Beams
Simply supported beam, central point load: bending stress and deflection
- where
- maximum bending moment (kNm)
- point load at midspan (kN)
- 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 beam carrying one load at midspan, such as a lifting beam or a post bearing on a beam.
A central point load gives a peak moment of PL/4. Stress is moment over section modulus, with the 1000 converting units. Deflection grows with the cube of the span and is resisted by E × I; the factors convert kN to N, m to mm, GPa to N/mm² and cm⁴ to mm⁴.
Standard simple beam theory for a central point load.
Sense check: 20 kN at the middle of a 4 m 203 × 133 × 25 UB gives 20 kNm, about 87 N/mm² and 5.4 mm deflection. Double the span and the deflection goes up eight times.
Watch out: a point load near a support or off-centre gives different results, and lifting beams need dynamic factors and a proof test. Get structural members designed.
Bending moment, stress and midspan deflection for a simply supported beam with a single load at the centre, such as a lifting beam or a beam carrying a post.
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. Self-weight is not included; for a heavy beam run the UDL calculator for self-weight and add the two deflections.
Common mistake
Applying the formula to an off-centre load. The central position gives the largest moment; for other positions use M = P·a·b/L and a proper deflection formula.
Limits
Elastic, prismatic beam, simple supports, load at midspan only. Not a structural design; check local web bearing under the load and lateral-torsional buckling.
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)
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.