PSI - Issue 84

4

Paolo Andrea Miglietta et al. / Procedia Structural Integrity 84 (2026) 1111–1118 P.A. Maglietta et al. / Structural Integrity Procedia 00 (2026) 000–000

1114

f

*

f

=

c

2

Xn



c

(5)

1

K

+

bars

b

c 

2

where f c = the uncracked concrete compressive strength, f * c = cracked concrete compressive strength, K = coefficient equal to 0.1 for medium rebars, X = corrosion penetration in mm, n bars = number of rebar in the compressive zone, b = cross- section width in mm and ε c2 = concrete strain at peak compressive strength. The compressive strength of the cracked concrete is applied to a circular area centred on the reinforcement, with a radius equal to the concrete cover. Lastly, the creep effect was considered by decreasing the concrete elastic modulus by a coefficient φ(t,t 0 ), estimated according to the formulation proposed by the Model Code (CEB-FIP, 2010). 3. Description of the analysis procedure In this study, a probabilistic framework is proposed to assess the seismic response of RC members, accounting for degradation phenomena and the inherent uncertainties associated with governing factors. Time-dependent bending moment–curvature and shear force–shear strain relationships are derived using the analytical formulations described in Section 2. The resulting relationships can be implemented in the numerical model of the structure to represent the cross-sectional behaviour, explicitly capturing degradation mechanisms as well as the related uncertainties. Firstly, both input random and deterministic parameters are selected. The former are represented through their statistical distributions, whereas the latter are characterized by their nominal (mean) value. Three representative values are sampled from each distribution, corresponding to the 16 th percentile, the mean value and the 84 th percentile, respectively. Subsequently, all feasible combinations of such percentile-based values were generated. Each resulting combination is then paired with the nominal values of the deterministic parameters. Once combinations of input parameters are established, degradation models are employed to compute flexural and shear cross-sectional performance over time. The flexural and shear cross-sectional behaviour are represented through bending moment-curvature (M- χ) and shear force-shear strain (V- γ) relationships, respectively. The M- χ diagram of the pier cross -section is determined by computing the first cracking moment M cr , the elastic moment M y , the ultimate moment M u and the corresponding curvatures χ cr , χ y , and χ u . Three different limit states are defined: first cracking, elastic limit and ultimate strength. The following assumptions are made: (i) Euler – Berno ulli’s assumption for the cross section of the beam; (ii) rigid -plastic bond behaviour at the concrete-rebar interface, with peak bond strength = R· τ bu ; (iii) piece-wise linear approximation of the Mander et al. model for concrete in compression (Mander et al., 1988) and bilinear hardening model for steel. Except for the first cracking stage, which is classically defined as the tensile strain of the concrete, ε ct , is attained, different failure mechanisms are considered for computing the resisting moment and the corresponding curvature at each limit state. The elastic limit stage is attained either when the steel yielding strain, ε sy , or the steel bond strain, ε bu , is reached in the reinforcement. The resisting moment and the corresponding curvature at this limit state is addressed as M y and χ y regardless of the failure mode obtained. The ultimate limit state can occur in two different scenarios, namely for compressed concrete crushing (at the attainment of ε cu ) or for rebar tensile failure (at the attainment of ε su ). The computation of the M- χ behaviour of the pier’s cross section also accounted for the effect of bar slip through a macro modelling fiber-based approach. The V- γ diagram is evaluated by computing the ultimate shear capacity, V u , and the corresponding shear strain, γ u . The shear-resisting mechanism is modelled using the classical variable-angle Ritter–Mörsch truss analogy. The ultimate shear capacity is assumed as the minimum between the tensile-controlled ultimate shear force (V u,rs ) and the compression-controlled ultimate shear force (V u,rc ), both computed according to the formulations provided in the Eurocode 2 (EN 1992-1-1, 2004). The ultimate shear strain is computed as the ratio between V u and the cross-sectional shear stiffness.