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In general, it seems accepted by many experts that the use of three dynamic models may be appropriate as follows:

. _{a model for a single pedestrian}

. _{a model for a group of pedestrians, for example from 10 to 15} . _{a model for a dense crowd.}

In the following, some background information is given concerning the first two models, but currently it is not possible to give a reliable model for a dense crowd. Many studies are being performed at the present time (2009), and results are expected in the future. The purpose of the following information is to give an idea of the directions adopted in current approaches.

With regard to comfort criteria, see Chapter 8 of this Designers’ Guide.

Model for a single pedestrian

The model for a single pedestrian can be directly used for some verifications, but it is mostly used to define the dynamic excitation due to a group of pedestrians. The most basic model, but often agreed by experts, is a harmonic load:

QpðtÞ ¼ G
sinð2ftÞ

where f is the fundamental frequency under consideration.

For the vertical excitation by a pedestrian who is not running, G is taken equal to 280 N: it is the result of the multiplication of 700 N (representing the average pedestrian weight) by 0.4

Structural type Structural damping, s

Steel bridgesþ lattice steel towers Welded 0.02 High-resistance bolts 0.03 Ordinary bolts 0.05

Composite bridges 0.04

Concrete bridges Prestressed without cracks 0.04 Prestressed with cracks 0.10

Timber bridges* 0.06–0.12

Bridges, aluminium alloys 0.02

Bridges, glass- or -reinforced plastic 0.04–0.08

Cables Parallel cables 0.006

Spiral cables 0.020

Note 1: The values for timber and plastic composites are indicative only. In cases where aerodynamic effects are found to be significant in the design, more refined figures are needed through specialist advice (agreed if appropriate with the competent authority).

Note 2: For cable-stayed bridges the values given in Table F.2 need to be factored by 0.75.

* In EN 1995-2 (Design of timber bridges) the logarithmic decrement of structural damping is in the range 0:01
2 ¼ 0:063 for structures without mechanical joints to 0:015
2 ¼ 0:094 for structures with mechanical joints.

which derives from the development in Fourier’s series of the action due to walking for f ¼ fv¼ 2 Hz and for a pedestrian velocity equal to 0:9fv.

For the horizontal lateral excitation, G varies from 35 to 70 N and, in the previous formula, the frequency is the relevant horizontal frequency.

More sophisticated dynamic models for the single pedestrian have been proposed by several authors: these models associate, in general, several harmonic functions introducing several vibration modes.

In Annex B to EN 1995-2 (Vibrations caused by pedestrians),5which is only applicable to timber bridges with simply supported beams or truss systems excited by pedestrians, formulae give directly the vertical and horizontal (lateral) accelerations of the bridge. (a) Vertical acceleration avert;1:

avert;1 ¼

200

M& for fvert 2:5 Hz 100

M& for 2:5 Hz fvert  5:0 Hz 8 > > < > > : ðB:1Þ where

M is the total mass of the bridge in kg, given by M¼ ml ‘ is the span of the bridge

m is the mass per unit length (self-weight) of the bridge in kg/m & is the damping ratio

fvert is the fundamental natural frequency for vertical deformation of the bridge.

(b) Horizontal acceleration ahor;1of the bridge:

ahor;1¼

50

M& for 0:5 Hz fhor 2:5 Hz

where fhoris the fundamental natural frequency for horizontal deformation of the bridge.

For example, in the formulae for vertical vibrations, the figure above M derives from 700
0:4 where  is the ratio between the structural response due to a pedestrian walking without moving forward and the structural response due to a pedestrian crossing the footbridge. This ratio depends on the structural response and it can only be given accept- able averaged values. For example, in the first case of vertical vibrations, 200ffi 280
0:7.

For a jogger, some figures may be different.

Model for a group of pedestrians

The forces exerted by several pedestrians in common circumstances are normally not synchronized and have somewhat different frequencies. However, if one of the natural frequencies of the deck is close to the frequencies of the forces normally exerted by pedestrians, it commonly happens that their perception of some movements of the bridge result in modifications of their gait: their steps tend to become synchronized with the vibrations of the bridge; resonance then occurs, increasing considerably the response of the bridge.

In the absence of significant vibration, the number of persons contributing to the resonance is highly random; beyond about 10 persons on the bridge, it is a decreasing func- tion of their number. For vertical vibrations, the resonance is in most cases mainly, but not solely, linked to the fundamental frequency of the bridge; for horizontal or torsional vibra- tions, the problem is more complex. However, correlation between forces exerted by pedes- trians may increase with movements.

For a group of pedestrians, the model is more sophisticated than for a single pedestrian, but the most simplified rules give a generic expression such as:

where

n is the equivalent number of pedestrians on the appropriate loaded surface

is the reduction factor, a function of the difference between the real frequency of the pedestrian excitation and the natural structural frequency under consideration: in fact, it is a mathematical function, varying between 0 and 1, equal to 1 when the natural structural frequency can be excited by pedestrians.

As an example, in EN 1995-2, the following expressions are proposed for a group of people crossing a timber bridge:

(a) Vertical acceleration a_vert;n: avert;n¼ 0:23avert;1nkvert ðB:2Þ

where

n is the number of pedestrians kvert is a coefficient according to Fig. 5.6

avert;1 is the vertical acceleration for one person crossing the bridge determined according

to Expression (B.1)

The number of pedestrians, n, should be taken as:

. _{n¼ 13 for a distinct group of pedestrians} . _{n¼ 0:6A for a continuous stream of pedestrians}

where A is the area of the bridge deck in m2.

It has to be noted that 0.23n is a good approximation ofpffiffiffinfor 12 < n < 20: 0:23nffipffiffiffin for nffi 19.

(b) Horizontal (lateral) acceleration ahor;n:

ahor;n¼ 0:18ahor;1nkhor ðB:5Þ

where khoris a coefficient according to Fig. 5.7. 1 0.5 0.33 0 0 1 2 3 4 5 fvert kve rt

Fig. 5.6. Relationship between the vertical fundamental natural frequency fvertand the coefficient kvert

1 0.5 0 khor 0 0.5 1 1.5 2 2.5 fhor

The number of pedestrians, n, should be taken as:

. _{n¼ 13 for a distinct group of pedestrians} . _{n¼ 0:6A for a continuous stream of pedestrians}

where A is the area of the bridge deck in m2.

Other models

Several other models have been proposed by authors or scientific associations. They all have qualities and inadequacies. The concept of critical number of pedestrians sometimes appears. For example, according to an Arup consultant (pers. comm.), the critical number of pedestrians leading to lateral instability may be expressed according to the formula:

nc¼

8; f_iM_i k where

 is the damping ratio

fi is the natural frequency (rad/s)

Mi is modal mass

k is the empirical factor equal, for example, to 300 Ns/m for frequencies in the range 0.5–1.0 Hz.

However, the concept of critical number of pedestrians still needs to be validated.6