User:Eas4200c.f08.wiki.f/11-24-08

Taking a Closer Look at Stringer 3


By using the Euler cut principal and cutting the webbing around the outside of the multi-celled "potato" we can analyze the shear flow around stringer 3.

$$\sum{F_x} = 0 = \int_{A_3}^{}{[\underbrace{\sigma_{xx}(x+dx)}_{[\tilde{q}_{23} - \tilde{q}_{43} + \tilde{q}_{31}]dx} - \sigma_{xx}(x)]dA_3}$$

Taylor Series: $$\frac{d\sigma_{xx}}{dx}dx + h.o.t. $$

$$\tilde{q}_{31} = \tilde{q}_{23} + \tilde{q}_{43} + q^{(3)}$$

$$q^{(3)} = -\int_{A_3}^{}{\frac{d\sigma_{xx}}{dx}dA_3}$$
 * q(3) is the contribution to the shear flow by stringer 3

Recall: $$V_y = \frac{dM_z}{dx},\;V_z = \frac{dM_y}{dx}$$

$$q^{(3)} = -(k_yV_y - k_{yz}V_z)Q_z^{(3)} - (k_zV_z - k_{yz}V_y)Q_y^{(3)}$$

$$Q_z^{(3)} = \int_{A_3}^{}{y dA_3}, \; Q_y^{(3)} = \int_{A_3}^{}{z dA_3}$$

Taking a Closer Look at Stringer 2
$$\tilde{q}_{23} = \underbrace{\tilde{q}_{12}}_{flow\; in} - \underbrace{\tilde{q}_{24}}_{flow\;out} + q^{(2)}$$
 * Notes:
 * $$\tilde{q}_{12} = \tilde{q}_{24} = 0 \; \mbox{because of the cuts}$$
 * flow into stringer (+)
 * flow out of stringer (-)

$$q^{(2)} = -(k_yV_y - k_{yz}V_z)Q_z^{(2)} - (k_zV_z - k_{yz}V_y)Q_y^{(2)}$$

$$Q_z^{(2)} = y_2A_2$$

$$Q_y^{(2)} = z_2A_2$$


 * y2 = y coordinate of stringer 2
 * z2 = z coordinate of stringer 2
 * A2 = Area of stringer 2

Taking a Closer Look at Stringer 4
$$\tilde{q}_{43} = \tilde{q}_{24} - \tilde{q}_{41} + q^{(4)}$$

$$q^{(4)} = -(k_yV_y - k_{yz}V_z)Q_z^{(4)} - (k_zV_z - k_{yz}V_y)Q_y^{(4)}$$

$$Q_z^{(4)} = y_4A_4$$

$$Q_y^{(4)} = z_4A_4$$

Superposition (Again)
$$q_{ij} = \tilde{q}_{ij} + q_k$$

$$q_{s2} = \underbrace{\tilde{q}_{12}}_{known} + \underbrace{q_1}_{unknown}$$

$$q_{23} = \tilde{q}_{23} + q_1$$

$$q_{31} = \tilde{q}_{31} + q_1$$

$$q_{24} = \underbrace{\tilde{q}_{24}}_{known} + \underbrace{q_2}_{unknown}$$

$$q_{43} = \tilde{q}_{43} + q_2$$

$$q_{41} = \tilde{q}_{41} + q_3$$

Solving Unknowns
Now there are 3 unknowns (q1, q21, and q3). We need 3 equations to solve them.


 * 1) Moment Equation: Take the moment of Vy, Vz, and {q12,...,q41} about and convenient point. (Usually this point where be where the lines of action of Vy and Vz intersect).


 * 1) $$\theta_1 = \theta_2$$


 * 1) $$\theta_2 = \theta_3$$