Simply Supported Beam Calculator (Reactions, Shear & Bending Moment)
Calculate the support reactions, maximum shear force and maximum bending moment of a simply supported beam with a point load anywhere on the span plus a uniform load, in kN and m or kip and ft.
How it works
RA = P(L − a)/L + wL/2 · RB = Pa/L + wL/2 · V(x) = RA − wx − P (for x > a) · M(x) = RA·x − wx²/2 − P(x − a) (for x > a) · the maximum moment is where the shear changes sign; for a single central load M = PL/4 and for a uniform load M = wL²/8.
A simply supported beam rests on a pin at one end and a roller at the other, so it is statically determinate: the two reactions come straight from equilibrium (sum of vertical forces and sum of moments equal zero). The shear force drops linearly under the uniform load and jumps down by P at the point load; the bending moment is the area under the shear diagram and peaks where the shear crosses zero. These are the textbook statics results tabulated in the AISC Steel Construction Manual beam diagrams and the American Wood Council Design Aid 6.
Worked example
A 6 m beam carries a 20 kN point load 2 m from the left support plus a 5 kN/m uniform load. RA = 20 × 4/6 + 5 × 6/2 = 13.33 + 15 = 28.33 kN and RB = 20 × 2/6 + 15 = 21.67 kN. Just left of the point load the shear is 28.33 − 5 × 2 = 18.33 kN and just right of it 18.33 − 20 = −1.67 kN, so the shear changes sign at the point load: the maximum bending moment is 28.33 × 2 − 5 × 2²/2 = 46.67 kN·m at x = 2.00 m. The maximum shear force is 28.33 kN at the left support.
Sources: Beam diagrams and formulas — AISC Steel Construction Manual, Table 3-23 (via Wikipedia, Euler–Bernoulli beam theory) · American Wood Council — Design Aid No. 6: Beam Design Formulas with Shear and Moment Diagrams
Frequently asked questions
What is a simply supported beam?
A beam that rests on two supports — one pinned and one roller — that resist vertical load but not rotation. The ends are free to rotate, so the bending moment is zero at both supports and largest somewhere in the span.
Where is the maximum bending moment?
Wherever the shear force changes sign. For a uniform load alone it is at mid-span (wL²/8); for a single point load it is under the load (Pab/L); with both, it is under the point load or somewhere between it and mid-span — this calculator checks each case.
How do I include the self-weight of the beam?
Add it to the uniform load w. A steel W-section weighs its designation in lb/ft (W12×26 = 26 lb/ft = 0.026 kip/ft), and a concrete beam weighs about 24 kN/m³ × width × depth.
Should I use factored or service loads?
Use factored loads (for example 1.2D + 1.6L) when you will compare the moment with a design strength such as φMn, and service (unfactored) loads when you check deflection or allowable stress.
Can I use this for a beam with several point loads?
Beams are linear, so you can superpose: run each point load separately and add the reactions and the moments at the same position. The maximum of the sum may sit at a different point, so check the moment under each load.
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This calculator is for educational purposes. Double-check important results before acting on them.