Propped Cantilever Calculator
Enter your span, your load and your section and get the fixed-end moment, the span moment, both reactions and the deflection of a propped cantilever (fixed at one end, simply supported at the other). Because this beam is statically indeterminate, a formula table only covers the standard cases — 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 = 8,358 cm⁴ · W_el = 557.2 cm³ · A = 53.82 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. Distributed load
Results
- Reaction at A, R_A
- 37.5 kN↑
- Reaction at B, R_B
- 22.5 kN↑
- Moment at A, M_A
- 45.0 kN·mhogging
- Maximum shear |V|
- 37.5 kNat x = 0 m
- Maximum sagging moment
- 25.3 kN·mat x = 3.75 m
- Maximum hogging moment
- 45.0 kN·mat x = 0 m
- Maximum deflection δ_max
- 4.00 mmL/1,500 · at x = 3.47 m
- Rotation at B, θ_B
- -0.00256 rad-0.147°
- Max bending stress σ = M/W
- 80.8 MPaσ/f_y = 0.344
- Plastic moment M_pl = W_pl·f_y
- 148 kN·mM/M_pl = 0.305
- W_el needed for σ ≤ f_y
- ≥ 191 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 | 37.50 kN | 37.50 kN | < 0.0001 % |
| Moment at A | 45.00 kN·m | 45.00 kN·m | < 0.0001 % |
| Reaction at B | 22.50 kN | 22.50 kN | < 0.0001 % |
| Rotation at B | -0.002564 rad | -0.002564 rad | < 0.0001 % |
| Maximum deflection | 3.999 mm | 3.999 mm | < 0.0001 % |
| Moment, worst FE station | -45.00 kN·m | -45.00 kN·m | < 0.0001 % |
| Shear, worst FE station | 25.50 kN | 25.50 kN | < 0.0001 % |
| Deflection, worst FE station | 3.999 mm | 3.999 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 × 8.358×10⁷ mm⁴ = 1.755×10¹³ N·mm²
Bending moment, slope and deflection
Support conditions
Solving them
- R_A
- 37,500 N
- M_A
- 4.5×10⁷ N·mm
- C₁
- 0 N·mm²
- C₂
- 0 N·mm³
- Equilibrium
- R_B = −ΣF − R_A = 60,000 N − 37,500 N = 22,500 N
Peaks along the span
- Maximum sagging moment
- V(x̄) = 0 ⇒ x̄ = 3.75 m → M(x̄) = 25.3 kN·m
- Maximum hogging moment
- most negative M: x = 0 m → M = -45.0 kN·m
- Maximum deflection
- θ(x̄) = 0 ⇒ x̄ = 3.47 m → δ_max = |EI·v(x̄)| / EI = 7.019×10¹³ N·mm³ / 1.755×10¹³ N·mm² = 4.00 mm
- Bending stress
- σ = M_max / W_el = 4.5×10⁷ N·mm / 557,228 mm³ = 80.8 MPa
- Stress ratio
- σ / f_y = 80.8 MPa / 235 MPa = 0.344
- Span / deflection
- L / δ_max = 6,000 mm / 3.999 mm = L/1,500
Formulas used
A propped cantilever is fixed (built in) at one end and propped on a simple support at the other. The largest moment is the hogging moment at the fixed end:
| Load case | Max moment | Reaction | Max deflection |
|---|---|---|---|
| Central point load P | M_fixed = 3·P·L / 16, M_span = 5·P·L / 32 | R_fixed = 11P/16, R_prop = 5P/16 | δ = 7·P·L³ / (768·E·I) |
| Uniformly distributed load w | M_fixed = w·L² / 8, M_span = 9·w·L² / 128 | R_fixed = 5wL/8, R_prop = 3wL/8 | δ_max ≈ w·L⁴ / (185·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 propped cantilever IPE 300 steel beam (I = 8.36×10⁻⁵ m⁴, E = 210 GPa) under a 10 kN/m uniformly distributed load.
- Fixed-end moment, w·L²/8
- 45.0 kNm
- Max sagging moment, 9w·L²/128
- 25.3 kNm
- Reaction at fixed end, 5wL/8
- 37.5 kN
- Reaction at prop, 3wL/8
- 22.5 kN
- Max deflection, w·L⁴/185EI
- 4.0 mm (≈ L/1500)
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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