Seismic Fragility Analysis of Steel Pipe Pile Wharves with Random Pitting Corrosion

Abstract

This paper investigates the seismic damage behavior of steel pipe pile wharves after pitting corrosion. The seismic intensity is treated as random, and a probabilistic strength model for randomly pitting corroded steel is utilized to assess the seismic response of a typical steel pipe pile wharf. By analyzing the internal force response of each pile and the deformation response of the deck and soil slope, the process of seismic failure in steel pipe pile wharves with different pitting corrosion ratios is investigated. The results demonstrate that pitting corrosion amplifies the internal force within the steel pipe piles, leading to more severe seismic damage. Additionally, probabilistic seismic demand functions are established for the most vulnerable row of piles affected by random pitting corrosion, and the seismic fragility of the pipe pile wharves considering different pitting corrosion ratios is evaluated. These findings provide valuable insights for the design and strengthening of steel pipe pile wharves.

1. Introduction

Wharves play a crucial role in the transportation of goods and raw materials in ports. However, they are susceptible to damage from seismic actions and seawater corrosion. The fragility of steel pipe pile wharves, particularly those exposed to seawater corrosion, has been highlighted in previous earthquakes such as the 1976 Tangshan earthquake in China, the 1995 Kobe earthquake in Japan, the 2004 Sumatra earthquake in India, and the 2010 Haiti earthquake [1,2,3,4].

Seismic damage to wharves can result in shipping disruptions and substantial economic losses [5]. Consequently, extensive research has been conducted to understand the damage mechanisms, fragility, and sensitivity of wharf structures under seismic actions [6,7,8]. It has been observed that steel pipe pile wharves typically experience damage, including pile foundation fractures, slope collapse, damage to pile-deck connections, and severe deformation of the overall wharf [9,10]. Additionally, steel pipe pile wharves are frequently exposed to environments with high concentrations of chloride ions. The corrosion caused by chloride ions leads to a reduction in the wall thickness of steel pipe pile foundations, resulting in declined strength and stiffness [11,12]. Pitting corrosion, which randomly attacks specific points on steel surfaces, aligns more closely with the corrosion mode in complex environments like seawater erosion compared to uniform corrosion [13]. Therefore, conducting seismic fragility analysis of steel pipe pile wharves with random pitting corrosion is crucial.

The seismic response and service life assessment of corroded pile foundation structures have been extensively studied by researchers [14,15,16,17,18]. For instance, Seung et al. [14] predicted the service life of corroded RC columns based on diffusion coefficients of chloride. Schmuhl et al. [15] proposed a residual strength model suitable for corroded RC pile foundations and effectively simulated the cracking failure of the foundation under reinforcement corrosion. Mirzaeefard et al. [16] obtained a time-dependent brittleness curve using incremental dynamic analysis and concluded that seawater erosion considerably exacerbated the fragility of pile foundations. Kiani et al. [17,18] considered the nonlinear stress characteristics of rust products and defined a corrosion damage model of RC structure based on the volume ratio of rust products. However, the randomness of pitting corrosion in marine environments was often overlooked. To address this, Kong et al. [19] proposed an autocorrelation function model for the pit depth of random pitting corrosion. Silva et al. [20] developed a normal distribution model for the residual strength of randomly pitted steel. The second author pointed out that the pit depth has a more significant degradation effect on the steel strength than the pit area and proposed a probability function of steel strength degradation with random pit area and depth [21]. In addition to the randomness of pitting corrosion, seismic actions themselves should also be treated as random [22]. Deep learning algorithms and artificial intelligence (AI) algorithms in solving engineering problems are effective, especially in solving nonlinear and complex problems in engineering problems. Currently, AI technology is being extensively applied in various domains [23,24,25,26,27]. It is evident that AI technology offers significant advantages in capturing more precise mechanical responses of structures, particularly for problems with high numbers of inputs and high dependency of the final results on the main input data. Wu et al. [28] expounded on the advantages and disadvantages of various deep neural network models in predicting engineering problems based on multiple time series. Du et al. [29] combined the statistical state space model and neural network to give a time series forecasting model. Zhao et al. [30] defined two pairs of approximation operators on the quotient space and gave a matrix representation that is easier to calculate. Chen et al. [31] proposed a novel matrix-based method for approximation sets of the neighborhood multigranulation decision-theoretic rough sets model and solved the decision domains through a series of matrix calculations. Ali et al. [32] proposed a machine learning model considering soil-structure interaction, in which seismic response parameters such as base shear and acceleration can be obtained by inputting seismic parameters and structural characteristics. Calabrese et al. [33] used artificial neural networks to establish the correlation between randomly input seismic records and residual deformation of the wharves and studied the fragility of wharves considering seismic randomness. Wang et al. [34] and Mohanty et al. [35] used deep learning methods to evaluate the vertical bearing capacity and damage behavior of piles under uncertain seismic parameters. Torkamani et al. [36] and Amirabadi et al. [37] established seismic demand models that considered the randomness of slope material and seismic intensity coupling to obtain the vulnerability curve. In general, there are few studies on the damage behavior of steel pipe pile wharves considering the coupling effects of random seismic action and random pitting corrosion, although most wharves are exposed to the risk of the simultaneous action of these two attacks.

In this paper, the seismic performance of steel pipe pile wharves after experiencing random pitting is investigated. The analysis takes into account the randomness of seismic intensity and utilizes a probabilistic strength model developed by the author to account for the strength degradation of random pitting corroded steel. The seismic failure mechanism of the wharf is investigated by studying the acceleration response, internal force of the pile body, and the residual displacement of the deck and slope. Furthermore, the damage thresholds of the most fragile position are determined, and the probabilistic seismic demand models for this position under different corrosion ratios are established. Finally, seismic fragility analyses of the wharf structure under different pitting corrosion ratios are conducted.

2. Finite Element Model

2.1. Details of the Wharf Structure Model

The seismic damage mechanism of the steel pipe pile wharves was determined by analyzing the seismic response of a three-dimensional finite element model. The internal force response of each pile and the deformation response of the deck and soil slope were analyzed. Based on pushover analysis, the seismic demand parameters for wharves with different corrosion ratios were obtained. Subsequently, the seismic fragility of the steel pipe pile wharf under various erosion ratios was evaluated using a probability density. The employed pile-supported structural and ground configuration is typical of such facilities in the U.S. Southwest. A typical steel pipe pile wharf structure with thirteen lines and five rows (A, B, C, D, and E) of vertical steel pipe piles was modeled by OpenSees, as shown in Figure 1 and Figure 2.

The piles are arranged with a line spacing of 6 m and a row spacing of 8 m. For simplicity, a representative segment with a width of 6.8 m, consisting of two lines of piles, was analyzed. Each steel pipe pile has a length of 40 m (elevation from 0.0 to 40.0 m), with an outer diameter of 0.8 m and a thickness of 0.02 m. The finite element seabed base has a length of 230 m, and the slope has an inclination angle of 18 degrees. The soil layers extend to depths of 18 m on the seaside and 40 m on the landside. According to Mirzaeefard et al. [16], the range of 1.0 m above to 2.0 m below the sea level is defined as the splash/tidal zone (elevation of 36–39 m). The soil profile consists of three layers: extremely dense soil (elevation of −1.0–9.0 m), normal density soil (elevation of 9.0–17.0 m), and medium density soil (17.0–39.0 m), as illustrated in Figure 2. The mechanical properties of the soil can be found in Section 2.3 below.

2.2. Steel Pipe Piles

The steel pipe piles were modeled using the DispBeamColumn element [16] and were divided into 32 elements along the circumferential direction and 2 elements along the radial direction. All steel pipe piles were divided into 10 segments along the pile length direction. The Steel02 material based on the Giuffré–Menegotto–Pinto model was used to simulate the properties of the steel pipe. The material properties of the uncorroded steel measured by the authors [21] were adopted, as shown in

Table 1. Steel pipe pile details.

Strength Label	Outer Diameter (m)	Thickness (m)	Length (m)	fy (MPa)	Es (GPa)	μs	b
Q345	0.8	0.02	40	399.86	206	0.29	0.01

. where f_y is the yield strength, E_s is the elastic modulus, μ_s is the Poisson’s ratio, and b is the strain hardening coefficient after yielding.

The pitting corrosion of steel pipe piles is considered to occur at the splash/tidal area cyclically [38]. The strength degradation due to random pitting corrosion is considered. The strength of the uncorroded steel follows a log-normal distribution, and the deterioration of the strength with the random pitting corrosion ratio follows a gamma distribution [21]. The probabilistic strength model of steel with random pitting corrosion proposed by the authors [21] was employed, as shown in Equation (1).

$$f_w{}_1(x_1,\alpha (\rho ),\beta )=\frac{{\left(0.041x_1\right)}^{23.94{\rho }^{1.045}}{e}^{-0.041x_1}}{x_1\cdot \Gamma (23.94{\rho }^{1.045})}$$

(1)

where x₁ is the degradation value of the yield strength, ρ is the corrosion ratio (volume loss ratio), α(ρ) and β are the shape parameter and scale parameter of the corrosion pits, respectively, and Γ is the gamma function [39].

In this paper, uncorroded (0% corroded), 5% corroded, 10% corroded, and 15% corroded conditions are adopted. For each corrosion ratio, 50 pitting corroded cases were randomly generated, taking x₁, α(ρ), and β as variables. The yield strengths obtained under the four pitting corrosion ratios are 400.06, 375.98, 342.28, and 323.53 MPa, respectively, as shown in Figure 3.

2.3. Soil Behavior

The soil is modeled using the 3Dbrick solid element, and the PressureDependMultiYield (PDMY) model is used to simulate the material behavior [40,41]. The PDMY material parameters for each soil layer are provided in

Table 2. Details of the PDMY material.

Parameter	Medium Density Sand	Normal Density Sand	Extremely Dense Sand
Relative density	35~65%	65~85%	85~100%
Porosity	0.7	0.55	0.45
Liquefaction coefficient lf3 (kPa)	1	1	0
Liquefaction coefficient lf2 (kPa)	0.01	0.003	0
Liquefaction coefficient lf1 (kPa)	10	5	0
Volume expansion coefficient d2	2	3	5
Volume expansion coefficient d1	0.4	0.6	0.8
Volume shrinkage ratio	0.07	0.05	0.03
Phase angle (°)	23	20	16
Effective restraint pressure (kPa)	80	80	80
Stress index	0.5	0.5	0.5
Peak shear strain	0.1	0.1	0.1
Internal friction angle φ (°)	33	37	40
Bulk modulus (kPa)	2.0 × 105	3.0 × 105	3.9 × 105
Shear modulus (kPa)	7.5 × 104	1.0 × 105	1.3 × 105
Density (kg/m3)	1900	2000	2100

. The constitutive property of the soil is defined using the multi-yield surface model [42], as shown in Figure 4, where $p^{\prime }$ is the effective average stress, $p_0{}^{\prime }$ is a small positive constant and was taken as 1.0 kPa in this paper, and ${\sigma }^{\prime }$ is the effective Cauchy stress tensor. The yield function of the soil is expressed in Equation (2).

$$f=3/2\left(s-p\alpha \right):\left(s-p\alpha \right)-m^2{p}^2=0$$

(2)

where s is the deviator stress tensor, which is determined by s = σ−pδ. p is determined by $p=p^{\prime }+p_0{}^{\prime }$; α is the deflection tensor of the center coordinates in the deviator stress subspace; and m is the size of the yield surface, which is defined by m = 6sinφ/(3−sinφ), where φ is the internal friction angle of the soil.

The outer normal vector of the yield surface is normalized to make it applicable to the three different soil layers in this paper, as shown in Equations (3)–(5).

$$Q=g\mathrm{r}\mathrm{a}\mathrm{d}f/\left|g\mathrm{r}\mathrm{a}\mathrm{d}f\right|$$

(3)

$$g\mathrm{r}\mathrm{a}\mathrm{d}f=\frac{\partial f}{\partial \sigma }=3\left(s-p\alpha \right)+\left[p\left(\alpha :\alpha -2/3m^2\right)-s:\alpha \right]\delta$$

(4)

$$Q:Q=Q^{\prime }:Q^{\prime }+\frac{1}{3{\left(3Q^{″}\right)}^2}=1$$

(5)

where $Q$ is the symmetric second-order tensor of the outer normal of the yield surface, $Q^{\prime }$ is the partial component of the outer normal of the yield surface, $Q^{″}$ is the expansion component of the outer normal of the yield surface, $\delta$ is a symbolic tensor, and “$:$” means the product of two tensors.

2.4. Pile–Soil Interaction

The pile–soil interaction significantly affects the seismic response of the overall wharf structures [43]. Herein, three types of nonlinear springs, namely P–y, T–z, and Q–z springs [16,44], are used to simulate the pile–soil interaction, as shown in Figure 5. Each spring is connected to the node at the intersection of the pile segments on one end and to the adjacent soil element node on the other end [45]. The P–y spring represents the horizontal spring, where y is the horizontal deformation of the spring and P is the horizontal reaction force of the piles. The T–z spring simulates the vertical friction around the piles, where z is the vertical displacement, and T is the friction resistance of the piles. The bearing of each pile end is modeled using the Q–z spring. The parameters of each spring are determined based on the API code [46]. Since the line spacing of piles in this simulation is less than 8 times the pile diameter, the influence of the pile group effect is considered [47]. The correction coefficient λ_h is used to calculate the horizontal resistance P of the P–y spring, and the calculation formula of λ_h is shown in Equation (6):

$${\lambda }_{\mathrm{h}}={\left(\frac{\frac{S_{\mathrm{o}}}{d}-1}{7}\right)}^{0.043\left(10-\frac{z}{d}\right)}$$

(6)

where S_o is the pile spacing (m), d is the pile diameter (m), and z is the depth of the pile embedded in the soil (m).

To enhance the overall efficiency and convergence of the model, sensitivity analyses are performed on the actual mesh size, particularly focusing on the number of spring elements and the size of soil elements at the slope. To determine the number of springs, it is based on the number of segments of the pile since the nodes at both ends of the spring are shared with the pile nodes and the soil nodes, respectively. The verifications demonstrate that dividing the pile body into 10 sections and employing a slope soil size of 50 mm ensure both simulation accuracy and calculation efficiency. As a result, the model is divided into a total of 46,990 elements.

2.5. Boundary Conditions and Seismic Excitation

In the initial state, the soil nodes at the bottom of the model are fixed in three directions. To simulate a viscous boundary and absorb seismic energy, soil columns with heavy mass are placed on both sides [48]. The out-of-plane deformation of the 6.8 m width slice is restricted. The proper shear deformation coordination within the soil layer is ensured through the use of equalDOF to simulate the free field boundary. When the static state is activated by applying gravity to the seabed soil, it remains in the elastic stage with a Poisson’s ratio of 0.4. Following this, the static soil gravity converts into an equivalent load for each soil node. The displacement is subsequently reset to zero using the OpenSees command InitialStateAnalysis to balance the ground stress [49].

In the case of the steel pipe piles, their bottoms are fixed due to their embedding in the seabed soil-bearing layer in actual engineering. An axial load is applied to each pile node to establish gravity equivalence, according to Chiaramonte [50]. The upper deck is rigidly connected to the pile’s top.

For the pile–soil interaction, zero-length elements in OpenSees are added to simulate the soil–pile interaction [8]. Initially, the soil spring is connected to the soil element. To establish a connection between one side node of the zero-length element and the corresponding adjacent soil node, the equalDOF constraint of OpenSees is employed [8] while removing the fixed constraint on the zero-length element node. The other side of the zero-length elements is connected to the pile elements.

In this study, the seismic excitation is applied at the base of the model. During shaking, the constraints on lateral displacement DOF in the shaking direction are released [8].

2.6. Verification of the FE Model

The quasi-static tests conducted by Wang et al. [51] are used to verify the rationality of the model. The simulated load–displacement (V–Δ) curves are compared with the tested ones. As shown in Figure 6, the tested maximum lateral forces observed in the push and pull directions are 64.5 kN and −64.6 kN, respectively, with corresponding maximum displacements of 109 mm and −110 mm, respectively. The maximum lateral forces at the push and pull directions of the FE model are 64.56 kN and −64.77 kN, respectively, and the corresponding displacement values are 97 mm and −97 mm, respectively. The comparison indicates that the FE model accurately simulates the mechanical characteristics of the pile foundation structure, considering the pile–soil interaction.

2.7. Applied Seismic Records

The 1994 Northridge seismic record [52] with a moment magnitude of 6.6 and a PGA value of 0.358 g, as shown in Figure 7, is used for the nonlinear response analysis of the structure. The Fourier spectrum of the Northridge wave shows that the frequency corresponding to its maximum amplitude is 1.3 Hz, and there is no significant amplitude attenuation in the frequency range of 1.1 to 6.3 Hz.

A total of 60 seismic records corresponding to multiple seismic events are used to consider the randomness of seismic intensity and acceleration attenuation. According to the Pacific Engineering Earthquake Research Center (PEER) [53], the peak accelerations (PGAs) of all the 60 seismic records are in the range of 0 to 0.6 g, as shown in Figure 8. Fifty of these seismic records are randomly selected to determine the ground motion parameter (IM) and used for fragility analysis in Section 4.3.

3. Nonlinear Response Analysis

3.1. Slope Response

The lateral displacements (δ) and acceleration (a) responses at different elevation positions of the slope under the non-corrosive condition are shown in Figure 9. The cumulative permanent displacements at elevations of 24.0 m and 36.0 m reach 0.208 m and 0.244 m, respectively. In contrast, the acceleration responses do not change significantly with altitude due to the consistent soil material of slopes at elevations of 24–36 m.

3.2. Deck Response

The FE model fixes the deck to the top of each pile to better reflect the seismic response of the overall wharf structure [36]. The lateral displacement and acceleration time history of the deck are analyzed, as shown in Figure 10. A permanent displacement of 0.227 m is observed on the wharf deck, with the peak displacement occurring at 12.16 s, reaching 0.301 m. The peak acceleration of the deck reaches 0.656 g, significantly amplified compared to the initial input seismic record. The maximum lateral displacement of the deck indicated that the wharf structure reached its limit.

3.3. Pile Deformations

The deformations of each pile at the maximum horizontal displacement and at the end of loading are shown in Figure 11. In Figure 11, the black lines denote the moment of maximum deformation of the pile body, while the red lines represent the end of the loading process. It can be seen that lateral displacements of the piles vary at different elevations, with the peak displacements being greater than the displacement at the end of loading. The displacements of the pile body exposed to the outside of the slope show a significant increase due to the absence of soil spring constraints.

3.4. Slope Deformations

The slope deformation at different loading stages was observed: t = 0.94 s (initial loading stage), t = 7.98 s (permanent displacement accumulation stage), t = 12.16 s (deck lateral displacement reaches the maximum value), and t = 40.00 s (loading terminal). As shown in Figure 12, the largest lateral displacement occurred in the uppermost layer of the soil, mainly due to shear deformation in the weak layer [40]. Significant lateral deformation was also observed in the ground slope area, causing the steel pipe piles to move towards the seaside. The top of each pile experienced the same lateral displacement due to the constraints of the rigid deck. The overall movement of the slope towards the seaside resulted in significant vertical collapse at the top of the slope.

3.5. Response of the Pile’s Internal Force

3.5.1. Shear Force

Under the attack of strong transverse seismic records, the steel pipe pile foundations experience significant shear forces. Figure 13 illustrates that the shear force of the exposed piles above the soil layer remains consistent, while the shear force of the pile body slightly decreases with increasing elevation. The embedded pile body below the soil layer shows irregular changes in shear force distribution, with extreme values basically occurring near the slope surface. The maximum shear force of the piles at the moment of maximum deck displacement is slightly larger than that at the end of loading.

3.5.2. Bending Moment

Performing a moment response analysis on the piles is crucial to gain a comprehensive understanding of the loading and failure modes of the wharf. Figure 14 illustrates the time history curves of the bending moment at the top of each row of piles during seismic action. The E-row piles experience the most severe bending moment response. From the land side to the seaside, the bending moment response of the steel pipe piles gradually decreases, indicating the significant influence of pile–soil interaction on the bending moment at the top of the pile. A deeper embedded depth results in higher peak bending moments at the top of the pile. This can be attributed to the fact that deeper embedded depth results in stronger soil confinement on the pile, thereby enhancing its horizontal stiffness [8,36]. After internal force redistribution, piles with greater embedded depth experience greater bending moments at their top.

Figure 15 displays that the maximum bending moment basically occurs at the intersection of the slope and the pile or at the connection between the top of the pile and the deck, indicating these two locations are most susceptible to damage. The simulated result illustrates the damage position consistent with most wharves after seismic action, such as Tianjin Port after the Tangshan earthquake in China and Kobe Port after the Hanshin earthquake in Japan.

3.5.3. Curvature

The curvature response in Figure 16 shows the law consistent with the bending moment response. The curvature distribution of the pile body is significantly affected by the pile–soil interaction. The elevations corresponding to the maximum negative curvature of each pile are below the slope surface and increase with the increase in embedded depth. The maximum positive curvature occurs at the top of the pile. As the embedded depth increases, the non-embedded part of the pile becomes shorter. Consequently, when the horizontal displacement of the pile top remains constant, a deeper embedded depth leads to a greater curvature of the non-embedded section. As the lateral displacement of the deck reaches its maximum amplitude, the curvature values gradually increase from pile A to pile E, with piles D and E exceeding the calculated yield curvature.

3.5.4. Axial Force

Figure 17 illustrates the response of pile axial force (N) at the maximum horizontal displacement and at the end of loading. The axial force of the pile is not only caused by deck weight and pile body weight but also significantly influenced by the pile–soil interaction under the seismic action. The pile–soil interaction causes the axial force, which should increase uniformly along the pile body, to increase sharply at the slope. From pile A to pile E, the axial force increases more greatly due to the increase in the embedded depth. On the other hand, the top of the pile even experiences axial tensile force due to severe bending deformation, such as pile E.

3.6. Dynamic Response of Structure with Corrosion Steel Pile

3.6.1. Time History Analysis

According to the comparison shown in Figure 18, as the steel piles were pitting corroded by 10%, the peak displacement and peak acceleration of the deck increased by 18.6% and 29.1%, respectively. In addition, the lateral residual displacement of the deck was increased by 9.7% due to greater plastic deformation of the pile body caused by pitting corrosion.

3.6.2. Effect of Corrosion on Internal Force of Pile

Figure 19 illustrates the shear response of each pile after corrosion, while Figure 20 shows the bending moment response. It can be observed that the extreme shear force and bending moment of the corroded piles have increased, whether in the positive or negative direction. Especially for the E-row pile with the largest embedded depth, 10% pitting corrosion caused the pile top shear force to increase by 53.6% and the pile top bending moment to increase by 19.2%. This has a detrimental effect on the structure.

4. Fragility Analysis

4.1. Determination of Damage Thresholds

According to PIANC’s definition of damage state [54], slight damage, moderate damage, and heavy damage are used to describe the damage degree of the wharf. The thresholds for these damage states are determined using the curvature ductility coefficients (μ_φ) of the E-row piles and the displacement ductility coefficients (μ_δ) of the deck. The displacement ductility coefficients μ_δ and μ_φ are calculated as μ_δ = δ_u/δ_y and μ_φ = φ_u/φ_y, respectively. These values are obtained from the pushover analysis results shown in Figure 21. It is considered as slight damage to the wharf structure when the deck displacement δ_m exceeds the yield displacement δ_y. Heavy damage is determined as the point at which the double plastic hinge is developed in pile E (at the cap and embedded portion of the pile), and the corresponding deck displacement is the limit displacement δ_u [36]. Moderate damage is accompanied by a certain degree of residual deformation within a controllable range, and the threshold μ_{δ, Moderate} is between μ_{δ, Slight} (the displacement ductility coefficients of slight damage) and μ_{δ, Heavy} (the displacement ductility coefficients of heavy damage) [36]. The moderate damage threshold is defined as the average value of the threshold of slight damage and heavy damage, as shown in Equation (7). The damage thresholds of the wharf under each corrosion ratio are listed in

Table 3. Damage thresholds under each corrosion ratio.

Ductility Parameters	Corrosion Ratio	Slight Damage	Moderate Damage	Heavy Damage	δy or φy	δu or φu
μδ	0%	1	1.90	2.80	0.169	0.473
	5%	1	1.79	2.58	0.161	0.415
	10%	1	1.73	2.46	0.151	0.371
	15%	1	1.68	2.37	0.141	0.334
μφ	0%	1	1.71	2.43	0.0053	0.0131
	5%	1	1.64	2.27	0.0048	0.0109
	10%	1	1.52	2.05	0.0044	0.0090
	15%	1	1.49	1.98	0.0041	0.0081

.

$${\mu }_{\mathsf{\delta },\mathrm{Moderate}}=\left({\mu }_{\mathsf{\delta },\mathrm{Slight}}+{\mu }_{\mathsf{\delta },\mathrm{Heavy}}\right)/2$$

(7)

4.2. Seismic Demand Models

Considering the randomness of seismic intensity, each corroded model is subjected to 50 random seismic responses. The peak ground acceleration PGA is used as the ground motion parameter (IM), and the simulated curvature ductility coefficients are used as the structural engineering demand parameters (EDPs) [55]. Cornell et al. [56] pointed out that the seismic demand of the structure obeys the assumption of normal distribution, and the relationship between EDP samples and IM is shown in Equation (8). The fitted seismic demand models of the E-row pile, which are severely affected by corrosion, are shown in Figure 22.

$$\ln \overline{EDP}=a+b\ln \left(IM\right)$$

(8)

4.3. Fragility Analysis

The fragility index of the wharf, that is, the probability of the wharf reaching critical damage under seismic action, is calculated according to reference [57], and the calculation method is shown in Equation (9):

$$F_R(x)=P_f=P\left[S_{\mathrm{D}}\ge S_{\mathrm{L}}\left|IM=x\right.\right]$$

(9)

where P_f is the conditional probability that the seismic demand of the structure exceeds the capacity, S_D is the estimate of the demand as a function of IM, and S_L is the estimated capacity threshold of the wharf.

The lognormal distribution model is used to describe the seismic fragility function [58]. The seismic fragility function takes into account the randomness of seismic intensity, as shown in Equation (10):

$$P_f=\mathrm{\Phi }\left(\frac{\ln S_{\mathrm{D}}-\ln S_{\mathrm{L}}}{\sqrt{{\beta }_{D\left|IM\right.}^2+{\beta }_{\mathrm{L}}^2}}\right)$$

(10)

where β_D|IM is the logarithmic standard deviation of the seismic demand of the structure under each IM, which is calculated by Equation (11). β_L is the logarithmic standard deviation of the threshold of the specified damage state, and $\sqrt{{\beta }_{D\left|IM\right.}^2+{\beta }_{\mathrm{L}}^2}$ is the combined logarithm standard deviation.

$${\beta }_{D\left|IM\right.}=\sqrt{\frac{{\displaystyle {\sum }_i^N{\left(\ln D_i-\ln S_{\mathrm{D}}\right)}^2}}{N-2}}$$

(11)

where D_i represents the seismic demand response obtained as the i-th seismic record was input; lnS_D is calculated by lnS_D = a + blnIM, where a and b are coefficients of the fitted probabilistic seismic demand model; and N is the number of nonlinear time history analysis. The obtained fragility curves of the wharf are shown in Figure 23.

The fragility curve demonstrates that the increase in corrosion ratio leads to the intensification of seismic damage in each part of the wharf. Random pitting corrosion not only reduces the material properties of the steel pipe in the splash/tidal zone but also indirectly increases the probability of seismic damage to the wharf. Figure 24 shows that the upper connections of E-row piles, being the most fragile position, have a significantly increased failure probability corresponding to PGA = 1.0 g with the increase in the corrosion ratio. In the reconstruction and reinforcement of actual marine engineering, it is important to emphasize the anti-corrosion measures for the splash/tidal zone. Additionally, substituting some of the straight piles with inclined piles can effectively reduce the shear force and bending moment experienced by the remaining straight piles. This is because inclined piles primarily resist seismic forces through axial forces rather than relying on bending moments and shear forces.

5. Conclusions

Three-dimensional FE models of steel pipe pile wharves with randomly pitting corrosion were established, and the random seismic response was applied. According to the dynamic response analysis and seismic fragility analysis results, the conclusions are as follows:

(1) Wharf failure mainly occurs as pile body damage and slope collapse. The joints connecting the pile top and the deck have the highest probability of damage. In addition, as the embedded depth increases, the stress concentration at this joint is intensified due to the increased horizontal stiffness of the pile.

(2) Pitting corrosion reduces the lateral resistance and the plastic deformation capacity of the steel pipe pile wharf by weakening the local strength of the pile body, resulting in more significant peak acceleration, displacement amplitude, and residual deformation of the wharf.

(3) The seismic damage possibility of each component of the steel pipe pile wharf increases with the increase in the pitting corrosion ratio. Among them, the seismic damage probability of the steel pipe pile with the largest embedded depth is significantly higher than that of other row piles under various pitting corrosion ratios.

The utilization of artificial intelligence and neural networks is immensely beneficial in addressing engineering challenges, particularly in tackling nonlinear and complex problems. Therefore, it is necessary to use artificial intelligence and neural networks to investigate the effects of random corrosion and random seismic action on the damage to wharf structure in future research.