PSI - Issue 78

Alfonso Ferdinando Coniglio et al. / Procedia Structural Integrity 78 (2026) 1389–1395

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2.2. Definition of independent variables

As explained before, independent variables are the ones for which a choice must be made during the optimization problem. Focusing now on the bracing systems composed of modules sub-parts and considering that all the seismic action acting on the buildings is absorbed by them, it is reasonable to assume as independent variables the stiffnesses of the sub parts composing the bracing systems. Indeed, as shown in Figure 3, the bracing systems global stiffness may be represented by the series sum of the stiffness , 1 , . . . , , being K the parallel sum of in-plane stiffnesses of bracing system sub-parts at the first story, 1 . . . coefficients introduced to investigate configurations with inter-story stiffness variation. Accordingly, the vector of the independent variables = ( , 1 , . . . , ) is defined as well as the boundary conditions, by limiting into a maximum and minimum value defined by considering different technical solutions and assuming 1≥ 1 ≥ 2 . . .≥ > . This last condition is defined assuming that the stiffness of the upper story levels cannot exceed that of the lower ones, also, the maximum stiffness variation between adjacent stories is governed by . Finally, another boundary condition is imposed to the period of the modular building T , assuming that its maximum value cannot exceed a given value T .

Fig. 3. (a) Independent variables related to bracing systems; (b) Parallel sum of in-plane stiffnesses of bracing system sub-parts; (c) Series sum of the in-plane stiffnesses of each story.

2.3. Definition of objective function: Working Index To maximize the number of scenarios in which the modules sub-parts guarantee multiple performance requirements, the objective function should be defined as a function of these requirements. In this work, it is deemed reasonable to detect the inter-story drift and the base shear force as performance respectively related to the structural damage and the foundation costs. It is wort to note that if the inter-story drift control is necessary for all the structural systems, the limitation of the base shear is a performance specifically detect to modular building, for respecting a fundamental core characteristic of this type of construction i.e. the reduction of construction cost. This last performance is thus justified considering that, if the base shear force is appropriately limited, a contained foundation cost is guaranteed for all the scenarios in which the limit is satisfied. On this basis, with reference to Figure 4, considering that the seismic action acts only in the Y direction, the following functions respectively related to the drift and to the base shear, are represented by equations 1 and 2. = ( ) (1) = ( ( ) ∙ ∙ ) ⁄ (2) In those equations, ( ) and ( ) are respectively the displacement and the pseudo-acceleration response of SDOF oscillator equivalent to the building of period and mass M, is the weight of the construction calculated with seismic combination, as defined by MIT (2018).