User:Eas4200c.f08.nine.s/Lecture 5

Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

 Group nine - Homework 5 

Stress-Strain Relation
$$\left\{\epsilon_{ij}\right\}_{6x1} = \begin{bmatrix}A_{3x3}&O_{3x3}\\ O_{3x3}&B_{3x3}\end{bmatrix}_{6x6}\left\{\sigma_{ij}\right\}_{6x1}$$   where the {6x6} determinant is denoted by  $$C$$

 $$\sigma-\epsilon$$  relation:

$$\left\{\epsilon_{ij}\right\}_{6x1} = \begin{bmatrix}A_{3x3}^{-1}&O_{3x3}\\ O_{3x3}&B_{3x3}^{-1}\end{bmatrix}_{6x6}\left\{\sigma_{ij}\right\}_{6x1}$$   where the {6x6} determinant is denoted by  $$C^{-1}$$

Verification of the Identity Matrix: $$C_{6x6}^{-1} C_{6x6} = I_{6x6}$$

$$I = \begin{bmatrix}A^{-1}A&0\\0&B^{-1}{B}\end{bmatrix}$$

$$\left\{\sigma_{ij} \right\} = \begin{bmatrix} \bar{A}^{-1} & 0 \\ 0 & \bar{B}^{-1} \end{bmatrix}\begin{Bmatrix} 0\\ 0\\ 0\\ 0\\ \epsilon_{31}\\ \epsilon_{12} \end{Bmatrix}$$

Stress Tensors

four zero stress components using this equation:

$$\sum_{i=1,j=1}^3\frac{\delta \sigma_{ij}}{\delta x_i}=0$$

gives:

$$\frac{\delta \sigma_{11}}{\delta x_{1}}+\frac{\delta \sigma_{12}}{\delta x_{2}}+\frac{\delta \sigma_{13}}{\delta x_{3}}=0$$

$$\frac{\delta \sigma_{21}}{\delta x_{1}}+\frac{\delta \sigma_{22}}{\delta x_{2}}+\frac{\delta \sigma_{23}}{\delta x_{3}}=0$$

$$\frac{\delta \sigma_{31}}{\delta x_{1}}+\frac{\delta \sigma_{32}}{\delta x_{2}}+\frac{\delta \sigma_{33}}{\delta x_{3}}=0$$

Bidirectional Bending




$$M_{y}=\int_{aA}^{}{z\sigma _{xx}dA}$$

$$M_{z}=\int_{aA}^{}{y\sigma _{xx}dA}$$

Moment of Inertia Tensors:

$$I_{yy}=\int_{A}^{}{z^{2}dA}$$

$$I_{zz}=\int_{A}^{}{y^{2}dA}$$

$$I_{yz}=\int_{A}^{}{zydA}$$

Hooke's Law

$$\sigma _{xx}=E\varepsilon _{xx}$$

$$\sigma=\frac{IyMz-IyzMy}{IyIz-(Iyz)^{2}}(y)+\frac{IzMy-IyzMz}{IyIz-(Iyz)^{2}}(z)$$

$$I=\begin{bmatrix}I_{11}&I_{12}&I_{13}\\|&I_{22}&I_{23}\\symm&-&I_{33}\end{bmatrix}$$

Nonuniform Stress Field
Nonuniform Stress Field in 3-D. $$\sum Fx=0=-\sigma(x)A+\sigma(x+dx)A+f(x)dx$$

0=A[ $$\sigma(x+dx)-\sigma(x)$$ ]+f(x)dx

$$\frac{d\sigma}{dx}+\frac{f(x)}{A}=0$$ were f(x) is the force per unit length and A is the applied load.

$$\sum{Fx}=0=dydz[-\sigma_{xx}(x,y,z)+\sigma_{xx}(x+dx,y,z)]+dzdx[-\sigma_{yx}(x,y,z)+\sigma_{yx}(x,y+dy,z)]

+dxdy[-\sigma_{zx}(x,y,z)+\sigma_{zx}(x,y,z+dz)]$$

$$0=(dxdydz)\left[\frac{\delta\sigma_{xx}}{\delta x}+\frac{\delta\sigma_{yx}}{\delta x}+\frac{\delta\sigma_{zx}}{\delta z}\right]$$

Similarly for the forces in the y direction:

$$\sum Fy =0= dy dz [-\sigma_{yx}(x,y,z)+\sigma_{yx}(x+dx),y,z)]+dzdx[-\sigma_{yy}(x,y,z)+\sigma_{yy}(x,y+dy,z)]+dxdy[\sigma_{yz}(x,y,z)+\sigma_{yz}(x,y,z+dz)]$$

which becomes:

$$0=(dx dy dz)\left[\frac{\delta \sigma_{yx}}{\delta_x}+\frac{\delta \sigma_{yy}}{\delta_y}+\frac{\delta \sigma_{yz}}{\delta_z}\right]$$

Z direction:

$$\sum Fy =0= dy dz [-\sigma_{zx}(x,y,z)+\sigma_{zx}(x+dx),y,z)]+dzdx[-\sigma_{zy}(x,y,z)+\sigma_{zy}(x,y+dy,z)]+dxdy[\sigma_{zz}(x,y,z)+\sigma_{zz}(x,y,z+dz)]$$

which becomes:

$$0=(dx dy dz) \left[\frac{\delta \sigma_{zx}}{\delta_x}+\frac{\delta \sigma_{zy}}{\delta_y}+\frac{\delta \sigma_{zz}}{\delta_z}\right]$$

Dimensional Analysis
$$[F] = \frac{F}{L} = \frac{force}{lenght} $$

$$[ \frac{F}{A}] = \frac{F}{L^3}$$

$$[A] = L^2 $$

$$[\sigma] = \frac{f}{L^2} \rightarrow [\frac{d \sigma}{dx}] = \frac{[F/L^2]}{[L]} = MLT^{-2}L^{-3} = ML^{-2}T^{-2} $$

$$[dx] = L $$

$$ [EI] = \frac{F}{L^3} \times L^4 = FL^2 $$

$$ [D] = [E][h^3]= \frac {F}{L^2} \times L^3 = FL $$

$$ \epsilon = \frac{du}{dx} = \frac{\Delta L}{L} \rightarrow [\epsilon] = \frac {[du]}{[dx]} = \frac {L}{L} = 1 $$

$$ \nu = \frac{\epsilon_{yy}}{\epsilon_{xx}} \rightarrow [\nu] = \frac{[\epsilon_{yy}]}{[\epsilon_{xx}]} = 1 $$

Sample Run of Code (NACA Plot)
NACA Airfoil calculation program Enter first digit of airfoil: 2 Enter second digit: 4 Enter the third and fourth digits: 15 Enter Py: 0 Enter Pz: 0 Enter number of segments: 60

The average area is: 0.103 The minumum number segments required to have the average area accurate within 1 percent is: 24.000

Figure 1 shows the cross-section of the NACA airfoil and the centroid line

Contributing Team Members
The following students contributed to this report:

Felix Izquierdo Eas4200c.f08.nine.F 18:34, 6 November 2008 (UTC) Ricardo Albuquerque Eas4200c.f08.nine.R 18:39, 6 November 2008 (UTC) Dave Phillips Eas4200c.f08.nine.D 18:46, 6 November 2008 (UTC) Stephen Featherman Eas4200c.f08.nine.S 18:49, 6 November 2008 (UTC)