Simply supported beam with a load in the middle
This is the most widely used model in structural engineering. A beam resting on two supports — one pin, one roller — with a concentrated load at midspan describes a floor joist, a bridge segment, a machine bed, a shelf between two uprights or a shaft on two bearings.
Because the beam is supported at both ends it is far stiffer than a cantilever of the same length: the midspan deflection formula carries a factor of 48 instead of 3, so the same load produces roughly sixteen times less deflection.
The calculator below is pre-set to this exact case with a 25 mm steel rod over a 2 m span carrying 500 N at the centre.
Calculator
This calculator is already set up for the problem above. Change the length, the load, the profile or the material and it recalculates immediately; the address bar keeps a shareable link to your exact configuration.
Input data:
Beam:
Material: Construction steel (Generic)
Profile:
| Ax: 78.54mm2 | Jxx: 981.75mm4 |
| Asy: 78.54mm2 | Iyy 490.87mm4 |
| Asz: 78.54mm2 | Izz: 490.87mm4 |
Simulation results:
Displacements (mm):
| # | x | y | zz |
|---|
Reactions (newtons):
| # | x | y | zz |
|---|
Direct link to this setup: https://beamcalc.net/en/solve/simply-supported-mid-load#v1-AQsAAPpEAADIQZQBAQIAAAAAAAD6QwAAAAA
How the beam is modelled
- The left support is a pin: it stops both translation and rotation.
- The right support is a roller: it stops vertical movement but allows the beam to lengthen and to rotate.
- A vertical point load acts at midspan, pointing down.
- The beam is a solid steel rod, 25 mm in diameter and 2000 mm long.
How to read the results
The largest deflection appears at midspan, right under the load, and the bending moment peaks there too — for a central load the maximum moment is F * L / 4.
Each support carries half of the load. Because the roller is free to move, the thermal expansion of the beam does not create axial force — this is exactly why real beams are built this way instead of being clamped at both ends.
If you move the load off centre, the maximum deflection moves with it and the larger support reaction follows the load. The closed-form formulas below cover any load position along the span.
Formulas
| Formula | What it gives |
|---|---|
delta_max = F * L^3 / (48 * E * I) | Deflection under the load at midspan. |
M_max = F * L / 4 | Bending moment under the load. |
R_left = R_right = F / 2 | Support reactions for a central load. |
delta_load = F * a^2 * b^2 / (3 * E * I * L) | Deflection at the load point when it sits at distance a from the left support and b from the right one. For a central load this is also the largest deflection. |
FAQ
What is the difference between a pin and a roller support?
A pin blocks translation in both directions and rotation, a roller blocks only vertical movement. Using one of each makes the beam statically determinate and lets it expand with temperature without building up axial force.
How much load does each support carry?
For a central load, half each. For an off-centre load, each support carries the share inversely proportional to its distance from the load, so the support nearer to the load is the heavily loaded one.
Is the bending moment diagram a triangle?
For a single central load it is: it rises linearly from zero at the supports to F * L / 4 under the load. With several loads it is the superposition of the individual triangles.
Does the calculator include the beam's own weight?
No, only the loads you enter. For long or heavy beams add the self weight as an extra load to keep the result realistic.
More beam examples
- Simply supported beam with an eccentric load A rectangular steel bar supported at both ends and loaded at the third point: where the load sits decides which support works harder.
- Simply supported beam with two point loads A steel tube on two supports carrying two equal loads: the middle section bends without shear, exactly like a four-point test.
- Simply supported beam with an inclined load A force pushing at an angle onto a supported beam: split it into bending and axial parts, and only the first one visibly bends the beam.
- Steel pipe loaded in the middle A steel pipe on two supports with a load at midspan: how much it bends, and which pipe dimensions actually matter.