Beam Deflection Calculator — Simply Supported with Uniform Load
The most common case in structural engineering is a simply-supported beam carrying a uniformly distributed load. This page explains how to calculate maximum deflection and bending moment, and gives you a free online calculator that does it in milliseconds.
Formulas
| Quantity | Formula |
|---|---|
| Maximum bending moment | M_max = wL² / 8 |
| Maximum deflection at midspan | δ_max = 5wL⁴ / (384·E·I) |
| End reactions (each support) | R = wL / 2 |
| Shear at distance x from left support | V(x) = w·(L/2 − x) |
How to use it
- Enter the uniform load w in kN/m. This is the dead+live load combined per meter of beam length.
- Enter the beam span L in meters between centerlines of supports.
- Enter the modulus of elasticity E in GPa (200 for structural steel, 25–35 for normal-weight concrete).
- Enter the cracked or gross moment of inertia I in cm⁴. Use gross I_g for service deflection checks per ACI 318 §24.2.3.
- Read M_max in kN·m and δ_max in mm. Compare δ_max to L/360 (live-load), L/240 (immediate), or L/480 (sensitive non-structural elements per ACI 318-19 Table 24.2.2).
Frequently Asked Questions
When should I use this calculator instead of finite-element software?
For preliminary sizing, code-compliance checks on isolated members, or cross-checks of FEA results. The closed-form 5wL⁴/384EI applies only to simply-supported, prismatic, linearly-elastic beams under uniform load — anything more complex (point loads, cantilevers, continuous beams, cracked sections) should be analyzed in dedicated software.
Should I use the gross or cracked moment of inertia?
For deflection checks under service loads in reinforced concrete, ACI 318-19 §24.2.3 requires the effective moment of inertia I_e (Branson's equation, or §24.2.3.5 simplified). For preliminary sizing or steel beams, use the gross I.
What deflection limit should I check against?
ACI 318-19 Table 24.2.2 gives L/180 (immediate, flat roofs), L/360 (immediate, floors with non-structural elements that are not damaged), L/240 (live load, members not supporting non-structural elements), and L/480 (live load + long-term, supporting non-structural elements likely to be damaged).
Does this account for long-term creep deflection?
No — it returns instantaneous elastic deflection only. For long-term deflection in reinforced concrete, multiply the sustained-load portion by the time-dependent factor λ_Δ from ACI 318-19 §24.2.4.1.1 (λ_Δ ≈ 2.0 for 5+ years, no compression reinforcement).
Can I use this for steel beams?
Yes. Use E = 200,000 MPa (200 GPa) and the section's I_x. The calculator returns elastic deflection only; for AISC limits, see Table B-1 of the AISC Specification Commentary.