Simplified Impact Load Model Analysis of Vehicle-to-Bridge Pier Collision

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1.It is recommended that the authors more accurately describe the "Simplified Mass-spring-pier Model", for instance, by referring to it as a "reduced-order dynamic model," and clearly articulate its fundamental difference from the response surface model in terms of application convenience.
2.The authors are advised to include a detailed discussion in the paper regarding the limitations of the simplified mass-spring model in addressing shear failure, and to specify the conditions under which this model is applicable.
3.It is recommended that the authors provide a detailed analysis and explanation for the significant 30.02% error in the simplified model's peak dynamic load prediction at the low-speed condition of 40 km/h (Table 1), and discuss the model's applicability within this low-speed range.
4.To significantly enhance the usability for readers and designers, it is recommended that the authors explicitly present the final regression equations for the Response Surface Model (RSM) in the main text or an appendix.
5.The authors are advised to supplement the manuscript with detailed results from the parameter sensitivity analysis
6.It is recommended that the authors add a section to the paper that directly compares and discusses their calculated equivalent static loads against the AASHTO LRFD specified load of 600 kips.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript is well written and considers a consistently hot topic within the field of impact and structural engineering. The authors need to address the following concerns to improve the quality of the manuscript:

1- Despite being known in the field, the concept of equivalent static force should be clearly defined and presented. 

2- Please use the full name of the codes when they are first mentioned. Also, being specific and including the chapter/section/equation numbers is needed. 

3- The following study should be included in the literature review section as it addresses the same topic in a very comprehensive manner:

Alomari, Qusai A., and Daniel G. Linzell. "Advanced analysis of intact, fire-damaged, and CFRP retrofitted bridge pier columns under vehicle collisions: Empirical equivalent static force equation and framework." Engineering Structures 314 (2024): 118250.

4- Where did the values of the parameters used in section 2, page 3, come from? Why were these specific values selected?

5- More details on the finite element analysis are needed. The modeling process is totally unclear. This also needs to be supported by sample results. 

6- How was the FE model validated? 

7- The assumption that is made on the effect of boundary conditions (Section 3.1, page 5) needs to be justified. Also, what previous studies were followed to show that the simplified mass-spring model can be effectively used?

8- Many of the values used in the analysis were selected without any rationale. For example, why was 4000 psi specifically considered?

9- Despite listing the 140 km/h as one of the studied speeds, no results/figures associated with that were presented!!

10- The results listed in Table 1 are coming from nowhere. Sample figures from each FE analysis should be presented to show how the ESF was derived from the dynamic impact forces. 

11- Table 2 needs to be placed on one page. 

12- For the parameters considered in the second approach, why was cargo but not vehicle weight considered? Vehicle weight might be way more critical. When "G" is zero, that means no mass effect is considered, which should not be true.

13- How do your results compare to previous similar and related studies? Are there any specific limitations?

14- Comment on the feasibility of applying your proposed approaches and why responsible agencies would use them. These agencies tend to avoid very long mathematical expressions and prefer simplified equations.   

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1. The abstract and introduction should be rewritten to more clearly and concisely state the research gap, the specific objectives of this study, and the novel contributions of the proposed models. The current motivation is buried in a descriptive literature review.
2. The manuscript requires thorough proofreading and editing to correct numerous grammatical errors, improve sentence structure, and enhance the overall academic tone. Awkward phrasing hinders comprehension and professionalism.
3. The simplified mass-spring model relies on several critical assumptions (e.g., ignoring the secondary impact from cargo mass, simplifying damping as a constant force). These assumptions need to be more rigorously justified, and their potential impact on the results should be discussed, perhaps with a brief sensitivity analysis or references to support their validity.
4. The manuscript would be significantly strengthened by a direct and explicit comparison of the proposed models' performance against other existing simplified methods from the literature (e.g., Consolazio & Cowan, 2005; Chen et al., 2016). This would better highlight the advantages and limitations of the new approaches.
5. While the AASHTO value is mentioned, the practical utility of the study would be clearer if the results (e.g., in Table 1) included a column for the corresponding AASHTO design load, allowing for a direct visual comparison of how the proposed models differ from the current code-based approach.
6. Figures are essential but are poorly integrated into the text. Each figure (e.g., Figure 1, Figure 9) should be explicitly referenced and discussed in the main body. The captions should be more descriptive, and the figures themselves should be checked for clarity (e.g., axis labels in Figure 9 are cut off).
7.  For the response surface model, the discussion should be expanded to address potential concerns about overfitting, given the 63 data points for 36 parameters. While R² values are high, a comment on the model's robustness and generalizability beyond the sampled parameter space would be beneficial.
8.  The manuscript should more explicitly acknowledge the limitations of both models. For instance, the mass-spring model's inability to capture shear failure—a common failure mode—is a significant limitation that should be clearly stated and its implications discussed.
9. The flow between sections is currently disjointed. The transition from the literature review to the finite element analysis, and then to the two simplified models, needs smoother bridging sentences to guide the reader.
10. The derivation of key parameters for the mass-spring model (e.g., damping constants D₁, D₂, and spring stiffness k₁) needs to be presented more clearly. The logic behind the chosen methods (e.g., using linear regression for k₁, the theorem of momentum for D₁) should be elaborated.
11.  The data and plots in the appendices are valuable. However, the main text should briefly summarize or interpret key findings from these appendices (e.g., the meaning of the 3D plots in Appendix B) rather than simply directing the reader there.
12. The conclusion should be more succinct and forceful. It should clearly summarize the key findings, reiterate the main contribution of the work, and provide a more specific outlook on future research or practical application of the proposed models.
13. Ensure consistent formatting of equations, variables, and units throughout the manuscript. Some references in the list have formatting inconsistencies (e.g., italics, capitalization) that should be corrected.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf