Methodology to Develop Fragility Curves of Glass Façades Under Wind-Induced Pressure

Abstract

High wind speeds produced by hurricanes or synoptic winds can cause considerable damage and the failure of structural and nonstructural elements. The use of glass façades in buildings has become very popular; in Mexico, a large number of buildings along the coast are designed with glass façades. Glass façades provide light, temperature control, and an esthetic view; however, this type of glass system is particularly vulnerable to high wind-induced pressures. A methodology to determine the fragility curves of glass façades under turbulent wind loading is proposed. This methodology could be used to select the appropriate glass thickness of a façade. The procedure employs an autoregressive and moving average model to simulate the wind field and Monte Carlo techniques to simulate the glass resistance of the windows. The methodology to construct the fragility curves is illustrated with a numerical example of a glass façade of a 96-m tall building. Three cases of glass resistance associated with coefficients of variation equal to 0, 10, and 20% were considered. The results of the numerical example show that the uncertainty in the glass resistance plays an important role in the development of the fragility curves of the glass façades for high mean wind speeds between 38 and 67 m/s at a height of 10 m.

Appendix

Normal polynomials were used to improve the fitting process of the fragility curves using the Hong and Lind method [39]. Further details of this method are described in the above-mentioned reference but are summarized below.

Considering that {ζ_i}, where i = 1,2,…,N, are independent random observations of variable Z arranged in ascending order, and Z are the data sets of the fragility curves related to different damage states. The cumulative distribution function for each value of ζ_i can be obtained using the following expression:

$$F({\zeta _i})=\frac{i}{{N+1}},\quad i=1, \ldots ,N.$$

(3)

Using the inverse of the standardized normal distribution, denoted by ${\Phi ^{ - 1}}$, in Eq. (3) yields

$$\eta ={\Phi ^{ - 1}}(F(\zeta )),$$

(4)

where the fractile constraint shown in Eq. (4) is mapped into a normal space using the normal polynomial as follows:

$$\zeta =\sum\limits_{{j=0}}^{{r - 1}} {{a_j}{\eta ^j},}$$

(5)

in which r < N is used to model the distribution of $\zeta$. The coefficient of the polynomial, a_j, where j = 0,…,r − 1, can be fixed to satisfy any r or the N constraints. An alternative to determine these coefficients is the least-square method, minimizing the error ε defined as follows:

$$\varepsilon ={\sum\limits_{{j \in {J_s}}} {\left( {{\zeta _j} - \sum\limits_{{i=0}}^{{r - 1}} {{a_i}{{\left( {{\eta _j}} \right)}^i}} } \right)} ^2},$$

(6)

where ${\eta _j}={\Phi ^{ - 1}}(F({\zeta _j}))$ and ${J_{\text{s}}}$ is an index set of selected constraints. The probability of $\zeta \leqslant {\zeta _{\text{o}}}$, $F({\zeta _{\text{o}}})$, is given by the following:

$$F({\zeta _{\text{o}}})=\Phi ({\eta _{\text{o}}}),$$

(7)

where ${\eta _{\text{o}}}$ is obtained by solving Eq. (7) after replacing $\zeta$ with ${\zeta _{\text{o}}}$.

In this study, the third-order normal polynomials were obtained to plot the fragility curves.