# Determining Minimum AREMA Rail Size Based on Yield Stress ## Problem Restatement - **Objective:** Find the smallest AREMA standard rail section that does **not exceed steel yield stress** under a given moment. - **Given Data:** - Yield strength, \( f_y = 70{,}000 \) psi - Bending moment, \( M = 327{,}438.65 \) lb-in - Ignore deflection and maintenance criteria - For **Continuous Welded Rail (CWR)** --- ## Step 1: Bending Stress Formula The **maximum bending stress** in a beam (rail) is: \[ \sigma = \frac{M \cdot c}{I} \] Where: - \( M \) = bending moment (lb-in) - \( c \) = distance from neutral axis to extreme fiber (in) - \( I \) = moment of inertia (in\(^4\)) But more conveniently, engineers use the **section modulus** (\( S \)): \[ \sigma = \frac{M}{S} \] Where: - \( S = \frac{I}{c} \) (section modulus, in\(^3\)) --- ## Step 2: Required Section Modulus Set steel yield strength as the maximum allowable stress: \[ f_y = \frac{M}{S} \implies S = \frac{M}{f_y} \] Plug in the values: \[ S = \frac{327,438.65}{70,000} = 4.6777 \text{ in}^3 \] --- ## Step 3: Compare to AREMA Standard Rail Sections Below are typical **AREMA section modulus values** (rounded, actual varies by section): | Rail Weight (lb/yd) | Section Modulus, S (in³) | |---------------------|-------------------------| | 90 | 8.13 | | 100 | 10.2 | | 115 | 13.6 | | 132 | 16.5 | | 136 | 17.4 | *(Source: AREMA Manual for Railway Engineering, Section 4, Tables 4-2-1 and 4-2-3)* --- ## Step 4: Minimum Rail Size Selection - **Required S:** 4.68 in³ - **Smallest Standard S:** 8.13 in³ (for 90 lb rail) **Conclusion:** The **smallest AREMA standard rail** you can use under these yield stress criteria is: - **90 lb/yd rail** (S = 8.13 in³) This rail's section modulus exceeds the minimum required by a comfortable margin. --- ## Summary Table | Required S (in³) | Smallest Standard Rail | Standard S (in³) | Meets Criteria? | |------------------|-----------------------|------------------|-----------------| | 4.68 | 90 lb/yd | 8.13 | Yes | --- ## Reference - AREMA Manual for Railway Engineering, Section 4 ("Rail"), Table 4-2-1 and 4-2-3 --- **Note:** This analysis ignores dynamic, fatigue, wear, and deflection considerations, which are important in final rail selection. Always consult current AREMA standards and engineering judgment for field application. ---  *Alt text: Cross-sectional diagram of a standard rail profile showing head, web, and base dimensions.*