Beam camber is calculated from the deflection it is meant to cancel. For a simply supported steel beam under a uniform load, the midspan deflection is Δ = 5wL⁴ / (384EI), where w is the load per unit length, L the span, E the elastic modulus (200 000 MPa for steel under CSA S16) and I the beam's moment of inertia. The engineer works out that deflection for the permanent load present when the floor or roof is built, typically self weight, deck and wet concrete, and then specifies camber as all or, more often, part of that number, rounded to a practical value.
This post walks through the arithmetic. Whether a beam should be cambered at all, and the reasons designers usually camber for less than the full deflection, are covered in when to specify pre-camber on a steel beam. Camber is a design decision by the engineer of record; the formulas here are to help you understand and check it, not to replace the design.
The deflection formulas behind camber
Camber is the mirror image of deflection, so the formulas are the standard elastic deflection equations for the load case. The common ones for a simply supported beam:
| Load case | Midspan deflection |
|---|---|
| Uniform load w over the full span | Δ = 5wL⁴ / (384EI) |
| Single point load P at midspan | Δ = PL³ / (48EI) |
| Two equal point loads P at the third points | Δ = 23PL³ / (648EI) |
| Two equal point loads P, each a distance a from its support | Δ = Pa(3L² − 4a²) / (24EI) |
Where a beam carries more than one of these, the deflections are added (superposition), as long as everything stays elastic.
Three things matter more than the formula itself:
- Which loads. Only the load the camber is meant to offset. On an unshored composite floor that is the steel beam's self weight, the deck and the wet concrete, all carried by the bare steel before the slab hardens. Superimposed dead load and live load act later on a stiffer composite section and are normally left out of the camber.
- Which section properties. For the wet-concrete stage, the bare steel moment of inertia, not the composite value.
- The support conditions. These formulas assume simple supports. Cantilevers, continuous beams and moment-connected beams behave differently and are generally poor candidates for camber.
Getting the units right
Most camber errors are unit errors, because the numbers span many orders of magnitude. In metric, the cleanest approach is to work in newtons and millimetres:
| Quantity | Typical unit on drawings | Convert to |
|---|---|---|
| Load w | kN/m | N/mm (1 kN/m = 1 N/mm, no change) |
| Span L | m | mm (multiply by 1000) |
| E | MPa | N/mm² (1 MPa = 1 N/mm², no change) |
| I | 10⁶ mm⁴ in section tables | mm⁴ (multiply by 1 000 000) |
The answer then comes out in millimetres.
In US customary units, use kips and inches: w in kip/in (divide kip/ft by 12), L in inches, E = 29,000 ksi and I in in⁴. The answer is in inches.
Worked example: a composite floor beam
Here is the calculation for a simply supported, unshored floor beam. The numbers are for illustration only.
- Beam: W310x52, Ix = 119 x 10⁶ mm⁴ (CISC section tables). Its US equivalent is W12x35.
- Span: 9.0 m
- Load on the bare steel during the pour, as worked out by the engineer from the beam's self weight, the deck and the wet concrete: 8.0 kN/m
Step 1. Convert to N and mm. w = 8.0 N/mm, L = 9000 mm, E = 200 000 N/mm², I = 119 000 000 mm⁴.
Step 2. Calculate the deflection.
Δ = 5 x 8.0 x 9000⁴ / (384 x 200 000 x 119 000 000)
- 9000⁴ = 6.561 x 10¹⁵
- Top: 5 x 8.0 x 6.561 x 10¹⁵ = 2.624 x 10¹⁷
- Bottom: 384 x 200 000 x 1.19 x 10⁸ = 9.139 x 10¹⁵
- Δ = about 28.7 mm
Step 3. Check it in imperial. W12x35 has Ix = 285 in⁴. The load is about 0.548 kip/ft, or 0.0457 kip/in, and the span is about 354 in. Then Δ = 5 x 0.0457 x 354⁴ / (384 x 29,000 x 285), which is about 1.13 in, or 28.7 mm. The two match, which is a good sign the units are right.
Step 4. Decide the camber. This is where the engineer's judgment comes in. Suppose, for this example, the engineer chose to camber for three quarters of the calculated deflection. That gives 0.75 x 28.7 = 21.5 mm, which would then be rounded to a practical value. Rounding down is the usual instinct, because a slight sag can be taken up with a little extra concrete while a hump cannot be taken away. The engineer might also decide the beam should not be cambered at all, for example if the slab is shored, or if the girders it frames into are deflecting too.
The span matters enormously. Deflection grows with the fourth power of the span: the same beam and load on a 4.5 m span (half of 9 m) would deflect one sixteenth as much, under 2 mm, which is not worth cambering.
Camber along the length of the beam
Camber is specified as a single value at midspan, but people often want to know what it is at other points, for example to check a profile or plan screed elevations. For a simply supported beam under uniform load, the deflected shape gives these fractions of the midspan value:
| Position along span (x/L) | Exact uniform-load shape | Parabolic approximation |
|---|---|---|
| 0.1 | 0.31 | 0.36 |
| 0.2 | 0.59 | 0.64 |
| 0.25 (quarter point) | 0.71 | 0.75 |
| 0.3 | 0.81 | 0.84 |
| 0.4 | 0.95 | 0.96 |
| 0.5 (midspan) | 1.00 | 1.00 |
The exact column comes from the elastic curve for uniform load, y = wx(L³ − 2Lx² + x³) / (24EI), divided by the midspan value. The parabolic column uses 4x(L − x) / L², a common simple approximation. In the example above, 21 mm of midspan camber would be about 15 mm at the quarter points.
In practice, a cambered beam's actual curve depends on how the shop induces it, and the check that counts is the midspan camber measured in the shop with the beam unloaded. Our post on camber tolerances and natural mill camber explains how that check is made and where the tolerances come from.
Common mistakes in camber calculations
- Using total load. Including live load or superimposed dead load produces a camber far larger than the beam will ever deflect under permanent load.
- Using the composite moment of inertia for the wet-concrete stage. The steel carries the pour on its own.
- Unit slips. Leaving I in 10⁶ mm⁴ or span in metres throws the answer off by a factor of a million or more. Do a quick imperial cross-check or a sanity check against typical deflection limits.
- Forgetting the girder. A beam framing into a girder rides down with the girder. If the girder deflects and is not cambered, the floor at midbay sits lower than either calculation suggests.
- Ignoring connection restraint. Even simple shear connections give some fixity, so the real beam deflects less than the formula. This is one reason engineers usually camber for less than the full calculated value.
- Cambering tiny amounts. Very small cambers are hard to induce and measure reliably, and are often within the beam's natural mill camber anyway.
For how cambered beams sit once they are in service, see do steel beams sag.
Putting the camber on the drawings
Once the camber is decided, the drawings should show:
- The camber value at midspan for each beam, on the framing plan or in the beam schedule.
- The direction, which is upward in normal use.
- Which beams are not to be cambered.
- Any tolerance different from the default.
- A note to coordinate cambered beams with the concrete pour, so the screed plan matches.
The shop then induces the camber, by heat or by machine, and records the measured value. Our post on heat cambering vs cold cambering explains how each method works.
Cambered beams fabricated to your drawings
We fabricate wide-flange beams with connections and camber as specified, and heat cambering of beams is one of our shop's specialty services. Include the camber values on your drawings when you send them. If you are not sure whether a beam needs camber, ask the engineer of record before fabrication; engineering services are also available for projects that need them.
Send your drawings for a quote, or see our services for fabrication, shop drawings and delivery across the GTA and Ontario.
Frequently asked questions
What is the formula for beam camber?
There is no separate camber formula. Camber is set from the calculated deflection under the load it is meant to offset. For a simply supported beam under uniform load that deflection is 5wL⁴/(384EI), and the engineer specifies camber as all or, more often, part of it.
What value of E do you use for steel?
CSA S16 assumes an elastic modulus of 200 000 MPa for steel. In US customary units the usual value is 29,000 ksi.
Do you camber for live load?
No. Camber offsets permanent load that is present when the floor or roof is finished, mainly self weight, deck and wet concrete on unshored composite floors. Live load comes and goes, so cambering for it would leave a hump most of the time.
How much camber is there at the quarter points?
For a simply supported beam under uniform load, the deflection at the quarter points is about 71 percent of the midspan value. A parabolic curve, often used as a simple approximation, gives 75 percent. Camber is specified and checked at midspan.
Who calculates beam camber?
The structural engineer of record. The fabricator puts in the camber shown on the drawings and checks it in the shop, but does not decide how much a beam should be cambered.
