1 Introduction

Quay walls are a typical mooring facility. Conventionally, their seismic performance is evaluated through simplified analysis based on the pseudo-static method, simplified dynamic analysis based on the sliding block method, or dynamic analysis using finite element or finite difference numerical techniques [2, 9, 12, 14, 22, 23, 37, 38, 40]. Since the Great Hanshin Earthquake-damaged Kobe Port in 1995, significant advances have been made in dynamic analysis, which can comprehensively consider soil–structure interaction and soil nonlinear behavior in response to an earthquake, and performance-based design has been introduced for port structures. However, challenges remain for dynamic analysis in that the results differ depending on the designer’s understanding of the numerical analysis and the selection of input parameters such as those for the constitutive models, mesh discretization, and material properties. Therefore, simplified analysis and simplified dynamic analysis, which offer the advantages of being capable of quickly and easily evaluating the seismic stability of structures, are still widely used in countries that apply prescriptive seismic design based on national seismic codes for the sake of practicality [9, 14, 22, 23, 40, 49].

In the simplified analysis, the active thrust is calculated by adopting the Coulomb earth pressure theory to determine the magnitude of the soil thrust acting on the wall and the Mononobe–Okabe (M–O) [42, 50] method to consider the seismic earth pressure. For the M–O method, defining an appropriate seismic coefficient (kh) is crucial for obtaining an analysis result close to the real seismic response because the earthquake motion is transduced to an equivalent pseudo-static inertial force through kh.

The conventional kh is simply defined as the peak ground acceleration (PGA) of the design ground motion, which is determined through a probabilistic seismic hazard analysis, divided by the acceleration of gravity (g). It does not sufficiently reflect the characteristics of real dynamic motions and is conservative. To improve the definition of kh to consider the seismic performance of a quay wall and site amplification effects, several methods have been considered, such as applying a correction factor according to the allowable deformation of the quay wall crown (Da) [14, 23, 37, 40, 60] or changing the selected PGA location for calculating kh according to the wall height (Hw) [14, 22, 23, 33, 40]. The representative definitions of kh are summarized in Table 1.

Table 1 Overview of the representative seismic coefficient (kh) definitions (modified from Lee et al. [33])
Full size table

However, the simplified analysis does not provide information on the performance of a structure when the force balance limit is exceeded. Various studies have focused on the effects of the frequency characteristics of the input earthquake [10, 17, 18, 44, 49], stiffness of the soil [20, 21, 43], and phase characteristics of the wall inertial forces and dynamic earth pressure [1, 8, 10, 45, 54, 57, 58] on the deformation of the quay wall (Dh) to overcome this limitation.

The simplified dynamic analysis adopts the Newmark sliding block theory [47] to evaluate Dh during earthquakes. This theory obtains Dh by double integration of the given design earthquake acceleration time history with the critical acceleration (acr) used as the reference datum. Here, acr is defined as the minimum horizontal acceleration resulting in a safety factor of 1 for sliding failure of the wall–backfill system. Because the M–O method is used to evaluate the sliding stability of the wall and backfill, simplified dynamic analysis adopts similar assumptions as simplified analysis.

Most simplified dynamic analysis methods are extensions of the Newmark theory to improve a certain functional relationship between the expected Dh, acr, and representative characteristic parameters of the earthquake record (MP). Representative simplified dynamic analysis methods are listed in Table 2. The main assumptions and limitations of each method are well described by Al-Homoud and Tahtamoni [2], Cai and Bathurst [9], Deyanova et al. [12], and Meehan and Vahedifard [36]. Simplified dynamic analysis methods do not include variables related to soil conditions or the geometric characteristics of the wall. MP consists of the PGA and peak ground velocity (PGV) and does not fully reflect the frequency components below 1 Hz, which are known to be primarily related to Dh induction, or the duration. Because the proposed simplified dynamic analysis methods cannot accurately predict Dh owing to earthquakes, various studies have focused on determining MP to consider the characteristics closely related to Dh induction [8, 12, 13, 16, 28, 30, 36, 44].

Table 2 Overview of the representative simplified dynamic analysis methods
Full size table

To consider the research trends described above, the Ministry of Land, Infrastructure and Transport (MLIT) [38] in Japan proposed the concept of a performance-based seismic coefficient (khk) to verify the performance of port structures exceeding the force balance limit in place of kh in the M–O method. This concept was derived from the work of Nagao and Iwata [44], who used the finite element analysis program (FLIP) to perform two-dimensional total stress analysis on quay wall models under various conditions for acr, Hw, frequency of the input motion, and stiffness of the soil. They numerically modeled all combinations of the above conditions and derived the PGA values for all the cases when Dh reached 20 cm by adjusting the intensities of sinewaves for various frequencies (i.e., 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 3.0, and 4.0 Hz). Through the derived results, they proposed the following b filter to make the filtered peak acceleration (af) converge to the target acr regardless of Hw, the frequency of the input motion (f), the initial natural period of the backfill ground (Tb), and the initial natural period of the subsoil underneath the wall (Tu). The b filter equation, which is flat below 1 Hz and rapidly attenuates at frequencies exceeding 1 Hz, has been derived using the regression analysis of b with three independent variables—Hw, Tu, and Tb—affecting the frequency characteristics of ground motion.

$$ a\left( f \right) = b, 0 < f \le 1.0 $$
(1a)
$$ a\left( f \right) = \frac{b}{{1 - \left( {\frac{f - 1.0}{1/0.34}} \right)^{2} + 6.8\left( {\frac{f - 1.0}{1/0.34}} \right)i}}, 1.0 < f $$
(1b)
$$ b = 1.05\frac{{H_{w} }}{{H_{wR} }} - 0.88\frac{{T_{b} }}{{T_{bR} }} + 0.96\frac{{T_{u} }}{{T_{uR} }} - 0.23 $$
(1c)
$$ 0.04H_{w} + 0.08 \le b \le 0.04H_{w} + 0.44 $$
(1d)

where a is the b filter, considering the frequency characteristics of the input earthquake, f is the frequency (Hz), Hw is the wall height (m), HwR is the standard wall height (15 m), Tb is the initial natural period of the backfill ground (s), TbR is the standard initial natural period of the backfill ground (0.8 s), Tu is the initial natural period of the subsoil underneath the wall (s), TuR is the standard initial natural period of the subsoil underneath the wall (0.4 s), and i is an imaginary unit. b should be set in a range determined by Eq. 1d with Hw. Regardless of the range in Eq. 1d, the lower limit should not be less than 0.28 in all cases.

Nagao and Iwata [44] used nine measured earthquake records to verify that the af values, which make the Dh of the quay walls with various target acr values to 20 cm, converged to the target acr values. Their analysis results indicated various differences between af and the target acr for each earthquake record. This difference was assumed to be due to the various durations of each earthquake record. By regression analysis, they derived the following reduction factor (P) that can be multiplied with af to obtain the duration-corrected peak acceleration (ac):

$$ P = 0.36\ln \left( {S/a_{f} } \right) - 0.29 $$
(2)

where P is the reduction factor (P ≤ 1.0), S is the root sum square of the acceleration time history after filtering (cm/s2), and af is the maximum acceleration obtained after filtering (cm/s2).

Finally, Nagao and Iwata [44] used the nine measured earthquake records to find all ac values that result in the other Da values (i.e., 5, 10, and 15 cm) for quay walls designed with various target acr values. Then, they performed a regression analysis with both the obtained ac values and the set values of Da to derive khk, which is the seismic coefficient corresponding to Da:

$$ k_{hk} = 1.78\left( {\frac{{D_{a} }}{{D_{r} }}} \right)^{ - 0.55} \frac{{a_{c} }}{g} + 0.04 $$
(3)

where khk is the characteristic value of the seismic coefficient for verification, ac is the maximum corrected acceleration (cm/s2), g is the gravitational acceleration (980 cm/s2), Da is the allowable deformation of the quay wall, and Dr is the standard deformation of the quay wall (10 cm).

The procedure for calculating khk was detailed by the MLIT [38] and is briefly summarized below:

  1. (1)

    The acceleration time history on the ground surface is calculated by performing an one-dimensional site response analysis using a level 1 earthquake ground motion determined through a probabilistic seismic hazard analysis as the input.

  2. (2)

    A fast Fourier transform (FFT) is performed on the surface acceleration time history to obtain the acceleration spectrum of the ground surface.

  3. (3)

    The filtered acceleration spectrum is obtained by filtering the surface acceleration spectrum with a b filter (Eq. 1a).

  4. (4)

    af is obtained from the acceleration time history following an inverse FFT operation on the filtered spectrum.

  5. (5)

    ac at the ground surface is obtained by multiplying af by P (Eq. 2).

  6. (6)

    Finally, khk is obtained by substituting ac and Da into Eq. 3. This method is applicable when Da = 5–20 cm.

The khk concept was validated by Fukunaga et al. [15] and Lee et al. [32], who used real case histories of gravity-type quay walls during earthquakes and dynamic centrifuge test results, respectively. Their results indicated that khk can be used to accurately predict the performance of port structures exceeding the force balance limit in general. However, they also confirmed the need for improvement because this approach evaluates Dh relatively conservatively for cases where low-frequency components are dominant.

This study analyzed incidents of earthquake-induced damage to quay walls in Japan and Korea and the dynamic centrifuge test results presented by Lee et al. [32] to quantitatively evaluate the performance of the representative simplified dynamic analysis methods and the khk concept. In addition, the dynamic centrifuge test results were used to improve the b filter included in the khk concept, and the improvements were verified according to actual cases of earthquake-damaged quay walls.

2 Methodology

2.1 Assessment of simplified dynamic analysis methods and the k hk concept using actual cases of earthquake-damaged quay walls

Over the last three decades, various methods (i.e., the simplified dynamic analysis methods summarized in Table 2 and the khk concept) have been proposed to improve the prediction accuracy for the performance of quay walls subjected to design earthquake motions. However, few efforts have been made to quantitatively evaluate the accuracy of these proposed methods by comparing them with actual measurements of Dh for quay walls damaged by earthquakes [2, 7, 9, 36]. Therefore, this study evaluated the prediction accuracy for Dh of the proposed methods by comparing the actual Dh values from real cases of gravity-type quay walls damaged by earthquakes with those obtained by inputting the real field conditions into the proposed methods.

Fukunaga et al. [15] recently summarized information on damage to quay walls caused by earthquakes that occurred throughout Japan: the structural details, geotechnical conditions, and acceleration time histories caused by earthquakes. Their data were provided by the National Institute for Land and Infrastructure Management (NILIM) and Port and Airport Research Institute (PARI). In the present study, this information was used for a quantitative analysis of the proposed methods. Cases that met the following conditions were selected for analysis: gravity-type walls, mainly sliding failure occurred, and Dh was within 30 cm. Furthermore, a moment magnitude (M) 5.4 earthquake occurred in Pohang City in the southeastern part of the Korean peninsula on 15 November 2017, and lateral spreading took place at a gravity-type quay wall in Youngil Bay Port approximately 6 km away from the main shock epicenter [25]. The Ministry of Oceans and Fisheries (MOF) investigated the damage to the quay wall in detail, and the reported data were also used in this study [41].

The information and backfill surface ground motion on the eight cases selected for the quantitative evaluation are summarized in Table 3 and Fig. 1, respectively. In Case 1, the backfill surface ground motion was recorded at the seismic monitoring station of the Korea Institute of Ocean Science and Technology, which was installed on the backfill surface of the damaged wall [41]. In Cases 2–8, the acceleration time histories on the ground surface were obtained by Fukunaga et al. [15], who performed a one-dimensional site response analysis on the bedrock ground motion measured at the seismic monitoring station of the nationwide strong-motion network (K-NET) closest to the damaged wall [29]. Either the north–south (N-S) or east–west (E-W) direction was used for the motions, depending on the direction that was closest to the perpendicular of the damaged wall.

Table 3 Details of the eight datasets used in quantitative evaluations [29, 41]
Full size table
Fig. 1
figure 1

Acceleration time histories and response spectra at the backfill surface of the eight datasets [15, 41]

Full size image

The Dh prediction accuracy of the proposed analysis methods for the performance-based design was quantitatively evaluated according to the following process:

  1. (1)

    The MP values included in the proposed methods were derived from the motions illustrated in Fig. 1, as summarized in Table 4.

    Table 4 MP values of the acceleration time histories at the backfill surface in the eight datasets
    Full size table
  2. (2)

    Dh was calculated for each event by substituting the acr values in Table 3 and MP values in Table 4 into the proposed simplified dynamic analysis methods (listed in Table 2) and the khk concept (Eq. 3). Here, the values of Dh calculated from the khk concept refer to the Da values derived by substituting the terms khk, ac, and Dr in Eq. 3 with the acr values in Table 3 divided by g, ac values in Table 4, and Dr value of 10 cm, respectively [15, 32, 44].

  3. (3)

    The calculated Dh values were then compared with the measured Dh values of the actual damage cases summarized in Table 3. The relative difference (RD) in percentage between the measured and calculated Dh values is plotted in Fig. 2 for all cases. To facilitate the analysis of the cause of the RD, this figure also presents the major MP values (i.e., PGA, PGV, Tp, Ia, acr/PGA, and PGV/PGA) for all cases.

    Fig. 2
    figure 2

    Relative difference in percentage between the measured and calculated Dh values and the major MP values for all cases

    Full size image

The eight cases are arranged in ascending order of the RD values. An RD converging to 0 indicates a high prediction accuracy, whereas negative and positive values indicate that the method underestimated or overestimated Dh, respectively. The analysis results were divided into three groups: Group 1 (i.e., upper bound [3, 9, 47, 53]), Group 2 (i.e., mean fit [61, 62]), and the khk concept.

The PGA is the most commonly used measure for the amplitude of a particular ground motion. Figure 2 indicates that the RD values of Groups 1 and 2 increased as PGA increased. The ratio acr/PGA is a key parameter of simplified dynamic analysis methods; the RD values of Groups 1 and 2 increased as acr/PGA decreased. In the case of Group 1, the absolute value of RD decreased as PGA increased up to 0.3 g and acr/PGA decreased up to 0.44. Meanwhile, Dh was gradually overestimated as PGA increased above 0.3 g and acr/PGA decreased below 0.44. In the case of Group 2, the absolute value of RD decreased as PGA increased and acr/PGA decreased, but Dh was underestimated overall. The predominant period (Tp) [52] and the ratio of PGV/PGA [56] are the MP reflecting the frequency content characteristics of the ground motion [30]. As they decreased, the RD values of each group exhibited the same trends with increasing PGA and decreasing acr/PGA. The Arias intensity (Ia) [5] represents the characteristic of the ground motion duration; Case 5 was exceptional because Ia was so small that it resulted in a relatively small Dh. The analysis results indicated that the simplified dynamic analysis methods were partially reliable when the MP values in the equations were within a certain range. Therefore, to accurately predict earthquake-induced Dh under various external conditions, simplified dynamic analysis methods should include the appropriate MP by considering the frequency characteristics and duration of the ground motion in their equations [8, 12, 16, 24, 30, 44].

The khk concept corrects the frequency characteristics and duration of the ground motion through the b filter and P, respectively. Thus, in most cases, it provided a better prediction accuracy of Dh than the simplified dynamic analysis methods. However, the khk concept was confirmed to predict Dh relatively conservatively in cases where the long-period components of the ground motion were dominant. This is similar to the results of Lee et al. [32], who verified the reliability of the khk concept through dynamic centrifuge tests. Therefore, the terms related to the low-frequency band should be improved for the design of the b filter.

2.2 Overview of the dynamic centrifuge tests performed by Lee et al. [32]

Centrifuge tests rotate a scaled model at high speed with a centrifugal acceleration far higher than that of gravity and can be used to simulate in situ stress conditions in soil models. They have been used in many prior studies to supplement the lack of recorded case histories to validate the reliability of existing design methods [6, 31, 33, 35, 45, 48, 59, 63] and develop seismic design techniques for port structures [11, 19, 31, 33, 63]. Several performance-based seismic design codes for port structures have recently recommended centrifuge testing as a seismic performance verification method [4, 22, 38, 40]. In this study, the dynamic centrifuge test results of Lee et al. [32] were used to quantitatively evaluate the proposed simplified dynamic analysis methods and the khk concept as well as to improve the b filter.

Lee et al. [32] performed four dynamic centrifuge tests on gravity-type quay wall models subjected to various conditions reflecting the primary variables of the khk concept (i.e., Hw, the thickness of the subsoil underneath the wall (Hu), and input earthquake motion) in order to validate the khk concept and assess the behavior of the model walls during earthquakes. Each of these test cases is summarized in Table 5. The effects of Hw, Hu, and the frequency characteristics of the input earthquake on the accuracy of predicting Dh using the khk equation were evaluated by comparing the results of Cases 1 and 2, Cases 2 and 3, and Cases 3 and 4, respectively. The tests were conducted at KAIST with an earthquake simulator mounted on the centrifuge [26, 27]. An equivalent shear beam (ESB) box, which minimized the boundary effect on the soil, was used as a model container [34]. The gravity-type quay wall models were made from aluminum alloy (T6061) and designed with khk = 0.13 based on the quay wall design procedure provided by MLIT [38]. The subsoil underneath the wall and the backfill soil behind the wall were constructed from poorly graded clean silica sandy soil (SP). The physical properties of the sand were reported by Lee et al. [32]. The subsoil was densified to a relative density of 86% by compaction to prevent overturning and bearing capacity failure. The soil in the backfill was prepared by sand pluviation at a relative density of 80%. Figure 3a shows the configurations of the test models with instrumentation for Cases 1 and 2, and Fig. 3b shows the configurations for Cases 3 and 4.

Table 5 Overview of the test cases (modified from Lee et al. [32])
Full size table
Fig. 3
figure 3

Schematic of the test instrumentation (all dimensions are in model scale): a Cases 1 and 2, b Cases 3 and 4, and c top view for Case 1 (modified from Lee et al. [32])

Full size image

In Cases 1 and 2, there were three pairs of bender elements, and in Cases 3 and 4, there were four pairs of bender elements. Tu and Tb were obtained by measuring the shear wave velocity (Vs) of models in flight. An accelerometer was attached to the bottom part of the ESB box parallel to the shaking direction to measure the input motion. Five of the eight horizontal accelerometers were buried in the soil, and the rest were attached to the model wall to measure the acceleration time histories at different soil heights and the gravity-type quay wall. Dh was measured with two potentiometers. Two laser sensors were used to measure the subsidence of the backfill and to check the bearing capacity failure of subsoil based on the settlement of the wall. The acceleration and Dh were positive in the active direction. The Ofunato earthquake motion recorded at Miyagi-Ken Oki, Japan, which has short-period components, and the Hachinohe earthquake motion recorded at Tokachi-Oki, Japan, which has long-period components, were used as the shaking events. Figure 4 presents the acceleration time histories and the response spectra of the input motions. The dynamic motions were inputted incrementally at the bottom of the ESB, beginning with a weak intensity. The centrifugal acceleration was set to 40 g for the 10 m high wall and 60 g for the 15 m high wall. All results presented herein are in prototype units unless otherwise stated according to the centrifuge scaling laws [55]. The details of the dynamic centrifuge tests are available in the paper by Lee et al. [32].

Fig. 4
figure 4

modified from Lee et al. [32])

Normalized acceleration time histories and response spectra of input motions to the earthquake simulator during the centrifuge tests at the prototype scale (

Full size image

3 Results and discussion

3.1 Assessment of the simplified dynamic analysis methods and k hk concept with the centrifuge test results

Similar to the analysis presented in Sect. 2.1, Lee et al. [32] evaluated the prediction accuracy of the khk concept at estimating Dh after the force balance limit was reached under various conditions (i.e., Hw, Hu, and earthquake input motion) by comparing the Dh values measured via the centrifuge tests with those calculated from ac based on the khk definition. This section presents a quantitative evaluation of the accuracy of the simplified dynamic analysis methods using the centrifuge test results. The process used was the same as that described in Sect. 2.1, except that the dynamic centrifuge test results were used instead of actual records of earthquake-damaged quay walls.

For each earthquake excitation, the acceleration time histories at the bedrock and soil and the displacement time histories of the wall were measured to deduce the MP values used to calculate Dh. Then, the actual Dh was obtained by subtracting the average value of the initial 500 samples from the average value of the last 500 samples from the displacement time histories. The deduced MP values (i.e., PGA, PGV, af, and ac) and the calculated Dh for all seismic events in every case are plotted with respect to the measured Dh in Figs. 5 and 6.

Fig. 5
figure 5

MP values of the acceleration time histories at the backfill surface with respect to the measured Dh values for all seismic events in every case (modified from Lee et al. [32])

Full size image
Fig. 6
figure 6

Comparison of the calculated and measured Dh values for all cases

Full size image

As summarized in Table 5, each case had different combinations of Hw, Hu, and input earthquake motions so that the average values of Tp obtained from the acceleration time histories on the backfill surface of each case were 0.15, 0.26, 0.55, and 0.60 s. As in Sect. 2.1, the analysis results were divided into three groups: Group 1 (i.e., upper bound [3, 9, 47, 53]), Group 2 (i.e., Mean fit [61, 62]), and the khk concept. In the case of Group 1, Fig. 6 indicates that the calculated Dh values were larger than the measured Dh values for Cases 1–3, and the difference between the calculated and measured Dh gradually decreased as the average Tp increased. The calculated Dh values were less than the measured Dh values in Case 4, which had the longest Tp in the test model. In the case of Group 2, the calculated Dh values were only larger than the measured Dh values for Case 1, which had the shortest Tp in the test model. In Cases 2–4, Group 2 increasingly underestimated Dh with increasing Tp. Finally, the khk concept slightly overestimated Dh in all cases, but it generally had a higher prediction accuracy than the other methods because it also considered the effects of external factors (e.g., wall geometry, stiffness of soil, and the frequency characteristics and duration of the ground motion). However, because the difference between the calculated and measured Dh increased for cases with more low-frequency components, the terms of the b filter related to the low-frequency band should be improved.

3.2 Time–frequency domain responses of the wall models

The analyses in Sects. 2.1 and 3.1 confirmed that the khk concept had a higher Dh prediction accuracy than the simplified dynamic analysis methods under various external conditions. This is because the khk concept adopts the b filter that corrects the frequency characteristics of the ground motion by considering the contribution of the waves of each frequency component comprising the ground motion to the Dh generation. Nagao and Iwata [44] confirmed through numerical analysis that the frequency components below 1 Hz of ground motion are mainly related to Dh generation and suggested the b filter (Eq. 1). In the present study, a time–frequency analysis was performed with the dynamic centrifuge test results to evaluate the suitability of the b filter shape experimentally. Time–frequency analysis is an effective method of investigating the changing frequency content of the dynamic response over time [46, 51, 59]. The dynamic displacement of the wall models was obtained by double integration of the acceleration signals measured from the accelerometer, which was attached to the top of the wall front (A(6) in Fig. 3). Figures 7 and 8 present the time histories of the dynamic displacement and permanent displacement, the frequency contents of the dynamic displacement, and the time–frequency domain of the dynamic response for the wall models of all cases during a weak earthquake (i.e., PGA at backfill surface, A(6) ≅ 0.1 g) and a strong earthquake (i.e., PGA at backfill surface, A(6) ≅ 0.3 g), respectively. The frequency energy of the dynamic response of the wall model was concentrated at 1 Hz or less in all cases. In particular, the permanent displacement occurred where the frequency energy below 1 Hz was dominant in all cases. Because the frequency components below 1 Hz were the main contributors to Dh generation, the suitability of b filter shape was experimentally reconfirmed. The results in Sects. 2.1 and 3.1 indicate that the khk concept generally predicts Dh reasonably well after the force balance limit is reached but is less accurate in cases containing many low-frequency components. To improve the performance, b (Eqs. 1c and 1d) was revised according to the dynamic centrifuge test results and actual cases of earthquake-damaged quay walls.

Fig. 7
figure 7

Frequency contents of the dynamic displacement, time histories of the dynamic and permanent displacements, and time–frequency domain of the dynamic response for the wall models of all cases during the weak earthquake (PGA at backfill surface, A(1) ≅ 0.1 g)

Full size image
Fig. 8
figure 8

Frequency contents of the dynamic displacement, time histories of the dynamic and permanent displacements, and time–frequency domain of the dynamic response for the wall models of all cases during the strong earthquake (PGA at backfill surface, A(1) ≅ 0.3 g)

Full size image

3.3 Revision of b using the centrifuge test results

Equations 1c and 1d were derived by multiple linear regression analysis of the correlation between the three independent variables of H, Tb, and Tu and the dependent variable of b based on the numerical analysis introduced by Nagao and Iwata [44].

To perform multiple linear regression analysis with the centrifuge test results, the target b values (btarget) corresponding to the values of Dh measured from the dynamic centrifuge tests were derived. Here, btarget represents the b value that makes ac from the measured acceleration time history produce the measured Dh of gravity-type quay wall models designed with khk = 0.13. Multiple linear regression analysis was performed using events where the measured Dh value was close to 5–20 cm, which is the applicable range of khk.

Figure 9 details the procedure for obtaining btarget using the measured signals in the centrifuge tests. The signals were obtained for Case 4, where the Hachinohe earthquake with PGA at the backfill surface equal to 0.2 g was applied to the test model.

Fig. 9
figure 9

Procedure for obtaining btarget from the measured signals

Full size image
  1. (1)

    During each earthquake excitation, the acceleration time history of the backfill soil surface (A(1)) and the displacement time history of the wall (P(top)) were obtained.

  2. (2)

    The ac value corresponding to each Dh (ac, target) was deduced by substituting the design khk value of 0.13 for the quay wall model and the measured Dh values into the khk and Da terms of Eq. 3.

  3. (3)

    To design the filter, b was initially set to 0.01. The filters for all seismic events were designed by using b and frequency characteristics of the measured acceleration time histories; the designed filters were applied to the acceleration spectra obtained by the FFT of the measured acceleration time histories.

  4. (4)

    The filtered acceleration time histories were obtained by the inverse FFT operation on the filtered acceleration spectra. Then, af and P were derived from the filtered acceleration time histories and multiplied with each other to calculate ac (ac, calculation), which was then compared with ac, target.

  5. (5)

    Steps 3 and 4 were repeated while b was increased from 0.01 to 1.5 in steps of 0.01, and the b value with the smallest error between ac, target and ac, calculation was found (btarget). Here, ac, calculation should be equal to or greater than ac, target.

The measured Dh, ac, target, btarget, Tb, and Tu values of the events for each case are summarized in Table 6. For the multiple linear regression analysis, the independent variables needed to be determined first. Various studies [20, 21, 43], including the results of the dynamic centrifuge test [32] and numerical analysis [44], confirmed that the low-frequency components of an earthquake motion in backfill soil become more amplified with increasing Tb, and a greater Dh is generated. Therefore, in order for the calculated b to increase with the amplification of low-frequency components in the backfill soil, the coefficients of the independent variables Hw, Tb, and Tu should be positive. However, in Eq. 3, the coefficients of Hw and Tu are positive whereas that of Tb is negative. According to Nagao and Iwata [44], there is no physical basis for estimating the coefficient of Tb to be negative, and this was merely done to improve the accuracy of the equation. In particular, because Tb represents a natural period that includes not only the backfill but also the subsoil, Eq. 3 considers Tu redundantly. Therefore, a multiple linear regression analysis was performed on b with Hw, Tb, and Tu as the independent variables. Then, additional analysis was conducted with only Hw and Tb as the independent variables. Following Nagao and Iwata [44], the reference values of Hw, Tb, and Tu were set to 15 m, 0.8 s, and 0.4 s, respectively, for non-dimensionalization and normalization.

Table 6 Measured Dh, ac, target, btarget, Tb, and Tu values for the events of each case used in the multiple linear regression analysis
Full size table

Through the multiple linear regression analysis, the following Eqs. 4 and 5 were derived. The adjusted R squared values, which indicated the suitability of the regression model, were 0.96 and 0.97, respectively.

$$ b_{1} = 0.13\frac{{H_{w} }}{{H_{wR} }} + 0.37\frac{{T_{b} }}{{T_{bR} }} - 0.06\frac{{T_{u} }}{{T_{uR} }} + 0.24 $$
(4)
$$ b_{2} = 0.18\frac{{H_{w} }}{{H_{wR} }} + 0.26\frac{{T_{b} }}{{T_{bR} }} + 0.25 $$
(5)

Here, Hw is the wall height (m), HwR is the standard wall height (15 m), Tb is the initial natural period of the backfill ground (s), TbR is the standard initial natural period of the backfill ground (0.8 s), Tu is the initial natural period of the subsoil underneath the wall (s), and TuR is the standard initial natural period of the subsoil underneath the wall (0.4 s).

The adjusted R squared values indicate that the suitability of b2 (Eq. 5), which excludes Tu as an independent variable, was the same or greater than that of b1 (Eq. 4), which does include Tu. In addition to the adjusted R squared values, the Dh values derived with Eqs. 4 and 5 were compared with the measured Dh values to evaluate the prediction accuracy. The results are presented in Fig. 10. Applying the revised b values (Eqs. 4 and 5) to obtain ac for calculating khk decreased the difference between the calculated and measured Dh values for all cases, compared to the original b value (Eqs. 1c and 1d). The revised b values considerably improved the Dh prediction accuracy for Cases 3 and 4, which included more low-frequency components, compared to Cases 1 and 2. In Cases 3 and 4, the Dh values calculated with b1 (Eq. 4) were slightly less than the measured Dh values, which means that Eq. 4 can cause khk to be underestimated. These results suggest that b2 (Eq. 5) is more suitable than b1 (Eq. 4) for calculating khk with regard to simplicity and safety.

Fig. 10
figure 10

Comparison between the Dh values calculated with ac filtered by b (Eqs. 1c and 1d), b1 (Eq. 4), and b2 (Eq. 5) and the measured Dh values for all cases

Full size image

Because the reliability of the revised b values (Eqs. 4 and 5) was evaluated with the dynamic centrifuge test results used to derive the two equations, it is natural for the results to be positive. Therefore, further verification using actual cases of earthquake-damaged quay walls was required to confirm the field applicability and reliability of the revised b values (Eqs. 4 and 5).

3.4 Assessment of the revised b values using actual cases of earthquake-damaged quay walls

The procedure and actual records of earthquake-damaged quay walls presented in Sect. 2.1 were applied to validate the field applicability and reliability of the revised b values (Eqs. 4 and 5). RD was determined between the calculated Dh values obtained with the different b values (Eqs. 1c, 1d, 4, and 5) and the measured Dh values summarized in Table 3. The calculated Dh values were obtained by substituting the acr values in Table 3 and the ac values derived from the different b values (Eqs. 1c, 1d, 4, and 5) into the real earthquake records in Fig. 1 into the khk and ac terms of Eq. 3. The revised b values for all cases used in the evaluation and the corresponding ac values are summarized in Table 7. The RD values for all cases are plotted in Fig. 11.

Table 7 Revised b values (Eqs. 4 and 5) for all cases and the corresponding ac values
Full size table
Fig. 11
figure 11

Relative difference in percentage between the Dh values calculated with the different b values (Eqs. 1c, 1d, 4, and 5) and the measured Dh values for all cases

Full size image

Similar to the evaluation in Sect. 3.3 with the dynamic centrifuge test results, Fig. 11 indicates that the revised b values (Eqs. 4 and 5) significantly improved the accuracy of the calculated Dh values, compared to the original b values (Eqs. 1c and 1d) in Cases 1–4, which contained more low-frequency components. In cases 5–7, where Hw was less than 7 m, using the revised b values caused Dh to be calculated slightly conservatively compared to when the original b values were used (Eqs. 1c and 1d). Particularly in case 5, using the original b values caused the calculated Dh to underestimate the measured Dh, but applying the revised b values (Eqs. 4 and 5) allowed Dh to be safely predicted. In addition, the variation in RD was confirmed to be small when the revised b values were used instead of the original b values. Thus, the revised b values (Eqs. 4 and 5) allowed Dh to be predicted consistently, regardless of external influences (e.g., Hw, Tb, earthquake motions).

The field applicability and reliability of the revised b values (Eqs. 4 and 5) were partially verified through the above evaluation using actual cases of earthquake-damaged quay walls. Because the RD values with b1 (Eq. 4) and b2 (Eq. 5) were similar, the better equation cannot be quantitatively determined. However, in terms of the simplicity of the equation and verification with the centrifuge test results in Sect. 3.3, Eq. 5 comprising the independent variables Hw and Tb can be concluded to be more effective.

4 Conclusions

This study assessed representative simplified dynamic analysis methods and the khk concept for the performance-based design of port structures. Incident records of earthquake-induced damage on quay walls in Japan and Korea and the results of dynamic centrifuge tests performed under various conditions were used for a quantitative evaluation and improvements. Unlike the simplified dynamic analysis methods, which simply reflect the frequency characteristics of the ground motion associated with the deformation of the quay wall as the ratio of PGV to PGA, the khk concept corrects the frequency characteristics of the ground motion by also considering the wall geometry, stiffness of the soil, main frequency range related to wall deformation, and duration of the ground motion. The khk concept was found to accurately and consistently predict the deformation of quay walls under various conditions, compared to the simplified dynamic analysis methods. However, the khk concept is relatively conservative when low-frequency components are dominant. Thus, the b filter was revised in relation to the low-frequency band. Multiple linear regression analysis was used to derive two revised equations for b, where the independent variables were the wall height and the natural periods of the subsoil and backfill soil or the wall height and natural period of the backfill soil only. The field applicability and reliability of the revised b values were partially verified through evaluations using the dynamic centrifuge test results and actual records of earthquake-damaged quay walls. The results of this study demonstrated how the khk concept can be improved and various methods of verifying the performance-based design of gravity-type quay walls. In the future, more accurate regression equations for verifying the performance of a quay wall should be derived by considering a few additional aspects.

First, the actual deformation of quay walls during an earthquake is a combined result of the shear deformation of the subsoil and the relative displacements between the subsoil and the quay wall. However, as the subsoil stiffness of the dynamic centrifuge tests and field records analyzed in this study were mostly high, the applicability of the equations with the revised value of b may be limited for liquefied or loose ground conditions. Therefore, for more comprehensive analyses of the deformation of quay walls, the effect of subsoil shear deformation should be investigated by considering additional experimental variables such as the shape, thickness, material properties, and relative density of the subsoil.

Second, it is necessary to derive appropriate independent variables for the regression analyses by conducting studies on additional influence factors that can affect the main frequency range related to wall deformation, in addition to existing variables constituting the b filter.

Lastly, the reliable databases from physical tests, numerical analyses, and field data for various variables such as the design seismic coefficient of the wall, ground stiffness, and input earthquakes should be constructed to improve the reliability of the regression analysis.