Beam deflection and bending stress
Point and distributed loads anywhere. Six sections, four support conditions, with a bending moment diagram.
Find the deflection and bending stress of a beam by choosing the section and the load condition. A point load can sit anywhere, a distributed load can cover any span, and you can apply both at once.
Six sections are available: rectangle (flat bar and square stock), round bar, round tube, square tube, channel and angle. Four support conditions: simply supported, cantilever, propped cantilever and fixed at both ends — so statically indeterminate beams are covered too.
Choosing a section narrows the material list to what is actually sold in that shape: STKM and STK for round tube, STKR for square tube, SS and SM for sections. The bending moment diagram and the section drawing redraw as you type.
Point and distributed loads can be applied together. Set the one you are not using to 0.
Only materials commonly sold in this section shape are listed. Yield and proof stress figures are representative values based on JIS specified minima. Check the mill certificate for the material you actually use.
| Second moment of area I | –mm⁴ |
|---|---|
| Section modulus Z (bottom) | –mm³ |
| Section modulus Z (top) | –mm³ |
| Area A | –mm² |
| Centroid (from bottom) | –mm |
| Mass per unit length | –kg/m |
| Max bending moment Mmax | –N·m |
|---|---|
| Position of Mmax (from left) | –mm |
| Position of δmax (from left) | –mm |
| Deflection / span | – |
| Reaction (left) | –N |
| Reaction (right) | –N |
| Fixed-end moment (left) | –N·m |
| Fixed-end moment (right) | –N·m |
| Material | – |
How this is calculated
The bending moment distribution M(x) comes from elementary beam theory (Euler–Bernoulli), and the deflection curve from numerically integrating EI·v'' = M twice. That is why a point load at any position and a distributed load over any span can be handled directly.
The propped cantilever (singly indeterminate) and the beam fixed at both ends (doubly indeterminate) cannot be solved from statics alone. They are solved here by the force method: a cantilever fixed at the left is taken as the primary system, the vertical reaction and fixing moment at the right end are taken as the redundants, and they are found from the compatibility conditions that the right end has zero deflection (and, for the fully fixed beam, zero slope). The results agree with the known solutions — for a uniformly distributed load on a fixed-fixed beam, fixed-end moments of −wL²/12 and a mid-span moment of +wL²/24; for a propped cantilever, a pinned reaction of 3wL/8.
For unsymmetrical sections (a channel with the opening up or down, and an angle) the section modulus differs at the top and bottom edges, so both values of Z are shown and the stress is evaluated with the smaller — the conservative one. Shear deformation, stress concentration, buckling and torsion are not included. An angle section has inclined principal axes and in reality couples bending with torsion; here it is treated as simple bending about the horizontal axis.
This is a simplified calculation intended for first-pass sizing. Stress concentration, buckling, fatigue, welds, dynamic loading and temperature effects are not included. Yield, proof and allowable stress figures are representative guide values and are not guaranteed to match the standards themselves. Always carry out your own verification before building and manufacturing real hardware. We accept no liability for loss arising from the results of this tool.
Other calculators
The same calculation, on your terms
The materials you actually use, the safety factors your company has settled on, the names your team gives things. We build the same kind of tool you see here, fitted to your own standards — in a browser, as a Windows or macOS desktop application, or as an add-in inside Office.
Beyond the calculation, bring the design to Simptech
We take on the whole run: concept design, 3D and 2D drawings, prototype sourcing and the ramp to production. A first-pass enquiry is welcome on its own.
Contact us