Crack Width Calculation Euro Code 2 Worked Examples

7.3.2 Minimum reinforcenlent areas

1. Euro Code Rocklin
2. Euro Code 3
3. Concrete Design To Euro Code 2 Worked Examples

(1)P If crack control is required, a minimum amount of bonded reinforcement is required to control cracking in areas where tension is expected. The amount may be estimated from equilibrium between the tensile force in concrete just before cracking and the tensile force in reinforcement at yielding or at a lower stress if necessary to limit the crack width.

(2) Unless a more rigorous calculation shows lesser areas to be adequate, the required minimum areas of reinforcement may be calculated as follows. In profiled cross sections like Tbeams and box girders, minimum reinforcement should be determined for the individual parts of the section (webs, flanges).

Calculation of crack width and crack spacing. The residual tensile strength which is used for crack width calculations should be obtained from tests. The book contains numerous examples. European Commission Joint Research Centre Institute for the Protection and Security of the Citizen Contact information Address: Joint Research Centre, Via.

where:

• A_s,min is the minimum area of reinforcing steel within the tensile zone
• A_ct is the area of concrete within tensile zone. The tensile zone is that part of the section which is calculated to be in tension just before formation of the first crack
• σ_s is the absolute value of the maximum stress permitted in the reinforcement immediately after formation of the crack. This may be taken as the yield strength of the reinforcement, f_yk. A lower value may, however, be needed to satisfy the crack width limits according to the maximum bar size or spacing (see 7.3.3 (2))
• f_ct,eff is the mean value of the tensile strength of the concrete effective at the time when the cracks may first be expected to occur:
  f_ct,eff = f_ctm or lower, (f_ctm(t)), if cracking is expected earlier than 28 days
• k is the coefficient which allows for the effect of non-uniform self-equilibrating stresses, which lead to a reduction of restraint forces
  = 1,0 for webs with h ≤ 300 mm or flanges with widths less than 300 mm
  = 0,65 for webs with h ≥ 800 mm or flanges with widths greater than 800 mm
  intermediate values may be interpolated
• k_c is a coefficient which takes account of the stress distribution within the section immediately prior to cracking and of the change of the lever arm:
  For pure tension k_c = 1,0
  For bending or bending combined with axial forces:
  - For rectangular sections and webs of box sections and T -sections:
  
  (7.2)
  
  - For flanges of box sections and T -sections:
  
  (7.3)
  
  where
  • σ_c is the mean stress of the concrete acting on the part of the section under consideration:
    
    (7.4)
  • N_Ed is the axial force at the serviceability limit state acting on the part of the cross-section under consideration (compressive force positive). N_Ed should be determined considering the characteristic values of prestress and axial forces under the relevant combination of actions
  • h* = h for h < 1,0 m
    h* = 1,0 m for h ≥ 1,0 m
  • k₁ is a coefficient considering the effects of axial forces on the stress distribution:
    k₁ = 1,5 if N_Ed is a compressive force
    k₁= 2h*/(3h) if N_Ed is a tensile force
  • F_cr is the absolute value of the tensile force within the flange immediately prior to cracking due to the cracking moment calculated with f_ct,eff

9.2 Beams

9.2.1 Longitudinal reinforcement

9.2.1.1 Minimum and maximum reinforcement areas

(1) The area of longitudinal tension reinforcement should not be taken as less than A_s,min.

Note 1: See also 7.3 for area of longitudinal tension reinforcement to control cracking.

Note 2: The value of A_s,min for beams for use in a Country may be found in its National Annex. The recommended value is given in the following:

where:

• b_t denotes the mean width of the tension zone; for a T-beam with the flange in compression, only the width of the web is taken into account in calculating the value of b_t
• f_ctm should be determined with respect to the relevant strength class according to Table 3.1:
  f_ctm = 0,30 × f_ck^(2/3), f_ck ≤ 50
  f_ctm = 2.12·Ln(1+(f_cm/10)), f_ck > 50/60
  with f_cm = f_ck+8 (MPa)

(2) Sections containing less reinforcement than A_s,min should be considered as unreinforced.

(3) The cross-sectional area of tension or compression reinforcement should not exceed A_s,max outside lap locations.

Note: The value of A_s,max for beams for use in a Country may be found in its National Annex. The recommended value is 0.04·A_c.

9.3 Solid slabs

(1) This section applies to one-way and two-way solid slabs for which b and l_eff are not less than 5h (member for which the minimum panel dimension is not less than 5 times the overall slab thickness).

9.3.1 Flexural reinforcement

9.3.1.1 General

(1) For the minimum and the maximum steel percentages in the main direction 9.2,1,1 (1) and (3) apply.

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(2) Secondary transverse reinforcement of not less than 20% of the principal reinforcement should be provided in one way slabs. In areas near supports transverse reinforcement to principal top bars is not necessary where there is no transverse bending moment.

(3) The spacing of bars should not exceed s_max,slabs.

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Note; The value of s_max,slabs for use in a Country may be found in its National Annex. The recommended value is:

- for the principal reinforcement, 3·h ≤ 400 mm, where h is the total depth of the slab;
- for the secondary reinforcement, 3.5·h ≤ 450 mm

In areas with concentrated loads or areas of maximum moment those provisions become respectively:
- for the principal reinforcement, 2·h ≤ 250 mm
- for the secondary reinforcement, 3·h ≤ 400 mm.

9.5 Columns

(1) This clause deals with columns for which the larger dimension h is not greater than 4 times the smaller dimension b.

http://lepetkowsnajs1972.mystrikingly.com/blog/add-a-blog-post-title. 9.5.1 General

9.5.2 Longitudinal reinforcement

(1) Longitudinal bars should have a diameter of not less than Φ_min.

Note: The value of ¢min for use in a Country may be found in its National Annex. The recommended value is 8 mm.

(2) The total amount of longitudinal reinforcement should not be less than A_s,min

Note: The value of A_s,min for use in a Country may be found in its National Annex. The recommended value is given by Expression (9.12N)

where:

• f_yd is the design yield strength of the reinforcement
• N_Ed is the design axial compression force

(3) The area of longitudinal reinforcement should not exceed A_s,max

Note: The value of A_s,max for use in a Country may be found in its National Annex. The recommended value is 0.04·A_c outside lap locations unless it can be shown that the integrity of concrete is not affected, and that the full strength is achieved at ULS. This limit should be increased to 0.08·A_c at laps.

(4) For columns having a polygonal cross-section, at least one bar should be placed at each corner. The number of longitudinal bars in a circular column should not be less than four.

9.6 Walls

9.6.1 General

(1) This clause refers to reinforced concrete walls with a length to thickness ratio of 4 or more and in which the reinforcement is taken into account in the strength analysis

9.6.2 Vertical reinforcement

(1) The area of the vertical reinforcement should lie between A_s,vmin and A_s,vmax.

Note 1: The value of A_s,vmin for use in a Country may be found in its National Annex. The recommended value is 0.002·A_c.

Note 2: The value of A_s,vmax for use in a Country may be found in its National Annex. The recommended value is 0.04·Ac outside lap locations unless it can be shown that the concrete integrity is not affected and that the full strength is achieved at ULS. This limit may be doubled at laps.

(2) Where the minimum area of reinforcement, A_s,vmin, controls in design, half of this area should be located at each face.

(3) The distance between two adjacent vertical bars shall not exceed 3 times the wall thickness or 400 mm whichever is the lesser.

9.6.3 Horizontal reinforcement

(1) Horizontal reinforcement running parallel to the faces of the wall (and to the free edges) should be provided at each surface. It should not be less than A_s,hmin.

Note: The value of A_s,hmin for use in a Country may be found in its National Annex. The recommended value is either 25% of the vertical reinforcement or 0.001·A_c, whichever is greater.

(2) The spacing between two adjacent horizontal bars should not be greater than 400 mm.

9.8 Foundations

9.8.1 Column and wall footings

(1) A minimum bar diameter Φ_min should be provided

Note: The value of Φ_min for use in a Country may be found in its National Annex. The recommended value is 8 mm.

Eurocode 2 part 1-1: Design of concrete structures 7.3 Crack control

The crack width, w_k, may be calculated as follows:

where:

s_r,max

is the maximum crack spacing

ε_sm

is the mean strain in the reinforcement under the relevant combination of loads, including the effect of imposed deformations and taking into account the effects of tension stiffening

ε_cm

is the mean strain in the concrete between cracks.

where:

with

E_cm

the secant modulus of elasticity of concrete

with

with

ξ

the ratio of bond strength of prestressing and reinforcing steel, according to Table 6.2

Φ_S

the largest bar diameter of the reinforcing steel

Φ_P

the diameter or equivalent diameter of prestressing steel:
Φ_p = 1,6⋅√A_P for bundles, where A_P is the area of a prestressing steel,
Φ_p = 1,75⋅Φ_wire for single 7 wire strands,
Φ_p = 1,20⋅Φ_wire for single 3 wire strands, where Φ_wire is the wire diameter.

• Where the bonded reinforcenlent is fixed at reasonably close centres within the tension zone (spacing ≤ 5(c + Φ/2), cf. Figure 7.2), the maximum crack spacing s_r,max may be calculated as follows:

where:

Euro Code Rocklin

where ε₁ is the greater and ε₂ is the lesser tensile strain at the boundaries of the section considered, assessed on the basis of a cracked section.

Euro Code 3

• Where the spacing of the bonded reinforcement exceeds 5(c + Φ/2) (cf. Figure 7.2), or where there is no bonded reinforcement within the tension zone, the maximum crack spacing s_r,max may be calculated as follows:

where:

h

is the overall depth of the section (see Figure 7.1)

x

is the neutral axis depth of the section (see Figure 7.1).

This application calculates the crack width w_k from your inputs. Intermediate results will also be given.

First, change the following option if necessary:

Concrete Design To Euro Code 2 Worked Examples