Since 1995, we have been measuring the
The great development of footbridges (also referred to as “pedestrian bridges”) in Portugal in the last 2 decades is related to the large amount of freeways or motorways constructed in recent years and the need for pedestrians to cross them. Also, footbridges were built over railways, in railway stations, near shopping centers and schools, and so forth. In a few cases, they were built as part of bicycle routes. Footbridges are in general quite different from viaducts over freeways, not only due to differences in the loading but also because the former can profit more easily from the advances in material developments and architectural creativity. In general, they are lighter, built with high strength materials, spanning quite large distances, and having a wide variety of structural designs. They tend to become slender structures with less mass, but they show more pronounced dynamic effects due to possible resonance in the passage of pedestrians. Large vibration amplitudes are in the range of discomfort, and so this phenomenon deserves much attention to understand the problems that may arise and the way they should be dealt with.
This paper is divided into two parts. In the first part we present and explore a database with
For several years, we have been building a database on the main dynamic characteristics of different types of footbridges built in Portugal. In this paper, we report a group of 79 footbridges of different typologies, setting correlations between the fundamental frequencies in the three orthogonal directions and their larger span length, (Table
Geometric characterization and first
Name | Place |
Constr. |
Height |
Width |
Deck |
Deck |
Largest |
Right |
Left |
Frequencies | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
|
||||||||||
Steel Box-Girders | ||||||||||||
Relógio_Gago_Coutinho | Lisbon | 1995 | 6.6 | 1.7 | slight curve | trapezoidal | 37 | L | L | 1.86 | 3.42 | 2.29 |
Entreposto Olivais | Lisbon | 2000 | 8.0 | 1.8 | slight curve | trapezoidal | 39 | T | T | 2.44 | 3.22 | 3.25 |
CP_Gago Coutinho | Lisbon | 1998 | 8.3 | 2.2 | trapezoidal | 28 | T | T | 2.15 | 2.16 | 3.42 | |
Relógio_Gomes Costa_West | Lisbon | 2000 | 8.0 | 2.0 | slight curve | trapezoidal | 42 | T | T | 2.70 | 3.71 | 3.12 |
Av. Padre Cruz | Lisbon | 2000 | 7.7 | 2.0 | trapezoidal | 37 | TL | TL | 2.25 | 5.08 | 3.42 | |
Acesso 25 Abril Alvito | Lisbon | 2000 | 8.0 | 2.0 | large curve | trapezoidal | 37 | N | N | 3.90 | 3.90 | 5.60 |
Belém (Café) | Lisbon | 2001 | 6.7 | 1.7 | large curve | trapezoidal | 43 | L | L | 3.42 | 5.20 | 5.05 |
2a Circular East Colegio Alemao* | Lisbon | 2000 | 5.2 | 2.0 | slight curve | trapezoidal | 26 | T | T | 6.70 | 2.54 | 3.52 |
Termas S. Pedro Sul | S. Pedro Sul | 1997 | 2.6 | 2.0 | trapezoidal | 53 | L | RC | 2.90 | 2.00 | 2.25 | |
2a Circular Colegio Alemao | Lisbon | 2000 | 5.0 | 2.0 | slight curve | rectangular | 41 | T | L | 1.95 | 3.67 | 2.25 |
Belém FIL Café-Bar | Lisbon | 2001 | 6.3 | 2.0 | rectangular | 14 | TL | T | 2.93 | 7.70 | 9.30 | |
Churrasqueira Campo Grande | Lisbon | 2000 | 5.0 | 2.0 | rectangular | 20 | L | L | 6.93 | 2.54 | 3.52 | |
Av. Ceuta South | Lisbon | 2002 | 5.3 | 2.1 | rectangular | 32 | T | T | 1.76 | 3.80 | 1.76 | |
Faro_Calouste Gulbenkian | Faro | 2003 | 3.5 | 2.0 | rectangular | 29 | T | T | 3.70 | 3.40 | 2.00 | |
Lagos-Nova CML | Lagos | 2004 | 4.6 | 2.0 | rectangular | 16 | T | T | 3.65 | 3.65 | 7.10 | |
Steel Truss I + I | ||||||||||||
Teatro Camões Parque Nações | Lisbon | 2010 | 7.0 | 2.5 | I + I | 21 | L | L | 6.80 | 4.30 | 4.30 | |
Coimbra 1A | Coimbra | 2004 | 5.5 | 2.0 | I + I | 21 | L | L | 2.00 | 5.00 | 5.20 | |
Coimbra 1B | Coimbra | 2004 | 5.5 | 2.0 | I + I | 28 | L | L | 2.00 | 5.00 | 4.08 | |
Coimbra 2A | Coimbra | 2004 | 4.5 | 2.0 | I + I | 17 | L | T | 3.36 | 11.00 | 9.60 | |
Pupilos Exercito South | Lisbon | 1999 | 10.0 | 1.7 | I + slab | 38 | T | T | 1.17 | 1.86 | 3.03 | |
Braga NE | Braga | 2002 | 5.0 | 2.0 | truss 2D | 25 | T | T | 4.79 | 2.64 | 4.77 | |
Av Gomes Costa (East) | Lisbon | 2002 | 5.6 | 2.8 | truss 2D | 30 | T | T | 2.73 | 2.73 | 3.42 | |
Rego | Lisbon | 1999 | 6.3 | 2.4 | truss 2D | 46 | Truss | Truss | 1.95 | 1.95 | 3.81 | |
Conde Almoster over railway | Lisbon | 2001 | 8.0 | 3.0 | n/straight | truss 2D | 50 | T | T | 2.25 | 2.93 | 2.93 |
IP3-Lameira | Coimbra | 2000 | 5.6 | 2.0 | truss 2D | 21 | L | L | 2.26 | 3.87 | 2.66 | |
Universidade Covilhã | Covilhã | 2010 | 6.0 | 1.4 | truss 2D | 20 | T | N | 2.07 | 2.07 | ||
IP3-CEPSA_Aguieira | Coimbra | 2000 | 5.8 | 2.0 | truss 2D | 22 | T | T | 5.03 | 3.22 | 7.72 | |
A1-Antuã | A1 km 260 | 1995 | 4.3 | 2.5 | truss 2D | 42 | RC | RC | 2.73 | 2.91 | 4.98 | |
Bela Vista-Feira Nova | Lisbon | 2008 | 12.0 | 2.5 | truss 3D | 48 | N | N | 2.20 | 3.00 | 5.60 | |
Massamá | Salg Maia | 4.7 | 2.5 | truss 3D | 26 | L | L | 5.40 | 5.40 | 6.54 | ||
EN-125_Chinicato | Lagos | 2001 | 5.0 | 2.0 | tubular | 20 | T | T | 3.20 | 3.20 | 7.40 | |
Pinhão-Douro | Pinhão | 2011 | 6 | 3.3 | light curve | tubular | 51 | N | N | 3.16 | 4.24 | 3.16 |
Exit Camacha | Madeira | 2002 | 4.0 | 2.0 | triangle | 28 | N | N | 2.44 | 1.86 | ||
Aeroporto | Madeira | 2002 | 4.0 | 2.0 | triangle | 19 | T | T | 3.71 | 4.32 | 2.50 | |
São Gonçalo | Madeira | 2002 | 4.0 | 2.0 | triangle | 28 | N | N | 3.32 | 2.64 | 1.86 | |
Conde Almoster | Lisbon | 2001 | 8.0 | 2.0 | triangle | 22 | T | T | 5.40 | 4.70 | 4.20 | |
Reinforced Concrete | ||||||||||||
Alfredo Bensaude (West) | Lisbon | 1997 | 5.0 | 2.1 | I + slab + I | 30 | T | T | 4.32 | 2.93 | 2.88 | |
Alfredo Bensaude (East) | Lisbon | 1997 | 5.0 | 2.1 | I + slab + I | 30 | T | T | 3.91 | 2.05 | 2.73 | |
Av. EUA_Chelas East | Lisbon | 2000 | 9.0 | 2.5 | I + slab + I | 25 | TL | T | 2.91 | 1.70 | 4.00 | |
Vale Formoso (Chelas) | Lisbon | 2000 | 7.2 | 2.5 | I + slab + I | 28 | T | T | 1.76 | 2.00 | 3.81 | |
Figueira_Foz_Lidl | Figueira Foz | 2002 | 6.0 | 2.5 | I + slab + I | 19 | T | T | 5.57 | 3.42 | 6.45 | |
Lusófona (Campo Grande)* | Lisbon | 2001 | 6.0 | 2.0 | I + slab + I | 26 | T | T | 3.73 | 2.77 | 3.79 | |
EN-125_Faro_Mercedes | Faro | 2003 | 6.3 | 1.5 | I + slab + I | 29 | T | T | 1.90 | 2.90 | 1.90 | |
Gambelas Rotunda Aeroporto | Faro | 2005 | 5.8 | 1.8 | I + slab + I | 38 | T | T | 2.06 | 2.51 | 2.30 | |
EN-125 Faro-Metalo-Farense | Faro | 2005 | 6.0 | 1.8 | I + slab + I | 32 | T | T | 1.80 | 2.20 | 3.10 | |
EN-125 Faro-Patacao (center)* | Faro | 2005 | 5.0 | 2.0 | I + slab + I | 22 | TL | TL | 2.60 | 2.60 | 3.58 | |
EN-125 Faro-Patacao (N part) | Faro | 2005 | 5.0 | 2.0 | I + slab + I | 13 | TL | TL | 2.60 | 2.35 | 10.70 | |
EN-125 Faro-Patacao (S part) | Faro | 2005 | 5.0 | 2.0 | I + slab + I | 11 | TL | TL | 2.60 | 2.50 | 13.50 | |
EN-125 Faro-Paga Pouco | Faro | 2005 | 5.0 | 2.0 | I + slab + I | 22 | T | T | 3.55 | 2.25 | 5.05 | |
EN-125 Faro-Coord Seguros Eng | Faro | 2005 | 5.0 | 1.5 | I + slab + I | 23 | T | T | 4.10 | 2.15 | 3.50 | |
Pupilos Exercito North | Lisbon | 1999 | 10.0 | 1.7 | I + slab | 31 | T | T | 1.37 | 1.86 | 4.49 | |
Conde Almoster | Lisbon | 2001 | 7.0 | 2.0 | n/straight | triangle | 22 | RC | RC | 5.40 | 4.70 | 4.20 |
Reinforced Concrete | ||||||||||||
IP25-Aveiro | Aveiro | 1999 | 4.0 | 2.0 | large curve | trapezoidal | 36 | L | L | 1.24 | 1.44 | 2.57 |
EN-125_Exit Aeroporto (Ibis) | Faro | 2003 | 5.0 | 2.0 | U | 37 | T | T | 2.10 | 1.65 | 1.90 | |
Circular Praia da Rocha | Portimão | 1970 | 5.0 | 2.0 | U | 21 | L | L | 6.90 | 5.10 | 6.90 | |
Entreposto Escola_Herc_Carvalho | Lisbon | 2001 | 5.7 | 2.3 | T + T | 20 | T | T | 1.60 | 3.12 | 3.50 | |
Amadora Cacém | Amadora | 1995 | 6.0 | 4.0 | rectangular | 22 | T | T | 6.30 | 6.20 | 3.71 | |
IC-19 Queluz | N117 | 2001 | 6.0 | 2.0 | U | 31 | T | T | 2.52 | 2.40 | 2.76 | |
IC-19 Estaçao Massamá-Barcarena | Exit 7 | 2001 | 6.0 | 2.0 | U/V | 32 | L | T | 2.52 | 4.12 | 3.13 | |
IC-19 Cacém West Cemitério) | Exit 9 | 2001 | 6.0 | 2.2 | U/V | 41 | T | T | 2.52 | 1.84 | 2.32 | |
IC-19 Cacém Leste | Exit 9 | 2001 | 6.0 | 2.4 | U/V | 46 | T | T | 2.07 | 1.80 | 2.12 | |
IC-19 Cacém Leste Gasolina BP | Exit 8 | 2001 | 6.0 | 2.0 | U/V | 39 | T | T | 2.76 | 2.34 | 2.58 | |
Guimarães | S-Exit A7 | 2001 | 6.0 | 2.0 | U/V | 25 | T | T | 3.00 | 2.12 | 3.76 | |
Steel variable cross-section | ||||||||||||
SMPC_Av Calouste Gulbenkian | Lisbon | 2010 | 8.6 | 4.0 | trapezoidal | 38 | N | N | 7.30 | 2.80 | 3.86 | |
Olaias-Bela Vista | Lisbon | 2011 | 15.0 | 4.0 | I + I | 56 | N | N | 2.16 | 3.04 | 1.90 | |
Other Types | ||||||||||||
Parque Nações near Vodafone | Lisbon | 2010 | 7.0 | 2.5 | Timber U | 25 | L | L | 5.80 | 5.80 | ||
Green park_Benavente | Benavente | 2007 | 3.0 | 2.5 | Steel ribbon | 48 | N | N | 1.12 | 1.76 | 2.36 | |
FEUP Campus* [ |
Oporto | 2001 | 5.0 | 2.8 | RC ribbon | 32 | N | N | 2.15 | 2.54 | 0.81 | |
CC Vasco Gama North* | Lisbon | 1998 | 8.0 | 2.4 | Steel arch | 75 | N | N | 1.83 | 0.94 | ||
CC Vasco Gama South | Lisbon | 1998 | 8.0 | 2.4 | Steel arch | 75 | N | N | 1.70 | 0.88 | ||
Guarda near railway station* [ |
Guarda | 2007 | 7.0 | 2.0 | Steel bow-string | 90 | N | N | 0.63 | 2.33 | ||
Microsoft Porto Salvo | Oeiras | 2001 | 6.0 | 2.0 | Steel bow-string | 25 | T | L | 3.22 | 5.66 | 3.22 | |
Infante D. Henrique | Lisbon | 1978 | 5.6 | 2.5 | Fiber-glass | 30 | L | L | 2.70 | 8.00 | 5.37 | |
Av. Marech Gomes Costa (East) | Lisbon | 1978 | 5.6 | 2.5 | Fiber-glass | 32 | T | T | 4.00 | 3.90 | 5.37 | |
Ribeira Carpinteira | Covilhã | 2009 | 39.0 | 4.4 | S-shape | Steel [+] 2D | 49 | N | N | 1.36 | 2.11 | 2.47 |
Pedro e Inês* [ |
Coimbra | 2006 | 10.0 | 4.0 | n/straight | S + RC arch-girder | 110 | N | N | 0.91 | 1.95 | |
Movable cable-stayed—Open* [ |
Viana Castelo | 2007 | 8.0 | 2.5 | Steel cantilever | 36 | N | N | 0.98 | 1.03 | ||
Movable cable-stayed—Closed* | Viana Castelo | 2007 | 8.0 | 2.5 | Steel cantilever | 36 | N | N | 4.09 | 3.17 | ||
Circular ribbon* [ |
Aveiro | 2009 | 5.0 | 2.0 | ring | Cable-stayed | ext diameter = 13 | L | N | 3.17 |
We summarize the work by Silva [
This work aims to contribute to the understanding of footbridge behavior under pedestrian loading, verifying the reliability of standard structural analysis programs to obtain a correct representation of these types of structures. This is in line of recent studies merging analytical modelling with
Although in many cases the footbridges do not show structural problems with pedestrian crossings, they may notice excessive vibration problems under specific loading, making the crossing uncomfortable and even scary. The case that called attention to this phenomenon was the Millennium Bridge, in London. In the inauguration’s day, when thousands of pedestrians crossed the footbridge, excessive horizontal vibrations were observed caused by a synchronized transverse movement of the crowd. This effect was known as “lock-in effect” [
The present database refers to 79 footbridges of several geometric layouts, structural types, deck cross sections, materials, and largest span length varying from 11 to 110 m, Figure
Largest span of analyzed footbridges (circular ribbon footbridge not included).
Main structural deck types for footbridges in Portugal.
Main typologies of footbridges in Portugal.
Even though more than 50% of the reported footbridges refer to the region of Lisbon, the author has selected a number of cases in each typology in other regions of the country to gain some statistical significance. Also, there is a set of recently built footbridges, which represent landmarks in the modern Portuguese scenario for their outstanding design, such as the stress-ribbon in the FEUP Campus (Oporto) [
The above most common (a) to (d) typologies built in Portugal are also seen in many other countries and, consequently, the results presented herein may be extrapolated outside Portugal. These footbridges, in a total of 63 cases, are essentially single span long over an entire free-way, supported in lateral columns or pillars, made of cylindrical hollow steel or of RC precast elements, with cylindrical or elongated cross sections. The access to the deck in an elevated level is made in different ways, commonly by RC stairways or ramps, running longitudinal with the axis of the footbridge (L), transversal to it (T), or at an angle (LT). For bicycle routes generally there is no ramp (N) and the deck sits directly in the abutments.
Foundations of columns vary from case to case but, in general, they are made of concrete blocks. Connections at the top of the columns and to the stairways or ramps also differ quite considerably. But as the majority of these structures are precast, these connections are weak points of the structure especially for seismic loadings.
The dynamic characterization of these structures is of most importance for a number of reasons. As they are slender structures, with continuous distribution of mass and stiffness, most of them spanning lengths of 20 to 60 m (Figure
Frequencies and damping characteristics are probably the most important parameters controlling the dynamic behavior, together with the frequency of walking pedestrians (number of people, velocity, synchronization of stepping, etc.). The frequencies of the structure depend on the geometry and mechanical properties. For geometry they vary with the number of spans, the type of connections, the lateral pathway (ramp and stairways in the longitudinal or transversal directions), the height of columns, the width of deck, and the curvature in elevation and its development in plant. For mechanical properties, the main characteristics to be accounted are the weight and the modulus of elasticity.
The main information contained in the database for each one of the 5 above mentioned typologies refers to the following parameters, obtained either from design drawings (a few cases) or from direct measurements of structural elements: (1) identification—name; (2) location—place; (3) construction year; (4) height of deck; (5) width; (6) deck development; (7) deck cross section; (8) largest span; (9) and (10) lateral access; (11) to (13) frequencies of first mode in the transversal, longitudinal, and vertical directions. Table
Data treatment consisted in analyzing peak acceleration values, predominant frequencies of vibrations in the three orthogonal directions through FFT techniques, and damping from amplitude decay in a few cases. A cross examination of results, together with the interpretation of modal shapes for the most simple geometrical layouts, led to the identification of first modal frequencies in the three directions of space (T: transversal; L: longitudinal; V: vertical). For all analyzed structures, damping is quite small with values varying in the interval 1% >
In several situations we repeat the
The experiments were of two types, with measurements performed with the accelerometric station located at mid-span and at quarter-span as follows: measurements for noise vibration produced by car traffic passing underneath, for mode identification; measurements for a set of typified tests with the passage of pedestrians at different velocities: (a) one person at slow walking; (b) one person at normal walking; (c) one person at fast walking; (d) one person jogging (slow running); (e) forced vibration caused by the movement of one person in resonance conditions; and (f) impulsive action derived from “jumping.” For details, see [
This technique, using a single instrument, can only be used with confidence for footbridges with simple geometric layouts, in which modes are easily separated into the three directions and no interaction is taking place. For more complex geometries, either in plan or in the vertical direction, multiple instrumentation with common time is required for identification of frequencies and modal shapes. The case of the Ribeira da Carpinteira (Covilhã) footbridge, a S-shape plan view with two lateral steel beams ([+] cross section) is the situation where modes are coupled in the 3 directions of space with an important participation of torsion of deck [
The main results are presented in Figure
Frequencies of 1st mode in the three directions (T, L, and V) as a function of largest span: (a) to (c) all typologies for V, T, L, and (d) V + L; (e) to (h) vertical for steel box-girders, truss structures, RC structures, and other structures, respectively. Red curves are explained in Section
Vertical
Transversal
Longitudinal
Vertical + longitudinal
Steel box-girders
Steel truss I + I
RC structures
Other typologies
Even though the steel structures are much lighter than RC, a comparison of their vertical frequencies shows similar results (Figure
Comparison of frequencies for steel box-girders and RC footbridges.
Figure
Correlation of V and L frequencies for all footbridges.
An empirical formula to compute the fundamental vertical frequency of a footbridge would be very useful for a designer to quickly assess its response to pedestrian crossing.
Let us consider a footbridge as a simple supported beam (length
The frequency (Hz) of the first mode is given by Clough and Penzien [
Expression (
We plotted the values produced by (
The dynamic loading in footbridges is essentially due to the passage of persons alone, in groups, randomly walking, jogging, running, or a combination of all previous cases. Also, sudden loads provoked by jumps, fall of objects, or rhythm action may arise. Sometimes the passage of bicycles or motorbikes may be observed. The “lock in effect” is another resonant effect, induced by the bridge itself, which influences the walking pattern. We will concentrate only on the passage of a single person walking, jogging, or running. Also excitation near to resonance by one single person was also performed. However, the response of footbridges for groups of pedestrians walking in rhythm or randomly walking was not analyzed.
There are three levels to be considered in the definition of pedestrian loading. The first one attends to the frequency of movement, resulting from a speed of 0.5 m/s to 0.8 m/s (for slow walk) to 3.5 m/s (for jogging-slow running) with a step size from 0.65 m for slow walk and not exceeding 1.7 m for fast running (Table
Frequency range (Hz) for different patterns of movement (adapted from [
Slow | Normal | Fast | Total | |
---|---|---|---|---|
Walk | 1.4–1.7 Hz | 1.7–2.2 Hz | 2.2–2.4 Hz | 1.4–2.4 Hz |
Jogging | 1.9–2.2 Hz | — | — | 1.9–2.2 |
Running | — | — | 3.0–3.4 Hz | 3.0–3.4 Hz |
The second and third levels are related to the contact form of the foot with the deck, with one vertical and two horizontal components. This contact form depends on the pattern of movement [
(a)
We generated a load curve of the form shown in Figure
Whereas the vertical component of the load always applys in each step due to the gravitational force in the same direction, the horizontal action introduces a force alternating to the right and left, according to the stepping foot.
These functions were programmed to be used with standard linear dynamic analysis software using the time history integration. For the L direction the intensity of loading is 50% of V, whereas in the T direction it is between 3 and 10% of V. The frequency in L direction is equal to the V direction, whereas in T direction the frequency is half of the V direction [
Table
Amplitude values in mg of the measured maximum motion at mid-span for a group of 10 footbridges in the transverse (T), longitudinal (L), and vertical (V) directions.
Name | Length |
Noise | 1 person walk | Jump | Excitation | Typology | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
T | L | V | T | L | V | T | L | V | T | L | V | |||
CP_Gago Coutinho | 28 | 0.5 | 0.5 | 0.9 | 2.5 | 2.5 | 12 | 8 | 5 | 109 | Steel BG | |||
Rel |
42 | 6.5 | 2.5 | 7 | 12 | 6 | 45 | 7 | 3 | 27 | Steel BG | |||
Braga NE | 25 | 1.5 | 1 | 7 | 4 | 1.4 | 17 | 24 | 9.6 | 138 | 19 | 18 | 40 | Steel truss |
Av Gomes Costa (Leste) | 30 | 1.8 | 2 | 7 | 5.2 | 5.4 | 32 | 36 | 7.1 | 79 | 6.7 | 7.6 | 97 | Steel truss |
Rego | 46 | 3.3 | 1.1 | 5.5 | 4.8 | 1.9 | 7.5 | 10.6 | 10 | 77 | 4 | 8.1 | 76 | Steel truss |
Av. EUA_Chelas Leste | 25 | 0.2 | 0.2 | 0.2 | 0.8 | 1.3 | 2.8 | 2.7 | 1 | 24 | RC | |||
Entreposto Escola_Herc_Carvalho | 20 | 2.6 | 4 | 31 | RC | |||||||||
Vale Formoso (Chelas) | 28 | 0.5 | 0.2 | 0.5 | 0.6 | 1 | 3.5 | 0.8 | 0.9 | 7.5 | 1.6 | 1.4 | 22 | RC |
Microsoft Porto Salvo | 25 | 2.9 | 0.9 | 10.8 | 11 | 4.8 | 3.8 | 35 | 9.6 | 155 | 24 | 7.6 | 90 | Other types |
CC Vasco Gama Norte | 75 | 1.6 | 1.9 | 4 | 5 | 5.2 | 12.5 | Other types |
Amplitude values in mg of the measured maximum motion at mid-span. Series 1: noise; Series 2: 1 person walking at “normal” pace (2 Hz); Series 3: 1 person jumping from 30 cm; Series 4: excitation close to resonant conditions.
Figure
Amplitude values in mg of the measured maximum motion at mid-span (T; L; V): 1 person walking at “normal” pace (2 Hz) in different typologies (Series 2).
As far as amplitudes are concerned, if we take the excitation amplitude given by the equivalent single degree of freedom under resonant conditions, the values are also similar to those indicated by measurements, for the low damping observed (1% >
Currently, there are two different design procedures for footbridges under dynamic loading, which are contemplated by the international standards.
They are essentially based on discomfort and resonance: (i) peak acceleration values in the vertical and horizontal directions should not surpass certain limits for given load patterns and (ii) fundamental frequencies should be outside the so-called
Acceleration limits recommended in some codes (Figure
Regulation | Vertical acceleration (m/s2) | Horizontal acceleration (m/s2) |
---|---|---|
BS5400 [ |
|
— |
OHBDC (1983) [ |
|
— |
Eurocode [ |
|
|
Hong Kong (2002) [ |
|
|
(a) Allowable vertical accelerations as a function of frequency of vertical mode [
Živanović et al. [
Four types of structures were the object of a detailed analytical study: two of them correspond to the most common types built recently in Portugal, (1) a steel box-girder and (2) a prefabricated RC I + I beam; the third (3) is a steel inclined arch with a large span; and the fourth (4) is a RC I + I beam, similar to (2) but with 3 spans and inclined stairways. Essentially, we were interested in checking how analytical models could be validated by
In model definition several parameter values are uncertain, such as the elastic material properties (
This structure, with 2.0 m wide and spanning 25.5 m at a height of 5.2 m, has a steel box-girder deck supported in two cylindrical, partially hollow columns and connected to 2 adjacent stairways, one at each side of the deck (Figure
Steel box-girder structure under study: (a) analytical model; (b) view of column, deck, and stairway.
The analytical model of this footbridge, including the stairways, was made with SAP2000 [
Comparison of frequencies (Hz) between measurements
Mode direction | Measurements |
Analytical model |
---|---|---|
Longitudinal | 2.54 | 2.55 |
Vertical | 3.42 to 3.52 | 3.52 |
Torsion about |
? | 5.27 |
Torsion about |
? | 5.74 |
Transversal | 6.74 to 6.84 | 6.38 |
This structure was also subjected to a set of
Peak acceleration amplitudes for different tests.
Walking load |
|
||
---|---|---|---|
Vertical | Longitudinal | Transversal | |
1 Person*—walk at |
18.9 | 2.7 | 3.8 |
1 Person—walk at |
29.1 | 5.2 | 6.1 |
1 Person—jogging at |
38.2 | 20.2# | 12.1 |
1 Person—jogging at |
251.3† | 16.5 | 21.4# |
1 Person—jogging at |
252.6† | 12.5 | 24.5# |
2 Person—walk at |
26.3 | 6.4 | 5.8 |
2 Person—walk at |
38.3 | 25.3# | 7.0 |
2 Person—jogging at |
74.4‡ | 28.9# | 19.0 |
3 Person—walk at |
71.5‡ | 29.1# | 14.1 |
7 or 8 kids (30 to 40 kg)—slow walk | 16.3 | 6.2 | 10.3 |
Noise | 6.8 | 0.9 | 2.6 |
Jump | 198.1† | 21.1# | 83.4# |
Excitation in transversal direction (resonance) | 27.9 | 3.2 | 15.5 |
Impulse in transversal direction | 39.8 | 3.8 | 17.0 |
Excitation in vertical direction (resonance) | 239.2† | 11.7 | 16.1 |
†Values of vertical accelerations exceeding the less stringent limit (93.8 mg); ‡the most stringent limit (66.7 mg); #values exceeding the code ones for horizontal vibrations (20 mg, EN 1990 [
Tests of the model feasibility were made to reproduce the walking of a person at different speeds, by comparing the peak acceleration amplitudes obtained in the model with the measurement
Comparison of amplitudes at mid-span (vertical acceleration) between
The analyzed structure spans 25.6 m, has a height of approximately 5-6 m, and is 2.0 m wide, Figure
RC structure under study: (a) analytical model; (b) deck cross section; (c) view of column, deck, and stairway.
Concrete is a B45.1 for the beams and B30.1 for the precast elements (plates, columns, and stairways). Stairways run perpendicular to the bridge axis and are supported in square columns at 1/3 height.
Similarly to what was done for the steel structure, we compare in Table
Comparison of frequencies (Hz) between measurements
Mode direction | Measurements |
Analytical model |
---|---|---|
Longitudinal | 2.77 | 2.77 |
Vertical | 3.79 | 3.34 |
Transversal | 3.73 | 3.73 |
Torsion about |
? | 5.71 |
Torsion about |
7.91 | 8.80 |
Peak acceleration amplitudes for different tests.
Walking load |
| ||
---|---|---|---|
Vertical | Longitudinal | Transversal | |
1 Person*—walk at |
7.3 | 1.7 | 2.9 |
1 Person*—walk at |
10.1 | 1.8 | 3.1 |
1 Person*—walk at |
8.3 | 4.9 | 8.0 |
1 Person*—walk at |
11.6 | 12.2 | 13.5 |
1 Person*—walk at |
15.5 | 7.8 | 15.8 |
1 Person*—walk at |
32.7 | 11.8 | 25.4† |
2 Jumps at mid-span | 61.1 | 34.2† | 90.2† |
Impulse in transversal direction | 8.5 | 5.2 | 15.6 |
†Values exceeding the code limits for horizontal vibration (20 mg). EN-1990 [
Comparison of amplitudes at mid-span (vertical acceleration) between
From the analysis of Tables
Comparing the RC structure with the steel structure, we see that the former is much more rigid, with peak values almost 1/2 to 1/3 below the latter, depending on the direction considered. This means that RC footbridges amplify much more the response for the same loading characteristics than the steel box-girders, as already mentioned in Section
This footbridge is located in Avenue D. João II in Lisbon and makes the connection between the Orient Station and the Shopping Centre Vasco da Gama. There are two identical footbridges, with length of 86.6 m and usable width of 2.4 m. Each one is constituted essentially by three tubular steel sections and thirty-seven pieces with variable “
(a) Vision of the CC Vasco da Gama (Calatrava) structure; and (b) analytical model.
The footbridge is simply supported in the two extremes, still having an intermediate support at
The analytical model of this footbridge was made with SAP2000 [
The frequencies of the first 15 modes obtained by the analytical model and by
Comparison of frequencies (Hz) between the analytical model and measurements
MODES | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Analytical | 0.84 | 1.23 | 1.48 | 2.23 | 2.59 | 3.30 | 3.56 | 4.22 | 4.89 | 5.27 | 5.49 | 6.04 | 6.53 | 6.98 | 7.52 |
|
0.88 | 1.7 | 3.12 | 3.55 | 4.52 | 5.06 | 5.66 | 6.59 |
The first vibration mode corresponds essentially to a vertical oscillation (Figure
First vibration mode with
It should be noted that, even though the two footbridges connecting the Orient Station to Vasco da Gama Shopping Centre are essentially identical, the
To simulate a pedestrian’s passage on the structure, the load functions as mentioned in Section
In general, it is verified that the results from the analytic model are conservative, because the acceleration is, in most cases, larger than the values measured experimentally. Table
Acceleration measured at 1/2 span for the normal walking.
Direction | Acceleration at 1/2 span [mg] | ||
---|---|---|---|
Measuring device | Analytic model | ||
133 | LN874 | ||
Longitudinal | 0.25 | 0.20 | 0.24 |
Lateral | 2.00 | 2.00 | 2.00 |
Vertical | 3.50 | 3.20 | 4.50 |
These acceleration obtained for one or two pedestrians is below the maximum values indicated by the design guidelines, so for one or two persons the footbridge accomplishes the regulations. However, it is foreseen that for people’s groups these limits are no longer respected. According to research related to the Millennium Bridge, there is a critical number
Considering that the vibration mode which has significant lateral oscillation is mode 4,
The solution to this type of problem can be found by introducing mass-tuning dampers, which will reduce significantly the vibration levels compatible to the estimated pedestrian flux [
The work developed by Oliveira [
In here we concentrate only on the comparison of frequency values from
RC footbridge in Faro: (a) front view; (b) cross-section with precast (I + slab + I); (c) central column (longitudinal view); (d) column-beam connection (units in m).
The connection between the beams and columns is provided by two vertical bars of 20 mm on top of these columns, crossing the ends of each beam through holes made during the molding of the pieces. The holes were filled during the assembly phase, by a grout. A neoprene plaque was interposed between the column and the beam to distribute the contact stresses.
The access to the deck is made through four flights of stairs in each side. The stairs are almost perpendicular to the deck. Each set of stairs is formed by an independent prefabricated part, supported only on the ends. The steel used in reinforcing concrete was A500NR. Three types of concrete were used in this footbridge: C20/25 for the foundations, C35/45 in the prestressed beams, and C25/30 in the other parts. The concrete cover is 2.5 cm in the columns and beams and 5 cm in the foundations elements. Each span of the deck is formed by two prestressed reinforced concrete I beams.
The deck was modulated as a single bar with cross section as given in Figure
The connection between the columns and other elements was made through steel bars. To simulate the contribution of these bars in the absorption of flexural moments, without giving excessive rigidity to allow for some flexibility, the modelling was done considering that on top of the columns there is an elastic support given by springs, which absorb some of the existing moments. These springs also help to simulate the effect of the slab, because the beams have no continuity above the columns, but the slab does.
The analysis of the
Comparison of frequencies (Hz) between measurements
Mode direction | Span 22 m | Span 12.5 m | Span 10.7 m | |||
---|---|---|---|---|---|---|
|
Analytical |
|
Analytical |
|
Analytical | |
Longitudinal | 2.2 | 2.25 | — | — | — | — |
Transversal | 2.6 | 2.69 | — | — | — | — |
Vertical | 3.58 | 3.51 | 10.7 | 10.64 | 13.5 | 14.5 |
Modes of vibration: (a) 1st longitudinal; (b) 2nd tranversal; (c) 3rd vertical, central span.
From Table
We have been creating a database with geometrical and mechanical properties of many footbridges built in Portugal in the last two decades, representing the most common structural types. These structures, with larger spans varying from 11 to 110 m long, tend to become light and slender structures made of diverse materials and cross sections. They may show a pronounced dynamic amplification with the passage of pedestrians due to possible resonant effects. These large vibration amplitudes may be excited to a point of discomfort or even structural failure, and so the phenomenon deserves special attention [
We present a simple method to derive the main dynamic properties of these structures based on experimental
The experimental results are essential to calibrate analytical studies (presented for four cases) and detect sources of errors, essentially due to the supporting connections and a few simplifications in the model. The localization and type of stairways influences drastically the frequencies and modal shapes, especially in the transversal and the longitudinal directions of the structures and, consequently, the vibration levels.
Once the analytical model has been calibrated for the first frequency, agreement of other higher frequencies between measured and modelling was very good.
In relation to amplitudes, the results are not always so good, especially for RC structures, due to difficulty in setting proper pedestrian loading, so that the loading model well represents the
Despite these small details which need improvements in both the experimental techniques (for measuring the dynamic applied forces) and in the analytical modelling, we can say that both the used
We believe that the information contained in this database, involving a number of different structural types, allows the extrapolation of results to similar footbridges in other countries.
As a final result of practical importance, the measured peak acceleration values under pedestrian loading for steel structures are well above the limits defined in the international recommendations, especially for the case of jogging. This aspect confirms the idea, already stated in previous studies [
For the RC structures the situation is reversed. The measured peak acceleration values under pedestrian loading are well contained within the limits of the international recommendations, causing a better sensation of comfort and structural safety.
Several topics should be developed in future work: develop standard experimental techniques for routine testing of footbridges based on simple methods as the one presented to obtain the transfer function of these types of structures, making use of use more parameters of the collected data to reduce dispersion on the correlations; from the convolution of the results obtained with the passage of a single person, obtain a simple analytical formula to estimate peak amplitude motion as function of number of people and velocity of crossing. Stochastic representation of pedestrian crossing can be analyzed in future work if we use the response of a footbridge produced by a single person at different speeds as a “green function” to obtain the stochastic formulation; develop dissipation systems to damp out the large amplitudes of vibration observed in steel and long spanned structures, as suggested by Caetano et al. [ estimate seismic vulnerability functions for these structures, as their collapse over main road access lines may be critical in case of earthquake emergency.
The author declares that there is no conflict of interests regarding the publication of this paper.
This paper was partially supported by the Programa Pluri-Anual of “Fundação para a Ciência e a Tecnologia” (FCT). A special acknowledgment is due to T. Nunes da Silva, Ana Chagas, and Rui Oliveira for the modelling and computations. An anonymous reviewer made substantial contributions to improve the initial paper. Dr. I. Viseu revised the final text.