Beams
Simply supported beam, uniformly distributed load: bending stress and deflection
- where
- maximum bending moment (kNm)
- uniformly distributed load (kN/m)
- span (m)
- maximum bending stress (N/mm²)
- elastic section modulus (cm³)
- midspan deflection (mm)
- Young's modulus (GPa)
- second moment of area (cm⁴)
- span to deflection ratio
Explain this equation
Checks a simply supported steel beam carrying an even load along its length: the bending moment, stress and deflection. Use it as a quick check on a lintel, rail or frame member.
For an even load the moment peaks at midspan at wL²/8. Stress is moment divided by section modulus; the 1000 converts kNm and cm³ to N/mm². Deflection grows with the fourth power of the span and is resisted by the stiffness E × I; the factors of 1000 and 10⁴ convert span to mm, GPa to N/mm² and cm⁴ to mm⁴. Span divided by deflection gives the familiar L/… figure.
Standard results from simple beam theory for a uniformly loaded span.
Sense check: 10 kN/m over 5 m on a 203 × 133 × 25 UB gives 31 kNm, about 136 N/mm² and 16.6 mm deflection, about span/300. Double the span and the deflection goes up 16 times.
Watch out: this is an elastic check only, without safety factors or lateral-torsional buckling. Structural members must be designed by a competent engineer.
Bending moment, stress and midspan deflection for a beam simply supported at both ends carrying an even load along its length.
Enter your values to see the result.
Good to know
E = 210 000 N/mm² (210 GPa) is the BS EN 1993-1-1 value for structural steel. Take I and Z from the Blue Book or the section property calculators here (cm⁴ and cm³). Common UK deflection limits under variable (imposed) load are span/360 for beams carrying brittle finishes and span/200 for general beams (UK National Annex guidance); check the project specification. The bending stress here is the plain elastic stress M/Z with no load factors. Unfactored stress well below yield does not prove the beam is adequate: a full design also applies partial factors and checks shear, bearing, web buckling and lateral-torsional buckling.
Common mistake
Using the factored (ultimate) load for deflection, or forgetting that deflection depends on the fourth power of span: doubling the span gives 16 times the deflection.
Limits
Elastic, prismatic beam, simple supports, small deflections. Not a structural design to BS EN 1993-1-1; unrestrained beams can fail by lateral-torsional buckling at a much lower stress.
Sources
- W. C. Young, R. G. Budynas, Roark's Formulas for Stress and Strain, 8th ed., Table 8.1 (simply supported beam: UDL M = wL²/8, δ = 5wL⁴/384EI; central point load M = WL/4, δ = WL³/48EI)
- BS EN 1993-1-1 Eurocode 3: Design of steel structures – General rules (clause 3.2.6: E = 210 000 N/mm², α = 12 × 10⁻⁶ per K)
- British Steel universal beams datasheet (UKB 203×133×25: h 203.2, b 133.2, tw 5.7, tf 7.8, r 7.6 mm; A 32.0 cm²; Iy 2340 cm⁴; Wel,y 230 cm³; iy 8.56 cm)
Results are calculations from the published methods and figures shown, and depend on the data you enter. Follow the welding procedure specification (WPS), BS EN ISO 15614/9606, BS EN 1090, BS EN 1993 and manufacturers’ data. Structural welds need qualified welders and engineered designs.