User:Eas4200c.f08.WIKI.E/hw6

=Mtg 34 Friday, November 14 =

Unsymmetrical thin walled cross sections
There will be a comparison drawn between cross sections with symmetry and cross sections of solids without symmetrical cross sections.

This is the generalized formula for a unsymmetrical cross section. (Equation 5.1)

$$ \int_{A_i}^{}{\frac{d\sigma _{xx}}{dx} dA}=-q_A$$



Case 1: Symmetrical about the Y-axis
The Stress factor is given under symmetrical cross sectional conditions as follows:

$$\sigma _{xx}=\frac{M_yZ}{I_y} $$

The shear flow factor $$q\;$$ is:

$$q(s)=\frac{-V_zQ_y}{I_y}$$

$$Q_y=\int_{A_S}^{}{ZdA}=Z_cA_s$$

Case 2:Unsymmetrical about the Y-axis
The Stress factor is given under unsymmetrical cross sectional conditions as follows:

$$\sigma _{xx}=(K_yM_z-K_{yz}M_y)y+(K_yM_y-K_{yz}M_z)z \;$$

$$K_y=\frac{I_y}{D}$$

$$K_{yz}=\frac{I_yz}{D}$$

$$K_z=\frac{I_Z}{D}$$

The stress in the x-direction can be solved for in matrix format.

$$\sigma _{xx}=\begin{vmatrix} y & z \end{vmatrix}_{1x2}\begin{vmatrix} K_y & -K_{yz}\\ -K_{yz} & K_z \end{vmatrix}\begin{Bmatrix} M_z\\M_y

\end{Bmatrix}\Leftrightarrow \sigma _{xx}=\begin{vmatrix} y & z \end{vmatrix}_{1x2}\begin{vmatrix} K_yM_Z & -K_{yz}M_y\\ -K_{yz}M_z & K_zM_y \end{vmatrix}$$

$$\sigma _{xx}=\begin{vmatrix} z & y \end{vmatrix}_{1x2}\begin{vmatrix} K_y & -K_{yz}\\ -K_{yz} & K_z \end{vmatrix}\begin{Bmatrix} M_y\\M_z

\end{Bmatrix}$$ It is more convenient for convention issues to use this version of the formula for $$\sigma _{xx}$$

Setting the moment about the z-axis to zero we are particularizing the to cross section of a symmetrical case

$$M_z=0\;$$

$$I_{yz}=0\Rightarrow D=I_yI_z$$

D being previously stated, $$K_y$$ and $$K_z$$ are as follows: $$K_y=\frac{1}{I_z}$$

$$K_y=0\;$$

$$K_z=\frac{1}{I_y}$$

The Shear flow for the particularize case is:

$$q(s)=-(K_yV_y-K_{yz}V_z)Q_z-(K_zV_z-K_{zy}V_y)Q_y\;$$

 $$Q_Z=\int_{A_s}^{}{ydA}\; \; \; \; \;

Q_y=\int_{A_s}^{}{zdA} $$