Finite Element Analysis of Membrane Structures 53 Table 1.Index map for Q array Indices Values a 123 1J1,12,21,2&2,1 b12 3 ij1,12,21,2&2,1 where A is the reference area for the triangular element. The variation of gis results in the values 6g11=2(62-6)T△221 6g12=(62-6)T△231+(63-6m)T△元21 (34) 6g22=2(63-621)T△231 At this stage it is convenient to transform the second order tensors to matrix form and write S11 7=6JSJ=[6D1 S22 -6ETS (35) S12 or for the alternative form 吉yw=吉[m n2ne】 811 (36) 2 822 812 Using(34)we obtain the result directly in terms of global cartesian components as gs-[eyr6eyey]rs =[6()T(2)T6(3)T]Q'S=6Es (37) where the strain-displacement matrir b is given by -(4221)T (△221)T0 b= -(4231)T 0 (431)T (38) -(△221+△231)T(4281)T(4221)T 3×9 Thus,directly we have in each element 6E=Qb62=80 (39) where 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=Qb (40)Finite Element Analysis of Membrane Structures 53 Table 1. Index map for Q array Indices Values a 12 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 the reference area for the triangular element. The variation of gij results in the values δg11 = 2  δx˜2 − δx˜1T ∆x˜21 δg12 =  δx˜2 − δx˜1T ∆x˜31 +  δx˜3 − δx˜1T ∆x˜21 δg22 = 2  δx˜3 − δx˜1T ∆x˜31 (34) At this stage it is convenient to transform the second order tensors to matrix form and write 1 2 δCIJ SIJ = δEIJ SIJ =  δE11 δE22 2 δE12  ⎡ ⎣ S11 S22 S12 ⎤ ⎦ = δET S (35) or for the alternative form 1 2 δgij sij = 1 2  δg11 δg22 2 δg12  ⎡ ⎣ s11 s22 s12 ⎤ ⎦ = 1 2 δgT s (36) Using (34) we obtain the result directly in terms of global cartesian components as 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 QT S = δET S (37) where the strain-displacement matrix b is given by b = ⎡ ⎣ −(∆x˜21) T (∆x˜21) T 0 −(∆x˜31) T 0 (∆x˜31) T −(∆x˜21 + ∆x˜31) T (∆x˜31) T (∆x˜21) T ⎤ ⎦
 3×9 (38) Thus, directly we have in each element δE = Q b δx˜ = 1 2 δC (39) 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)