Improving FEMA P-58 non-structural component fragility functions and loss predictions

Abstract

Fragility functions are an important tool in earthquake engineering, used to compute the probabilities of different damage states as a function of seismic response. They can be developed for large systems like buildings and bridges, as well as for individual structural and non-structural components, such as those used in the FEMA P-58 Seismic Performance Assessment Procedure. There are currently a number of problems associated with some P-58 non-structural mechanical component fragility functions and related loss predictions, including non-convergence when fitting the fragility functions in some cases and non-monotonic loss predictions. In this study, we recommend improvements to these fragility functions and loss predictions. Firstly, we recommend using the maximum likelihood method to fit the fragility functions to the underlying empirical data. This mitigates the non-convergence problems when fitting and makes predictions that better align with damage observed in past events. To compute predicted losses for anchored mechanical components, it is necessary to additionally consider anchorage damage, which can be predicted using fragility functions based on building code provisions. We recommend refining the current FEMA P-58 method for predicting anchored mechanical component losses, such that component and anchorage damage are calculated directly according to their corresponding fragility functions. The proposed method yields more intuitive loss predictions that vary monotonically with anchorage capacity. It also leads to better predictions of losses relative to damage observed in previous events. If implemented, the recommendations made in this paper would enhance the FEMA P-58 Seismic Performance Assessment Procedure.

Appendix

1.1 Comparing mechanical component damage predictions with observed damage

See Table 3.

Table 3 Comparing mechanical component damage probabilities (calculated using the proposed and current P-58 fragility function fitting procedures) with component damage observed in the SMIP Information System for Northridge buildings, as described in Sect. 2.4.2

1.2 Sample calculations of anchored component repair cost predictions

Let $EDP = 1$ g and $\theta _a = 1$ g for both components. All other variable values are summarized in Table 2 of Sect. 3.3.

1.2.1 Chiller

Proposed procedure

First, calculate the probability of anchorage damage with Eq. 1, using the anchorage fragility parameters:

$$\begin{aligned} P(DS\ge a|EDP=1) = \varPhi \left( \frac{\log (1/1)}{0.5}\right) = 0.50 \end{aligned}$$

(11)

Then, calculate the probability of equipment damage with Eq. 1, using the equipment fragility parameters:

$$\begin{aligned} P(DS\ge e|EDP=1) = \varPhi \left( \frac{\log (1/0.75)}{0.2}\right) = 0.92 \end{aligned}$$

(12)

Finally, compute the expected repair cost using Eq. 10:

$$\begin{aligned} RC = 0.50 \times [0.70 \times 1100 + (1-0.70) \times 51{,}920] + 0.92 \times \left[ 1 - (1-0.70) \times 0.50 \right] \times 50{,}820 = {\mathbf{\$48{,}121 }} \end{aligned}$$

(13)

$$\begin{aligned} RC \% = \frac{RC}{RC_{ae}} \times 100 = \frac{48{,}121}{51{,}920} \times 100 = {\mathbf{93\% }} \end{aligned}$$

(14)

Current P-58 procedure

Since $0.3\times \theta _e \le \theta _a$ and $0.3\times \theta _a \le \theta _e$, select combined failure damage mode. First, calculate the probability of occurrence of the damage mode with Eq. 1, using the anchorage fragility parameters:

$$\begin{aligned} P(DS\ge c|EDP=1) = \varPhi \left( \frac{\log (1/1)}{0.5}\right) = 0.50 \end{aligned}$$

(15)

Then, compute the expected repair cost using Eq. 9:

$$\begin{aligned} RC = RC_c = 0.50 \times [0.35 \times 1100 + 0.15 \times 51{,}920 + 0.50 \times 50{,}820] ={\mathbf{\$16{,}792 }} \end{aligned}$$

(16)

$$\begin{aligned} RC \% = \frac{RC}{RC_{ae}} \times 100 = \frac{16{,}792}{51{,}920} \times 100 = {\mathbf{32\% }} \end{aligned}$$

(17)

Even though the probability of equipment failure is significant in this case, the expected repair cost predicted using the current P-58 procedure is low since it is restricted by the lower vulnerability of the anchorage.

1.2.2 Distribution panel

Proposed procedure

First, calculate the probability of anchorage damage with Eq. 1, using the anchorage fragility parameters:

$$\begin{aligned} P(DS\ge a|EDP=1) = \varPhi \left( \frac{\log (1/1)}{0.5}\right) = 0.50 \end{aligned}$$

(18)

Then, calculate the probability of equipment damage with Eq. 1, using the equipment fragility parameters:

$$\begin{aligned} P(DS\ge e|EDP=1) = \varPhi \left( \frac{\log (1/3.40)}{0.6}\right) = 0.02 \end{aligned}$$

(19)

Finally, compute the expected repair cost using Eq. 10:

$$\begin{aligned} RC = 0.50 \times [0.70 \times 1440 + (1-0.70) \times 63{,}120] + 0.02 \times \left[ 1 - (1-0.70) \times 0.50 \right] \times 61{,}680 = {\mathbf{\$11{,}057 }} \end{aligned}$$

(20)

$$\begin{aligned} RC \% = \frac{RC}{RC_{ae}} \times 100 = \frac{11{,}057}{63{,}120} \times 100 = {\mathbf{18\% }} \end{aligned}$$

(21)

Current P-58 procedure

Since $0.3\times \theta _e \le \theta _a$ and $0.3\times \theta _a \le \theta _e$, select combined failure damage mode. First, calculate the probability of occurrence of the damage mode with Eq. 1, using the anchorage fragility parameters:

$$\begin{aligned} P(DS\ge c|EDP=1) = \varPhi \left( \frac{\log (1/1)}{0.5}\right) = 0.50 \end{aligned}$$

(22)

Then, compute the expected repair cost using Eq. 9:

$$\begin{aligned} RC = RC_{c} = 0.50 \times [0.35 \times 1440 + 0.15 \times 63{,}120 + 0.50 \times 61{,}680] ={\mathbf{\$20{,}406 }} \end{aligned}$$

(23)

$$\begin{aligned} RC \% = \frac{RC}{RC_{ae}} \times 100 = \frac{20{,}406}{63{,}120} \times 100 = {\mathbf{32\% }} \end{aligned}$$

(24)

Even though the probability of equipment failure is extremely low in this case, the repair cost predicted using the current P-58 procedure is relatively large as it is inflated by the higher vulnerability of the anchorage.

1.3 Comparing anchored mechanical component repair cost predictions with observed damage

1.3.1 Calculating anchorage capacity

To calculate anchorage capacity, it is first necessary to obtain the anchorage system design resistance ($\phi R_n$), which is calculated according to ASCE/SEI 7-10 equations 13.3-1 to 13.3-3 as follows:

$$\begin{aligned} \phi R_n = \frac{F_p}{W_p} = \frac{0.4 \times a_p \times S_{DS} \times (1+2\frac{z}{h})}{\frac{R_p}{I_p}} \end{aligned}$$

(25)

$$\begin{aligned} \phi R_n\le 1.6 \times S_{DS} \times I_p \end{aligned}$$

(26)

$$\begin{aligned} \phi R_n \ge 0.3 \times S_{DS} \times I_p \end{aligned}$$

(27)

where $a_p$ is a component amplification factor (assumed to be 1 for all components examined since we are calculating capacity), $S_{DS}$ is the short period spectral acceleration value (see Table 5 of the Appendix for building-specific values), h is the height of the building relative to the ground, z is the height of the component in the building relative to the ground (equal to h in this case, since we are considering roof-level components), $I_p$ is the component importance factor (equal to 1 for all components examined in this study) and $R_p$ is a component response modification factor (equal to 2.5 or 6 for components examined in this study, depending on the component of interest). We assume that the anchorage has the brittle failure modes typical for concrete anchorage (FEMA 2012b), so the relevant equation to calculate median capacity is equation 3-2 in FEMA (2012a):

$$\begin{aligned} \theta = C_q \times e^{(2.81 \beta )} \times \phi R_n \end{aligned}$$

(28)

where $\theta$ is the median anchorage capacity, $C_q$ is an adjustment coefficient for construction quality, and $\beta$ is the $\beta$ parameter for the anchorage fragility function. $C_q$ and $\beta$ are both set equal to 0.5, in accordance with FEMA (2012b).

1.3.2 Component repair cost data

See Table 4.

Table 4 Repair cost data for the components examined in Northridge buildings

1.3.3 Comparing predictions and observations

See Table 5.

Table 5 Comparing expected component repair cost percentage predictions (calculated using the proposed and current P-58 procedures) with component damage observed in the SMIP Information System for Northridge buildings, as outlined in Sect. 3.4.2