❓ What is Crack Width?

Cracking in reinforced concrete section occurs when the tensile stress induced by bending, shear, torsion, or restrained shrinkage exceeds the tensile strength of the concrete.

Because concrete is weak in tension, reinforcement is provided to carry these tensile forces. However, for the reinforcement to take up the tensile stress, the surrounding concrete must stretch and eventually crack. To protect reinforcement from corrosion and ensure structural integrity and appearance, crack widths must be limited to acceptable values depending on the environmental exposure class.

Crack width control is the design process of ensuring that any cracks that form are kept within acceptable limits (often 0.3 mm or 0.4 mm) under quasi-permanent load combinations.

Common symbol in eurocode:

w_k

The general formula for crack width is:

Characteristic Crack Width

\begin{aligned}w_{k} &= s_{r,max} (\epsilon_{sm} - \epsilon_{cm})\end{aligned}

Eq. 7.8, BSEN 1992-1-1:2004

❓ What crack analysis is considering?

To understand what exactly happening inside the crack analysis, let's briefly understand the major components: 1. mean strain different at tension reinforcement level and 2. maximum crack spacing.

1. Mean Strain different at tension reinforcement level, ε_sm - ε_cm

When a load is applied, tensile stress is induced in the tension zone of the section. The steel reinforcement carries most of the tensile stress and therefore undergoes strain. Although the surrounding concrete also experiences tensile stress, its strain and elongation differ from those of the reinforcement. This difference in deformation leads to the formation of cracks

The graph below illustrates the difference in strain at the level of the tension reinforcement.

In calculation of the strain, the strain is obtained by dividing the stress by Young's modulus: $ε = \frac{σ}{E}$ .

Strain-Stress-Young Modulus

\begin{aligned}ε = \frac{σ}{E}\end{aligned}

Therefore, the strain difference can be calculated using the formula below, as recommended in Eurocode 2. The equation is based on first principles, with empirical coefficients introduced to account for the actual behaviour of reinforced concrete.

Mean Strain Difference

\begin{aligned}\epsilon_{sm} - \epsilon_{cm} &= \frac{\sigma_s - k_t \frac{f_{ct,eff}}{\rho_{p,eff}} (1 + \alpha_e \rho_{p,eff})}{E_s} ≥ 0.6 \frac{\sigma_s}{E_s}\end{aligned}

Eq. 7.9, BSEN 1992-1-1:2004

2. Maximum Crack Spacing, s_r,max

The maximum crack spacing depends on the geometry of the reinforcement within the section, as suggested in below diagram:

For rebars spacing within section < (within 5(c + ϕ/2)): he maximum crack spacing is calculated using an empirical equation based on the concrete cover, bar diameter (ϕ), and the effective reinforcement ratio.

Maximum Crack Spacing for rebar spacing < 5(c + ϕ/2)

\begin{aligned}s_{r,max} &= k_3 c + k_1 k_2 k_4 \frac{\phi}{\rho_{p,eff}}\end{aligned}

Eq. 7.11, BSEN 1992-1-1:2004

For rebars spacing within section > (within 5(c + ϕ/2)):, the crack spacing is assumed to depend primarily on the geometry of the concrete section.

Maximum Crack Spacing for rebar spacing > 5(c + ϕ/2)

\begin{aligned}s_{r,max} &= 1.3(h - x)\end{aligned}

Eq. 7.14, BSEN 1992-1-1:2004

🧩 Dependence of Crack Width from Section Properties

From the above consideration an eurocode 2, we can conculded that the crack width is depending on .. From Eurocode 2, w_k mainly depends on: Cl. 7.3.4, BSEN 1992-1-1:2004

Concrete cover, c
Bar diameter, ϕ
Effective tension area of concrete, A_c,eff
Steel stress under quasi-permanent load, σ_s
Effective reinforcement ratio, ρ_p,eff = $\frac{A_{s}}{A_{c,eff}}$

If you want to reduce the crack width, can consider:

reduce the stress (moment at the section)

incrase the section depth and breath

reduce concrete cover

increase the size of steel rebar and also the area

🔍 Formula of Crack Width wk

The characteristic crack width can be calculated using the below formulas Eq. 7.8, 7.9, 7.11 & 7.14, BSEN 1992-1-1:2004.

Characteristic Crack Width

\begin{aligned}w_{k} &= s_{r,max} (\epsilon_{sm} - \epsilon_{cm})\end{aligned}

Eq. 7.8, BSEN 1992-1-1:2004

Mean Strain Difference

\begin{aligned}\epsilon_{sm} - \epsilon_{cm} &= \frac{\sigma_s - k_t \frac{f_{ct,eff}}{\rho_{p,eff}} (1 + \alpha_e \rho_{p,eff})}{E_s} ≥ 0.6 \frac{\sigma_s}{E_s}\end{aligned}

Eq. 7.9, BSEN 1992-1-1:2004

where
f_ct,eff = mean value of tensile strength of the concrete effective
σ_s = stress in tension reinforcement assuming a cracked section
α_e = E_s / E_cm
k_t = a factor dependent on the duration of load

Maximum Crack Spacing for rebar spacing < 5(c + ϕ/2)

\begin{aligned}s_{r,max} &= k_3 c + k_1 k_2 k_4 \frac{\phi}{\rho_{p,eff}}\end{aligned}

Eq. 7.11, BSEN 1992-1-1:2004

Maximum Crack Spacing for rebar spacing > 5(c + ϕ/2)

\begin{aligned}s_{r,max} &= 1.3(h - x)\end{aligned}

Eq. 7.14, BSEN 1992-1-1:2004

where
ϕ = bar diameter
k₁ = coeff. which takes account of bond properties of the bonded reinforcement [0.8 for high bond bar, 1.6 for bars with effectively plain surface]
k₂= coeff. take account of the distribution of strain [0.5 for bending, 1.0 for pure tension]
k₃ = 3.4
k₄ = 0.425
c = cover to longitudinal reinforcement
ρ_p,eff= ( A_s + ξ₁ A'_p)/ A_c,eff

Given a section with an applied quasi-permanent moment M_Ed, find the crack width w_k

Calculate Steel Stress (Cracked Section)

x = depth to neutral axis of the cracked section

I_cr = cracked second moment of area

σs = (M_Ed / I_cr) ⋅ (d - x) ⋅ α_e

Determine Effective Tension Area

Calculate effective height: h_c,ef= min(2.5(h-d), (h-x)/3, h/2)

Calculate area: A_c,eff = h_c,ef ⋅ b

Calculate effective ratio: ρ_p,eff = As / A_c,eff

Calculate Strain Difference

ε_sm - ε_cm = [σs - kt(f_ct,eff / ρ_p,eff)(1 + α_e ⋅ ρ_p,eff)] / Es

Check minimum limit: must be ≥ 0.6(σ_s / E_s)

Calculate Maximum Crack Spacing

Determine bond factors: k₁ = 0.8, k₂ = 0.5

Apply EC2 constants: k₃ = 3.4, k₄ = 0.425

s_r,max = 3.4c + 0.8 ⋅ 0.5 ⋅ 0.425 ⋅ (ϕ / ρ_p,eff)

Determine Final Crack Width

w_k = s_r,max ⋅ (ε_sm - ε_cm)

Ensure w_k ≤ w_max (e.g., 0.3mm)

📝Summary & Key Takeaways

Concrete cracking in tension is an expected part of reinforced concrete behavior. Crack control ensures these cracks remain small enough to protect the reinforcement from corrosion and maintain appearance.
The width of a crack is primarily determined by the maximum crack spacing and the difference in strain between the elongating steel and the surrounding concrete.
Common crack width calculation procedures are demonstrated in this article, guiding you through determining the effective tension area, calculating steel stress, and establishing the final characteristic crack width step-by-step.

#rc design #crack width calculation #formula #structural

CivilSimple Team

The CivilSimple Team writes practical engineering guides for the profession and the curious. All articles are reviewed for technical accuracy before publication.