Simply Supported Beam Reactions and Bending Moment Calculator
Support reactions, maximum bending moment and where it occurs, shear and moment at any point, and the bending stress, for a beam with a point load and a uniform load — with the moment diagram.
Results
The dashed line marks the value you entered.
What this tool does
Two supports and the equilibrium of forces and moments give the reactions at once: each takes its share of the point load by lever arm, plus half the uniform load. The bending moment is greatest where the shear force changes sign — under the point load or at the point where the uniform load has been carried — and that is what sizes a beam: stress = M ÷ Z. For a single central load the maximum is PL/4, for a uniform load wL²/8. This is a quick estimate of the forces; deflection, shear, lateral buckling and the load combinations of the building codes still have to be checked by whoever designs the member.
Formula
R_A = P (L − a) ÷ L + wL ÷ 2 · R_B = P a ÷ L + wL ÷ 2 · M(x) = R_A x − w x² ÷ 2 − P (x − a) for x > a · σ = M ÷ Z
Variables
| Symbol | Meaning | Unit |
|---|---|---|
l | Span between the supports | m |
p | Point load | kN |
a | Distance of the point load from the left support | m |
w | Uniform load along the whole span | kN/m |
z | Elastic section modulus of the beam | cm³ |
x | Position where you want shear and moment | m |
MM | Maximum bending moment | kN·m |
RA | Reaction at the left support | kN |
RB | Reaction at the right support | kN |
XM | Where the maximum moment is, from the left support | m |
SX | Shear force at your position | kN |
MX | Bending moment at your position | kN·m |
SG | Largest bending stress in the section | MPa |
Worked example
- Span between the supports6 m
- Point load20 kN
- Distance of the point load from the left support2 m
- Uniform load along the whole span5 kN/m
- Elastic section modulus of the beam400 cm³
- Position where you want shear and moment3 m
- Maximum bending moment46.67 kN·m
- Reaction at the left support28.33 kN
- Reaction at the right support21.67 kN
- Where the maximum moment is, from the left support2.00 m
- Shear force at your position-6.67 kN
- Bending moment at your position42.50 kN·m
- Largest bending stress in the section116.7 MPa
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.