PSI - Issue 44

Sourav Das et al. / Procedia Structural Integrity 44 (2023) 1680–1687 Das and Tesfamariam/ Structural Integrity Procedia 00 (2022) 000–000 stochastic process ( ) is calculated for each value of Y EEV . The GDEEs are solved for evaluating the PDF of ( ) using the finite difference method combined with the TVD scheme. Fig. 3(a) shows the PDF of ( ) . The PDF of Y EEV is obtained by setting τ to 1, as illustrated in Fig. 3(b). Once PDF is obtained, the probability of failure is estimated using Eq. 18. The failure probability is 2.5×10 -3 . In comparison, a Monte Carlo simulation with 10 6 samples yields a failure probability of 2.43×10 -3 , which is close to the value obtained using stochastic spectral embedding. 1685 6

Fig. 3. PDF of (a) virtual stochastic process and (b) equivalent extreme event

4.2. Problem 2 As a final example, a shape memory alloy (SMA)-based damped outrigger tall timber building is considered. As shown in Fig. 4, the outrigger is a connecting member between the column and shear wall which helps to reduce structural deformation of the tall building while the structure is subjected to external dynamic loads. The excessive load demand is often seen on the column connected with outrigger. To reduce load demand, a SMA spring is attached in between the outrigger and the column, which dissipates the energy due to external loads. With this in view, the initial aim is devoted to the formulation of the governing equation of motion of the SMA based damped outrigger structure. The core of the structure is modeled as a cantilever beam, and each floor mass is considered as a discrete mass that acts at the junction between the floor and the core of the structure. The outrigger beam is modeled as a torsional spring having constant rotational stiffness, and the moment developed due to the torsional spring is concentrated at the junction of the core of the structure and the outrigger, as shown in Fig. 4. The governing equation of motion is derived using the Lagrange formulation, which is given by (Das and Tesfamariam, 2022): (20) where u is displacement of the structure which is a function of time and position i.e., u ( x , t ) = Φ ( x )q( t ) = ∑ ( ) ( ) . φ i and q i denote the i th mode-shape of the system and displacement of the i th mode shape, respectively. H is the total height of the building; EI ( x ) and m ( x ) are the continuous beam's flexural stiffness and the mass per unit height, respectively; the j th equivalent rotational stiffness due to outrigger is denoted by K r,j ; n is the total number of outriggers considered in the structure. Eq. 20 is converted into an ODE which is given by (Das and Tesfamariam, 2022): (21) where ̈ represents the ground acceleration, I g the influence vector, and F SMA the force induced by SMA spring. The details of the derivation of the equation of motion and load-deformation hysteresis of the SMA spring are found in (Das and Tesfamariam, 2022). 2 ∂ ∂  ∂ ∂  ∂  u u EI t x x 2 2 2 2 2 0    ∂ + − = F m ; ( ) ( ) 2 2 1 1 , u H t α , u H t ˆ M 2 0 = = = x H      ∂ ∂  ∂  ∂ + − = ∂ ∂  ∑ ∑ N j n j k r k j k EI u K x t x [ ][ ( )] [ ][ ( )] [ ][ ( )] + +   M q t C q t K q t ( ) 2 − [ ][ ( )] 2       d       = +  I mHx t T q t φφ g g SMA r F K