PSI - Issue 44

Sourav Das et al. / Procedia Structural Integrity 44 (2023) 1680–1687 Das and Tesfamariam/ Structural Integrity Procedia 00 (2022) 000–000

1686

7

Fig. 4. Schematic representation of SMA-based damped outrigger structure

To illustrate this problem, a 20-storey timber building is considered. The architectural plan of the building can be found in Das and Tesfamariam (2022). The beams and columns of the building are designed using glulam, and the core shear wall is cross-laminated timber (CLT). The material properties of glulam and CLT, the cross-sectional details of the beam, column, and shear wall are taken from (Das and Tesfamariam, 2022). Two SMA-based outriggers are considered. The objective of this study is to estimate the optimal positions of outriggers and tuning parameters of SMA spring by minimizing the probability of failure of the system. One of the design parameters is the normalized outrigger location, α , which ranges from 0 to 1 (Fig. 4). Other design parameters associated with SMA hysteresis parameters include the initial austenite stiffness ( k a ), the maximum design displacement ( x s,max ), and the ductility ratio ( μ s ) of SMA. Also, the initial austenite stiffness is estimated from the normalized transformation strength of SMA, which is F 0 = k a x ys /( mH ), and ranges 0 [0.1, 0.5] ∈ F , where x ys is the yield displacement of SMA. The mean value of the other parameters, x s,max and μ s are taken as 0.2m and 20, respectively. The coefficient of variation of these two variables is assumed to be 20%, which follows a uniform distribution. The bounds are applied to the design parameters to avoid numerical instability during optimization. For this reason, the bounds are taken as , [0.13, 0.27] ∈ s max x and [13, 27] ∈ s µ , respectively. This study also considers the material uncertainties, which are the modulus of elasticity of CLT in longitudinal and transverse directions, i.e., E 1 and E 2 , and the modulus of elasticity of glulam beam and column ( E g ). Therefore, it is seen that there are two sets of random variables, i.e., θ = [ α 1 , α 2 , F 0 , μ s , x s,max ] which are design variables and ε = [ E 1 , E 2 , E g ]. The variables relating to CLT and glulam material properties, namely E 1 , E 2 , and E g , are assumed to have Gaussian distributions with mean values of 11.7, 9 and 12.4 ( ) 9 10 Pa × , respectively, and coefficient of variation is 10%. The objective function is taken as the maximum value of the inter-storey drift ratio of the floors. To compute that, an ensemble of ground motions is considered, which are selected based on the conditional mean spectrum-based record selection method for the Vancouver site (Canada) at a time period of 2.0s. The probability of failure of the system is defined in Eq. 8, where Y EEV is given as:

{

}

g N

( 1 20 [0, ] 1 1 max max IDR , , ≤ ≤ ∈ = ∑  i j i t T N θ ,

)

Y

t

=

(22)

EEV

g j

where N g denotes the total number of ground acceleration time histories and IDR the inter-storey drift ratio. Y Thres is the allowable inter-storey drift ratio, which is taken as 2.5% according to NBCC (2015). The procedure to evaluate the PDF of Y EEV is same as before. Once PDF is obtained corresponding to each value of θ , the probability of failure is estimated. The obtained value of failure probability corresponding to each set of design variables, θ are used to carry out the optimization problem. The objective function is defined as follows: ( ) * argmin   =   f P θ θ θ s.t. ≤ ≤ ll ul θ θ θ (23)