2025 Volume 66 Issue 4 Pages 233-240
We have proposed a method for evaluating the restorability of railway structures. In the proposed method, all earthquake motions expected within a structure's design service life are used as the design of earthquakes. In addition, the recovery time after an earthquake, which is directly related to early recovery, is used as the verification index. We also propose a more practical method of expressing structural conditions with the same recovery time as a nomogram by performing calculations under various conditions in advance. The proposed method allows us to construct structures that are easy to recover in the same procedure as the conventional seismic design, and it is expected to shorten the recovery time after an earthquake.
In addition to ensuring safety, infrastructure facilities developed as foundational structures for industrial and residential areas are required to ensure restorability during earthquakes. For example, railway structures need to “maintain in a state where they can be restored functionally in a short period of time by limiting damage to within a certain range determined by difficulty of structural repair in response to expected seismic action” [1]. One approach to confirming the restorability of these structures is to verify that the recovery period and expenses are within a reasonable range when subjected to multiple seismic motions expected during their useful lives, considering initial costs and earthquake loss costs [2]. Various facilities have undergone examinations that consider the total cost [3, 4, 5], and there are cases where this has been introduced into seismic design [6]. We previously proposed a design method for minimizing the total cost of railway RC piers [7].
Following the trends described above, the restorability of railway structures after seismic damage is, in principle, verified according to the concept [1]. On the other hand, there have been moderate earthquakes in recent years, such as the 2018 Northern Osaka Earthquake, the 2021 Earthquake off the Coast of Fukushima Prefecture, and the Northwestern Chiba Earthquake. Although structural damage in these earthquakes was limited, determining the extent of damage and undertaking post-earthquake restoration work took time. Issues regarding the early resumption of operations and post-earthquake restoration have been highlighted [8]. Methods to address such issues may include cost-based restorability verification and the explicit calculation of the post-earthquake recovery time, which can be used as an indicator for structural design. From this perspective, we previously calculated the relationship between the damage level caused by the earthquake and the recovery time required for various railway structures. We then prepared a database [9],which makes it relatively easy to calculate the recovery time for each structure after an earthquake. However, implementing these methods requires specialized design techniques and knowledge, as well as relevant information such as the probability of earthquakes and the concept of loss costs. Analytical techniques for large-scale numerical calculations are also necessary. Therefore, similar to when calculating the total cost of a structure, it is expected that implementing this method in practice will be difficult.
In this paper, we propose a method for verifying the restorability of railway structures [10]. In Section 2, we propose a verification method for railway structure restorability in which the recovery time is used as the verification index. We propose a basic procedure and present a display method called a restorability verification nomogram. This nomogram enables implementation in practical designs. In Section 3, we perform trial calculations to verify the restorability of reinforced concrete (RC) rigid-frame viaducts. In Section 4, we evaluate and validate the restorability verification nomogram for the structure that was subject to trial calculations in Section 3.
First, we propose a method for verifying the restorability of railway structures using post-earthquake recovery time as the verification index. Figure 1 shows the proposed verification procedure. The general flow process involves setting the required performance and design earthquake motion, calculating the response value of the structure and verifying its performance of the structure. This process is equivalent to the seismic design procedure of normal railway structures [1]. Meanwhile, the design method proposed here has several features.
The “post-earthquake recovery time” is set as the required performance of the structure. This directly addresses the issue of post-earthquake recovery time, which has become a serious concern in recent medium- to large-scale earthquakes.
To correspond to the above-mentioned verification index, the design earthquake motion must be set as “multiple earthquake motions with a wide range of characteristics expected at the construction location.” These seismic actions are represented by a set of waveforms, and its occurrence probabilities [11] are represented by the results of probabilistic earthquake risk analysis [12, 13].
The method used to calculate the response value of the structure is based on the method used for normal seismic design. It should be noted that the current seismic design of railway structures aims to accurately evaluate the response to L2 earthquake motion. The structures are modeled to respond relatively safely to earthquake motions with smaller amplitudes than an L2 earthquake, such as the L1 earthquake and other earthquakes. However, the restorability verification method proposed in this paper requires calculating the appropriate earthquake response values for small- and medium-sized earthquakes. Therefore, it can effectively adopt a structural modeling method and response value calculation method that considers such aspects [14].
Finally, the recovery time of the structure is evaluated. This requires setting a recovery time that corresponds to the response value of the structure. However, the time required for recovery can naturally vary greatly depending on the part of the structure that is damaged and the degree of damage. Recovery is known to vary greatly depending on circumstances, such as the structural type and surrounding environment. Therefore, the recovery time must be appropriately evaluated in accordance with the earthquake response value and the situation at the location. We previously conducted a basic examination of the relationship between earthquake response value and recovery time under standard railway structure conditions [9]. The results of this examination are used in the following calculations.
The above procedure enables us to calculate the expected recovery time for a group of earthquake motions. In this procedure, the design earthquake motion is set as multiple earthquake motions acting on the structure at the target location. We plan to verify the performance by determining if it meets the required recovery time. Meanwhile, performance verification is conducted using the following equation, which is based on the limit state design method - the standard design method for railway structures.
| (1) |
where IRD, ILD, and γi represent the design response value (expected recovery time), design limit value (required recovery time), and structural factor (1.0 in this study), respectively.
2.2 Restorability verification nomogramThe proposed method requires a large amount of work to set the design earthquake motion, calculate the response value, and verify the performance. Consequently, implementing this method for all structures in actual designs is difficult. Therefore, we propose a more practical method.
To verify practical restorability, it is necessary to simplify each stage of work, and the strength demand spectra [1] used in the seismic design of railway structures can then be a useful reference here. Therefore, we propose a method to evaluate the recovery time under a wide range of conditions in advance, similar to the strength demand spectra. We display this as a restorability verification nomogram. Figure 2 presents the calculation procedure for the restorability verification nomogram. An overview of each step is given below.
Step 1: Set the target location. Then, evaluate the design earthquake motion at the location. This involves considering multiple earthquake motions with a wide range of characteristics expected at construction location. This is expressed as a group of earthquake motions and their respective probabilities.
Step 2: Calculate the response value of the structure against the design earthquake motion. First, build a group of structure models with different yield seismic coefficients khy for conditions with Teq and µ. Conduct a dynamic analysis on this structure by inputting the earthquake motion waveforms at the target point. Then, calculate the response ductility factor and occurrence probability of the degree of damage for each structure.
Step 3: Evaluate the recovery time corresponding to the damage obtained in Step 2. Then, multiply it by the occurrence probability to evaluate the expected recovery time.
Step 4: Set the required performance of the structure (target recovery time). Then, determine the structure that satisfies the required performance based on the relationship of the expected recovery time obtained in Step 3. The structure is a combination of Teq, µ, khy.
Step 5: Repeat Steps 2-4 by varying the Teq and µ. Using these results, connect the conditions that result in the same recovery time. This will display the combination of the structure's vibration characteristics that satisfies a certain recovery time via a nomogram.
This procedure displays the same dimensions corresponding to the strength demand spectra for a location and recovery time. The calculation conditions for creating the above restorability verification nomogram are compared with those based on the fundamental method proposed in the previous section. In addition, research has confirmed that appropriate earthquake response values and damage occurrence probabilities can be calculated for railway bridges and viaducts, even if the entire structure is replaced with an equivalent single-degree-of-freedom (SDOF) system [15]. Thus, an analysis model can be used to obtain equivalent results for both. Moreover, the restorability verification nomogram displays the characteristics of the structure based on the recovery time. Therefore, it can be used for any required performance and recovery time. The proposed restorability verification nomogram uses the same seismic action as the fundamental method described in Section 2.1. The same results are obtained for the structural response value and recovery time. This suggests that the work required for verification has been considerably reduced and that an appropriate restorability verification has been achieved.
In Step 2, the “recovery time according to each earthquake response value of the structure” needs to be calculated. Since this varies considerably depending on the damaged part and the surrounding environment, multiple nomograms need to be prepared as required. Further study is needed on how to create and display a simple nomogram that considers this aspect appropriately. In this paper, however, we plan to estimate a nomogram using the relationship between earthquake response values and recovery time under standard conditions [9].
2.3 Design procedure using restorability verification nomogramThis restorability verification nomogram simplifies the determination of the seismic yield coefficient demand according to various conditions, such as the earthquake seismicity of the construction location, the vibration characteristics of structure, deformation performance, damaged areas, ease of restoration, and required performance. We summarize the restorability verification procedure for a structure using the restorability verification nomogram. Figure 3 shows the specific flow process. The differences from the basic restorability verification method proposed in Section 2.1 (Fig. 1) are listed below.
● In “design earthquake motion setting,” the basic method uses a group of earthquake motions with occurrence probabilities for each region. However, in our examination, we select a restorability verification nomogram based on various conditions.
● In the “structure response value calculation and recovery time evaluation,” the basic method uses each waveform to evaluate the response value, damage level, and recovery time. However, in our examination, we calculate the structure's yield seismic coefficient khy is from the results of the push-over analysis.
● In the “performance verification,” the basic method uses Eq. (1) to verify the recovery time. However, our study confirmed that khy of the target structure is equal to or greater than the required yield seismic coefficient calculated by the restorability verification nomogram.
As previously described, an advanced preparation of the restorability verification nomogram based on various conditions enables verification using recovery time as the verification index because its function is similar to performance verification using the strength demand spectra. Therefore, this method is considered a design procedure that can be applied to practical designs. The validity of these results is confirmed in Section 4.
We verify the effectiveness of the proposed method by applying the basic procedure of the restorability verification method to an actual railway structure. The ground conditions shown in Fig. 4 are used as prerequisites for the calculations. The target structure is a rigid-frame viaduct, which has a height of 12.2 m from the ground to the track. The outcome of this method varies depending on the seismicity of the assumed area. Therefore, a construction location needs to be set. For this study, the Sendai area was selected as the location. We set a cross-section to meet the required recovery time at this location.
The proposed method set the “expected post-earthquake recovery time” as the required performance of the structure. In this case, the expected recovery time is set to five days. Although there is room for debate on how to set this value, the average recovery time is five days according to trial calculations conducted in major regions across the country for multiple structures designed according to current railway standards (rigid viaducts with pile foundations, where the upper structure yields first). Therefore, we adopted this value in our study, considering the perspective of code calibration. This recovery time of five days corresponds to the design limit value ILD in Eq. (1).
3.2 Setting the design earthquake motionIn the proposed method, the earthquake occurrence probability and design earthquake motion are set based on the construction location. For this trial calculation, we conducted a probabilistic earthquake hazard analysis in the Sendai area, which was set as the location. The return period for the calculation is set to 100 years, which corresponds to the design working life [1] of the structure. The specific implementation procedure of the earthquake hazard analysis and the information used are based on reference [13], which includes the calculation method for the earthquake motion waveform group described later. Figure 5 shows the final evaluation results of the earthquake occurrence probability.
This result was used as the basis for synthesizing the group of earthquake motions by occurrence probability. For this examination, we divided the amplitude into 15 levels with 100 Gal increments from 100 to 1,500 Gal (“Gal” refers to cm/s2). Twenty waves were evaluated for each amplitude level for a total of 300 waves. Figure 6 presents an example of the final calculated waveform. Naturally, the magnitude Mw and epicenter distance R assumed for each earthquake motion waveform differ, affecting not only the amplitude but also the time and frequency characteristics. A group of earthquake motion waveforms is set as the design earthquake motion.
The dimensions and sectional reinforcement of the rigid-frame viaduct were determined based on the various conditions at the construction site. The cross-section was designed to satisfy the verification of restorability using our proposed method, as well as normal structural safety. Finally, we set the structure dimensions and section reinforcement shown in Fig. 7. Note that only the columns and piles are shown for the cross-sectional reinforcement for subsequent discussions. We created a model to calculate the earthquake response value of this structure. The model is created using two-dimensional beam and spring elements in accordance with various design standards for railway structures [1]. The elastic and nonlinear characteristics of each element are modeled in accordance with railway standards. Figure 8 shows the results of a push-over analysis perpendicular to the track. This analysis reveals the structure's equivalent natural period is Teq = 1.14 s and its yield seismic intensity is khy = 0.33. We note that points Y, M, and N in this figure are damage control points used to evaluate the number of days required for structural restoration.
A nonlinear dynamic analysis that uses the detailed model of the structure can be conducted to calculate the response value. However, we replaced this model with an equivalent SDOF system, considering the number of seismic waves used [1, 14, 15]. We conducted a dynamic analysis by comprehensively inputting all 300 earthquake motion waveforms calculated in the previous section into the analysis model of this structure. Then, we calculated the response value for each waveform. Figure 9 shows the results of organizing the relationship between the response ductility factor of the structure and exceedance probability from the maximum response displacement of each waveform. This figure shows the control points (µy, µm) of each form of damage obtained by the push-over analysis of the target structure. However, in our examination of structures and earthquake motions, there was no response that exceeded µn, which corresponds to the collapse of the structure.
We calculated the recovery time of the structure based on the relationship with the standard recovery time corresponding to the structure type and degree of damage that we described in a previous study [9]. As shown in Table 1, the maximum response displacement of the structure and corresponding number of days to recovery were set to the damage level for the rigid-frame viaduct. The details of the calculation conditions and method for recovery time are based on reference [9]. The damage level of the structure is the same as the definition of seismic design for railway structures [1]. We assume ideal conditions for the surrounding environment of the structure, including sufficient workspace and the ability to bring materials and equipment in and from the side road. The expected recovery time of the structure is calculated by combining the relationship between the structural response and recovery time with the occurrence probability of each degree of damage shown in Fig. 9. The results showed that the expected recovery time of the target structure was 3.0 days. This corresponds to the design response value IRD in Eq. (1).
| Damage level | Response ductility factor µ | Recovery time (days) |
| 1 | 0≦µ<µy | 1 |
| 2 | µy≦µ<µm | 8 |
| 3 | µm≦µ<µn | 23 |
| 4 | µn≦µ | 28 |
We verified the restorability of the structure using Eq. (1). Assuming the structural factor γI = 1.0, using the expected required performance and recovery time from the study in the previous section, the value is obtained by the following equation.
| (2) |
This indicates that the structure shown in Fig. 7 satisfies the performance requirements. However, if Eq. (2) were to fail to meet the performance requirements, then the structural cross-section would be reviewed, as shown in Fig. 1. The recovery time would then be calculated using the same procedure.
We confirmed that using the proposed method to evaluate the design earthquake motion, calculate the response value, and conduct the performance verification enabled us to design a structure that satisfied the required recovery time at the relevant location.
We confirm the effectiveness of using the nomogram for the structure in this section. The target examination area is set as the Sendai area, and the target recovery time is set to “5 days.”
The equivalent natural period of the structure was set to Teq = 0.5 s, and a response analysis was conducted under conditions where the M-point ductility factor (µm) and the yield seismic coefficient (khy) were both changed. Then, we used the same procedure as in the previous section to calculate the expected recovery time of each structure using. Figure 10 shows the results. It is now conceivable to use the N-point ductility ratio, which defines damage level 4, as a structural parameter for estimating recovery time. However, in designing railway structures, structural details are focused on ensuring safety against Level 2 (L2) seismic motions, which are the largest anticipated ground motions at a given site. In this study, the N-point ductility ratio is not included as a parameter since the damage level 4 will not be reached under the design seismic motion. In the seismic design of railway structures, events with an extremely low probability exceeding the L2 level are usually addressed within the framework of “resilience against catastrophic events.”
As shown in Fig. 10, the expected recovery time decreases as the khy of the structure increases. Furthermore, the recovery time shows low sensitivity to µm when the ductility factor µm of the structure is 2 or greater. Figure 9 shows that this can be attributed to the relatively small probability of a structure suffering major damage. In terms of the recovery time, information on the yield seismic coefficient becomes more important when µ exceeds 1. This figure can be used to easily calculate the khy of a structure with the target performance of a recovery time of five days. For example, if µm = 1, then a yield seismic coefficient khy approximately 0.6 satisfies the recovery time of five days.
A similar examination was conducted for various Teq values. Figure 11 shows the relationship between Teq and khy resulting in a recovery time of five days for each ductility factor µm. This is the restorability verification nomogram proposed in Section 2.
As shown in Fig.11, the restorability verification nomogram can be used to easily determine the combination of Teq, khy, and µm of a structure that satisfies the required recovery time in the region (five days in this case). If the khy of a structure is equal to or greater than the vertical axis of the nomogram, then the expected recovery time of that structure will be five days or less. Therefore, the vertical axis of the nomogram is labeled “yield seismic coefficient demand” referring to the yield seismic coefficient required to achieve a recovery time of a certain length or less. In addition, the restorability verification nomogram calculated in this examination showed that the demand of the yield seismic coefficient is high for structures with µm = 1. However, the demand is similar for structures with µm ≥ 2. Figure 10 shows that differences in damage and deformation performance do not significantly affect the recovery time of structures undergoing large deformation.
Furthermore, Fig. 11 shows the restorability verification nomogram, which lots the conditions of the structure with the trial design from Section 3 (Teq = 1.14 s, khy = 0.33) as a red circle. The M-point ductility factor of the structure is µm = 4.43, However, according to the nomogram, the yield seismic coefficient of the structure is slightly larger than the yield seismic coefficient demand. Therefore, the restorability verification nomogram can be used to appropriately verify the performance design of the structure section. It can be said that the proposed nomogram can be used in the general design procedure when a recovery time is considered as the verification index.
In this paper, we proposed a method for verifying the restorability of railway structures that uses the post-earthquake recovery time as the verification index. The results of our study were as follows:
● The proposed method uses “multiple earthquake motions with a wide range of assumed characteristics during the design working life” as the design earthquake motion, and the “recovery time” as the verification index. Recovery time is directly related to the speed of restoration after an earthquake. This allows for the seismic design of structures that consider the issue of recovery time.
● A trial design of a rigid-frame viaduct was conducted using the proposed method. The results indicate that the method allows for the seismic design of structures with recovery time as a direct verification index. However, since multiple dynamic analyses and damage evaluations are required each time the specifications of the structure change, this method of restorability verification during seismic design requires a large amount of effort.
● A more practical design method was proposed by conducting calculations under various conditions in advance and displaying the structural conditions resulting in the same recovery time in a nomogram. Additionally, trial calculations for the above-mentioned rigid-frame viaduct showed that the performance of structures can be verified using this restorability verification nomogram. This method can be used to design structures that can be easily restored using the same procedure as conventional seismic design.
The developed method can be used to design new structures that are easier to restore. Furthermore, identifying the parts and members of existing structures that require restoration in advance enables the implementation of targeted inspections and measures, ultimately shortening the post-earthquake recovery time. Additionally, evaluating existing structures based on their future service life can help determine the appropriate level of measures, set the same performance requirements, and optimize the priority of measures for special structures that require more time to recover.
However, it should be noted that this examination has limitations. First, it is based on a proposed method. Second, it uses trial calculations based on limited conditions, such as regions and structures. Moreover, recovery time for damaged structures can vary considerably depending on various conditions. While trial calculations in this examination consider the uncertainty and variance of earthquake occurrence and motion through probabilistic earthquake hazard analysis, they ignore the uncertainty and variance of structural response values and recovery time. Resolving these issues requires a more in-depth examination. This includes improving the evaluation of recovery time associated with structural damage, correcting the restorability verification nomogram according to the structural characteristics, and considering structural responses uncertainty. Furthermore, standardizing the nomogram requires future evaluations under a wide range of conditions.
This work was financially supported by the Japanese Ministry of Land, Infrastructure, Transport and Tourism. We would like to express our gratitude to those involved here.
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Kimitoshi SAKAI, Dr.Eng. Senior Chief Researcher, Head of Structural Dynamics and Response Control Laboratory, Center for Railway Earthquake Engineering Research Research Areas: Earthquake Engineering, Seismic Design of Railway Structure |
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Kazunori WADA Senior Researcher, Structural Dynamics and Response Control Laboratory, Center for Railway Earthquake Engineering Research Research Areas: Earthquake Engineering, Seismic Design of Railway Structure |
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Akihiro TOYOOKA, Dr.Eng. Director, Head of Center for Railway Earthquake Engineering Research Research Areas: Earthquake Engineering, Seismic Design of Railway Structure |
