Solving a BC Differential equation, (Buckling) any way possible

Question

I am trying to verify my manual solution for this problem by any way, so I tried NDSolve, and DSolve, with no success. Can some one help, or even give me the final numbers :D I need the first 3 alphas for a buckling

Here is my interpretation (yields the NDSolve::ivone error)

xDim = 1;
yDim = 1/2;

alpha = 4 Pi^2;

eqn2D = D[w[x, y], {x, 4}] + 2 D[w[x, y], {x, 2}, {y, 2}] + 
   D[w[x, y], {y, 4}] + alpha D[w[x, y], {x, 2}];

buck2D = NDSolve[{eqn2D == 0,
   w[-xDim, y] == 0,
   w[xDim, y] == 0,
   w[x, -yDim] == 0,
   w[x, yDim] == 0,
   Derivative[2, 0][w][-xDim, y] == 0,
   Derivative[2, 0][w][xDim, y] == 0,
   Derivative[0, 2][w][x, -yDim] == 0,
   Derivative[0, 2][w][x, yDim] == 0},
   w, {x, -xDim, xDim}, {y, -yDim, yDim}, Method -> {"MethodOfLines"}]

Even the 1D solution seems incorrect:

youngs = 2*^5;(*N/mm^2, steel*)

sideLength = 1.78;(*mm*)
areaMomInertia = sideLength^4/12;(*mm^2, square section*)

load = 42;(*N*)

alpha = load/(youngs areaMomInertia);

eqn1D = D[w[x], {x, 4}] + alpha D[w[x], {x, 2}];

buck1D = NDSolveValue[{eqn1D == 0,
   w[0] == 0,
   w[200] == 0,
   (D[w[x], {x, 1}] /. x -> 0) == 0,
   (D[w[x], {x, 1}] /. x -> 200) == 0},
  w, {x, 0, 200}]

Plot[buck1D[x], {x, 0, 200}, PlotRange -> All]

References:

https://uta-ir.tdl.org/uta-ir/bitstream/handle/10106/300/umi-uta-1041.pdf http://www.arpnjournals.com/jeas/research_papers/rp_2007/jeas_0207_35.pdf

Answer 1

Since you just want to verify your manual solution by any way, let me show you a solution based on finiteFourierSinTransform.

dim@x = 1;
dim@y = 1/2;

Clear[alpha]

With[{w = w[x, y]}, 

 eqn = D[w, {x, 4}] + 2 D[w, {x, 2}, {y, 2}] + D[w, {y, 4}] + alpha D[w, {x, 2}] == 0;

 help = ({w == 0, D[w, {#, 2}] == 0} /. List /@ {# -> -dim@#, # -> dim@#} // Flatten) &;

 {bc@x, bc@y} = help /@ {x, y}]

Then make finite Fourier sine transform with respect to $x$, and substitute bc@x in:

Format@finiteFourierSinTransform[f_, __] := Subscript[\[ScriptCapitalF], s][f]

ffst = finiteFourierSinTransform;
{teq, tbc@y} = ffst[{eqn, bc@y}, {x, -dim@x, dim@x}, n] /. (Rule @@@ bc@x /. y -> y_)

Once again, make finite Fourier sine transform. This time we eliminate derivatives with respect to $y$:

tteq = ffst[teq, {y, -dim@y, dim@y}, m] /. 
     ffst[ffst[a_, b__], c__] :> ffst[ffst[a, c], b] /. (Rule @@@ tbc@y /. 
      x -> x_) /. (Rule @@@ bc@x /. y -> y_) // Simplify

At this point, it's clear that $w$ only has a trivial solution, unless the coefficient (-4 alpha n^2 + (4 m^2 + n^2)^2 Pi^2) equals zero, which is relevant to buckling. It's clear that m and n won't be that large for first 3 alphas, so let's find them by brute force:

alphavalue = Solve[tteq[[1, 1]] == 0, alpha][[1, 1, -1]]
Sort@Union@Flatten[Table[{alphavalue, {n, m}}, {m, 5}, {n, 5}], 1][[1 ;; 3]]
(* {{4 π^2, {2, 1}}, {(169 π^2)/36, {3, 1}}, {(25 π^2)/4, {1, 1}}} *)

The first one is 4 π^2, which is exactly the value tested in your question.