About Beam Reactions Calculator
The beam reactions calculator solves a single-span beam by static equilibrium and returns the support reactions, the maximum shear force, the maximum bending moment, and sampled shear-force and bending-moment diagrams. It handles simply supported beams, cantilevers fixed at the left end, and overhanging beams with the two supports placed anywhere inside the span.
Add any number of loads: concentrated point loads at a given position, uniformly distributed loads (UDLs) over a range, and applied moments. The solver sums vertical forces and takes moments about a support to find the reactions, then integrates the loading from the left end to build the SFD and BMD so you can read the critical sections directly.
How It Works
- Choose the beam type (simply supported, cantilever, or overhanging) and enter the span length.
- For two-support beams, set the positions of support A and support B along the beam.
- Add loads: point loads (magnitude and position), UDLs (intensity, start, and end), and applied moments.
- The calculator applies sum of vertical forces equals zero and sum of moments equals zero to solve the reactions, then samples the shear and moment along the member for the diagrams.
Worked Example
A simply supported beam of span L = 10 m carries a central point load P = 100 kN at midspan. By symmetry each reaction is R = P/2 = 100/2 = 50 kN. The maximum bending moment is at midspan, M_max = P*L/4 = 100*10/4 = 250 kN.m, and the maximum shear force is V_max = P/2 = 50 kN, constant between each support and the load. The bending-moment diagram is a triangle peaking at 250 kN.m and the shear diagram steps from +50 kN to -50 kN across the load.
Formulas
- Vertical equilibrium
sum Fy = 0: R_A + R_B = sum(downward loads)- Moment equilibrium (about support A)
R_B = ( sum(P_i * (x_i - a)) - sum(M_applied_ccw) ) / (b - a)- Simply supported, central point load
R = P/2, M_max = P L / 4, V_max = P / 2- Simply supported, full-span UDL
R = w L / 2, M_max = w L^2 / 8, V_max = w L / 2- Cantilever (fixed at the left end)
point P at tip: M_fixed = P L, V = P | full UDL w: M_fixed = w L^2 / 2, V = w L
Standards & References
- Static equilibrium (statics)
- Hibbeler, Structural Analysis
- Gere & Goodno, Mechanics of Materials
Frequently Asked Questions
Which beam configurations does the beam reactions calculator support?
It supports single-span simply supported beams, cantilevers fixed at the left end, and overhanging beams where the two supports can be placed anywhere within the span. Each can carry any mix of point loads, distributed loads, and applied moments.
How are the support reactions calculated?
By static equilibrium. The tool sums the vertical forces (sum Fy = 0) and takes moments about one support (sum M = 0) to find the other reaction, then back-substitutes for the first. A cantilever instead develops one vertical reaction and a fixing moment.
What sign convention is used for the diagrams?
Downward loads and upward reactions are positive. Shear is positive when the net force to the left of the cut is upward, and the bending moment is sagging-positive. Applied moments are counter-clockwise positive.
Does it handle multiple loads at once?
Yes. You can add as many point loads, uniformly distributed loads, and applied moments as you need, and the solver superimposes them when computing the reactions, the maximum shear, and the maximum moment.