PSI - Issue 8

V.G. Belardi et al. / Procedia Structural Integrity 8 (2018) 368–378

376

V.G. Belardi et al. / Structural Integrity Procedia 00 (2017) 000–000

9

5. Results

The results obtained by means of Galerkin method, applied to the governing equation (12) previously defined, are presented in comparison with corresponding FE analyses ones, performed with refined FE models featuring 4 noded layered shell finite elements with 6 DOFs per node. The loading condition acting on the theoretical reference model was reproduced setting a fully clamping constraint to the nodes at the outer radius, whereas the rigid inner core was realized connecting the nodes at the inner edge with a radial set of rigid beams to node present at the plate geometrical center where the external transversal load P is applied. Quasi-isotropic composite annular plates were considered for the case-study and di ff erent geometrical features were tested. The quasi-isotropic stacking sequence hereafter considered is [45 / 0 / -45 / 90] 5 s and is taken from litera ture, McCarthy et al. (2005); the unidirectional fiber-reinforced layer sti ff ness properties are listed in Table 2.

Table 1: Unidirectional fiber-reinforced layer sti ff ness properties McCarthy et al. (2005).

E 11 [GPa] E 22 [GPa] E 33 [GPa] G 12 [GPa] G 13 [GPa] G 23 [GPa]

ν 12 [ − ] ν 13 [ − ] ν 23 [ − ] 0.3 0.3 0.5

140

10

10

5.2

5.2

3.9

Table 2: Mid-surface deflection percentage error, evaluated at ρ = β , between Galerkin method and FEA results.

t a

Err β 1 [%] Err β 2 [%] Err β 3 [%]

0.02 0.03 0.04 0.05

1.88 1.94 4.06 6.30

0.43 3.24 3.03 6.45

0.98 2.35 4.20 7.57

The loading condition considered consists in a force P applied to the plate axis, with unitary intensity, acting along the axis and transversally with respect to the plate mid-surface; the distributed load q ( r ) is not present. The rectilinear orthotropic composite annular plates analyzed are realized with 40 layers, each layer has thickness t lay = 0 . 13 mm; consequently, the plate thickness results t = 5 . 2 mm. In the order of ideas of evaluating the reliability and an applicability range of developed Galerkin approach to the transversally loaded rectilinear orthotropic composite annular plate, di ff erent slenderness ratios between the thickness t and the external radius a are studied. Numerical results are reported in dimensionless form. The dimensionless deflection, evaluated as the ratio between mid-surface deflection w ( r ) and its value at the inner edge w ( β ) taken from FEA, with β = b a , is reported in Fig. 2 against the dimensionless radius ρ = r a . Additionally, the inner hole radius e ff ect on the mid-surface deflection is assessed providing results for di ff erent values of the β parameter for a fixed slenderness ratio. The β values investigated are: β 1 = 0 . 05, β 2 = 0 . 1, β 3 = 0 . 2. The results in Fig. 2 outline the dimensionless mid-surface deflection as a function of the dimensionless radial coordinate; it can be highlighted that the approximation functions are adequate to satisfy the boundary conditions. Moreover, the outcomes provided by Galerkin method and those obtained through the FE analyses presents a high level of matching, indicating that the presented method represents a reliable instrument for the analysis of rectilinear orthotropic composite plates. Furthermore, it should be noted that the accuracy of the results is not significantly a ff ected by β value for all the slenderness ratios analyzed. About the validity range of the proposed procedure, at the increasing of slenderness ratio the plate shear deforma bility becomes not negligible. As a result, the diagrams show a bigger misfit between the two mid-surface deflection curves for higher slenderness ratios; this shows the limit of applicability of this approach based on the Kirchho ff -Love theory.