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y 12

y 12

Completing the matrix multiplication, reversing the sequence of some of the coordinates so that all subscripts are in descending order, gives the required element stiffness matrix as yj2 +xj2(l - v)/2.

»i»2 + *2l*32(l - v)/2, - »i»i - *2l*3l(l - v)/2, y\x + ( 1 - v)/2

-V*32»2 - »2*32(1 - v)/2, V*32»1 + »2*3lU - v)/2, -1**32»! ~ »2*2l(l - »0/2,

»«31 »2 + »1*32(1 -v)/2, — WJC3iy31 ~»l*3l(l — v)/2, U*31»l +»1*21(1 — v)/2,

-VJC21W2 - »1*32(1 - v)/2, U*2I»I + »1*31 (1 - »0/2, -V*2!»l ~ »1*21 ( 1 - »0/2, Symmetric

*21*32 + » 1 »2 ( 1 - »0/2, *2I*3I - »l»l(l - v)/2, -I- (1 - v)/2-

Example 9.6

(a) Evaluate the element stiffness matrix, in global coordinates, for the three-node triangular membrane element, labelled a in Fig. 9.43. Assume plane stress conditions, Young's modulus, E = 200 GN/m2, Poisson's ratio, v = 0.3, thickness, t = 1 mm, and the same displacement functions as Example 9.5.

Fig. 9.43.

(b) Evaluate the element stiffness matrix for element b, assuming the same material properties and thickness as element a. Hence, evaluate the assembled stiffness matrix for the continuum.

Solution

(a) Figure 9.44 shows suitable node labelling for a single triangular membrane element. The resulting element stiffness matrix from the previous Example, 9.5, can be utilised. A specimen evaluation of an element stiffness term is given below for k\\. The rest are obtained by following the same procedure.

Substituting = 4 x 2(i _ 0 32) [(2 ~ 0) + (0 " 2> (1 ~ 0-3)/2]

Evaluation of all the terms leads to the required triangular membrane element stiffness matrix for element a, namely

0 0

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