PSI - Issue 8

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

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

The circumferentially variable sti ff ness terms are:

N k = 1

N k = 1

1 2

1 3

Q ( k )

Q ( k )

2 k − z

2 k − 1 )

3 k − z

3 k − 1

B i j ( θ ) =

i j ( θ )( z

D i j ( θ ) =

i j ( θ )( z

)

(9)

where B i j ( θ ) are the bending-extension coupling sti ff nesses and D i j ( θ ) the bending sti ff nesses, expressed in the cylin drical coordinate system, for a rectilinear orthotropic composite circular plate; whereas, z k and z k − 1 are the oriented distances to the bottom and the top, respectively, of the k th layer.

3. Analytical definition of governing equation

A quasi-isotropic stacking sequence, symmetric respect to the plate mid-surface, was taken into account in this work. The transversal loading condition was realized through the application of a unitary force P acting on the plate axis, alongside the z -axis of the cylindrical coordinate system and with the same verse, orthogonally with respect to the mid-surface of the rectilinear orthotropic composite circular plate. The derivation of the governing equation for bending of rectilinear orthotropic laminate circular plates is performed within the framework of classical lamination theory and, as a consequence, Kirchho ff -Love hypotheses are considered to outline the plate behavior.

Fig. 1: Stress resultants acting on a plate element (up) and theoretical reference model (down).

According to these assumptions and to the quasi-isotropic stacking sequence considered, the mid-surface deflec tion w and the radial and circumferential bending moments M r and M θ are assumed to depend only on the radial coordinate r ; furthermore, the torque moment M r θ vanishes. The stress resultants that must be considered in writing the equilibrium equations are the radial and the circumferential bending moments M r and M θ , and the shear force Q r . In addition, no in-plane stress resultants act on the plate because no membrane-shell coupling e ff ects are considered and consequently no mid-surface strains that induce in-plane stress resultants are present. Moreover, being the quasi isotropic stacking sequence symmetrical respect to the plate mid-surface, the bending-extension coupling sti ff nesses matrix B ( θ ) is null and therefore no in-plane stress resultants are produced by the curvature variations.