updated repeatedly until the optimization goal was achieved, and then the optimal initial weights and thresholds used in BPNN were obtained.

$$L=\sqrt{m+n}+a$$

(27)

where l, m, and n are the neuron nodes of the hidden, input, and output layers, respectively; a is a constant between 1 and 10.

In the obtained dataset, 70% of the data were randomly selected as the training dataset. Of the remaining data, 50% was selected as the validation set and 50% as the test set. The service age T, design parameters (h/b, b_f/t_f, h_w/t_w, n, and L/h), and damage states (DSi) were used as input variables, and the residual capacities K_res/K, M_res/M_max, and E_res/E were used as output variables. Based on the empirical formula (Equation (27)) [47] and a trial-and-error analysis, the hidden layer neuron nodes l = 8. Finally, referring to previous studies [47,48,49] and repeated debugging, the parameters of the PSO-BPNN model were as follows: (1) For the PSO algorithm, the particle swarm size was 40, the inertial-weight factor was 0.6, all learning factors were 2, the maximum flying velocity of the particles was 0.8, and the maximum number of iterations was 50. (2) For the BP algorithm, the neuron nodes of the input, output, and hidden layers were 7, 1, and 8, respectively. The maximum number of iterations was 1000, the learning rate was 0.01, the allowable error was 0.001, and the momentum constant was 0.5.

(2) PSO-BPNN mode validation

Figure 25 presents an example of the regression results of the PSO-BPNN model for the severe damage state (DS3), and the explicit expressions are derived in Appendix A. The correlation coefficients R for K_res/K, M_res/M_max, and E_res/E on the training, validation, testing, and all datasets were above 0.96. 96% of the samples in the dataset fell within the regression interval with a 95% confidence level, indicating good simulation accuracy.

Table 9. Performance of PSO-BPNN network mode at severe damage state (DS3).

Performance Indicators		Kres/K	Mres/Mmax	Eres/E
R2	Training set	0.9638	0.9191	0.9437
	Validation set	0.9732	0.8601	0.9642
	Testing set	0.9599	0.9430	0.9550
	All dataset	0.9653	0.9225	0.9518
RMSE	Training set	0.0129	0.0291	0.0200
	Validation set	0.0115	0.0331	0.0170
	Testing set	0.0130	0.0269	0.0190
	All dataset	0.0127	0.0294	0.0194

shows the performance of the PSO-BPNN model, which is reflected by the determination coefficient R² and root mean square error (RMSE). These results demonstrate that the constructed PSO-BPNN model can effectively avoid overfitting and has good accuracy and prediction.

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _i{({\widehat{y}}_i-y_i)}^2}}{{\displaystyle \sum _i{({\overline{y}}_i-y_i)}^2}}}$$

(28)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{(y_i-{\widehat{y}}_i)}^2}}$$

(29)

where $y_i$, ${\widehat{y}}_i$ and ${\overline{y}}_i$ are the measured values, predicted values and the average of measured values, respectively; N is the number of data.

5.4. Residual Seismic Behavior Model

Based on the proposed time-dependent fragility model for existing multi-aged steel frame columns (see Equation (20)), the exceedance probability of all damage states at a given drift ratio (i.e., the damage probability matrix) can be mathematically expressed as:

$$P_{\mathrm{matrix}}\left[{\mathrm{DS}}_i\left|\theta \right.\right](T)=\left\{\begin{array}{l}1-P\left[{\mathrm{DS}}_{i+1}\left|\theta \right.\right](T)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=0\\ P\left[{\mathrm{DS}}_i\left|\theta \right.\right](T)-P\left[{\mathrm{DS}}_{i+1}\left|\theta \right.\right](T)\hspace{1em}0<i<4\\ P\left[{\mathrm{DS}}_i\left|\theta \right.\right](T)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=4\end{array}\right.$$

(30)

Subsequently, by combining the residual capacities K_res/K, M_res/M_max and E_res/E predicted by the PSO-BPNN model (Equation (21)) using the damage probability matrix (Equatoin (30)), the expected value of residual seismic performance for existing multi-aged steel frame columns at a given drift ratio can be obtained in a probabilistic manner (see Equations (31)–(33)). These values can be used to predict the corresponding seismic capacity safety reserve. For example, for the 50-year-old steel frame columns in urban offshore atmospheric environment (with the design parameter n = 0.3, b_f/t_f = 8.61, h_w/t_w = 24.67, h/b = 1, and L/h = 6), at the elasto-plastic drift limit of 2% in GB 50011-2010 [28], K_res/K, M_res/M_max, and E_res/E were 0.82, 0.91, and 0.80, respectively. At the drift ratio limit of 4% in AISC 341-10 [46], K_res/K, M_res/M_max, and E_res/E were 0.66, 0.79, and 0.63, respectively.

$$E\left(K_{\mathrm{res}}\right)={\displaystyle \sum _{i=0}^4{P}_{\mathrm{matrix}}\left[{\mathrm{DS}}_i\left|\theta \right.\right]\left(T\right)}\cdot {\displaystyle \frac{K_{\mathrm{res}}}{K}}\left({\mathrm{DS}}_i\right)$$

(31)

$$E\left(M_{\mathrm{res}}\right)={\displaystyle \sum _{i=0}^4{P}_{\mathrm{matrix}}\left[{\mathrm{DS}}_i\left|\theta \right.\right]\left(T\right)}\cdot {\displaystyle \frac{M_{\mathrm{res}}}{M_{\max }}}\left({\mathrm{DS}}_i\right)$$

(32)

$$E\left(E_{\mathrm{res}}\right)={\displaystyle \sum _{i=0}^4{P}_{\mathrm{matrix}}\left[{\mathrm{DS}}_i\left|\theta \right.\right]\left(T\right)}\cdot {\displaystyle \frac{E_{\mathrm{res}}}{E}}\left({\mathrm{DS}}_i\right)$$

(33)

In this study, based on the PSO-BPNN model, nonlinear mapping relationships between the design parameters, service ages, earthquake damage levels, and residual seismic capacity of steel frame columns were constructed to quantitatively characterize the actual seismic capacity of existing steel frame columns after earthquakes. Combined with time-dependent fragility functions, the expected values of the residual seismic capacity for existing multi-aged steel frame columns at a given drift ratio can be presented directly in a probabilistic sense, achieving a scientific and efficient evaluation of the residual seismic behavior.

These findings could compensate for the low efficiency, poor quantization accuracy, and reliance on expert knowledge of traditional seismic performance-assessment methods. The results can also improve the intelligence and reliability of revealing the evolution law of the seismic performance of existing steel structures before earthquakes and assessing the residual seismic capability of damaged steel structures after earthquakes. This provides theoretical support for the seismic performance evaluation and post-earthquake safety identification of existing steel structures in earthquake-prone areas and a technical basis for engineers and technicians.

6. Conclusions

(1)

With an increase in the corrosion level or axial compression ratio, the damage phenomena of steel column yielding, plate local buckling, and plastic hinge formation occurred prematurely, the bearing capacity decreased, and ductility and energy dissipation gradually deteriorated. Furthermore, an FEA method for steel frame columns considering corrosion damage and ductile metal damage criteria was developed and validated.

(2)

Considering the impacts of environmental erosion and aging, a classification criterion and theoretical characterization of the damage states for existing steel frame columns were proposed. Thus, the probability distribution of the fragility function parameters (median and logarithmic standard deviation) was constructed, thereby presenting a time-dependent fragility model for existing steel frame columns in offshore atmospheric environments. At the drift ratio limit of 4% for SMFs in AISC 341-10, the probability matrices of complete damage to columns with 40, 50, 60, and 70-year service ages were 35.2%, 43.3%, 53.4%, and 66.9%, respectively, with increases of 18.1%, 45.3%, 79.2%, and 124.5%, respectively, compared with columns within 30-year service age (probability of 29.8%). These results indicate that existing steel components affected by environmental corrosion and aging undergo significant deterioration in seismic performance, and a feasible anti-corrosion measure with periodic maintenance should be introduced in practical seismic design.

(3)

Using the PSO-BPNN model, nonlinear mapping relationships between the characteristic variables, in terms of service age T, design parameters (n, b_f/t_f, h_w/t_w, L/h, and h/b), damage level, and residual seismic capability (K_res/K, M_res/M_max, and E_res/E) for earthquake-damaged columns were constructed. Then, combined with the damage probability matrix of time-dependent fragility, the expected values of the residual seismic capacity for existing multi-aged steel frame columns at a given drift ratio can be presented directly in a probabilistic sense, achieving a scientific and efficient evaluation of the residual seismic behavior. For example, for 50-year-old steel frame columns (with the design parameters n = 0.3, b_f/t_f = 8.61, h_w/t_w = 24.67, h/b = 1, and L/h = 6), at a drift ratio limit of 4% for SMFs in AISC 341-10, the residual capacities K_res/K M_res/M_max and E_res/E were 0.66, 0.79, and 0.63, respectively.

The application range of the proposed time-dependent fragility function and residual seismic behavior model was for Q235B hot-rolled H-shaped steel frame columns with 0 ≤ T ≤ 70 years, 0.1≤ n ≤ 0.4, 8.61 ≤ b_f/t_f ≤ 15, 10 ≤ h_w/t_w ≤ 30, 4.8 ≤ L/h ≤ 8, and 1 ≤ h/b ≤ 2. This study was conducted based on numerical simulations; therefore, more refined analyses and experimental verification will be carried out in the future. Additionally, the time-dependent seismic fragility and residual seismic performance of other cross-sectional forms, steel grades, and categories of steel components should be further investigated, ultimately supporting rapid seismic resilience assessment of aging steel structures in earthquake-prone offshore areas.