Acta Polytechnica CTU Proceedings


doi:10.14311/APP.2018.15.0114
Acta Polytechnica CTU Proceedings 15:114–119, 2018 © Czech Technical University in Prague, 2018

available online at http://ojs.cvut.cz/ojs/index.php/app

RELATION BETWEEN STATIONARY AND MOVING
PEDESTRIAN LOAD MODELS

Jan Štěpánek∗, Jiří Máca

Czech Technical University in Prague, Faculty of Civil Engineering, Thákurova 7, 166 29 Prague 6, Czech
Republic

∗ corresponding author: jan.stepanek@fsv.cvut.cz

Abstract. The dynamic analysis of footbridges became common in recent years, but the approach
to modelling pedestrian induced load is still inconsistent. This paper compares two different ways
of modelling single pedestrian load. The response of the structure to stationary pulsating force is
compared to the results given by the same force moving along simply supported beam. 300 randomly
generated beams were used in this parametric study to describe the relation among these two approaches
and parameters of the structure. The proposed formula was successfully applied to get more realistic
estimation of response of the structure to a single walking pedestrian using stationary pulsating force.

Keywords: Footbridges, pedestrian load models, forced vibration, response of structure.

1. Introduction
Footbridges designed and built during recent years
are very slender and flexible structures in many cases.
Such footbridges may be very susceptible to pedestrian
induced load which may cause an uncomfortable level
of vibration. Natural frequencies of such structures are
often very close to walking frequency of pedestrians,
and therefore the resonance effect may occur.

A number of studies have been made in past decades
to define the procedure of effective and safe design of
footbridges. A footbridge designed this way has to
satisfy the criteria of serviceability limit state (SLS).
To be able to detect the risk of unacceptable vibration
during design phase the dynamic analysis is often
necessary. The European codes define the maximal
level of acceleration of a bridge deck which can be
reached to keep the sufficient comfort of pedestrians.
Numerical simulation can reveal the sensitivity of the
structure due to pedestrians but unfortunately there
is still a lack of pedestrian load models in current
Eurocodes.
Many single pedestrian load models have been de-

veloped [1, 2] but most of them assume the numerical
analysis using moving forces or moving biodynamic
models. Nevertheless, most of the commercial pro-
grams used for static design of bridges do not provide
the option of calculation with force moving along the
structure.

Stationary harmonic pulsating force was presented
in guidebook [3] to simulate single walking pedestrian.
The advantage of this estimation is its simplicity when
it comes to numeric analysis in static software. The
main disadvantage of this load model is that it does
not consider the time that one spends on the bridge.
The stationary pulsating force without time limita-
tion may cause overestimation of the response of the
structure which leads to unnecessary design changes
and potential increase of cost of the structure.

This paper focuses on comparison of approaches of
modelling single pedestrian load by stationary and
moving pulsating force. The goal is to find a simply ap-
plicable formula which combines these two approaches.
The formula can be used to find realistic estimation of
response of a simply supported beam using stationary
pulsating force.

2. Pedestrian load models
Pedestrian load can be divided into several categories.
Normal traffic on the bridge can be defined as a single
pedestrian load, group of walking people or stream
of pedestrians. Exceptional load situations (jumping,
running, kneeling or vandalism) must not excite the
intolerable vibrations either [2]. To be able to evaluate
the response of the structure in normal operation, it
is necessary to define a simple but accurate dynamic
single pedestrian load model.

2.1. Single pedestrian load
Dynamic force produced by a single person can be
represented by three components in perpendicular di-
rections: vertical, horizontal lateral and horizontal
longitudinal components [1]. Horizontal longitudi-
nal force is usually considered negligible. The lateral
component is caused by moving centre of gravity hori-
zontally while walking. The frequency of such force is
half than the stride frequency. A vertical component
with the stride frequency is the cause of most of the
documented problems with uncomfortable vibrations
of footbridges. Only the vertical component is further
investigated in this paper.
Vertical pedestrian load, assuming to be perfectly

periodical force, can be formulated by Fourier series
in Equation (1). Numerous different approximations
of such formulation were developed in the past years
[1, 4].

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vol. 15/2018 Relation between stationary and moving pedestrian load models

Fv (t) = G + G
n∑

i=1
αi sin(2πifpt + ϕi) (1)

where:

Fv (t) - vertical pedestrian load
G - load corresponding to pedestrian’s weight
αi - Fourier coefficient of the respective member
fp - step frequency
ϕi - phase shift
n - total number of included members

Various coefficients αi and ϕi can be found in litera-
ture [1, 4, 5]. In general, it can be said that including
more than three or four harmonic members has no
effect on the final load due to low values of αi for
higher components.

The worst-case scenario, when the stride frequency
is equal to one of the natural frequencies of the bridge,
is usually adopted for simulating a single pedestrian
walking along the structure [1, 4, 6]. Pedestrian walk-
ing velocity vs also plays an important role in service-
ability verification of the structure. The speed vs can
be taken by Eq. (2)

vs = fsls (2)

Where fs is step frequency and ls is step length.
The recommended mean value for ls also varies in
different studies. Guidebook [4] recommends ls =
0.9 m. According to guidebook [3] ls is dependent on
step frequency and for 2 Hz (defined as normal walk)
the typical step length ls = 0.75 m is considered.

2.2. Stationary pedestrian load model
The stationary pulsating force can be a simple way to
find out the response of structure to dynamic pedes-
trian load. The guidebook [6] recommends to apply
stationary harmonic force with amplitude 280 N to
the most adverse position of the structure. The value
of the force is taken as 700×0.4=280 N which is in ac-
cordance with average pedestrian weight 70 kg and the
first Fourier coefficient 0.4. Other approaches take the
movement of a person in account. The amplitude of
applied force is multiplied by the reduction constant R.
Allen And Murray give R=0.7 while Grundman takes
R=0.6 [1].
The formula for reasonable reduction factor R for

simply supported beam is set in further parametric
study.

2.3. Moving pedestrian load model
The moving load models can be used to define the
response of the structure to pedestrian load in a very
realistic way [5, 7]. Examples of more sophisticated
load models, including biodynamic models and in-
verted pendulum can be found in [8]. Many force
models are summarized in [1, 4] and most of them
contains first Fourier coefficient α1 ≈ 0.4. For this rea-
son, one of the simplest models proposed by Charles

and Hooprah was used in this study to simulate a
walking pedestrian. This model consists only of static
force G and first harmonic member for α1 = 0.4. This
simplified approach is also tolerated in guidebook [2]
for simulation of a walking pedestrian.

3. Parametric study
In this chapter a numerical analysis of 300 simply
supported beams with randomly generated parameters
was performed to find the formula for the reduction
factor R. Both stationary a moving pulsating force
were used to find the peak acceleration in vertical
direction. Value of R defined in Eq. (3) was taken
for every single structure. Finally, the function of
R was found using the least-squares approximation
with parameters of the tested beams used as the basis
functions.

Rcal =
amov
ast

(3)

Where amov is peak acceleration all over the struc-
ture calculated using moving force model and ast is
acceleration induced by stationary force model with-
out time limitation.

3.1. set of simply supported beams
All the tested structures share the scheme of simply
supported beam. Every beam was generated so that
one of the three lowest natural frequencies equals to
2 Hz. The i-th natural frequency of simply supported
beam can be calculated from analytical equation (4).

ωi = π2i2
√
EIy
µl4

(4)

Where:

ωi - i-th natural circular frequency
E - modulus of elasticity
Iy - cross sectional second moment of area
µ - mass of one meter of length
l - span length

Iy can be expressed:

Iy =
µl4ω2i
π4i4E

(5)

and µ is defined in Eq. (6).

µ = ρA + µad (6)

If ωi = 2π × 2 Hz, every other parameter excluding
Iy can be randomly generated and Iy is found using
Eq. (5). This procedure allows to generate simply
supported beams with i-th natural frequency equals
to 2 Hz.
Assuming all the theoretical structures have rect-

angular cross-section, its width b and height h can be
found by solving system of two equations (7) and (8).

A = bh (7)

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Jan Štěpánek, Jiří Máca Acta Polytechnica CTU Proceedings

Iy =
bh3

12
(8)

Where:

A - cross sectional area
ρ - material density
µad - mass added on each meter of length without

static effect

A single material – reinforced concrete C 30/37 was
chosen for all the theoretical beams.
The ranges of randomly generated parameters:

ρ = 2500 kg/m3 - standard density of concrete
C 30/37

E = 32 GPa - modulus of elasticity of concrete
C30/37

ωi = 12.5664 - circular frequency for fi = 2 Hz.
l = 10-50 m - The typical span length of foot-

bridges with static scheme of simply supported
beam.

A = 0.03-3 m2 - Wide range of cross sectional
area which allows sufficient diversity of struc-
tures in the set.

µad = 0 - 300 kg/m
i = 1-3 - One of the first three natural frequen-

cies equals to 2 Hz
ξ = 0.008-0.02 - damping ratio of concrete foot-

bridges [1]

The goal of this chapter is to assemble a set of sim-
ply supported beams with wide range of parameters
so that the statistical analysis can be done using the
leas-squares method. It means that some of the struc-
tures could exist just as mathematical model, not a
real structure, because of their slenderness or high
susceptibility to vibration. Some other conditions of
the properties of generated beams are set to reduce
the amount of such structures in the list:

• h > 0.1 m, b > 0.1 m - minimal height and width
of the cross section

• 0.1 < h/b < 10 - minimal and maximal ratio of
height and width

• h/l < 0.1 - maximal ratio of cross sectional height
and span length

• b/l < 0.5 - maximal ratio of width and span length

The final set of the structures is composed of three
parts. Each part contains 100 beams with i-th natural
frequency 2 Hz for i = 1-3. Every structure is modelled
using finite element method (FEM) in MATLAB and
consists of 50 beam elements. The shear deformation
of elements is neglected according to Bernoulli-Navier
hypothesis. This simplification can be applied thanks
to condition h/l<0.1 .

3.2. Response of the structure
3.2.1. Response to stationary pulsating

force
The i-th mode shape with corresponding natural fre-
quency 2 Hz must be found at first. The vertical
harmonic force Fv (t) with amplitude of 280 N and
frequency fv = 2 Hz (9) is then applied in the most
adverse position of the bridge.

Fv (t) = 280 sin(2πfvt) N (9)

Maximal displacement of the structure is found
using Newmark-β method for solving equation of mo-
tion:

Kr(t) + Cṙ(t) + Mr̈(t) = p(t) (10)

Where:

K - stiffness matrix
C - damping matrix
M - mass matrix
r(t) - vector of displacement
ṙ(t) - vector of velocity
r̈(t) - vector of acceleration
p(t) - load vector created from Fv (t)

Maximal acceleration can also be found solving the
system of 2N linear equations (11) for system of N
degrees of freedom. This approach is technically much
easier but it does not provide any information about
time history.

[
(K − ω2M) −ωC

ωC (K − ω2M)

]{
rs
rc

}
=
{
fs
fc

}
(11)

Where ω is the circular frequency of pacing force
and r(t) and p(t) are expressed by two components:

r(t) = rs sin(ωt) + rc cos(ωt) (12)

p(t) = ps sin(ωt) + pc cos(ωt) (13)

The amplitude of acceleration r̈ of such harmonic
motion is given by Eq. (14).

r̈ = ω2
√
r2s + r2c (14)

The reference value ast is finally found as maximal
value of acceleration vector r̈ in vertical direction.

3.2.2. Response to moving pulsating force
The pedestrian load model, used in this section, was
adopted from [1]. It consists only of static weight of
pedestrian G = 70 kg and the first member of Fourier
series α1 = 0.4 and is expressed by Eq. (15). The
model is moving along the structure. To define the
velocity of its movement, the step length needs to
be defined. The step length ls = 0.75 m which is
appropriate for step frequency was adopted from [3].
The walking speed was set according to Eq. (2).

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vol. 15/2018 Relation between stationary and moving pedestrian load models

Figure 1. Separate contributions of basis functions.

Fv (t) = 700 + 280 sin(2πfvt) N (15)

The peak acceleration amov in vertical direc-
tion reached on each structure was analysed using
Newmark-β method.

3.3. Evaluation
The peak accelerations ast induced by stationary force
model and amov induced by the moving force model
were evaluated for all the structures which resulted in
300 values of reduction coefficient Rcal Eq. (3)

The least-squares method was used for determina-
tion of the relation between reduction coefficient R
and the parameters of the structure. Eleven basis
functions bj were defined:

(1.) l - span length
(2.) µ - mass for 1 meter of the beam
(3.) Iy - cross sectional second moment of area
(4.) µl - total mass of the structure

(5.)
1
l
- multiplicative inverse of span length

(6.)
1
µ

- multiplicative inverse of mass for 1 meter of
the beam

(7.)
1
Iy

- multiplicative inverse of cross sectional sec-

ond moment of area
(8.) i - the order of frequency which equals to 2 Hz.
(9.) constant
(10.) ξ1 - damping ratio of the first mode shape

(11.)
1
ξ1

- multiplicative inverse of damping ratio of
the first frequency

The coefficients βj of basis functions were found
solving system of equations (16).



X11 X12 · · · X1n
X21 X22 · · · X2n

· · · · · ·
...

...
Xm1 Xm2 · · · Xmn





β1
β2
...
βn


 =



y1
y2
...
ym


 (16)

Where Xij =
300∑
k=1

bi,kbj,k and yi =
300∑
k=1

bi,kRcal,k

and k stands for number of the structure in the list.
The first calculation was primarily performed in

order to define the parameters which have the great-
est influence on the value of reduction factor. Four
functions (no. 5,8,9 and 10) were chosen for further
analysis.
The second calculation was performed using only

four selected functions and the respective coefficients
β were found.
The function Rcal can be approximated by Rapp

which is expressed as a linear combination of basis
functions (17). The Figure 1 shows the contributions
of chosen basis functions.

Rcal ≈ Rapp =
∑

bjβj (17)

The formula for R is finally set by equation:

R =
−6.248

l
+ 0.0495i + 14.42ξ1 + 0.726 (18)

and is usable in reduction of the stationary pulsating
force model of single walking pedestrian according to
Eq. (19).

Fv (t) = R280 sin(2πfvt) N (19)

As can be seen in Figure 2, all the values of Rcal falls
in range between 0.5 - 1 thus the range of meaningful
values is defined as:

0.5 ≤ R ≤ 1 (20)

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Jan Štěpánek, Jiří Máca Acta Polytechnica CTU Proceedings

Figure 2. Calculated and approximated values of R.

The average relative error of the approximation
∆R calculated according to Eq.(21) is equal to 2.3 %.
Taking all the uncertainties of pedestrian load models
into account, the value of relative error is sufficiently
small.

∆R =
∣∣∣∣Rcal − RappRcal

∣∣∣∣ (21)
4. Verification of the proposed

formula
Verification of the proposed formula has been per-
formed on numerical model of experimental structure
presented in [5]. The experimental footbridge is pre-
stressed concrete beam with T cross section and short
overhangs (Figure 5). The tuned mass damper (TMD)
was installed under the midspan of the beam to show
its influence on the susceptibility to vibration. TMD
allows the options to be locked, which is performed
by stiff connection to the structure.

The moving load model used in section 3 provided
very accurate results of response of the structure in
comparison to load tests [5].

Calculations were performed with both locked and
free TMD to verify the formula for reduction factor
R (18).
The first natural frequency of the structure f1 =

1.98 Hz, and therefore the step length of 0.75 m can
be applied according to guidebook [3]. Several pa-
rameters of the structure are required to estimate the
reduction factor R:

• l = 19 m
• i = 1 (the first natural frequency is close to 2 Hz)
• ξ1, damping ratio of the first mode shape = 0.0143
(locked TMD), 0.06 (free TMD)

4.1. locked TMD
The peak accelerations at midspan amov using moving
pedestrian load model according to Eq. (19) was

Figure 3. Response of the structure - stationary
pedestrian load, locked TMD.

Figure 4. Response of the structure - moving pedes-
trian load, locked TMD.

evaluated and can be seen in Figure 4. The value for
reduction coefficient was calculated from Eq. (18).

Calculation of R for locked TMD:

R =
−6.248

l
+ 0.0495i + 14.42ξ1 + 0.726

R =
−6.248

17.4
+ 0.0495 × 1 + 14.42 × 0.0143 + 0.726

R = 0.623

The amplitude of the stationary pulsating force is
then R × 280 = 174 N. Applying this force according
to Eq. (19) in the midspan of the beam, the maximal
acceleration ast was evaluated.

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vol. 15/2018 Relation between stationary and moving pedestrian load models

Figure 5. Static scheme of experimental footbridge [5].

TMD R acceleration - moving model [ms−2] acceleration - stationary model [ms−2]

locked 0.623 1.100 1.073
free 1 0.264 0.261

Table 1. Maximal acceleration at midspan.

4.2. free TMD
The evaluation of acceleration with free TMD was
performed using the same method as with locked
TMD. Due to high structural damping (≈6%), the
reduction factor R exceeded limit value 1 which was
presented in Eq. (20). Therefore, the value of R is
set to 1. Comparison of the results can be seen in
Table 1.

5. Conclusions
The formula for the reduction coefficient R, which can
be used for simulation of single pedestrian load, was
proposed. This formula simplifies the pedestrian load
from moving force model to stationary harmonic force
model but keeps the accuracy of moving model. The
formula is applicable on footbridges with static scheme
of simply supported beams and with one of the first
three natural frequencies close to 2 Hz. The proposed
method could be a very effective way to evaluate
response of the structure to a single pedestrian load,
because it does not require a time analysis and can
be simply performed in commercial static software.
Parametric study in section 3 proved it is mainly

length of the bridge and structural damping that influ-
ences the reduction coefficient. The numerical analysis
in section 4 proved the estimation of reduction factor
R = 0.6 - 0.7 [1] for low damped structures. However,
it also showed that these approximations can lead to
underestimation of load, which was demonstrated on
the structure with TMD in this case.

Verification of the proposed formula was performed
with very accurate results. Nevertheless, further ex-
perimental verification and investigation in this field
shall be performed.

Acknowledgements
The authors gratefully acknowledge support from
the Czech Technical University in Prague, project
SGS18/037/OHK1/1T/11 Development and application of
advanced algorithms for numerical analysis and modeling
in mechanics of structures and materials

References
[1] Štimac Grandić. Serviceability verification of
pedestrian bridges under pedestrian loading. Tehnički
vjesnik 22:527–537, 2015.
doi:10.17559/TV-20131030105641.

[2] SETRA. Footbridges: Assessment of vibrational
behaviour of footbridges under pedestrian loading, 2006.

[3] FIB Bulletin 32: Guidelines for the design of
footbridges, 2005.

[4] S. Živanović, A. Pavić, P. Reynolds. Vibration
serviceability of footbridges under human-induced
excitation: a literature review. Journal of Sound and
Vibration 279(1-2):1–74, 2015.
doi:10.1016/j.jsv.2004.01.019.

[5] A. Dazio. Fundamentals of structural dynamics.
An-Najah National University, 2013.

[6] Guide to basis of bridge design related to eurocodes
supplemented by practical design - handbook 4.
Leonardo da Vinci pilot project, 2004.

[7] E. Caetano, A. Cunha, W. Hoorpah, J. Raoul.
Footbridge Vibration Design. CRC Press, 2009.

[8] E. Shahabpoor, A. Pavic, V. Racic. Interaction
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http://dx.doi.org/10.1016/j.jsv.2004.01.019
http://dx.doi.org/10.1155/2016/3430285

	Acta Polytechnica CTU Proceedings 15:114–119, 2018
	1 Introduction
	2 Pedestrian load models
	2.1 Single pedestrian load
	2.2 Stationary pedestrian load model
	2.3 Moving pedestrian load model

	3 Parametric study
	3.1 set of simply supported beams
	3.2 Response of the structure
	3.2.1 Response to stationary pulsating force
	3.2.2 Response to moving pulsating force

	3.3 Evaluation

	4 Verification of the proposed formula
	4.1 locked TMD
	4.2 free TMD

	5 Conclusions
	Acknowledgements
	References