Equivalence and Comprehensive Guidance for Vertical Curves on Railroads
Bibliographic record
Abstract
This paper aims to resolve confusion among track professionals regarding the shape of vertical curves on railroads, specifically whether they should be considered parabolic or circular. Through mathematical, geometric demonstrations and CAD drawing, the paper establishes the equivalence between parabolic and circular vertical curves on railroads. It derives a general formula for vertical curves that serves as the foundational source for existing formulas in reputable literature, such as the Transit Cooperative Research Program (TCRP) report and the American Railway Engineering and Maintenance-of-Way Association (AREMA) manual. The paper reviews these vertical curve formulas and recommends a maximum acceptable value for vertical acceleration. It also provides installation criterion of a vertical curve. The minimum radius required to ensure stability of a vertical curve under thermal load under unloaded condition is analyzed; this important safety aspect is ignored in literature. The paper then suggests the absolute minimum values for the radius of a vertical curve on both ballasted and direct fixation track. By addressing these aspects, the paper aims to offer a comprehensive guidance for professionals involved in designing and implementing vertical curves on railroads, ensuring safety and efficient operation of the rail network.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".