Untitled

54 Robert L. Taylor, Eugenio Oñate ˜ and Pere-Andreu Ubach
A residual form for each element may be written as
⎧
⎨ R
⎩
1
R 2
⎫ ⎧
⎬ ⎨ f
=
⎭ ⎩
1
f 2
⎫ ⎧
⎬ ⎪⎨⎪
⎨
¨˜x
− [M e]
⎭ ⎪⎩⎪
1
¨˜x 2
⎫
⎪⎬⎪
⎬
⎪⎭⎪
− [C ⎧
⎪⎨⎪
⎨
˙˜x
e]
⎪⎩⎪
1
˙˜x 2
⎫
⎧
⎪⎬⎪
⎬
⎨
− hA[B]T
⎪⎭⎪
⎩
R 3
f 3
¨˜x 3
˙˜x 3
⎨ S11
where [M e] and where [C e] are the element mass and damping matrices given by
⎡ ⎤
⎡ ⎤
11 12 13
11 12 13
M M M C C C
with
[M e] = ⎣ M 21 M 22 M 23
M 31 M 32 M 33
⎦
M αβ
=
3.1 Pressure Follower Loading
Ω
ρ0 hξα ξβ dΩ I and C αβ
=
S22
S12
and [C e] = ⎣ C 21 C 22 C 23
C 31 C 32 C 33
⎦
Ω
⎫
⎬
⎭
(41)
(42)
c0 h ξα ξβ dΩ I (43)
For membranes subjected to internal pressure loading, the finite element nodal forces
must be computed based on the deformed current configuration. Thus, for each
triangle we need to compute the nodal forces from the relation
δ˜x α,T f α = δ˜x α,T
ξα (p n) dω (44)
For the constant triangular element and constant pressure over the element, denoted
by pe, the normal vector n is also constant and thus the integral yields the nodal
forces
ω
f α = 1
3 pe n Ae
We noted previously from Eq. (6) that the cross product of the incremental vectors
∆˜x 21 with ∆˜x 31 resulted in a vector normal to the triangle with magnitude of twice
the area. Thus, the nodal forces for the pressure are given by the simple relation
by
where
f α = 1
6 pe ∆˜x 21 × ∆˜x 31
(45)
(46)
Instead of the cross products it is convenient to introduce a matrix form denoted
∆˜x 21 × ∆˜x 31 21
= ∆˜x
ij
∆˜x
⎡
= ⎣
∆˜x ij
3
−∆˜x ij
2
0 −∆˜x ij
3
∆˜xij
1
∆˜x 31
∆˜x ij
2
0 −∆˜x ij
1
0
⎤
⎦
(47)
. (48)