Deflection and crack control

A beam must not only be strong. It must also be stiff enough not to sag too much, and its cracks must stay fine. This lesson checks both with the quick SANS 10100-1 rules: span/effective depth and bar spacing.

1. Learning outcomes

After this lesson you can:

explain why deflection and crack width matter, and why they grow with time;
check a beam's span/effective depth ratio with the basic ratio and its modification factors;
work out the service stress in the steel and the tension steel factor;
apply the flange factor to T-beams;
check the bar spacing for crack control, and read STATIX's long-term deflection and crack width results.

2. Before you start

Lesson 1 (the rectangular beam) and lesson 2 (the T-beam): this lesson checks both.
The free STATIX RC beam calculator, open in a second tab.
A pocket calculator and about 50 minutes.

3. Why deflection matters

Lessons 1 to 3 made the beam strong enough at the ultimate load. That is the safety check. This lesson is about the beam in normal use, at the service load: the serviceability limit state.

Why deflection matters. A beam can be strong enough and still sag too much. The sag δ cracks the partition below, jams doors and windows, and on a roof lets water pond. These are the faults clients notice first.

A beam that sags a lot is not unsafe, but it is a nuisance. The sag also grows with time, because concrete creeps and shrinks.

Concrete keeps moving after the load goes on. Under a steady load it creeps, and as it dries it shrinks; both make a cracked beam sag more for years. That is why the span/depth rules and STATIX's long-term check aim at the final sag, not the first one.

4. The span/depth method (SANS 10100-1, 4.3.6)

Working out the sag of a cracked concrete beam by hand is long. For normal beams the code offers a short cut: limit the ratio of span to effective depth.

The idea behind span/depth. Stiffness grows very fast with depth (with d³), so the ratio of span to effective depth is a good guide to how much a beam will sag. The code gives an allowed L/d; if the beam's actual L/d is below it, the sag will be acceptable without a calculation.

The allowed ratio is built from four factors:

The span/depth check is one number built from four factors (SANS 10100-1, 4.3.6). Start from the basic ratio for the support conditions, then let the steel and the shape move it up or down.

allowed L/d = basic ratio × tension steel factor × compression steel factor × flange factor pass when L/d (actual) ≤ allowed L/d

For spans over 10 m there is a further reduction (SANS 10100-1, 4.3.6), and cantilevers need more care. This lesson stays with normal simply supported spans.

The tension steel factor

The bottom bars stretch as the beam bends. The harder they work at service load, the more the beam curves and sags. So the factor depends on two things: the steel's service stress f_s, and how hard the section works, M/(b d²), with the ultimate moment M (SANS 10100-1, Table 11):

tension steel factor = 0.55 + (477 − f_s) / (120 (0.9 + M/(b d²))) ≤ 2.0

The tension steel factor from its formula, 0.55 + (477 − f_s)/(120 (0.9 + M/bd²)), capped at 2.0. Two things raise it: a lower service stress in the steel f_s, and a lightly loaded section (small M/bd²). A beam that works hard (large M/bd², high f_s) gets a factor below 1.

The service stress f_s is the stress in the bars under the service load. It starts from the design stress and is scaled down twice:

How STATIX and the code find the service stress f_s in the tension steel. Start from 0.87 f_y, scale it down to the service load, and down again when you provided more steel than needed (with moment redistribution there is one more factor). Bar lengths show the idea, not a real beam.

f_s = 0.87 f_y × (M_SLS/M_ULS) × (A_s,req/A_s,prov) (no redistribution)

This is a useful lesson in itself: adding a little more steel than needed lowers f_s and lets the beam span further. A deeper beam does even better, because d appears in L/d and in M/(b d²).

The compression steel factor

Bars in the compression zone resist creep of the concrete there, so they reduce the long-term sag. SANS 10100-1 Table 12 gives a factor of 1.0 or more against 100A_s′/(b d). If you ignore small top bars, use 1.0: that is on the safe side.

The flange factor for T and L beams

A T-beam is very strong in bending for the size of its web, but its cracked web is not as stiff. So for flanged beams the allowed ratio is cut: by 0.8 when b_w/b_eff ≤ 0.3, rising to 1.0 for a rectangle (SANS 10100-1, 4.3.6.5). In M/(b d²) STATIX uses the web width b_w, which is on the safe side.

5. Crack control by bar spacing

Cracks are normal in reinforced concrete: the steel only starts to work once the concrete around it has cracked. What matters is that the cracks stay fine.

At service load the concrete below the neutral axis is cracked, and the bars carry the tension. The bar grips the concrete next to it, so a crack is narrow at the bar and opens up towards the face. Cracks wider than about 0.3 mm look bad and let water reach the steel.

For normal beams the code does not ask you to calculate the crack width. Instead it limits how far apart the tension bars may be (SANS 10100-1, 4.11.8.2):

The clear gap between bars near the tension face must not be more than the value in Table 25. It depends on f_y (and on any moment redistribution).
From a corner of the beam to the nearest bar, the clear distance must not be more than half of that value.
Beams deeper than 750 mm also need side bars on the faces of the web.

Crack control by bar spacing. The same steel area spread over more, smaller bars gives many fine cracks instead of a few wide ones. SANS 10100-1 (4.11.8.2, Table 25) limits the clear gap between tension bars by f_y, and the gap from a corner to the nearest bar to half of that.

6. What STATIX checks

What the STATIX RC beam calculator adds. Besides the span/depth check, it works out the long-term deflection of the cracked beam (BS 8110-2 method, with creep and shrinkage) and the crack width, and compares them with span/250 and 0.3 mm.

When you run the RC beam calculator you get both: the quick span/depth check of this lesson, and the long-term deflection and crack width worked out from the cracked section. If they disagree, the calculation is the better answer; the span/depth rule is a safe short cut for normal beams.