User:Eas4200c.f08.blue.a/Lecture 32



Fig.1 to the right shows an unsymmetric thin-walled section. $$\displaystyle y_c$$ and $$\displaystyle z_c$$ are measured from the centroid of the shaded area. As labeled in the figure, the thickness $$\displaystyle t_s$$ can be a function of s, the curvilinear coordinate.

A general equation for thin-walled sections is the following (5.1):
 * $$\iint_{A_s} \frac{d\sigma_{xx}}{dx} dA = -q_s$$

A comparison of equations for symmetric and unsymmetric sections: (Note: in the following symmetric cases, the section is symmetric about the y-axis)

Symmetric: $$\displaystyle \sigma_{xx} = \frac{M_yz}{I_y}$$

Unsymmetric: $$\displaystyle \sigma_{xx} = (k_yM_z - k_{yz}M_y)y + (k_zM_y - k_{yz}M_z)z$$

where
 * $$\displaystyle k_y = \frac {I_y}{D}$$
 * $$\displaystyle k_{yz} = \frac {I_{yz}}{D}$$
 * $$\displaystyle k_z = \frac {I_z}{D}$$
 * $$\displaystyle D = I_yI_z - I_{yz}^2$$

The above equation for an unsymmetric section can be rewritten in matrix form to become
 * $$\displaystyle \sigma_{xx} = \begin{bmatrix}

y & z \end{bmatrix} \begin{bmatrix} k_y & -k_{yz} \\ -k_{yz} & k_z \end{bmatrix} \begin{Bmatrix} M_z \\ M_y \end{Bmatrix} = \begin{bmatrix} z & y \end{bmatrix} \begin{bmatrix} k_z & -k_{yz} \\ -k_{yz} & k_y \end{bmatrix} \begin{Bmatrix} M_y \\ M_z \end{Bmatrix} $$

The equation for an unsymmetric section can also be particularized to become the equation for a symmetric section (considering $$\displaystyle M_z$$ to be equal to zero). With a symmetric section, $$\displaystyle I_{yz}$$ is equal to zero.


 * $$\displaystyle M_z = 0 $$
 * $$\displaystyle I_{yz} = 0 $$
 * $$\displaystyle\therefore D = I_yI_z$$
 * $$\displaystyle k_y = \frac {I_y}{I_yI_z} = \frac {1}{I_z}$$
 * $$\displaystyle k_{yz} = \frac {I_{yz}}{I_yI_z} = 0$$
 * $$\displaystyle k_z = \frac {I_z}{I_yI_z} = \frac {1}{I_y}$$
 * $$\displaystyle\therefore \sigma_{xx} = (k_yM_z - k_{yz}M_y)y + (k_zM_y - k_{yz}M_z)z = (0-0)y + (k_zM_y - 0)z = \frac{M_yz}{I_y}$$

Continuing with our comparisons of equations between symmetric and unsymmetric sections:

Symmetric: $$\displaystyle q(s) = -\frac {V_zQ_y}{I_y}$$
 * $$Q_y = \int_{A_s} z\,dA = z_cA_s$$

Unsymmetric: $$\displaystyle q(s) = -(k_yV_y - k_{yz}V_z)Q_z - (k_zV_z - k_{yz}V_y)Q_y$$
 * $$Q_y = \int_{A_s} z\,dA$$
 * $$Q_z = \int_{A_s} y\,dA$$

The above equation for an unsymmetric section can be rewritten in matrix form to become
 * $$\displaystyle q(s) = -\begin{bmatrix}

Q_z & Q_y \end{bmatrix} \begin{bmatrix} k_y & -k_{yz} \\ -k_{yz} & k_z \end{bmatrix} \begin{Bmatrix} V_y \\ V_z \end{Bmatrix} = -\begin{bmatrix} Q_y & Q_z \end{bmatrix} \begin{bmatrix} k_z & -k_{yz} \\ -k_{yz} & k_y \end{bmatrix} \begin{Bmatrix} V_z \\ V_y \end{Bmatrix} $$

As with the equation for $$\displaystyle \sigma_{xx}$$, the equation for $$\displaystyle q(s)$$ with an unsymmetric section can be particularized to become the equation with a symmetric section. Similar to the case for $$\displaystyle I_{yz}$$ proven above, $$\displaystyle Q_z$$ is equal to zero due to symmetry about the y-axis. Again, $$\displaystyle I_{yz}$$ is also equal to zero.
 * $$\displaystyle Q_z = \int_{A_s} y\,dA = 0$$
 * $$\displaystyle k_y = \frac {I_y}{I_yI_z} = \frac {1}{I_z}$$
 * $$\displaystyle k_{yz} = \frac {I_{yz}}{I_yI_z} = 0$$
 * $$\displaystyle k_z = \frac {I_z}{I_yI_z} = \frac {1}{I_y}$$
 * $$\displaystyle \therefore q(s) = -(k_yV_y - k_{yz}V_z)Q_z - (k_zV_z - k_{yz}V_y)Q_y = -(k_yV_y-0)0 - (k_zV_z - 0)Q_y = -\frac {V_zQ_y}{I_y}$$

When dealing with stringer-web sections, it is assumed that the thickness $$\displaystyle t$$ of the skin and spar web is very small, therefore their areas are neglected in the computation of $$\displaystyle I_y, I_z, I_{yz}, Q_y, Q_z$$. The only areas used are the areas of the stringers (as shown in Fig. 2 to the left).

With a section that is symmetric about the y-axis, the areas will be set such that $$\displaystyle A_3 = A_2$$ and $$\displaystyle A_4 = A_1$$. However, for the unsymmetric case, the areas will be set such that $$\displaystyle A_1 = A,\,\, A_2 = 2A,\,\, A_3 = 3A,\,\, A_4 = 4A$$.