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In many occasion we have received queries on the topic of obtaining the cracking moment and the cracked inertia of a concrete cross section in FAGUS.
The obtention of these results is necessary for the calculation of the instantaneous deflections on cracked elements with constant cross section:
p = linear load of the permanent actions
l = length of the beam
E = elastic modulus of concrete
I = Ie = equivalent inertia
The value of the instantaneous deflection that is used, for example, for the calculation of the natural frequency of beams that, takes part at the same time in the calculation of the impact coefficient according to the IAPF-07.
The calculation of the natural frequency is carried out by the following formula:
In the calculation of the equivalent inertia is where the cracking moment and the cracked inertia come are needed, being part of the formula below (art. 50.2.2. EHE-08):
Mf = Mcrk = Nominal cracking moment of the cross section
Ma = Maximum moment applied to the characteristic combination
Ib = Inertia of the gross section
If = Icrk = Inertia of the cracked section
The obtention of Mf and If is not straightforward in FAGUS but it is fairly simple to calculate both parameters from values that can be obtained immediately from this program.
- CRACKING MOMENT
The cracking moment of a concrete cross section is defined as:
Mcrk = fct,m * Ih/yh
fct,m = average tensile strength of concrete.
Ih = inertia of the homogenised cross section.
yh = distance from the neutral axis of the homogenised cross section to the most stressed fibre.
Taking into account that fct,m is given (it depends on the concrete, the cross section is made with):
Lastly we have to see where FAGUS gives us the mechanical characteristics of the homogenised cross section.
Once we have introduced the geometry of the cross section and its reinforcements, we must go to the ‘variants’ tab and access the third tab on the dialog box of the material properties.
Clicking on the highlighted option on the previous figure (taking the reinforcements into account), we indicate the program that we want to obtain the homogenised characteristics of the cross section instead of the gross ones that we would obtain had we not checked that option.
Accessing the input legend, the values we will need will be available.
NOTE: zs is the distance from the y axis to the centre of gravity, therefore, if the base of our cross section does not meet with the line z=0, this value will not coincide with yh, instead, that distance will have to be measured graphically. For example, for a positive moment where the most stress fibre would be the inferior one:
- CRACKED INERTIA
For the computation of the secant stiffnesses, Neither the value of E (elastic modulus of concrete) nor the cross section inertia (I) are taken into account by FAGUS. The stiffnesses are obtained from the forces that act (N,M) and the strains (ε, χ).
Nevertheless, if any of the forces is zero, those parameters are taken into account and the tangent stiffness is obtained.
Looking at the second equation it seems obvious that to obtain the crack inertia one simply has to divide the value of the stiffness for the crack section given by FAGUS by the elastic modulus of concrete, that is:
Icrk = (My/χy) / E
The value of the elastic modulus that must be taken is the one defined in the ‘material classes’ table.
The stiffness My/χy can be found on the numerical results output for a stress analysis given the forces, defining, in the analysis parameters, a behaviour of concrete at a type 0 tension (without any tensile strength):
NOTE: the value of the stiffness depends on the applied moment. This implies that, if we exceed the value of the moment that make the reinforcement in tension to reach its elastic limit (Point A in the figure), the stiffness we will obtain will not be the cracked one but the one that is obtained when that reinforcement is in the plastic regime. When this occurs, the strain of the reinforcement increases without an increase in stress until it reaches its yield strength (point B), which is specified in the analysis parameters.
Having said that, to obtain the cracked stiffness of the cross section lower moments must be entered on point A.
To sum up: