What are the shear stress components on an arbitrary plane in a 3D stress state?

[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?

 

[PerennialII]

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'.

 

[Xinyue]

thanks

You are right. Thanks.

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'.

 

[PerennialII]

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.

 

[litters95]

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

 

[PerennialII]

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?

 

[litters95]

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

 

[PerennialII]

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 $\alpha,\beta,\gamma$).

 

[litters95]

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