# 3d stress on arbitrary plane

• Xinyue
In summary: Hi again, I am not sure if you're referring to experimental data or a theoretical problem, but if you're dealing with experimental data you can basically rotate the reference frame so that the plane in question becomes aligned with one of the axes, and from there it's just a matter of plugging in the values. If it's a theoretical problem, then it'll be a bit more complicated but the same principles apply. Just make sure you're using consistent units throughout and that you're using the correct formulas for the direction cosines. In summary, the shear stress components on an arbitrary plane in a cubic under 3D stress state can be derived by using a generalized 3D stress transformation, which involves using 9 direction cosines obtained from

#### Xinyue

I have a question about the 3D stress distribution. I need to know the shear stress components on a arbitrary plane in a cubic under 3d stress state. But it seems not possible to derive them. I haven't found a book about this. Anybody knows something about it?

Am I understanding it correctly that you'd essentially need a generalized 3D stress transformation, from one coordinate system to another ? Essentially doing the transform using e.g. the 9 resulting direction cosines arising from 3 rotations (naturally depending on how complex is the orientation of your plane compared to the initial state). If so that can sure be done, the transformation matrix is somewhat lengthy but not too 'bad'.

thanks

You are right. Thanks.
PerennialII said:
Am I understanding it correctly that you'd essentially need a generalized 3D stress transformation, from one coordinate system to another ? Essentially doing the transform using e.g. the 9 resulting direction cosines arising from 3 rotations (naturally depending on how complex is the orientation of your plane compared to the initial state). If so that can sure be done, the transformation matrix is somewhat lengthy but not too 'bad'.

Hi Xinyue, the tensor form is way more compact but this is probably clearer, between the original system and $^'$ system :

$$\left( \begin{array}{c} \sigma_{xx}^'\\ \sigma_{yy}^'\\ \sigma_{zz}^'\\ \sigma_{yz}^'\\ \sigma_{xz}^'\\ \sigma_{xy}^' \end{array} \right) =[T_{\sigma}] \left( \begin{array}{c} \sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \sigma_{yz}\\ \sigma_{xz}\\ \sigma_{xy} \end{array} \right)$$

where

$$[T_{\sigma}] = \left( \begin{array}{cccccc} l_{1}^2 & m_{1}^2 & n_{1}^2 & 2m_{1}n_{1} & 2n_{1}l_{1} & 2l_{1}m_{1}\\ l_{2}^2 & m_{2}^2 & n_{2}^2 & 2m_{2}n_{2} & 2n_{2}l_{2} & 2l_{2}m_{2}\\ l_{3}^2 & m_{3}^2 & n_{3}^2 & 2m_{3}n_{3} & 2n_{3}l_{3} & 2l_{3}m_{3}\\ l_{1}l_{3} & m_{1}m_{3} & n_{1}n_{3} & (m_{1}n_{3}+m_{3}n_{1}) & (l_{1}n_{3}+l_{3}n_{1})& (l_{1}m_{3}+l_{3}m_{1})\\ l_{2}l_{3} & m_{2}m_{3} & n_{2}n_{3} & (m_{2}n_{3}+m_{3}n_{2}) & (l_{2}n_{3}+l_{3}n_{2})& (l_{2}m_{3}+l_{3}m_{2})\\ l_{1}l_{2} & m_{1}m_{2} & n_{1}n_{2} & (m_{1}n_{2}+m_{2}n_{1}) & (l_{1}n_{2}+l_{2}n_{1})& (l_{1}m_{2}+l_{2}m_{1}) \end{array} \right)$$

where the direction cosines are

$$l=cos\alpha$$
$$m=cos\beta$$
$$n=cos\gamma$$

and $\alpha$ is the angle between $x,x^'$, $\beta$ is the angle between $y,y^'$, $\gamma$ is the angle between $z,z^'$ where you'll get the direction cosine components.

Hi I need to rotate stresses as above but I am not sure exactly what L1,L2 and L3 are , same with m1 ect and n1 etc can some one please help
thanks

Hi litters95 and welcome to Pf! You're referring to the components of the direction cosines, these might be of use (be careful with the notation, this is a tad more complex than the 2D cases typically presented since it's the "general" 3D transformation):

http://www.electromagnetics.biz/DirectionCosines.htm
http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node52.html
http://en.wikipedia.org/wiki/Direction_cosines

if you need a general form what's in #4 will do, but if you need something which works for example in 2D it can be clarified a whole lot ... what sort of a problem you're working with?

Hi
I am trying to rotate 3 d stresses like the matrix above but I am not sure what L1..L3, n1.. N3 and m1 m2 and m3 are .
thanks

Ok, so we've the primed and unprimed systems between which the transformation is being made. $l_{i}$ are the direction cosines between the $x$ and $x^{'}$, $y^{'}$, $z^{'}$. $m$ and $n$ are defined similarly, so you've 9 different direction cosines in a general 3D transformation. It is quite a bit simpler if you can simplify your system a bit, but actually if you do it systematically and consider the rotations with respect to each axis one by one it'll be fairly straightforward ($l=cos(\alpha), m=cos(\beta), n=cos(\gamma)$ if consider a system where the axes are rotated by $\alpha,\beta,\gamma$).

PerennialII said:
Ok, so we've the primed and unprimed systems between which the transformation is being made. $l_{i}$ are the direction cosines between the $x$ and $x^{'}$, $y^{'}$, $z^{'}$. $m$ and $n$ are defined similarly, so you've 9 different direction cosines in a general 3D transformation. It is quite a bit simpler if you can simplify your system a bit, but actually if you do it systematically and consider the rotations with respect to each axis one by one it'll be fairly straightforward ($l=cos(\alpha), m=cos(\beta), n=cos(\gamma)$ if consider a system where the axes are rotated by $\alpha,\beta,\gamma$).

thanks again i think I am nearly there just having a few probelms now with the rotated shear stresses

## 1. What is meant by "3d stress on arbitrary plane"?

"3d stress on arbitrary plane" refers to a type of stress analysis technique used in engineering and materials science. It involves calculating the stress distribution on a specific plane within a three-dimensional object or structure.

## 2. Why is it important to understand 3d stress on arbitrary plane?

Understanding 3d stress on arbitrary plane allows engineers and scientists to accurately predict the stress and strain on various components of a structure. This information is crucial in designing and building safe and durable structures.

## 3. How is 3d stress on arbitrary plane calculated?

To calculate 3d stress on arbitrary plane, a combination of mathematical equations and computer simulations are used. This involves determining the forces acting on the structure, as well as the material properties, and using them to calculate the stresses on the specific plane.

## 4. What factors can affect the 3d stress on arbitrary plane?

The 3d stress on arbitrary plane can be affected by various factors such as the material properties, external forces, and the geometry and shape of the structure. Changes in any of these factors can alter the stress distribution on the plane.

## 5. How is 3d stress on arbitrary plane used in real-world applications?

3d stress on arbitrary plane is used in a wide range of real-world applications, including building construction, aerospace engineering, and materials testing. It allows engineers to analyze the structural integrity of various components and make necessary design improvements for safety and efficiency.