Ramifications of blind adoption of load and resistance factors in building codes: reliability-based assessment

Abstract

This paper presents a probabilistic approach to reveal the negative consequences of overlooking code calibration when adopting the provisions of nonindigenous building codes in design. A taxonomy of the national building codes of various countries whose provisions are adopted from one or more original standards is first presented. Next, the reliability-based approach to quantify the lack of safety that may arise due to the absence of code calibration is presented. This approach is then applied to the case study of the National Building Code of Iran, whose load and resistance factors are directly adopted from ASCE/SEI 7. To this end, seven steel building archetypes with various structural systems at various regions with different seismicity levels and site classes are designed according to the Iranian loading and steel design codes. Subsequently, 2510 structural elements, including beams, columns, and braces, from these archetypes are subjected to reliability analysis to compute their reliabilities. In this analysis, the uncertainty of material properties are quantified using local laboratory test data and the uncertainties associated with manufacturing tolerances of steel elements are quantified through extensive field surveys. The analysis results in reliability indices that are markedly smaller than the targets set by the original standards the Iranian code has adopted from, which exposes the notable lack of safety in the ensuing designs. Subsequently, the load and resistance factors are calibrated to achieve the target reliabilities. Among other changes, the results indicate the need for a 25% increase in the earthquake load factor, which reveals the lack of safety that is implicit in the current practice. Reliability sensitivity analysis is then employed to determine the sources of uncertainty that are most influential on the resulting reliabilities. The presented approach is immediately applicable to the assessment of the building codes of other countries.

Appendices

Appendix 1: FORM

FORM linearizes the limit-state function in the space of standard normal variables using a first-order Taylor series expansion. To this end, Nataf transformation (Liu and Der Kiureghian 1986) is employed to transform the variables from the original space, x, to standard normal variables, y. Nataf first transforms the original variables, x, to auxiliary variables z, which are correlated and normally distributed with a mean of zero and a standard deviation of unity using the following one-to-one transformation:

$$z_{i} = \Phi^{ - 1} \left( {F_{i} (x_{i} )} \right)$$

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where z_i is ith variable in vector z, Ф⁻¹(.) is standard normal inverse cumulative distribution function (CDF), and F_i(.) is CDF of the ith original variable, x_i. Given the known correlation between each pair of the original variables x_i and x_j, denoted by ρ_xi,xj, the correlation between the corresponding pair of auxiliary variables z_i and z_j, denoted by ρ_zi,zj, is computed using the following integral equation:

$$\rho_{{x_{i} ,x_{j} }} = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {\left( {\frac{{x_{i} - \mu_{i} }}{{\sigma_{i} }}} \right)\left( {\frac{{x_{j} - \mu_{j} }}{{\sigma_{j} }}} \right)\varphi_{2} \left( {z_{i} ,z_{j} ,\rho_{{z_{i} ,z_{j} }} } \right){\text{d}}z_{i} {\text{d}}z_{j} } }$$

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where φ₂(.) is bivariate correlated standard normal probability density function (PDF). These computed correlations form the correlation matrix of the auxiliary variables, R_zz. Next, Nataf transforms the auxiliary variables, z, into uncorrelated standard normal variables, y, using the following linear transformation:

$${\mathbf{y}} = {\mathbf{L}}^{ - 1} {\mathbf{z}}$$

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where L is Cholesky decomposition of R_zz matrix.

In summary, Eq. (2) is transformed to the following integral in the standard normal space using the Nataf transformation:

$$p_{f} = \int_{{\mathop {G({\mathbf{y}}) \le 0}\limits^{ \ldots } }} {\int {\varphi_{n} ({\mathbf{y}}){\text{d}}{\mathbf{y}}} }$$

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where G(y) is the limit-state function in terms of the standard normal random variables, y, φ_n(y) is n-variate standard normal PDF, and n is the number of random variables.

Now that the reliability problem is redefined in the standard normal space, FORM searches for the most probable point (MPP), defined as the point with the highest probability density on the limit-state surface, i.e., on G(y) = 0. To this end, FORM employs an optimization algorithm, here, the iHLRF algorithm (Hasofer and Lind 1974; Rackwitz and Flessler 1978; Zhang and Der Kiureghian 1995), to find the nearest point of the limit-state surface to the origin. Upon finding the MPP, denoted by y^*, the reliability index, β, and the ensuing probability of failure are computed as follows:

$$\beta = \left\| {{\mathbf{y}}^{*} } \right\|$$

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$$p_{f} = \Phi ( - \beta )$$

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Appendix 2: System reliability

In accordance with Der Kiureghian (2005), the probability of failure for a series system is computed as follows:

$$p_{f} = 1 - \Phi_{k} \left( {{\varvec{\upbeta}},{\mathbf{R}}} \right)$$

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$${\varvec{\upbeta}} = \left\{ {\begin{array}{*{20}c} {\beta_{1} } & {\beta_{2} } & \cdots & {\beta_{k} } \\ \end{array} } \right\}^{T}$$

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$${\mathbf{R}} = \left[ {\begin{array}{*{20}c} 1 & {{\varvec{\upalpha}}_{1}^{\text{T}} {\varvec{\upalpha}}_{2} } & \cdots & {{\varvec{\upalpha}}_{1}^{\text{T}} {\varvec{\upalpha}}_{k} } \\ {{\varvec{\upalpha}}_{1}^{\text{T}} {\varvec{\upalpha}}_{2} } & 1 & \cdots & {{\varvec{\upalpha}}_{2}^{\text{T}} {\varvec{\upalpha}}_{k} } \\ \vdots & \vdots & \ddots & \vdots \\ {{\varvec{\upalpha}}_{1}^{\text{T}} {\varvec{\upalpha}}_{k} } & {{\varvec{\upalpha}}_{2}^{\text{T}} {\varvec{\upalpha}}_{k} } & \cdots & 1 \\ \end{array} } \right]$$

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where k is the number of limit-state functions, which here equals five, Ф_k is k-variate correlated standard normal CDF, β is the vector of reliability indices corresponding to all limit-state functions, R is correlation matrix, and α_i is the negative normalized gradient of the ith limit-state function at MPP, defined as

$${\varvec{\upalpha}}_{i} = - \frac{{\nabla G_{i} ({\mathbf{y}}^{*} )}}{{\left\| {\nabla G_{i} ({\mathbf{y}}^{*} )} \right\|}}$$

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