User:Eas4200c.f08.wiki.c/homework report 7

=Meeting 35=



Mean (Average) Value Theorem (MVT)

$$\int_{A}^{}{\bar{y}dA} = \bar{y}_{c}\int_{A}^{}{dA}$$

$$=\bar{y}A$$

$$\int_{A}^{}{\bar{z}dA} = \bar{z}_{c}A$$

With $$\bar{z}_{c}$$ denoting the average z value.

$$A = \sum_{i=1}^{4}{A_{i}}$$

(Neglect skin and spar webs)

This is a similar problem as was done in homework 5.

Equation 5.5 in the book is given as:

$$q(s) = (k_{yz}Q_{z}(s) - k_{z}Q_{y}(s))V_{z}$$

Since: $$1) V_{z} 2) k_{yz},k_{z} 3) Q_{z}, Q_{y}$$ are all concentrated in the stringers and are all independent of s, this implies that the shear flow q(s) is constant between 2 stringers.

However, q(s) would increment (jump) when crossing a stringer.

Now, to find q(s), the following recipe is given:

Step 1:

Find $$(\bar{y}_{c}, \bar{z}_{c})$$ by the following formulas:

$$\bar{y} = \sum{\frac{yA}{A}}$$

$$\bar{z} = \sum{\frac{zA}{A}}$$

Step 2:

Find $$I_y, I_z, I_{yz}$$

Step 3:

Find $$k_y, k_z, k_{yz}$$

Step 4:

Follow the path of 's' to find $$q_{12}, q_{23}, q_{34}$$

Where the subscripts denote the shear flow from one stringer to the next.

It should be noted that:

$$Q_{z}^{23} = y_{1}A_{1} + y_{2}A_{2}$$

and so on. In other words, each Q is the sum of the Q's before it.