PSI - Issue 66

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

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

Table 1: Validation tests: Obtained results

Case

Ref. Sol.

This Study

w ( L / 2) φ (3 L / 4)

0 . 24721 cm − 0 . 00474 rad − 8459 . 21 kNm 14561 . 7 kNm − 7459 . 21 kNm 2645 . 49 kNm

0 . 24721 cm − 0 . 00474 rad − 8459 . 21 kNm 14561 . 7 kNm − 7459 . 21 kNm 2645 . 49 kNm

M (0)

1

M − ( L / 5) M + ( L / 5)

M ( L )

− 0 . 00879 cm 0 . 00016 rad

− 0 . 00879 cm 0 . 00016 rad

w ( L / 2)

2

φ ( L )

E U = 70 GPa (metallic matrix - aluminium) and p = 1. The Poisson’s ratio was taken as ν = 0 . 17. The variation of the cross-section height was assumed as h ( x ) = h 0 (1 − x / (2 L )) 2 , with h 0 = h (0) = 20 cm , and the cross-section width was set to b = 10 cm . The adopted shear correction factor was f s = 5 / 6. The crack coefficients were set to β 1 = 1 / 10 (corresponding to d 1 / h (¯ x 1 ) = 0 . 198216) and β 2 = 1 / 8 (corresponding to d 2 / h (¯ x 2 ) = 0 . 375305). The length of the beam was assumed as L = 50 cm and the applied load was set to ¯ V = 100 kN .

¯ V

L / 2

h ( x )

C 2

C 1

b

L / 5

L / 5

L

Fig. 3: Case 3 : Clamped-clamped multi-cracked non-uniform FGM beam.

There are no reference solutions available in the literature for this specific problem. A convergence study was performed, in which the internal forces and bending moments were computed using 2, 4, 8 and 16 uniform finite element meshes. The required integrations along the axial coordinate were performed using a 5-point Gauss quadrature rule, whereas in the z -axis all integrations were exact. The obtained results are provided in Table 2. As it can be observed, the convergence is clear for all quantities, with no changes of the results from the 8- to the 16-element mesh considering a 10-digit precision. Therefore, it can be concluded that the proposed formulation can produce accurate solutions on relatively coarse meshes. The obtained axial force, shear force and bending moment fields obtained on the 16-element mesh are displayed in Figure 4. It is worth noting the appearance of the non-zero axial force, even though no axial load is applied to the beam, which is due to the functionally graded material response assumed for the beam. As it can be observed, all the obtained static fields are C 0 -continuous, apart from the shear force field at x = L / 2, which is a consequence of the applied force.