PSI - Issue 64

Sayandip Ganguly et al. / Procedia Structural Integrity 64 (2024) 757–765 Ganguly and Roy/ Structural Integrity Procedia 00 (2023) 000 – 000 3 deviation . Reliability of the estimated damage, denoted by ( ) , is then assessed as ( ) = 1 − . Two established closed-form equations are considered in this study to evaluate their reliability in presence of uncertainties. For any amount of stiffness degradation between p th and ( p-1 )th floors due to damage, Chaudhary et al. (2021) derive the relationship between mode shape and damage intensity using spectral element method as, = − ̂ = ( ̂ − ̂ −1 )−( − −1 ) ̂ −1 − ̂ −2 (3) where, the subscripts p , ( p-1 ), and ( p-2 ) of modal value ( ) denote corresponding degree-of-freedom (DoF) of the shear building and ( ̂) is used to refer damaged state. In another study, Roy (2022) formulates damage quantity in terms of mode shape as, = − ̂ =1− − −1 ̂ − ̂ −1 (4) 3. Numerical Analysis The impact of uncertainty propagation on estimated damage quantity is evaluated for an N-story (N=10) shear building (Fig. 1(a)) with concentrated mass =200 kg at each floor. Stiffness of each story is assigned with 10 4 numbers of samples distributed about mean value ̅ =2.5 x 10 5 N/m. The random numbers are generated by Monte-Carlo simulation in MATLAB. Reliability analysis of damage intensity obtained using Eq. (4) is considered as Case-I. Case II analyses reliability of damage quantity computed using Eq. (3). Each sample of estimated damage is further subjected to decision function ( , ) and subsequently, reliability is evaluated. The damage location also plays a significant role in modal parameter based damage quantification as discussed in Roy (2022). Hence, effect of damage location is examined by consecutively inflicting damage at 3rd, 5th, and 7th stories. Damage is numerically modeled at a particular story by reducing the story stiffness. Normalized mode shape slope of the first mode for 30% damage is provided in Fig. 1(b). Due to stiffness degradation, modal displacements derived from corresponding stiffness values also alter at damaged story (Roy (2017)). As a result, the slope of mode shape becomes discontinuous at the damage location. It is graphically demonstrated in Fig. 1(b). The normalization of the first mode shape slope is performed with respect to the maximum value among all stories. Hence, the range of X axis in Fig. 1(b) is set as 0 to 1. 759

Breakage line a

N th floor

m N

k N

p+1 th floor

b

Damage-adjacent upper story

m p+1

k p k p+1

p th floor

m p

Damaged p th story

p-1 th floor

m p-1

k p-1

3 rd floor 2 nd floor

m 1 m 2 m 3

k 1 k 2 k 3

Damage-adjacent lower story

1 st floor

Fig. 1. (a) N-story shear building model, normalized mode shape slope of (b) 1st, (c) 2nd, and (d) 3rd modes for 30% damage.