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Finite Element Analysis of Membrane Structures 53
Table 1. Index map for Q array
Indices Values
a 1 2 3
I,J 1,1 2,2 1,2 & 2,1
b 1 2 3
i,j 1,1 2,2 1,2 & 2,1
where A is thereference area for the triangular element.
The variation of gij results in the values
δg11 = 2 δ˜x 2 − δ˜x 1 T ∆˜x 21
δg12 = δ˜x 2 − δ˜x 1 T ∆˜x 31 + δ˜x 3 − δ˜x 1 T ∆˜x 21
δg22 = 2 δ˜x 3 − δ˜x 1 T ∆˜x 31
At this stage it is convenient to transform the second order tensors to matrix
form and write
1
2 δCIJSIJ = δEIJSIJ = ⎡ ⎤
S11
δE11 δE22 2 δE12 ⎣ S22 ⎦ = δE T S (35)
or for the alternative form
1
2 δgijsij = 1
⎡
δg11 δg22 2 δg12 ⎣
2
as
⎡ s11
s22
s12
⎤
⎦
S12
(34)
= 1
2 δgT s (36)
Using (34) we obtain the result directly in terms of global cartesian components
1
2 δgT s = δ(˜x 1 ) T δ(˜x 2 ) T δ(˜x 3 ) T [b] T s
= δ(˜x 1 ) T δ(˜x 2 ) T δ(˜x 3 ) T [b] T Q T S = δE T S (37)
where the strain-displacement matrix b is given by
⎡
−(∆˜x
b = ⎣
21 ) T
(∆˜x 21 ) T
0
−(∆˜x 31 ) T
0 (∆˜x 31 ) T
−(∆˜x 21 + ∆˜x 31 ) T (∆˜x 31 ) T (∆˜x 21 ) T
⎤
⎦
Thus, directly we have in each element
3×9
(38)
δE = Q b δ˜x = 1
δC (39)
2
where ˜x denotes the three nodal values on the element. It is immediately obvious
that we can describe a strain-displacement matrix for the variation of E as
B = Q b (40)