# Exploring the Effects of Scale Analysis on Structural Mechanics Problems

• c.teixeira
In summary, scale analysis is a method used in structural mechanics to study the behavior of a structure at different sizes or scales. It allows for a better understanding of a structure's behavior under different conditions and helps in optimizing its design. This method is widely used in various fields, such as aerospace engineering and biomechanics, but it does have limitations, such as assuming linear and elastic behavior of materials and not taking into account other factors that may influence a structure's behavior.
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.

c.teixeira

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!

## 1. What is scale analysis in structural mechanics?

Scale analysis is a method used in structural mechanics to study the behavior of a structure at different sizes or scales. It involves analyzing the effects of changing the dimensions of a structure on its overall behavior and performance.

## 2. Why is scale analysis important in structural mechanics?

Scale analysis allows engineers and scientists to better understand how a structure will behave under different loading conditions and at different sizes. It also helps in optimizing the design of a structure to ensure its safety, efficiency, and cost-effectiveness.

## 3. How does scale analysis impact structural design?

Scale analysis plays a crucial role in the structural design process by providing insights into the structural behavior and performance at different scales. This information is used to make informed decisions about the size, shape, and materials to be used in the design of a structure.

## 4. What are some common applications of scale analysis in structural mechanics?

Scale analysis is widely used in various fields, including aerospace engineering, civil engineering, and biomechanics. It is used to study the behavior of structures such as bridges, buildings, and aircraft, as well as biological structures like bones and tissues.

## 5. What are some limitations of scale analysis in structural mechanics?

One limitation of scale analysis is that it assumes linear and elastic behavior of materials, which may not always be the case in real-life structures. Additionally, scale analysis may not take into account other factors such as material properties, boundary conditions, and environmental effects, which can also impact the behavior of a structure.

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