Issue 70

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

1

1





0 , I

I

f

f

(18)





1

2

0

b b 

b b 

1 2

1 2

b

f ˆ





g g m m k r x c r x    3 1 3 ( ) 0 1

v ( ) ,

2

1

1

1 b b b 

1 2

f ˆ



g g m m k r x c r x    4 2 4 ( ) g g b b m m k r x c r x   2 3 2 3 ( ) ( ) t t t t v 0 2 1 2     0 1 

v ( ) ,

(19)

2

2

f ˆ

3

and f 1  and f 2  are forces due to the moment of rotational inertia of the tractor. f 1 ˆ , f 2 ˆ and f 3 ˆ are forces due to the inertia of tractor and trailer axles. In the Eqns. (14), (15) and (16) the sub-matrices [ M b ] and [ K b ] are obtained from [67]. The Rayleigh type of damping considered for the beam is defined as follows: [ ] [ ] [ ] b b b     C M K (20)

  and   are calculated according to the following relations:

1 2 1 2 2 1 2 2 2 1 )       

2 (





(21)

2 2 1 1 2 2 2 1      

2(

)





,

(22)

the first and second frequencies of the beam are given by   and   , respectively [67]. Modal analysis method

In this section, TT and bridge interaction equations needed for the modal analysis are derived. TT vibration equation is represented in Eqn. (3). Vehicle-bridge interaction relationships in the modal analysis are derived using the relations described in [68], which were for train-bridge interaction. Modified relationships have been derived for the case of TT passing a bridge. The bridge vibration due to the moving TT is given as:

f n

 

w w w ]{}[ ]{}[ ]{}   K  

F { } bk 

[

(23)

M C

b

b

b



k

1

where, [ M b ], [ C b ], and [ K b ] represent the mass, stiffness, and damping of the beam, n f is the number of contact points of the TT with the bridge,   denotes the Dirac delta function, and w is the vertical displacement of the beam. The contact points x 1 , x 2 , and x 3 , which are shown in Fig. 4b, denote the TT contact points from the beginning of the beam; they are expressed as:

(24)

v x t b x t   ,

v     ( b b b

v x t 

),

1

3

2

1

2

3

3

(25)

F R F bk wk wk  

In the above equation, F bk represents the forces that apply to the beam by the TT, and F bk consists of two forces of weight ( R wk ) and the force of inertia ( F wk ) applied by the TT.

32