User:Eas4200c.f08.wiki.c/homework report 6

=Meeting 30=

The torsional analysis previously started is continued by looking at the dimensions of the derivative of stress:

$$[\frac{\partial \sigma _{ij}}{\partial x_{i}}] = \frac{F}{L^3} = \frac{force}{volume}$$

Recalling - D. Prandtl stress function - from the road map previously constructed in class we see that:

$$\sigma _{yx} = \frac{\partial \phi }{\partial z}$$

and

 $$ \sigma _{zx} = -\frac{\partial \phi }{\partial y}$$

where $$\phi$$ plays the role of a potential function and $$(\sigma _{yx}, \sigma _{zx}$$ are composed of the "gradient" of $$\phi$$ with respect to (y,z).

As a review of the gradient operators, recall that for scalar functions f(x,y,z):  $$\bigtriangledown f(x,y,z) = \frac{\partial f}{\partial x}\vec{i} +\frac{\partial f}{\partial y}\vec{j} +\frac{\partial f}{\partial z}\vec{k}$$

Where f is the potential function.

Now, substituting the values for $$\sigma _{yx}$$ and $$\sigma _{zx}$$ into the previously given Laplace's Equation we get:

$$\frac{\partial }{\partial y}(\frac{\partial \phi}{\partial z}) + \frac{\partial }{\partial z}(-\frac{\partial \phi}{\partial y}) = 0$$

Which can be rearranged to show that:  $$\frac{\partial ^2 \phi}{\partial y \partial x} = \frac{\partial ^2 \phi}{\partial z \partial y}$$

Due to $$\phi$$ being a continuous and smooth function, the second derivatives are interchangeable in the order they are performed.

Likewise:

$$\frac{\partial ^2 \phi}{\partial y^2} + \frac{\partial ^2 \phi}{\partial z^2} = -2G\theta $$

The lefthand side can also be written as $$\bigtriangledown ^2 \phi$$ which is the Laplacian of $$\phi$$.

$$[t]_{3x1} = [\sigma]_{3x3}[n]_{3x1}$$

Where t is the component of the traction force t, $$\sigma$$ is the component of the stress tensor, and n is the component of the normal vector $$\vec{n}$$

A graphical depiction of the 2-D case is shown below:



=MATLAB=



Above is a graph of $$\sigma_{xx}$$ at all y positions in the middle wing section on the top skin. Due to the graph being concave up, it can be seen that the load is compressive on the top skin of the wing section.



Above is a graph of $$\sigma_{xx}$$ at all y positions in the middle wing section on the bottom skin. Due to the graph being concave down, it can be seen that the load is tensile on the top skin of the wing section.

The average compressive stress was found to be 1.4423e+008.