PSI - Issue 8

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

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8

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

In (14) 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 both essential and natural boundary conditions. On the contrary, ϕ 0 is the approximation function needed to satisfy the non-homogeneous 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. The theoretical reference model considered in this work (that is the basis of the custom composite bolted joint finite element) consists in a rectilinear orthotropic composite annular plate featuring a rigid core at the inner radius and an outer radius fully clamped (Fig. 1). Subsequently, these displacement constraints require the approximation functions to satisfy: ( I ) null deflection at the outer radius, ( II ) null rotation in radial plane at the outer radius and the ( III ) states null rotation in radial plane at the inner edge. In terms of the unknown mid-surface deflection w ( r ):

∂ w ( b , θ ) ∂ r

∂ w ( a , θ ) ∂ r

= 0

( III )

= 0

(15)

( I ) w ( a , θ ) = 0

( II )

According to the discretized form defined in (14), the mid-surface deflection w ( r ) becomes:

N j = 1

w ( r ) ≈ W N ( r ) =

c j ϕ j ( r )

(16)

in this expression ϕ 0 is null since all the boundary conditions are homogeneous. The approximation functions utilized are polynomials and the three boundary conditions defined in (15) are re spected by the polynomial series:

c j ( r − a )

( b − a )( r − a ) j + 1

N j = 1

j + 2 j + 1

j + 2 −

(17)

W N ( r ) =

The weight coe ffi cients c j can be found by means of a linear system where the N algebraic equations state the annulment of the weighted residual form of the governing equation (12): N j = 1 G i j c j − F i = 0 , i = 1 , 2 , . . . , N (18)

a b

a b

G i j =

F i =

π

π

ϕ i L ( ϕ j ) d θ rdr

ϕ i f d θ rdr

(19)

− π

− π

In the next section results obtained by means of the proposed method are presented, 7 terms of the series W N ( r ) were taken for the discretization of the mid-surface deflection w ( r ), considering the governing equation (12).