User:Eas4200c.f08.wiki.f/9-22-08

Curved Panels
Below is an image of a curved panel in the (x,y,z) coordinates.



Note: Sometimes (y,z) coordinates are used for the axes in plane of the cross section of the panel instead of (x,y) coordinates.

Below is an image of another curved panel with thickness t. A force will be applied to the panel which can be seen in the next image below.



Below is a cross sectional view of the same curved panel with force F acting on it at an angle above the horizontal (y-axis). The cross section goes from point A to point B. It's height is b and it's width is a.



$$d\overrightarrow{F} = q\overrightarrow{dl} = q(dl_yj + dl_zk)$$

$$d\overrightarrow{F} = q(dl\cos\theta \overrightarrow{j} + dl\sin \theta \overrightarrow{k})$$


 * $$dl\cos\theta = dy$$


 * $$dl\sin\theta = dz$$


 * $$ q = \tau t $$ = shear flow

Here's a closer look at the panel where the force is acting. You can see it's acting on and parallel with section dl.



Closed Thin-Walled Cross Section
$$\overrightarrow{T} = T\overrightarrow{i} $$

Below is the cross section of a closed thin-walled area. There is a force acting on section dl.

$$\overrightarrow{r}$$ = the distance from the origin to section dl

$$\rho$$ = the distance from the origin to the force that is perpendicular to the force vector.



$$ d\overrightarrow{T} = \overrightarrow{r}xd\overrightarrow{F} $$

$$ dT = \rho dF $$

Take a closer look where the force acts on on section dl.

$$dA = \frac{1}{2}\rho dl $$

$$T = \oint_{}^{}{dT} = q\oint_{}^{}{\rho dl} $$


 * $$ \rho dl = 2dA $$

$$ T = 2q\int_{\overline{A}}^{}{dA}$$

$$T = 2q\overline{A}$$

$$\overline{A}$$ = Average Area

Below is an image of the average area of the cross section of a curved panel: