Reliability Index of a Multi-story Building Considering the Uncertainties of Monitoring System

Abstract

This paper aims to study the effects of measuring errors during the process of structural health monitoring (SHM) on the reliability index (RI) of a building. A six-story plane frame model of a flexible long-span building is studied in which defects are randomly occurred both in the superstructure and its foundation. The response of the building due to different damage patterns would be collected, analyzed, and the outputs as raw data sending to the measuring the system would be deliberately changed to obtain a perturbed data set of signals. The RI of the system would be computed using the first-order reliability method (FORM), and the effects of modeling and measuring errors (noises) on the index would be examined. These uncertainties both in aleatoric aspect (load, section, limit stress and quality of material) and epistemic (modeling device instrumentation and signal transformation method) aspect affect remarkably the reliability analysis. Findings are the order of importance of each factor involved, the percentage of contribution to the RI and an approach for computing the RI of the structural system using the Taylor series expansion. Some recommendations for the measuring procedure, the variation in frequency of the response, and the change in the performance of the building concerning various kinds of defects are studied and the RI in terms of different damage patterns and variables (i.e., kinds of errors) are suggested.

Appendix

Partial derivatives of the performance functions and variances with respect to specific variable, reinforced concrete structure:

1. Flexural Beams: $G(X_{i} ) = 0.85f_{y}^{'} (A_{s} - A'_{s} )\left( {d - \frac{kd}{2}} \right) - \left( {q\frac{{L_{s}^{2} }}{8}} \right)$; $\frac{\partial G}{{\partial L_{s} }} = q\frac{{L_{s} }}{4}$$\frac{\partial G}{{\partial f'_{y} }} = 0.85(A_{s} - A'_{s} )\left( {d - \frac{kd}{2}} \right);$ $\frac{\partial G}{\partial d} = f_{y}^{'} (A_{s} - A'_{s} )\left( {1 - \frac{k}{2}} \right)$; $\frac{\partial G}{\partial q} = - \frac{{L_{s}^{2} }}{8}$; $\sigma_{X} = \left[ {\left( {\frac{\partial G}{{\partial f'_{y} }}} \right)^{2} \cdot Var\left( {f'_{y} + \left( {\frac{\partial G}{\partial d}} \right)} \right)^{2} \cdot Var\left( {d \, + \, \left( {\frac{\partial G}{\partial q}} \right)^{2} } \right) \cdot Var\left( q \right) + \left( {\frac{\partial G}{{\partial L_{s} }}} \right)^{2} \cdot Var\left( {L_{s} } \right)} \right]^{0. 5}$

2. Eccentric Compression Columns (ACI 318): $H(X_{i} ) = \frac{{P_{u} }}{{\varphi P_{n} }} + \frac{{M_{u} }}{{\varphi M_{n} }} - 1 \le 0$

$P_{n} = 0.8[0.85f_{c}^{'} (A_{g} - A_{s} ) + (A_{s} f'_{y} )$; $M_{n} = 0.85f_{c}^{'} W$ where P_u chosen w.r.t Max M_u $\frac{\partial H}{{\partial f'_{c} }} = - \frac{{P.A_{b} }}{{\varphi (A_{b} f'_{c} + A_{s} f'_{y} )^{2} }};$$\frac{\partial H}{\partial d} = - \frac{{bf'_{c} P}}{{\varphi (A_{b} f'_{c} + A_{s} f'_{y} )^{2} }} - \frac{M}{{\varphi \left( {0.85 \cdot f'_{y} \cdot A_{s} \left( {1 - \frac{k}{2}} \right)d} \right)^{2} }}$$\frac{\partial H}{{\partial f'_{y} }} = - \frac{{P \cdot A_{s} }}{{\varphi (A_{b} f'_{c} + A_{s} f'_{y} )^{2} }} - \frac{{0.85 \cdot M \cdot A_{s} \left( {d - \frac{kd}{2}} \right)}}{{\varphi \left( {0.85 \cdot f'_{y} \cdot A_{s} \left( {1 - \frac{k}{2}} \right)d} \right)^{2} }};$

$\sigma_{X} = \left[ {\left( {\frac{\partial H}{{\partial f'_{c} }}} \right)^{2} \cdot Var\left( {f'_{c} } \right) \, + \left( {\frac{\partial H}{{\partial f'_{y} }}} \right)^{2} \cdot Var\left( {f'_{y} } \right) \, + \left( {\frac{\partial H}{\partial d}} \right)^{2} \cdot Var\left( d \right)} \right]^{0. 5}$

3. Horizontal Displacement: $I(X_{i} ) = \Delta_{\text{Limit}} - \Delta = \frac{H}{500} - \Delta \ge 0$

4. Vertical Deflection of beam: $J(X_{i} ) = f_{\text{Limit}} - f = \frac{{L_{s} }}{250} - \frac{{5qL_{s}^{4} }}{{384EI_{cr} }} \ge 0$

$\frac{\partial J}{{\partial L_{s} }} = \frac{1}{250} - \frac{{20qL_{s}^{3} }}{{384EI_{cr} }} = \frac{1}{250} - \frac{{qL_{s}^{3} }}{{19.2EI_{cr} }};$ $\frac{\partial J}{\partial q} = - \frac{{5qL_{s}^{4} }}{{384EI_{cr} }};$ $\sigma_{X} = \left[ {\left( {\frac{\partial J}{{\partial L_{s} }}} \right)^{2} \cdot Var\left( {L_{s} } \right) \, + \left( {\frac{\partial J}{\partial q}} \right)^{2} \cdot Var\left( q \right)} \right]^{0.5}$

5. Settlement: $K(X_{i} ) = s_{\text{Limit}} - \frac{N}{k} \ge 0;$ $\frac{\partial K}{\partial N} = - \frac{1}{{k^{2} }};\frac{\partial K}{\partial k} = - N$