Supports
mm
Section
Material
Reactions, shear force, bending moment, slope and deflection for beams on four support types under any mix of point loads, distributed loads and moments. Steel IPE, HEA and HEB sections, tubes and bars or your own I and W, with bending stress and a deflection limit check.
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Downward loads and clockwise moments are positive. Positions from the left end.
| Type | Value | From x | To x (UDL) |
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The beam is a prismatic Euler–Bernoulli beam (plane sections, small deflections, linear-elastic, constant EI). It is solved with cubic beam finite elements – about 200 elements plus a node at every load point – which give exact nodal deflections for point loads and moments and exact consistent loads for distributed loads, so fixed and propped beams are solved from compatibility rather than from special-case formulas. Reactions come from the solved system; shear and bending moment are then built by statics from the left end, and the deflection between nodes from the element shape functions. Maximum values are searched along the whole span, not only at mid-span or under the loads.
Section properties for IPE, HEA and HEB are the nominal values of the European tables (EN 10365); tubes and bars are calculated from their dimensions. The bending stress is an elastic stress only. It is not a design-code check: lateral-torsional buckling of open sections, local buckling, shear, web crippling under concentrated loads, combined actions and connections can govern well before the stress reaches yield, so check them with the applicable code (for example EN 1993-1-1). Shear deformation is neglected; it matters for short, deep beams (L under about 10 times the depth).
This calculator solves a straight beam on one of four supports – simply supported, cantilever, fixed at both ends or propped cantilever – under any combination of point loads, distributed loads (full or partial span) and point moments. Choose a steel IPE, HEA or HEB section, a rectangular or round tube, a solid bar or enter your own I and W, and the material. It returns the support reactions and end moments, the shear force, bending moment and deflection diagrams, the true maximum deflection and where it occurs, the maximum slope and the elastic bending stress, and compares the deflection with a limit L/n you choose.
Reactions 9.667 kN and 6.333 kN, maximum moment 8.667 kN·m (54.2 MPa) under the load, maximum deflection 4.127 mm at 1406 mm – inside an L/500 limit. These are the numbers of the independent reference case BEAM-001 on the verification page.
Where the slope of the deflected shape is zero. For a single central load or a full-span uniform load on a simply supported beam that is mid-span, but for an off-centre load or a mix of loads it moves towards the heavier side, and for a cantilever it is at the free end. The calculator searches the whole span for it instead of assuming mid-span.
It depends on what the beam carries. L/250 to L/300 is common for general floor and roof members, L/360 where brittle finishes such as plaster are carried, L/500 or stricter for machine frames, crane runways and anything that must stay aligned. Your national code or the client specification sets the actual value, so the calculator lets you choose it.
These beams are statically indeterminate: the reactions cannot be found from equilibrium alone, because the extra support adds a compatibility condition (zero slope at a fixed end, zero deflection at a prop). The calculator solves the full stiffness system, which handles any mix of loads; for a propped cantilever under a uniform load it gives the classic 5wL/8 and 3wL/8 reactions and a fixed-end moment of wL²/8.
No. An open I-section bent about its strong axis can buckle sideways (lateral-torsional buckling) at a stress well below yield unless the compression flange is restrained. Shear, web crippling under point loads, local buckling of thin plates and connections must also be checked to EN 1993-1-1 or your code. Use this calculator to choose a sensible section and check stiffness, then do the code checks.
Loads acting downward and clockwise applied moments are positive, sagging bending moment (tension at the bottom) is positive and deflection downward is positive. Reaction moments are shown clockwise-positive, so the wall moment of a cantilever loaded downward appears negative (anticlockwise on the beam).
Tick “Add self-weight” and the mass per metre of the chosen section is added as a uniform load over the full span. For tubes and bars the mass is calculated from the area and the material density; for a custom section enter the area.
What you need to build, measure or check the result.
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