Issue 62

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

e ij M a unitary acceleration is applied in the direction j . Thus, the inertia force that results in i is calculated to

determine

keep all the accelerations in other zero directions. Therefore, the consistent mass matrix is given by

  

2 L L L L L 156 22 54 L 13 4 13 3 e

      

      

e

22 54



e

e

e

e

e

e

 M AL

(6)

13 156 L

L

22

420

e

e

   2 3 L L L L 13 22 4 , 2

e

e

e

e

where

e L is the length of the element.

Dimensionless equation The Timoshenko beam design is performed to determine dimensionless variables and the parameters that govern the problem. Considering the equilibrium equations of bending and shear moments (1) of a Timoshenko beam with constant section, the equation of motion is given by:

  Au kGA u         ¨ '' I EI   ¨

    '

f

''

(7)



   

kGA u

'

0



The second equation when taking derivatives with respect to space becomes

     '



  

    EI

  kGA u  



I

(8)

0

  f



Au



where  is the material density, A the cross-section area, u the transversal displacement,  the rotation, k the shear correction factor, G the shear modulus, f the external force, I the second moment of area and E the Young’s modulus. Then, Eqn. (8) can rewritten as,               ' 0 I EI f Au (9)

that is,

     I

'



    ''

  





EI

Au f

0

(10)

where we can introduce 

   ' '' ' . u From this,

 ' ''           Au f     I  '

 EI u

''

0

(11)

 

in which EI and  A are constant. Thus,

  

 

¨

¨

¨

IV EIu EI

      ''' Au f

    '



(12)

I u

''

0

Recall that,

    ' Au kGA f   

(13)

352