Issue 62

J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25

experimental tests since it is more practical to simulate damage as mass added which does not require the destruction of the sample under study. Also, it was evaluated for different levels and positions of damages, and the results were presented as maps.

T IMOSHENKO BEAM FORMULATION WITH MASS DISCONTINUITY

I

n the case of free vibration, the motion equation for uniform beams with a roving mass

a m positioned at the damage

d L is described by [56–58]:

position

 

2





u x

u

 

  

  





      

 A m x L 

 

kGA

 

    



a

d

2

   x





t

(1)

2





  





u x

  

  

 

   I



 

EI

kGA

 

2







x

x



t

where u and  are the beam’s transverse displacement and rotation, respectivetly, E is the Young’s modulus, I the second moment of area,  the density of material, A the cross-sectional area, k a shear correction factor (  5 / 6 k for rectangular sections), and G the shear modulus, L the length of the beam and  is de Dirac delta function. According to Lee and Park [58], the dynamic equilibrium equation based on the principle of virtual work can be written as

      





u x

u x

     

  

     

  

L

L









 

   dx

EI

dx

kGA





x x

0

0

(2)

¨

¨

¨

, L

L

L











  u Au dx



 



 

 m u x L dx

I dx

d

d

0

0

0

which L the beam length and  denotes the virtual terms. The equation of motion in the frequency domain in matrix form is given by

  2 n K M

 Φ 0

(3)

   / 2 n n f

where Φ is a set of displacement-type amplitude at the control points otherwise known as the model vector, the natural frequency   Hz , K and M are global stifness and mass matrices that can be explicitly written as:

T





N

N

     

     

     

     

, a p

, b p





0

0

EI

0

  

  





x

x

L



 

K K

dx

(4)

ab



 kGA N

N

0

0

, a p

, b p

N

N

, a p

, b p





x

x

, L

L





 M M N AN dx  



 N IN dx , a p , a p

(5)

ab

, a p

, a p

0

0

ab M are stiffness and mass matrices, respectively, to control points a and b ; and

, a p N and

, b p N are the

ab K and

where

shape functions. The elementary mass matrix

e M can be defined as consistent or concentrated. In the first case, it considers the acceleration effects in i and the inertia force in j . In other words, the mass matrix coefficients are not null when  i j . Briefly, to

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