Cracking

AdSec calculates the onset of #cracking and, where applicable, #crack-width for reinforced concrete sections. This page describes how these calculations are performed and the approaches used for the supported design codes.

• Cracking moment
• Crack width
  • EN1992-1-1:2004
  • EN1992-1-1:2004+PD6687:2010
  • BS8110 (and Hong Kong Code of Practice 2013)
  • BS5400, IRS Bridge and HK Structures Design Manual
  • CSA-S6

Cracking moment

The programme calculates the total axial force and moment from the loads in the analysis case definition. It uses the axial force and the angle of the applied moment to define the cracking moment analysis task. The programme then searches for a strain plane that gives the cracking strain as shown below.

Integrating the stresses from this strain plane over the section will give an axial force equal to the applied force, and a moment which is parallel to the applied moment. The value of this moment is the cracking moment. Short-term material properties are always used for the cracking moment calculation.

Crack width

EN1992-1-1:2004

Code approach

The code approach considers the section as a whole from which a single crack width is derived. It is really only suitable for rectangular sections subject to uniaxial forces.

The expression for the crack width is expression 7.8

$$w_k=s_{r,max}(\varepsilon_{sm}-\varepsilon_{cm})$$

In this $s_{r,max}$ is given by expression 7.11 when centre to centre bar spacing is less than $5(c + \phi/2)$

$$s_{r,max}=k_3c+k_1k_2k_4\phi/\rho_{p,eff}$$

The code gives values for $k_1$ as 0.8 for high bond bars and 1.6 for plain bars or pre-stressing tendon. The code gives values for $k_2$ as 0.5 for bending and 1.0 for pure tension. In AdSec we interpolate between these values using

$$k_c=\frac{(\varepsilon_{max}+max(\varepsilon_{min},0))}{2\times max (|\varepsilon_{max}|,|\varepsilon_{min}|)}$$

But limited to the range [0.5:1]. $k_3$ and $k_4$ are nationally determined parameters which default to 3.4 and 0.425 respectively.

Where the spacing of bars is large, $>5(c+\phi/2)$, then expression 7.14 applies for $s_{r,max}$:

$$s_{r,max}=1.3(h-x)$$

$\rho_{p,eff}$ is the ratio of reinforcement to concrete in the cracking zone where the area of concrete is a function of $h_{c,ef}$ defined as follows: for a rectangular section subject to uniaxial loads:

$$h_{c,ef}=min(2.5(h-d),(h-x)/3,(h/2))$$

For a rectangular section subject to uniaxial loads then:

$$A_{c,eff}=bh_{c,ef}$$

A bar is only considered in $\rho_{p,eff}$ if the centre of the bar falls inside $h_{c,ef}$. Hence, reinforcement is either "in" or "out" of $A_{c,eff}$ so slight shifts in neutral axis or section geometry can cause sudden jumps in crack width.

$(\varepsilon_{sm}-\varepsilon_{cm})$ is defined by expression 7.9

$$\varepsilon_{sm}-\varepsilon_{cm}=\frac{\sigma_s-k_t\frac{f_ct{,eff}}{\rho_{p,eff}}(1+\alpha_e\rho_{p,eff})}{E_s}\ge0.6\frac{\sigma_s}{E_s}$$

If no creep factor is defined then $k_t$ is taken as 0.6, else as 0.4 as long term loading is then assumed.

Local approach

When the local check is carried out, AdSec finds the two bars (or tendons) with the highest strain (excluding prestress strain). If more than two bars have the same strain then the two with the lowest bar numbers are selected. A local value for $A_{c,eff}$ is allocated to each of these bars, which allows a value of $\rho_{p,eff}$ to be calculated and used to predict the crack width. The $A_{c,eff}$ areas allocated to the bars are plotted on the section view.

For each local area, $h_{c,ef}$ considers the code-based limits, with the following additional constraints:

• $(h-d)$ replaced by $(c+\phi/2)$
• upper limit of $(c+\phi/2)$ + (vertical spacing)/2 when more than one layer of bars present

The width of each zone is calculated as a minimum of the following three values for each side bar:

1. $2.5(c + \phi$/2)$
2. the distance to the edge of the section in this direction
3. half of the spacing to the next bar in this direction

$k_t$ is always taken as 0.4 for long term loading.

$s_{r,max}$ is always taken from expression 7.11

$$s_{r,max}=k_3c+k_1k_2k_4\phi/\rho_{p,eff}$$

The remainder of the calculation is as per the Code Approach.

EN1992-1-1:2004 + PD6687-1:2010 cl.2.22

This is the option implemented if the National Annex is selected as "PD6687-1 (2010)".

PD6687-1: 2010 provides a method under cl.2.22 for calculating crack widths for non-rectangular tension zones and irregular bar layouts. Crack widths are based on the equation below, with concrete tensile stress assumed to vary linearly from zero at the neutral axis to 0.7 $N/mm^2$. PD6687-1: 2020 provides the same method under cl.4.21.4.

$$w_k=\{3\alpha_{cr}\varepsilon_m/[1+2(\alpha_{cr}-c)/(h-x)]\}$$

where

$\alpha_{cr}$ is the distance from the point calculated to the surface of the nearest longitudinal bar;

$\varepsilon_m$ is the average strain at the level at which cracking is determined.

Hong Kong Code of Practice 2013 (and BS8110)

Crack width formulae to BS8110 are based on a weighted interpolation between two effects. Close to a bar, crack width is a function of the bar cover, $c_{min}$. Between bars, it is a function of the depth of tension zone, $h-x$.

Crack widths can be calculated to BS8110 using any of the three available tension stiffening options.

After calculating the cracking moment, AdSec will search for a strain plane which gives forces and moments within tolerance of the applied forces and moments. The resulting strain distribution is used to calculate the crack width.

The maximum crack width output is related to the given resultant moment orientation. This is particularly important for circular sections, as the maximum strain may not occur between two bars.

This would give a lower crack width value than may occur in reality.

The sides of the section are divided into small segments and the crack width calculated for each segment. The crack width formula

$$w=\frac{3 a_{c r} \varepsilon_m}{1+2\left(\frac{a_{c r}-c_{\min }}{h-x}\right)}$$

is given in BS8110, section 3.8, equation 12.

Crack width calculations involve a large amount of engineering interpretation for faceted sections, sections with voids, sections with re-entrant corners, and multi-zone sections. Depending on the situation, a different definition of cover is required.

The programme stores the minimum cover to each bar and uses this in the calculation of crack width. This means that the cover used in the calculation may not relate to the side being checked (it will always give a conservative result). The reason for this approach is that curved sections are analysed as a multifaceted polygon, and there may be no bars present parallel to a small facet. This is because the number of facets may be greater than the number of bars.

The crack width calculation is done on a zone by zone basis using the zonal strain plane (resulting strain plane + component strain plane + concrete only component strain plane). This component strain plane applied to the whole section is used to calculate neutral axis depth and section height (x and h) relative to the whole section -- using all the section coordinates.

The crack width includes the term $(a_{c r}-c_{\min })$. If $c_{\min}$ is smaller than $a_{c r}$ the crack width is increased.

For each division on the concrete outline the closest bar is found (minimum $a_{c r}$). For a re-entrant corner, and a bar which is on the outside of the section with reference to the side being checked a warning flag is generated, a conservative crack width can still be calculated using the minimum cover to the bar.

If the cover is greater than half the depth of the tension zone, the crack width in both codes is invalid.

The term for concrete only cracking should be used instead. This is $1.5(h-x)\varepsilon_m$. This is included in AdSec.

These are warnings in the crack width calculation. They do not necessarily mean that the answer is incorrect, but that the graphical results should be checked for engineering interpretation.

• $c_{\min}$ < controlling bar diameter -- crack width not valid

• Controlling bar is remote from crack location

• Controlling bar and crack are located on either side of re-entrant corner

• Cover to controlling bar measured to different side from crack location

BS5400, Hong Kong Structures Design Manual and IRS Bridge Code

The BS8110 specification above is valid for BS5400 plus some additional points.

BS5400 includes an adjustment for effective tension stiffening. So for crack widths to be correct using the BS5400 formula, SLS analysis must use no tension stiffening in calculation of the strains around the section.

This adjustment requires calculation of the level of tension steel. This is re-calculated on a component by component basis using a similar method to BS8110-2 tension stiffening.

The tension steel is identified as the steel which is in the tension zone when the zonal strain plane is extended across the whole section (the zonal strain plane is the resulting strain plane + zonal locked in plane + zonal concrete only plane). From the steel bars identified, the centroid of steel force is calculated using the actual stress in the bars ignoring prestress and ignoring any bars in compression.

Once the level of centroid of tension steel is found, the width term $b_t$ needs to be calculated. This includes interrogation of all the section coordinates (as for x and h). If more than 2 sides cross the level of centroid of tension steel, the width is taken as the distance between the two extreme dimensions.

This needs to be checked for sections with large voids, or channels with thin legs, as the term $b_t\times h$ is used to make an approximation for the force in the tension zone and assess the area of tension steel versus the area of tension concrete. It may be appropriate to substitute a smaller value of $b_t$.

BS5400 includes a notional surface a distance $c_{nom}$ from the bars. AdSec will look at all bars to define this surface excluding any with negative cover. This should be reviewed, particularly for sections with sharp acute angles and re-entrant corners.

In the example below the adjustment to sides A and C may not be the adjustment that would be chosen by engineering judgement. In this situation, the cracking parameters output for the relevant sides can be extracted from the output, and the results recalculated, substituting the corrected values.

The crack width equations in BS5400-4, which are offered by AdSec, are either equation 24

$$w=\frac{3 a_{c r} \varepsilon_m}{1+2(a_{c r}-c_{nom})/{(h-d_c)}}$$

where the strain $\varepsilon_m$ is given by equation 25

$$\varepsilon_m=\varepsilon_1-\left[\frac{3.8 b_t h\left(a'-d_c\right)}{\varepsilon_s A_s\left(h-d_c\right)}\right]\left[\left(1-\frac{M_q}{M_g}\right) \times 10^{-9}\right]$$

or the alternative equation 26

$$w={3 a_{c r} \varepsilon_m} \\$$

CSA S6

Code approach

The equation for the crack width is given in section 8.12.3.2 as

$$w=k_b\beta_cs_{rm}\varepsilon_{sm}$$

The parameters $k_b$ and $\beta_c$ depend on the section and the cause of cracking. The code defines $s_{rm}$ in mm as

$$s_{rm}=50+0.25k_c \frac{d_b}{\rho_c}$$

Where

$$\rho_c=\frac{A_s}{A_{ct}}$$

and $A_{ct}$ is the concrete area excluding the reinforcement. The code gives values for $k_c$ as 0.5 for bending and 1.0 for pure tension. In AdSec we interpolate between these values using

$$k_c=\frac{(\varepsilon_{max}+max(\varepsilon_{min},0))}{2 \times max(|\varepsilon_{max}|,|\varepsilon_{min}|)}$$

But limited to the range [0.5:1].

The $d_b$ term is taken as the average bar diameter in the tension zone, and the area of steel in $\rho_c$ is taken as the weighted area of the bar in the tension zone

$$A_s=\Sigma \space a_i \frac{\pi \space d_i^2}{4}$$

And the weighting is based on the stress in the bar compared with the stress in the extreme bar.

$$a=\frac{\sigma_i}{\sigma_{extreme}}$$

The area of concrete in $\rho_c$ is

$$A_c=min(A_c(2.5h_t),A_c(h_b/3))$$

where c is the centroid of reinforcement in tension. The strain term is given in the code as

$$\varepsilon_{sm}=\frac{f_s}{E_s}\Bigr[1-\Bigl(\frac{f_w}{f_s}\Bigl)^2\Bigr]$$

Where $f_s$ is the stress in the reinforcement at the serviceability limit state and $f_w$ is the stress in the reinforcement at initial cracking. In AdSec this is implemented as

$$\varepsilon_{sm}=\frac{\sigma_s}{E_s}\Bigr[1-\Bigl(\frac{f_{c,rup}}{\sigma_{ct}}\Bigl)^2\Bigr]$$

Where $\sigma_s$ is the stress in the most tensile reinforcement at the serviceability for a fully cracked calculation, $f_{c,rup}$ is the rupture strength of the concrete and $\sigma_{ct}$ is the maximum tensile stress in the concrete at the extreme fibre assuming an uncracked material.

Local approach

The above approach becomes difficult to justify when the section is not in uniaxial bending. In these cases the alternative local approach can be used. This assumes there is a local relationship between a bar and the surrounding concrete.

The first stage is to identify the most tensile bar and determine the cover $c$ to this bar. We then define $h_t$ as the cover plus half the bar diameter.

$$h_t=c+\frac{d_b}{2}$$

The depth from the neutral axis the most tensile bar $b$ is calculated, and $h_b$ is then defined as

$$h_b=b+c+\frac{d_b}{2}$$

Then the concrete area is based on a dimension

$$h_c=min(2.5h_t,h_b/3,(h_t+s_v/2))$$

The width associated with this is

$$w_c=min(5h_t,(w_1+w_2))$$

So that

$$A_c=h_c\times w_c$$