[탄성학] 3차원 응력변환

(x,y,z) 좌표계에서의 응력요소를 (x',y',z')좌표계에서의 응력요소로 바꾸는 방법에 대해 알아봅시다. 유도과정은 생략하고 결과만 알아볼 것입니다. 두 좌표계 간의 방향코사인은 아래와 같습니다. 

 

	x	y	z
x'	$l_{1}$	$m_{1}$	$n_{1}$
y'	$l_{2}$	$m_{2}$	$n_{2}$
z'	$l_{3}$	$m_{3}$	$n_{3}$

 

변환 공식은 아래와 같습니다. 

 

$\sigma_{x'}=\sigma_{x}l^2_{1} +\sigma_{y}m^2_{1}  +\sigma_{z}n^2_{1}  +2( \tau_{xy} l_{1}m_{1}  + \tau_{yz} m_{1}n_{1}  +  \tau_{xz} l_{1}n_{1})$

 

$\sigma_{y'}=\sigma_{x}l^2_{2} +\sigma_{y}m^2_{2}  +\sigma_{z}n^2_{2}  +2( \tau_{xy} l_{2}m_{2}  + \tau_{yz} m_{2}n_{2}  +  \tau_{xz} l_{2}n_{2})$

 

$\sigma_{z'}=\sigma_{x}l^2_{3} +\sigma_{y}m^2_{3}  +\sigma_{z}n^2_{3}  +2( \tau_{xy} l_{3}m_{3}  + \tau_{yz} m_{3}n_{3}  +  \tau_{xz} l_{3}n_{3})$

 

$\tau_{x'y'}=\sigma_{x} l_{1}l_{2} + \sigma_{y} m_{1}m_{2}  + \sigma_{z} n_{1}n_{2}  + \tau_{xy}(l_{1}m_{2}+m_{1}l_{2})  + \tau_{yz}(m_{1}n_{2}+n_{1}m_{2})  + \tau_{xz}(n_{1}l_{2}+l_{1}n_{2})$

 

$\tau_{y'z'}=\sigma_{x} l_{2}l_{3} + \sigma_{y} m_{2}m_{3}  + \sigma_{z} n_{2}n_{3}  + \tau_{xy}(l_{2}m_{3}+m_{2}l_{3})  + \tau_{yz}(m_{2}n_{3}+n_{2}m_{3})  + \tau_{xz}(n_{2}l_{3}+l_{2}n_{3})$

 

$\tau_{x'z'}=\sigma_{x} l_{1}l_{3} + \sigma_{y} m_{1}m_{3}  + \sigma_{z} n_{1}n_{3}  + \tau_{xy}(l_{1}m_{3}+m_{1}l_{3})  + \tau_{yz}(m_{1}n_{3}+n_{1}m_{3})  + \tau_{xz}(n_{1}l_{3}+l_{1}n_{3})$

 

행렬 형태로 나타내면 아래와 같습니다. 

 

$\begin{bmatrix} \sigma_{x'} \\  \sigma_{y'} \\  \sigma_{z'} \\  \tau_{x'y'} \\  \tau_{y'z'} \\  \tau_{x'z'} \\  \end{bmatrix} =\begin{bmatrix} l^2_{1} & m^2_{1}  &n^2_{1}  & 2l_{1}m_{1}  & 2m_{1}n_{1}  & 2l_{1}n_{1} \\  l^2_{2} & m^2_{2}  &n^2_{2}  & 2l_{2}m_{2}  & 2m_{2}n_{2}  & 2l_{2}n_{2} \\  l^2_{3} & m^2_{3}  &n^2_{3}  & 2l_{3}m_{3}  & 2m_{3}n_{3}  & 2l_{3}n_{3} \\ l_{1}l_{2} & m_{1}m_{2}  & n_{1}n_{2}  & l_{1}m_{2}+m_{1}l_{2}  & m_{1}n_{2}+n_{1}m_{2}  & n_{1}l_{2}+l_{1}n_{2} \\ l_{2}l_{3} & m_{2}m_{3}  & n_{2}n_{3}  & l_{2}m_{3}+m_{2}l_{3}  & m_{2}n_{3}+n_{2}m_{3}  & n_{2}l_{3}+l_{2}n_{3} \\ l_{1}l_{3} & m_{1}m_{3}  & n_{1}n_{3}  & l_{1}m_{3}+m_{1}l_{3}  & m_{1}n_{3}+n_{1}m_{3}  & n_{1}l_{3}+l_{1}n_{3} \\ \end{bmatrix} \begin{bmatrix} \sigma_{x} \\  \sigma_{y} \\  \sigma_{z} \\  \tau_{xy} \\  \tau_{yz} \\  \tau_{xz} \\  \end{bmatrix}$