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Non-stationary Response of a Bridge Due to Moving Vehicle with Random Arrival Rate

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Vibration Engineering and Technology of Machinery, Volume I (VETOMAC 2021)

Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 137))

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Abstract

Moving vehicle excites the bridge with dynamic force which is realised as a stationary process when the vehicle velocity is constant. However, this condition is not always true when the vehicle speed varies with time while travelling over the bridge. In this paper, the bridge response to non-stationary excitation has been studied considering speed variation, uneven pavement and also random arrival rate of the vehicle. The bridge vehicle interaction has been modelled using continuum approach and the solution has been obtained using orthogonal polynomial expansion method. The generalised co-ordinates of the system response are expressed in terms of orthogonal polynomial series, which offered certain advantages to arrive at the expression of first and second order statistics of system response using the properties of the polynomial. The movement of multiple vehicles has been considered in different time windows assuming their arrival rate follows a Poisson process. Response statistics- mean and standard deviation has been studied for a single cell box girder section of single span bridge in different time windows to observe the effect of vehicle arrival rate, vehicle speed and acceleration and pavement unevenness. The amplification of maximum static flexural stress due to dynamic effect has been obtained incorporating the standard error of the mean. Sequence of accelerating vehicles is found to cause higher stress in a bridge with poor maintenance of surface. The segment of response history in an optimal time window is found to decrease the computational cost since the presence of total number of vehicles over the bridge were dependent on the vehicle speed and their arrival rate.

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Abbreviations

DAF:

Dynamic Amplification Factor

SEM:

Standard Error of the Mean

DI:

Dynamic Increment

As:

Amplitude of cosine wave

cs:

Suspension dam**

cw:

Tyre dam**

C:

Dam** matrix

Cmean:

Mean values of dam** matrix

F:

Force vector

Fdynamic:

Maximum dynamic response on the bridge

Fmean:

Mean values of force vector

Fstatic:

Maximum static response of the bridge

h(\(\tilde{x }\)):

Bridge deck profile

hmean(\(\tilde{x }\)):

Deterministic mean surface profile

hroad(\(\tilde{x }\)):

Random road roughness of the pavement

ks:

Suspension stiffness

kw:

Trye stiffness

K:

Ztiffness matrix

Kmean:

Mean values of stiffness matrix

L:

Span of the bridge

\({L}_{l}^{n}\left(\stackrel{\sim }{\lambda }{t}_{n}\right)\):

Orthogonal function considered

ms:

Sprung mass

mw:

Unsprung mass

M:

Mass matrix

n:

Shape parameter of Gamma distribution and represents number of vehicle arrivals

nd:

Number of degrees of freedom

N:

Number of terms used to construct the road surface roughness

Ns:

Number of samples

N1:

Number of basic functions with respect to \(\stackrel{\sim }{\lambda }{t}_{n}\)

ptn(t):

Probability density function of the arrival time

Qil(t):

Time variation of displacement

\(\tilde{x }\):

Spatial distance

tn:

Vehicle arrival time on the bridge

v:

Velocity of vehicle

\({\text{y}}\left( {\tilde{x},{\text{t}}} \right)\):

Displacement of the bridge at time instant, t at location, \(\tilde{x }\)

z1:

Displacement of sprung mass

z2:

Displacement of unsprung mass

δlk:

Kronecker delta function

Г:

Gamma function

\(\stackrel{\sim }{\lambda }\):

Mean arrival rate

µ(\(\stackrel{\sim }{\lambda }{t}_{n}\)):

Mean arrival time

\(\mu_{{\text{f}}} \left( {\tilde{x},{\text{t}}} \right)\):

Mean of bridge response

\(\sigma_{{\text{f}}} \left( {\tilde{x},{\text{t}}} \right)\):

Standard deviation of bridge response

θs:

Independent random phase angle uniformly distributed from 0 to 2Ï€

ΩL:

Lower cut off frequencies of spatial unevenness

Ωs:

Spatial frequency (c/m)

ΩU:

Upper cut off frequencies of spatial unevenness

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Authors and Affiliations

  1. Department of Civil Engineering, IIT Guwahati, Assam, India

    Anjaly J. Pillai & Sudip Talukdar

Authors
  1. Anjaly J. Pillai
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  2. Sudip Talukdar
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Correspondence to Anjaly J. Pillai .

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Editors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, India

    Rajiv Tiwari

  2. Department of Mechanical Engineering, B.M.S. College of Engineering, Bengaluru, Karnataka, India

    Y. S. Ram Mohan

  3. Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India

    Ashish K. Darpe

  4. Department of Mechanical Engineering, B.M.S. College of Engineering, Bengaluru, Karnataka, India

    V. Arun Kumar

  5. Department of Mechanical Engineering, Indian Institute of Technology Patna, Patna, Bihar, India

    Mayank Tiwari

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Pillai, A.J., Talukdar, S. (2023). Non-stationary Response of a Bridge Due to Moving Vehicle with Random Arrival Rate. In: Tiwari, R., Ram Mohan, Y.S., Darpe, A.K., Kumar, V.A., Tiwari, M. (eds) Vibration Engineering and Technology of Machinery, Volume I. VETOMAC 2021. Mechanisms and Machine Science, vol 137. Springer, Singapore. https://doi.org/10.1007/978-981-99-4721-8_7

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  • DOI: https://doi.org/10.1007/978-981-99-4721-8_7

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