# Step-by-Step Solution: Smallest AREMA Rail Size Based on Yield Stress ## Problem Restatement - **Goal:** Find the smallest AREMA standard rail size that will not exceed the steel yield stress under a moment \( M \) for continuous welded rail (CWR). - **Given:** - Yield strength, \( f_y = 70,000 \) psi - Bending moment, \( M = 327,438.65 \) lb-in - Ignore deflection and maintenance - Use AREMA standard rail sections --- ## Step 1: Yield Stress Criterion For a beam (rail) under bending: \[ \sigma_{\text{max}} = \frac{M \cdot c}{I} \] Where- \( M \) = Maximum moment - \( c \) = Distance from neutral axis to extreme fiber (half the height) - \( I \) = Moment of inertia But more conveniently, **section modulus** \( S \): \[ \sigma_{\text{max}} = \frac{M}{S} \] \[ S = \frac{I}{c} \] To **not exceed yield**: \[ \sigma_{\text{max}} \leq f_y \implies \frac{M}{S} \leq f_y \] \[ S \geq \frac{M}{f_y} \] --- ## Step 2: Calculate Required Section Modulus Plug in values: \[ S_{\text{required}} = \frac{M}{f_y} = \frac{327,438.65\,\text{lb-in}}{70,000\,\text{psi}} \] \[ S_{\text{required}} = 4.6777\,\text{in}^3 \] --- ## Step 3: Compare to AREMA Rail Section Properties AREMA standard rail sizes (partial list, values approximate): | Rail Weight (lb/yd) | Section Modulus \( S \) (in³) | |---------------------|-------------------------------| | 90 | 5.14 | | 100 | 6.11 | | 115 | 7.63 | | 132 | 9.85 | | 136 | 10.26 | (Source: AREMA Manual for Railway Engineering, Section 4) --- ## Step 4: Find the Smallest Rail Satisfying \( S \geq 4.68 \) in³ - **90 lb/yd rail:** \( S = 5.14 \) in³ > 4.68 in³ - **85 lb/yd rail:** \( S = 4.44 \) in³ < 4.68 in³ So, **90 lb/yd rail** is the smallest AREMA standard section that satisfies the criterion. --- ## Step 5: Final Answer **Final Answer:** > **The smallest AREMA standard rail size that can be used without exceeding the steel yield stress is the 90 lb/yd rail section, which has a section modulus of 5.14 in³, exceeding the required 4.68 in³.** --- ### Table: AREMA Rail Section Comparison | Rail Size | Section Modulus (in³) | Meets Requirement? | |-----------|-----------------------|--------------------| | 85 lb/yd | 4.44 | No | | **90 lb/yd** | **5.14** | **Yes** | | 100 lb/yd | 6.11 | Yes | --- **Note:** Always verify with the latest AREMA manual for exact dimensions and properties for rail selection. ---  *Alt text: Illustration of various AREMA rail cross section profiles*