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

Setting to 0 the derivatives of the discretized form of the augmented Lagrangian L h c with respect to all the unknown parameters gives rise to a governing system of linear equations that involve the element constants and the Lagrange multipliers as unknowns as follows

F A T

A O

s λ

=

0 0

(20)

where s is the global vector of element parameters ( c λ is the global vector of Lagrange multipliers and F and A are the global flexibility and global equilibrium matrices, respectively. The solution of this system of equations represents the most compatible possible statically admissible solution to the boundary-value problem under analysis. i constants),

5. Numerical Results and Discussion

5.1. Validation Tests

In order to validate the proposed formulation, two problems analysed in [10] were first addressed, in particular, a clamped-clamped cracked thin beam (Case 1) and a clamped-pinned cracked thick beam (Case 2), both with uniform cross-sections and a homogeneous material response, as depicted in Figure 2.

q

¯ V

¯ M

d = 0 . 5 h

L / 2

L / 5

L

L

Case 2 : Clamped-pinned cracked uniform and homogeneous thick beam

Case 1 : Clamped-clamped cracked uniform and homogeneous thin beam

Fig. 2: Validation tests.

In both cases, the Young’s and shear moduli were taken as E = 210 GPa and G = 80 . 77 GPa , while the length of the beam was assumed as L = 150 cm . In Case 1, the cross-section dimensions and the crack depth were taken as h = b = 5 cm and d = 2 . 5 cm . The applied shear force and bending moment were set to ¯ V = 70 kN and ¯ M = 20 kNm . In Case 2, the cross-section dimensions and the crack depth were taken as h = 45 cm , b = 15 cm and d = 0 . 5 h = 22 . 5 cm . The adopted shear correction factor was f s = 5 / 6. The applied distributed load was set to q = 5 kN / cm . Both cases were tackled by discretizing the beams into two equal-length elements. All the required integrations were performed exactly. The obtained results are displayed in Table 1. The reference solutions were taken from [10]. As it can be observed, all the obtained results match exactly the reference ones.

5.2. Clamped-Clamped Multi-Cracked Non-Uniform FGM Beam

Finally, a clamped-clamped non-uniform FGM beam with two cracks ( C 1 and C 2 ), located at ¯ x 1 = 4 L / 5, subjected to a transverse tip load as illustrated in Figure 3 was analysed. A power-law function was assumed for the material behaviour, E ( z ) = E L + ( E U − E L ) ( z / h 0 + 1 / 2) p with E L = 380 GPa (ceramic inclusion - alumina), = L / 5 and ¯ x 2