Shear Force and Bending Moment Diagram Calculator
Draw SFD and BMD diagrams for simply supported, overhanging and cantilever beams with point loads, partial or triangular UDLs and moments. Free.
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About this toolWhat is the Shear and Bending Moment Calculator?
Every beam design starts with two diagrams: shear force along the span and bending moment along the span. The peak moment sizes the section for bending, the peak shear checks the web and the connections, and the points where shear crosses zero tell you where the moment peaks. This calculator draws both diagrams for a statically determinate beam and reports the support reactions, maximum and minimum values with their positions, and every zero-shear point.
Loads can be mixed freely: up to twelve point loads, uniform loads over the whole span or part of it, triangular and trapezoidal loads (set a different intensity at each end), and applied moments such as a bracket bolted to the side of a beam. Supports can be a pin and roller anywhere on the beam, which covers overhanging beams, or a cantilever fixed at either end.
Units are labels only: choose metres or feet and kN, N, kip or lbf, and the moment comes out in the matching force times length. A sketch of the beam with its loads sits above the diagrams so you can confirm you described the right problem.
- Point loads, full or partial UDLs, triangular and trapezoidal loads and applied moments
- Simply supported beams with movable supports (overhangs) and cantilevers fixed at either end
- Reactions, maximum shear, maximum sagging and hogging moments with locations
- Zero-shear points located numerically, shear and moment diagrams and a value table
- Load sketch, unit labels for SI or US, copy summary and shareable link
How to use the Shear and Bending Moment Calculator
- Enter the beam length and pick the supports. For a simply supported beam you can move either support inward to create an overhang.
- Choose length and force units. They only label the inputs and outputs; the maths works in any consistent set.
- Add loads. Point loads need a position and value; distributed loads need a start, an end and an intensity at each end; moments need a position and a clockwise-positive value.
- Check the beam sketch, then read the reactions, the maximum shear, the largest sagging and hogging moments and the zero-shear points.
- Use the diagrams and the table of values at tenths of the span for design checks, then copy the summary or share the link.
Worked examples
Point load plus partial UDL (default)
Input
L = 8 m, pin at 0 and roller at 8 m, 10 kN at 2 m, 5 kN/m from 4 m to 8 m
Output
RA = 12.5 kN, RB = 17.5 kN; zero shear at 4.5 m; Mmax = 30.625 kN·m at 4.5 m
The peak moment sits inside the UDL, where the shear line crosses zero, not under the point load.
Overhanging beam
Input
L = 10 m, supports at 0 and 8 m, 2 kN/m over the full length
Output
RA = 7.5 kN, RB = 12.5 kN; +14.06 kN·m at 3.75 m; −4 kN·m over support B
An overhang creates a hogging moment over the support and a second zero-shear point there. Both must be checked, and the hogging region needs top reinforcement in concrete.
Cantilever with a UDL and tip load
Input
Fixed at left, L = 3 m, 4 kN/m full length, 2 kN at the tip
Output
Reaction 14 kN, fixed-end moment −24 kN·m
The root moment is wL²/2 + PL = 18 + 6 = 24 kN·m, hogging, and that is where the cantilever is sized.
Bridge-style loading in US units
Input
L = 20 ft, 12 kip at 5 ft, 8 kip at 12 ft, 0.5 kip/ft self-weight
Output
RA = 17.2 kip, RB = 12.8 kip; Mmax = 87.04 kip·ft at 10.4 ft
With the distributed self-weight the zero-shear point moves off the point loads into the span.
Background and insight
The calculation follows the method taught in every statics course. The reactions come from the two equilibrium equations, the sum of vertical forces and the sum of moments about one support. Then a cut at any position x gives the shear as the net upward force to the left of the cut and the moment as the sum of each force times its lever arm, plus any applied couples to the left. Distributed loads are integrated exactly, including linearly varying ones, so a triangular load gives the correct parabola and cubic rather than a stepped approximation.
Sign conventions vary between textbooks, so this tool states its own: downward loads are positive, upward reactions are positive, an applied moment is positive when clockwise, shear is the upward force on the left-hand piece, and sagging moment (tension on the bottom) is positive. Hogging moments over supports and at cantilever roots therefore show as negative, which matches Hibbeler, Gere and most design codes.
Because dM/dx = V, the bending moment reaches a local maximum or minimum wherever the shear passes through zero, either smoothly under a distributed load or by jumping across zero at a point load or support. The calculator finds these points numerically to better than a millionth of the span and evaluates the moment there.
Common use cases
Beam and lintel design
Get the design moment and shear for a steel, timber or concrete beam before choosing a section.
Statics and mechanics homework
Check reactions and diagram shapes for textbook problems with mixed point, distributed and moment loads.
Equipment on a floor beam
Model a machine, tank or hoist as point loads on a beam that already carries a floor UDL.
Overhangs and cantilevers
See the hogging moment over a support or at a wall, where cracks and top-flange buckling start.
Tips and limitations
- Only statically determinate beams are handled. Continuous beams over three or more supports and propped cantilevers need compatibility equations or software.
- Factor your loads first if you are designing to LRFD or Eurocode, for example 1.2D + 1.6L or 1.35G + 1.5Q, then enter the factored values.
- For a uniform load the moment peak is at midspan only when the loading and supports are symmetric. Always read the zero-shear point rather than assuming L/2.
- An applied clockwise moment makes the moment diagram jump up by that amount at that point. Brackets and eccentric connections often introduce such couples.
- The SFD is drawn with vertical jumps at point loads; read the values on each side of the jump from the table when you need the shear at a support face.
Frequently asked questions
How do you draw a shear force and bending moment diagram?
Find the reactions from ΣFy = 0 and ΣM = 0. Then walk along the beam from left to right: the shear jumps by each point load and slopes by the distributed load intensity, and the moment changes by the area under the shear diagram. Point moments make the moment diagram jump.
Where does the maximum bending moment occur on a beam?
Where the shear force is zero or changes sign. Under a distributed load that is a smooth crossing; at a point load the shear jumps through zero and the moment peaks right under the load. For overhangs and cantilevers, check the supports too, where the hogging moment is largest.
What is the relationship between shear force and bending moment?
The slope of the moment diagram equals the shear, dM/dx = V, and the slope of the shear diagram equals minus the load intensity, dV/dx = −w. So the change in moment between two points equals the area under the shear diagram between them.
What is the sign convention for sagging and hogging moments?
Sagging (positive) bends the beam into a smile with tension on the bottom face, typical at midspan. Hogging (negative) bends it into a frown with tension on top, typical over supports and at the root of a cantilever.
How do I handle a triangular distributed load?
Enter it as a distributed load with w1 = 0 at one end and the peak at the other. Its resultant is half the base times the peak intensity, acting one third of the length from the heavy end. On a simple span the maximum moment is wL²/(9√3) at L/√3 from the light end.
Can this calculator solve continuous or propped beams?
No. It covers statically determinate beams only: simple spans, overhangs and cantilevers. Continuous beams need moment distribution, the three-moment equation or a stiffness-method program, because the reactions depend on the beam stiffness.
Is there an in-depth guide to the Shear and Bending Moment Calculator?
Yes. “How to Draw Shear Force and Bending Moment Diagrams, Step by Step” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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