PSI - Issue 78

Raffaele Laguardia et al. / Procedia Structural Integrity 78 (2026) 678–685

681

where λ CP and λ ∗ CP are the MAF of collapse limit state of the retrofitted building and its acceptable threshold, respec tively.

3. Numerical Example

The design procedure exposed in Section 2 has been applied to low code Reinforced concrete building already studied in Laguardia and Franchin (2022). Figure 1 shows the plan of the building and two peripheral Frames where the braces are placed (i.e., Frame 1 and Frame D). The floor plan of the building is 330 m 2 whereas the height of each storey is 3.5m with the exception of the first storey whose height is 4.3m, moreover the columns are tapered between third and fourth storeys, further details on the structural characteristics are given in Laguardia and Franchin (2022). In order to assess the environmental and economic losses due to seismic events there is the need to define its components and their related fragilities and consequence functions. Table 1 resumes the quantity of the components along with the references for fragility and consequence functions adopted for each of them. Please note that the quantities are subdivided according to their EDP sensitivity, thus considering components sensitive to IDR along X or Y axis (i.e., X and Y columns in Table 1) and component sensitive to PFA (i.e., XY column in Table 1). As far as the cost of demolition and reconstruction of the building is concerned, for the economic part we adopted the values proposed in Cardone and Perrone (2017) equal to 730 e / m 2 for reconstruction 44 e / m 3 for demolition; whereas for the environmental part we used the value of 577kgCO 2 eq / m 2 according to the value provided for an italian multi-storey o ffi ce by Fenner et al. (2018) and Asdrubali et al. (2013). The demand dispersions have been considered equal to 0.6 and 0.5 for economic and environmental part, respectively, whereas the capacity dispersion has been considered null. For the λ CP assessment instead, demand and capacity dispersions have been considered equal to 0.3 and 0.5, respectively. The optimization problem of Eq.6 is solved by the means of a Global Pattern Search (GPS) optimization Algorithm (Torczon, 1997) with no ”Opportunistic Strategy” implemented. Ten Independent Variables (IV) are considered in the problem, that are the area of dissipative devices (i.e., A D ) shown in Figure 1. For the IV formalization some further details are given in Laguardia et al. (2023). Figure 2 (A) shows the values of the OF and the λ CP (i.e., the MAF of collapse adopted as constraint) along the optimization or, in other words, against function counts (FCs). It can be noticed that λ CP increases until it reaches values around the constraint threshold (i.e., λ ∗ C = 2 × 10 − 4 ). On the contrary, the OF decreases uniformly during optimization and only in the last iterations its value increases, probably due to the need of satisfying the constraint within tolerance. Figure 2(B) shows the IV values (i.e., the dissipative area of braces, A D ) with the function counts, it can be noticed that almost all the variables reduce their values along optimization, thus consistently with the reduction of the OF exposed in Figure2(A). Figure 3 shows the initial values of braces (i.e., d [1] ) and the final values of them at the end of optimization (i.e., d [ f ] ), it can be noticed how, the distribution along the height is di ff erent between the two directions and highly di ff erent from initial one. Figures 4 (A) and (B) show the values of the di ff erent components of LCCI along optimization for economic part and environmental part, respectively. Looking at Figure 4 (A) it can be noticed that C I is significantly higher than C S . Figure 4 (A) shows also the components of C I (i.e., Steel, BRB devices, Infills, Foundation) and it can be noticed that

Table 1. Component List considered for the case study structure: Quantities, fragility and consequence functions adopted. Component Type U.o.m. X Y XY Fragility Economic

Environmental

Internal Joint External Joint

each 20 3 each 7 6

0 Cardone and Perrone (2017) Cardone and Perrone (2017) Aljawhari et al. (2024) 0 Cardone and Perrone (2017) Cardone and Perrone (2017) Aljawhari et al. (2024)

m 2 m 2

0 21 0 Cardone and Perrone (2017) Del Vecchio et al. (2020) 166 19 0 Cardone and Perrone (2017) Del Vecchio et al. (2020) 166 195.2 0 Cardone and Perrone (2017) Del Vecchio et al. (2020)

Aljawhari et al. (2024) Aljawhari et al. (2024) Aljawhari et al. (2024) Aljawhari et al. (2024) Hammond G. (2019) Hammond G. (2019) Aljawhari et al. (2024) Hammond G. (2019)

Infillw / oopening Infill w opening

Partition w / o opening m 2 Partition w opening m 2

166 0 0 0 0 0

0 Cardone and Perrone (2017) Del Vecchio et al. (2020)

m 2 m 2 m 2

330 FEMA P-58 C3034.002 330 FEMA P-58 C3032.001a

Abruzzo (2023) Abruzzo (2023)

Lighting Ceilings

Pipes / Electric

165 165 0 Cardone and Perrone (2017) Del Vecchio et al. (2020)

Sprinkler System m 0 0

216 FEMA P-58 D4011.021a

Abruzzo (2023)