Universität Karlsruhe (TH) - am Institut für Baustatik

3.2 Interpolation of the Green–Lagrangean strains and associated
variations
The element has to fulfil membrane and bending patch test, see e.g. [27]. As has been shown
in appendix B the bending patch test – when using the present mixed interpolation for the
stress resultants and shell strains – can be fulfilled with substitute shear strains defined in [23]
but not with the bilinear displacement interpolation inserted in the transverse shear strains
(5) 3 . Thus the finite element interpolation of the Green–Lagrangean strains reads
⎡
ε h ⎤ ⎡
1
11
2 (xh , 1 ·x h , 1 −X h , 1 ·X h ⎤
, 1 )
ε h 1
22
2 (xh , 2 ·x h , 2 −X h , 2 ·X h , 2 )
2ε h 12
x h , 1 ·x h , 2 −X h , 1 ·X h , 2
ε h κ h 11
x h , 1 ·d h , 1 −X h , 1 ·D h , 1
G =
κ h =
22
x h , 2 ·d h , 2 −X h , 2 ·D h (19)
, 2
2κ h 12
x h , 1 ·d h , 2 +x h , 2 ·d h , 1 −X h , 1 ·D h , 2 −X h , 2 ·D h , 1
⎧ ⎢ γ1
h ⎥ ⎢ ⎨ 1
⎣ ⎦ ⎣ J [(1 − η) ⎫
−1 2 γB ξ +(1+η) γξ
D ⎬
⎥
⎦
γ2
h ⎩ 1
[(1 − ξ) 2 γA η +(1+ξ) γη C ] ⎭
.
The strains at the midside nodes A, B, C, D of the element are specified as follows
γξ M = [x, ξ ·d − X, ξ ·D] M M = B,D
γη L = [x, η ·d − X, η ·D] L (20)
L = A, C ,
where the following quantities are given with the bilinear interpolation (12) and(17)
d A = 1 (d 2 4 + d 1 ) D A = 1 (D 2 4 + D 1 )
d B = 1 (d 2 1 + d 2 ) D B = 1 (D 2 1 + D 2 )
d C = 1 (d 2 2 + d 3 ) D C = 1 (D 2 2 + D 3 )
d D = 1 (d 2 3 + d 4 ) D D = 1 (D 2 3 + D 4 )
x A , η = 1 (x 2 4 − x 1 ) X A , η = 1 (X 2 4 − X 1 )
x B , ξ = 1 (x 2 2 − x 1 ) X B , ξ = 1 (X 2 2 − X 1 )
x C , η = 1 (x 2 3 − x 2 ) X C , η = 1 (X 2 3 − X 2 )
x D , ξ = 1 (x 2 3 − x 4 ) X D , ξ = 1 (X 2 3 − X 4 ) .
Accordingly the interpolated virtual strains read
⎡
δε h ⎤ ⎡
11
δx h , 1 ·x h ⎤
, 1
δε h 22
δx h , 2 ·x h , 2
2δε h 12
δx h , 1 ·x h , 2 +δx h , 2 ·x h , 1
δε h δκ h 11
δx h , 1 ·d h , 1 +δd h , 1 ·x h , 1
G =
δκ h =
22
δx h , 2 ·d h , 2 +δd h , 2 ·x h , 2
2δκ h 12
δx h , 1 ·d h , 2 +δx h , 2 ·d h , 1 +δd h , 1 ·x h , 2 +δd h , 2 ·x h , 1
⎧ ⎢ δγ1
h ⎥ ⎢ ⎨ 1
⎣ ⎦ ⎣ J [(1 − η) ⎫
−1 2 δγB ξ +(1+η) δγξ
D ⎬
⎥
⎦
δγ2
h ⎩ 1
[(1 − ξ) 2 δγA η +(1+ξ) δγη C ] ⎭
8
(21)
(22)