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Particular Solution of 4th degree differential equation (column buckling)

Posted 7 years ago
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I'm trying to use Mathematica to solve the Euler Column buckling problem (http://www.continuummechanics.org/cm/columnbuckling.html), but its not working out. Mathematica only shows me the zero solution. Is there any way of getting the no zero displacements solutions? For now, what I have is basically this:

equation = EIy''[x] + Py[x] == 0

DSolve[{equation, y[0] == 0, y[L] == 0}, y[x], x]

Out[143]= {{y[x] -> 0}}

Thanks, Luís Valarinho

6 Replies

Replace y[L] ==0 with y'[0] == yp0 and then combine y[L] ==0 with the solution you get.

The solution given is correct but it isn't the only solution. Yes, you'll have to handle the boundary condition at L separately.

I recommend not using capital letters for your constants (or at least give them unique names) because for example I is the symbol for Sqrt[-1] and E is the euler number Here is how I would solve it:

In[13]:= Clear["Global`*"]

In[14]:= sol = 
 DSolve[{y''[x] + ps/(es is) y[x] == 0, y[0] == 0}, y[x], x]

Out[14]= {{y[x] -> C[2] Sin[(Sqrt[ps] x)/(Sqrt[es] Sqrt[is])]}}

In[15]:= Solve[Sqrt[ps] l/Sqrt[es is] == n Pi, ps]

Out[15]= {{ps -> (es is n^2 \[Pi]^2)/l^2}}

So, if I understood Mathematica doesn't solve the equation automatically.

The last boundary condition will have to be me adding it, right?

Ok, thanks!

Hello Kay!

Thanks for the suggestion. What I did actually was using the nomenclature that we use in Civil Engineering But you're right, it can lead to mistakes. Thanks for the other suggestions too. I was actually wondering if it is possible to do it in one step. We cannot ask Mathematica to give all the possible solutions? Or adding an assumption that C[2] should be different than zero? (something that is not necessarily truth).

Thanks, Luís Valarinho

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