Beam Shear, Moment & Deflection Calculator
Analyses a simply-supported or cantilever beam under any mix of point and distributed loads — reactions, live shear-force and bending-moment diagrams, the maxima and their positions, and the deflected shape from EI.
Beam & loads
| Type | Magnitude | From (m) | To (m) | |
|---|---|---|---|---|
| kN | — |
Reactions & maxima
Diagrams
Sagging moment is positive, hogging negative; the bending-moment and deflection diagrams are drawn with the sagging/downward side down. Deflection uses EI = E·I. A study and preliminary-design aid only — real structural design must follow the relevant code and be checked by a qualified engineer.
Before you rely on this: First-pass guide only. Verify safety-critical or regulated work against the relevant standards, your project requirements and a qualified professional.
How to use this calculator
- Pick the beam type (simply supported or cantilever) and enter the span.
- Add point loads (kN at a position) and distributed loads (kN/m over a range); add as many as you need.
- Read the reactions and maxima, and — to get deflection — tick 'Compute deflection' and enter the modulus E and second moment of area I.
How it works
Reactions come from statics: for a simply-supported beam the two vertical reactions balance the total load and its moment about a support; for a cantilever the fixed end carries the whole load plus a fixed-end moment equal to the load's moment about the support.
The shear force at any section is the sum of the transverse forces to one side of it, and the bending moment is the sum of their moments about that section — sampled along the beam to draw the shear-force (SFD) and bending-moment (BMD) diagrams. Sagging moment is taken as positive, hogging as negative.
Deflection is found by integrating the curvature M/EI twice along the beam and applying the support conditions (zero deflection at both supports for a simply-supported beam; zero deflection and slope at the fixed end of a cantilever). EI = E·I, so the shape scales inversely with the stiffness you enter.
Worked example
Worked example. A 6 m simply-supported beam with a 10 kN point load at midspan: reactions are 5 kN each, the maximum shear is 5 kN and the maximum sagging moment is PL/4 = 15 kN·m at midspan. With E = 200 GPa and I = 100×10⁶ mm⁴ (EI = 20,000 kN·m²) the midspan deflection is PL³/48EI ≈ 2.25 mm.
Common mistakes
- Mixing up units — enter point loads in kN, distributed loads in kN/m, the span in metres, E in GPa and I in ×10⁶ mm⁴ (so EI comes out in kN·m²).
- Placing a load outside the span, or giving a distributed load an end position that is not past its start.
- Using this for an indeterminate beam — it solves determinate beams (a single simple span or a cantilever), not propped or continuous beams.
Frequently asked questions
What is a shear force and bending moment diagram?
They plot how the internal shear force and bending moment vary along the beam. The bending-moment diagram peaks where the shear passes through zero, and its maximum sets the required section size, so the two diagrams together drive the design.
How do I get the deflection?
Tick 'Compute deflection' and enter the modulus of elasticity E (about 200 GPa for steel, 30 GPa for concrete) and the second moment of area I of the section in ×10⁶ mm⁴. The tool forms EI = E·I and integrates the curvature to give the deflected shape and the maximum deflection.
Which beams can it solve?
Statically-determinate beams: a single simply-supported span (pin and roller) or a cantilever fixed at one end, under any number of point and uniformly distributed loads. Propped, fixed-fixed and continuous beams are statically indeterminate and need a different method.
Related tools
- Simply Supported Beam Calculator
- Cantilever Beam Calculator
- Beam Reaction Calculator
- Beam Deflection Calculator
- Combined Axial & Bending Check
- Fixed-Fixed Beam Moment Calculator
Explore more in Structural, Materials, Mechanical & Workshop.
Tip: Enter any known values to calculate the remaining results.
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