PSI - Issue 62

Andrea De Flaviis et al. / Procedia Structural Integrity 62 (2024) 871–878 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

875

5

Fig. 3. (a) conventional modes; (b) extensional modes; (c) shear modes.

2.2. Member analysis The member analysis, i.e. the resolution of the Eq. (5), has been conducted with 2 different numerical approaches, both leading to approximate but accurate solutions. The first one is based on the finite differences Matlab algorithm bvp4c, which solves a boundary value problem constituted by a system of n first-order differential equations, starting from a trial solution; in free dynamics there is a further unknown which is the natural frequency, i.e. the eigenvalue. Starting from Eq. (5), considering only conventional modes, the boundary conditions of a simply supported beam are imposed, i.e. u(s,z) and v(s,z) are null at the two ends, so φ is zero while φ’ is in general different from zero, since rotations are allowed. The system of equations in the dynamic case, with φ 1 = φ , φ 2 = φ’ , φ 3 = φ’’ , φ 4 = φ’’’ , is that of Eq.(6), where a normalization condition has to be added to b.c when eigenvalue analysis is addressed.

0 0 0

I

0

0 0

      

      

' ' ' '

4 3 2 1    

4 3 2 1    

      

      

      

      

0 0

I



         s ΩΩ f f T Q D D D D 0 I 0 λ t M ΩΩ   



  









 





a 1 ΩΩ

f

e

a 1 ΩΩ

 C C M B B 0 C C λ   



f

f

(6)

0 0 0 0

(0,L) (0,L) (0,L) (0,L)

   

      

      

      

      

4 3 2 1

 I 0 D 0 C C 0 a ΩΩ f f 0 0

   

   



This method gives good results in case of global restraints at the end sections but it is not suitable in case of punctual restraints. That is why the second method, based on Finite Elements, always inside the framework of GBT, has also been used in this work, making it possible to apply punctual restraints, or to consider the simplest case where all the points of the end section are restrained (as in the first method). In this case, always referring to conventional modes, using the index notation, the unknown amplitude function φ k is expressed through a linear combination of (Hermitean or Lagrangian) polynomials Ψ α (ξ) , where ξ = z/L e , where L e is the length of the finite element along the longitudinal direction z , with generalized displacements d kα e , where d k1 e = φ k,z (0) , d k2 e = φ k (0) , d k3 e = φ k,z (L e ) , d k4 e = φ k (L e ) . The equations are those of the well-known Finite Element Method, so they are not here reported.