PSI - Issue 78

Riccardo Maurizio Ambrogio Baltrocchi et al. / Procedia Structural Integrity 78 (2026) 9–16

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Curvature [1/km]

Fig. 3. Non-linear moment-curvature: (a) Ondal section; (b) I-shaped 25 m section.

3.3. Spectral Response Analysis

From the sectional analysis, bending moment values associated with each EDP for each section are obtained (Figure 3). The additional moment M add value for reaching each EDP is then calculated from the bending moment M Ed value obtained from LSA. In Straus7, the vertical elastic response spectrum calculated according to Eurocode 8 is entered, using a damping value ξ of 2% (Figure 1c) considering the little influence of non-structural elements on the vibration of the roof. From the software, it was thus calculated the maximum bending moment M MAX which, related to each M add value, gives the vertical acceleration a vg value for each limit condition considered. The acceleration values obtained from the SRA are then plotted as a function of the three EDPs. All elements reach first cracking in sag, with Ondal member at 0.409 g and beam at 0.279 g. Subsequently, yielding and cracking of the sections in hog are attained. The Ondal roof member and the I-shaped beam 25 m fail at 0.868 g and 0.756 g, respectively. Thus, the beam element results the more vulnerable from the analysis in uncoupled configuration. Analyzing the coupled configuration, this tendency is reversed, and the roof elements have slightly lower boundary accelerations, while the beam undergo a significant increase. In particular, Ondal cracks in sag at 0.392 g, and fails at 0.831 g. The beam cracks at 0.385 g and failure occurs at 1.111 g. Thus, the Ondal elements result the more vulnerable from the analysis in coupled configuration. 3.4. Non-Linear Time-History Analysis Once the linear analyses had been performed, non-linear analyses were carried out in order to better investigate the realistic behaviour over time of the elements under vertical seismic acceleration. The previously described non-linear moment-curvature diagrams were implemented in the software in a simplified three-linear backbone curve with the hysteretic Takeda model. Values of limit curvature and moment of Ondal and I-shaped beam section for both sagging and hogging conditions are calculated associated with the different EDPs. In hogging, Ondal cracks with curvature of -0.90 1/km and a bending moment of -120.70 kNm, yields at -1.66 1/km and -161.10 kNm, and fails at -5.46 1/km and -199.90 kNm. In sag, cracking occurs at 2.16 1/km with a moment of 1502 kNm, yielding at 5.96 1/km and 2385 kNm, and failure at 13.46 1/km and 2603 kNm (Figure 3a). On the other hand, the I-shaped beam section cracks at -0.83 1/km and -450.20 kNm, yields at -1. 01 1⁄km and -500 kNm, and fails at -1. 19 1⁄km and -490 kNm in hog. In sag, cracking begins at 1. 45 1⁄km and 7861 kNm, yielding at 3. 73 1⁄km and 13930 kNm, and failure at 8 . 12 1⁄km and 14930 kNm (Figure 3b). The NLTHA requires not only the use of simplified moment-curvature, but also the generation of 10 artificial accelerograms spectrum-compatible with the vertical elastic spectrum of Eurocode 8. Subsequently, the analysis involves incrementally increasing the acceleration value by 0.15 g, starting from 0.15 up to 1.50 g and 3.00 g if required, following Incremental Dynamic Analysis (IDA). The maximum and minimum bending moment values associated with each acceleration increment for each accelerogram are obtained, thus being able to calculate the acceleration value required to reach the EDPs.