Skip to the content
EurocodeCalculator

Design guide

Deflection and crack control to Eurocode 2: span/effective depth and Tables 7.2N and 7.3N

How Eurocode 2 checks deflection by the span to effective depth ratio (7.16a, 7.16b) and cracking by bar size and spacing, with worked examples.

Open the slab calculator

The underside of a newly cast concrete floor slab, still propped, in a building under construction
Reading time
4 minutes
Standard
EN 1992-1-1
Clauses
7.3.2, 7.3.3, 7.4.2
Calculator
Slab
Updated
On this page 3 sections
  1. Deflection: the span to effective depth ratio
  2. Cracking: limiting the crack width
  3. Points to remember

At the serviceability limit state a slab or beam must not sag too much, and its cracks must stay narrow enough not to spoil its look or let the steel corrode. EN 1992-1-1 lets both be checked without calculating deflections or crack widths directly: deflection by a limit on the span to effective depth ratio (7.4.2), and cracking by limits on the bar size or spacing (7.3.3).

Two members, worked below

Slab, basic l/d
60.13
lightly reinforced: Expression 7.16a
Beam, basic l/d
16.13
ρ close to ρ0: a tighter limit
Beam steel stress
226.0 MPa
σs under the quasi-permanent load
Largest bar for cracking
23.9 mm
from Table 7.2N at that stress

Deflection: the span to effective depth ratio

The limiting ratio depends on how heavily the section is reinforced. A lightly reinforced slab can be much more slender than a heavily reinforced beam, because its concrete is barely stressed. With as a reference ratio:

where is the tension steel ratio needed at mid-span, the compression steel ratio, and depends on the structural system (Table 7.4N): 1.0 for a simply supported span, 1.3 for an end span of a continuous member, 1.5 for an interior span and 0.4 for a cantilever.

The basic ratio is then adjusted:

  • by (Expression 7.17), approximately , when more steel is provided than needed. The National Annex can cap this factor; the Sri Lanka annex caps it at 1.5;
  • by for spans over 7 m carrying brittle partitions.

The whole check, from the steel ratio to the verdict:

Checking deflection by the span/effective depth ratio
  1. Steel ratio ρ = As,req / (b d) needed at mid-span
  2. Is ρ ≤ ρ0 = √fck / 1000?

    Yes

    1. Basic ratio from Expression 7.16a

    No

    1. Basic ratio from Expression 7.16b
  3. Times K for the structural system (Table 7.4N)
  4. Times 310/σs, about As,prov / As,req, within the annex's cap
  5. Times 7/leff for spans over 7 m with brittle partitions
  6. Is the actual leff / d within the limit?

    Yes

    1. Deflection needs no further check

    No

    1. Deepen the section, or add steel
  7. On to crack control

How steeply the limit falls with the steel ratio is easiest to see drawn out. Each curve below is Expression 7.16a and 7.16b as the calculator works them, for a simply supported span:

Basic span/depth ratio against the steel ratio

Simply supported spans, K = 1, from Expressions 7.16a and 7.16b. Lightly reinforced slabs may be far more slender than heavily reinforced beams, and a stronger concrete allows a little more.

  • C25/30
  • C30/37
  • C40/50
0 25 50 75 100 125 (l/d)_basic 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Steel ratio ρ (%) 7.16b from ρ = 0.55 % (C30/37) Worked slab: ρ = 0.23 % 0 25 50 75 100 125 (l/d)_basic 0.5 1.0 1.5 Steel ratio ρ (%) 7.16b from ρ = 0.55 % (C30/37) Worked slab: ρ = 0.23 %
Show the numbers
Steel ratio ρ (%)C25/30C30/37C40/50
0.2059.173.7105.3
0.3526.231.342.8
0.5018.520.525.8
0.6516.817.920.2
0.8015.716.618.5
0.9514.915.717.3
1.1014.415.116.5
1.2514.014.615.8
1.4013.714.215.3
1.5013.514.015.0

For the single span slab of the slab guide, 175 mm thick, the steel ratio is low, so Expression 7.16a gives a generous limit:

EN 1992-1-1 §7.4.2(2), Table 7.4N

Structural system factor — simply supported span

= 1.0
EN 1992-1-1 §7.4.2(2), Eq. (7.16.a)

Reinforcement ratios

ρ ≤ ρ₀, so Expression (7.16a) applies.

= 0.00227
EN 1992-1-1 §7.4.2(2), Eq. (7.16.a)

Basic limiting span/effective depth

ρ = 0.227 % is far below the ρ = 0.5 % of the "lightly stressed" column of Table 7.4N (l/d = 20), and Expression (7.16a) rises steeply as ρ falls, so this limit is large. It follows the standard; for long or slender slabs a calculated check (7.4.3) is more reliable.

= 60.13
EN 1992-1-1 §7.4.2(2), Eq. (7.17)

Steel stress modification

= 1.024
EN 1992-1-1 §7.4.2(2)

Span/effective depth check

✓ OK

For the 300 × 500 mm beam of the beam example, the ratio is close to and the check is tighter:

EN 1992-1-1 §7.4.2(2), Eq. (7.16.a)

Reinforcement ratios

ρ > ρ₀, so Expression (7.16b) applies.

= 0.00877
EN 1992-1-1 §7.4.2(2), Eq. (7.16.b)

Basic limiting span/effective depth

= 16.13
EN 1992-1-1 §7.4.2(2), Eq. (7.17)

Steel stress modification

= 1.038
EN 1992-1-1 §7.4.2(2)

Span/effective depth check

✓ OK

Cracking: limiting the crack width

Cracks are expected in reinforced concrete; the aim is to keep them narrow. Table 7.1N recommends a maximum crack width of 0.3 mm under the quasi-permanent load for most exposure classes, and 0.4 mm for dry, protected concrete (X0 and XC1). The National Annex can change these, and the calculator shows the value it uses. Clause 7.3.3 then gives two tables, so that the width does not have to be calculated:

  • Table 7.2N: the largest bar diameter for a given steel stress;
  • Table 7.3N: the largest bar spacing for a given steel stress.

Either one is enough. The steel stress is found for the cracked section under the quasi-permanent combination, so the beam's long-term (creep-adjusted) modular ratio is used:

EN 1992-1-1 §7.3.1(5), Table 7.1N

Crack width limit

= 0.3 mm
EN 1992-1-1 §7.4.3(5), Eq. (7.20)

Effective modulus for long-term loading

= 20.64
EN 1992-1-1 §7.3.4(2)

Neutral axis of the cracked section

Elastic cracked section, tension concrete ignored.

= 208.5 mm
EN 1992-1-1 §7.3.3(2)

Steel stress under the quasi-permanent load

= 226.0 MPa
EN 1992-1-1 §7.3.3(2), Table 7.2N
EN 1992-1-1 §7.3.3(2), Eq. (7.6N)

Maximum bar diameter (Table 7.2N, adjusted by 7.6N)

φ*s from Table 7.2N for σs = 226.0 MPa and wmax = 0.3 mm, interpolated linearly in σs; hcr is the depth of the tension zone just before cracking.

= 23.9 mm ✓ OK
EN 1992-1-1 §7.3.3(2), Table 7.3N

Maximum bar spacing (Table 7.3N)

From Table 7.3N for σs = 226.0 MPa and wmax = 0.3 mm; s is the centre spacing of the tension bars.

= 218 mm ✓ OK
EN 1992-1-1 §7.3.3(2)

Crack control check

Satisfied by the bar size (Table 7.2N); either table is sufficient.

✓ OK
Cracked section — Span 1h = 500x = 208.5Stresses (elastic)σcσs = 226.0 MPaLever armd − x/3 = 390.5Mqp = 110.88 kNm on As = 1256.6 mm² (bw = 300). Tension concrete ignored. Lengths in mm.
Cracked section. Elastic cracked section under the quasi-permanent load: the steel stress σs sets the bar size and spacing limits.

The minimum steel for crack control (7.3.2) must also be present, so that the steel does not yield when the concrete first cracks:

EN 1992-1-1 §7.3.2(2), Eq. (7.1)
EN 1992-1-1 §7.3.2(2), Eq. (7.2)

Minimum reinforcement for crack control

kc = 0.4 for the web in bending without axial force (7.2); k for h = 500 mm (≤ 300 mm: 1.0; ≥ 800 mm: 0.65). σs = fyk; fct,eff = fctm.

= 149.5 mm² ✓ OK

Thin slabs

Clause 7.3.3(1) exempts slabs up to 200 mm thick in bending without significant axial tension from these measures, provided the minimum steel is there and the bar spacing rules of 9.3 are met. The calculator applies this to the example slab:

EN 1992-1-1 §7.3.1(5), Table 7.1N

Crack width limit

= 0.3 mm
EN 1992-1-1 §7.3.3(1)

Crack control

The slab is 175 mm deep (≤ 200 mm), in bending without significant axial tension, and detailed to 9.3, so no specific crack-control measures are needed.

✓ OK

Points to remember

  • Use the effective span and the effective depth to the centre of the tension steel.
  • For a two-way slab, check the shorter span.
  • The quasi-permanent load, not the ultimate load, sets the steel stress for cracking. For an office with , it is well below the ultimate load.
  • If cracking fails, smaller bars at a closer spacing usually solve it, with the same steel area.

Open the slab calculator to try a thinner slab and watch the deflection check tighten.