PSI - Issue 8

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

374

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

7

The only non-null moments, the radial and the circumferential bending moments M r and M θ in (8), assume the form, considering Eqs. (9): M r M θ = D 11 ( θ ) D 12 ( θ ) D 12 ( θ ) D 22 ( θ ) κ r κ θ (10) Furthermore, the only unknown displacement component of the composite annular plate is the mid-surface deflec tion w ( r ) as its radial u and circumferential v displacements are not present. In the end, combining the equilibrium of forces along the z -axis and the equilibrium of moments about the θ -axis, the resulting general equilibrium equation can be found: M r − M θ + r ∂ M r ∂ r + r ∂ Q r ∂ r + q ( r ) = 0 (11)

The analytical expression of the governing equation can be identified replacing the bending moments with their ex pression in (10) and the curvatures in (7):

∂ w ∂ r −

+ q ( r ) = 0

∂ 2 w ∂ r 2

12 ( θ ) d θ 2

22 ( θ ) d θ 2 −

r

∂ 3 w ∂ r 3

d 2 D

d 2 D

1 r

1 r 2

∂ Q r ∂ r

D 22 ( θ ) +

(12)

D 11 ( θ ) +

D 11 ( θ ) +

+

This is a third order di ff erential equation where the unknown displacement component w ( r ) compares explicitly.

4. Galerkin method

One of the most widespread weighted residual method is Galerkin method (see, for example, Reddy (2006)). This weighted residual method was applied to the governing equation (12) that can be rewritten in terms of a functional L , a linear di ff erential operator, acting on the mid-surface transversal displacement w ( r ) and f , i.e. a known term determined by the radial shear force Q r and the distributed load q ( r ). It is defined in the analytical integration region of the rectilinear orthotropic composite annular plate Ω as:

L ( w ) − f = 0 in Ω

(13)

It must be solved in compliance with all the boundary conditions defined on the boundary of the integration region Ω , that in this case are all essential. The Galerkin method transfers the searching for the solution from the unknown displacement functions to a limited number of unknown coe ffi cients c j present in the discretized definition of the same displacement functions; this implies that, from the analytical point of view, the problem to solve is shifted from a third order di ff erential equation to an algebraic equations system. As a consequence, every unknown displacement component s must be written as a finite linear combination of approximation functions:

N j = 1

s ≈ S N = ϕ 0 +

c j ϕ j

(14)