PSI - Issue 12

Valerio G. Belardi et al. / Procedia Structural Integrity 12 (2018) 281–295 V.G. Belardi et al. / Structural Integrity Procedia 00 (2018) 000–000

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6

The circumferentially variable sti ff ness terms are:

N k = 1

N k = 1

N k = 1

1 2

1 3

Q ( k )

Q ( k )

Q ( k )

2 k − z

2 k − 1 )

3 k − z

3 k − 1

A i j ( θ ) =

i j ( θ )( z k − z k − 1 )

B i j ( θ ) =

i j ( θ )( z

D i j ( θ ) =

i j ( θ )( z

) (10)

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

3. Principle of virtual displacements and Ritz method

The presented solution methodology exploits the Ritz method Reddy (2006) that is applied in conjunction with the principle of virtual displacements: δ W = δ W I + δ W E = 0 = Ω σ r δε r + σ θ δε θ + τ r θ δγ r θ d Ω − Ω f · δ u d Ω + Γ σ T · δ u d Γ (11) where δ W I and δ W E are, respectively, the internal and the external virtual works. The symbol δ is employed to identify virtual displacements and strains, Ω is the analytical integration region defining the rectilinear orthotropic composite circular plate, f denotes the body forces per unit volume, T the surface tractions per unit area acting on the external boundary portion Γ σ . Additionally, the expression of the internal virtual work of rectilinear orthotropic composite circular plates can be further specialized considering the strains dependence on the displacement components and performing an integration along the thickness direction: δ W I = a b π − π N r ∂δ u ∂ r + N θ δ u r + 1 r ∂δ v ∂θ + N r θ 1 r ∂δ u ∂θ + ∂δ v ∂ r + − M r ∂ 2 δ w ∂ r 2 − M θ 1 r ∂δ w ∂ r + 1 r ∂ 2 δ w ∂ 2 θ − 2 M r θ ∂ 2 ∂ r ∂θ δ w r d θ rdr (12) Furthermore, Ritz 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. Consequently, every unknown displacement component s must be written as a finite linear combination of approxi mation functions:

N j = 1

s ≈ S N = ϕ 0 +

c j ϕ j

(13)

In (13) the terms ϕ j represents approximation functions. These approximation functions must be selected as a contin uous, linearly independent and complete set of functions fulfilling the homogeneous form of the essential boundary conditions. On the contrary, ϕ 0 is the approximation function needed to satisfy the non-homogeneous essential bound ary conditions if any is present, meanwhile the weights of the N approximation functions, i.e. the c j coe ffi cients, are the problem unknowns.