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I am trying to write code for the buckling of composite plates using the Ritz approximation but I can't understand something. According to Reddy (Mechanics of laminated composites), this is the equation of the principle of minimum potential energy (since we are concerned with buckling, q = 0 and the frequency omega = 0):

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And the Ritz approximation

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But after substituting 6.6.5 in 6.6.4 we get

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It seems that delta w_0 has been substituted with X and Y but with indexes p and q, but it seems we do not sum over them. If that is the case, how do we go around solving this? Do we iterate over them but for some reason it is not written explicitly?

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  • $\begingroup$ Welcome on the Engineering SE! This site supports Latex, type $\sum_{i=1}^{M} and you get $\sum_{i=1}^{M}$. Using it is more than strongly adviced, but lesser than expected. $\endgroup$
    – peterh
    Nov 7 '18 at 16:05
  • $\begingroup$ What you have is a set of equations. The indexes p and q are used for the variation (test function) to distinguish from the i and j indices of the trial function. The test and trial functions appear to have been assumed to be identical. The variations are arbitrary except at the boundaries (where they are zero). That allows you to get rid of $c_{pq}$ and $\sum_p, \sum_q$. You can form a matrix equation from what remains and solve that. $\endgroup$ Nov 7 '18 at 19:22
  • $\begingroup$ Thanks for your answer. What do you mean get rid of? Shall I just construct the matrix equation as if they do not exist? $\endgroup$ Nov 8 '18 at 12:22
  • $\begingroup$ The best way to see what's happening is to assume a situation where the problem is one-dimensional (e.g., $\phi_{ij}(x,y) = X_i(x)$) and work out the algebra. What happens is that all the equations are independently equal to zero because of the arbitrariness of the variations. $\endgroup$ Nov 8 '18 at 19:21

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