Shear and Moment Diagram Calculator
Describe a beam with its supports and loads and get the support reactions, the shear force diagram, the bending moment diagram and the maximum values with their locations. Handles simply supported beams with overhangs, cantilevers, propped cantilevers and fixed-fixed beams.
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Loads
x is measured from the left end. Forces act downward when positive; moments are clockwise when positive.
Values at key points
How to use the Shear and Moment Diagram Calculator
- Choose units, the beam length and the support type. For a simply supported beam you can move either support inward to create overhangs.
- Add loads. Point loads and distributed loads are positive downward; applied moments are positive clockwise. Set w₁ and w₂ differently for a triangular or trapezoidal load.
- Press Calculate diagrams (results also update as you type).
- Read the cards for the reactions, the maximum moment and shear and where they occur, then hover the diagrams for values at any point.
- Copy the results, download the CSV of shear and moment along the beam, or save the diagrams as SVG for a report.
How shear and moment diagrams are built
Cut the beam at a distance x from the left end and look only at the part to the left of the cut. The internal shear V(x) is the sum of the vertical forces on that part (upward positive), and the internal moment M(x) is the sum of their moments about the cut. The three diagrams are linked by two simple rules:
dV/dx = −w(x): the slope of the shear diagram equals the downward distributed load. A uniform load makes V a straight sloping line; a triangular load makes it a parabola.dM/dx = V(x): the slope of the moment diagram equals the shear. Where V = 0, M has a peak; the change in M between two points equals the area under the shear diagram.- A point load makes V jump by the size of the load; an applied moment makes M jump by the size of the moment.
The reactions come first from equilibrium: ΣFy = 0 and ΣM = 0 about one support. For a simply supported beam with supports at A and B, R_B = Σ(load × distance from A) ÷ (B − A) and R_A = total load − R_B.
Common beam formulas for checking
| Beam and load | Reactions | Maximum moment |
|---|---|---|
| Simply supported, uniform load w | wL/2 each | +wL²/8 at midspan |
| Simply supported, point load P at midspan | P/2 each | +PL/4 under the load |
| Simply supported, P at distance a from A (b = L − a) | R_A = Pb/L, R_B = Pa/L | +Pab/L under the load |
| Cantilever, uniform load w | R = wL, M = wL²/2 | −wL²/2 at the fixed end |
| Cantilever, point load P at the tip | R = P, M = PL | −PL at the fixed end |
| Propped cantilever, uniform load w | R_A = 5wL/8, R_B = 3wL/8 | −wL²/8 at the fixed end, +9wL²/128 at 5L/8 from it |
| Fixed both ends, uniform load w | wL/2 each | −wL²/12 at the ends, +wL²/24 at midspan |
| Fixed both ends, point load P at midspan | P/2 each | −PL/8 at the ends, +PL/8 at midspan |
Tips for accurate inputs
- Include the beam's self-weight as a uniform load over the full length if it matters, for example 0.3 kN/m or 20 lb/ft for a typical steel joist.
- Use a negative value for loads that push upward (uplift) or for counterclockwise applied moments.
- An overhang is created by placing a support inside the span; the moment over that support is usually hogging (negative).
- Keep units consistent: with kN and m the moment is in kN·m; with lb and ft it is in lb·ft. Divide by 12 or multiply by 12 when moving between lb·ft and lb·in.
Frequently asked questions
What is a shear and moment diagram?
It is a pair of graphs showing how the internal shear force V and the internal bending moment M change along a beam. Engineers use them to find the most heavily stressed sections, because bending stress depends on M and shear stress on V.
What sign convention does this calculator use?
Shear is positive when the net force to the left of a section acts upward. Bending moment is positive when it causes sagging, meaning the bottom of the beam is in tension, and negative (hogging) when the top is in tension. Loads are positive downward and applied moments positive clockwise.
Where does the maximum bending moment occur?
Because dM/dx = V, the bending moment reaches a peak where the shear force is zero or changes sign, for example under a point load or partway along a distributed load. Peaks can also occur at fixed supports and over the supports of overhanging beams, so the calculator checks every one of those points.
What are the standard results for a simply supported beam?
With a uniform load w over the full span L, each reaction is wL/2 and the maximum moment is wL²/8 at midspan. With a single point load P at midspan, each reaction is P/2 and the maximum moment is PL/4. You can load those cases to check the calculator.
Can it solve statically indeterminate beams?
Yes, for propped cantilevers and beams fixed at both ends. The extra reactions are found from compatibility of deflection and slope using the unit-load method, assuming the bending stiffness EI is constant along the beam. For a fixed-fixed beam with a uniform load this gives end moments of wL²/12.
What is a point of contraflexure?
It is a point where the bending moment changes sign between sagging and hogging, so the beam's curvature reverses. Overhanging, propped and fixed beams usually have one or more of them, and they are often good places for splices.
Does it calculate deflection or stress?
No. This tool focuses on reactions, shear and bending moment. Use the beam deflection calculator for deflections, and divide the maximum moment by the section modulus to estimate bending stress.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.