Abstract
The pedestrian-induced lateral vibration of footbridges is essentially classified as a nonlinear stochastic vibration. Accordingly, bridge vibration stability falls within the field of nonlinear stochastic vibration stability. At present, the Lyapunov method is mainly used to analyze such stability. However, this method is qualitative, and it cannot quantitatively analyze the vibration stability probability. In this study, a new analytical method based on a comparison of the input energy and the variation of intrinsic energy (IEVIE) is used to analyze the nonlinear stochastic vibration stability of the lateral vibration of the footbridge. The improved Nakamura model is used to describe the lateral nonlinear stochastic vibration of the footbridge. A combination of the IEVIE method and the probability density evolution (PDE) method is then proposed, in which the IEVIE method is utilized to determine vibration stability. The PDE method is used to obtain the reliability of vibration stability. The proposed method is successfully applied to the Millennium Bridge, and its effectiveness is verified by comparing the Monte Carlo and Lyapunov methods. The proposed method can obtain the dynamic probability of the vibration as stable or instable and provide a reference for quantitative analysis of lateral nonlinear stochastic vibration stability of footbridges.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bayat M and Pakar I (2012), “Accurate Analytical Solution for Nonlinear Free Vibration of Beams,” Structural Engineering and Mechanics, 43(3): 337–347.
Brownjohn JMW, Fok P, Roche M, et al. (2004a), “Long Span Steel Pedestrian Bridge at Singapore Changi Airport — Part 1: Prediction of Vibration Serviceability Problems,” Structural Engineer, 82(16): 21–27.
Brownjohn JMW, Fok P, Roche M, et al. (2004b), “Long Span Steel Pedestrian Bridge at Singapore Changi Airport — Part 2: Crowd Loading Tests and Vibration Mitigation Measures,” Structural Engineer, 82(16): 28–34.
Brownjohn JMW, Pavic A and Omenzetter P (2004c), “A Spectral Density Approach for Modelling Continuous Vertical Forces on Pedestrian Structures Due to Walking,” Canadian Journal of Civil Engineering, 31(1): 65–77.
Chen JB and Li J (2005), “Extreme Value Distribution and Reliability of Nonlinear Stochastic Structures,” Earthquake Engineering and Engineering Vibration, 4(2): 275–286.
Chen JB and Li J (2007), “Joint Probability Density Function of the Stochastic Responses of Nonlinear Structures,” Earthquake Engineering and Engineering Vibration, 6(1): 35–47.
Chen JB and Li J (2005), “Dynamic Response and Reliability Analysis of Non-Linear Stochastic Structures,” Probabilistic Engineering Mechanics, 20(1): 33–44.
Chen JB, Ghanem R and Li J (2009), “Partition of the Probability-Assigned Space in Probability Density Evolution Analysis of Nonlinear Stochastic Structures,” Probabilistic Engineering Mechanics, 24(1): 27–42.
Dallard P, Fitzpatrick T, Flint A, et al. (2001a), “The London Millennium Footbridge,” Structural Engineer, 79(171): 17–33.
Dallard P, Fitzpatrick T, Flint A, et al. (2001b), “London Millennium Bridge: Pedestrian-Induced Lateral Vibration,” Journal of Bridge Engineering, 6(6): 412–417.
Deng Y, Ding YL and Li AQ (2010), “Structural Condition Assessment of Long-Span Suspension Bridges Using Long-Term Monitoring Data,” Earthquake Engineering and Engineering Vibration, 9(1): 127–135.
Fang KT and Wang Y (1994), Application of Number-Theoretic Methods in Statistics, CRC Press, London, UK.
Fujino Y, Pacheco BM, Nakamura SI, et al. (1993), “Synchronization of Human Walking Observed During Lateral Vibration of a Congested Pedestrian Bridge,” Earthquake Engineering and Structural Dynamics, 22(9): 741–758.
Hoorpah W FO and Cespedes X (2008), “The Simon De Beauvoir Footbridge in Paris. Experimental Verification of the Dynamic Behaviour Under Pedestrian Loads and Discussion of Corrective Modifications,” Proceedings of the Proceedings of Footbridge 2008, Porto, Portugal.
Huang ZL, Zhu WQ, Ni YQ, et al. (2002), “Stochastic Averaging of Strongly Non-Linear Oscillators Under Bounded Noise Excitation,” Journal of Sound and Vibration, 254(2): 245–267.
Ingólfsson ET and Georgakis CT (2011), “A Stochastic Load Model for Pedestrian-Induced Lateral Forces on Footbridges,” Engineering Structures, 33(12): 3454–3470.
Ingólfsson ET, Georgakis CT and Jönsson J (2012a), “Pedestrian-Induced Lateral Vibrations of Footbridges: A Literature Review,” Engineering Structures, 45(15): 21–52.
Ingólfsson ET, Georgakis CT, Ricciardelli F, et al. (2012b), “Experimental Identification of Pedestrian-Induced Lateral Forces on Footbridges,” Journal of Sound & Vibration, 330(6): 1265–1284.
Jia BY, Yu XL, Yan QS, et al. (2017), “Nonlinear Stochastic Analysis for Lateral Vibration of Footbridge under Pedestrian Narrowband Excitation,” Mathematical Problems in Engineering, 2017(22): 1–12.
Jia BY, Yu XL and Yan QS (2018), “Effects of Stochastic Excitation and Phase Lag of Pedestrians on Lateral Vibration of Footbridges,” International Journal of Structural Stability & Dynamics, 18(7): 1–17.
Kozin F (1969), “A Survey of Stability of Stochastic Systems,” Automatica, 5(1): 95–112.
La Salle J and Lefschetz S (2012), Stability by Liapunov’s Direct Method with Applications by Joseph L Salle and Solomon Lefschetz, Elsevier, New York.
Li J and Chen JB (2008), “The Principle of Preservation of Probability and the Generalized Density Evolution Equation,” Structural Safety, 30(1): 65–77.
Li J (2016), “Probability Density Evolution Method: Background, Significance and Recent Developments,” Probabilistic Engineering Mechanics, 44(2016): 111–117.
Li J and Xu J (2016), “A Quantitative Approach to Stochastic Dynamic Stability of Structures,” Chinese Journal of Theoretical and Applied Mechanics, 48(3): 702–713. (in Chinese)
Li YC and Wang Z (2016), “Unstable Characteristics of Two-Dimensional Parametric Sloshing in Various Shape Tanks: Theoretical and Experimental Analyses,” Journal of Vibration and Control, 22(19): 4025–4046.
Macdonald JHG (2009), “Lateral Excitation of Bridges by Balancing Pedestrians,” Proceedings of the Royal Society a Mathematical Physical & Engineering Sciences, 465(2104): 1055–1073.
Nakamura S (2004), “Model for Lateral Excitation of Footbridges by Synchronous Walking,” Journal of Structural Engineering, 130(1): 32–37.
Nawrotzki P and Eller C (2000), “Numerical Stability Analysis in Structural Dynamics,” Computer Methods in Applied Mechanics and Engineering, 189(3): 915–929.
Piccardo G and Tubino F (2008), “Parametric Resonance of Flexible Footbridges under Crowd-Induced Lateral Excitation,” Journal of Sound & Vibration, 311(1–2): 353–371.
Racic V and Brownjohn JMW (2011a), “Stochastic Model of near-Periodic Vertical Loads Due to Humans Walking,” Advanced Engineering Informatics, 25(2): 259–275.
Racic V and Brownjohn JMW (2011b), “Mathematical Modelling of Random Narrow Band Lateral Excitation of Footbridges Due to Pedestrians Walking,” Computers & Structures, 90–91(1): 116–130.
Ricciardelli F, Mafrici M and Ingólfsson ET (2014), “Lateral Pedestrian-Induced Vibrations of Footbridges: Characteristics of Walking Forces,” Journal of Bridge Engineering, 19(9): 1265–1266.
Shu CW (1988), “Total-Variation-Diminishing Time Discretizations,” SIAM Journal on Scientific and Statistical Computing, 9(6): 1073–1084.
Sun WL, Chen JB and Li J (2011), “Stochastic Harmonic Functions of Second Kind for Spectral Representations,” Journal of Tongji University, Natural Science, 39(10): 1413–1419. (in Chinese)
Wait I, Yang ZJ, Chen G, et al. (2019), “Wind-Induced Instabilities and Monitoring of Wind Turbine,” Earthquake Engineering and Engineering Vibration, 18(2): 475–485.
Wang R and Ru CQ (1985), “An Energy Criterion for the Dynamic Plastic Buckling of Circular Cylinders under Impulsive Loading,” Metal Forming and Impact Mechanics. Elsevier, New York, 213–223.
Xie WC (2003), “Moment Lyapunov Exponents of a Two-Dimensional System Under Bounded Noise Parametric Excitation,” Journal of Sound and Vibration, 263(3): 593–616.
Xu J and Li J (2015), “An Energetic Criterion for Dynamic Instability of Structures Under Arbitrary Excitations,” International Journal of Structural Stability and Dynamics, 15(02): 2496–2606.
Yuan XB (2006), “Research on Pedestrian-Induced Vibration of Footbridge,” Tongji University, Shanghai. (in Chinese)
Zhou GD, Yi TH and Chen B (2017), “Innovative Design of a Health Monitoring System and Its Implementation in a Complicated Long-Span Arch Bridge,” Journal of Aerospace Engineering, 30(2): 4016001–4016017.
Zhou GD, Yi TH, Zhang H, et al. (2015), “Energy-Aware Wireless Sensor Placement in Structural Health Monitoring Using Hybrid Discrete Firefly Algorithm,” Structural Control & Health Monitoring, 22(4): 648–666.
Zhu WQ (1992), Nonlinear Stochastic Dynamics and Control—Hamiltonian Theoretical Framework, Science Press, Beijing. (in Chinese)
Acknowledgment
This research was supported by the National Natural Science Foundation of China (No. 51608207), the Natural Science Foundation of Guangdong Province, China (No. 2019A1515011941) and the China Scholarship Council (No. 201806155102, No. 201906155028).
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by
National Natural Science Foundation of China under Grant No. 51608207, the Natural Science Foundation of Guangdong Province, China under Grant No. 2019A1515011941, and China Scholarship Council under Grant Nos. 201806155102 and 201906155028
Rights and permissions
About this article
Cite this article
Jia, B., Yu, X. Random stability for lateral vibration on footbridge based on IEVIE-PDE method. Earthq. Eng. Eng. Vib. 20, 981–992 (2021). https://doi.org/10.1007/s11803-021-2063-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11803-021-2063-2