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Beam Deflection Calculator

Deflection, bending moment, stress and safety factor for six standard load cases.

Calculate maximum deflection and bending stress for simply supported, cantilever and fixed-fixed beams carrying a centre point load or a uniformly distributed load. Pick an I-beam, rectangular tube, round tube, solid bar or enter your own moment of inertia.

  • Live results
  • Full step-by-step working
  • PASS / FAIL checks
  • Free PDF report

Inputs

Results

Max deflection
2.891 mm
PASS
Allowable (L/500)
4 mm
Max bending stress
86.73 MPa
Safety factor
2.88
PASS
Step-by-step calculation (7 steps)
  1. 1Second moment of area
    FormulaI = [B·H³ − (B−2t)(H−2t)³] / 12
    SolutionI = [50 × 100³ − 42 × 92³] / 12
    ResultI = 1441259 mm⁴ (144.1 cm⁴)
  2. 2Maximum deflection
    Formulaδ = F·L³ / (48·E·I)
    Solutionδ = 1/48 × 5000 × 2000³ / (200000 × 1441259)
    Resultδmax = 2.891 mm
  3. 3Deflection check
    Formulaδallow = L / 500
    Solutionδallow = 2000 / 500 = 4 mm; δmax / δallow = 0.723
    Result2.891 mm ≤ 4 mm (L/692)PASS
  4. 4Maximum bending moment
    FormulaM = F·L / 4
    SolutionM = 1/4 × 5000 × 2000
    ResultMmax = 2500 N·m (2500000 N·mm)
  5. 5Section modulus
    FormulaZ = I / c
    SolutionZ = 1441259 / 50
    ResultZ = 28825 mm³
  6. 6Maximum bending stress
    Formulaσ = M / Z
    Solutionσ = 2500000 / 28825
    Resultσmax = 86.73 MPa
  7. 7Safety factor on yield
    Formulan = Sy / σmax
    Solutionn = 250 / 86.73
    Resultn = 2.88 ≥ 1.5 requiredPASS

Engineering takeaways

  • Both deflection and strength are within limits for this load case.
  • Self-weight, shear deflection, connection flexibility and lateral-torsional buckling are not included. Check open sections (I-beams) for buckling if the compression flange is unrestrained.
  • Machine frames are usually stiffness-limited: design to deflection first, then confirm stress.

Rectangular tube 100 × 50 × 4 mm (sharp corners)

The PDF report contains every input, formula, substitution and check shown above, in LTM calculation-sheet format.

Worked Example

Machine frames and gantries are usually limited by stiffness, not strength. The calculator checks deflection against a span limit from L/250 up to L/2000 for precision axes, then confirms the bending stress against yield.

The calculator above opens with this example. Change any input to run your own case.

Inputs

Support and load caseSimply supported, centre point load
Section profileRectangular / square tube
Width / flange width B50 mm
Height H100 mm
Wall thickness t4 mm
Span / length L2000 mm
Point load F5000 N
MaterialStructural steel IS 2062 E250 / S235 (E 200 GPa, Sy 250 MPa)
Deflection limitL/500 (machine guards, frames)
Required safety factor on yield1.5

Results

Max deflection2.891 mm PASS
Allowable (L/500)4 mm
Max bending stress86.73 MPa
Safety factor2.88 PASS

Beam Deflection Formulas Used

Simply supported, centre load

δmax=FL348 E I,Mmax=FL4\delta_{max} = \frac{F L^3}{48\,E\,I},\qquad M_{max} = \frac{F L}{4}

Simply supported, uniform load

δmax=5 wL4384 E I,Mmax=wL28\delta_{max} = \frac{5\,w L^4}{384\,E\,I},\qquad M_{max} = \frac{w L^2}{8}

Cantilever, end load

δmax=FL33 E I,Mmax=FL\delta_{max} = \frac{F L^3}{3\,E\,I},\qquad M_{max} = F L

Cantilever, uniform load

δmax=wL48 E I,Mmax=wL22\delta_{max} = \frac{w L^4}{8\,E\,I},\qquad M_{max} = \frac{w L^2}{2}

Fixed both ends

δF=FL3192 E I,δw=wL4384 E I\delta_{F} = \frac{F L^3}{192\,E\,I},\qquad \delta_{w} = \frac{w L^4}{384\,E\,I}

Bending stress

σ=MZ=M cI,n=Syσ\sigma = \frac{M}{Z} = \frac{M\,c}{I},\qquad n = \frac{S_y}{\sigma}
  • Eelastic modulus (200 GPa steel, 69 GPa aluminium)
  • Isecond moment of area
  • cdistance from neutral axis to outer fibre
  • Syyield strength

Basis: Euler-Bernoulli beam theory; standard beam tables. Results are for preliminary design and should be confirmed against supplier data and the governing standard.

Beam Deflection Calculator FAQ

What is an acceptable beam deflection?

Building codes commonly use L/250 to L/360. Machine guards and frames are typically L/500, machine bases and gantries L/1000, and precision linear axes L/2000 or stiffer. Pick the limit that matches the function of the beam.

How do I reduce beam deflection?

Increase the section height first: the second moment of area grows with the cube of height, so a deeper section is far more effective than a wider or thicker one. Shortening the span or adding fixed end supports also helps a lot.

Does this include the beam self-weight?

No. Add the self-weight as a uniform load w in N/m. Use the steel weight calculator to find kg/m and multiply by 9.81.

Which elastic modulus should I use for steel?

Use 200 to 210 GPa for carbon and alloy steel, 193 GPa for stainless 304 and 69 GPa for aluminium 6061. Heat treatment does not change E noticeably, so a stronger steel does not reduce deflection.

Is the RHS moment of inertia exact?

The calculator treats tube corners as sharp. Real cold-formed RHS has rounded corners, so the table value is a few percent lower. For final design use the section property from the manufacturer or IS 4923 / EN 10219 tables.

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