PSI - Issue 64

Sayandip Ganguly et al. / Procedia Structural Integrity 64 (2024) 757–765 Ganguly and Roy/ Structural Integrity Procedia 00 (2023) 000 – 000

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1. Introduction Implementation of structural health monitoring (SHM) leverages in extending the safety of a structure. The efficiency of SHM in attaining this purpose greatly depends on its robustness to uncertainty (Lopez and Klijn (2010)). Typically, structural uncertainties are integrated into an analysis through probabilistic methods (Leonel et al. (2012)) with uncertain parameters represented as random variables. Reynders et el. (2008) propose a methodology to measure uncertainty in modal parameters extracted from ambient vibration data. Chandrasekhar and Ganguli (2009) model the uncertainty introducing 15% measurement noise uniformly distributed on the variation of healthy and damaged natural frequencies. Mendler et al. (2021) propose a reliability-based approach to examine minimum detectable changes in dynamic parameters due to damage, in presence of various uncertainties. In another study, the effect of model error as a source of uncertainty in supervised learning-based damage identification technique is investigated by Seventekidis and Giagopoulos (2023). A more detail discussion about the sources of uncertainty and its modeling techniques can be found in the review of Jiang et al. (2017). In most of the literature (Machado and DosSantos (2015)), these uncertainties are defined by continuous probability density function (PDF). The Gaussian distribution is one of the commonly used PDFs employed for this purpose (Ibrahim and Pettit (2005)) in the past. The powerful aspect of the Gaussian distribution is its relationship to the central limit theorem. It states that the sum of a large number of independent random variables tends to follow a Gaussian distribution, regardless of the original distribution of the variables. With real acceleration data of a 195m tall building, Zhou et al. (2023) show that due to operational and environmental variability, natural frequency of the structure statistically varies following normal distribution. Ge and Lui (2005) suggest that error in the measurement of mode shapes affects significantly in the estimation of damage severity than localization of damage. However, from the past literature, it can be observed that most of the studies are conducted either to quantify uncertainty or to examine the effect of uncertainties on the reliability of detection and localization of damage. Also, most of these damage indicators are based on time-consuming and complex iterative or machine learning-based methods. A closed-form equation is rather simple for application as it explicitly defines the relation between damage intensity and associated changes in modal response. Hence, the present study aims to measure the confidence bound of closed-form equations of damage quantification in a shear frame through reliability analysis, incorporating material and measurement uncertainties as two predominant sources of variability. To demonstrate the effect of uncertainties on the reliability of damage quantification, a 10-story shear frame is first numerically modeled. Material uncertainty is simulated by assuming standard normal distribution for stiffness value of each story using mean and specific coefficients of variation (COV); 1%, 5%, and 10%. Similarly, measurement uncertainty is introduced by varying associated modal values in the closed-form equations (Chaudhary et al. (2021) and Roy (2022)) considered in this study. The reliability of the closed-form expressions in quantifying damage intensity, located at different stories, is then separately evaluated for intended acceptable limits. In this way, the result quantifies the impact of uncertainty propagation in material and measured data on the quantification of damage values. 2. Mathematical background The free-vibration equation of motion of an N-story shear building can be expressed by the following equation as ( − ){Ф ( ) } = { }, =1,2,….., (1) where and are mass and stiffness matrices respectively at healthy state of the structure, is the eigen value, and Ф ( ) is mass normalized eigen vector of i th mode. Material uncertainty is introduced in the p th story by varying stiffness as ̅ (1+ ) ; whereas uncertainty in modal displacement is presented as ( ) = ̅ ( ) (1+ ) . ̅ and ̅ ( ) are the mean values of p th story stiffness and modal displacement of i th mode respectively. is a standard Gaussian random variable bounded between -1 to 1 with zero means and corresponding coefficient of variation (Rahman and Jahanbin (2021)). The probability of failure ( ) is then computed for number of samples by the following (Chowdhury and Rao (2009)) = 1 ∑ ( , ) =1 (2) Here, ( , ) represents a decision function. ( , ) =1 for , < ( − ) and , > ( + ); ( , )=0 elsewhere. In the Eq. (2), , is the estimated damage for n th sample defined by actual damage and allowable