PSI - Issue 44

Mariateresa Guadagnuolo et al. / Procedia Structural Integrity 44 (2023) 766–773 Guadagnuolo et al. / Structural Integrity Procedia 00 (2022) 000–000

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The mechanism M C seems analogous to the vertical bending failure mechanism of masonry walls but differs in both the different location of the roller (reproducing the roof) and the conspicuous structural portion above the roller itself. In the chimneys, the roller is placed on the side towards which they tend to overturn, while in the usual vertical bending mechanism it is located on the opposite side. The changed position of the roller and the upper portion of the chimney masonry involve a modification of the equations of kinematic analysis. The mechanism M S is similar to the overturning of a customary masonry wall if, among the external constraints, a spring is introduced for simulating the reaction of the roof structure loaded in its plane. The mechanism M P does not differ from the classic overturning mechanism of masonry walls, except for the different hollow cross-sections. 2.2. Failure mechanism M C An intermediate crack is assumed to occur at an unknown position in the chimney portion below the roofline. This implies a kinematic mechanism composed of two rigid blocks A and B, joined by the hinge C. Block A is considered hinged at the outer edge O. The roller D prevents translations in the horizontal plane, constraining the displacements of block B. Therefore, vertical and horizontal displacements due to the rotations around the cylindrical hinges O and D are allowed (Fig. 4). The following shows the differences with respect to the vertical bending mechanism of masonry walls. The coordinates of the key points are computed referring to the Oxy system; the rotations are assumed positive if counter-clockwise. 2.2.1. Static force-based approach (Linear kinematic procedure) The virtual displacements (Fig. 4b) of point A due to an infinitesimal virtual rotation Ψ are for immediate evaluation, while the virtual displacements of point B, belonging to the upper block, require computing the angle Φ as a function of Ψ :

h

(1 ) β −β

1 2

(1)

Φ = Ψ =

Ψ

h

The probable height of the crack is obtained by minimizing the load multiplier λ determined through the Principle of Virtual Work (P.V.W.).

Fig. 4. Mechanism M C of chimney stacks.