About this calculator
This tool computes reactions, the shear-force and bending-moment diagrams, and the elastic deflection curve for a single-span simply-supported or cantilever beam carrying point loads, uniform loads, linearly varying (trapezoidal) loads, and applied moments. It's a quick hand-check tool for students and engineers doing preliminary framing analysis on statically determinate beams.
Method & formulas
Reactions come from basic statics — sum of vertical forces and sum of moments about a support:
Simply supported: ΣMA = 0 → Rb; ΣFy = 0 → Ra = W − Rb
Cantilever (fixed at left): Ra = ΣW ; Mfixed = net moment of loads about the fixed end
Shear and moment are found by sectioning the beam at x and summing everything to the left:
V(x) = Ra − Σ(loads left of x)
M(x) = Ra·x − Σ(moment of loads left of x about x) + applied moments
Deflection follows Euler–Bernoulli beam theory. The tool numerically double-integrates curvature to get slope and deflection, then enforces the boundary conditions (y = 0 at both supports for a simple beam; y = θ = 0 at the fixed end for a cantilever):
EI·y″(x) = M(x) → θ(x) = ∫M/EI dx → y(x) = ∫θ dx
Worked example
Tool defaults: simply-supported beam, L=6 m, E=200,000 MPa, I=1×10⁸ mm⁴, a full-span UDL w=10 kN/m plus a 20 kN point load at midspan (x=3 m).
Ra = Rb = (10·6 + 20)/2 = 40 kN (symmetric loading)
Max |V| = 40 kN at the supports
M(midspan) = Ra·3 − w·3²/2 = 120 − 45 = 75 kN·m
EI = (200×10⁹ Pa)(1×10⁻⁴ m⁴) = 2×10⁷ N·m²
δUDL = 5wL⁴/(384EI) = 5(10000)(6⁴)/(384·2×10⁷) ≈ 8.44 mm
δpoint = PL³/(48EI) = (20000)(6³)/(48·2×10⁷) ≈ 4.50 mm
δmax ≈ 8.44 + 4.50 ≈ 12.9 mm (downward, at midspan)
Assumptions & limitations
- Linear-elastic, prismatic (constant EI), small-deflection theory only — no shear deformation and no cracked-section stiffness reduction.
- Single span only; continuous or otherwise statically indeterminate beams are outside its scope.
- Sign convention: downward loads and downward deflection are both positive; applied moments are positive counter-clockwise.
- Deflection is from numerical integration over 200 stations — a reasonable approximation, but not an exact closed-form solution.
- Verify results against the governing code and a licensed engineer's judgment, especially for serviceability deflection limits and cracked-section (effective) I for reinforced concrete members.
FAQ
Q: Does the tool include the beam's self-weight?
A: No — add it yourself as a full-span UDL if self-weight needs to be part of the loading.
Q: Can I analyze a multi-span continuous beam?
A: No, only single-span simply-supported and cantilever beams are supported. Continuous beams are statically indeterminate and need moment distribution, the stiffness method, or dedicated software.
Q: What I should I enter for reinforced concrete deflection checks?
A: Enter an effective (cracked-section) moment of inertia rather than the gross section Ig if you want deflections that reflect concrete cracking — the tool applies no automatic reduction.