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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Furthermore, the strains related to the displacement field (6) are:

    1 r

   

∂ 2 w ∂ 2 r

  

   1 r

   − z

   1 r

∂ u r ∂ r

∂ u ∂ r

   ε r ε θ γ r θ

   =

=    ε 0 r ε 0 θ γ 0 r θ

  

+ z    κ r κ θ κ r θ

   =

∂ 2 w ∂ 2 θ

∂ u θ ∂θ

u r

∂ v ∂θ

1 r

u r +

∂ w ∂ r +

1 r

1 r

(7)

r +

∂ u ∂θ +

∂ v ∂ r −

v r

∂ u θ ∂ r −

u θ r

∂ u r ∂θ +

2 ∂ r ∂θ

w r

2 ∂

in which every strain component features two contributions, the first one represents the mid-surface strains and the second one is related to the mid-surface curvatures. Stress resultants per unit width are obtained by means of the

Fig. 2: Stress resultants acting on a plate element.

integration of elemental forces and moments in the laminate thickness t , over the N layers composing the composite circular plate. Thus, the resulting in-plane forces are (see Fig. 2):

   =

Q ( θ )

   N r N θ N r θ

   σ r σ θ τ r θ

   dz =

  

+    κ r κ θ κ r θ

  

   ε 0 r ε 0 θ γ 0 r θ

  

   κ r κ θ κ r θ

  

   ε 0 r ε 0 θ γ 0 r θ

z    

   

t / 2

z k

N k = 1

dz = A ( θ )

+ B ( θ )

(8)

− t / 2

z k − 1

whereas the resulting radial and tangential bending moments and the torque moment turn out to be (see Fig. 2):

=

Q ( θ )

  

  

   κ r κ θ κ r θ

  

z +    κ r κ θ κ r θ

  

   σ r σ θ τ r θ

  

  

   M r M θ M r θ

   ε 0 r ε 0 θ γ 0 r θ

   ε 0 r ε 0 θ γ 0 r θ

z 2    

   

t / 2

z k

N k = 1

+ D ( θ )

dz = B ( θ )

(9)

zdz =

− t / 2

z k − 1