Fatigue Durability Analysis for Suspenders of Arch Bridge Subjected to Moving Vehicles in Southwest China
Round 1
Reviewer 1 Report
This paper investigated the fatigue problem of suspenders on arch bridges. The subject is valuable, while according to the reviewer, there are some questions/problems that should be explained/justified, given as follows:
1. There are plenty of errors on the cross-references (table and figures) e.g. Line 130, 149, 164, etc. It results in a lot of difficulties in understanding the results and discussions in this paper.
2. It is curious how the vehicles are categorized in Table 1. It seems that some types are of high standard deviations. For example, V4 & V5 are both 3-axle vehicles, while with the standard deviation of the weight, one vehicle can be categorized to both types. Not to mention that the subtypes, e.g. 3 axle -2 and 3 axle -3 both in V5, are more difficult to distinguish.
3. How is Equation (7) proposed? In the context, there are no references to literatures, and no illustrations on how is it obtained.
4. Fig.10 shows the S-N curve for fatigue assessment. It is suggested to use the logarithmic axis in it, as the common way in literatures.
5. Table 8 uses capitalized letters on all columns, making no differences between the target and actual value.
6. Line 421, it is addressed that the mean stress is about 98 MPa. However, there is no clue on how this value is obtained. Mostly importantly, it is used in Equation (13) to conduct the fatigue assessment. By doing so, it should be noted that, according to the form of the equations for S-N curves, sigma_m is usually the mean stress for the constant amplitude, i.e. the mean value of the maximum and minimum stress. For the realistic load, it is surely inducing variable amplitude, which means it may not be applicable.
7. The similarity of the traffic load on Xijiang Bridge and Dafeng Bridge should be explained and discussed. Southwest China is quite large, thus it is not a solid assumption that the traffic load measured on one site can be used directly on the other.
8. It is necessary to polish the languages and formats of the paper. Apart from the cross-references, other problems include the improper word representation, some words should/ should not be capitalized (e.g. the title), using different fonts in figures, incomplete legend in figures, etc.
Author Response
We would like to express our sincere thanks to the reviewers for the constructive and positive comments which will be answered below one by one.
1.Response to the reviewer 1’s remarks:“There are plenty of errors on the cross-references (table and figures) e.g. Line 130, 149, 164, etc. It results in a lot of difficulties in understanding the results and discussions in this paper”
Response: Thanks very much for your suggestion.
We are sorry for making plenty of errors on the cross-references by mistake. According to the reviewer’s recommendations. all the errors on the cross-references (table and figures) have been corrected. Such as Table 1、Table 2...
Corresponding changes have been marked in yellow in the revised manuscript.
2.Response to the reviewer 1’s remarks:“It is curious how the vehicles are categorized in Table 1. It seems that some types are of high standard deviations. For example, V4 & V5 are both 3-axle vehicles, while with the standard deviation of the weight, one vehicle can be categorized to both types. Not to mention that the subtypes, e.g. 3 axle -2 and 3 axle -3 both in V5, are more difficult to distinguish’’
Response: Thanks very much for your suggestion.
This is a very good suggestion and we are sorry for not mentioning the classification method in the paper. So, add the classification method before Table 1 in the revised manuscript (line 128).
The specific classification method is as follows: Firstly, the vehicle is classified by the number of axles, wheelbase Di and weight W. Calculate the average and standard deviation of wheelbase and weight after classification. Secondly, we combine vehicles with similar average weights and wheelbase in the first classification, which reduces the number of categories and reduces the calculation amount.
Therefore, according to the above classification method. 3 axle -1,2and 3 are classified according to wheelbase Di. all cars of 3 axle-2 wheelbase D1 are close to 3.6m, and all cars of 3 axle-3 wheelbase D1 are close to 4.2m. There is a significant difference between them, so they are divided into two categories. Then it is found that the average weight of 3 axles -2 and 3 are similar, so they are combined to form V5. The large standard deviation of weight is due to the difference in weight of the 3-axle car with a similar wheelbase.
In this paper, the problem of the excessive standard deviation of weight is also taken into account when classifying. However, to simplify the calculation and reduce the number of classifications, the classification is not too detailed. Subsequently, the classification standard of vehicles will be further refined or optimized.
Corresponding changes have been marked in yellow in the revised manuscript.
3. Response to the reviewer 1’s remarks:“How is Equation (7) proposed? In the context, there are no references to literatures, and no illustrations on how is it obtained.”
Response: Thanks very much for your suggestion.
This is a very good suggestion and we are sorry for not mentioning this in the paper. According to your suggestion, we have added the derivation process of Equation (7) ,which is Eq .(11) in the revised manuscript. (line 256)
The S-N curve of the material is modified by the mean stress[39].
Paulson et al. of the United States put forward the S-N fatigue curve of grade 270 (equivalent to grade 1860 in China) steel strand with an average stress of 1050 MPa as Eq. (7).
lgN=13.84-3.5lgσa (7)
where: σa is the stress amplitude; N is the number of cycles.
Therefore, based on Eq. (7), the mean stress is corrected. Because the suspenders may be corroded, Goodman, the stress line suitable for notched specimens, is selected for correction. Goodman formula is as Eq. (8).
σa’=σar(1－σm/σb) (8)
Where: σa’ is the ultimate fatigue stress amplitude; σar is the fatigue limit under symmetrical cycle; σm is the mean stress; σb is the tensile strength.
In engineering, the stress corresponding to 107 cycles is considered the fatigue limit, so N=107 cycles are brought into Eq. (7) to obtain the ultimate fatigue stress amplitude σa'. Then, σa' is brought into Eq. (8) to obtain the σar corresponding to the average stress of 1050MPa. Thus, the ultimate fatigue amplitude is deduced as Eq. (9).
σa'=206.69(1－σm/σb) (9)
The slope of the S-N curve of the material remains unchanged under different ultimate strengths. Therefore, the revised Eq. (7) should be lgN=a-3.5lgσa, bring in N=107, σa=σa', the formula for obtaining a is as Eq. (10).
a=lg107-3.5lg[206.69(1－σm/σb)] (10)
Therefore, the S-N fatigue curve after the mean stress correction is as follows
lgN=7+3.5lg[206.69(1－σm/σb)]－3.5lgσa=15.1+3.5lg[1/σa－σm/(σbσa)] (11)
Corresponding changes have been marked in yellow in the revised manuscript.
4.Response to the reviewer 1’s remarks:“10 shows the S-N curve for fatigue assessment. It is suggested to use the logarithmic axis in it, as the common way in literatures”
Response: Thanks very much for your suggestion.
This is a very good suggestion, according to your recommendation, the S-N curve for fatigue assessment, which is Fig.11 in the revised manuscript, has been changed to use the logarithmic axis . Please see the attachment.
Corresponding changes have been marked in yellow in the revised manuscript.
5. Response to the reviewer 1’s remarks:“Table 8 uses capitalized letters on all columns, making no differences between the target and actual value”.
Response: Thanks very much for your suggestion.
We are sorry for not making differences between the target and actual value. According to the reviewer’s recommendations, Table 8 has been changed as follows:
Table 8. Random data test of vehicle distance.
|
Data set |
Capacity |
U(m) |
u(m) |
D(m) |
d(m) |
Error |
|
daytime lane No. 2 |
29 |
2758.6 |
2759 |
40 |
40.62 |
1.55% |
|
daytime lane No. 3 |
59 |
1355.9 |
1356 |
20 |
20.11 |
0.55% |
|
night lane No. 2 |
10 |
8000 |
8000 |
120 |
119.41 |
0.494% |
|
night lane No. 3 |
20 |
4000 |
4000.1 |
60 |
60.83 |
1.33% |
Where U and D represent the set target mean and variance, respectively; u and d represent the actual mean and variance of the generated random numbers, respectively.
Corresponding changes have been marked in yellow in the revised manuscript.
6.Response to the reviewer 1’s remarks:“Line 421, it is addressed that the mean stress is about 98 MPa. However, there is no clue on how this value is obtained. Mostly importantly, it is used in Equation (13) to conduct the fatigue assessment. By doing so, it should be noted that, according to the form of the equations for S-N curves, sigma_m is usually the mean stress for the constant amplitude, i.e. the mean value of the maximum and minimum stress. For the realistic load, it is surely inducing variable amplitude, which means it may not be applicable”
Response: Thanks very much for your suggestion.
We have also considered this problem in the process of doing this research. However, due to the irregular stress spectrum of the vehicle load, it is difficult to calculate the fatigue life of the suspender, so this paper refers to the method of references [38,39]. According to reference[38,39], the irregular stress spectrum of the vehicle is equivalent to the sinusoidal stress spectrum by obtaining the equivalent stress amplitude σae. That is, the mean stress is the mean value obtained from the stress spectrum, and the irregular stress spectrum is changed to vibrate σae up and down on the basis of the mean stress. The equivalent stress amplitude load σae is calculated by the rain flow counting method. Then the mean stress σm and σae are used to calculate the life of suspenders. Therefore, due to the calculation of σae, changing irregular stress amplitude into sinusoidal stress amplitude, we think Equation (13), which is Eq. (16) in the revised manuscript, can be applied.
In the follow-up, we will rely on more practical projects to test the correctness and feasibility of the methods used in this paper, and more research will be made on the life of suspenders and the stress spectrum under vehicle load.
7.Response to the reviewer 1’s remarks:“The similarity of the traffic load on Xijiang Bridge and Dafeng Bridge should be explained and discussed. Southwest China is quite large, thus it is not a solid assumption that the traffic load measured on one site can be used directly on the other”
Response: Thanks very much for your suggestion.
This point is also taken into account when studying the influence of different bridge vehicle loads on the life of suspenders. However, considering that every bridge needs to re-count the vehicles, the workload is too heavy. Therefore, referring to the established methods of the large-scale standard fatigue vehicle in China and the United States[Chinese General Code for Design JGD60-2015 and AASHTO LRFD Bridge Design Specifications], establish the standard fatigue vehicle in Southwest China, according to the traffic flow data of Xijiang Bridge, a transportation hub in Southwest China.
In this paper, the calculation method of suspender life is to use the bridge health monitoring system to count the traffic flow across the bridge, and then regard all the vehicles over 3 tons as the established standard fatigue vehicle. Therefore, the life of suspenders of different bridges is calculated by using the same standard fatigue vehicle but different traffic flows. Such as Dafeng River Bridge As for the life calculation of the suspender of Dafeng River Bridge, this paper uses the health monitoring system installed on Dafeng River Bridge to monitor the traffic flow, regards the vehicles, which weigh above 3 t, as the standard fatigue vehicle to calculate the fatigue life of the bridge suspender. Therefore, we think even though southwest China is quite large, it can be calculated by using different traffic flows of different bridges.
We will further explore and verify the applicability of the standard fatigue vehicle established in the simulation of other bridge vehicles in Southwest China.
8. Response to the reviewer 1’s remarks: “It is necessary to polish the languages and formats of the paper. Apart from the cross-references, other problems include the improper word representation, some words should/ should not be capitalized (e.g. the title), using different fonts in figures, incomplete legend in figures, etc”
Response: Thanks very much for your suggestion.
According to your recommendation, the language of the revised manuscript has been reorganized and some sentences have been modified. all the errors in the paper have been corrected. We hope these revisions could meet the requirements of your journal. Some changes are listed as follows:
In Abstract:
- “To study the influence of vehicle loads on the fatigue life of local arch bridge suspenders in Southwest China, vehicle statistics were made on typical road sections in Southwest China, and the standard fatigue car model was established.”is changed to “An analysis of the effect of vehicle loads on the fatigue life of local arch bridge suspenders in Southwest China was conducted by generating vehicle statistics, creating a fatigue vehicle model, and developing a fatigue life prediction method”
...
In Introduction
- “In the past fatigue researches, researchers conducted various fatigue tests to predict the fatigue life”is changed to “In the past, researchers conducted various fatigue tests to predict fatigue life”
- “ Roffey et al. [16,17] carried out the tensile and fatigue cyclic test on the cable, studied the fracture mechanism of the cable, tried to alleviate the steel wire fracture rate and prolong the fatigue life of the cable”is changed to “P. Roffey et al. [16,17] studied the fracture mechanism, conducted tensile and fatigue tests on the cable to reduce the steel wire fracture rate and lengthen the fatigue lifespan of the cable”
...
In Establishment of standard fatigue car model
- “Therefore, the Third Xijiang Bridge is selected as a typical section in Southwest China, and vehicle statistics are made”is changed to “Therefore, in this study, vehicle statistics are made in the Third Xijiang Bridge, which is selected as a typical section in Southwest China”
- “Reasonable arrangement of the lane closure”is changed to “Closure of lanes in a reasonable manner”
...
In Engineering verification and application
- “of which 1 lane is a small car (V1-type car) mainly used in the fast lane”is changed to “Lane No. 1 is mainly used by small cars (V1-type cars)”
- “roughly defines the variance of vehicle distance, which is 40 and 20 in the lanes 2, 3 during the day, and 120 and 60 at night, respectively.”is changed to “Roughly define the variance of vehicle distance, taking 40 and 20 for lanes No. 2 and 3 in daytime and 120 and 60 at night, respectively.”
...
Corresponding changes have been marked in yellow in the revised manuscript.
We tried our best to improve the manuscript and made corresponding changes in revised manuscript according to the reviewer’s remarks. We appreciate for editors and reviewers’ warm work earnestly and hope that the correction will meet with approval.
Once again, thank you very much for your remarks and suggestions.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Interesting study. Thansks Authros. I have some question regardign the study below:
1) Considered only fn, the fundamental frequency of the structure (Equation 6)? What about other higher frequencies? Not considered?
2) Mean stress corrections were consdiered?
3) What fatigue category was used for the suspenders?
4) Where in the suspender was measured for stress or fatigue life?
5) Not all stresses influence the fatigue. What is the constant fatigue limit for the suspender or material of the suspender?
6) Cross section view of the suspender?
7) Young's modulus in line 302 unit?
8) Figure 10 needs to be a log scale.
9) Which AASHTO code of the United States the authors used? There is no reference.
Why AASHTO is used? The bridge is designed based on the Chinese code?
10) The authors condiered other load (i.e., wind) for fatigue?
Author Response
We would like to express our sincere thanks to the reviewers for the constructive and positive comments which will be answered below one by one.
1.Response to the reviewer 2’s remarks:“Considered only fn, the fundamental frequency of the structure (Equation 6)? What about other higher frequencies? Not considered?”
Response: Thanks very much for your suggestion.
In the process of doing this research, this paper also considered this problem. However, due to the limitation of time and practical engineering, the stress sensor is not installed on the real bridge. Therefore, only through theoretical calculation, the stress spectrum under vehicle load can be calculated. Because of the uneven pavement and other problems, the impact coefficient should be considered in the calculation. At present, According to JTG D60-2004 "General Code for Design of Highway Bridges and Culverts", the impact coefficient of the reinforced concrete arch bridge is Eq. (5). The impact coefficient is determined by the bridge fundamental frequency fn. So we only consider the fundamental frequency to obtain the impact coefficient.
In the follow-up, we will install the actual stress sensor on the suspender to compare and verify the correctness of the stress spectrum obtained in this paper, and further study the influence of high frequency on the stress spectrum under vehicle load.
2.Response to the reviewer 2’s remarks:“Mean stress corrections were consdiered?”
Response: Thanks very much for your suggestion.
This is a very good suggestion and we are sorry for not mentioning this in the paper. The S-N curve of the material is modified by the mean stress[39].(line 256)
Paulson et al. of the United States put forward the S-N fatigue curve of grade 270 (equivalent to grade 1860 in China) steel strand with an average stress of 1050 MPa as Eq. (7).
lgN=13.84-3.5lgσa (7)
where: σa is the stress amplitude; N is the number of cycles.
Therefore, based on Eq. (7), the mean stress is corrected. Because the suspenders may be corroded, Goodman, the stress line suitable for notched specimens, is selected for correction. Goodman formula is as Eq. (8).
σa’=σar(1－σm/σb) (8)
Where: σa’ is the ultimate fatigue stress amplitude; σar is the fatigue limit under symmetrical cycle; σm is the mean stress; σb is the tensile strength.
In engineering, the stress corresponding to 107 cycles is considered the fatigue limit, so N=107 cycles are brought into Eq. (7) to obtain the ultimate fatigue stress amplitude σa'. Then, σa' is brought into Eq. (8) to obtain the σar corresponding to the average stress of 1050MPa. Thus, the ultimate fatigue amplitude is deduced as Eq. (9).
σa'=206.69(1－σm/σb) (9)
The slope of the S-N curve of the material remains unchanged under different ultimate strengths. Therefore, the revised Eq. (7) should be lgN=a-3.5lgσa, bring in N=107, σa=σa', the formula for obtaining a is as Eq. (10).
a=lg107-3.5lg[206.69(1－σm/σb)] (10)
Therefore, the S-N fatigue curve after the mean stress correction is as follows
lgN=7+3.5lg[206.69(1－σm/σb)]－3.5lgσa=15.1+3.5lg[1/σa－σm/(σbσa)] (11)
Corresponding changes have been marked in yellow with an underline in the revised manuscript.
3.Response to the reviewer 2’s remarks:“What fatigue category was used for the suspenders?”
Response: Thanks very much for your suggestion.
We have also considered this problem in the process of doing this research. However, due to the irregular stress spectrum of the vehicle load, it is difficult to calculate the fatigue life of the suspender, so this paper refers to the method of references [38,39]. According to reference[38,39], the irregular stress spectrum of the vehicle is equivalent to the sinusoidal stress spectrum by obtaining the equivalent stress amplitude σae. That is, the mean stress is the mean value obtained from the stress spectrum, and the irregular stress spectrum is changed to vibrate σae up and down on the basis of the mean stress. The equivalent stress amplitude load σae is calculated by the rain flow counting method. Then the mean stress σm and σae are used to calculate the life of suspenders. Therefore, due to the calculation of σae, change irregular stress amplitude into sinusoidal stress amplitude.
We will further monitor the life of suspenders through more practical projects and compare them with this method to verify the correctness and feasibility of this method.
4.Response to the reviewer 2’s remarks:“Where in the suspender was measured for stress or fatigue life?”
Response: Thanks very much for your suggestion.
In the process of doing this research, we also considered this problem. However, due to the limitation of time and practical engineering, the stress sensor is not installed on the real bridge. Therefore, we use the bridge health monitoring system to monitor the forces of different suspenders in real time. The stress of suspender is calculated by dividing the suspender force by the area. Through theoretical calculation, the stress spectrum under vehicle load can be calculated, and the fatigue life of the suspender is calculated by the method proposed in this paper, through the calculated standard fatigue vehicle and the traffic flow on the monitored bridge.
In the follow-up, we will rely on more practical projects to test the correctness and feasibility of the methods used in this paper, and more research will be made on the life of suspenders and the stress spectrum under vehicle load.
5.Response to the reviewer 2’s remarks:“Not all stresses influence the fatigue. What is the constant fatigue limit for the suspender or material of the suspender?”
Response: Thanks very much for your suggestion.
This is a very good suggestion. In the process of doing this research, we also considered this problem. However, due to the limitation of time and practical engineering. At present, the fatigue test has just been carried out. Therefore, we refer to previous practical projects and the S-N curve of materials. We think in engineering, the stress corresponding to 107 cycles is considered the fatigue limit. Concerning [39], we derive the modified S-N curve formula as Eq. (11). So N=107 cycles are brought into Eq. (11) can obtain the equivalent ultimate fatigue stress amplitude σa'=206.2MPa. Therefore, we think that adding the equivalent ultimate fatigue stress amplitude to the mean stress is the constant fatigue limit for the suspender.
In the follow-up, we will conduct fatigue tests to verify the correctness of the present theory, and further study the constant fatigue limit for the suspender.
6.Response to the reviewer 2’s remarks:“Cross section view of the suspender?”
Response: Thanks very much for your suggestion.
We are sorry for not putting the section view of the suspender in the paper. According to your recommendation, add the cross section view of the suspender in the revised manuscript. (Fig. 9) Please see the attachment.
Corresponding changes have been marked in yellow with an underline in the revised manuscript.
7.Response to the reviewer 2’s remarks: “Young's modulus in line 302 unit?”
Response: Thanks very much for your suggestion.
We are sorry for not adding the unit of Young's modulus by mistake. According to your recommendation, add the unit of Young's modulus in the revised manuscript.
Young's modulus is 2.95×1011 Pa
Corresponding changes have been marked in yellow with an underline in the revised manuscript.
8.Response to the reviewer 2’s remarks:“Figure 10 needs to be a log scale.”
Response: Thanks very much for your suggestion.
This is a very good suggestion, according to your recommendation, the S-N curve for fatigue assessment, which is Fig.11 in the revised manuscript, has been changed to use the logarithmic axis . Please see the attachment.
Corresponding changes have been marked in yellow with an underline in the revised manuscript.
9.Response to the reviewer 2’s remarks: “Which AASHTO code of the United States the authors used? There is no reference.Why AASHTO is used? The bridge is designed based on the Chinese code?”
Response: Thanks very much for your suggestion.
This is a very good suggestion. In the process of doing this research, we also considered this problem. So, we only adopted AASHTO's method of establishing a wide range of standard fatigue vehicle in the United States to establish a standard fatigue vehicle in Southwest China. As we have established the standard fatigue vehicle through the vehicle data of Xijiang Bridge, a transportation hub in Southwest China, we believe that the standard fatigue vehicle can be applied in Southwest China. (AASHTO LRFD Bridge Design Specifications-2007 is used. )
According to your recommendation, The bridge is designed based on the Chinese code, Therefore, we have made the following modifications. In the conclusion, we will change the comparison with the American standard fatigue vehicle to the comparison with the standard fatigue vehicle in China's existing specifications, which may better explain the situation of this paper. we made the following changes in the revised manuscript.
Compared our model with the standard fatigue vehicle in the Chinese General Code for Design JGD60-2015, we established that model’s weight is lower.
Corresponding changes have been marked in yellow with an underline in the revised manuscript.
10.Response to the reviewer 2’s remarks: “The authors condiered other load (i.e., wind) for fatigue?”
Response: Thanks very much for your suggestion.
This is a very good suggestion. In the process of doing this research, we also considered this problem. However, due to the time limit and the difficulty of wind simulation to calculate the stress curve of the suspender. According to references [4,5], during the service period of bridges, suspenders are susceptible to repeated fluctuating fatigue loads, mainly vehicle loads. It is considered that the vehicle load plays a decisive role in the life of suspenders, and the wind has little effect on the life of suspenders. So in this paper, we only consider the influence of vehicle load on the life of the suspender.
In the future, we will conduct further research on the life of suspenders and increase the influence of other loads such as wind. Establish windmill coupling theory, to more accurate calculation of suspenders life.
We tried our best to improve the manuscript and made corresponding changes in revised manuscript according to the reviewer’s remarks. We appreciate for editors and reviewers’ warm work earnestly and hope that the correction will meet with approval.
Once again, thank you very much for your remarks and suggestions.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors have answered all questions in a proper way. No further questions from me.
Reviewer 2 Report
The authors addressed all the comments appropriately and accordingly.
