PSI - Issue 44

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

770

5

2.2.2. Displacement-based approach (Nonlinear kinematic procedure) The rotation Φ in large displacements of the upper block as a function of the angle Ψ is given by (Fig. 5a): ( ) ( ) ( ) ( ) ( ) 2 2 2 1 x x 1 x x 1 x x 2h 2h 4 h sin L cos L h sin L cos L 2 arctan 2 h sin L cos L   − − − Ψ − Ψ − Ψ − Ψ +   Φ =   Ψ − Ψ −     (2) The application of the P.V.W. to each configuration defined by Ψ and Φ needs also computing the virtual angle Φ as a function of the virtual rotation Ψ (Fig. 5b): ( ) ( ) 1 x 2 x h cos L s in h cos L sin Ψ + Ψ Φ = Ψ Φ + Φ (3) The capacity of mechanisms until the collapse (i.e. the blocks lose their equilibrium in the static condition) is described by the relationship between the load multiplier λ and the horizontal displacement d CP of a control point that could be indifferently located inside one of blocks A and B. Usually, the center of masses C M is used as the control point. In chimneys, C M could lie close to the roofline, and when the blocks rotate and the roller D moves upwards, C M could move from above to below the roller, with a consequent inversion of the sign of its horizontal displacement. This has an unreliable effect on the acceleration-displacement curve describing the chimney mechanism. It is then more suitable to assume a different control point, like the point G located on the top of the stack (Fig. 4). 2.3. Failure mechanism M S and M P The P.V.W. of mechanism M S must be written considering the force F D that reproduces the reaction due to the roof structure. Frequently, the flue crosses the roof where only timber boards and tiles are present. Their presence can be simulated by a spring with rigid-plastic behavior with strength and maximum deformation computed according to the mechanical characteristics of the roof elements. In most cases, the aforementioned spring increases the load multiplier λ in linear kinematic analyses, whereas is irrelevant in non-linear kinematic analyses. In the latter case, the roof structure shares in the chimney seal as long as the spring does not exceed the maximum allowable deformation. If the contribution is annulled before the loads' multiplier becomes zero, the curve λ -d CP describing the mechanism development comes back to that of an analogous block without spring and cannot be considered in the evaluation of the chimney safety. This means that the chimney would lean against a damaged roof and not collapse only if the roof is still in the condition of supporting it.

Fig. 5. Models for computing angles Φ and Φ of mechanism M C .