Deflection and crack control

How a beam is checked in service to EN 1992-1-1, on the Eurocode 2 route of the STATIX RC beam calculator: span over depth with K and the steel ratio (7.4.2), long-term deflection by calculation (7.4.3), the bar size and spacing rules (7.3.3) and the crack width (7.3.4).

1. Learning outcomes

After this lesson you can:

check a beam’s deflection by span over depth with eq 7.16 and the 7.17 adjustment;
explain how creep, shrinkage and cracking make a beam sag over time;
work out the minimum steel for crack control;
control cracks by bar size or spacing, and calculate a crack width;
read the calculator’s serviceability results, and know when the rule and the calculation disagree.

2. Before you start

Lesson 1: the beam and its bars.
The RC beam calculator, set to Eurocode 2 (EN 1992-1-1).
A pocket calculator and about 55 minutes.
The same lesson on the SA code: RC Design 1, lesson 4 (SANS 10100-1).

The values this lesson uses are those of the calculator’s Eurocode 2 route: Table 7.4N K factors and eq 7.16 with the 7.17 factor A_s,prov/A_s,req (up to 1.5); the quasi-permanent load with ψ₂ (0.3 by default) for the crack width and the calculated deflection; default creep and shrinkage from 3.1.4 and Annex B; w_max 0.3 mm; k_t 0.4 and the recommended k₁ to k₄ in eq 7.11.

3. Deflection

Deflection limits. EN 1992-1-1 7.4.1 suggests span/250 for the total sag under the quasi-permanent load, for appearance, and span/500 for the part that happens after partitions and finishes are fixed. The calculator’s default limit is L/250 (total).

Span over depth. Most beams and slabs are checked for deflection without a calculation: the span over the effective depth must not exceed a limit. K allows for the support conditions (Table 7.4N gives the values).

The basic ratio. Lightly reinforced members may be much more slender than heavily reinforced ones. Eq 7.16a applies when the tension steel ratio ρ is below ρ₀, eq 7.16b above it (with compression steel ρ′ added). The curve is for C30/37 with no compression steel.

Adjusting the ratio. Extra steel lowers the steel stress, so the beam cracks less and sags less: the ratio may be raised by A_s,prov/A_s,req, up to 1.5. Wide flanges and long spans lower it (7.4.2(2)).

Long-term deflection. Under a sustained load the concrete creeps, and as it dries it shrinks; both add sag for years. The calculator works with an effective modulus E_c,eff = E_cm/(1 + φ) and a shrinkage curvature, with default φ and ε_cs from 3.1.4 and Annex B that you can change.

Between cracked and uncracked. A real beam is cracked near mid-span and uncracked near the supports, and the concrete between the cracks still helps. EN 1992-1-1 7.4.3 interpolates the curvature between the two states with ζ. The calculator takes M_cr of the first loading (the uncracked section with E_cm) and counts the beam as cracked when the quasi-permanent or the characteristic moment passes it (7.4.3(3)).

4. Cracks

Crack control. Reinforced concrete always cracks in tension; the aim is to keep the cracks fine, for durability and appearance. Table 7.1N recommends limits by exposure class; the calculator checks against 0.3 mm for every class, which for X0 and XC1 is stricter than the recommended 0.4 mm.

Minimum steel for cracking. When the concrete cracks, the tension it carried jumps into the steel; there must be enough steel not to yield at once, or a few wide cracks form. This is a separate minimum from the strength minimum of lesson 1.

Control without calculation. If the bars are small enough or close enough together for the steel stress they work at, the cracks will be within the limit (7.3.3). Either table is enough. The calculator’s checks use the spacing table, with the steel stress from the characteristic load (on the safe side).

Calculating the crack width. The widest crack is the largest crack spacing times the difference between the steel strain and the concrete strain. The calculator does this for each span under the quasi-permanent load.

The effective tension area. Only the concrete close to the bars is held together by them: an area h_c,ef deep around the tension steel. ρ_p,eff = A_s/A_c,eff measures how well that area is reinforced.