CTR K

Beam Deflection Calculator — Deflection, Moment, Shear & Fatigue (EN 1993-1-1)

Governing standard: EN 1993-1-1· EN 1993-1-1 (Eurocode 3: Design of steel structures, Part 1-1) — bending §6.2.5, shear §6.2.6 and serviceability deflection §7.2 folded into one governing verdict · Euler-Bernoulli closed-form beam theory · Goodman fatigue criterion with mean-stress-corrected Basquin life · exact Euler-Bernoulli natural frequencies with a Rayleigh added-mass correction

Page last updated

How EN 1993-1-1 works — the method explained

Video walkthrough (3:31)

  1. 0:00 SAFE at 1 m, FAIL on deflection at 3 m
  2. 0:31 Bending: strength and stiffness
  3. 0:48 Steps 1–4 — Load case, section, material, loads
  4. 1:21 Step 5 — Checks: bending, shear, deflection
  5. 1:34 What if: 3 m and 10 kN, then the fix
  6. 2:17 I-beams: lateral-torsional buckling
  7. 2:56 Step 6 — Deep analysis and what is not checked
  8. 3:13 PDF report
Read the transcript

Here's an 80 mm steel bar, spanning 1 m with 3 kN in the middle. The tool says it's SAFE, with a safety factor of 15.74.

Now make the span 3 m and the load 10 kN. It fails, but not on strength. It sags too much.

This is the MechanixCalc beam calculator. It works out bending, shear and deflection for six load cases, and it checks lateral-torsional buckling too. Let's go through it, step by step.

A beam carries its load by bending. The top is squeezed and the bottom is stretched, so the stress peaks at the outer fibres.

Two things can go wrong. The stress can pass the yield strength, or the beam can bend too much to use. Strength and stiffness are separate checks.

Step one is the load case. Here it's simply supported, with a point load in the middle. There are six cases, from cantilevers to fixed ends.

Step two is the span and the section. A span of 1 m, and a solid round bar, 80 mm across.

Step three is the material. Steel, with a modulus of 210 GPa and a yield strength of 250 MPa.

Step four is the load. 3 kN at mid-span, plus the bar's own weight, which the tool adds by default.

Step five shows the checks. Bending gives a safety factor of 15.74, and the bar sags only 0.16 mm. Everything is green, so the verdict is SAFE.

Now make the span 3 m, and raise the load to 10 kN.

Deflection now fails. The bar sags 14.29 mm, and the limit at L over 300 is 10. Bending is only a CHECK, with a safety factor of 1.58.

So stiffness governs here, not strength. A stronger steel would not help at all, because stiffness depends on the modulus and the section.

The fix is a slightly bigger bar. At 90 mm, the second moment of area rises by 60 %.

The deflection drops to 9.08 mm, that's L over 330. Bending reaches 2.22, and the verdict turns green.

I-beams have one more way to fail. Click Use on an IPE 240, a rolled steel beam, and span it 6 m with a distributed load.

Bending looks fine, with a safety factor of 2.05. But the verdict is FAIL. The beam buckles sideways before it yields.

That is lateral-torsional buckling. The unrestrained top flange twists out of plane, and its safety factor is only 0.61.

If a floor deck holds the top flange along the span, set the compression flange to restrained.

Now it cannot buckle sideways, and the verdict turns green. Same beam, with the restraint made real.

Step six is optional deep analysis. Moving loads, natural frequency, fatigue and a two-span continuous beam.

What the tool does not check is listed too: web buckling, patch loading and connections. So silence never reads as a pass.

Every input, every formula and every check goes into a PDF report. A report your reviewer can follow.

That's a beam, checked. Try it on your own design, on MechanixCalc.com.

The MechanixCalc beam deflection calculator computes maximum deflection, bending moment, shear force and bending stress for six standard load cases — simply supported and cantilever beams under point loads or uniformly distributed loads, plus a fixed-fixed configuration — using closed-form Euler-Bernoulli solutions that are consistent with the serviceability and strength checks in EN 1993-1-1 (Eurocode 3). Enter the span, cross-section geometry, material and load, and the tool returns the full result in one pass, including a bending moment diagram, shear force diagram and deflection curve.

It is designed for structural and mechanical engineers who need a defensible, standards-cited beam calculation — whether screening a mezzanine floor beam, a machine frame member or a shaft-like bar (a crane runway girder needs wheel patch loading and EN 1993-6 fatigue, which are not covered) — and who need to provide a reviewer with a traceable, fully worked calculation rather than a back-of-envelope estimate.

What this calculator does

  • Deflection, bending moment and shear for 6 closed-form load cases (EN 1993-1-1 / Euler-Bernoulli)
  • One governing verdict folding bending (§6.2.5), shear (§6.2.6) and serviceability deflection (§7.2)
  • Selectable deflection limit — L/200, L/250, L/300, L/360 or L/500 per your National Annex
  • Bending and shear stress with safety factors for solid, hollow, rectangular and I-section cross-sections
  • Natural frequency of the first three vibration modes (SS, cantilever, fixed-fixed) with a Rayleigh added-mass correction
  • Fatigue safety factor using the Goodman criterion for cyclic loading, with a mean-stress-corrected Basquin S-N life
  • Two-span continuous beam analysis via the 3-moment (Clapeyron) equation with bending moment diagram
  • Section efficiency comparison — second moment of area for equal-weight cross-sections
  • Combined loading superposition (simultaneous point load + UDL)
  • Branded PDF engineering report with the full method and diagrams shown

Method & formulas

Closed-form deflection and bending (Euler-Bernoulli)

The calculator solves the Euler-Bernoulli beam differential equation EI·d²y/dx² = M(x) in closed form for each of the six supported load configurations. This produces exact analytical expressions for deflection, bending moment and shear at every section along the beam, with no numerical integration error. The peak deflection and peak bending moment — and the section where each occurs — are identified from the full distribution sampled at 61 stations, with the load position itself added to the sample set for an offset load so the peak moment is exact.

Bending stress follows from the flexure formula σ = M·c / I, where c is the distance from the neutral axis to the outermost fibre and I is the second moment of area of the cross-section. Shear stress follows from τ = V·Q / (I·b) and is checked against the shear yield strength σy/√3 per EN 1993-1-1 §6.2.6. The deflection is checked against a serviceability limit you select (L/200 to L/500 — EN 1993-1-1 §7.2.1 defers the number to the National Annex).

All three checks — bending, shear and deflection — are folded into a SINGLE governing verdict, so the on-screen hero, the header status pill, the sketch and the PDF report can never disagree with one another. A check that cannot be evaluated (for example the bending stress of a custom section whose extreme-fibre distance was not supplied) is reported as NOT CHECKED, never as a pass.

Simply supported beam — central point load (maximum deflection at midspan)
δ_max = F · L³ / (48 · E · I)

where δ_max = maximum deflection (mm); F = point load (N); L = span (mm); E = Young's modulus (N/mm²); I = second moment of area (mm⁴)

Bending stress (flexure formula)
σ = M · c / I

where σ = bending stress (MPa); M = bending moment (N·mm); c = distance from neutral axis to extreme fibre (mm); I = second moment of area (mm⁴)

Natural frequency (exact eigenvalues + Rayleigh added mass)

The first three natural frequencies of the beam in bending come from the exact closed-form eigenvalue expression for uniform beams under the three common boundary conditions (βL = π, 1.8751 and 4.7300 for simply supported, cantilever and clamped-clamped respectively). Any added lumped mass is then folded in with a Rayleigh effective-mass correction using the appropriate equivalent-mass fraction (48/π⁴ ≈ 0.493 for SS, 0.24 for cantilever, 192/4.73⁴ ≈ 0.384 for fixed-fixed — each matched to the beam's static stiffness at the mass, so the frequency is exact for a very light and a very heavy mass alike). The resonance advice always quotes the frequency WITH the added mass, since that is the one the installed beam actually has.

First natural frequency — Euler-Bernoulli beam
f_n = (β·L)² / (2π·L²) · √(E·I / ρ_lin)

where f_n = natural frequency (Hz); β·L = first mode coefficient (π for SS, 1.8751 for cantilever, 4.730 for fixed-fixed); E·I = flexural rigidity (N·mm²); ρ_lin = mass per unit length (kg/mm)

Fatigue safety factor (Goodman criterion)

For beams under cyclic loading, the calculator applies the Goodman mean-stress criterion to the alternating and mean bending stresses derived from the maximum and minimum load amplitudes. The modified endurance limit uses Shigley's Se′ = 0.5·Sut estimate, capped at 700 MPa for Sut above 1400 MPa, corrected by a surface factor Csurf. A compressive mean stress receives no mean-stress benefit (the criterion reduces to σa ≤ Se), which is the conservative reading of the Goodman diagram.

The fatigue LIFE is read at the equivalent completely-reversed stress σrev = σa / (1 − σm/Sut) (Shigley 11e §6-14), not at the raw amplitude — so a large tensile mean stress properly shortens the life instead of being ignored. Above the endurance limit the Basquin power-law (Shigley 11e §6-7, high-cycle 10³–10⁶ fit) gives a finite N_f; at or below it the life is reported as infinite (run-out); below 10³ cycles the result is flagged as low-cycle and outside the fit's validity rather than extrapolated. Only the surface Marin factor is applied — size, loading, temperature and reliability factors are not, and the panel carries an EstimateBadge saying so.

Goodman fatigue safety factor
1 / SF_f = σ_a / S_e + σ_m / S_u

where SF_f = fatigue safety factor; σ_a = alternating bending stress (MPa); σ_m = mean bending stress (MPa); S_e = modified endurance limit (MPa); S_u = ultimate tensile strength (MPa)

Worked example

A simply supported steel beam of span L = 3000 mm carries a central point load F = 10 kN. The cross-section is a solid circular bar of diameter d = 120 mm in S355 steel (E = 210 GPa, density 7850 kg/m³). Find the maximum deflection, peak bending stress and yield safety factor, including the bar's own weight.

Given

  • Span L3000 mm
  • Central point load F10 000 N (10 kN)
  • Bar diameter d120 mm
  • Young's modulus E210 000 N/mm² (210 GPa)
  • Yield strength Sy295 MPa (S355, EN 10025-2 at 100 < t ≤ 150 mm — the 355 MPa nominal applies only up to 16 mm)
  • Self-weightincluded (ρ = 7850 kg/m³)

Result

  • Maximum deflection δ_max3.06 mm incl. self-weight (limit L/300 = 10.0 mm — pass)
  • Peak bending moment M_max8.48 kN·m
  • Peak bending stress σ_max50.0 MPa
  • Yield safety factor SF5.90 at Sy = 295 MPa (pass)
  • Peak shear stress τ_max0.74 MPa (SF 229 — pass)
  • Governing verdictPASS — bending, shear and deflection all within limits
  1. Compute the second moment of area: I = π × d⁴ / 64 = π × 120⁴ / 64 = 10 178 760 mm⁴ ≈ 1.018 × 10⁷ mm⁴. The extreme-fibre distance is c = d/2 = 60 mm. Area A = πd²/4 = 11 310 mm².
  2. Self-weight: w_sw = ρ·A·g = 7850 kg/m³ × 11 310 mm² × 9.81 m/s² = 0.871 N/mm, a uniform load superposed on the point load.
  3. Maximum deflection (midspan): δ_max = F·L³ / (48·E·I) + 5·w_sw·L⁴ / (384·E·I) = 2.63 mm (point load: 2.70 × 10¹⁴ / 1.026 × 10¹⁴) + 0.43 mm (self-weight) = 3.06 mm.
  4. Peak bending moment (at midspan): M_max = F·L / 4 + w_sw·L² / 8 = 7 500 000 + 979 800 = 8 479 800 N·mm = 8.48 kN·m.
  5. Maximum bending stress: σ_max = M_max × c / I = 8 479 800 × 60 / 10 178 760 = 50.0 MPa.
  6. Yield strength: a 120 mm bar is far thicker than 16 mm, so EN 10025-2 lowers S355 to 295 MPa (100 < t ≤ 150 mm). Picking S355 J2 from the material library fills that banded value for the section and says so beside the box.
  7. Yield safety factor: SF = Sy / σ_max = 295 / 50.0 = 5.90. The beam is well within the elastic limit.
  8. Shear check (EN 1993-1-1 §6.2.6): V_max = F/2 + w_sw·L/2 = 5 000 + 1 306 = 6 306 N. For a solid circle τ_max = 4V / (3A) = 0.74 MPa against a shear yield strength of 295/√3 = 170 MPa — a shear safety factor of 229.
  9. Serviceability check (EN 1993-1-1 §7.2): with the limit set to L/300, δ_limit = 3000/300 = 10.0 mm. The actual 3.06 mm is well inside it (the beam achieves L/980), so the deflection check passes.
  10. Governing verdict: all three checks pass, so the tool reports PASS. Had the span been 6 m instead of 3 m, the deflection would rise to 27.9 mm (self-weight included) against a 20.0 mm limit and the verdict would flip to FAIL on serviceability — even though the bending safety factor would still be 2.65.

Illustrative example with round inputs — verify all results against your actual geometry, loads and material standard. A solid circular bar of 120 mm diameter is unusually heavy for a structural beam; an I-section of similar depth would achieve a much higher I and lower deflection for the same mass.

Frequently asked questions

Which standard does this beam calculator use?

The deflection, bending moment and shear calculations use closed-form Euler-Bernoulli beam theory. Three elastic screening checks after EN 1993-1-1 (Eurocode 3, Part 1-1) — characteristic values, no partial factors — are then applied and folded into one governing verdict: bending resistance (§6.2.5), shear resistance against σy/√3 (§6.2.6) and the serviceability deflection limit you select (§7.2, which defers the numeric limit to the National Annex). The fatigue panel applies the Goodman mean-stress criterion with a mean-stress-corrected Basquin S-N life (Shigley 11e §6-7 and §6-14). Natural frequencies use the exact Euler-Bernoulli eigenvalues (Blevins 1979) with a Rayleigh added-mass correction. The two-span continuous beam uses the 3-moment (Clapeyron) equation. The cross-section is classified to EN 1993-1-1 Table 5.2 (I-section compression flange and web, circular hollow d/t); a Class 4 section's bending check reads NOT CHECKED, because its effective section (EN 1993-1-5) is not computed, unless the gross section already fails. The beam's own weight (ρ·A·g) is included by default. Lateral-torsional buckling (§6.3.2.2, general case) is checked for I-sections and rectangles deeper than wide, with the load on the top flange, C1 = 1 and the span as the buckling length (or skipped when you declare the compression flange restrained). Lateral-torsional buckling with intermediate restraints, web shear buckling, web patch loading under a concentrated load or reaction (EN 1993-1-5 §6), the Class 4 effective section and axial-load interaction are NOT covered — check those separately. The governing method and formulas are reproduced in the generated PDF report.

Does it check deflection as well as stress?

Yes, and the deflection check carries the same weight as the stress check. Choose your serviceability limit — L/200, L/250, L/300, L/360 or L/500 — and if the calculated deflection exceeds it the overall verdict reads FAIL, naming deflection as the governing check. A long, lightly loaded beam is very often stiffness-governed rather than strength-governed: it can carry the load with a large margin against yield while sagging far past what the floor, cladding or machine alignment can tolerate.

What happens if I enter a custom second moment of area?

You also need to enter the extreme-fibre distance c (h/2 for a symmetric section). Without it the flexure formula σ = M·c/I cannot be evaluated, so the tool reports the bending and shear checks as NOT CHECKED rather than assuming they pass — the deflection is still computed from I alone. Enter c and the full check runs. Every IPE row in the built-in section table lists its c value for exactly this purpose.

Which load cases are supported?

Six standard cases are covered: simply supported beam with a central point load; simply supported beam with a uniformly distributed load (UDL); simply supported beam with an offset (eccentric) point load; cantilever with an end point load; cantilever with a UDL; and a fixed-fixed (clamped-clamped) beam with a central point load. A combined-loading panel handles simultaneous point load and UDL by superposition.

Can I check a two-span continuous beam?

Yes. The two-span continuous beam panel solves the interior support moment and all three reactions using the 3-moment (Clapeyron) equation for UDL on each span, then draws the full bending moment diagram. The span lengths and UDL magnitudes can differ between the two spans.

What cross-sections does it support?

You can choose from solid circle, hollow circle, solid rectangle, or I-section (defined by flange width, total height, flange and web thickness). A custom-I option lets you enter the second moment of area directly. The section efficiency comparison panel shows how six cross-section shapes compare in bending stiffness for the same cross-sectional area (same weight per unit length).

Is the beam deflection calculator free?

You can use it during a free 30-minute preview with no sign-up, and a free 14-day account trial unlocks every calculator with no credit card. The branded PDF engineering report and saved calculations are included in the free 14-day trial and in every paid plan.

Related calculators

Run the Beam Deflection on your own numbers

Free 30-minute preview — no sign-up. A free 14-day account trial unlocks every tool and the branded PDF report, no credit card required.

Start free

Using MechanixCalc at work? See plans & pricing — one subscription unlocks all 50+ calculators, PDF reports and saved projects.