Calculator · Structural Mechanics
Beam Reactions Calculator
Support reactions of a determinate beam. The three planar equilibrium equations are solved once, by the same rank-based solver the Statics Lab uses, so a redundant or insufficient support set is refused rather than quietly averaged away.
Beam
Scroll the schematic sideways to see the whole beam; every result is also listed in the tables below.
Beam and loads
The beam runs from x = 0 to x = L.
Supports
Point loads
Point moments
Uniform distributed loads
Downward loads are negative (q < 0, Fy < 0); an applied couple is positive counter-clockwise. Internally, M > 0 is sagging, v > 0 is upward and θ > 0 is counter-clockwise.
Reactions
| A — position | 0 m |
|---|---|
| A — vertical reaction Ry | 5 kN (upward) |
| B — position | 6 m |
| B — vertical reaction Ry | 5 kN (upward) |
| Configuration | simply supported |
| Equilibrium residuals ΣFx / ΣFy / ΣMz | 0 N / 9.09e-13 N / 0 N·m — satisfied |
Assumptions
- · Statically determinate planar beam; reactions come from the shared equilibrium solver.
- · Small displacements, linear elasticity, plane sections remain plane (Euler-Bernoulli); shear deformation is neglected.
- · Uniform E and I along the beam when slope and deflection are reported.
Open in the Lab
The Lab carries this beam over and adds shear, bending moment, stress and deflection.
Technical reference
ΣFx = 0
ΣFy = 0
ΣMz = 0 about any point
- Fy
- vertical load component, negative downward[N]
- Mz
- applied couple, positive counter-clockwise[N·m]
- q
- distributed intensity, negative downward[N/m]
- Ry
- vertical support reaction, positive upward[N]
Support idealisations: a pin restrains horizontal and vertical translation (two reactions), a roller on a horizontal surface restrains vertical translation only (one reaction), and a fixed support restrains both translations and the rotation (three reactions, including a couple).
Determinacy boundary: three independent equilibrium equations determine at most three independent reactions. A cantilever, a pin-and-roller beam and the same pin and roller placed inside the beam ends (an overhanging beam) are all determinate. Fixed plus roller, two pins, or three vertical restraints are statically indeterminate and are refused — equilibrium alone cannot answer them.
Reactions are computed on the undeformed geometry under small-displacement assumptions, with rigid supports and a planar load system.