Universität Karlsruhe (TH) - am Institut für Baustatik

To avoid numerical difficulties the series expansion of the coefficients should be used if ω I
approaches zero. Hence the finite element formulation of the linearized virtual membrane
strains and curvatures considering (34) reads
Δδε αβ =
Δδκ αβ =
4∑
4∑
I=1 K=1
4∑ 4∑
I=1 K=1
1
2 (N I, α N K , β +N I , β N K , α ) δu I · Δu K
{ 1 2 (N I, α N K , β +N I , β N K , α ) δu I · Δd K
+ 1 2 (N I, α N K , β +N I , β N K , α ) δd I · Δu K
(35)
+δ IK
1
2 [δw I · (N I , α M I (x, β )+N I , β M I (x, α )) Δw K ] }
where δ IK denotes the Kronecker delta.
Finally, we specify the product Δδε hT
G σ h with the independent stress resultants
σ h =[n 11 ,n 22 ,n 12 ,m 11 ,m 22 ,m 12 ,q 1 ,q 2 ] T using (32) -(35)
Δδε hT
G σ h =
=
4∑
I
4∑
I
4∑
δvI T k σIK Δv K
K
[ ] ⎡
4∑
T
δuI ⎣
δβ
K I
The matrix k σIK is determined with
ˆn IK 1
(ˆm IK +ˆq uw
IK)T K
(ˆm IK +ˆq wu
IK)T T I δ IK ˆMI (h I )
ˆn IK = n 11 N I , 1 N K , 1 +n 22 N I , 2 N K , 2 +n 12 (N I , 1 N K , 2 +N I , 2 N K , 1 )
ˆm IK = m 11 N I , 1 N K , 1 +m 22 N I , 2 N K , 2 +m 12 (N I , 1 N K , 2 +N I , 2 N K , 1 )
ˆq IK uw = 1 2 (qξ N I , ξ fIK 1 + q η N I , η fIK)
2
ˆq IK wu = 1 2 (qξ N K , ξ fIK 1 + q η N K , η fIK)
2
ˆM I = T T 3I H T I M I (h I ) H I T 3I
⎤
⎦
[
ΔuK
] (36)
Δβ K
h I = m 11 N I , 1 x h , 1 +m 22 N I , 2 x h , 2 +m 12 (N I , 2 x h , 1 +N I , 1 x h , 2 )
(37)
+ q ξ N I , ξ ξ I x M , ξ +q η N I , η η I x L , η
[f 1 IK] =
⎡
⎢
⎣
1 1 0 0
1 1 0 0
0 0 1 1
0 0 1 1
⎤
⎥
⎦
⎡
[fIK] 2 = ⎢
⎣
1 0 0 1
0 1 1 0
0 1 1 0
1 0 0 1
⎤
⎥
⎦
[
q
ξ
q η ]
= J −T [
q
1
q 2 ]
.
where M I (h I )isdefinedin(34) and the allocation of the midside nodes M,L to the corner
nodes in (30).
3.4 Interpolation of the stress resultants and shell strains
Regarding the requirements on the interpolation functions to fulfil patch test and to ensure
stability of the discrete equations according to appendix B the independent field of stress
11