PSI - Issue 44

Maria Maglio et al. / Procedia Structural Integrity 44 (2023) 550–557 M. Maglio, R. Montuori, E. Nastri, V. Piluso / Structural Integrity Procedia 00 (2022) 000–000 ≤ � . (13) where is the virtual work of the external forces and the second term represents the potential virtual work developed by internal resistant bending moments . of columns. V is the external forces, δ is the displacement imposed in a kinematically acceptable yield mechanism, θ is the rotation of plastic hinges. In a force based analysis, taking into account the elastic strain energy, we have the roof top displacement under 1 equal to 1 = , where 1 is the global shear up to which the structure remains elastic and the roof top displacement demand imposed by the earthquake = , and δ is calculated as = . In a structure which fails with a soft-storey mechanism, the drift in the storey interested by the mechanism is equal to the roof displacement in the post-elastic phase. Therefore, it can be stated that: = = 1 → = , = , = ( − ) , (14) where , is the displacement at the top of the building under the calculated design shear action . The virtual work of the external force in the plastic phase in the soft storey is thus as given by = ( − ) , . For simplicity, it is considered that the representative value of the external force in the plastic phase is that is calculated as = . The internal work is evaluated at each storey for the plastic part of the chord rotation at Significant Damage limit state: 2 � . . ( ) =1 (15) n is the number of the columns. It is supposed that for soft-storey mechanism, plastic hinges develop at top and bottom of columns. . . ( ) is the plastic moment of resistance of columns taking into account the interaction of axial force. So, the soft-storey mitigation criterion to avoid soft-storey mechanism is given by Eq. (11). It is observed that this formulation comes from some preliminary assumptions that present several criticalities, for example it set: ≤ . In other words, it is stated that the virtual external work can be lower than the virtual internal work, but this statement has not correspondence in the physic principles. Moreover, the displacement at the roof top is smaller for large beams thus enlarging the gap between the first and second member of Eq. (11). Finally, no beam to column hierarchy criteria is included and no second-order effects are directly considered. It can be said that the starting point has a strong draw back because it is neither a static nor a kinematic approach. In fact, the collapse mechanism is essentially obtained deriving the load multiplier as the maximum among all the statically admissible multipliers (static approach) or as an alternative as the minimum one among all the kinematically admissible multiplier (kinematic approach). So, the Eq. (11) is not able to assure any plastic mechanism control. Moreover, the second order effects are very important when considering soft-storey mechanism because in plastic range they govern the slop of the mechanism equilibrium curve which attains a minimum value in the case of the global mechanism, and this is strongly amplified in case of storey mechanism. It means that any equation to control the occurrence of storey mechanism should take directly and explicitly into account the second order work due to the gravity loads. In addition, it is not possible to have a rule devoting to control of plastic mechanism where the plastic resistance of the beams is not explicitly considering as typically occurs when very simple beam-column hierarchy criteria are used. 3.5. Design of beams Beams should be verified to have sufficient resistance against lateral and lateral torsional buckling in accordance with prEN1993-1-1 (2004), 6. In DC2, the spacing and the resistance of lateral-torsional restraints should comply with prEN1993:1-1 (2004), 6.3.5.3. Section classes 1 and 2 should be used for beams of MRFs in DC2 and Formulae (16) should be satisfied (prEN1998-1-2, 2021): ≤ , ≤ 0.15 , ≤ 0.5 , (16) , and are the bending moment, the axial force, and the shear force, respectively, in the seismic design situation. , , , and , are the corresponding design resistance of the cross sections of the beams to be determined in accordance with prEN1993-1-1 in function of their class. 555 6