PSI - Issue 64

Alba Hyseni et al. / Procedia Structural Integrity 64 (2024) 246–253 Alba Hyseni / Structural Integrity Procedia 00 (2019) 000 – 000

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Therefore, detecting the II-stage related strain rate may give information on the residual service life and sufficient time for the design and installation of the retrofitting. In the present work , the creep’s law was simulated with the rheological model according to the Burgers theory, whose adopting the Kelvin and Maxwell components for the I-stage (having stiffness E K and strain ε K ) and II-stage (having stiffness E M and strain between ε M,el and ε M,visc ), respectively. The III-stage ( ε D ) was described with a damage model based on a damage index , Fig. 4b. To accurately catch the behavior of the case-study masonry, the parameters proposed in Verstrynge et al. (2011) for the B-type masonry were considered. In fact, the B-type referred to masonry tower with large openings and high longitudinal elastic modulus (up to 3000 MPa) which reflect the state of-play the herein case-study. a b

Fig. 4. schematic presentation of the rheological Burgers model including damage: (a) indication of primary (I), secondary (II) and tertiary (III) creep stages; (b) representation of the model. If the acting stress is assumed constant (at an initial time, t=0 ), a closed-form one-dimensional formulation can be assessed for the general -estimation according to Eq. (1). ( , ) = + 1 (1 − (− )) + (1− ) (1) Where τ M/K are time constants of the Maxwell ( M ) and Kelvin ( K ) models, respectively. If the stress level or internal variables do not remain constant, the constitutive equations can be integrated in closed form within small time increments, which results in an incremental formulation. The viscous damage, , evolves in function of time from zero (no damage) to 1 (total material failure), according to Eq. (2). ̇ V = ( σ ∗ 1−D V ) n (2) The damage parameter has a positive, non-decreasing value and damage is initiated when a certain relative stress level is exceeded. The value of the damage parameter is described in function of the relative stress, according to the linear relation in Eq. (3). = σ ∗ + (3) With σ ∗ being the relative stress level, obtained by dividing the absolute stress by the average compressive strength of the considered material. ̇ V is the first derivative of the damage with respect to time and c, n, A and B are experimentally obtained coefficients, elaborated in Verstrynge et al. (2011). At any point, the value of the damage parameter is the maximum of Eqs. (2) and (3). The outcome is so illustrated in Fig. 5.