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There is an interesting debate in Germany about how the position of the neutral fibre (xc) should be calculated to carry out the cracking analysis of a concrete cross section.
On the one hand there is a current that thinks that xc should be calculated using the standard stress analysis method on concrete and consider the cross sections that coincide with the position of the cracks (xc will be the value of x2 of the cross section A shown in the following figure):
This is equivalent to selecting the cracked concrete option on the analysis parameters:
On the other hand there is the group that believes that among the crack cross sections there are others that are uncracked and, hence, xc should be calculated with the homogenised cross section (xc will be the value of x1 on section B shown in the following figure):
This is equivalent to selecting the uncracked concrete option on the analysis parameters:
Let’s look at a simple example in which the difference between considering one option or the other is evaluated. We will carry out the crack verification for the presented beam, subjected to an axial force of 100 kN and a quasi-permanent moment of 650 kNm. The concrete is C25730 class and has a total reinforcement area of 3770 mm2. It will be supposed that the coefficient of creep is 2.63.
- If we check the ‘Cracked concrete’ option then we are considering that cross section A (the cracked one) and, therefore, we have a minor compressed zone, with a lower xc and a greater effective height of the tensile zone (hc,eff) and, thus, an also greater tensile effective area, Ac,eff.
In the results output a value of xc = 478.07 mm can be seen. It is important to know that choosing ‘cracked concrete’ is equivalent to indicating the program that we will use the values of the cross section in which there is a crack and so we will obtain poorer results (greater crack widths and smaller distances between them).
- Analysis with the Uncracked concrete option: Cross section B ➡ greater xc ➡ lower hc,eff ➡ smaller tensile zone Ac,eff ➡ lower crack width and greater distance.
In the results output a value of xc = 512.82 mm can be seen, slightly higher than in the first case, as a result of taking into account the linear behaviour of the material. This value can be easily checked in the section ‘Stresses in the homogeneous cross section (Linear Material) ‘:
10+9.5/h = 10/xc -> xc=0.5128 h = 512.8 mm
That is, considering a tensile stress of 9.5 MPa on the bottom corner fibre and a compressive stress of 10 MPa on the top corner fibre, which come from the N-M combination introduced, it all results in a neutral fibre that goes through the xc position indicated.