Development of fragility surfaces for reinforced concrete buildings under mainshock‐aftershock sequences

Abstract

A single intensity measure (IM) is commonly used to develop fragility curves for structures subjected to mainshock‐aftershock (MS‐AS) scenarios. This study aims to propose a framework to develop MS‐AS fragility surfaces by characterizing the MS‐AS sequences with two IMs. To do this, the well‐known Park & Ang damage index, which increases monotonically with the ground motion duration, is adopted to represent the accumulative structural damage during sequential excitations. The MS‐AS accumulative damage of structures is thus represented by the combination of the initial damage caused by MS and the incremental damage caused by AS. Based on total probability theorem, the process of assessing MS‐AS fragility is consequently divided into two parts, namely the MS fragility assessment and the AS fragility assessment, respectively, where the cases of dynamic instability are considered. The proposed framework is illustrated using a typical four‐story reinforced concrete frame building. The peak ground velocity values of both MS and AS are taken as the representative IMs for the MS‐AS sequences. The fragility results for MS only and MS‐AS sequences are compared to scrutinize the effects of AS on structural damage. It is found that the MS‐AS sequence can cause significantly larger exceedance probabilities of limit states compared to the MS only. The MS‐AS sequence also leads to a reduction of the MS‐AS fragility median conditioned on a given AS intensity. Such reduction increases with the increase of the AS intensity.

Probabilistic seismic performance assessment of steel moment‐resisting frames considering exposed column‐base plate connections with ductile anchor rods

“…Let P GMS [BCF | S a ( T 1 ) GM1 , S a ( T 1 ) GM2 ] be the BCF probability given a GMS, and GMS is assembled by GM1 with S a ( T 1 ) GM1 and GM2 with S a ( T 1 ) GM2 . Similar to the formulation of limit‐state exceedance probability due to MS‐AS sequence, 64,65 the probability of BCF due to GMS can be further broken down as follows: the base connection fails due to GM1 with the probability P GM1 [BCF | S a ( T 1 ) GM1 ], or it fails with probability P GM2 [BCF | S a ( T 1 ) GM1 , S a ( T 1 ) GM2 ] during GM2, given that the base connection does not fail due to GM1. By using the total probability theorem, P GMS [BCF | S a ( T 1 ) GM1 , S a ( T 1 ) GM2 ] can be expressed as: $$$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {P^{{\mathrm{GMS}}}}\left[ {{\mathrm{BCF}}\left| {{S_a}{{\left( {{T_1}} \right)}_{{\mathrm{GM1}}}},{S_a}{{\left( {{T_1}} \right)}_{{\mathrm{GM2}}}}} \right.}$$…”

“…It is noted that Equation () is valid only if the base connections do not fail during GM1, which implies that (1) the considered IM of GM1 (i.e., S a ( T 1 ) GM1 ) would also affect the derivation of Equation (11); and (2) only the portion of cloud data (obtained from NLTHAs for the GMS suite) which does not cause BCF during GM1 is applied to determine these terms. A bivariate power‐law model 65 is used to determine them, and it is expressed as follows: $$$$\begin{equation}{D_*} = {b_0} \cdot {\left[ {{S_a}{{\left( {{T_1}} \right)}_{{\mathrm{GM1}}}}} \right]^{{b_1}}} \cdot {\left[ {{S_a}{{\left( {{T_1}} \right)}_{{\mathrm{GM2}}}}} \right]^{{b_2}}}\end{equation}$$$$and it can be re‐written in a natural logarithmic form as follows: $$$$\begin{equation}\ln \left( {{D_*}} \right) = \ln \left( {{b_0}} \right) + {b_1} \cdot \ln \left[ {{S_a}{{\left( {{T_1}} \right)}_{{\mathrm{GM1}}}}} \right] + {b_2} \cdot \ln \left[ {{S_a}{{\left( {{T_1}} \right)}_{{\mathrm{GM2}}}}} \right] + e\end{equation}$$$$where, D * has the same meaning of it in Equation (), e is a zero‐mean random variable representing the variability of ln( D * ) given S a ( T 1 ) GM1 and S a ( T 1 ) GM2 , and b 0 , b 1 , b 2 are the parameters of the linear logarithmic regression. According to Equations () and (), the necking/ULCF fragility function due to GM2, given non‐BCF in GM1 condition, that is, P GM2 [ D * > 1 | S a ( T 1 ) GM1 , S a ( T 1 ) GM2 ], can be expressed as a lognormal function: …”

“…Let P GMS [BCF|S a (T 1 ) GM1 ,S a (T 1 ) GM2 ] be the BCF probability given a GMS, and GMS is assembled by GM1 with S a (T 1 ) GM1 and GM2 with S a (T 1 ) GM2 . Similar to the formulation of limit-state exceedance probability due to MS-AS sequence, 64,65 the probability of BCF due to GMS can be further broken down as follows: the base connection fails due to GM1 with the probability P GM1 [BCF|S a (T 1 ) GM1 ], or it fails with probability P GM2 [BCF|S a (T 1 ) GM1 ,S a (T 1 ) GM2 ] during GM2, given that the base connection does not fail due to GM1. By using the total probability theorem, P GMS [BCF|S a (T 1 ) GM1 ,S a (T 1 ) GM2 ] can be expressed as:…”

“…It is noted that Equation ( 11) is valid only if the base connections do not fail during GM1, which implies that (1) the considered IM of GM1 (i.e., S a (T 1 ) GM1 ) would also affect the derivation of Equation (11); and (2) only the portion of cloud data (obtained from NLTHAs for the GMS suite) which does not cause BCF during GM1 is applied to determine these terms. A bivariate power-law model 65 is used to determine them, and it is expressed as follows:…”

Structural Damage Prediction of a Reinforced Concrete Frame Under Single and Multiple Seismic Events Using Machine Learning Algorithms

“…Also, more recently, Kavvadias et al [12] and Zhou et al [13] investigated the correlation between aftershock related Intensity Measures (IMs) and final structural damage indices. Additionally, multiple researchers [14][15][16][17] have been evaluating the fragility of buildings and infrastructures against seismic sequences, in the past.…”