## Cracked stiffness in concrete and masonry

Clauses 4.3.1(6), A fundamental assumption underlying the provisions of Eurocode 8 for design for energy

4.3.1 (7) dissipation and ductility is that the global inelastic response of a structure to monotonic lateral forces is bilinear, close to elastic-perfectly-plastic. The elastic stiffness used in analysis should correspond to the stiffness of the elastic branch of such a bilinear global force-deformation response. This means that the use of the full elastic stiffness of uncracked concrete or masonry in the analysis is completely inappropriate. For this reason, Section 4 of EN 1998-1 requires that the analysis of concrete, composite steel-concrete or masonry buildings should be based on member stiffness, taking into account the effect of cracking. Moreover, to reflect the requirement that the elastic stiffness corresponds to the stiffness of the elastic branch of a bi-linear global force-deformation response, Section 4 of EN 1998-1 also requires that the stiffness of concrete members corresponds to the initiation of yielding of the reinforcement. Unless a more accurate modelling of the cracked member is performed, it is permitted to take that stiffness as equal to 50% of the corresponding stiffness of the uncracked member, neglecting the presence of the reinforcement. This default value is quite conservative: the experimentally measured secant stiffness of typical reinforced concrete members at incipient yield, including the effect of bar slippage and yield penetration in joints, is on average about 25% or less of that of the uncracked gross concrete section.53 The experimental values are in good agreement with the effective stiffness specified in Eurocode 2 for the calculation of second-order effects in concrete structures:

• a fraction of the stiffness ECIC of the uncracked gross concrete section equal to 20% or to 0.3 times the axial load ratio vi = N/Acfcd, whichever is smaller, plus the stiffness ESIS of the reinforcement with respect to the centroid of the section, or

• if the reinforcement ratio exceeds 0.01 (but its exact value may not be known yet), 30% of the stiffness ECIC of the uncracked gross concrete section.

When an estimate of the effective stiffness on the low side is used in the analysis, second-order effects increase, which is safe-sided in the context of Eurocode 2. In contrast, within the force- and strength-based seismic design of Eurocode 8 it is more conservative to use a high estimate of the effective stiffness, as this reduces the period(s) and increases the corresponding spectral acceleration(s) for which the structure has to be designed. The use of 50% of the uncracked section stiffness serves exactly that purpose. However, lateral drifts and P-A effects computed on the basis of overly high stiffness values may be seriously underestimated.

Torsion in beams, columns or bracings is almost immaterial for their earthquake resistance. In concrete buildings the reduction of torsional rigidity when the member cracks diagonally is much larger than that of shear or flexural rigidity upon cracking. The effective torsional rigidity, GCei, of concrete members should be assigned a very small value (close to zero), because torsional moments due to deformation compatibility drop also with torsional rigidity upon cracking, and their overestimation may be at the expense of member bending moments and shears, which are more important for earthquake resistance. The reduction of member torsional rigidity should not be effected through reduction of the concrete G value, as this may also reduce the effective shear stiffness GAsh, and unduly increase member shear deformations.

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