Crack width and Stress limitation
EUROCODE-2

Crack width and Stress limitation

DATA
Dimensions (?)
b(cm)     h(cm)     c(mm)
  • b: width section. Valid values from 10 to 150
  • h: depth section. Valid values from 10 to 150
  • c: cover to the longitudinal reinforcement. Valid values from 10 to h/3
Materials
fck(MPa)   (?)   fyk(MPa)
  • fck: Characteristic compressive cylinder strength of concrete. Valid values from 20 to 90
  • fyk: Characteristic strength of tension reinforcement. Valid values from 400 to 600

REINFORCEMENT
Tensile reinforcement (?)
As(cm2)    Φeq(mm)    s(cm)
  • As1: Cross-sectional area of tensile reinforcement. Valid values from As,min (2Φ8) to As1,max (layer Φ32 with bar spacing s=5 cm).
  • Φeq: is the bar diameter, or an equivalent diameter (Expression 7.12). Valid values from 5 to 40.
  • s: The maximum spacing between tension bars. Valid values from 4 to b-6cm.

You can enter values directly or through the following data:

typeNum Φ
1
2
Compression reinforc.
As2 (cm2 (?)
  • As2: Cross-sectional area of compression reinforcement. Valid values from 0 to As2,max (layer Φ32 with bar spacing s=5 cm).

You can enter the area directly or through the following data:

typeNum Φ
1
2

LOADS
Duration of load  (?)

It affects the coefficient kt:
kt = 0,6 for short term loading
kt = 0,4 for long term loading

Bending Mk(Kn·m) (?)

Characteristic bending moment in the section considered (Mk). Valid values from 0 to 2·104


VALUES FOR USE IN A COUNTRY
Use values recommended
Stress limitation
Value k1   (?)   Value k3
  • k1: (see clause 7.2(2)). The recommended value is 0.6. Valid values from 0.3 to 1
  • k3: (see clause 7.2(5)). The recommended value is 0.8. Valid values from 0.4 to 1
Crack width control
Value k3   (?)   Value k4
  • k3: (see clause 7.3.4(3)). The recommended value is 3.4. Valid values from 2 to 7
  • k4: (see clause 7.3.4(3)). The recommended value is 0.425. Valid values from 0.2 to 1

 

RESULT

Maximum compressive stress in concrete

Concrete compressive stress Compression stress limit Requirement
σc (MPa) σc,max (MPa) σc < σc,max
10.88 18 OK

Maximum tensile stress in reinforcement

Reinforcement tensile stress Tensile stress limit Requirement
σs1 (MPa) σs1,max (MPa) σs1 < σs1,max
303.39 400 OK

Crack width

Maximum crack spacing Mean strain Crack width
sr,max (cm) εsm - εcm (‰) Wk (mm)
32.2 0.91 0.29



DETAILS OF CALCULATION

Notation and methodology according to clause 7 of EC2

Stress calculation assuming cracked section

  • Stress in the tension reinforcement
    σs1 = n·σc·(d-X)/X = 6.09·10.88·(25.9-4.64)/4.64 = 303.39 MPa
  • Maximun reinforcement tensile stress
    σs1,max = k3 · fyk = 0.8 · 500 = 400 MPa
  • Compressive stress in the most compressed concrete fibre
    σc = Mk · X / Icr = 10.88 MPa
  • Maximum compressive stress
    σc,max = k1 · fck = 0.6 · 30 = 18 MPa

where

  • n= Es / Ecm = 200 / 32.84 = 6.09
    Ecm = 22·[fcm/10]0.3 = 22·[38/10]0.3 = 32.84 GPa
    fcm = fck + 8 = 30 + 8 = 38 MPa
  • d (effective depth) = h – c - Φ/2 = 30 – 3.5 – 1.2/2 = 25.9 cm
  • X (Depth of the neutral fibre) = 4.64 cm
    depth of the neutral fibre
    with:
    • d′ = h - d = 30 - 25.9 = 4.1 cm
    • ρ1 = As1 / (b·d) = 3.39 / (40·25.9) = 0.0033
    • ρ2 = As2 / (b·d) = 2.36 / (40·25.9) = 0.0023
  • Icr (cracked inertia) = 10668.66 cm4
    Icr = n·As1·(d-X)·(d-X/3) + n·As2·(X-d′)·(X/3-d′)
    Icr = 6.09·3.39·(25.9-4.64)·(25.9-4.64/3) + 6.09·2.36·(4.64-4.1)·(4.64/3-4.1) = 10668.66 cm4

Calculation of crack width

wk = sr,msx·(εsm - εcm) = 322.5 · 0.00091 = 0.29 mm

where

εsm - εcm (difference between mean strains) = 0.00091
εsm - εcm = max(εm,1 ; εm,2) = max(0.0006 ; 0.00091)
with:

  • εm,1 = [σs1 - kt·(fct,effp,eff)·(1+αe·ρp,eff)] / Es
    εm,1 = [303.39 - 0.6·(2.9/0.01)·(1+6.09·0.01)] / 200000 = 0.0006
    • kt = 0.6 (Long term loading)
    • fct,eff = fctm = 2.9 MPa
      fctm = 0,30 × fck(2/3) = 0,30 × 30(2/3) = 2.9 MPa
    • ρp,eff = As / Ac,eff = 3.39 / 338.1 = 0.01
      Ac,eff = b · hc,ef = 40 · 8.45 = 338.1 cm2
      hc,ef = min[2.5(h-d) ; (h-X)/3 ; h/2] = min[10.25 ; 8.45 ; 15] cm
    • αe = Es / Ecm = 6.09
  • εm,2 = 0.6 · σs / Es = 0.6 · 303.39 / 200000 = 0.00091

sr,msx (Maximum crack spacing) = 32.25 cm
(case spacing = 170 mm ≤ 5(c+Φ/2) = 205 mm)
sr,msx = k3·c + k1·k2·k4·Φ/ρp,eff
sr,msx = 3.4·35 + 0.8·0.5·0.425·12/0.01 = 322.46 mm
with:

  • k1 = 0.8 (high bond bars)
  • k2 = 0.5 (bending)