Beam calculator
Shear, moment, slope and deflection for a single-span beam of uniform section — the classical beam tables, solved from the beam equation rather than looked up, so any combination of supports and loads works, not just the rows someone printed in 1968.
Loads (down = positive; moments CCW = positive)
| Type | Magnitude | a | b |
|---|
Euler–Bernoulli beam, uniform EI, linear elastic, small deflections; no shear deformation. Loads superpose. Positions a, b are measured from the left end. Guided = zero slope and zero shear (a sliding clamp).
Need the same for a built-up frame, a variable section, or a beam-column? That is the model-building service — or the frame and strut calculators coming next.
Twenty-one stations along the span.
How the numbers are produced
Every case comes from the Euler–Bernoulli beam equation \(EI\,y'''' = q(x)\). The applied loads and the unknown support reactions are written as singularity functions — a point force \(W\) at \(a\) is \(W\langle x-a\rangle^{-1}\), a distributed load starting at \(a\) is \(w\langle x-a\rangle^{0}\), a couple is \(M_o\langle x-a\rangle^{-2}\) — and the equation is integrated four times: \(V=\int q\), \(M=\int V\), \(EI\theta=\int M + C_3\), \(EI\,y=\int EI\theta + C_4\). The reactions and the two constants are then solved from the support conditions (\(y\) and \(\theta\) at pinned, fixed and guided ends) together with whole-beam equilibrium. The result is an exact closed form for \(V, M, \theta, y\) along the whole span; what you see is that expression evaluated at 2001 stations, with several loads simply added.
Support pairs not derived directly (fixed–free, fixed–pinned, fixed–guided, pinned–guided) are solved as the mirror image of a derived pair and reflected. The symbolic derivation was checked against the 13 classical entries of a 1968 Grumman beam table; all agree, and two of the printed coefficients (15.6 for \(9\sqrt3\), and \(wL^4/185EI\) for the propped cantilever, whose exact coefficient is 1/184.6) turn out to be the table's rounding, not ours.
- Linear elastic, uniform section, small deflections, no shear deformation (Timoshenko), no axial load (no beam-column effects) — for those, see the strut calculator or ask.
- Sign convention on screen: loads and deflection positive downward; shear \(V\) = net upward force to the left of the section; moment sagging positive; slopes in radians; reactions positive upward and counter-clockwise.
- Runs entirely in your browser; nothing is uploaded. Results are first-order engineering values, not a substitute for a checked analysis.
- Something wrong, or a case you want added (spring supports, overhangs, multi-span)? Tell us.