PSI - Issue 78

Emanuele Rizzi et al. / Procedia Structural Integrity 78 (2026) 1420–1427

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its longitudinal axis. In the OOP models, the top of the column and the nave wall were restrained against in-plane deflection and rotations, as well as torsion about their longitudinal axis. Additionally, in OOP R model, horizontal springs, spaced every 1750 mm were introduced at the top of the nave wall, at the rafter locations, to represent the braced nave roof, The stiffness of these equivalent springs (1.06 kN/mm) was calibrated by developing a simplified model of the entire nave roof structure, idealized as a simply supported truss beam composed of truss elements (representing the timber members) and pairs of diagonal tension-only springs (representing the X-bracings). The in plane deformability of this roof diaphragm was evaluated and the most unfavorable portion of the nave arcade - the one at mid-span of the roof, was considered.

(a)

(b)

(c)

(d)

Fig. 5.Deformed shapes of numerical models: (a) IP U,C2 ; (b)IP R,C2 ; (c) OOP U,C2 ; (d) OOP R,C2 .

A non-linear static analysis, also considering geometric nonlinearities, was performed. The load and work convergence criteria were considered, along with the arc-length method, to solve the unstable static equilibrium state after the limit point, with parameters reported in Table 2. The analysis was conducted in two steps: first, applying gravitational loads in the Z (vertical) direction; then, applying loads proportional to masses in the X and Y directions for the IP and OOP models, respectively, and concurrently disabling the vertical springs where toe crushing is expected in C2 cases. The horizontal displacement of the centers of mass of the two blocks (column and nave wall portion) in the loading direction are plotted against the IP or OOP horizontal load multiplier, given by the ratio between the horizontal load and the total weight. The initial stiffness of the mechanisms is calculated as the ratio between the horizontal load and the displacement of the centers of mass of the two blocks. The stiffness ratio between the unstrengthened and strengthened models is presented in the results discussion.

Table 2.Analysis convergence criteria.

Parameter

Value

Load error tolerance Work error tolerance

1e-3

1e-6 Minimum arc-length adjustment ratio 0.5 Maximum arc-length adjustment ratio 8.0

5.1.1. In-plane models results (IP) The unstrengthened models IP U show a typical trilinear response (dashed lines in Fig. 6). In the C1 model, the initial linear phase ends with the onset of the uplift at the column base, at a load multiplier of 0.163. This is followed by a stiffness reduction due to the progressive loss of contact between the top of the column and the base of the arcade. The curve reaches a peak load multiplier of 0.187, after which the kinematic mechanism fully develops. In contrast,