PSI - Issue 78

570 Alessandro Pisapia et al. / Procedia Structural Integrity 78 (2026) 568–575 vertical frames, is the in-plane flexural stiffness of the diaphragm and is the lateral stiffness of the vertical frames.

Fig. 1. Theoretical model of a diaphragm with three vertical seismic-resisting frames.

According to Fig.1, it is well known that, under the assumption of the rigid diaphragm, the displacements of the vertical frames are the same and they are equal to: ∞ = 2 3 (1) In order to evaluate the influence of the ratio between the in-plane flexural stiffness of the floor system and the lateral stiffness of the vertical frames, the following relations can be derived through some mathematical steps: ∞ = 3 4 � 1 − 1 4 1 − 12 1+3 +9 � ∞ = 3 2 � 1+ 1 4 1 − 12 1+3 +9 � (2) where: = 3 = 2 (3) The first one represents the relative flexural stiffness of the in-plane floor system, while the second one is the ratio between the flexural stiffness and the shear stiffness. In fact, is the tangential modulus, represents the cross sectional area of the diaphragm and is the shear factor. It is worth noting that the expressions given in Eq. (2) also correspond to the ratio between the seismic force acting on a vertical seismic-resisting system, accounting for the in-plane flexural stiffness of the diaphragm, and the seismic force that would act on the same macro-element in the case of an infinitely rigid diaphragm. In other words, these expressions quantify the influence of the diaphragm’s relative in-plane flexural stiffness on the distribution of seismic forces among the vertical seismic-resisting systems. Fig. 2 shows the influence of the in-plane flexural stiffness of the diaphragm on the seismic effects of the vertical frames. It is immediate to recognize that when increases ∞ ⁄ and ∞ ⁄ values tend to 1.00. Moreover, for a given value of the non-dimensional stiffness , it is observed that the aforementioned deviation increases with increasing non-dimensional shear deformability , which therefore plays an important role in defining a threshold value for the dimensionless stiffness .