User:Eas4200c.f08.blue.a/Lecture 39

dRefer to the course page for more information with regards to this lecture. Equation numbers referenced in this lecture come from the aforementioned page.

Plotting buckling shapes under shear stress:

Express $$\displaystyle\{C_{22},C_{13},C_{31},C_{33} \}$$ in terms of $$\displaystyle C_{11}$$ for $$\displaystyle \vartheta = 1.5$$ 1) Find $$\displaystyle \lambda$$ for $$\displaystyle \vartheta = 1.5$$ using equation (30).
 * $$\displaystyle \lambda = \left[\frac{\vartheta^4}{81 (1 + \vartheta^2)^4}\left\{1 + \frac{81}{625}+\frac{81}{25}\left(\frac{1 + \vartheta^2}{1 + 9 \vartheta^2}\right)^2+

\frac{81}{25}\left(\frac{1 + \vartheta^2}{9 + \vartheta^2}\right)^2\right\}\right]^{1/2}$$ 2) Evaluate $$\displaystyle \mathbf K_{5 \times 5}$$ numerically in equation (26).
 * $$\displaystyle \left[\begin{array}{lllll}\frac{\lambda (1 + \vartheta^2)^2}{\vartheta^2} &\frac{4}{9}&0&0&0\\\frac{4}{9}&\frac{16 \lambda (1 + \vartheta^2)^2}{\vartheta^2}

&- \frac{4}{5}&- \frac{4}{5}&\frac{36}{25}\\0&- \frac{4}{5}&\frac{\lambda (1 + 9 \vartheta^2)^2}{\vartheta^2} &0&0\\0&- \frac{4}{5}&0& \frac{\lambda (9 + \vartheta^2)^2}{\vartheta^2} &0\\0&\frac{36}{25}&0&0&\frac{\lambda (9 + 9 \vartheta^2)^2}{\vartheta^2}  \end{array} \right]  \left\{      \begin{array}{l}C_{11}\\C_{22}\\C_{13}\\C_{31}\\C_{33} \end{array}\right\} =  \left\{ \begin{array}{l}0\\0\\0\\0\\0 \end{array}  \right\}$$ 3) $$\displaystyle \mathbf K = [K_{ij}]$$
 * $$\displaystyle \begin{bmatrix} K_{22} & K_{23} & K_{24} & K_{25}\\K_{32} & K_{33} & K_{34} & K_{35}\\K_{42} & K_{43} & K_{44} & K_{45}\\K_{52} & K_{53}  & K_{54}  & K_{55} \end{bmatrix} \begin{Bmatrix}C_{22}\\C_{13}\\C_{31}\\C_{33} \end{Bmatrix} = \begin{Bmatrix} -\frac{4}{9}C_{11}\\0\\0\\0 \end{Bmatrix}$$

Solve for $$\displaystyle\{C_{22},C_{13},C_{31},C_{33} \}$$ in terms of $$\displaystyle C_{11}$$
 * $$\displaystyle \{C\}= [K]^{-1}\begin{Bmatrix} -\frac{4}{9}C_{11}\\0\\0\\0 \end{Bmatrix}$$
 * $$\displaystyle u_z = C_{11}\sin\left(\frac{\pi x}{a}\right)\sin\left(\frac{\pi y}{b}\right) + C_{22}\sin\left(\frac{2\pi x}{a}\right)\sin\left(\frac{2\pi y}{b}\right) + C_{13}\sin\left(\frac{\pi x}{a}\right)\sin\left(\frac{3\pi y}{b}\right) + C_{31}\sin\left(\frac{3\pi x}{a}\right)\sin\left(\frac{\pi y}{b}\right) + C_{33}\sin\left(\frac{3\pi x}{a}\right)\sin\left(\frac{3\pi y}{b}\right)$$

Set $$\displaystyle C_{11} = 1$$, plot $$\displaystyle u_z$$.

Results:
 * $$\displaystyle \lambda = 0.0288$$
 * $$\displaystyle \mathbf K = \begin{bmatrix}   2.1597  & -0.8000 &  -0.8000   & 1.4400\\

-0.8000 &  5.7706     &    0    &     0\\   -0.8000    &     0  &  1.6174    &     0\\    1.4400     &    0    &     0  & 10.9334\end{bmatrix}$$
 * $$\displaystyle \begin{Bmatrix}C_{22}\\C_{13}\\C_{31}\\C_{33} \end{Bmatrix}= \mathbf K^{-1}\begin{Bmatrix} -\frac{4}{9}\\0\\0\\0 \end{Bmatrix} = \begin{Bmatrix}  -0.3037\\

-0.0421\\  -0.1502\\    0.0400\end{Bmatrix}$$ Setting a = 3, b=2 (to keep the plate aspect ratio = 1.5).
 * $$\displaystyle u_z = \sin\left(\frac{\pi x}{3}\right)\sin\left(\frac{\pi y}{2}\right) -0.3037\sin\left(\frac{2\pi x}{3}\right)\sin\left(\frac{2\pi y}{2}\right) -0.0421\sin\left(\frac{\pi x}{3}\right)\sin\left(\frac{3\pi y}{2}\right) -0.1502\sin\left(\frac{3\pi x}{3}\right)\sin\left(\frac{\pi y}{2}\right) + 0.0400\sin\left(\frac{3\pi x}{3}\right)\sin\left(\frac{3\pi y}{2}\right)$$

Note: (continued from lecture on 12/1/08) Answer: in 2 parts Part 1: Equilibrium of isolated stringer Part 2: Closed cell, equilibrium of stringers Stringer 1:
 * $$\displaystyle \tilde{q_{12}} = \tilde{q_{31}} + \tilde{q_{41}} + q^{(1)}$$
 * $$\displaystyle \tilde{q_{31}} = 0$$
 * $$\displaystyle \Rightarrow \tilde{q_{12}} = \tilde{q_{41}} + q^{(1)}$$

Stringer 2:
 * $$\displaystyle \tilde{q_{24}} = \tilde{q_{12}} - \tilde{q_{23}}+q^{(2)}$$
 * $$\displaystyle \tilde{q_{23}} = 0$$
 * $$\displaystyle \Rightarrow \tilde{q_{24}} = \tilde{q_{12}} +q^{(2)}$$

Stringer 4:
 * $$\displaystyle \tilde{q_{41}} = \tilde{q_{24}} + \tilde{q_{34}} + q^{(4)} $$
 * $$\displaystyle \tilde{q_{34}} = 0$$
 * $$\displaystyle \tilde{q_{24}} = \tilde{q_{12}} +q^{(2)} = \tilde{q_{41}} + q^{(1)}+q^{(2)}$$
 * $$\displaystyle \Rightarrow \tilde{q_{41}} = \tilde{q_{41}}+q^{(1)}+q^{(2)}+q^{(4)}$$