1 Introduction

The port serves as the hub for water-land transshipment, which has immense impacts on the economic and social activities in the region and even the entire country. Once the interruption in the operation of port occurs, the economic loss would be tremendous and widespread. The economists have quantified the economic consequences of closure for a port on the west coast of United States. The results show that a twenty days’ port shutdown scenario would lead to a 2.5 billion dollars’ loss to the GDP (Gross Domestic Product) for United States [1]. There are many reasons causing port closure, such as labor strike, terrorist attack, natural disaster, etc., one of which should be the earthquake. In recent years, significant damages to port facilities resulting from earthquakes in U.S. and Haiti have obviously illustrated that strong earthquake can severely affect port facilities, especially wharves [2]. The 1989 Loma Prieta, California, earthquake caused considerable damages to pile-supported wharf at 7th Street Terminal of the Port of Oakland, and the damages mainly occurred at the pile-deck connections [3]. Two pile-supported wharves in Port-au-Prince suffered extensive damages during 2010 Haiti earthquake [4]. One wharf mainly collapsed, and about one third of the other one collapsed. Thus, above historical cases have shown that wharves are vulnerable to earthquake damages, and there is a necessity to conduct seismic performance assessment for wharves. Due to the uncertainty of earthquake ground motion, probabilistic seismic assessment has drawn the attention of researchers in port industry, and fragility analysis has been the most popular assessment procedure. The analysis can provide the failure probability of occurrence for different prescribed damage states (performance requirements) as a function of a seismic intensity measure (IM) representative of the earthquake loading, and the geometric expression of probability is fragility curve. The occurrence of a structure failing to meet a damage state is defined by the case where seismic demand (D) exceeds seismic capacity (C), and the nonlinear time-history analysis is recognized as the most accurate demand analysis method.

Many researchers have conducted analyses to develop fragility curves of pile-supported wharves. Shafieezadeh [5] conducted fragility analysis for a wharf on the west coast of United States based on nonlinear time-history analysis, and concluded that peak ground velocity is the optimal intensity measure. Chiou et al. [6] developed fragility curves for a wharf in Taiwan, China by employing a two-dimensional modeling, and the deck displacement and peak ground acceleration were used as performance index and intensity measure in the study, respectively. Yang et al. [7] derived fragility curves for a wharf in U.S. based on the two-dimensional modeling, and the curvature of pile-deck connection was utilized as performance index. Thomopoulos and Lai [8] also employed the curvature to conduct fragility analysis for wharf. Nevertheless, Heidary-Torkamani et al. [9] took the displacement ductility factor as the performance index, and spectral acceleration was used as intensity measure. Amirabadi et al. [10] have investigated the impact of intensity measure on fragility curves, and concluded that spectral intensity measures are optimal. Shah [11] conducted fragility analyses for typical wharves in India, and the transverse displacement of deck was utilized as performance index. Su et al. [12] studied the influence of soil-pile interaction on fragility curves, and Su et al. [13] investigated the impact of soil permeability. Apparently, it should be pointed out that the deck displacement is the most commonly used performance index, for which displacement-based seismic design method has been adopted in current practice [14], and nonlinear time-history analysis is the main method to determine displacement in the above-mentioned researches.

Aforementioned studies highlight the significance of dynamic time-history analysis in fragility analysis. However, due to the presence of soil-pile interaction and pile material nonlinearity, the dynamic analysis of wharf is very computationally expensive for seismic demand analysis, especially in fragility analysis where demand analyses should be carried out for a large number of earthquake records. It is noteworthy that the most commonly used seismic performance index, i.e., deck displacement, is the global displacement of wharf, rather than the local deformation of pile. In this case, it is of great value to simplify the actual structure with an equivalent single-degree-of-freedom (SDOF) model. This approach has been adopted in seismic demand analysis for several years [15]. Thus, for dynamic time-history analysis, a SDOF model could be subjected to a relatively large quantity of earthquake records to provide a good representation of the uncertainty associated with global displacement demand due to the variability of the ground motion. For the knowledge of the authors, an equivalent SDOF model has not been proposed for dynamic analysis of pile-supported wharf.

Therefore, an equivalent SDOF model is established in this study by adopting reasonable restoring model to mimic hysteresis characteristic for wharf with soil-pile interaction, and then the SDOF model is applied to seismic fragility analysis. To validate the applicability, two case studies are conducted to compare the difference of fragility curves from actual wharves and their corresponding SDOF model. In the next section, the framework for constructing equivalent SDOF model for wharf is presented, and Sect. 3 shows the basic methodology of seismic fragility for wharf. Then we present the validation of proposed SDOF model in fragility analysis in Sect. 4.

2 Framework for constructing equivalent SDOF model for wharf

2.1 Modeling of wharf and illustration of SDOF model

For seismic analyses, the practical pile-supported wharf is commonly modeled by the beam on nonlinear Winkler foundation method, as illustrated in Fig. 1a. The soil-pile interaction is modeled by the soil springs, and the pile nonlinearity is represented by the plastic hinges along the length of piles. Taking into account the fact that the major seismic mass is concentrated at the deck level, an equivalent SDOF system is intended to be an appropriate representation of the actual wharf, as shown in Fig. 1b. Once the SDOF system has the same dynamic characteristics (i.e. restoring force characteristics, seismic mass and dam**) with the practical wharf, they will possess the identical displacement response. Then, it is recommended that the demand displacement of wharf can be evaluated by time history analysis of the equivalent SDOF system, whose governing equation of motion can be given as follows:

$$m\ddot{\Delta } + c\dot{\Delta } + F_{S} (\Delta ,\dot{\Delta }) = - m\ddot{\Delta }_{g} (t)$$
(1)
Fig. 1
figure 1

Analytical model of pile-supported wharf and corresponding SDOF model

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where \(\Delta\), \(\dot{\Delta }\), \(\ddot{\Delta }\) are displacement, velocity and acceleration, respectively; \(m\) and \(c\) are mass and viscous dam** coefficient, respectively; \(F_{S} (\Delta ,\dot{\Delta })\) is the restoring force, and \(\ddot{\Delta }_{g} (t)\) is the acceleration of ground motion.

To solve Eq. (1), a nonlinear SDOF dynamic time-history analysis is commonly used. In the analysis, the restoring force can be determined with the cyclic pushover analyses of actual wharves, which is discussed in the following paragraphs.

2.2 Restoring force model for SDOF system

2.2.1 Backbone curve

Restoring force model is composed of backbone curve and hysteresis rule. For concrete structures, the strength loss always occurs after the peak has been reached. In order to incorporate that phenomenon, a trilinear backbone curve is adopted in this study for ease, which is composed of three branches, i.e. elastic, hardening and degrading, as illustrated in Fig. 2. The first, second and third branches represent elastic stiffness K, hardening stiffness \(r_{1} K\) and degrading stiffness \(- r_{2} K\) (\(r_{2} > 0\)), respectively.

Fig. 2
figure 2

Backbone curve and Pivot hysteresis rule

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Common sense would seem to suggest that the structure subjected to monotonic loading possesses greater strength than that subjected to cyclic loading, which has been demonstrated by experimental and numerical investigations for the effect of loading history on the response of reinforced concrete frames [16]. Consequently, it is recommended that an envelope curve of hysteretic loops for pile-supported wharf should be approximated to obtain the trilinear backbone curve. However, sometimes the aforementioned strength deterioration is so slight that can be ignored. In addition, the monotonic pushover analysis is always utilized to determine the displacement capacity in displacement-based design. Therefore, the trilinear backbone curve is determined from the approximation of pushover curve in the paper for simplicity, as shown in Fig. 3, where the elastic stiffness K is taken as the slope of the line that starts from the pushover curve origin point to the point of the first plastic hinge formed in a pile in accordance with reference [14].

Fig. 3
figure 3

Determination of backbone curve

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2.2.2 Hysteresis rule

Upon the backbone curve has been determined, an appropriate hysteresis rule is need to be established based on the shape of hysteresis loops for wharves. Gao et al. [17] concluded that Pivot hysteresis model proposed by Dowell et al. [18] can represent the hysteresis characteristic of wharf, as shown in Fig. 2, and determined the values of model parameters \(\alpha\) and \(\beta\). It is suggested that Eqs. (2) and (3) from [17] can be utilized to calculate \(\alpha\) and \(\beta\) for cast-in-situ pile-supported wharf, respectively. Furthermore, Eqs. (4) and (5) can be applied to prestressed high-strength concrete (PHC) pipe pile. For piles different from those, Pivot model parameters can be determined in accordance with procedure in reference [17]. In Eqs. (2)–(5), \(\rho_{sl}\) is the ratio of longitudinal reinforcing bar for cast-in-situ pile with the unit of percentage %, and \(\rho_{sp}\) is the ratio of prestressed bar.

$$\alpha { = }\left\{ {\begin{array}{*{20}c} {6.464\rho_{sl} + 4.206} & {\text{for dike with sand}} \\ {8.241\rho_{sl} + 11.176} & {\text{for dike with clay}} \\ \end{array} } \right.$$
(2)
$$\beta { = }\left\{ {\begin{array}{*{20}c} {0.703\rho_{sl}^{0.284} } & {\text{for dike with sand}} \\ {0.627\rho_{sl}^{0.251} } & {\text{for dike with clay}} \\ \end{array} } \right.$$
(3)
$$\alpha { = }\left\{ {\begin{array}{*{20}c} {3.784\rho_{sp} - 0.452} & {\text{for dike with sand}} \\ {6.159\rho_{sp} + 2.702} & {\text{for dike with clay}} \\ \end{array} } \right.$$
(4)
$$\beta { = }\left\{ {\begin{array}{*{20}c} {0.347\rho_{sp}^{1.065} } & {\text{for dike with sand}} \\ {0.324\rho_{sp}^{0.795} } & {\text{for dike with clay}} \\ \end{array} } \right.$$
(5)

It should be pointed out that above equations are limited to homogeneous dike, which is conflict to the actual situation to some extent. However, the soils that located at the range of 10 times of the pile diameter below dike surface generally have significant impact on seismic response [14]. Thus, if clay or sand is predominant in this range, aforementioned formulas are recommended, otherwise cyclic pushover analyses should be carried out to determine Pivot model parameters with methodology in reference [17].

2.2.3 Mass and viscous dam** coefficient for SDOF system

In current seismic design practice, ASCE [14] recommends that the seismic mass for the seismic analysis shall include the mass of the wharf deck, permanently attached equipment, 10% of the design uniform live loads or 100 psf. In addition, 1/3 of the pile mass between the deck soffit and 5 times of the pile diameter below the dike surface shall be considered as additional mass lumped at the deck. Apparently, it is inadequate that the percentage of pile mass incorporated in seismic mass is one constant value for various piles and soil types. For determining the mass m of SDOF system reasonably, the following equation is utilized:

$$m = \frac{{KT^{2} }}{{4\pi^{2} }}$$
(6)

where T is the elastic period of the wharf mode with the maximum modal participating factor in transverse direction. Thus, the viscous dam** coefficient c for SDOF system can be defined as following:

$$c = 2m\omega \xi$$
(7)

in which \(\omega\) is the nature frequency corresponding to T, and is denoted as \(\omega { = }2\pi {/}T\), \(\xi\) is the equivalent dam** ratio, and is taken as a value of 5% in this study.

Consequently, an equivalent SDOF model is proposed in this study to represent the actual wharf for displacement demand analysis. A framework for develo** SDOF model for a wharf is summarized as follows:

  1. (1)

    Establish the the two-dimensionally numerical model of a target pile-supported wharf and conduct pushover analysis. The force–deformation relation of the practical wharf can be determined by the pushover curve, and the curve can be approximately represented by trilinear backbone curve in accordance with Fig. 3 for ease, which will be used as the backbone curve of SDOF oscillator.

  2. (2)

    Determine the Pivot model parameters (i.e. \(\alpha\) and \(\beta\)) of wharf based on the soil type of dike and ratio of longitudinal reinforcing bar of concrete pile based on Eq. (2) to Eq. (5). Thus, the restoring force model of SDOF system is obtained, which means the SDOF system will approximately possess the same restoring force characteristics with the actual wharf.

  3. (3)

    Conduct modal analysis of wharf to acquire the elastic period T with the maximum modal participating factor, and calculate seismic mass m and viscous dam** coefficient c for SDOF system in terms of Eq. (6) and Eq. (7), respectively. Thus, the SDOF system has the same inertial mass and dam** property with the actual wharf.

3 Seismic fragility analysis

3.1 Fragility function

The seismic fragility (or vulnerability) can be defined as the conditional probability of a structure reaching a prescribed damage limit state for an intensity measure (IM) representative of the earthquake loading, and is generally expressed as the fragility function [19]. Then, the geometric representation of the function is fragility curve. Due to the fact that deck displacement has been used as performance index in current seismic practice, the displacement is adopted as engineering demand parameter to develop fragility curves in this study. Assuming that both displacement demand and capacity can be described by lognormal distributions, the probability of the wharf exceeding a particular damage state for a given IM is expressed as following [19]:

$$P\left( {C < D\left| {IM} \right.} \right) = 1 - \Phi \left( {\frac{{\ln \mu_{C} - \ln \mu_{D} }}{{\sqrt {\beta_{C}^{2} + \beta_{D}^{2} } }}} \right)$$
(8)

in which C and D are the displacement capacity and demand of wharf, respectively; \(\Phi ( \cdot )\) is the cumulative distribution function of standard normal distribution; \(\mu_{C}\) is the average displacement capacity under a prescribed damage state; \(\mu_{D}\) is the median displacement demand for a given IM; \(\beta_{C}\) and \(\beta_{D}\) are the logarithmic standard deviations associated with capacity and demand, respectively. Furthermore, a value of 0.3 is recommended for \(\beta_{C}\) when pushover analysis is employed to determine \(\mu_{C}\) [19].

There are many methods for deriving probabilistic seismic demand analysis, among which the computation cost of cloud method is relatively small [20]. Thus, the method is incorporated in this paper to determine the parameters \(\mu_{D}\) and \(\beta_{D}\). During analysis, displacement demand can be expressed as a power function of the following form:

$$D\left( {IM} \right) = aIM^{b} \varepsilon$$
(9)

where ε is a lognormal random variable with mean value of one, whose logarithmic standard deviation is \(\beta_{D}\); a and b are constants, which can be estimated by using least square regression analysis for demand model in transformed logarithmic space as follows:

$$\ln \left[ {D\left( {IM} \right)} \right] = \ln a + b\ln \left( {IM} \right) + \ln \left( \varepsilon \right)$$
(10)

Therefore, the estimated values of a and b, denoted by \(\widehat{a}\) and \(\widehat{b}\) are unbiased estimators by assuming that \(\ln \left( \varepsilon \right)\) is normally distributed. Thus, the natural logarithm of \(\mu_{D}\) for a given IM is denoted as:

$$\ln \mu_{D} = \ln \hat{a} + \hat{b}\ln \left( {IM} \right)$$
(11)

By assuming that the dispersion of displacement demand does not depend on intensity measure, the uncertainty of demand, i.e.\(\beta_{D}\), can be determined as following:

$$\beta_{D} = \sqrt {\frac{{\sum\limits_{i = 1}^{n} {\left[ {\ln \left( {D_{i} } \right) - \ln \mu_{D} } \right]^{2} } }}{n - 2}}$$
(12)

in which \(D_{i}\) is the displacement demand of nonlinear time-history for i-th earthquake wave; n is the total number of earthquake records incorporated in fragility analysis. It is important to notice that the contribution of the uncertainty in material is not considered in this paper, for which it is found to be negligible compared to the uncertainty of ground motion on the overall response variability [21].

3.2 Damage states

The displacement capacity is based on the capacities of structural components of wharf, which are defined in terms of limit state models. The limit states for components are defined by qualitative damage states, such as low, moderate, and high, as shown in Table 1, which are adopted from reference [14]. The low damage state possesses a high probability of occurrence during the service life in which minor structural damage is allowed. On the contrary, the moderate state has a lower probability of occurrence, where only reparable damages are permitted. All other damage states that are less probable and more serious than above two states shall be assigned a classification of high. Each damage state is associated with pile plastic hinge zone strain limits defined in Table 1, in which \(\varepsilon_{c}\) is concrete compression strain, \(\varepsilon_{s}\) is steel tensile strain and \(\varepsilon_{p}\) is total prestressing steel tensile strain. The deep in-ground hinge in the table refers to the one occurs at more than ten pile diameters below dike surface. The evaluation of displacement capacity requires a pushover analysis of wharf by monitoring the hinge material strains during analysis. The displacement can be determined at the instant when one of the materials reaches the strain limit value listed in Table 1.

Table 1 Damage states for wharves
Full size table

4 Case studies

4.1 Ground motions

To generate probabilistic seismic demand model for deck displacement, representative ground motions as the input for nonlinear time-history analysis should be selected. There are various selection procedures of ground motions at present. As this study merely aims to validate the applicability of proposed SDOF model in fragility analysis, a suit of 60 ground motions was selected with simple criteria, i.e., (1) the average shear wave velocity for top 30 m of soil profile should be no less than 180 m/s, and (2) minimum peak ground acceleration (PGA) is 0.20 g. For more information about ground motions, the readers are referred to reference [17]. Considering that this study is just to validate the reasonability of proposed procedure, the PGA is used as IM in the study for its popularity in fragility analysis. The applicability of different IMs, e.g., peak ground velocity, peak ground displacement, and spectral acceleration, is not evaluated in the study.

4.2 Case study I

4.2.1 Geometry and modeling

The transverse section of wharf is shown in Fig. 4. The beam and slab system consists of transverse cap beams with the width of 1.5 m and composite slabs with the thickness of 0.45 m. The spacing of bent is 6.3 m. The widths of generally longitudinal beams are 1.5 m, while the crane beams possess a width of 1.6 m. A typical value of 40 kPa is used for uniform live load. The segment length is 47.1 m. The properties of piles and dike soils are listed in Table 2, in which Dp is the pile diameter, s is center-to-center spacing of confining steel along pile axis, γ is effective unit weight and ϕ is internal friction angle. The C40 concrete that has a compressive strength of 26.8 MPa. Moreover, the individual pile is reinforced with 20 HRB400 (a Chinese steel grade with a yield strength of 400 MPa) bars.

Fig. 4
figure 4

Transverse section of wharf

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Table 2 Properties of piles and dike soil for case study I
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Two-dimensional model was constructed by the widely used program SAP2000 due to the uniformity and symmetry of the wharf in accordance with Fig. 1a. The backbone curves of springs were determined by P-y curves, in which P refers to soil resistance per unit pile length, y is the deformation of pile. The pile group effect is ignored for ease due to the fact that the pile spacing is generally larger than six times of the pile diameter. The distributed plastic hinge model recommended by Chiou et al. [22] is utilized to represent the nonlinearity of pile without calculating the hinge lengths. As the axial forces of piles vary when subjected to whether monotonic or cyclic loading, the fiber hinges in SAP2000 are adopted.

4.2.2 Static pushover analysis

Two-dimensional static pushover analysis was conducted to obtain the pushover curve and the sequence of formation for plastic hinges. Figure 5 shows the pushover curve and its trilinear approximation, which is used as the backbone curve of equivalent SDOF system. The Pivot model parameters were determined by Eqs. (2) and (3), resulting in \(\alpha = 12.29\) and \(\beta = 0.75\). Thus, the restoring force model for equivalent SDOF system has been established. During analysis, the material strains were monitored to determine displacement capacity. Consequently, the deck displacements for low, moderate and high damage states are 1.64, 4.42 and 5.69 cm, respectively.

Fig. 5
figure 5

Pushover curve for cast-in-situ pile-supported wharf

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4.2.3 Modal analysis

Modal analysis is performed to determine the elastic period of the wharf mode with the maximum modal participating factor in transverse direction, resulting in T = 0.5469 s. The elastic stiffness K is determined as a value of \(5.5369 \times 10^{4} {\text{kN/m}}\) from pushover curve. Then mass and viscous dam** coefficient for the SDOF system are calculated as \(4.195 \times 10^{5} {\text{kg}}\) and \(481.94{\text{kN}} \cdot {\text{s/m}}\), respectively, by using Eqs. (6) and (7). So far, the equivalent SDOF system has been developed. Then, the SDOF model can also be constructed in SAP2000 with the elements in the program.

4.2.4 Fragility analysis

The approach in aforementioned section is utilized to develop fragility curves for actual wharf and its equivalent representation. The term “actual wharf” is referred to the refined finite element model established by SAP2000, which is the numerical model of practical wharf. Thus, seismic demand analyses were conducted for both models by nonlinear time-history analyses with above ground motions. Since the two-dimensional model cannot address the torsional effect under bidirectional ground motion, the dynamic magnification factor (DMF) recommended by the current practice [14] was used to consider this effect by multiplying the transverse displacement demand with DMF to obtain the total demand. The equation for DMF is listed as:

$$DMF = \sqrt {1 + \left[ {0.3\left( {1 + {{20e_{0} } \mathord{\left/ {\vphantom {{20e_{0} } {L_{l} }}} \right. \kern-\nulldelimiterspace} {L_{l} }}} \right)} \right]^{2} }$$
(13)

in which \(L_{l}\) is the segment length; \(e_{0}\) is the eccentricity between the center of rigidity and mass. For this study, the DMF is calculated as 1.696 by the above formula.

The total displacement demands of actual wharf and its equivalent SDOF model under different ground motions are illustrated in Fig. 6, in which \(\Delta_{{{\text{SDOF}}}}\) and \(\Delta_{{{\text{Wharf}}}}\) are the displacements of SDOF model and actual wharf, respectively. It is observed that \(\Delta_{{{\text{SDOF}}}}\) is in good agreement with \(\Delta_{{{\text{Wharf}}}}\), which validates the applicability of equivalent SDOF model in demand analysis. Furthermore, Fig. 7 shows the displacement time histories of SDOF system and actual wharf under a certain ground motion. It should be pointed out that the results in Fig. 7 are the transverse displacement demands without multiplication by DMF. It can be found that the displacement time history of SDOF model is consistent with that of actual wharf, especially at their maximum. Moreover, it is worthy of explaining the time efficiency and disk usage of the SDOF model during nonlinear time-history analysis. The analysis of SDOF model under one ground motion only elapsed several minutes regardless of the magnitude of time duration for earthquake wave to some extent, and took up hundreds of megabytes of disk space. However, the elapsed time for actual wharf was dependent on the magnitude of time duration for earthquake record with a minimum value of about 1 h. Especially for the usage of disk, an analysis could take up dozens of gigabytes due to the fiber hinges of SAP2000, which are used for refined modeling of pile nonlinearity. Thus, multiple analyses with different waves cannot be running simultaneously on ordinary computer, which makes the demand analyses time-consuming.

Fig. 6
figure 6

Displacement demands for case study I

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Fig. 7
figure 7

Displacement time histories of SDOF system and actual wharf under a certain ground motion for case study I

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The seismic demand models for actual wharf and its SDOF model were developed in terms of PGA, as shown in Fig. 8. As described before, regression analyses were utilized to determine parameters of fitted lognormal distribution for demand models in Eq. (10). The estimated values of a and b are presented in Eqs. (14) and (15) for actual wharf and SDOF model, respectively. Furthermore, the values for \(\beta_{D}\) of these two models are 0.4550 cm and 0.4316 cm. Then the resulting fragility curves for low, moderate and high damage states are developed by Eq. (8), as shown in Fig. 9.

$$\ln \left( {\mu_{D} } \right) = 0.{4149}\ln \left( {PGA} \right) + {2}{\text{.6387}}$$
(14)
$$\ln \left( {\mu_{D} } \right) = 0.{4500}\ln \left( {PGA} \right) + 2.{6689}$$
(15)
Fig. 8
figure 8

Probabilistic seismic demand models for actual wharf and SDOF model in case study I

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Fig. 9
figure 9

Fragility curves for case study I

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From the suit of fragility curves in Fig. 9, little difference can be found between the curves for actual wharf and those for SDOF model. Furthermore, moderate and high damages of SDOF model are more probable than the occurrence of the identical damage states of actual wharf for PGA more than 0.2 g, which means the SDOF model may provide a conservative assessment for fragility analysis.

4.3 Case study II

4.3.1 Geometry and modeling

The transverse section illustrated in Fig. 4 is also utilized for case study II, and the properties for pile and soil are summarized in Table 3, where t is wall thickness for hollow prestressed pile, \(\sigma_{{{\text{con}}}}\) is the stretching control stress of tendon. The individual pile is reinforced with 48 prestressed tendons, and the ratio of longitudinal reinforcing steel (i.e.\(\rho_{sp}\)) is 0.87%, and the C80 concrete has a compressive strength of 50.2 MPa. Moreover, the modeling of wharf is identical to case study I.

Table 3 Properties of piles and dike soil for case study II
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4.3.2 Static pushover analysis

Two-dimensional static pushover analysis was performed, resulting in the pushover curve and its trilinear approximation shown in Fig. 10. The Pivot model parameters were calculated by Eqs. (4) and (5) with the result of \(\alpha = 2.84\) and \(\beta = 0.30\). As well, the displacement capacity for low, moderate and high damage states were determined as 2.53, 3.82 and 4.64 cm, respectively.

Fig. 10
figure 10

Pushover curve for case study II

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4.3.3 Modal analysis

The elastic period of the wharf mode with the maximum modal participating factor in transverse direction was determined as a value of 0.4055 s by modal analysis. The elastic stiffness K was determined as a value of \(7.3728 \times 10^{4} {\text{kN/m}}\) from pushover curve. Then the mass and viscous dam** coefficient for the SDOF system were determined as \(3.071 \times 10^{5} {\text{kg}}\) and \(475.82{\text{kN}} \cdot {\text{s/m}}\), respectively. Thus, the equivalent SDOF model can be constructed.

4.3.4 Fragility analysis

Displacement demand analyses were conducted, and the corresponding results are shown in Fig. 11. Moreover, the displacement time histories of SDOF model and actual wharf under a certain ground motion are plotted in Fig. 12. Similarly, favorable consistency and correlation can be found in the displacement responses of these two models.

Fig. 11
figure 11

Displacement demands for case study II

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Fig. 12
figure 12

Displacement time histories of SDOF system and actual wharf under a certain ground motion for case study II

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The seismic demand models for case study II were developed as shown in Fig. 13, and the estimated values of a and b are listed in Eqs. (16) and (17) for actual wharf and SDOF model, respectively. As well, the values for \(\beta_{D}\) of these two models were determined as 0.4230 cm and 0.3837 cm. Thus, the fragility curves for various damage states are illustrated in Fig. 14. Likewise, the fragility curves of SDOF model can be found above those of actual wharf, which signifies that a conservative result can be provided by the SDOF representation. Moreover, the difference between curves for two models increases a bit compared to case study I, however, the difference is still tolerable.

$$\ln \left( {\mu_{D} } \right) = 0.{8788}\ln \left( {PGA} \right) + {2}{\text{.5647}}$$
(16)
$$\ln \left( {\mu_{D} } \right) = 0.{8487}\ln \left( {PGA} \right) + 2.{5738}$$
(17)
Fig. 13
figure 13

Probabilistic seismic demand models for actual wharf and SDOF model in case study II

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Fig. 14
figure 14

Fragility curves for case study II

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5 Conclusion

This study presents the derivation of an equivalent SDOF model of actual wharf, and its applicability in fragility analysis are validated with two case studies. The SDOF system is developed to capture the characteristics of stiffness and strength degradation of the actual wharf. During probabilistic seismic demand analysis, nonlinear time history analyses with a set of 60 ground motions were conducted for both actual wharf and its SDOF substitute. It should be reminded that the results of this investigation are limited to regular wharves. The primary conclusions are as follows:

  1. 1.

    The seismic demand analysis for SDOF model can undoubtedly reduce the computation cost compared to the analysis of actual wharf at the case where tremendous ground motions are incorporated in analysis. It is important to notice that the engineering demand parameter must be the total displacement of wharf. Coincidentally, the displacement has been the performance index of current seismic design, and pushover analysis is used to determine displacement capacity in design. Fortunately, the pushover analysis is needed in the construction of SDOF model, which means the constructing framework can coordinate with current design practice.

  2. 2.

    The validation implies the preferable applicability and accuracy of proposed SDOF model in fragility analysis, which indicates that the trilinear backbone curve and Pivot hysteresis rule are qualified to simulate the nonlinear characteristics of concrete pile-supported wharves. Furthermore, the SDOF representation seems to provide a conservative assessment result to some extent.

  3. 3.

    Unfortunately, nonlinear time history analyses still need to be conducted with some complexity for design engineer who wants to know the vulnerability of wharf. Consequently, parametric analyses should be conducted for proposed SDOF model in further study to gain standardized fragility curves of wharves with appropriate selection of ground motions.