Shear force and bending moment diagram calculator
Enter your values. The answer and the full worked solution update as you type.
Result
| x (m) | V⁻ (kN) | V⁺ (kN) | M (kN·m) |
|---|---|---|---|
| 0 | — | 21 | 0 |
| 2 | 21 | 1 | 42 |
| 4 | 1 | 1 | 44 |
| 4.167 (V = 0) | 0 | 0 | 44.08 |
| 8 | −23 | — | 0 |
Sign convention: shear force is positive when the forces left of the section act upward; a sagging bending moment is positive. Positive values are drawn above the axis.
Draw the shear force diagram (SFD) and bending moment diagram (BMD) of a statically determinate beam. Pick a simply supported beam, a cantilever or a beam with an overhang, add up to three point loads, a distributed load and a concentrated moment, and the calculator finds the support reactions, the largest shear force and bending moment with their positions, the point of zero shear and any points of contraflexure. Every number is worked out step by step, so you can check your own hand calculation line by line.
Shear force and bending moment diagrams
Step-by-step solution
Resultant of the distributed load
Support reactions from equilibrium
Shear force at the key sections
Bending moment at the key sections
Where the bending moment is largest
Check at the right end
Key formulas
- Shear and moment
- V = ΣF on one side, M = ΣF·d on one side
- Simply supported, uniform load
- Mmax = wL²/8
How to draw shear force and bending moment diagrams
- Draw the free-body diagram of the whole beam. Replace a distributed load by its resultant: the intensity times the loaded length, acting at the middle of that length.
- Find the support reactions. Take moments about one support to get the reaction at the other, then use the sum of vertical forces. A cantilever’s fixed end takes the whole load plus a reaction moment.
- Mark the key sections: the supports, every point load and concentrated moment, and the start and end of each distributed load. Between two key sections both diagrams have a simple shape.
- Shear force: start at zero at the left end and move to the right. An upward force makes the diagram jump up by its value and a downward point load makes it jump down. Under a uniform load the line slopes down by w per unit length; with no load it stays level.
- Bending moment: the change in moment between two sections equals the area of the shear force diagram between them, because dM/dx = V. The moment is a straight line where V is constant and a parabola under a uniform load. A clockwise concentrated moment makes it jump up by its value.
- Read off the results. The moment peaks where the shear force passes through zero. It is zero at a free end and at an end pin or roller, and where it changes sign there is a point of contraflexure.
Questions and answers
What sign convention does the calculator use?
Shear force is positive when the forces to the left of a section add up to an upward force. Bending moment is positive when it sags the beam, with the bottom face in tension. Positive values are drawn above the axis. Enter downward loads and clockwise moments as positive numbers, and use a negative number for an upward load or an anticlockwise moment.
Where is the bending moment largest?
Because dM/dx = V, the bending moment peaks where the shear force is zero or changes sign. Under a uniform load that is inside the loaded length, a distance V/w from the start of the segment. With only point loads it is directly under one of the loads. On a cantilever with downward loads it is at the fixed support. Also check both ends and every point where a concentrated moment is applied: the moment jumps there, and the largest absolute value can be at such a point.
What is a point of contraflexure?
A section where the bending moment changes sign, from sagging to hogging or back. The curvature of the beam reverses there. Beams with an overhang usually have one between the supports, which matters for where reinforcement goes in a concrete beam.
Why does the shear force diagram jump at a point load?
A point load acts on a very short length of beam, so the shear force just left and just right of it differs by the whole load. The table of key sections gives both values, V⁻ and V⁺, at every load and support.
Can I use kip and ft?
Yes. Switch the units to US customary and every result is given in kip, ft and kip·ft. The method is the same in both systems.
Which beams does it solve?
Statically determinate beams: simply supported, cantilever, and simply supported with an overhang on the right. Continuous beams and propped cantilevers are statically indeterminate and need the deflections as well, so they are not covered.
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