Beam Deflection & Bending Calculator

What this calculator does

It gives you the four numbers that decide whether a beam works — maximum deflection, maximum bending moment, maximum shear and maximum bending stress — for the five load cases that cover most of what a designer checks by hand. Pick the case, enter the span, the load, the material stiffness E and the section’s I, and it returns the deflection both in millimetres and as the span/N ratio that serviceability limits are written in. Span/593 means something to an engineer; 10.13 mm on its own does not, because whether that is acceptable depends entirely on how far the beam spans.

This is a first-principles elastic check, not a code design — the calculation you do to size a member before running it properly, or to sanity-check an analysis result that looks wrong.

The formula

Everything here is Euler–Bernoulli beam theory — the classical engineer’s beam equation:

EI · d⁴y/dx⁴ = q(x)

Integrated for each standard case, with L = span, P = point load, w = distributed load, E = modulus of elasticity, I = second moment of area about the bending axis:

Simply supported + UDL        δ = 5wL⁴ / (384EI)     M = wL²/8     V = wL/2
Simply supported + P at mid   δ = PL³ / (48EI)       M = PL/4      V = P/2
Simply supported + P at a     δ = P·b·(L²−b²)^1.5 / (9√3·L·EI)
                              M = P·a·b / L          V = P·max(a,b)/L
Cantilever + UDL              δ = wL⁴ / (8EI)        M = wL²/2     V = wL
Cantilever + P at free end    δ = PL³ / (3EI)        M = PL        V = P

In the arbitrary-position case, a is the distance from the left support to the load, and b is the shorter of a and L−a — the peak deflection always falls in the longer segment, at x = √((L²−b²)/3) from the far support. Put the load at mid-span and that expression collapses exactly to PL³/48EI, which is a useful check that you have entered it correctly.

The bending stress comes from the flexure formula:

σ = M·c / I  =  M / S

where c is the distance from the neutral axis to the extreme fibre and S = I/c is the elastic section modulus. If your section table gives you S directly, enter c = I/S.

These closed forms are not this tool’s invention — they are the standard beam diagram tables reproduced in every handbook (Roark’s Formulas for Stress and Strain, Table 8.1; the beam tables in AISC and the steel designers’ manuals).

Reading the result

Deflection and stress fail differently. Bending stress is a strength check — exceed it and the member yields. Deflection is a serviceability check — exceed it and nothing breaks, but doors bind, plaster cracks, floors bounce and glazing gets pinched. Long shallow spans in steel and timber are almost always governed by deflection, not stress. If your stress result sits comfortably below the material’s design strength while the span/N ratio is failing, you need a deeper section, not a stronger one: deflection scales with 1/I, and I goes with depth cubed.

The span/N number is the one to quote. Divide the span by the deflection and you get the ratio the code is written in. A larger N is stiffer. Span/250 and span/360 are the values you will see most often, but the limit is not universal — it depends on the code in force and on what the beam carries. Check yours; do not assume.

Moment tells you where to look. wL²/8 peaks at mid-span for a simply supported UDL; a cantilever’s wL²/2 peaks at the root and is four times larger for the same length and load. That is why cantilevers punish an over-run so hard — the moment goes with L² and the tip deflection with L⁴.

What this does not cover

Be blunt about the boundaries. The results assume a prismatic (constant section), isotropic, homogeneous, linearly elastic member undergoing small deflections, with no shear deformation and ideal supports. It therefore does not cover:

• Buckling of any kind — flexural, local/web, or lateral-torsional. An unrestrained steel beam frequently fails by LTB long before the extreme fibre reaches yield, and nothing in this calculation sees that coming.
• Composite action — a steel beam acting with a concrete slab, or a flitch beam, has an effective EI this tool cannot derive for you.
• Cracked-section behaviour in reinforced concrete, and long-term creep and shrinkage deflection, which routinely exceed the elastic value by a factor of two or more.
• Code load factors and combinations. Enter the load you actually want the answer for. Deflection is checked at serviceability loads; strength is checked at factored loads. They are different numbers.
• Continuity. A beam continuous over several supports has smaller mid-span moments and larger support moments than the simply supported case, and different deflections.
• Shear deflection, which matters once the beam becomes deep relative to its span. The tool warns you when span/depth drops below about 10.

Typical values

Deflection limits in common use run from span/180 (roofs with no ceiling below) through span/250 (general total load) and span/360 (live load, floors with brittle finishes) to span/500 or tighter under glazing, masonry or precise machinery. Cantilevers are usually assessed on twice the projection, or against a tighter ratio, because a given tip movement is far more visible. Take the number from the standard applicable in your jurisdiction — the Eurocodes (EN 1990 with EN 1993/EN 1995), AS/NZS 1170, or IBC/ASCE 7 with AISC 360 — not from this page.

Worked example

A 6.0 m simply supported steel beam carries a uniformly distributed load of 12 kN/m, including its own weight. The section has I = 100 × 10⁶ mm⁴ and an overall depth of 300 mm, so c = 150 mm. Steel, so E = 200 000 MPa. The serviceability limit is span/250.

Work in newtons and millimetres. L = 6000 mm, and 12 kN/m = 12 N/mm exactly.

δ = 5wL⁴ / (384EI)
  = 5 × 12 × 6000⁴ / (384 × 200 000 × 100 000 000)
  = (60 × 1.296×10¹⁵) / (7.68×10¹⁵)
  = 7.776×10¹⁶ / 7.68×10¹⁵
  = 10.125 mm

As a ratio: 6000 / 10.125 = span/593. The span/250 limit allows 6000/250 = 24.00 mm, so the beam passes with a wide margin — 10.13 mm against 24.00 mm.

M = wL²/8 = 12 × 6² / 8 = 54.00 kN·m
V = wL/2  = 12 × 6 / 2  = 36.00 kN
σ = M·c/I = (54×10⁶ N·mm × 150 mm) / (100×10⁶ mm⁴) = 81.0 MPa

81 MPa in a steel with a yield around 275–355 MPa is a lightly stressed beam. Both checks pass comfortably, which usually means the section is bigger than it needs to be — try the next size down and re-run.

FAQ

Which load field do I fill in? The one the case uses. UDL cases read the line load in kN/m; the three point-load cases read the point load in kN. The other field is ignored, so you can leave both populated and switch cases freely.

The position field — does it apply to every case? No. It is read only by “simply supported — point load at any position”, where it is the distance a from the left support and must be strictly between zero and the span. The other four cases ignore whatever is in it.

Do I include self-weight? Yes, in the load you enter — the calculator has no idea what your section weighs. Negligible for a rolled steel section under heavy load; not negligible for concrete.

Why does the span/depth warning appear? Because Euler–Bernoulli theory assumes plane sections stay plane and ignores shear deformation entirely. That is an excellent assumption for slender members and a progressively worse one as the beam gets stubby. Below a span/depth of roughly 10 the real deflection is meaningfully larger than the value here, and the member may need a deep-beam or strut-and-tie treatment instead.

Can I use this for a multi-span or continuous beam? Not directly. Continuity redistributes moment to the supports and stiffens the spans. Modelling one span of a continuous beam as simply supported is conservative for mid-span moment and deflection, and unconservative for the support moment — which is often where the section is actually governed.

This tool provides indicative figures for preliminary design and checking. Final member sizes must be verified against the design standard applicable in your jurisdiction, including all strength, stability, buckling and serviceability checks.