Doubly reinforced and flanged beams

What to do when K exceeds K′, and how a slab cast with its beams makes T- and L-beams, to EN 1992-1-1 on the Eurocode 2 route of the STATIX RC beam calculator: compression steel, its yield check, the effective flange width and where the neutral axis lies.

1. Learning outcomes

After this lesson you can:

recognise when K > K′ and design the compression and extra tension steel;
check that the compression steel yields, and hold it with links;
find the effective flange width of a T- or L-beam to 5.3.2.1;
decide whether the neutral axis is in the flange or the web, and design the beam;
say what the calculator’s Eurocode 2 route does and does not design.

2. Before you start

Lesson 1: beams in bending, K, K′ and z.
The RC beam calculator, set to Eurocode 2 (EN 1992-1-1).
A pocket calculator and about 50 minutes.
The same lesson on the SA code: RC Design 1, lesson 2 (SANS 10100-1).

The values this lesson uses are those of the calculator’s Eurocode 2 route, as in lesson 1: EN 1990 equation 6.10 with the recommended factors, α_cc 0.85 for bending (so K′ 0.168 and z with 1.134), the 5.5(4) limit x_u/d ≤ 0.45, and the recommended values elsewhere. For the effective width the calculator’s full check takes l₀ from Figure 5.2 for each span (l when simply supported, 0.85 l for an end span, 0.7 l for an interior span), and it does not cap z at 0.95d.

3. Compression steel

Compression steel. When K exceeds K′, the neutral axis would have to go below 0.45d and the beam would fail by crushing before the steel yields. Instead the neutral axis is held at the limit and bars near the top face take the extra compression.

Two parts. The section carries the moment M′ that it can take with tension steel alone, at the lever arm z for K′. The rest of the moment is carried by a couple: compression bars A_s2 at the top and the same area of extra tension bars at the bottom, a lever arm d − d₂ apart.

Does the compression steel yield? Its strain falls in a straight line from 0.0035 at the top to zero at the neutral axis. If the bars sit too low, they do not reach the yield strain f_yd/E_s and give less than A_s2f_yd. For B500 that happens when d₂/x is more than about 0.38.

Holding the compression bars. A bar in compression can buckle out through the cover. Bars counted in the resistance must be held by links at a spacing of not more than 15 times their diameter.

4. T- and L-beams

T- and L-beams. Where a slab is cast with its beams, part of the slab acts as the beam’s top flange. An internal beam becomes a T-beam, an edge beam an L-beam. In sagging the flange is in compression, so the stress block can be very wide and shallow.

The effective width. Only part of the slab works with the beam, because the slab shears and lags away from the web. Each side adds b_eff,i, never more than half the clear distance to the next beam, b_i.

The length l₀. The effective width depends on the distance between the points of zero moment: the whole span for a simply supported beam, shorter lengths in continuous spans and over supports. The calculator’s full check takes them from Figure 5.2 for each span: l for a simply supported span, 0.85 l for an end span, 0.7 l for an interior one.

Where is the neutral axis? If the moment is no more than the flange can take with the block filling it, f_cd b_eff h_f(d − h_f/2), the block stays in the flange and the beam is designed as a rectangle b_eff wide. Most T-beams in floors are like this.

The block in the web. Then the flange outstands carry a fixed force at mid-flange, and the web carries the rest of the moment as a rectangular beam b_w wide. This is how the calculator designs a flanged beam when the moment is large.

Hogging. Over an interior support the moment reverses: the slab is in tension and the narrow web in compression. A T-beam there is a rectangular beam b_w wide, and its top steel spreads into the slab. The calculator uses the flange only where it is in compression.

The choices. Compression steel lets a beam stay shallow, but it costs steel and close links. Often a slightly deeper beam is simpler and cheaper; a T-beam usually needs neither.