(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, ν = 0.3, thickness, t = 1 mm, and the same displacement functions as Example9.5

Question

(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, ν = 0.3, thickness, t = 1 mm, and the same displacement functions as Example 9.5.

(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.

Accepted Answer

(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 [latex]k_{11} [/latex]. The rest are obtained by following the same procedure

[latex]k_{11}=\frac{Et}{4a(1- u ^{2})} [y_{32}^{2}+x_{32}^{2}(1- u )/2][/latex]
[latex]=\frac{Et}{4a(1- u ^{2})}[(y_{3}-y_{2})^{2}+(x_{3}-x_{2})^{2}(1- u )/2][/latex]
Substituting [latex]=\frac{200 imes 10^{9} imes 1 imes 10^{-3}}{4 imes 2(1-0.3^{2})} [(2-0)^{2}+(0-2)^{2}(1-0.3)/2][/latex]
[latex]=14.835 imes 10^{7} N/m[/latex]

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

[latex][k^{a}]=10^{7}[N/m]\begin{bmatrix}14.835 & & & & & \ -10.989 & 10.989 & & symmetric & & \ -3.846 & 0 &3.846 & & & \ 7.143 & -3.297 & -3.846 & 14.835 & & \-3.846 & 0& 3.846 & -3.846 & 3.846 & \ -3.297 &3.297 & 0& -10.989 & 0 & 10.989 \end{bmatrix}[/latex]

(b) Element b can temporarily also be labelled with node numbers 1,2 and 3, as element a. To avoid confusion, this is best done with the elements shown "exploded", as in Fig. 9.45. The alternative is to re-number the subscripts in the element stiffness matrix result from Example 9.5.

Performing the evaluations similar to part (a) leads to the required stiffness matrix for element b, namely

[latex][k^{a}]=10^{7}[N/m]\begin{bmatrix}3.846 & & & & & \ -3.846 & 14.835 & & symmetric & & \ 0 & -10.989 &10.989 & & & \0 & -3.297 & 3.297& 10.989 & & \-3.846 &7.143& -3.297 & -10.989 &14.835 & \ 3.846 &-3.846 & 0& 0 & -3.846 & 3.846 \end{bmatrix}[/latex]

With reference to §9.10, the structural stiffness matrix can now be assembled using a dof. correspondence table. The order of the structural stiffness matrix will be 8 ×8, corresponding to four nodes, each having 2 dof. The dof. sequence,[latex] u_{1} , u_{2}, u_{3}, υ_{1}, υ_{2}, υ_{3},[/latex] adopted for the convenience of inverting matrix [A], covered in §9.9, can be converted to the more usual sequence, i.e. [latex]u_{1} υ_{1}, u_{2}, υ_{2}, u_{3}, υ_{3},[/latex] with the aid a dof. correspondence table. Whilst this re-sequencing is optional, the converted sequence is likely to result in less rearrangement of rows and columns, prior to partitioning the assembled stiffness matrix, than would otherwise be needed.
If row and column interchanges are to be avoided in solving the following Example, 9.7, and therefore save some effort, then the dof. labelling of Fig. 9.46 is recommended. This implies the final node numbering, also shown.

The dof. correspondence table will be as follows:

Row/column in [latex][k^{(e)}][/latex]		1	2	3	4	5	6
Row/column in [K]	a	5	3	7	6	4	8
Row/column in [K]	b	3	1	7	4	2	8

Assembling the structural stiffness matrix, gives

Summing the element stiffness contributions, and writing the structural governing equations ,gives the result as

[latex]\begin{bmatrix} X_{1} \ Y_{1} \ X_{2} \ Y_{2}\X_{3}\Y_{3}\X_{4} \Y_{4}\end{bmatrix} =10^{7}[N/m] \begin{bmatrix} 14.835 &7.143&-3.846&-3.297&0&0&-10.989&-3.846 \7.143 &14.835&-3.846&-10.989&0&0&-3.297& -3.846\-3.846 &-3.846&14.835&0&-10.989&-3.297&0&7.143 \ -3.297&-10.989&0&14.835&-3.846&-3.846&7.143&0\0&0&-10.989&-3.846&14.835&7.143&-3.846&-3.297\0&0&-3.297&-3.846&7.143&14.835&-3.846&-10.989\-10.989&-3.297&0&7.143&-3.846&-3.846&14.835&0\-3.846&-3.846&7.143&0&-3.297&-10.989&0&14.835\end{bmatrix} \begin{bmatrix} u_{1} \ υ_{1} \ u_{2} \ υ_{2}\u_{3}\υ_{3}\u_{4} \υ_{4}\end{bmatrix}[/latex]

i.e. {P}=[K]{p}

where [K] is the required assembled stiffness matrix.

Question 9.6: (a) Evaluate the element stiffness matrix, in global coordin...

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