Exploring the Effects of Scale Analysis on Structural Mechanics Problems

[c.teixeira]

Hi there,

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 $\frac{\partial w}{\partial x} = \varphi$

0 = kAG$\frac{\partial(\frac{\partial w}{\partial x} - \varphi)}{\partial x}$

0=P$\frac{\partial w}{\partial x} + EI \frac{\partial^{2} \varphi}{\partial x^2} + kAG(\frac{\partial w}{\partial x} -\varphi)$

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 $\frac{\partial w}{\partial x} = \varphi$, then I ixpected the system of ODE'S to understang under which circustances that would happen.
I divided the second equation by KAG:

0=$\frac{P}{kAG}$$\frac{\partial w}{\partial x} + \frac{EI}{kAG} \frac{\partial^{2} \varphi}{\partial x^2} + (\frac{\partial w}{\partial x} -\varphi)$

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 $w_{s}$ for w and $\varphi_{s}$ for $\varphi.$

Hence:

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

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

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

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


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

 

[jvicens]

Hi c.teixeira,

Thank you for sharing your problem and approach with us. Your scale analysis does make sense and is a valid approach to understanding the behavior of the two theories under different conditions.

From your equations, it seems like the parameter P represents a load applied to the beam. As this parameter increases, the solutions from the two theories start to diverge. This can be explained by the fact that the Timoshenko beam theory takes into account shear deformation, while your theory neglects it. As the load increases, the effects of shear deformation become more significant, causing the solutions from the two theories to differ.

Your scale analysis shows that under certain conditions (small P and appropriate choice of scale factors), the two theories should provide similar solutions. This is because the effects of shear deformation become negligible in these conditions, and your simplified theory is able to accurately describe the behavior of the beam.

In conclusion, your scale analysis is a valid approach to understanding the behavior of the two theories and can help explain the divergence of solutions under different conditions. It is important to carefully choose the scale factors and understand the limitations of your simplified theory in order to make accurate predictions. I hope this helps you in your research. Best of luck!