PSI - Issue 62

Rossella Venezia et al. / Procedia Structural Integrity 62 (2024) 796–808 Rossella Venezia and Alessio Lupoi / Structural Integrity Procedia 00 (2019) 000 – 000

798

3

= 1 1 = ∑ ∑

2 = ∗ ∑

2

(2) Note that Γ depends on the choice of a displacement shape Φ , that can be any reasonable shape (Fajfar and Eeri 2000). Moreover, the N2 procedure requires the approximation of the capacity curve to a simplified elastic-perfectly plastic curve. The elastic period of the idealized bilinear system T ∗ is given by Eq. (3), where F y∗ and D y∗ are the yield strength and displacement, respectively. ∗ =2 √ ∗ ∗ ∗ (3) The capacity curve is turned in acceleration-displacement (AD) format by dividing the forces F ∗ by the equivalent mass m ∗ and getting S a . The elastic demand spectra (design earthquake) have to be plotted in the acceleration displacement (AD) format, S ae - S de , as well. Both demand spectra and bilinear capacity curve have to be plotted in the same graph. The intersection of the radial line corresponding to the elastic period of the idealized bilinear system T ∗ with the elastic demand spectrum defines the acceleration demand required for elastic behaviour, S ae (T ∗ ) , and the corresponding elastic displacement demand S de (T ∗ ) . If the elastic period T ∗ is larger than or equal to T C , the equal displacement rule applies, i.e., the displacement demand of inelastic system, S d , is equal to the displacement demand of the corresponding elastic system with the same period S de (T ∗ ) . The reduction factor due to ductility, R μ , is determined as the ratio between the acceleration corresponding to the elastic, S ae (T ∗ ) , and inelastic system, S ay . The ductility demand, μ , is defined as the ratio between the elastic displacement demand S de (T ∗ ) and the yield displacement D y∗ . From the triangles it follows that μ is equal to R μ . In brief, the inelastic displacement demand corresponds to the intersection point of the capacity bilinear curve with the demand spectrum corresponding to the ductility demand μ (Fajfar and Eeri 2000). The inelastic displacement demand for the SDOF model, S d ,is transformed by the constant Γ into the maximum displacement of the MDOF system. It is called target displacement and provides an estimate of the global displacement the structure is expected to experience in design earthquake (Krawinkler and Seneviratna 1998). In the last step, structural performance can be assessed by comparing the displacement capacity with the displacement demand. The displacement capacity is a function of the local achievement of the first element limit condition of the structural system. Hence, the inelastic static pushover technique (SPO) consists in evaluating the expected structural performance by comparing the displacement capacity with the displacement demand on the pushover curve. Eurocode 8 Part 2 (EC8-2, CEN 2005), which deals with the seismic design of bridges, prescribes that the horizontal loads should be increased until a target displacement is reached at a reference point. The main objective of the analysis is the assessment of the force-displacement curve of the structure (capacity curve) and of the deformation demands of the plastic hinges up to the target displacement. The load patterns are intended to represent the distribution of inertia forces in a design earthquake. In the traditional pushover analysis, invariant load patterns are used. The assumptions are that the distribution of inertia forces will be constant throughout the earthquake and that the maximum deformations obtained will be comparable to those expected in the design earthquake. Hence, a relevant consideration that affects the accuracy of seismic demand prediction is the selection of load patterns that are supposed to deform the structure in a manner similar to that experienced in a design earthquake (Krawinkler and Seneviratna 1998). Eurocode 8 Part 2 (EC8-2, CEN 2005), which deals with the seismic design of bridges, refers that the horizontal load increments ΔP i,j assumed acting on lumped mass m i , in the direction investigated, at each loading step j, is given by Eq. (4). , = (4) In this formula, Δp j is the horizontal force increment, normalized to the weight g m i , applied in step j , while Φ i is a shape factor defining the load distribution along the structure. Eurocode 8 Part 2 (EC8-2, CEN 2005) prescribes that both of the following distributions should be investigated. First load distribution is constant along the deck. In this 2.1. Load distribution and displacement shape