5.5 Sensitivity from Nonlinear Time-History Analyses
5.5.1 Sensitivity of Response using Radar Charts
Radar charts, which graphically display multivariate data in the form of a two- dimensional chart, are presented in Figures 5.2 - 5.4, in which the three charts in each row represent the variation in displacement response of bare, OGS, and FI frames considering different samples (values) of an input parameter. The scatter in each chart represents the frame responses obtained from the nonlinear time-history analysis for the seven ground motions after normalization. Normalization of frame response is carried out with respect to the maximum value of displacement response obtained for each ground motion. The numbers at the bottom of each chart represent the coefficient of variation (COV) in frame response for the seven ground motions due to uncertainty in the considered variable. A higher value of COV means a higher variation in the response, and hence a higher sensitivity of the output. The numbers at the circumference of the radar charts represent the sample numbers of each uncertain parameter. The outermost circle of the chart represents the maximum frame response for a given ground motion with a value equal to one after normalization. A perfect circle formed by the scatter of the data points in the radar means there is a negligible variation in the output response due to any variation in the input parameter. In other words, it can be said that the frame response is insensitive
0.8 0.9 1.0 1.1 1.2
0 2 4 6 8 10 12 14 16
1st Mode Period (s)
Sample No.
BF
fck fy ξ Bc
Db γc γm
0.6 0.7 0.8 0.9 1.0 1.1 1.2
0 2 4 6 8 10 12 14 16
1st Mode Period (s)
Sample No.
OGS
fck fy ξ Bc
Db γc γm
0.1 0.2 0.3 0.4 0.5 0.6
0 2 4 6 8 10 12 14 16
1st Mode Period (s)
Sample No.
FI
fck fy ξ Bc Db
γc γm fm' Ws εm
5.5 Sensitivity from Nonlinear Time-History Analyses
to any variation in the input data. On the other hand, if the scatter in the radar forms a spiral, it shows that the response varies significantly with respect to variation in the input, i.e., the response is highly sensitive to the uncertain parameter.
(a) (b) (c)
Figure 5.2 Displacement sensitivity radar charts for uncertain geometric input parameters with their COV for seven ground motions: (a) bare frames, (b) OGS frames, and (c) FI frames.
It is observed from the radar charts that for Bc, the scatters form spirals that tends to move inwards towards the center for several ground motions. As already discussed, formation of a spiral shows that the response is sensitive to the change in the variable.
Scattering (formation of spirals) is highest in the OGS frames and least in the FI frames.
A change in the value of width of column changes the sectional capacity, i.e., the yield and ultimate rotational capacities of the column members. This implies that the member section properties, i.e., the yield and ultimate curvature are sensitive parameters that yield a significant amount of variation in the response of OGS frames. There is no change in frame response due to a change in the input variable Db for any ground motion in bare and OGS frames, i.e., the scatter forms a perfect circle. In FI frame, there is a slight variation in the response, however, the COV suggests that the values are negligible.
0.0 0.2 0.4 0.6 0.8 1.0
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Bc: COV
0.02 0.12 0.05 0.04 0.03 0.06 0.14
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16
Bc: COV
0.08 0.19 0.07 0.03 0.01 0.16 0.07
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Bc: COV
0.01 0.06 0.01 0.01 0.04 0.01 0.02 1
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16
Db: COV
0.01 0.02 0.02 0.00 0.02 0.02 0.01
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Db: COV
0.00 0.01 0.00 0.00 0.00 0.01 0.00
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Db: COV
0.01 0.04 0.00 0.00 0.02 0.00 0.01
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Comparing the sensitivity of Db and Bc, it is inferred that the frame (bare, OGS or FI) response is more sensitive to any change in dimension of column rather than in beams.
The yield and ultimate rotation in columns are important as influencing parameters.
(a) (b) (c)
Figure 5.3 Displacement sensitivity radar charts for uncertain material input parameters with their COV for seven ground motions: (a) bare frames, (b) OGS frames, and (c) FI frames.
Significant variation in frame response (spirals in radar charts) is observed in the bare and OGS frames with a change in fck (Fig. 5.3). The value of COV ranges from 1% to 12% and 1% to 16%, respectively, in bare and OGS frames, indicating that there is little to moderate sensitivity in response of bare and OGS frames to change in fck. In fully infilled
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16
fck: COV
0.01 0.12 0.06 0.03 0.01 0.05 0.05
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fck: COV
0.06 0.16 0.04 0.01 0.00 0.13 0.04
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fck: COV
0.02 0.06 0.02 0.02 0.06 0.03 0.03 1
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16
fy: COV
0.01 0.01 0.00 0.00 0.05 0.03 0.00
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5
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16
fy: COV
0.01 0.00 0.02 0.00 0.00 0.00 0.03
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16
fy: COV
0.01 0.03 0.00 0.00 0.05 0.00 0.00 1
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16
ξ: COV
0.03 0.05 0.05 0.02 0.00 0.01 0.05
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5
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16
ξ: COV
0.06 0.02 0.04 0.01 0.01 0.05 0.07
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16
ξ: COV
0.06 0.19 0.04 0.05 0.24 0.06 0.04
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5.5 Sensitivity from Nonlinear Time-History Analyses
frames, the COV is found to be negligible in the range of 2% to 6%. In case of fy also, the scatters form almost perfect circle indicating that there is no sensitivity of bare, OGS or FI frame responses due to the change in fy (the COV lies in the range of 1% to 5%). Lesser influence of fy, as compared to fck, on sensitivity of seismic response of RC frames with different storey heights is also reported by Celik and Ellingwood (2010). The change in ξ shows the highest influence in response output of FI frames with a coefficient of variation as high as 24%. However, the effect of change in damping is negligible in response of BF and OGS frames with COV lying in the range of 1% to 7% only. There is a minor variation in the frame responses with respect to a change in the input parameter γc (Fig. 5.4). The response to infill load (γm or IL) variation also shows minor to moderate sensitivity with maximum COV of 3%, 9% and 12% in BF, OGS and FI frame, respectively.
(a) (b) (c)
Figure 5.4 Displacement sensitivity radar charts for uncertain load input parameters with their COV for seven ground motions: (a) bare frames, (b) OGS frames, and (c) FI frames.
Three additional parameters (fm′, Ws, and εm) are considered in case of FI frames only, for which the variation in response due to input uncertainty are plotted as radar charts in Fig. 5.5. There occurs a large variation in the response due to uncertainty in fm′
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γc: COV
0.00 0.04 0.02 0.01 0.01 0.00 0.02
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γc: COV
0.03 0.06 0.01 0.00 0.00 0.05 0.02
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γc: COV
0.04 0.08 0.03 0.03 0.10 0.02 0.05 1
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IL: COV
0.00 0.06 0.02 0.01 0.02 0.01 0.04
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IL: COV
0.04 0.09 0.02 0.00 0.00 0.08 0.03
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IL: COV
0.06 0.06 0.06 0.08 0.12 0.03 0.08
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and Ws as seen from the formation of spiral in Fig. 5.5(a) and Fig. 5.5(b), respectively.
Individual COVs for different ground motions are also very high with a maximum of 58%
and 52% for fm′ and Ws, respectively. On the other hand, the variation in response due to a change in εm is negligible for most ground motions, except for the two cases where the COV is 11%. Thus, the effect of fm′ and Ws should be emphasized in the sensitivity of displacement response of FI frames.
(a) (b) (c)
Figure 5.5 Displacement sensitivity radar charts for uncertain input parameters in FI frames:
(a) fm′, (b) Ws, and (c) εm, with their coefficient of variation (COV) for seven ground motions.