Simply Supported Beam Calculator
Find the maximum bending moment, support reactions and deflection of a simply supported beam in seconds. Enter the span, load and section, and the result is checked live against the FERS finite-element solver — so you can see the closed-form answer and the FE answer agree.
Bending moment, reactions and deflection are all free, with no sign-up. A free account adds the downloadable HTML and PDF report.
Beam
E = 210.0 GPa
I = 3,893 cm⁴ · W_el = 324.4 cm³ · A = 39.13 cm²
Loads
Positions from end A · downward loads and clockwise moments are positive- 1. Point load
Results
- Reaction at A, R_A
- 5.00 kN↑
- Reaction at B, R_B
- 5.00 kN↑
- Maximum shear |V|
- 5.00 kNat x = 0 m
- Maximum sagging moment
- 15.0 kN·mat x = 3.00 m
- Maximum deflection δ_max
- 5.50 mmL/1,090 · at x = 3.00 m
- Rotation at A, θ_A
- 0.00275 rad0.158°
- Rotation at B, θ_B
- -0.00275 rad-0.158°
- Max bending stress σ = M/W
- 46.2 MPaσ/f_y = 0.197
- Plastic moment M_pl = W_pl·f_y
- 86.2 kN·mM/M_pl = 0.174
- W_el needed for σ ≤ f_y
- ≥ 63.8 cm³
Euler–Bernoulli beam theory: linear elastic, small deflections, shear deformation neglected. Lateral-torsional buckling, local buckling and connections are not checked. σ/f_y is an elastic stress ratio, not a design check. Check a steel beam to Eurocode 3 →
Checked by the FERS finite-element solver
✓ matches the closed form| Quantity | Closed form | FERS FE | Difference |
|---|---|---|---|
| Reaction at A | 5.000 kN | 5.000 kN | < 0.0001 % |
| Reaction at B | 5.000 kN | 5.000 kN | < 0.0001 % |
| Rotation at A | 0.002752 rad | 0.002752 rad | < 0.0001 % |
| Rotation at B | -0.002752 rad | -0.002752 rad | < 0.0001 % |
| Maximum deflection | 5.504 mm | 5.504 mm | < 0.0001 % |
| Moment, worst FE station | 12.00 kN·m | 12.00 kN·m | < 0.0001 % |
| Shear, worst FE station | 5.000 kN | 5.000 kN | < 0.0001 % |
| Deflection, worst FE station | 5.425 mm | 5.425 mm | < 0.0001 % |
Precomputed with FERS 0.2.68 for this example; edit any input to re-run it live in your browser.
Lighter sections that meet your limits
Screens a family of catalogue sections against your deflection limit and σ ≤ f_y, each with its own self-weight, and lists the lightest that meet both.
Diagrams
Hover or tap a diagram — or type x — to read every value at that point.
Step-by-step solution
Macaulay's method: one bending-moment expression for the whole span, integrated twice; the support conditions give the reactions and the integration constants. Hand-calculation units: N and mm, x in mm.
Bending stiffness
- EI
- EI = E × I = 210,000 N/mm² × 3.893×10⁷ mm⁴ = 8.175×10¹² N·mm²
Bending moment, slope and deflection
Support conditions
Solving them
- R_A
- 5,000 N
- C₁
- -2.25×10¹⁰ N·mm²
- C₂
- 0 N·mm³
- Equilibrium
- R_B = −ΣF − R_A = 10,000 N − 5,000 N = 5,000 N
Peaks along the span
- Maximum sagging moment
- V(x̄) = 0 ⇒ x̄ = 3.00 m → M(x̄) = 15.0 kN·m
- Maximum deflection
- θ(x̄) = 0 ⇒ x̄ = 3.00 m → δ_max = |EI·v(x̄)| / EI = 4.5×10¹³ N·mm³ / 8.175×10¹² N·mm² = 5.50 mm
- Bending stress
- σ = M_max / W_el = 1.5×10⁷ N·mm / 324,420 mm³ = 46.2 MPa
- Stress ratio
- σ / f_y = 46.2 MPa / 235 MPa = 0.197
- Span / deflection
- L / δ_max = 6,000 mm / 5.504 mm = L/1,090
Formulas used
A simply supported beam has a pin at one end and a roller at the other. For the two most common load cases the maximum bending moment, support reaction and maximum deflection are:
| Load case | Max moment | Reaction | Max deflection |
|---|---|---|---|
| Central point load P | M_max = P·L / 4 | R = P / 2 | δ_max = P·L³ / (48·E·I) |
| Uniformly distributed load w | M_max = w·L² / 8 | R = w·L / 2 | δ_max = 5·w·L⁴ / (384·E·I) |
where L is the span, P a point load, w a distributed load, E Young's modulus and I the second moment of area. The calculator above runs the FERS finite-element solver on your inputs and shows the analytical and finite-element answers agreeing live.
Worked example
Worked example: a 6 m simply supported IPE 240 steel beam (I = 3.89×10⁻⁵ m⁴, E = 210 GPa) with a 10 kN point load at midspan.
- Maximum bending moment, P·L/4
- 15.0 kNm
- Reaction at each support, P/2
- 5.0 kN
- Midspan deflection, P·L³/48EI
- 5.5 mm (≈ L/1090)
Frequently asked questions
Is this beam calculator free?
Yes. It runs entirely in your browser with no installation and no sign-up. Bending moments, reactions and deflections are all free; a free account adds the downloadable report (HTML and PDF).
How accurate is it?
Every result is computed two ways: from the closed-form beam equations and, live, by the FERS finite-element solver. You can watch the bending moment and reactions from the two methods agree on screen as you change the inputs — the finite-element result reproduces classical beam theory to numerical precision.
What units does it use?
SI units: length in metres (m), point loads in kilonewtons (kN), distributed loads in kN/m and Young's modulus in gigapascals (GPa). Results are shown in kNm, kN and mm.
Does it include the beam's self-weight?
No. Only the loads you enter are applied, so you can match a hand calculation exactly. Add a distributed load equal to the self-weight if you want to include it.
Can I analyse continuous beams, 3D frames or Eurocode checks?
Yes — open the full FERS Cloud app for multi-span and continuous beams, 3D frames and trusses, load combinations and Eurocode EC3 steel checks. This page covers the single-span cases most people need quickly.
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