PSI - Issue 66

H.A.F.A. Santos et al. / Procedia Structural Integrity 66 (2024) 195–204

203

9

H.A.F.A. Santos et al. / Structural Integrity Procedia 00 (2025) 000–000

Table 2: Case 3: Obtained results

N [ kN ]

V (0)[ kN ]

V ( L )[ kN ]

M (0)[ kNm ]

M ( L )[ kNm ]

n e

− 44968 . 07992 − 45109 . 73280 − 49468 . 99477 − 49468 . 99477

− 6898 . 829517 − 6895 . 131458 − 5143 . 519739 − 5143 . 519739

− 4382 . 869475 − 4449 . 997856 − 4878 . 017123 − 4878 . 017123

2 3217 . 629078 55031 . 92008 4 3249 . 152573 54890 . 26720 8 3895 . 012370 50531 . 00523 16 3895 . 012370 50531 . 00523

4

40

3

20

N [ kN ]

V [ kN ]

2

0

- 40 - 20

1

0

0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.2

0.3

0.4

0.5

x [ m ]

x [ m ] (b) Axial force distribution

(a) Shear force distribution

8

6

4

2

M [ kNm ]

- 4 - 2 0

0.0

0.1

0.2

0.3

0.4

0.5

x [ m ] (c) Bending moment distribution

Fig. 4: Case 3: Internal forces and bending moment distributions for the 16-element mesh.

6. Conclusions

A new finite element formulation for the quasi-static analysis of functionally graded multi-cracked non-uniform Timoshenko beams was presented. The non-propagating cracks, assumed to be open during the loading process, were modelled using a discrete spring approach in which Dirac’s delta generalized functions are introduced into the bending flexibility of the beams. The formulation was derived from a complementary-energy-based variational principle and can be used to obtain statically admissible solutions, i.e. , solutions that satisfy all equilibrium equations of the associated boundary-value problem in a strong form. Its application to various numerical problems demonstrated