Universität Karlsruhe (TH) - am Institut für Baustatik

with b Iα = T T I x, α , b M = T T I x M , ξ
to the corner nodes is given by
and b L = T T I x L , η . The allocation of the midside nodes
(I,M,L) ∈{(1,B,A); (2,B,C); (3,D,C); (4,D,A)} . (30)
To alleviate the notation the subscript h is omitted in the matrix.
3.3 Second variation of the functional
Assuming conservative external loads ¯p and ¯t the second variation of the functional yields
∫
Dg · Δθ h = [δε hT (C Δε h − Δσ h )+δσ hT (Δε h G − Δε h )+δε hT
G Δσ h +Δδε hT
G σ h ] dA (31)
(Ω)
with C := ∂ 2 εW . The linearized strains Δε h G are defined with (22) replacing the operator δ
by Δ whereas the linearized virtual shell strains are given with
⎡
Δδε h ⎤ ⎡
11
δx h , 1 ·Δx h ⎤
, 1
Δδε h 22
δx h , 2 ·Δx h , 2
2Δδε h 12
δx h , 1 ·Δx h , 2 +δx h , 2 ·Δx h , 1
Δδκ h Δδε h 11
δx h , 1 ·Δd h , 1 +δd h , 1 ·Δx h , 1 +x h , 1 ·Δδd h , 1
G =
Δδκ h =
22
δx h , 2 ·Δd h , 2 +δd h , 2 ·Δx h , 2 +x h , 2 ·Δδd h , 2
(32)
2Δδκ h δx h ,
12
1 ·Δd h , 2 +δx h , 2 ·Δd h , 1 +δd h , 1 ·Δx h , 2 +δd h , 2 ·Δx h , 1
+x h , 1 ·Δδd h , 2 +x h , 2 ·Δδd h , 1
⎧ ⎢ Δδγ1
h ⎥ ⎢ ⎨ 1
⎣ ⎦ ⎣ J [(1 − ⎫
−1 2 η)ΔδγB ξ +(1+η)Δδγξ
D ⎬
⎥
⎦
Δδγ2
h ⎩ 1
[(1 − 2 ξ)ΔδγA η +(1+ξ)Δδγη C ] ⎭
with
Δδγξ M = [δx, ξ ·Δd +Δx, ξ ·δd + x, ξ ·Δδd] M M = B,D
Δδγη L = [δx, η ·Δd +Δx, η ·δd + x, η ·Δδd] L (33)
L = A, C
The second variation of the current orthogonal base system has been derived in [30], see
appendix A. In the following representation the constants c i introduced in [30] are simplified
and the Taylor series expansion is given. With an arbitrary vector h I ∈ R 3 and b I = d I × h I
we obtain
h I · Δδd I = δw I · M I Δw I
M I (h I ) = 1 2 (d I ⊗ h I + h I ⊗ d I )+ 1 2 (t I ⊗ ω I + ω I ⊗ t I )+c 10 1
t I = −c 3 b I + c 11 (b I · ω I ) ω I c 10 = ¯c 10 (b I · ω I ) − (d I · h I )
c 3 = ω I sin ω I +2(cosω I − 1)
ω 2 I (cos ω I − 1)
¯c 10 =
sin ω I − ω I
2ω I (cos ω I − 1)
c 11 = 4(cosω I − 1) + ω 2 I + ω I sin ω I
2 ω 4 I (cos ω I − 1)
10
= 1 6 (1 + 1 60 ω2 I)+O(ω 4 I)
= 1 6 (1 + 1 30 ω2 I)+O(ω 4 I)
= − 1
360 (1 + 1 21 ω2 I)+O(ω 4 I)
(34)