On Pedestrian’s Walking Load Identification

*Ming Wang1) and Jun Chen2)

1), 2)Department of Structural Engineering, Tongji University, Shanghai, China

1)1431980wangming@tongji.edu.cn

ABSTRACT

Pedestrian’s walking loads may induce severe vibrations to structures likefootbridges, long-span floors and corridors leading to the so-called structural vibrationserviceability issues. When designing structures subjected to pedestrian loads，reliable walking load model is crucial for accurate prediction of structural responses.However, it is not an easy task to measure the walking load directly. Inspired by themethod for identifying moving load, the paper suggests an approach to identifypedestrian’s walking load using structural responses. The method is based on modalsuperposition principle. With assumed modal shape functions for the line-like structure,analytical solution of the structural response due to pedestrian was derived andidentification function was then established. In general, the identification matrix is ill-conditioned and the identification results are sensitive to measurement noise. To tacklethis problem, the general regularization method is adopted. For the scenario of crowdmoving load, it was simplified as a concentrated load or a moving load for identificationpurpose. Numerical examples show that accurate loads can be obtained by thesuggested method for noise-free measurement. For noise-polluted case, theidentification accuracy of walking load is acceptable for low noise level.

1. INTRODUCTION

Currently moving load identification is mainly focused on vehicle loading, forhuman-induced excitations acted on the footbridge or long-span floors have not beenstudied from the perspective of load identification. With the development of socialeconomy, new structure form and high-strength lightweight materials have been widelyused in public buildings, causing structural natural frequencies to be lower and lower.These structures are prone to vibrations under human-induced excitation, so thestructural response under pedestrian’s walking loads can’t be ignored, especially forfootbridges in urban or tourism area. Accurate estimate of pedestrian’s walking load isvery important for the structural design, control and vibration serviceability assessment.However, it is not an easy task to measure the walking load directly. Inspired by themethod for identifying moving load, this paper suggests an approach to identifypedestrian’s walking load using structural responses.

1)Graduate Student

2)Professor

In this paper, an attempt is made for pedestrianbasis of modal superpositionline-like structure (Law 1997pedestrian was derived andexamples are given to show the appl

2. THEORY OF WALKING LOAD IDENTIFICATION

Suppose one person walking on a simply supported beam at a constant speed, asshown in Fig.1. The beam is assumed to be of constant crossmass per unit length, having linear, viscous proportional damping with small deflections,and the effects of shear deformation and rotary inertia are not

Fig.1 Simply supported beam subjected to pedestrian’s walking load

The equation of motion can be written as

2 4

2 4

( , ) ( , ) ( , )u x t u x t u x t

t t xρ δ∂ ∂ ∂

+ + = −∂ ∂ ∂

where u(x,t) is the beam deflection at pointis the viscous damping parameter,second moment of inertia of thethe time varying load point, f(t)and δ(t) is the Dirac delta function.

Based on modal superposition, the dynamic deflectionfollows:

where n is the mode number,qn(t) are the nth mode amplitudes. Substituting

2

2

( ) ( )n nd q t dq t

dt dt M+ + =

n this paper, an attempt is made for pedestrian’s walking load identificationbasis of modal superposition principle, and assuming modal shape functions for the

1997), analytical solution of the structural response due topedestrian was derived and identification function was then established.examples are given to show the application of the proposed method.

THEORY OF WALKING LOAD IDENTIFICATION

uppose one person walking on a simply supported beam at a constant speed, ashe beam is assumed to be of constant cross-section with constant

mass per unit length, having linear, viscous proportional damping with small deflections,and the effects of shear deformation and rotary inertia are not considered.

imply supported beam subjected to pedestrian’s walking load

The equation of motion can be written as:

2 4

2 4

( , ) ( , ) ( , )( ( )) ( )

u x t u x t u x tC EI x a t f t

t t xρ δ∂ ∂ ∂

+ + = −∂ ∂ ∂

is the beam deflection at point x and time t, ρ is the mass per unit length,is the viscous damping parameter, E is the Young’s modulus of the material,second moment of inertia of the beam cross-section, l is the length of the beam,

f(t) is the time varying walking load, c is the walking speed,is the Dirac delta function.

Based on modal superposition, the dynamic deflection u(x,t) can be

1

( , ) ( ) ( )n nn

u x t x q t∞

=

= Φ∑

is the mode number, Фn(x) is the mode shape function of theth mode amplitudes. Substituting Eq. (2) into Eq. (1), we obtain

2( ) ( ) 12 ( ) ( )n n

n n n n n

n

d q t dq tw w q t p t

dt dt Mξ+ + =

identification, on themodal shape functions for the

, analytical solution of the structural response due todentification function was then established. Numerical

uppose one person walking on a simply supported beam at a constant speed, assection with constant

mass per unit length, having linear, viscous proportional damping with small deflections,considered.

imply supported beam subjected to pedestrian’s walking load

(1)

is the mass per unit length, Cis the Young’s modulus of the material, I is the

is the length of the beam, a(t) isis the walking speed,

can be described as

(2)

is the mode shape function of the nth mode, and the, we obtain

(3)

where wn is the modal frequency of the nth mode, ξn is the damping ratio of the nthmode, and Mn is the modal mass of the nth mode, and pn(t) is the modal fore. Based onassumptions for the beam, the modal parameters of the beam can be calculated asfollows:

2( / ) / ,nw n l EIπ ρ= ( ) sin( / )n x n x lπΦ = (4)

/ 2,nM lρ= ( ) ( )sin( ( ) / )np t f t n a t lπ= (5)

For real structures, the modal parameters can be obtained from the finite elementmodel or modal testing.

Eq. (3) can be solved in the time domain:

0

1( ) ( ) ( )

t

n n n

n

q t h t p dM

τ τ τ= −∫ (6)

where ' '( ) (1 / ) sin( ),n nw tn n nh t w e w tξ−= ' 21n nw w ξ= − (7)

Substituting Eq. (4) and Eq. (6) into Eq. (2), the dynamic deflection of the beam atpoint x and time t can be obtained (Chan 1999).

( ) '

' 01

2 ( )( , ) sin( ) sin ( )sin ( )n n

tw t

nn n

n x n a tu x t e w t f d

lw l lξ τπ π

τ τ τρ

∞− −

=

= −∑ ∫ (8)

Eq. (8) can be rewritten in discrete terms as

( ) '

'1 0

1 ( )( ) ( ) sin( ( ))sin( ) ( )n n

iw t i j

n nn jn n

n a ju i x e w t i j f j t

M w lξ π∞

− ∆ −

= =

= Φ ∆ − ∆∑ ∑ (9)

0,1,2 ti N= L

where t∆ is the sampling interval and Nt+1 is the number of sample points.The response of mode n is

( ) '

'0

1 ( )( ) ( ) sin( ( ))sin( ) ( )n n

iw t i j

n n njn n

n a ju i x e w t i j f j t

M w lξ π− ∆ −

=

= Φ ∆ − ∆∑ (10)

Let

'

2sin( )xn

n

n xD

lw l

π

ρ= (11)

'( ) sin( )n nw k tn nH k e w k t tξ− ∆= ∆ ∆ (12)

( )( ) sin( )n

n a kS k

l

π= (13)

Arranging Eq. (10) into matrix form,

(0) (0) (0) 0 0 0

(1) (1) (0) (0) (1) 0 0

(2) (2) (0) (1) (1) (0) (2) 0

(3) (3) (0) (2) (1) (1) (2) 0

( ) ( ) (0) ( 1) (1) ( 2) (2) ( )

n n

n n n n

n n n n n n

n n n n n n

t n t n n t n n t n n t Bn

u H S

u H S H S

u H S H S H S

u H S H S H S

u N H N S H N S H N S H N N S

=

− − −

L

L

L

L

M M M M O M

L

(0)

(1)

(2)

(3)

( ) ( )n B B

f

f

f

f

N f N

M

(14)

Assuming the initial condition as

(0) 0,f = ( ) 0tf N = ， (0) 0,u = (1) 0u = (15)

Eq. (14) can be condensed as

(2) (1) (1) 0 0 0

(3) (2) (1) (1) (2) 0 0

(4) (3) (1) (2) (2) (1) (3) 0

(5) (4) (1) (3) (2) (2) (3) 0

( ) ( 1) (1) ( 2) (2) ( 3) (3) (

n n

n n n n

n n n n n n

xn

n n n n n n

t n t n n t n n t n n tn

u H S

u H S H S

u H S H S H SD

u H S H S H S

u N H N S H N S H N S H N

=

− − −

L

L

L

L

M M M M O M

L

(1)

(2)

(3)

(4)

1) ( 1) ( 1)B n B B

f

f

f

f

N S N f N

− + − −

M

(16)

Eq. (16) is simply rewritten as

( 1) ( 1)( 1) 1 ( 1) 1t Bt BN NN N

u B f− × −− × − ×

= (17)

3. REGULARIZATION

The natural frequencies and mode shapes obtained from modal testing and modalanalysis are subject to measurement error. Noise contamination in the test data hasadverse effect on the accuracy of the identified walking load. In general, the walkingload can be identified from Eq. (16) by the least-squares method, but the result wouldbe unbound. A regularization technique can be used to solve the ill-conditionedproblem in the form of minimizing the function parameter (Y. Xiao 2013). Thegeneralized cross-validation and L-curve method are employed to determine theoptimal regularization parameter (Zhu 2002).

2 2( , ) *J P B P Pλ ε λ= − + (18)

where λ is the non-negative regularization parameter

4. SIMULATION AND RESULTS

To check the correctness and effectiveness of this method, there is a simplysupported beam subject to walking load. The parameters of the beam are as followed:EI=1×109(N*m2), ρ=600kg/m, l=20m, ξ=0.02, N=3, human walking speed v=1.5m/s, steplength Δl=0.75m, and suppose that the first step is 0.5m from the left support.Nt=NB=1300, and the responses of 0.25*l, 0.45*l and 0.8*l are used in the identification.(G. Ding 2016) studied the single step walking load, so this paper assumes that eachstep load is the same and adopts one of the cases.

2 4 61 600(0.81 0.1632sin( ) 0.3442sin( ) 0.1470sin( )

2 2 2f t t t

T T T

π π π π π π= + − + − + − (19)

Fig.2 Single step load of walking

Fig.3 Simulated time varying walking load f(t)

0.0 0.1 0.2 0.3 0.4 0.50

150

300

450

600

750

900

sing

lest

eplo

ad(N

)

time (s)

0 2 4 6 8 10 120

150

300

450

600

750

900

wal

king

load

(N)

time (s)

Fig.4 The displacement of 0.25*l (Ep=0)

Fig. 5 The displacement of 0.45*l (Ep=0)

Besides, white noise is added to the calculated responses to simulate the pollutedmeasurements.

calculated p calculated oiseu u E u N= + × × (20)

where Ep is a specified error level, Noise is a standard normal distribution vector. Theestimation error can be calculated by the followed equation:

100%identified true

true

f fError

f

−= × (21)

Eq. (18) and Eq.(19) are used to identify the walking load and the following resultsare obtained for different cases.

(1) If Ep=0, it means no noises are added to the measured displacements (Fig.4 andFig. 5), accurate results are obtained. This confirms that using displacements to identifythe walking load is feasible.

0 2 4 6 8 10 12 14

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

disp

lace

men

t(m

)

time (s)

10-4

0 2 4 6 8 10 12 14

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

dis

pla

cem

ent

(m)

time (s)

10-4

(2) If Ep=1%, the simulation of the displacements are showed in the Fig.6 and Fig.7,and the walking load can be identified as Fig.8 , Error=10.6%.

Fig.6 The displacement of 0.25*l with 1% noise

Fig.7 The displacement of 0.45*l with 1% noise

Fig.8 The identified load form polluted displacement (Ep=0.01)

0 2 4 6 8 10 12

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

disp

lace

men

t(m

)

time (s)

1% noiseno noise

10-4

0 2 4 6 8 10 12

-0.4-0.20.00.20.40.60.81.01.21.41.61.82.0

dis

plac

emen

t(m

)

time (s)

1% noiseno noise

10-4

0 2 4 6 8 10 12-200

0

200

400

600

800

1000

disp

lace

men

t(m

)

time (s)

simulated loadidentified load

5. CONCLUSIONS

From theoretical studies and computation simulation, some conclusions can beobtained. First it is feasible to use measured responses such as displacement toidentify pedestrian’s walking load in the time domain. Although the identification matrixis ill-conditioned, the general regularization method is a good solution to this problem.When measured responses are polluted, the identified result is satisfied bypreprocessing the data and least square method.

REFERENCES

Law, S.S., Chan, T.H.T., and Zeng, Q.H. (1997), “Moving Force Identification: a timedomain Method”, J. Sound Vib., 201(1), 1-22.

Chan, T.H.T., Law, S.S., et al. (1999), “an interpretive method for moving forceidentification”, J. Sound Vib., 219(3), 503-524.

Y. Xiao, J. Chen, et al. (2013), “A joint method of denoising correction andregularization preconditioned iteration for dynamic load identification in timedomain”, Journal of Vibration Engineering, 26(6):854-863.（in Chinese）

Zhu, X.Q., Law, S.S. (2002), “Moving loads identification through regularization”,Journal of Engineering Mechanics, 128(9): 989-1000.

G. Ding, J. Chen. (2016), “Influences of walking load randomness on vibrationresponses of long-span floors”, Journal of Vibration Engineering, 29(1):123-131.(in Chinese)