6.7 Estimation of Uncertainties in EDPs
6.7.1 Epistemic Uncertainty
Epistemic uncertainty depends on how well the material and sectional properties of the members of a model and various loads are simulated. The properties, such as, weight density, the strength of concrete or rebars are random characteristics that vary from source to site. Such randomness can be physically reduced and quantitatively
0 0.2 0.4 0.6 0.8
0 0.5 1 1.5 2 2.5
TL Disp (m)
PGA(g)
BF 50%
84%
16%
0 0.2 0.4 0.6 0.8
0 0.5 1 1.5 2 2.5
TL Disp (m)
PGA(g) 50% OGS 84%
16%
0 0.2 0.4 0.6 0.8
0 0.5 1 1.5 2 2.5
TL Disp (m)
PGA(g) 50% FI 84%
16%
0 0.2 0.4 0.6 0.8
0 0.5 1 1.5 2 2.5
GL Disp (m)
PGA(g)
BF 50%
84%
16%
0 0.2 0.4 0.6 0.8
0 0.5 1 1.5 2 2.5
GL Disp (m)
PGA(g) 50% OGS 84%
16%
0 0.2 0.4 0.6 0.8
0 0.5 1 1.5 2 2.5
GL Disp (m)
PGA(g) 50% FI 84%
16%
6.7 Estimation of Uncertainties in EDPs
identified through uncertainty analysis of the random input variables by using either nonlinear static or dynamic analysis. In the present study, nonlinear response history analysis is carried out in order to determine the displacement demand on the frames. Any variation in the obtained displacement demand for the frames is due to the randomness in the input parameters. This variability is determined as the lognormal standard deviation considering all the Nsim models, and is called the epistemic uncertainty or the uncertainty in capacity (βC).
The epistemic uncertainty arising due to variation in input parameters is shown for the three frames in Fig. 6.4 and Fig. 6.5 with respect to PGA and median displacement, respectively. It is observed that even though all the three frames are categorized as RC frames, the uncertainty in their response vary significantly from each other. The uncertainty in response increases with increasing PGA or median displacement. At lower values of PGA (here up to about 1g), the uncertainty in the estimation of displacement response of all the three frames at both TL and GL is very low. At higher values of PGA, however, the value of βC for the three frames increases considerably with highest βC for the FI frame (about 0.5) and lowest for the OGS frame (about 0.25). At a PGA value of 3g, the maximum uncertainty in BF is 0.4 and 0.3, respectively, considering TL and GL displacement as EDP. An uncertainty value of 0.25 for code-designed buildings and 0.3 for pre-code buildings is recommended to account for the variability in the capacity curves as defined in HAZUS obtained from pushover analysis of the building models.
However, there is neither any categorization of building category for OGS frames in HAZUS nor it does mention any local EDPs.
The median displacement and the corresponding logarithmic standard deviation (or uncertainty) is plotted in Fig. 6.5 for the three frames. The median displacement is estimated as the median of all the displacement response obtained from the 25 different models for each frame type at different PGA. The basic observation is that the uncertainty in response of FI frame at any displacement value is always higher compared to that observed in case of the BF or OGS frame. The maximum uncertainty in the response of FI frame considering both TL and GL displacement is approximately equal to 0.6. In OGS frame also, the maximum uncertainty for TL and GL displacement is similar, and is nearly equal to 0.4. This is not so in case of BF, where the uncertainty related to the TL-EDP is higher (0.5) compared to GL-EDP (0.4).
(a) (b)
Figure 6.4 Epistemic uncertainty in response with respect to PGA at (a) top storey level and (b) ground storey level.
(a) (b)
Figure 6.5 Epistemic uncertainty in response with respect to median displacement at (a) top storey level and (b) ground storey level.
To estimate the seismic fragility based on different damage state levels as prescribed in literature (eg., HAZUS) or for developing continuous fragility flow plots (Choudhury and Kaushik 2018a), the uncertainty in capacity estimation for a chosen displacement threshold (i.e., the median value) can be obtained directly from Fig. 6.5. It is clear from Fig. 6.4 and Fig. 6.5 that the epistemic uncertainty in both TL and GL displacement response is always higher for FI frames considering either PGA or median displacement as intensity measure. It is also important to note that the epistemic uncertainty in response estimation of OGS frame is mostly the lowest except for some intermediate values of median TL dispalcement where the uncertainty in response of BF is slightly lesser. Clearly, the uncertainty is a function of frame typology and it depends more on the characteristics and distribution of masonry infills in the frame.
In Chapter 5 sensitivity analyses was carried out considering variability in the uncertain input parameters and highly sensitive parameters having highest swing values
0 0.2 0.4 0.6
0 1 2 3
βC|PGA
PGA(g)
TL BF
OGS FI
0 0.2 0.4 0.6
0 1 2 3
βC|PGA
PGA(g)
GL
0 0.2 0.4 0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 βC|Disp
Median Disp (m)
TL BF
OGS FI
0 0.2 0.4 0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 βC|Disp
Median Disp (m)
GL
6.7 Estimation of Uncertainties in EDPs
were identified. For BF and OGS frames, fck and Bc were found to have highest swing values, and for FI frame, Ws and fm′ were the parameters showing highest sensitivity. A separate set of uncertainty analysis is, therefore, carried out considering only these parameters to understand their influence on the estimated uncertainty. Again, 25 different sets of models are prepared by varying only the selected variables for carrying out nonlinear time-history analyses considering El Centro ground motion. The comparison of epistemic uncertainty considering all variables and only selected variables is shown in Fig. 6.6 plotted with respect to PGA.
(a) (b)
Figure 6.6 Epistemic uncertainty in response with respect to PGA considering all random variables (RV) and selected random variables for the three frames at (a) top storey level and (b) ground storey level.
0 0.2 0.4 0.6
0 1 2 3
βC|PGA
PGA(g) BF-TL
All RV Selected RV
0 0.2 0.4 0.6
0 1 2 3
βC|PGA
PGA(g) BF-GL
0 0.2 0.4 0.6
0 1 2 3
βC|PGA
PGA(g) OGS-TL
0 0.2 0.4 0.6
0 1 2 3
βC|PGA
PGA(g) OGS-GL
0 0.2 0.4 0.6
0 1 2 3
βC|PGA
PGA(g) FI-TL
0 0.2 0.4 0.6
0 1 2 3
βC|PGA
PGA(g) FI-GL
Fig. 6.6 shows that there is no definite pattern in the difference in uncertainty when only selected parameters are considered versus when all the parameters are considered. In case of BF, the estimated uncertaintyconsidering selected parameters is always less (about one-third of the value obtained by considering all the random variables). In case of OGS frames, the uncertainty obtained by considering all the variables is significantly higher at higher PGA (beyong 1g). For lower values of PGA the uncertainty follow the trend of BF. In case of FI frames, the uncertainty obtained considering only selected variables is even more than that obtained considering all the variables upto about PGA of 1.3g, beyond which the trend reverses. Thus, epistemic uncertainty increases the overall uncertainty by significant amount even if only selected random variables are considered in the analyses. Thus, it is extremely important to consider the epistemic uncertainty correctly and it cannot be simply ignored in the process of seismic fragility assessment.