Issue 70

S.K. Shandiz et alii, Frattura ed Integrità Strutturale, 70 (2024) 24-54; DOI: 10.3221/IGF-ESIS.70.02

The bridge vibration equation as an Euler-Bernoulli beam with two simple ends and 3-point contact with TT is given as:

d [ ]{}[ ]{}[ ]{} [ ]    2 3 M C K N N N   d d f [ ] [ ] T T   1

T

f 1

f

(4)

b

b

b

2

t





b b   b b  

0 2 m b I

m b

g d 0 1

d



g d 1 3      1 m m b b 1 2

0 2 2

f

(5)

1

1 2





0 1 m b I

g d d 0 1 0 1 2 m b



g d 2 4      2 m m b b 1 2

f

(6)

2

1 2

g d 1 0 5 t t m m   

f t

(7)

where f 1 , f 2 , and f t are the interaction forces at the rear, front, and trailer axle, respectively. The displacement of the beam u at an arbitrary point such as x is obtained using the shape function [ N ] and the displacement of the node d [66]:

(8)

d [ ]{ }  N

u

The beam element shape function is given as:   1 2 3 4 [ ] N N N N  N

(9)

where shape function components are defined by the following well-known relations:

l        



2

3

2

2

3

2

x             x l l 

x

x             x l l 

x x

  

 

 

 



 

 

N

N x

N

N x

1 3

2 ,

1 , 

3

2 ,

(10)

1

2

3

4

l

l



The derivative of u with respect to time is defined as:



u u 

(11)

x     x

( , ) x t

u 



t

Given that [ N ] is a location-dependent function and d is time-dependent, according to Eqn. 8:

 

u

(12)



d [ ] x N

x

By placing Eqn. (11) and Eqn. (12) into Eqn. (3) and Eqn. (4), the trailer-tractor and bridge interaction equation can be reached as: d d d F [ ]{}[]{}[ ]{}{}    M C K   (13)

The matrices [ M ], [ C ], and [ K ] are defined as follows:

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