Geometric Optimization of Pedestrian Bridges

Abstract

Advancements in structural engineering increasingly lead to more slender and architecturally challenging footbridge designs, characterized by reduced height-to-span ratios and lower self-weight. These designs, often serving as landmarks with aesthetic and functional purposes, are becoming more susceptible to human-induced vibrations. The increased live-to-dead load ratio and reduced eigenfrequencies increase the risk of resonance when pedestrian step frequencies align with the structure’s natural frequencies. This resonance amplifies deck accelerations, compromising pedestrian comfort and freedom of movement. Synchronization effects are at risk of further intensifying amplification and obstructing movement. No studies have demonstrated that human-induced vibrations cause structural failure in the ultimate limit state (ULS); rather, they primarily affect user comfort in the serviceability limit state (SLS).

Over the past three decades, substantial research has focused on understanding and mitigating humaninduced vibrations in footbridges. The temporary closures of iconic structures such as the Passerelle Solférino in Paris (1999) and the London Millennium Bridge (2000) have potentially accelerated findings and highlighted the significant challenges posed by pedestrian-structure interaction. In particular, a focus is drawn to the lateral lock-in phenomenon, which resulted in excessive lateral vibrations and discomfort. Lateral lock-in and other human-induced incidents resulted in extensive testing, leading to the development of new guidelines and advancing the study of lightweight footbridges across Europe through both in-situ testing and numerical simulations. However, the literature reveals a considerable variation in assessment methods and verification techniques, which complicates the accurate evaluation of a footbridge’s dynamic behaviour by structural engineers. Nevertheless, a consensus is reached in the literature that design situations must be carefully considered in every footbridge design. Pedestrian comfort and dynamic response are crucial design factors, requiring a thorough understanding of expected traffic patterns and the structure’s dynamic behaviour. Design situations encompass a range of conditions, such as daily pedestrian use or special events, to establish realistic performance limits under ranging circumstances. Studies show that higher pedestrian density leads to reduced walking speeds and restricted movement, which in turn influences the dynamic loads on the structure. Comfort is assessed through acceleration measurements during loading, with predefined ranges to categorize acceptable performance levels. These evaluations stress the importance of a comprehensive analysis of dynamic effects, rather than relying on a single limit criterion.

To mitigate the issues observed in the aforementioned bridge designs, external control devices are applied to offer additional damping and reduce vibrations to acceptable levels. These control devices are applied after footbridge construction, enabling thorough testing of the bridge to determine the structure’s dynamic properties and ensuring the damping system is optimized and properly tuned. External damping has various forms applied in civil engineering structures. Most notably there are three categories to be distinguished, namely: tuned mass/liquid, viscoelastic and viscous fluid dampers. Tuned mass dampers (TMDs) are most commonly used due to their ease of application, allowing for effective control of vibrations post-installation. By tuning the TMD’s eigenfrequency to match the primary structure’s critical natural frequency, energy dissipation is achieved through the mass of the damper and its motion relative to the structure to which it is attached. TMD design is highly effective in controlling the target frequency. However, it should be noted that this localized damping primarily addresses the response of a single frequency, rather than the total response of the structure. If eigenfrequencies are closely spaced, a shift in frequency could lead to a new resonant response in the overall structure, as damping the initially critical mode may consequently amplify nearby modes.

These imposed challenges raise the question if reducing excessive human-induced vibrations within footbridge design can be achieved through other means. A promising method regards the geometric modification of the structural design, providing a change of dynamic characteristics and reducing resonance effects. Modern-day advancements enable engineers to perform more complex problem solving, namely through computational power by utilising optimisation techniques which require many iterations. An optimisation is characterised by its objective function, constraints, design variables and requirements it should satisfy. Evolutionary algorithms, such as genetic behaviour from groups observed by animals in nature provide effective results for optimisation.

To provide context to such an optimisation algorithm, a case study is presented, showcasing how the concept can be utilised. The optimization primarily targets reducing the structure’s mass as an objective, improving cost and sustainability while influencing dynamic performance. Minimizing accelerations is likewise pursued to evaluate the extent of reduction possible and identify the most influential geometric parameters. The structure must meet ultimate limit state requirements in the optimized design to ensure feasibility. Data from the original design, including FEM models, analysis reports, and TMD specifications, informs the optimization process. A parametric model is developed to support geometric optimization. Key design variables, objectives and constraints are carefully selected to maximize the effectiveness of the optimization and achieve a design that either mitigates or eliminates the need for external damping devices.

When footbridge design deviates significantly from conventional girder bridge design will the effectiveness of assessment methods drop, requiring more extensive analysis to address dynamic behaviour. The presented case study shows the closest adherence to measured results via direct time integration, being most costly in time whilst requiring a substantial level of engineering judgment. Furthermore, does the correct assessment of damping in footbridge design play a major role, showing agreement with the proposed mean damping values addressed in the literature.

Optimisation to exclude the need for external damping devices through evolutionary algorithms by conducting a geometric parameter study is a feasible approach. However, it requires a deep understanding of the structural behaviour of footbridges and a robust parametric model capable of performing both static and dynamic analyses to account for geometric changes. In the case study, significant improvements were achieved through this optimization process, resulting in a new design that eliminates the need for TMDs.