Whilst footfall induced vibrations on buildings and bridges is not normally significant in terms of structural integrity, footfall induced vibration can be a critical serviceability condition. This publication presents a method of evaluating the vibration due to a single pedestrian walking on a flat surface, such as a floor slab or bridge deck. Eurocode 1 BS Standards for precast concrete. Innovative concrete 3D printed concrete Bioreceptive concrete Carbon-Capture Algae Robotic formwork Self-healing concrete Visual concrete Exposed concrete floors Visual blockwork and masonry Visual in-situ concrete Visual precast concrete. Planning National planning policy framework.

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Whilst footfall induced vibrations on buildings or bridges is not normally signicant in terms of structural integrity, footfall vibration can be a critical serviceability condition. This publication presents a new method for evaluating the vibration due to a single pedestrian walking on a at surface, such as a oor slab or bridge deck. The method was developed by Arup, and has been calibrated and rened with verication measurements taken on completed structures over a period of ten years.

This publication provides an informative description of the factors effecting footfall induced vibration and guides the engineer through the process of designing for vibration. It includes owcharts for calculation procedures and a useful glossary. It also includes worked examples on a concrete footbridge, a low frequency multispan post tensioned concrete oor and a high frequency ribbed slab. Michael Willford and Peter Young have over 30 years combined experience in the area of structural dynamics.

The methods presented in this guide have been developed by them over the last 10 years and have been used extensively within Arup for the past 5 years. Acknowledgements The authors would like to acknowledge the contribution of their colleagues at Arup to the validation of the methods described in this guide, in particular Caroline Field, Kathy Gubbins, Kubilay Hicyilmaz and Mark Arkinstall.

Professor Aleksander Pavic of Shefeld University also provided additional validation data from his own independent research projects. The authors would like to thank Dr Stuart Kerr for providing the data which is the basis of the footfall forces used here. Finally, special thanks are due to Professor Tom Wyatt of Imperial College London, who was the independent peer reviewer for this document. No liability including that for negligence for any loss resulting from such advice or information is accepted by The Concrete Centre or its subcontractors, suppliers or advisors.

Readers should note that publications are subject to revision from time to time and should therefore ensure that they are in possession of the latest version.

Introduction Understanding footfall induced vibration Quantifying vibration Predicting footfall induced vibration of structures Worked examples 2 5 6 13 22 41 61 62 Evaluating modal properties of a structure Procedures for measuring oor response Validation 66 75 Introduction This guide describes a reliable methodology for predicting the vertical vibration induced by pedestrians crossing structures such as oors and bridges. Many methods already exist1, 2, 3,4, 5 but each of these is signicantly limited in some respect.

A full discussion of these limitations can be found elsewhere6, 7. The method presented here addresses all of the issues described below in a consistent manner: n It is applicable to any type of structure on which people walk, including floors and bridges.

The method described in this guide is based on the well-established principles of modal analysis. This enables rst principles calculations to be made, and unlike most other methods, it does not require the introduction of arbitrary or empirical factors.

This makes it a robust approach for the assessment of any type of structure of any construction material. The method is appropriate for calculating the vibration caused by a single pedestrian walking on a at surface on any structure which is signicantly heavier by at least a factor of 10 than the individual. Whilst very simple regular structures can be assessed entirely by hand or spreadsheet calculation, it is envisaged that the method will be used principally in conjunction with nite element analysis as a means of estimating the modal properties of oor and bridge structures.

Whilst this might be seen as an added complexity, in practice the additional accuracy that nite element analysis of less regular structures brings to the assessment more than outweighs the modest additional effort associated with building and analysing a model. The method was developed within Arup and rened by reference to the measured performance of completed structures over a period of ten years.

It is published here as a Cement and Concrete Industry publication but it is applicable to structures of any construction material. Understanding footfall induced vibration When people walk on oors, staircases, bridges or other structures they apply dynamic forces that cause the structures to vibrate. The vibration displacement amplitude is normally very small micrometres rather than millimetres and is not perceptible visually or signicant in terms of structural integrity.

However it can often be felt by other people and, if excessive, can be distracting and detrimental to the perceived quality of the structure. Vibration can also impair the function of sensitive equipment in buildings such as laboratories and precision manufacturing facilities. In order to assess how a structure responds to forces induced by walking persons it is necessary to understand: n The forces applied to the structure.

Each of these is discussed in the following sections. The dynamic forces applied to the ground by a walking person can be measured directly by walking experiments using instrumented force plates or platforms. Four example measured footfall time histories11 are shown in Figure 2. These show the total dynamic. In normal walking there are typically between 1. The static weight of the individual has been subtracted from the measurements as it does not vary with time and so does not induce any dynamic response.

The four time histories are very different in both shape and magnitude. Extensive research10 has shown that factors which contribute to this variation include height, gender, footwear, weight, walking surface and walking speed. Given the extent of this variability, there is no single correct force that can be used to calculate structural response, but it is instead appropriate to adopt values that are statistically representative and have an arbitrary but known probability of exceedence.

In this guide, mean dynamic forces and their coefcients of variation are presented which have been derived from measured footfall time histories All structures have what are known as natural modes of vibration, and each of these modes has a unique spatial distribution of displacement known as the mode shape , natural frequency of vibration, modal mass and modal damping associated with it.

The natural modes are a structures preferred patterns of vibration the modes in which it will vibrate if excited by a sudden impact. These concepts are dened formally in the glossary and illustrated by example below. Consider the short bridge shown in Figure 2. Some of its lowest modes of vibration could be excited by someone performing a single jump at mid-span. This would cause the bridge to vibrate, and the displacement history at mid-span in the rst mode would be as shown in Figures 2.

The vibration displacement, velocity and acceleration all vary sinusoidally with time at the natural frequency of the mode, which depends on the stiffness, mass and span of the bridge. The amplitude of vibration decays with time at a rate dependent on the damping, but during this decay the frequency and spatial distribution of the vibration do not change.

Neither the frequency nor mode shape of the vibration depends on how high the person jumps assuming linear elastic response but the initial amplitude of vibration is affected by the jump height.

The motion in one vibration mode following a single impact is known as damped simple harmonic motion, and a rigorous mathematical description of this can be found in dynamics textbooks In fact, the bridge has many modes of vertical vibration, the rst three of which are shown in Figure 2. A mode cant be excited by forces applied at a position of zero displacement in the mode. In this case, therefore, whilst the rst and third modes would be excited by a jump at mid-span, the second would only respond if the person jumped elsewhere on the.

Jumping at the quarter point, for instance, would excite all three modes, but would excite the rst mode also known as the fundamental mode less effectively than would a jump at mid-span. Figure 2. The methodology described in this guide requires an assessment of the modal properties that is, frequency, mode shape, modal mass and damping of all the relevant modes of the structure. Details of how this can be done are described in Appendix A.

Whilst for some simple and regular structures e. Rather than jumping on the bridge, the person described above is more likely to want to walk across it. In so doing, he or she applies a periodically varying dynamic force such as shown in Figure 2. Initially, let us assume that the bridge is long and that the point of application of the load can be considered as stationary at mid-span i. The response of the bridge to the repeated application of this force depends on several factors, the most important being: n the stiffness and mass of the bridge.

The inuence of the frequency ratio r can be illustrated using one of the force time histories described in section 2. Consider one in which the person is walking at 2.

The velocity time histories calculated at the centre of the bridge are as shown in Figure 2. It is seen that the vibration response of the two structures is signicantly different. For the 4Hz bridge, the vibration builds up over time such that the response after several footfalls is signicantly greater than after the rst. This is a resonance phenomenon, in which the vibrations induced by each footfall reinforce the vibration generated by previous footfalls.

For the 3Hz bridge this is not the case, and the response is similar at each footfall. The maximum response of the 4Hz stiffer bridge is signicantly greater than that of the less stiff, but otherwise identical, structure. A natural extension of this analysis is to calculate such velocity time histories for many more values of r and to graph the variation of maximum response velocity with r.

This is similar to the development of a velocity response spectrum under seismic excitation. Figures 2. In these graphs it can be seen that, in general, the vibration response reduces as r the ratio of a structures modal frequency to footfall rate increases. However, there are peaks in the response spectra at particular whole number values of r. If r is close to 1. If r is close to 2. These resonances lead to signicantly greater response when r is close to 1.

Inspection of the time histories shows that the response does not build up over time, and the response to each individual footfall is comparable. At high values of r the velocity response is characterised by an initial peak response such as that produced by a single impulse followed by a decaying sinusoid similar to that illustrated in Figure 2. The non-resonant response can therefore be treated as repeating impulsive responses to individual foot impacts.

It can be seen that the magnitude of resonant response for whole number values of r is strongly dependent on both the damping of the structure and the harmonic number. The above generic analyses show that if a person can walk at a footfall rate that is a factor of 1, 2, 3 or 4 lower than the natural frequency of a mode, then resonance is possible, and this will lead to vibration levels greater than walking at slightly different footfall rates. For values of r greater than approximately 4.

As footfall rates typically vary between 1. For structural modes with natural frequencies sufciently high that they cannot be excited by the fourth harmonic i. If the damping is very low, then in theory resonance to the fth and sixth harmonics could produce a higher response, but this is not usually of practical concern because successive footfalls are not completely identical, and this reduces the magnitude of the higher harmonics.

Many structures have several modes that may simultaneously experience signicant responses to footfalls, and practical prediction methods must allow for this. Whilst the determination of modal properties of the structure is the same, it is convenient to have different approaches for the calculation of resonant and impulsive vibration responses. Therefore structures with vertical natural frequencies less than 4. Quantifying vibration Before attempting to predict vibration levels, it is necessary to understand how vibration is described quantitatively.

Generic terms, and those that refer specically to footfall induced vibration, are described below. Typical performance targets are also given for structures of differing usage. At least two parameters are required to dene vibration quantitatively. Formal denitions of some of these are included in the Glossary but the metrics generally refer either to the magnitude of vibration e. Consider rst a system which vibrates such that its displacement is a sinusoidal function of time, as shown in Figure 3.

This is known as simple harmonic motion.

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## A design guide for footfall induced vibration of structures

This publication provides an informative description of the factors effecting footfall induced vibration and guides the engineer through the process of designing for vibration. Michael Willford and Peter Young have over 30 years combined experience in the area of structural dynamics. The methods presented in this guide have been developed by them over the last 10 years and have been used extensively within Arup for the past 5 years. A cement and concrete industry publication. A tool for designers to engineer the footfall vibration characteristics of buildings or bridges. M R Willford.

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## A Design Guide for Footfall Induced Vibration of Structures

Whilst footfall induced vibrations on buildings or bridges is not normally signicant in terms of structural integrity, footfall vibration can be a critical serviceability condition. This publication presents a new method for evaluating the vibration due to a single pedestrian walking on a at surface, such as a oor slab or bridge deck. The method was developed by Arup, and has been calibrated and rened with verication measurements taken on completed structures over a period of ten years. This publication provides an informative description of the factors effecting footfall induced vibration and guides the engineer through the process of designing for vibration. It includes owcharts for calculation procedures and a useful glossary. It also includes worked examples on a concrete footbridge, a low frequency multispan post tensioned concrete oor and a high frequency ribbed slab. Michael Willford and Peter Young have over 30 years combined experience in the area of structural dynamics.

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Latest version of document. A methodology, based on modal analysis, for predicting the vertical vibration induced by pedestrians crossing structures like floors and bridges, enabling first principles calculations without the need for arbitrary or empirical factors. Explains footfall-induced vibration as well as how to quantify and predict vibration, with worked examples. The Concrete Centre is the central development organisation for the UK concrete sector and provides material, design and construction guidance.