Crack Calculation - background

 

Overview

 

While the long-term deflection calculations for a slab require an advanced calculation methodology to be employed to account for non-linearity in terms of material properties and section modification due to cracking, the calculation of crack widths is assessed separately, as per the code methodology as given in Clause 7.3.4 of BS EN 1991-1-1:2004. The calculated crack widths produced from this calculation are not related to the crack ratios which are calculated as part of the deflection analysis.

The crack widths in concrete are proportional to the difference in the strain in the reinforcement and the strain in the uncracked concrete between cracks. Factors that affect the strain at service are:

The Young's modulus of the concrete (adjusted to account for creep for long term loading)

Reinforcement grade

Tension stiffening of the concrete between the cracks

Service stresses

Cover to reinforcement

Reinforcement bar diameter

The effective area of concrete

The area of reinforcing

Many of these parameters are not time dependent, but the Young’s modulus of the concrete does vary with time and has an effect on the stress in the reinforcement. The service stress will also be affected by the section properties and stiffness of the slab which will be influenced by the cracking that occurs in the concrete.

 

Calculation of crack widths

Crack widths are calculated using the methodology given in Section 7.3.4 of BS EN 1991-1-1:2004 and the UK National Annex.

The crack width wk is given by

          

                                                                                                                                                                                                                                      (6)

 

Where

          wk          crack width

          Sr,max           maximum crack spacing

                    mean strain in the reinforcement

                    mean strain in the concrete between cracks

 

The difference in the strains in the concrete and steel is calculated from the expression

 

          

                                                                                                                                                                                                        (7)

 

Where

                    stress in the tension reinforcement

                    modular ratio  (accounting for creep)

                    ratio of As/Ac,eff

                    factor depending on the load duration (kt = 0.4 for long term loading)

 

The maximum crack spacing is given by

          

                                                                                                                                                                                                                                       (8)

 

Where

                    the reinforcing bar diameter

          c           cover to the longitudinal reinforcement

                    coefficient for reinforcement bond ( 0.8 for high bond reinforcement)

                    coefficient for distribution of strain

                    UK NA defined parameter with value 3.4

                    UK NA defined parameter with value 0.425

 

In the case that the reinforcement is composed of a mixture of bar diameters, an equivalent diameter is used in place of , which is given by

          

                                                                                                                                                                                                                                                (9)

 

For the calculation of crack width in reinforced concrete slabs where the stress and strains are dependent on the stress state in two directions, the stresses are converted to principal stresses. Where the axes of the principal stresses are significantly different from the local axes of the reinforcement, the crack spacing calculation needs to be modified to account for the differences in the orientation of the axes. Clause 7.3.4(4) defines the difference in principal and reinforcement axes to be significant when the difference is greater than 15 degrees. In this case, the maximum crack spacing is then given by

          

                                                                                                                                                                                                                                      (10)

 

Where

θ            the angle between the reinforcement in the y-direction and the axis of principal tensile stress

                    crack spacing in the x- and y-directions

 

Calculating crack widths using the FE analysis

 

A key component in the calculation of crack widths is the determination of the service stress in the reinforcement. With the use of a finite element analysis, it is possible to more accurately determine the stress in the reinforcement at any point in the slab, which can also take account of the effects of creep, cracking and shrinkage strains on the slab, by using the results of the non-linear analysis used for the determination of the long-term deflections. The effects of axial forces can also be explicitly taken into account. By utilising the non-linear analysis, the impact of early loading cracking, which affects the stiffness of the structure and so the distribution of stress in the slab, is also taken into account.

As part of the deflection calculations, the stresses within each finite element are used to determine if the section has cracked, with the extent of cracking between uncracked and fully cracked determined from Eq (7.18) of BS EN 1992-1-1:2004. In effect, this determines the depth of cracking and is used to determine the modified section properties of each finite element. However, this analysis does not identify the position or size of cracks and so is separate from the calculation of crack widths as determined from Section 7.3.4 Calculation of crack widths of BS EN 1992-1-1:2004.