Simple Beam Calculator — Multiple Loads, Shear/Moment/Deflection Diagrams
When a beam carries more than just one uniform load — say, two point loads from columns plus a triangular load from soil — closed-form formulas stop being practical. This calculator handles any combination using superposition and gives you all four diagrams (load, shear, moment, deflection) at once.
Formulas
| Quantity | Formula |
|---|---|
| Sum of forces (vertical equilibrium) | ΣF_y = 0 → R_A + R_B = ΣP_i + ΣW_i |
| Sum of moments about A | ΣM_A = 0 → R_B = (ΣP_i·a_i + ΣW_i·x̄_i) / L |
| Shear at section x | V(x) = R_A − Σ(loads to the left of x) |
| Moment at section x | M(x) = ∫₀ˣ V(s) ds |
| Deflection at section x (Macaulay) | y(x) = (1/EI) · ∫∫ M(x) dx² + C_1·x + C_2 |
How to use it
- Set the beam span and EI (or just E and I separately).
- Add as many loads as needed: P (point load), w (UDL), or w_1, w_2 (UVL — uniformly varying load).
- Specify each load's position and length along the beam.
- Hit Calculate. The calculator shows reactions, max shear/moment/deflection, and full diagrams.
- Use the values for member design — compare M_max against ɸM_n, V_max against ɸV_n, and δ_max against your serviceability limit.
Frequently Asked Questions
Does it support continuous (multi-span) beams?
Not in this version. For continuous beams, model each span and apply moment redistribution from ACI 318-19 §6.6.5, or use full structural analysis software.
What sign convention is used?
Sagging (smile) moments are positive. Loads pointing down are positive. Reactions pointing up are positive.
What units are accepted?
Loads in kN and kN/m, distances in m, E in GPa, I in cm⁴ — all standard SI for civil engineering.
Is the deflection computation reliable?
Yes — it's exact (closed-form integration of the moment diagram). For materials with non-linear behaviour, use specialized software.