Beam Stress and Deflection Calculator

Formula

• Max moment M = P × L ÷ 4 (simply supported, centre point load)
• Bending stress σ = M × 1000 ÷ Z (M in kN·m, Z in cm³, σ in MPa)
• Deflection δ = P × L³ ÷ (48 × E × I) (P in N, L in m, E in Pa, I in m⁴, result in mm)

About this calculator

Part of our Mechanical Calculators collection. Beams, bolts, springs, gears, bearings, and core ME calculations.

How it works

For a simply supported beam with a central point load, maximum bending moment equals PL divided by 4. Bending stress uses sigma equals M divided by Z. Deflection uses delta equals PL cubed divided by 48EI. Verify critical designs with full section properties and code checks.

Enter your values in the inputs above and click CALCULATE. Results appear on the right without a page reload. No login and no server upload.

Worked example

10 kN centre load on a 2 m span, Z = 50 cm³, E = 200 GPa, I = 500 cm⁴.

1. M = 10 × 2 ÷ 4 = 5 kN·m
2. σ = 5 × 1000 ÷ 50 = 100 MPa
3. δ = 10,000 × 2³ ÷ (48 × 200×10⁹ × 500×10⁻⁸) = 8.33 mm

Result: Max stress 100 MPa, deflection 8.33 mm, span/deflection ratio 240.

When to use

• Preliminary beam check before detailed section selection
• Coursework and hand-calculation verification
• Comparing deflection against a serviceability target

Limitations

• Single centre point load only
• Prismatic beam, linear elastic material
• Not a code-compliant structural design tool

FAQ

What is section modulus Z?

Z is a geometric property of the cross-section that links bending moment to stress: σ = M/Z. Larger Z means lower stress for the same moment.

What deflection ratio is acceptable?

Floor beams often target L/360 for live load. Compare the span/deflection ratio output to your project requirement.

Can I use this for a uniformly distributed load?

Not yet. UDL uses a different formula (5wL⁴/384EI). This tool is for a single centre point load only.