Call For Paper - Upcoming Conferences
Research Article | Open Access | Download PDF
Volume 13 | Issue 6 | Year 2026 | Article Id. IJCE-V13I6P114 | DOI : https://doi.org/10.14445/23488352/IJCE-V13I6P114

Numerical Simulation of Crowd-Induced Vibrations Using Duhamel's Integral

Albert Jorddy Valenzuela Inga, Jesus Eugenio Depaz Huertas, Pamela Rodríguez Pérez, Giancarlo Fernando Meza Terbullino

Received Revised Accepted Published
16 Mar 2026 15 Apr 2026 14 May 2026 30 Jun 2026

Citation :

Albert Jorddy Valenzuela Inga, Jesus Eugenio Depaz Huertas, Pamela Rodríguez Pérez, Giancarlo Fernando Meza Terbullino, "Numerical Simulation of Crowd-Induced Vibrations Using Duhamel's Integral," International Journal of Civil Engineering, vol. 13, no. 6, pp. 205-217, 2026. Crossref, https://doi.org/10.14445/23488352/IJCE-V13I6P114

Abstract

Crowd-induced vibrations in footbridges represent a relevant issue in the design of lightweight structures. The numerical simulation performed in this research, which is based on Duhamel’s integral, allows for the estimation of the dynamic response to pedestrian loads. Within this simulation, the structure is initially modeled as a Single-Degree-of-Freedom (SDOF) system that is subjected to ramp, harmonic, and pedestrian-type excitations, which allows for validating the numerical implementation against already known analytical solutions. After this, a simplified modal model of a supported beam is adopted, considering only the first vibration mode. The footfall force is represented through harmonic functions adjusted by walking speed and the Dynamic Load Factor (DLF), including the random distribution of speeds in individual walks and in crowds. The simulations, developed in Python, show that the system responds with increasing amplitudes when the step frequency approaches the natural frequency. In the case of crowds of 200 pedestrians, maximum displacements close to 2.50 mm and amplification of the modal envelope were observed during the minutes of greatest overlap. The model is efficient and applicable in preliminary design stages. Its limitations include structural linearity and the absence of pedestrian–structure interaction, although it provides results consistent with the reviewed literature.

Keywords

Duhamel’s Integral, Pedestrian Load, Structural Resonance, Numerical Simulation.

References

  1. C. Maraveas, Z.C. Fasoulakis, and K.D. Tsavdaridis, “A Review of Human Induced Vibrations on Footbridges,” American Journal of Engineering and Applied Sciences, vol. 8, no. 4, pp. 422-433, 2015.
    [    CrossRef] [Google Scholar] [Publisher Link]
  2. F. Lucà et al., “Human-Structure Interaction: Convolution-Based Estimation of Human-Induced Vibrations Using Experimental Data,” Mechanical Systems and Signal Processing, vol. 167, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  3. Edwin Lerner, London Millennium Footbridge, Also Known as the Wibbly Wobbly Bridge, Guide London, 2023. [Online]. Available: http://www.guidelondon.org.uk/blog/around-london/london-millennium-footbridge/
  4. Haikuan Liu et al., “Dynamic Amplification Analysis of the Main Girder of Cable-Stayed Bridges After Cable Rupture Using the Modal Superposition Method,” Buildings, vol. 15, no. 4, pp. 1-19, 2025.
    [    CrossRef] [Google Scholar] [Publisher Link]
  5. Liangkun Wang, Ying Zhou, and Weixing Shi, “Random Crowd-Induced Vibration in Footbridge and Adaptive Control using Semi-Active TMD Including Crowd-Structure Interaction,” Engineering Structures, vol. 306, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  6. Jiecheng Xiong et al., “A Review of Evaluation Methods of Standards for Structural Vibration Serviceability under Crowd Walking,” Buildings, vol. 14, no. 3, pp. 1-19, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  7. Simone Turrisi, Emanuele Zappa, and Alfredo Cigada, “Experimental Validation of a Vision-Based Technique to Estimate the Crowd Loading on Stadium Grandstands,” IEEE Open Journal of Instrumentation and Measurement, vol. 1, pp. 1-12, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  8. Dong Cao, Zuanfeng Pan, and Yu Fang, “Dynamic Response Analysis of the Floor Structure under Random Crowd Excitation,” Shock and Vibration, vol. 2024, no. 1, pp. 1-19, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  9. Jun Chen, Bingqian Yu, and Haoqi Wang, “Time-Dependent Synchronization Factor of Crowd Rhythmic Motion and its Application on Intelligent Structural Monitoring,” Engineering Structures, vol. 308, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  10. Xinxin Wei, Jin-Cheng Liu, and Sifeng Bi, “Uncertainty Quantification and Propagation of Crowd Behaviour Effects on Pedestrian-Induced Vibrations of Footbridges,” Mechanical Systems and Signal Processing, vol. 167, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  11. Christian Gallegos-Calderón et al., “A Frequency-Domain Approach to Model Vertical Crowd-Structure Interaction in Lightweight Footbridges,” Journal of Sound and Vibration, vol. 557, pp. 1-17, 2023.
    [    CrossRef] [Google Scholar] [Publisher Link]
  12. Haoqi Wang et al., “Human-Induced Vibration Serviceability: From Dynamic Load Measurement towards the Performance-Based Structural Design,” Buildings, vol. 13, no. 8, pp. 1-23, 2023.
    [    CrossRef] [Google Scholar] [Publisher Link]
  13. Dongjun Zeng, Haoqi Wang, and Jun Chen, “Dynamic Reliability Analysis of Large-Span Structures under Crowd Bouncing Excitation,” Buildings, vol. 12, no. 3, pp. 1-18, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  14. Yiwen Dong et al., “Context-Aware Crowd Monitoring for Sports Games Using Crowd-Induced Floor Vibrations,” Data-Centric Engineering, vol. 5, pp. 1-24, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  15. Javier Naranjo-Pérez et al., “Vertical Crowd–Structure Interaction Modeling: Numerical and Experimental Assessment of a Cable-Stayed Footbridge,” Journal of Bridge Engineering, vol. 29, no. 4, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  16. Michael Fouli, and Alfredo Camara, “Human–Structure Interaction Effects on Lightweight Footbridges with Tuned Mass Dampers,” Structures, vol. 62, pp. 1-16, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  17. Seyedmilad Komarizadehasl et al., “Development of a Low-Cost System for the Accurate Measurement of Structural Vibrations,” Sensors, vol. 21, no. 18, pp. 1-22, 2021.
    [    CrossRef] [Google Scholar] [Publisher Link]
  18. Chase Hibbard et al., “Probabilistic Assessment of Footfall Vibration,” Model Validation and Uncertainty Quantification, Conference Proceedings of the Society for Experimental Mechanics Series, pp. 125-133, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  19. K. Van Nimmen, A. Pavic, and P. Van den Broeck, “A Simplified Method to Account for Vertical Human-Structure Interaction,” Structures, vol. 32, pp. 2004-2019, 2021.
    [    CrossRef] [Google Scholar] [Publisher Link]
  20. Liangkun Wang et al., “Experimental Study on Adaptive-Passive Tuned Mass Damper with Variable Stiffness for Vertical Human-Induced Vibration Control,” Engineering Structures, vol. 280, 2023.
    [    CrossRef] [Google Scholar] [Publisher Link]
  21. Edwin Hidalgo Banda Roque, and William Conya Ascue, “Determination of the Dynamic Vibration Properties and Calibration of the Numerical Model of the Tupac Amaru Avenue By-Pass Bridge Using Oma Tests, Cusco – 2023,” National University of San Antonio Abad of Cusco, pp. 1-187, 2024.
    [    Google Scholar] [Publisher Link]
  22. Córdoba Acosta, and Nicolás Daniel, “Publication: Analysis of Vibrations in Pedestrian Bridges and Determination of the Comfort Level for Pedestrians,” Thesis, University of Antioquia, 2023.
    [    Google Scholar] [Publisher Link]
  23. Jesús de Sebastián Sanz, “Serviceability Limit State Analysis and Vibration Control in Pedestrian Walkways,” Thesis, University of Valladolid, Spain, pp. 1-306, 2014.
    [    CrossRef] [Google Scholar] [Publisher Link]
  24. Kenyo Eul Loa López, “Control of the Dynamic Response in the Los Próceres Pedestrian Bridge through the Incorporation of Viscous and Tuned Mass Dampers,” Thesis, Pontifical Catholic University, pp. 1-112, 2022.
    [    Google Scholar] [Publisher Link]
  25. Mohamad Emami, Morteza Eskandari-Ghadi, and Amir K. Ghorbani-Tanha, “Generalization of Duhamel’s Integral to Multi-Degree-of-Freedom Systems,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 478, no. 2259, pp. 1-22, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  26. Mohamed Guesmi, Nassira Belkheiri, and Mohamed Lakhder Guesmi, “Time History Analysis of Structures Under Multi-Support Excitation by State-Space Method,” Studies in Engineering and Exact Sciences, vol. 5, no. 1, pp. 209-222, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  27. Xiaohan Wang et al., “Structural Dynamics and Vibration Analysis of the MDOF Systems,” Journal of Physics: Conference Series, vol. 2381, pp. 1-8, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  28. Fang-Yin Shen et al., “Vision-based Estimation Method of Crowd-Induced Vertical Dynamic Loading and Vibration Analysis,” Measurement, vol. 236, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  29. Sampo Laine et al., “OpenTorsion: Python Library for Torsional Vibration Analysis,” SoftwareX, vol. 29, pp. 1-7, 2025.
    [    CrossRef] [Google Scholar] [Publisher Link]
  30. Le-Hung Tran et al., “An Analytical Model to Calculate the Forced Vertical Vibrations of Two Rails Subjected to the Dynamic Loads of Ballasted Railway Track,” Structures, vol. 68, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  31. Michał Różański et al., “On Theoretical and Practical Aspects of Duhamel’s Integral,” Archives of Control Sciences, vol. 31, no. 4, pp. 815-847, 2021.
    [    CrossRef] [Google Scholar] [Publisher Link]
  32. Jinbao Yao et al., “Pedestrian-Induced Bridge Vibration Driven by Behavioral Preferences,” Buildings, vol. 15, no. 22, pp. 1-24, 2025.
    [    CrossRef] [Google Scholar] [Publisher Link]
  33. Anil K. Chopra, Structural Dynamics, 4th ed., Pearson, pp. 1-756, 2014.
    [    Google Scholar] [Publisher Link]
  34. Anil K. Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering, 4th ed., SI Units, Harlow, England: Pearson Education, 2020.
    [    Google Scholar] [Publisher Link]
  35. Paz Mario, and William Leigh, Structural Dynamics: Theory and Computation, 5th ed., Boston, MA, USA: Springer Science & Business Media, 2006.
    [    Google Scholar] [Publisher Link]
  36. Fiammetta Venuti, Luca Bruno, and Paolo Napoli, “Pedestrian Lateral Action on lively Footbridges: A New Load Model,” Structural Engineering International, vol. 17, no. 3, pp. 236-241, 2008.
    [    CrossRef] [Google Scholar] [Publisher Link]
  37. Charles R. Harris et al., “Array Programming with NumPy,” Nature, vol. 585, pp. 357-362, 2020.
    [    CrossRef] [Google Scholar] [Publisher Link]
  38. John D. Hunter, “Matplotlib: A 2D Graphics Environment,” Computing in Science & Engineering, vol. 9, no. 3, pp. 90-95, 2007.
    [    CrossRef] [Google Scholar] [Publisher Link]
  39. National Annex of AN/UNE-EN 1990, Eurocode 0: Basis of Structural Design and Annex A2: Application to Bridges, 2026. [Online]. Available: https://cdn.mitma.gob.es/portal-web-drupal/carreteras/normativa/AN_UNE-EN1990.pdf
  40. Emma Moliner, Pedro Museros, and Reza Allahvirdizadeh, “Track–Bridge Interaction Effects in the Acceleration and Displacement Response of High-Speed Railway Bridges: Simplified vs Refined Modelling,” Engineering Structures, vol. 314, pp. 1-17, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  41. A.E. Martínez-Castro, P. Museros, and A. Castillo-Linares, “Semi-Analytic Solution in the Time Domain for Non-Uniform Multi-Span Bernoulli–Euler Beams Traversed by Moving Loads,” Journal of Sound and Vibration, vol. 294, no. 1, pp. 278-297, 2006.
    [    CrossRef] [Google Scholar] [Publisher Link]