Crack width and deflection by calculation

How a beam or slab is checked in service with the STATIX crack width and deflection calculator on its Eurocode 2 route. It covers the quasi-permanent load and the limits of EN 1992-1-1, minimum steel and crack control by bar size or spacing, the crack width of 7.3.4, and the long-term deflection of 7.4.3 with tension stiffening, creep and shrinkage. It ends with a question that decides the answer: does the member count as cracked?

1. Learning outcomes

After this lesson you can:

choose the serviceability combination (EN 1990 6.5.3) and the limits: w_max by exposure class (Table 7.1N), span/250 and span/500 (7.4.1);
find the minimum steel for crack control (7.3.2) and control cracking by bar size or spacing (7.3.3);
calculate a crack width by 7.3.4: σ_s, h_c,ef, ρ_p,eff, ε_sm − ε_cm and s_r,max;
work out the creep coefficient and the shrinkage strain (3.1.4, Annex B);
calculate a long-term deflection by 7.4.3 with ζ, E_c,eff and the shrinkage curvature, and the part after the partitions;
decide whether a member counts as cracked (7.4.3(3)), and say which checks the page does and which you do by hand.

2. Before you start

RC Design 1 (Eurocode): bending (lesson 1, the beam of Example A) and deflection and cracking (lesson 4).
The crack width and deflection calculator, set to Eurocode 2 (EN 1992-1-1), and its guide.
A pocket calculator and about 60 minutes.
The same lesson on the SA code: RC Design 2, lesson 6 (SANS 10100-1 Annex A).

What the crack width and deflection page’s Eurocode 2 route uses.
The member. One span: simply supported or a cantilever, with a UDL or a point load, or one or both ends continuous with a UDL. A point load shape also carries the member’s own weight (25 kN/m³) as a UDL on the same supports. A rectangle, or a T with the flange at the top. One layer of tension bars (a second bar size is allowed) and optional compression bars. It checks one section: mid-span, or the support of a cantilever. For the continuous shapes it takes M = wL²/10 or wL²/12 on the bottom bars; the crack width over a continuous support (top bars) is not checked.
Actions. The quasi-permanent combination G_k + ψ₂Q_k (EN 1990 6.16b), the default on this route, or the characteristic G_k + Q_k (6.14b). There is no frequent combination. ψ₂ comes from a list (EN 1990 Table A1.1) or your own value.
Materials. E_cm and f_ctm by Table 3.1, with E_cm scaled for the aggregate (3.1.3(2)); E_s 200 GPa. The creep coefficient φ and the shrinkage strain ε_cs are inputs (default 2.0 and 300 × 10⁻⁶): the page does not work them out from 3.1.4 or Annex B.
Crack width. 7.3.4: eq. 7.8 to 7.11, with eq. 7.12 for mixed bar sizes and eq. 7.14 where the bars are further apart than 5(c + φ/2). σ_s comes from the cracked section with α_e = E_s/E_cm and z = d − x/3; h_c,ef by 7.3.2(3). k₁ 0.8, k₂ 0.5, k₃ 3.4, k₄ 0.425 and k_t 0.4 are inputs with the recommended values. f_ct,eff = f_ctm (there is a flexural option; it only changes the cracking decision, and the crack width and ζ keep f_ctm, as 7.1(2) asks). The cover input is the c of eq. 7.11: the cover to the main bars. w_max comes from the exposure list or your own value.
Deflection. 7.4.3: ζ by eq. 7.19 (β 0.5, an input) between the uncracked and the cracked transformed sections, both with E_c,eff of eq. 7.20. M_cr,1, which decides cracking and goes into ζ, comes from f_ctm and the uncracked section at first loading, with E_cm (7.4.3(3)). The shrinkage curvature is eq. 7.21, interpolated with the same ζ. The deflection is K l²(1/r), with ζ of the checked section used over the whole span (7.4.3(7)). The coefficients K for the six shapes are the page’s own; the simply supported and cantilever values are the elastic ones. The total is checked against span/250 (7.4.1(4)). The part after the partitions is checked against span/500 (7.4.1(5)), span/350 or your own value. It is the total minus the instantaneous deflection under the self-weight (25 kN/m³, E_cm, β 1.0) or under a load you give, or minus a deflection you give. Creep and shrinkage before the partitions are not deducted, which is on the safe side.
Cracked or not. The page takes the member as cracked when the moment it checks, or the characteristic moment, exceeds M_cr,1 (7.4.3(3)); the characteristic moment also covers the frequent one. Where only the characteristic moment cracks it and ζ comes out 0, a NOT CHECKED row asks for the higher service load (section 5 and Example B).
Not checked: span/d (7.4.2), A_s,min (7.3.2), the bar size and spacing tables (7.3.3), the stress limits (7.2), skin steel in deep beams (7.3.3(3)), restraint and early thermal cracking, the frequent combination, two-way slabs and vibration. The examples do the first four by hand (the stress limits in Example A). Other annexes may use other values.

3. Serviceability on the Eurocodes

The serviceability combinations. EN 1990 6.5.3 gives three combinations, all with partial factors of 1.0. The characteristic load is reached rarely, the frequent load often, the quasi-permanent load most of the time. Creep, shrinkage and the look of a member depend on the load that stays on, so EN 1990 A1.4.3(4) and (6) use the quasi-permanent combination for appearance and long-term effects. For reinforced members, Table 7.1N and 7.4.1(4) and (5) use it too. The page offers the quasi-permanent combination (its default on the Eurocode 2 route) and the characteristic one, not the frequent one.

Crack limits and ψ₂. Table 7.1N gives the recommended w_max for each exposure class; your National Annex may change it. For X0 and XC1 the crack width does not affect durability, so 0.4 mm is set for appearance and may be relaxed where appearance does not matter. Members with bonded tendons use the frequent load and decompression instead. The page’s exposure list offers 0.4 mm (‘Wide’), 0.3 mm (‘XC/XD/XS’), 0.2 and 0.1 mm and your own value. For XC1 choose 0.4 mm: the 0.3 mm entry lists XC together with XD and XS, which is stricter than Table 7.1N for XC1. ψ₂ sets how much of the imposed load counts as quasi-permanent.

Deflection limits. 7.4.1(4) keeps the sag under the quasi-permanent load, measured from the supports, to span/250, for appearance and use. 7.4.1(5) keeps the deflection after construction to span/500 where it could damage partitions, finishes or cladding; other limits may suit less sensitive parts. EN 1990 A1.4.3(3) explains what ‘after’ means: the part of the deflection that happens after the partition or finish is built. That part is the total less what came before: the instantaneous deflection under the loads already on, and the creep and shrinkage up to that day.

Two routes. 7.4.1(6) lets you check deflection either by span/d (7.4.2) or by calculating it (7.4.3) and comparing it with a limit. 7.3.1(9) lets you control cracks either by the bar size or spacing of 7.3.3 or by calculating the crack width (7.3.4). The rules are quick and cover ordinary members, but they give no number, and they rest on assumptions about the loads, creep and shrinkage (the text after Table 7.4N describes the study behind eq. 7.16). Calculate where the rules do not reach, or where you need the number. The page calculates; it does not check span/d or the tables, so do those by hand (RC Design 1 (Eurocode), lesson 4).

4. Crack control

Minimum steel. When the first crack forms, the tension the concrete carried jumps into the steel. 7.3.2 asks for enough bonded steel to take that force without yielding, so the crack stays fine instead of opening wide: A_s,minσ_s = k_c k f_ct,eff A_ct. A_ct is the tension zone just before cracking, about half the depth in bending. k_c allows for the stress shape (0.4 in bending). k allows for self-balancing stresses in thick parts, from 1.0 at 300 mm to 0.65 at 800 mm. f_ct,eff is f_ctm, or less if the member cracks young. In T-beams, check web and flange separately (7.3.2(2)). The page does not check A_s,min; the examples do it by hand.

Crack control without calculation. With at least A_s,min in place, 7.3.3(2) lets you limit either the bar size (Table 7.2N) or the bar spacing (Table 7.3N), for the steel stress in the cracked section under the quasi-permanent load. The higher the stress, the smaller or closer the bars must be. Table 7.2N was set up for a cover of 25 mm and h − d = 0.1h; eq. 7.6N corrects the bar size with k_ch_cr/(2(h − d)), so a thin slab with a large cover is allowed smaller bars. Slabs up to 200 mm deep that follow 9.3 need nothing more (7.3.3(1)). The page does not check the tables; the examples do.

The effective tension area. Only the concrete close to the bars is gripped by them through bond. 7.3.2(3) and Figure 7.1 define it by its depth h_c,ef: the least of 2.5(h − d), (h − x)/3 and h/2, with x the depth of the cracked neutral axis. In beams 2.5(h − d) usually governs, in slabs (h − x)/3. The steel ratio in that area, ρ_p,eff, sets both the crack spacing and how much the concrete stiffens the bar between cracks. The page works it out for the rectangle (or the flange of a hogging T).

Crack width by calculation. A crack is the steel stretching more than the concrete around it, summed over the length between two cracks. 7.3.4 multiplies the largest crack spacing s_r,max by the mean strain difference. The strain is the steel strain under σ_s (cracked section, quasi-permanent load) less the help of the concrete between cracks (k_t 0.4 for long-term load), but never less than 60 % of the bare steel strain. The spacing grows with the cover c and with φ/ρ_p,eff: thick cover and few large bars give wide cracks. k₃ and k₄ are nationally set. Where the bars are far apart, eq. 7.14 gives an upper bound.

Restraint and early thermal cracking (a wall cast on a stiff base, a slab tied between cores) follow the same 7.3.2 and 7.3.4, with the steel stress just after cracking and 7.3.4(5) for walls. The page checks cracks from load only, and this lesson does the same.

5. Deflection by calculation

Tension stiffening. Below M_cr the member is uncracked. Above it, the concrete between cracks still carries some tension, so the member is stiffer than the bare cracked section. 7.4.3(3) takes any deformation parameter α (a strain, a curvature or, as a simplification, a deflection) between the uncracked value α_I and the fully cracked value α_II with the distribution factor ζ of eq. 7.19. Its note lets M_cr/M replace the steel stress ratio in bending. β is 0.5 for sustained load, which is the case for a long-term deflection. ζ = 0 for an uncracked section.

Creep. Under a load that stays on, concrete keeps shortening for years. 7.4.3(5) allows for it with the effective modulus of eq. 7.20, so a creep coefficient of 2 cuts the modulus to a third. Figure 3.1 gives φ(∞, t₀) for inside (RH 50 %) and outside (RH 80 %) air, against the notional size h₀ = 2A_c/u (u the perimeter that dries) and the age at loading t₀. Annex B gives the same as formulas: B.3 for humidity and size, B.4 for strength, B.5 for age at loading, B.9 for the cement class. At t = ∞ the time factor of B.7 is 1. The page does not work φ out: type in your own.

Shrinkage. Concrete shrinks as it dries (ε_cd, eq. 3.9 with Table 3.2 or eq. B.11, and k_h of Table 3.3) and as it hardens (ε_ca, eq. 3.12). Bars near one face hold that face back, so the member curves. 7.4.3(6) gives the curvature as ε_csα_eS/I, with S the first moment of the steel about the centroid and α_e the long-term modular ratio. It is worked for the uncracked and the cracked section and combined with ζ. Shrinkage often adds a fifth or more to the long-term deflection. The page takes ε_cs as an input (default 300 × 10⁻⁶).

From curvature to deflection. 7.4.3(7) prefers curvatures at many sections, integrated along the span; it also accepts working the whole member uncracked and fully cracked and combining the two with eq. 7.18. With ζ from the mid-span section that is the K l²(1/r) of the page, a little on the safe side because the ends crack less. The shrinkage curvature is nearly even along the span, so K_cs = 1/8. For the limit of 7.4.1(5) the page deducts only the instant deflection under the self-weight, which happens before the partitions. The real part before them also holds some creep and shrinkage; you can type your own estimate into the a_pre input.

Cracked or not. 7.4.3(3) treats a member as uncracked only if it is not expected to be loaded, anywhere, beyond the point where the concrete cracks. A crack does not close for good when the load drops. A member that cracks under its frequent, characteristic or construction load is cracked under the quasi-permanent load too. Use eq. 7.18 and 7.19 at the quasi-permanent moment. The page compares M_cr,1 of the first loading (the uncracked section with E_cm) with the larger of the checked and the characteristic moment, and takes the same M_cr,1 in ζ (Example B). M_cr itself is uncertain: the long-term section gives a higher value, and shrinkage tension lowers it (7.4.3(4)). Near M_cr a calculated deflection is least reliable (the note to 7.4.1(6)).