Efficient Moment-Independent Sensitivity Analysis of Uncertainties in Seismic Demand of Bridges Based on a Novel Four-Point-Estimate Method
Abstract
:1. Introduction
2. An Efficient Algorithm for Moment-Independent Importance Analysis
2.1. Moment-Independent Importance Index
2.2. The Modified 4PEM
2.3. Moment-Independent Sensitivity Analysis Based on 4PEM and SGLD
2.4. The Procedures of the Proposed Method
- Set the input vector composed of the uncertain parameters from structural and material properties.
- Implement 4PEM to the structural uncertain parameter vector to obtain the locations and weights .
- Change the uncertain parameters in the structural model according to and obtain the samples of seismic demand through NTHA.
- Calculate the first four moments of by Equation (6).
- Evaluate through SGLD.
- Fix at and implement 4PEM to to obtain the new locations and weights . Repeat step (2)~(4) to obtain .
- Calculate the expectation by Equations (12) and (11) and obtain the MII index by Equation (4).
- Repeat step (5) and (6) until the MII index calculation of all the variables of is finished.
3. Numerical Example
3.1. Bridge Model Description
3.2. Uncertain Parameters
3.3. Characterization of Bridges’ Seismic Demands
3.4. Moment-Independent Sensitivity Results
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
the expectation of * | |
the marginal PDF of | |
the unconditional PDF of | |
the conditional PDF by fixing at one implementation value | |
the conditional PDF under the condition of fixing at its 4PEM locations | |
nonlinear performance function of the system | |
the area difference between and | |
the n-dimensional vector of the structural input variables | |
(n − 1)-dimensional transformation of by removing | |
realization of the random variable | |
the output seismic demand | |
the realization of by the point-estimated method | |
the realization of when | |
the r-th central moment of * | |
the location of for the point-estimate method | |
the moment-independent importance index | |
the ratio of to the -th power of | |
the mean value of | |
the point-estimated method’s sampling weight | |
the point-estimated method’s sampling coefficients |
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Uncertain Parameters | Symbol | Distribution | Mean | COV | Ref. |
---|---|---|---|---|---|
Mass per-length of superstructure (kg/m) | Normal | 362 | 0.10 | [26,33] | |
Post-yield stiffness of LRB (kN/m) | Normal | 96 | 0.14 | [33] | |
Peak compressive strength of unconfined concrete (MPa) | Lognormal | 26.1 | 0.14 | [32,33] | |
Strain at the peak strength of unconfined concrete | Lognormal | 0.002 | 0.20 | [32,33] | |
Strain at the crushing strength of unconfined concrete | Lognormal | 0.005 | 0.20 | [32,33] | |
Peak compressive strength of confined concrete (MPa) | Lognormal | 33.6 | 0.21 | [32,33] | |
Strain at the peak strength of confined concrete | Lognormal | 0.004 | 0.2 | [32,33] | |
Strain at the crushing strength of confined concrete | Lognormal | 0.015 | 0.3 | [32,33] | |
Young’s modulus of steel rebar (MPa) | Lognormal | 2E5 | 0.02 | [32,34] | |
Initial yield stress of steel rebar (MPa) | Lognormal | 378 | 0.07 | [32,34] | |
Strain hardening ratio of steel rebar | Lognormal | 0.01 | 0.20 | [34] |
MPTD | MBD | |||||||
---|---|---|---|---|---|---|---|---|
4PEM | 10MC | 100MC | Rank | 4PEM | 10MC | 100MC | Rank | |
0.0303 | 0.0585 | 0.0335 | 5-3-5 * | 0.0494 | 0.0172 | 0.0376 | 5-10-5 | |
0.0165 | 0.0289 | 0.0175 | 10-7-10 | 0.0374 | 0.0435 | 0.0191 | 9-4-9 | |
0.0505 | 0.1147 | 0.0354 | 3-1-3 | 0.0385 | 0.0590 | 0.0211 | 8-1-8 | |
0.0569 | 0.0648 | 0.0525 | 1-2-1 | 0.0579 | 0.0379 | 0.0434 | 3-5-3 | |
0.0487 | 0.0529 | 0.0336 | 4-5-4 | 0.0478 | 0.0369 | 0.0346 | 6-6-6 | |
0.0244 | 0.0126 | 0.0214 | 8-8-8 | 0.0045 | 0.0303 | 0.0179 | 11-7-11 | |
0.0246 | 0.0106 | 0.0223 | 7-10-7 | 0.0550 | 0.0130 | 0.0431 | 4-11-4 | |
0.0006 | 0.0115 | 0.0164 | 11-9-11 | 0.0395 | 0.0246 | 0.0213 | 7-8-7 | |
0.0249 | 0.0084 | 0.0248 | 6-11-6 | 0.0235 | 0.0180 | 0.0188 | 10-9-10 | |
0.0167 | 0.0568 | 0.0207 | 9-4-9 | 0.0703 | 0.0528 | 0.0542 | 1-2-1 | |
0.0551 | 0.0474 | 0.0381 | 2-6-2 | 0.0641 | 0.0484 | 0.0505 | 2-3-2 |
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Li, X.; Lei, Y.; Liu, L. Efficient Moment-Independent Sensitivity Analysis of Uncertainties in Seismic Demand of Bridges Based on a Novel Four-Point-Estimate Method. Appl. Sci. 2021, 11, 10405. https://doi.org/10.3390/app112110405
Li X, Lei Y, Liu L. Efficient Moment-Independent Sensitivity Analysis of Uncertainties in Seismic Demand of Bridges Based on a Novel Four-Point-Estimate Method. Applied Sciences. 2021; 11(21):10405. https://doi.org/10.3390/app112110405
Chicago/Turabian StyleLi, Xingyu, Ying Lei, and Lijun Liu. 2021. "Efficient Moment-Independent Sensitivity Analysis of Uncertainties in Seismic Demand of Bridges Based on a Novel Four-Point-Estimate Method" Applied Sciences 11, no. 21: 10405. https://doi.org/10.3390/app112110405
APA StyleLi, X., Lei, Y., & Liu, L. (2021). Efficient Moment-Independent Sensitivity Analysis of Uncertainties in Seismic Demand of Bridges Based on a Novel Four-Point-Estimate Method. Applied Sciences, 11(21), 10405. https://doi.org/10.3390/app112110405