Deflection: Why the Cracked Section Changes Everything

Deflection is the check that most often gets done last, done quickly, and done wrong. Strength calculations have a satisfying finality to them: a section either resists the moment or it does not. Deflection has no such comfort. The answer depends on how much of the beam has cracked, how long the load has been sitting there, and how the concrete has crept and shrunk in the meantime, and every one of those is an estimate.

Get the estimate wrong in the safe direction and you have an oversized beam. Get it wrong the other way and you have a floor that ponds water, doors that stick, partitions that crack, and a client who is entirely right to complain even though the structure is nowhere near failure.

The two mistakes, and they compound

There are two independent ways to under-estimate deflection, and engineers in a hurry make both at once.

Using the gross section. The uncracked second moment of area of a rectangular beam is $bh^3/12$, a familiar and reassuring number. But a working reinforced concrete beam is cracked over much of its length, and a cracked section has lost the entire tension half of its concrete. What is left is the compression zone plus the transformed steel, and that is a much smaller section.

Using the short-term modulus. Concrete under sustained load creeps: it keeps deforming at constant stress, for years. The code handles this by reducing the modulus rather than modelling the creep directly:

$$E_{c,eff} = \frac{E_{cm}}{1 + \varphi(\infty,t_0)}$$

where $\varphi$ is the creep coefficient, typically between 1.5 and 3 depending on humidity, member size and loading age. A creep coefficient of 2.0 means the effective modulus is one third of the short-term value, so the long-term deflection is three times the short-term one from the same load.

These two errors multiply. That is the whole story of this article.

One beam, four answers

Take a 300 by 600 mm beam, effective depth 550 mm, C30/37 concrete, 3 No. 20 mm bars, simply supported over 7 m, carrying a quasi-permanent load of 16.33 kN/m. That load produces a mid-span moment of 100 kN.m.

The uncracked section:

$$I_g = \frac{bh^3}{12} = \frac{300 \times 600^3}{12} = 5.40 \times 10^9\ \mathrm{mm^4}$$

The cracked section. The neutral axis of the cracked transformed section sits 126.9 mm from the compression face, and:

$$I_{cr} = \frac{bx^3}{3} + \alpha_e A_s (d-x)^2 = 0.204 \times 10^9 + 1.022 \times 10^9 = 1.23 \times 10^9\ \mathrm{mm^4}$$

Cracking has removed 77 percent of the stiffness. The ratio $I_g/I_{cr}$ is 4.4. That single number is the one to carry away.

Now the four deflections, all from $\delta = 5wL^4/384EI$:

The limit for this beam, at span over 250, is 28 mm.

The first row passes with 90 percent to spare. The last row fails. Same beam, same load, same code, and the only difference is whether the assumptions match what the concrete is actually doing.

The real answer sits between, and EC2 says where

A beam is not uniformly cracked. It cracks where the moment exceeds the cracking moment and stays uncracked elsewhere, so the true stiffness is somewhere between the two extremes. EN 1992-1-1 interpolates with a distribution coefficient:

$$\zeta = 1 - \beta \left(\frac{M_{cr}}{M}\right)^2$$

where $\beta$ is 0.5 for sustained or cyclic loading, and $\zeta$ is zero for an uncracked member.

For this beam the cracking moment is:

$$M_{cr} = \frac{f_{ctm} I_g}{h/2} = \frac{2.9 \times 5.40 \times 10^9}{300} = 52.2\ \text{kN.m}$$

With an applied moment of 100 kN.m:

$$\zeta = 1 - 0.5 \left(\frac{52.2}{100}\right)^2 = 0.864$$

So the beam behaves 86 percent like a fully cracked member, and:

$$\delta = \zeta \, \delta_{cracked} + (1-\zeta) \, \delta_{uncracked} = 0.864 \times 37.9 + 0.136 \times 8.6 = 33.9\ \text{mm}$$

33.9 mm against a limit of 28 mm. The beam fails, and it fails by 21 percent, having appeared to pass by a factor of ten under the naive calculation.

Note how quickly $\zeta$ saturates. At an applied moment double the cracking moment, the member is already 86 percent cracked. Interpolation only rescues members that are barely cracked at all, and a beam designed efficiently in bending is never in that category.

What is still missing from 33.9 mm

Two things, both of which make it worse.

Shrinkage curvature. Concrete shrinks as it dries. In a singly reinforced beam the steel is all near the bottom and restrains shrinkage there while the top shrinks freely, so the beam curves downward with no load on it at all. On a lightly loaded, heavily reinforced member this can be a significant fraction of the total.

Construction loading. If the beam was struck early and carried construction loads before the concrete reached full strength, it cracked at a lower moment and stayed cracked. The section does not heal. A beam that was overloaded once at seven days behaves as a cracked beam for the rest of its life, regardless of what the design load is.

That second point is worth emphasising because it is invisible in any calculation. Cracking is a one-way process. The stiffness you get is set by the worst load the member has ever seen, not the load it is carrying now.

Why span-to-depth ratios exist, and their limits

The span-to-depth check in EC2 §7.4.2 exists because the above is tedious and, done at scheme stage, mostly a waste of time. The tables encode a calculation like the one above for common cases, and staying within them means the deflection check will pass without being done.

They are a reasonable scheme-design tool. They are not a substitute for the calculation when anything is unusual: long spans, high imposed loads, brittle finishes, transfer structures, or a client specification tighter than span over 250. And they say nothing about the span over 500 limit that applies after installation of partitions, which is the one that actually generates complaints, because it governs the incremental deflection that cracks a plasterboard wall rather than the total.

The distinction between total and incremental deflection is where most real disputes end up. A slab can sag 30 mm and cause no trouble at all if 25 mm of it happened before the partitions went in.

How to fix a failing deflection check

Look at $\delta \propto wL^4/EI$ and the leverage is obvious.

• Depth beats everything. $I$ goes with $h^3$, so 10 percent more depth is roughly 33 percent more stiffness, for a small increase in concrete and no extra steel. It is almost always the cheapest fix.

• Compression steel helps the long-term part specifically. Steel in the compression zone restrains creep and shrinkage. It does little for short-term deflection but can cut the long-term component appreciably.

• Tension steel helps less than you would think. Adding area raises $I_{cr}$ only through the transformed term, and it is a weak lever compared with depth.

• A higher concrete grade is a weak lever. $E_{cm}$ rises roughly with the cube root of strength, so going from C30 to C40 buys under 10 percent stiffness.

• Change the structural system. Continuity over a support is worth far more than any section change: a fixed-end beam deflects one fifth of the simply supported case under the same load.

• Precamber if the deflection is acceptable in itself but not in appearance.

Takeaways

• The cracked second moment of area was 23 percent of the gross value in this example. That factor of 4.4 is typical, not extreme.

• Creep with $\varphi = 2$ triples the long-term deflection. The two effects multiply.

• The example beam computed at 2.9 mm on gross and short-term assumptions and 33.9 mm correctly. The limit was 28 mm.

• EC2's $\zeta$ interpolation saturates fast: at twice the cracking moment a member is already 86 percent cracked.

• Cracking never reverses. Early construction overload permanently sets the stiffness.

• Shrinkage curvature and construction history are both missing from the standard calculation and both make it worse.

• Increase depth first. It is the only variable that enters as a cube.

• Total deflection and incremental deflection are different limits, and the incremental one causes the complaints.

#ec2 #eurocode #concrete #deflection #serviceability #creep