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c.teixeira

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I am trying to solve a structural mechanics problem. I am doing so by two methods. On one hand, I am using a F.E.A software (ANSYS) to get me the solutions. At the same time I am solving the problem analytically. The issue is that ANSYS is solving the problem using a diferent theory than I am. ANSYS solves the problem based on the TImoshenko beam theory which consists in 2 uncoupled ODE's. I am solving a theory that altough not as precise, is simpler, thus easier to solve analytically, Regardless, the two theories are expected to provide similar solutions under certain conditions, for instance, on the analysis of a slender beam.

Below, you have the 2 un-coupled ODE's ANSYS uses.

**The theory I am using is the special case**when [itex]\frac{\partial w}{\partial x} = \varphi[/itex]

0 = kAG[itex]\frac{\partial(\frac{\partial w}{\partial x} - \varphi)}{\partial x}[/itex]

0=P[itex]\frac{\partial w}{\partial x} + EI \frac{\partial^{2} \varphi}{\partial x^2} + kAG(\frac{\partial w}{\partial x} -\varphi) [/itex]

Indeed, the solutions I get are similar. However as the parameter P below becomes bigger the solutions start to diverge. This is what I want to explain. I want to use scale analysis to explain that a bigger P means that the aproximation starts to become innapropriate.

Here is my reasoning:

If in my theory [itex]\frac{\partial w}{\partial x} = \varphi[/itex], then I ixpected the system of ODE'S to understang under which circustances that would happen.

I divided the second equation by KAG:

0=[itex]\frac{P}{kAG}[/itex][itex]\frac{\partial w}{\partial x} + \frac{EI}{kAG} \frac{\partial^{2} \varphi}{\partial x^2} + (\frac{\partial w}{\partial x} -\varphi) [/itex]

I then procedeed to use the leght of the beam L as a scale for the x coordinate. Also I used an unkonw scale factor [itex]w_{s}[/itex] for w and [itex]\varphi_{s}[/itex] for [itex]\varphi.[/itex]

Hence:

0=[itex]\frac{P w_{s}}{kAG L}\frac{\partial \hat{w}}{\partial \hat{x}}[/itex] +[itex] \frac{EI \varphi_{s}}{kAG L^2}\frac{\partial \hat{\varphi}^2}{\partial \hat{x}^2}+(\frac{\partial w}{\partial x} -\varphi) [/itex]

If the scale factors are choosen properly it can be assumed that [itex]\frac {\partial \hat{w}}{\partial \hat{x}} and \frac{\partial \hat{\varphi}^2}{\partial \hat{x}^2}[/itex] are O(1) correct?

If that is the case, can I say that under the circunstances:

[itex]\frac{P w_{s}}{kAG L}[/itex]→0 and [itex]\frac{EI \varphi_{s}}{kAG L^2}[/itex] →0, then [itex](\frac{\partial w}{\partial x} -\varphi) = 0[/itex]?

What do you make of this? Does my "scale" analysis make any sense to you? I am allowed to do this?

I hope I was clear enough.

Thank you in advance.

c.teixeira