Bending Stress Calculator - All Sections — σ = My / I
Rectangular Section
Section Properties: I, ymax and Z
| Shape | I | ymax | Z = I/ymax | σmax | Notes |
|---|---|---|---|---|---|
| Rectangle (b×d) | bd³/12 | d/2 | bd²/6 | 6M/(bd²) | d = depth in bending direction |
| Solid Circle (D) | πD⁴/64 | D/2 | πD³/32 | 32M/(πD³) | Least efficient for bending |
| Hollow Circle (Do,Di) | π(Do⁴−Di⁴)/64 | Do/2 | π(Do⁴−Di⁴)/(32Do) | MDo/(2I) | Tubes are efficient in bending |
| I-Section (symm.) | [BD³−(B−tw)hw³]/12 | D/2 | 2I/D | MD/(2I) | Most efficient standard section |
| T-Section | Parallel-axis theorem | max(ytop,ybot) | I/ymax | Mymax/I | N.A. NOT at mid-depth; find centroid first |
| Square (a×a) | a⁴/12 | a/2 | a³/6 | 6M/a³ | Rectangle with b = d |
Calculation Explanation
Use the calculator above — the full derivation appears here automatically.
Bending Stress Theory & Derivation
The Flexure Formula: σ = My/I
Bending stress varies linearly from zero at the neutral axis to maximum at the extreme fibres. For a downward load on a simply supported beam: top fibres compress (σ negative), bottom fibres are in tension (σ positive).
Sign Convention
y is measured positive upward from the neutral axis:
- y > 0 (above N.A.): Compression, σ is negative for positive moment M
- y = 0 (at N.A.): Zero stress always
- y < 0 (below N.A.): Tension, σ is positive for positive moment M
- |y| must not exceed ymax — any larger value lies outside the cross-section
Engineering Validity Rules
- |y| ≤ ymax always. If y exceeds ymax, the point is outside the section — physically meaningless.
- Formula assumes linear-elastic material (Hooke's Law applies, σ ≤ fy).
- Allowable stress (IS 800 ASD): 0.66fy for compact sections; (0.60fy for non-compact).
- For asymmetric sections (T-beam): ytop ≠ ybot; compute both and use larger for design.
Section Modulus Z
Z = I/ymax (mm³). σmax = M/Z. Higher Z = lower peak stress for same moment. I-beams maximise Z by placing flanges far from the N.A. IS 808 lists steel sections by Z for direct selection.
Frequently Asked Questions
1. What is the bending stress formula?
σ = My/I (Navier's flexure formula). M in N·mm, y in mm, I in mm⁴ gives σ in MPa. Maximum stress at the extreme fibre: σ_max = M/Z where Z = I/y_max is the section modulus (mm³).
2. Can y be greater than half the depth?
No. y is the distance from the neutral axis to a point inside the cross-section. For a symmetric section y_max = d/2. Any |y| > y_max lies outside the section and is physically invalid. This calculator flags such entries as errors.
3. What sign does bending stress have?
For positive bending moment (concave-up beam): top fibres (y > 0) are in compression (σ negative); bottom fibres (y < 0) are in tension (σ positive). Neutral axis: σ = 0. The stress diagram is a straight line through the origin.
4. What is the section modulus Z?
Z = I/y_max (mm³). σ_max = M/Z. Higher Z = more efficient section. I-sections are efficient because flanges are placed far from the N.A., maximising Z per unit area. IS 808 lists steel sections by Z for direct beam selection.
5. How do I find I for a T-section?
Use the parallel-axis theorem: (1) find centroid ȳ = Σ(A_i·y_i)/ΣA_i from any reference. (2) For each sub-area: I_i = bd³/12 + A_i·d_i² where d_i = distance from sub-area centroid to overall centroid. (3) I_total = ΣI_i. The calculator does this automatically for the T-section option.
6. What safety factor should I use?
IS 800:2007 (steel, LSM): γ_m0 = 1.10, design strength = f_y/1.10. Working stress (ASD): allowable = 0.66f_y for compact sections. As a conservative rule: keep σ ≤ 0.60f_y under service loads (SF ≈ 1.67).
7. Why are compression and tension different colours in the diagram?
Green shading = compression zone (above N.A., y > 0). Red shading = tension zone (below N.A., y < 0). The straight diagonal line through the origin (N.A.) is the stress distribution. The amber dot marks your specific point of interest with its stress value.
8. What is the neutral axis?
The N.A. is the line through the centroid of the cross-section where bending stress is zero. For symmetric sections it is at mid-depth. For asymmetric sections (T-beam, angle) it must be located by finding the centroid: ȳ = Σ(A_i·y_i)/ΣA_i. Shear stress, by contrast, is maximum at the N.A.
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