Untitled

140 Antonio J. Gil
element in the initial prestressed state is thus defined by a plane of uniform thickness
t bounded bystraight lines which intersects at three points called nodes.
A fixed local coordinate system Oξ pret
1 ξ pret
2 ξ pret
3 , in addition to the global reference
frame, is established for each element. The behaviour of each element is to
be first described independently in terms of its local coordinates and is then to be
transformed into global coordinates. For the sake of simplicity, it is assumed that
each element lies in the Oξ pret
1 ξ pret
2 plane of its local coordinate system. The shape
functions are evaluated to obtain:
a I = ξ pretJ
1 ξ pretK
2
N I (ξ pret
1 , ξ pret
2 ) = 1
− ξ pretK
1 ξ pretJ
2
2Γ pret (a I + b I ξ pret
1
b I = ξ pretJ
2
− ξ pretK
2
+ cIξ pret
2 )
c I = ξ pretK
1
− ξ pretJ
1
Γ pret = 1
2 (cKb J − c J b K ) I,J,K = 1, 2, 3
(A.1)
where Γ pret is the area of the initial prestressed triangle. Note that the quantities
a I , b I and c I -named Zienkiewicz’s coefficients, see for instance [24]- are independent
of the deformation of the membrane and are computed directly from the geometry
of the initially prestressed shape. According to the Direct Core Congruential Formulation,
the vectorized displacement gradient tensor is given in its transpose form
as:
g T =
g1 g2 g3 g4 g5 g6 g7 g8 g9
g T =
b I u I 1
2Γ pret
b I u I 2
2Γ pret
b I u I 3
2Γ pret
c I u I 1
2Γ pret
c I u I 2
2Γ pret
c I u I 3
2Γ pret (A.2)
0 0 0
The vector of global internal forces is particularized for the triangular flat element
by using the DCCF as:
where:
fint f = tΓ pret B T φint = t
2
b 1 I3 c 1 I3
b 2 I3 c 2 I3
b 3 I3 c 3 I3
φ int
⎛
⎞
s1(1+g1) + s3g4
⎜ s1g2 + s3(1+g5) ⎟
⎜
⎟
⎜ s1g3 + s3g6 ⎟
φint = sihi + siHig = ⎜
⎟
⎜ s2g4 + s3(1+g1) ⎟
⎝
⎠
s2(1+g5)+s3g2
s2g6 + s3g3
(A.3)
(A.4)
The total tangent stiffness matrix may be computed by means of the congruential
transformation and thus the submatrix due to the contribution of the nodes I and
J is depicted as:
where:
K IJ =
t
4Γ pret (bI b J S11 + c I c J S22 + b I c J S12 + c I b J S21
Sij = S geo
ij
(A.5)
+ Smat
ij i, j =1, 2 (A.6)