Beam Deflection Calculator
Work out how much a simply supported or cantilever beam bends under an even or point load, with the bending moment, for timber, steel and more.
New: please check your results.
For guidance only. This is not a substitute for a qualified engineer, and important designs should be checked by one.
- Maximum deflection
- 3.3 mm
- Deflection as a share of span
- Span ÷ 909
- Maximum bending moment
- 1.125 kN·m
- Maximum bending stress
- 3.777 MPa
- Largest support force
- 1.5 kN
What it does
Estimate how much a beam bends under load, for the four standard cases: a beam resting on two supports or a cantilever, with a load spread evenly along it or a single point load. Choose the material and the beam's size, and see the maximum deflection, how it compares with the span, the bending moment and the support force.
How to use it
- How it's supported
A joist or lintel resting on a wall at each end is simply supported. A beam built firmly into a wall at one end with the other end free, like a balcony, is a cantilever.
- Load
Floors, roofs and walls load a beam evenly. A single heavy item is a point load: it's placed in the middle of a simply supported beam, or at the free end of a cantilever, which are the worst cases.
- Span
The clear length between the supports, or from the wall to the free end for a cantilever.
- Load per length
The load along the beam, including its own weight. For a floor joist: the floor load per m² × the joist spacing. A domestic floor is often designed for about 1.5 kN/m² of people and furniture plus the floor's own weight.
- Point load
The single load in kilonewtons or pounds-force. 1 kN is roughly the weight of 100 kg.
- Material
Sets the stiffness (modulus of elasticity, E). C24 is the common grade of structural softwood in the UK; C16 is the lower grade.
- Modulus of elasticity (E)
From the manufacturer or a materials table. For example oak is roughly 10 to 12 GPa, and stainless steel about 200 GPa.
- Beam section
Choose a rectangle for sawn timber or glulam. For steel I-beams, channels and hollow sections, enter I from the section tables.
- Width (breadth)
The thinner side, across the beam. UK joists are often 47 mm wide.
- Depth
The height of the beam, top to bottom, with it standing on its edge. Depth matters most: doubling it makes the beam 8 times stiffer.
- Second moment of area (I)
From steel section tables (often listed as Iyy or Ix, about the strong axis), for example about 2,340 cm⁴ for a 203 × 133 × 25 universal beam.
Common uses
- Getting a first idea of whether a timber joist or beam is stiff enough.
- Comparing how much stiffer a deeper beam would be.
- Studying or checking hand calculations.
- Working out the pitch of a roof. Try the Roof Pitch Calculator
The formula
Simply supported, even load: δ = 5wL⁴ ÷ 384EI, M = wL² ÷ 8Simply supported, point load in the middle: δ = PL³ ÷ 48EI, M = PL ÷ 4Cantilever, even load: δ = wL⁴ ÷ 8EI, M = wL² ÷ 2Cantilever, point load at the end: δ = PL³ ÷ 3EI, M = PLRectangle: I = b × d³ ÷ 12, bending stress = M ÷ (b × d² ÷ 6)Where:
- δ
- maximum deflection
- w, P
- load per length, or the point load
- L
- span
- E
- the material's modulus of elasticity (stiffness)
- I
- the section's second moment of area
- M
- maximum bending moment
Standard elastic beam theory for a straight beam of one size along its length. It doesn't check strength, shear, bearing, notches, holes, lateral buckling, long-term creep (which can add a lot to timber and concrete deflection), safety factors or Building Regulations. Use it to understand and compare, not to design: anything structural should be designed or checked by a qualified structural engineer, and most structural work in the UK needs Building Control approval.
Worked example
A 47 mm × 195 mm C24 joist spanning 3 m with an even load of 1 kN/m:
- I = b × d³ ÷ 12 = 2,904.16 cm⁴, and E = 11 GPa
- Deflection: 5wL⁴ ÷ 384EI = 3.3 mm, or Span ÷ 909
- Bending moment: wL² ÷ 8 = 1.125 kN·m
Questions people ask
How much deflection is acceptable?
It depends on the use and the design code. Limits such as span ÷ 250 or span ÷ 360 are common starting points, and UK timber floor guidance has often used the smaller of 0.003 × span and 14 mm. A structural engineer will apply the right limit for your project, along with strength checks this calculator doesn't do.
Why does beam depth matter so much?
Stiffness goes up with the cube of the depth. A joist twice as deep is 8 times stiffer and bends an eighth as much, while doubling the width only halves the bending. That's why joists stand on their edge.
Can I use this to size a beam for knocking through a wall?
No. Removing a load-bearing wall needs a structural engineer's calculations and Building Control approval. The loads from floors, roofs and walls above are hard to judge, and a beam also needs checking for strength, bearings and supports, not just deflection.
Last checked 26 October 2026