Cracking stress

A section is assumed to be in cracked state when the maximum tensile stress, calculated on basis of a uncracked section, exceeds the tensile strength of the concrete. At bending the crack condition is then expressed as:

σ_n + σ_m ≤ f_ctm

where:

• σ_n is stress of normal force
• σ_m is stress of moment
• f_ctm is the mean tensile design strength.

Cracking moment

Cracking moment is the moment that barely causes cracking. With section forces N and M and belonging stresses σ_n and σ_m we find the factor c that induces  cracking at forces N and c M by:

c = (f_ctd – σ_n ) / σ_m

and cracking moment by:

M_crack = c M

Crack width calculation

The crack width w_k may be calculated from:

w_k = S_r,max (ε_sm - ε_cm)

where:

• S_r,max is the maximum crack spacing
• ε_sm is the mean strain in the reinforcement
• ε_cm is the mean strain in the concrete between cracks
• ε_sm - ε_cm may be calculated from:

• ε_sm - ε_cm = [σ_s - k_t f_ct,eff / ρ_p,eff (1 + α_e ρ_p,eff)] / E_s ≥ 0,6 σ_s / E_s

where:

• σ_s is the stress in the tension reinforcement assuming cracked section
• α_e is the ratio E_s / E_cm
• f_ct,eff is the mean value of the tensile strength of the concrete when the first crack occur
• f_ct,eff = f_ctm
• ρ_p,eff = A_s / A_c,eff
• A_c,eff is the effective area of concrete as calculated below
• k_t is a factor dependent on the duration of the load
  • k_t = 0,6 for short term loading
  • k_t = 0,4 for long term loading.

Effective area of A_c,eff

A_c,eff is the effective area of concrete of depth h_c,ef

where

• h_c,ef is the lesser of:
  • 2,5 (h - d)
  • (h - x) / 3
    or
  • h/2, see figure below:

Crack spacing S_r,max

For bonded reinforcement with spacing ≤ 5(c + ø / 2) the crack spacing is calculated as:

• S_r,max = k₃ c + k₁ k₂ k₄ ø/ ρ_p,eff

where:

• ø is the bar diameter in mm. If more than one bar size is present an average bar size ø_eq should be used,
• ø_eq = (n₁ ø₁² + n₂ ø₂²) / (n₁ ø₁ + n₂ ø₂)
• c is the cover to the longitudinal reinforcement,
• k₁ = 0,8 for high bond bars, 1,6 for plain bars
• k₂ = 0,5 for bending, 1,0 for pure tension,
• k₃ = 3,4
• k₄ = 0,425
• ρ_p,eff as above.

 Swedish annex

k₃ = 7ø /c

 Danish annex

k₃ = 3.4 (25/c)^2/3

For not bonded reinforcement or reinforcement with spacing > 5 (c + ø / 2) the crack spacing is calculated as:

• S_r,max = 1,3 (h - x)

where:

• x is the neutral axis depth.

The crack spacing should be calculated in the direction of the principle tensile stress as:

• S_r,max = 1 / (cos θ / S_r,max,y + sin θ / S_r,max,z)

where:

• θ is the angle between the reinforcement in the y-direction and the direction of the principal tensile stress
• S_r,max,y and S_r,max,z are the crack spacing’s calculated in the y and z directions respectively.