Bending Stress Calculator
Estimate maximum elastic bending stress and section properties for common beam cross-sections.
Beam inputs
Live results
Calculation breakdown
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How to use this bending stress calculator
What this calculator does
This tool estimates the maximum normal stress caused by a bending moment in a prismatic beam. It calculates the area moment of inertia, the distance from the neutral axis to the most distant fiber, the elastic section modulus, and the resulting maximum bending stress. The model is intended for preliminary engineering checks and education. It does not determine whether a real member is safe, because design also depends on material strength, load combinations, stability, connections, defects, fatigue, and the governing building or mechanical code.
When to use it
Use it to compare candidate beam sizes, check a hand calculation, study how section depth affects stress, or create a reproducible calculation record for a report. It is especially useful when you already know the applied bending moment from a statics or structural-analysis model and want a quick elastic stress estimate.
How to calculate
- The calculator opens with a ready-to-use rectangular example: 200 mm wide, 300 mm high, and subjected to 10 kN·m. The displayed results and Excel workbook are available immediately.
- Select the Cross-section. The visible dimension fields update to match the chosen shape.
- Replace the sample dimensions with positive millimetre values. Enter the Applied bending moment (M) in kN·m. Results update as you type.
- Read the maximum stress and section properties, then use Download Excel to save the current validated model. Reset clears the demonstration values and results; Excel download remains unavailable until a complete valid state is entered again.
Input guide
Cross-section is required and selects the geometry formula. Width (b) and Height (h) are positive decimal dimensions in millimetres for rectangles and rectangular tubes. For a square, the width field is used as the side length. Wall thickness (t) is required only for a rectangular tube and must be smaller than half both outer dimensions. Outer diameter (D) is required for circular sections. Inner diameter (d) is required only for a hollow circle and must be smaller than the outer diameter. Applied bending moment (M) is required, accepts a decimal in kN·m, and may be positive or negative; the sign indicates bending direction, while the reported maximum stress uses its magnitude. Scientific notation and decimal-comma entries are rejected to avoid ambiguous parsing.
Output guide
Maximum bending stress (σmax) is reported in MPa and equals the moment divided by the elastic section modulus. Area moment of inertia (I), in mm⁴, measures the section's resistance to curvature about the selected axis. Extreme-fiber distance (c), in mm, is the farthest distance from the neutral axis to the section edge. Section modulus (S), in mm³, combines geometry through S = I/c; a larger section modulus produces less stress for the same moment. The calculation table repeats the exact formulas and canonical values used by both the screen and workbook.
Worked example
For the startup rectangle, b = 200 mm and h = 300 mm. The centroidal area moment of inertia is I = bh³/12 = 200 × 300³ / 12 = 450,000,000 mm⁴. The extreme-fiber distance is c = h/2 = 150 mm, so the section modulus is S = I/c = 3,000,000 mm³. A 10 kN·m moment equals 10,000,000 N·mm. Therefore σmax = M/S = 10,000,000 / 3,000,000 = 3.333 MPa, matching the first-open result.
Learn more
The underlying relationship is the elastic flexure formula used in elementary beam theory. For supporting background, review MIT OpenCourseWare's solid mechanics course materials, NIST's SI unit guidance, and the Engineering LibreTexts explanation of shear and bending-moment diagrams.
How the formulas work
For elastic bending, σ = Mc/I = M/S. Geometry controls I and S. For a rectangle, I = bh³/12, so increasing depth is particularly effective because height is cubed. A solid circle uses I = πD⁴/64. Hollow shapes subtract the inner void's inertia from the outer shape. These formulas assume a homogeneous section, small deflection, a straight member, and loading that produces bending about the stated centroidal axis.
Interpreting the result
Compare the calculated stress with an appropriate allowable or design resistance only after applying the correct load factors, material standard, safety format, and limit-state checks. A low bending stress does not rule out shear failure, lateral-torsional buckling, local buckling, bearing, connection failure, or serviceability problems. The American Institute of Steel Construction publishes additional context through its standards and specifications portal.