FACTA UNIVERSITATIS

Series: Architecture and Civil Engineering Vol. 2, No 4, 2002, pp. 281 - 289

SOME RELEVANT ASPECTS OF FOOTBRIDGE VIBRATIONS

UDC 624.073.33(045)

Ana Spasojević1, Djordje Dordjević2,

Marija Spasojević2, Novak Spasojević2
1
Ecole National Fédérale de Lausane, Suisse
2
Faculty of Civil Engineering and Architecture,
A. Medvedeva 14, 18000 Niš, Serbia and Montenegro

Abstract. Considering the contemporary structural materials that are becoming more
resistant, having higher strength to weight ratio, and the fact that live load of
footbridges is low, the design based on static analysis only, respecting ultimate limit
states requirements, leads to slender bridge structures for pedestrian and cycle track
use. As a consequence, stiffness and masses decrease, facing lively, easy to excite
structures, with smaller natural frequencies. The excitation of a footbridge by a
pedestrian passing over it can be unpleasant for a person walking or standing on the
bridge, but usually not destructive for the structure itself. Recent experiences regarding
dynamic behavior of slender footbridges have especially shown that vibration
serviceability limit states are very important requirements in any structural design. We
are presenting a general algorithm for analytical testing of dynamic parameters of
structures, calculation of deflection, thus speed and acceleration of superstructure
under human-induced excitation, as predicted by Eurocode, British and Canadian
standards in use, since no Yugoslav code deals with the problem. The evaluated system
is a footbridge in a system of a simply supported concrete girder. The presented model
is used to show correspondence of results, obtained by the algorithm, with the results
obtained using the simplified methods suggested by the Codes of Practice, since the
latter exists only for certain structural systems.
Key words: Footbridge, human-induced excitation, forced vibrations,
serviceability limit states.

1. INTRODUCTION
The design of up-to-date bridges is a challenge, having in mind a trend of building
elegant structures. Considering footbridges, regarding the fact that their live load is
relatively low and that modern materials enable larger bearing power, by design based on

Received April 30, 2004

282 A. SPASOJEVIĆ, DJ. DORDJEVIĆ, M. SPASOJEVIĆ, N. SPASOJEVIĆ

static analyses, controlling ultimate limit states, one gets the structures with small stiffness
and mass, which, therefore, are easy to excite. The excitations caused by walk can be
classified as a question of user's comfort, thus serviceability, regarding sensibility of
human body to vibrations, not large enough to cause structural damage, but large enough
to cause walking disturbance. Therefore, it is especially important to fulfill the
requirements of serviceability limit states of structural elements, in particular regarding
vibrations induced by pedestrians, in vertical, as well as in horizontal transversal
direction. The actual Yugoslav Codes do not treat this phenomenon, so that this analysis
is proceeded according to some current codes in this field: Eurocode, British Standards,
and Canadian standards.
In principle, there are two approaches in avoiding excessive vibrations of structures,
caused by pedestrians:
− tuning fundamental natural frequency of the structure,
− calculation of forced vibrations, i.e. limitation of vertical acceleration of any part of
superstructure under dynamic load caused by walking pedestrian.
Here, as a prerequisite for previously noted procedures, one can raise the following
questions:
− establishing a relevant mathematical model of exciting force due to walking,
− defining acceptability limit for vibrations level.

1.1. Modeling of dynamic force due to walking

Observing the walk of a pedestrian, one can conclude that every step can be treated as
one impulse, and series of steps as impulses along the way and shifted in time (Fig. 1).
Therefore, load induced by walking can be assumed as sum of loads caused by continual
steps, which further can be simulated with moving pulsating point load. With accurate
assumptions (see [5]) that the load applied by every step is approximately of the same
value, and that the time needed for transmission of pressure is constant for given walking
pace, one can assume that this load is of periodic nature.

Fig. 1. Walking pedestrian load – change along trace and in time

Consequently, vertical load due to walking, as a sum of static components presenting

weight G of pedestrian, with additional periodic components, can be presented in the form
of Fourier series:
 Some Relevant Aspects of Footbridge Vibrations 283

Fp(t) = G + G⋅α1⋅sin(2⋅π⋅fp⋅t) +G⋅α2⋅sin(4⋅π⋅fp+ϕ2) +… (1.1)

where: t - time, fp – walking frequency, αn - so known factor of dynamic load for nth load
harmonics, ϕn – phase angle of nth harmonic in relation to the first harmonic. One
can get the factor of dynamic load usually from the analysis of response signal in
the frequency domain, isolating the contribution of each harmonic.
The research has shown that during normal walk the pedestrians take approximately
two steps in second, what has as a result the frequency of the first Fourier harmonic of
about 2 Hz (see [2]). Therefore, theoretically speaking, the footbridges with natural
frequency close to the first (≈ 2 Hz) or higher (≈ 4 Hz, ≈ 6 Hz, etc.) excitation harmonic's
frequency, can experience too high vibrations under normal conditions of exploitation,
assuming sufficient energy in particular harmonic. Although possible, it is generally
assumed that other kinds of excitations, like running or jumping, are not occurring
frequently enough to be taken into account during calculation of these structures.
In Eurocode, British, and Canadian standards, the load applied by pedestrian due to
walking is described as harmonic pulsating point load F(t) moving over the bridge with
constant velocity. This load has a frequency fp, which has to coincide the first natural
frequency f0 of unloaded bridge. The force F(t), in Newton, is given as
F(t) = 180⋅sin(2⋅π⋅fp⋅t), (1.2)
where: t – time. In equation 1.2 factor 180 (N) is amplitude of first harmonic of the load, i.e.
product of pedestrian weight (assuming value 700 N) and dynamic factor α = 0.257.
The expression 1.2 has to be applied to footbridges which have fundamental natural
frequency lower than 5 Hz (frequency range of first two harmonics). The
comparison of expressions 1.1 and 1.2 shows that noted standards are taking in
considerations only first dominant harmonic of pedestrian load.

1.2. Problem acceptability limit for vibrations level

The criterion of acceptability of vibrations level is more a problem of psychological
effects on humans (mechanical, optical, acoustical effects), i.e. problem of serviceability,
than a problem of exceeding ultimate limit states of structure (stresses, strains, fatigue).
As such one, this criterion is rather difficult to define precisely. Because of that, different
standards propose different values for acceptable level of vertical or horizontal vibrations
(see [5]). Some aspects that have to be taken in consideration in criterions application
are: frequency of appearance of certain vibrations (frequent, exceptional, rare
appearance), desirable level of comfort, expected tolerance level of user.
Basically, the criterion of vibration acceptability is function of frequency and
displacements, usually expressed in acceleration units. In case of vertical vibrations,
accelerations from 0.5 do 1 m/s2, i.e. 5-10 % gravitation acceleration of Earth, g, is
acceptable. The humans are more sensitive to horizontal vibrations, so that the acceptable
accelerations are of order 1 – 2 % g. It is also to be mention that amplitudes larger than 10
mm in vertical direction and 2 mm in horizontal direction can cause the phenomenon
known as «lock-in» effect – synchronization in walk of certain number of pedestrians,
having as a result significant excitation of structure.
284 A. SPASOJEVIĆ, DJ. DORDJEVIĆ, M. SPASOJEVIĆ, N. SPASOJEVIĆ

Acceptability limit of vibrations adopted in Eurocode and British standard (BSI

1978), based on many experiments, is defined by the following expression:

alim = 0.5 f v (1.3)

where: fv (Hz) –frequency of structure, alim (m/s2) – maximal acceleration. The standards
cite that one can assume that fv is fundamental natural frequency f0 of the structure.

2. SYSTEM VIBRATIONS
2.1. Natural frequency tuning
The approach accepted by European Commission for Concrete (Comité Euro-
International du Beton - CEB 1993) considering problems of serviceability regarding
vibrations is avoiding critical frequency range: the avoidance of vertical natural frequency
of footbridges between 1.6 and 2.4 Hz (range of normal distribution of walk frequency)
and between 3.5 and 4.5 Hz (range of second harmonic) is recommended.
Deriving natural frequency of structure and functions of oscillation modes is the very
first step in algorithm. This procedure is very known in Structural Dynamics. Natural
frequencies, i.e. eigenfrequencies are obtained as eigenvalues of matrix of frequency
equation, and oscillation forms as their eigenfunctions, by means of programming
procedure written in programming package - system Mathematica.

2.2. Dynamic response analysis

More precise approach to verify the state of serviceability of footbridges regarding
vibrations is by means of dynamic response analysis of structure subjected to designed load
(problem of forced vibrations), i.e. load causing the largest response, exciting the structure
close to its fundamental natural frequency f0 in vertical direction. As already mentioned in
1.1, it is of general opinion that normal walk, as considered in actual standards and described
by expression 1.2, is the most appropriate form of excitation in order to find out the actual
state of serviceability regarding the vibrations of footbridges. The response to dynamic load
depends on factors as stiffness and damping of structure, and relation between frequency of
applied force and fundamental natural frequency of structure.
With a possibility to be measured easily, the acceleration became the most width
accepted parameter for check-out of vibration level. As a general approach, the
acceleration has to be calculated in the design phase of structure or to be measured on the
real bridge.
Forced vibrations of beam systems
In this part of the paper the theoretical background of the problem of forced vibrations is
presented. Beam system excited by moving concentrated deterministic force is treated,
suggesting the algorithm for its resolution, involving some appropriate assumptions (see [3]).
 Some Relevant Aspects of Footbridge Vibrations 285

2.2.1. Theoretical background

Consider vertical vibrations of beam of length l (Fig. 2), with constant bending stiffness
EI and constant mass µ per unit length. With corresponding assumptions, this vibrations can
be described with the following differential equation 2.2.1.1:

∂ 2 υ( x, t )  ∂υ ∂ 5 υ  4
µ + Fd  , 4  + EI ∂ υ( x, t ) = Q ( x, t ) (2.2.1.1)
∂t 2  ∂t ∂ x∂t  ∂4x
 
With the boundary conditions, respecting the type of support:
υ( x,0) = υ0 ( x); υ! ( x,0) = υ! 0 ( x)
(2.2.1.2)

Fig. 2. Mathematical model

In equation 2.2.1.1 υ(x,t) is deflection at the point x in time t; Q(x,t) is

deterministic excitation force; Fd is damping according to 2.2.1.3, as follows.
 ∂υ ∂5υ   ∂υ( x, t ) d  ∂ 4υ( x, t )  ∂υ( x, t ) d  ∂ 4υ( x, t )  (2.2.1.3)
Fd  , 4  = 2βµ or = EIα  or = 2βµ + EIα 
 ∂t ∂x ∂t   ∂(t ) 
dt  ∂x 4  ∂(t ) dt  ∂x 4 
   

Constants α and β are to be determinate experimentally, and they usually represent the
weak point in the analysis before the building process, since not being easy predictable.
Applying modal analysis, υ(x,t) is modeled as series of natural mode functions Vr(x),
according to relations 2.2.1.4:
∞ l l
1
υ( x, t ) = ∑ηr (t )Vr ( x) "(a) ηr (t ) = υ( x, t )Vr ( x)dx "(b), Hr = ∫Vr2 ( x)dx (2.2.1.4)
r =1 Hr 0∫ 0

Equation 2.2.1.1 can be presented as a system of r-independent equations of the

following form:
1
!! r (t ) + 2ξr ωr η! r (t ) + ω2r ηr (t ) =
η Fr (t ) (r = 1,2,...,∞) (2.2.1.5)
Mr

where: ηr(t) is the principal coordinate of mode r, ωr is natural circular frequency of mode
r, Mr and Fr(t) are generalized mass of mode r
286 A. SPASOJEVIĆ, DJ. DORDJEVIĆ, M. SPASOJEVIĆ, N. SPASOJEVIĆ

l
M r = µH r = µ ∫ Vr2 ( x)dx (2.2.1.6)
0
Fr(t) generalized force of mode r
l
Fr (t ) = ∫ Q ( x, t )Vr ( x)dx (2.2.1.7)
0

2.2.2. Problem solution

∞
υ( x, t ) = ∑ ηr (t )Vr ( x)
r =1
The basic solution is the deflection function υ(x,t), expressed as in 2.2.1.4(a), and as
its first and second derivative, velocity and acceleration functions can be obtained, for the
optional exciting deterministic force F(t) moving across the span at a constant speed c.
In order to simplify the solution, talking about slender structure, it is also acceptable
to negligee the damping, since it is relatively small. In that case ηr(t) can be found as the
solution of differential equation:
1
!! r (t ) + ωr2 ηr (t ) =
η Fr (t ) (r = 1,2,..., ∞) (2.2.2.1)
Mr

where: Mr is generalized mass of mode r, according to 2.2.1.6, Fr(t) generalized force of

mode r, determinate in 2.2.1.7, and Q(x,t) is excitation force. For the observed
pulsating point load F(t) moving across the span of the structure at the constant
speed c, exciting function Q(x,t) can be presented with the relation:
Q(x,t)= F(t) ⋅ δ(x − ct) (2.2.2.3)
using Dirac delta function, δ, which is determinate as:
0 za x ≠ ct
δ( x − ct ) = 
1 za x = ct
l
∫ δ( x − ct )dx = 1 0 ≤ ct ≤ ∞ (2.2.2.4)
0
l
∫ F ( x) ⋅ δ( x − ct )dx = F (ct ) 0 ≤ ct ≤ ∞
0
implying:
l
Fr (t ) = ∫ F (t ) ⋅ δ( x − ct ) ⋅ Vr ( x)dx ⇒ Fr (t ) = F (t ) ⋅ Vr (ct ) (2.2.2.5)
0
According to Eurocode the exciting load representing pedestrian is pulsating point load:
F(t) = F0⋅sin(2πfpt) i.e. F(t) = F0⋅sin(νt) (2.2.2.6)
moving over the span with the speed of c = 0,9⋅f0 .
 Some Relevant Aspects of Footbridge Vibrations 287

In the (2.2.2.6), F0 is amplitude of the exciting force, F0 = α ⋅ G = 0.257 ⋅ 700 = 180 N,

fp is load frequency, assumed to equal to f0 the first natural frequency of the system.
In British and Canadian Codes there is a simplified method for deriving maximal
acceleration, applicative for single span or two-or-three-span continuous symmetric
superstructures, of constant cross section. This calculation method is obtained form the
numerical studies, including the application of equation 1.2 in respect of varying bridge
structure parameters.

3. DESIGN EXAMPLE

Single span concrete footbridge

The evaluated system is a single span concrete footbridge (shown in figure 3), 15.0 m
in span, constant box cross-section, and constant weight of 33.0 kN/m. The natural
frequency of the system is f0 = 3.75 Hz. According to the fact that structural system of
single span bridge may have certain enclosed form solutions for solutions for dynamic
properties, the aim of this design example is to show the accordance of the algorithm.

Fig. 3. Appearance and cross-section of the bridge

Dynamic response analysis
Applying the Mathematica procedure, written according to algorithm explained in part
2.2.2, the obtained maximal acceleration value is a = 0.22 m/s2.
288 A. SPASOJEVIĆ, DJ. DORDJEVIĆ, M. SPASOJEVIĆ, N. SPASOJEVIĆ

Simplified calculation method

According to British Standard BS 5400, Part 2, maximal vertical acceleration should
be calculated as: a = 4π2f02yskψ, where f0 = 3.75 Hz, is fundamental natural frequency, ys
= 0.0000517, is static deflection at the half span, caused by point force of 0.7 kN in the
same cross-section, k = 1, is the structural system factor for single span girder, ψ = 0.6, is
dynamic response factor for adopted logarithmic damping decrement δ = 0.05 for
reinforced concrete structures. The calculated acceleration value is a = 0.172 m/s2.
The requirement a < alim ,where alim is determined according to equation 1.3, should
be satisfied.
⇒ a = 0.172 m/s2 < alim = 0.968 m/s2

4. CONCLUSION
The paper presents algorithm for calculation of important dynamic parameters of
pedestrian bridges subjected to human induced loading, allowing to predict serviceability
behavior. The authors are aware of assumptions made in this approach, considering them
acceptable. The numerical example of the simply supported beam structure shows
satisfying accordance of results obtained by this algorithm with the results obtained by
using simplified methods presented in relevant Codes of practice.

REFERENCES
1. Brčić, V.: Dinamika konstrukcija, Gradjevinska knjiga, Beograd, 1981 (in Serbian).
2. Bachmann, H.: "Lively" Footbridges – a Real Challenge, International Conference Footbridge 2002,
Paris 2002.
3. Milićević, M., Spasojević, N.: Some Aspects of vibration Analysis of Concrete Bridges, Modern
Concrete Structures, Faculty of Civil Engineering of University of Belgrade, 1994.
4. Kreuzinger, H.: Dynamic design strategies for pedestrian and wind actions, International conference
Footbridge 2002, Paris 2002.
5. R. L. Pimentel, A. Pavic and P. Waldron: Evaluation of design requirements for footbridges excited by
vertical forces from walking, Published on the NRC Research Press Web site on August 16. 2001,
University of Sheffield.
6. Milovanović, G.V., Djordjević, Dj. R.: Programming of Numerical Methods in FORTRAN (in Serbian).
University of Niš, Niš, 1981.

NEKI RELEVANTNI ASPEKTI VIBRACIJA PEŠAČKIH MOSTOVA

Ana Spasojević, Djordje Dordjević,
Marija Spasojević, Novak Spasojević
Prateći trend izgradnje elegantnih konstrukcija, posebno mostovskih, susrećemo se sa novim
izazovima u pogledu projektovanja i proračuna. Kada je reč o pešačkim mostovima vibracije
prouzrokovane ljudima mogu se kalsifikovati kao pitanje stanja upotrebljivosti iz razloga što je
ljudsko telo vrlo osetljivo na vibracije, a nivo vibracija koji prouzrokuje smetnju pešacima je
nedovoljan da prouzrokuje konstrukcijska oštećenja. Imajući u vidu činjenice da je korisno
opterećenje pešačkih mostova relativno malo, a da savremeni materijali omogućavaju veću
 Some Relevant Aspects of Footbridge Vibrations 289

nosivost, proračunom zasnovanim na isključivo statičkoj analizi, poštujući granična stanja

nosivosti, dobijamo konstrukcije male krutosti i mase, koje se, dakle, lako pobuđuju. Stoga je
naročito važno ispuniti uslove graničnih stanja upotrebljivosti konstrukcijskih elemenata, posebno
granična stanja upotrebljivosti u pogledu vibracija indukovanih pešacima, kako u vertikalnom
tako i u bočnom pravcu. Važeći jugoslovenski propisi ne tretiraju ovaj fenomen, te je u radu
sprovedena analiza prema nekoliko aktuelnih svetskih standarda za ovu oblast: Evrokod, Britanski
standardi i Kanadski standardi. Prikazan je algoritam za određivanje dinamičkih karakteristika
konstrukcija i sračunavanje ugiba, odnosno brzine i ubrzanj, tačaka konstrukcije pod pokretnim
harmonijskim opterećenjem. Testiran je model pešačkog mosta sistema proste grede, sa ciljem
provere tačnost algoritma sobzirom na postojeća približna rešenjia za ovaj sistem u navedenim
standardima.
Ključne reči: Pešački most, pobudjenje pešacima, prinudne vibracije, granična stanja.