Fixed Beam Calculator (Built-in / Fixed-Fixed)
Enter your span, your load and your section and get the support moment, the midspan moment, the reactions and the deflection of a fixed (built-in) beam. Both ends are rigidly restrained against rotation, so the beam is statically indeterminate — a formula table only covers the standard cases, while here every result is solved live by the FERS finite-element solver and shown next to the closed-form answer.
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
Advanced
For a connection that rotates under moment: the end turns by M/k_φ. A very stiff spring behaves as a fully fixed end; a very soft one as a pin.
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↑
- Moment at A, M_A
- 7.50 kN·mhogging
- Moment at B, M_B
- 7.50 kN·mhogging
- Maximum shear |V|
- 5.00 kNat x = 0 m
- Maximum sagging moment
- 7.50 kN·mat x = 3.00 m
- Maximum hogging moment
- 7.50 kN·mat x = 6.00 m
- Maximum deflection δ_max
- 1.38 mmL/4,360 · at x = 3.00 m
- Max bending stress σ = M/W
- 23.1 MPaσ/f_y = 0.0984
- Plastic moment M_pl = W_pl·f_y
- 86.2 kN·mM/M_pl = 0.0870
- W_el needed for σ ≤ f_y
- ≥ 31.9 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 % |
| Moment at A | 7.500 kN·m | 7.500 kN·m | < 0.0001 % |
| Reaction at B | 5.000 kN | 5.000 kN | < 0.0001 % |
| Moment at B | -7.500 kN·m | -7.500 kN·m | < 0.0001 % |
| Maximum deflection | 1.376 mm | 1.376 mm | < 0.0001 % |
| Moment, worst FE station | -7.500 kN·m | -7.500 kN·m | < 0.0001 % |
| Shear, worst FE station | 5.000 kN | 5.000 kN | < 0.0001 % |
| Deflection, worst FE station | 1.374 mm | 1.374 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
- M_A
- 7.5×10⁶ N·mm
- C₁
- 0 N·mm²
- C₂
- 0 N·mm³
- Equilibrium
- R_B = −ΣF − R_A = 10,000 N − 5,000 N = 5,000 N
- Equilibrium
- M_B = M(6,000) = -7.5×10⁶ N·mm = -7.5×10⁶ N·mm
Peaks along the span
- Maximum sagging moment
- V(x̄) = 0 ⇒ x̄ = 3.00 m → M(x̄) = 7.50 kN·m
- Maximum hogging moment
- most negative M: x = 6.00 m → M = -7.50 kN·m
- Maximum deflection
- θ(x̄) = 0 ⇒ x̄ = 3.00 m → δ_max = |EI·v(x̄)| / EI = 1.125×10¹³ N·mm³ / 8.175×10¹² N·mm² = 1.38 mm
- Bending stress
- σ = M_max / W_el = 7.5×10⁶ N·mm / 324,420 mm³ = 23.1 MPa
- Stress ratio
- σ / f_y = 23.1 MPa / 235 MPa = 0.0984
- Span / deflection
- L / δ_max = 6,000 mm / 1.376 mm = L/4,360
Formulas used
A fixed (built-in / encastré) beam is clamped against rotation at both ends. The largest moment occurs at the supports (hogging), with a smaller sagging moment at midspan:
| Load case | Max moment | Reaction | Max deflection |
|---|---|---|---|
| Central point load P | M_support = P·L / 8, M_mid = P·L / 8 | R = P / 2 | δ_max = P·L³ / (192·E·I) |
| Uniformly distributed load w | M_support = w·L² / 12, M_mid = w·L² / 24 | R = w·L / 2 | δ_max = 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 fixed (built-in) IPE 240 steel beam (I = 3.89×10⁻⁵ m⁴, E = 210 GPa) with a 10 kN point load at midspan.
- Support moment, P·L/8
- 7.5 kNm
- Midspan moment, P·L/8
- 7.5 kNm
- Reaction at each support, P/2
- 5.0 kN
- Midspan deflection, P·L³/192EI
- 1.4 mm (≈ L/4360)
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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