User:Eas4200c.f08.blue.a/Lecture 78

Continuing Torsional Analysis:

Dimensional analysis, from lecture 29, is applied to the differential stress formula from lecture 27 to show that the differential of stress over distance x has the dimensions of force over volume.

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

In equation 3.14 in Mechanics of aircraft structures by C.T. Sun, it is shown that the of partial derivatives of shear force in the x and y directions is equal to zero.

$$\frac{\partial \tau _{xz}}{\partial x}+\frac{\partial \tau _{yz}}{\partial y}=\frac{\partial \sigma _{xz}}{\partial x}+\frac{\partial \sigma _{yz}}{\partial y}=0$$

From the roadmap described in lecture 16, the Prandtl stress function $$\phi $$ can be found using the following formulas (Sun 3.15):

$$\sigma _{yx}=\frac{\delta \phi }{\delta z}$$ $$\sigma _{zx}=\frac{-\delta \phi }{\delta y}$$

As can be seen, the Prandtl stress function plays the role of a potential function. $$\left(\sigma _{yx},\sigma _{zx} \right)$$ is a component of the "gradient" of the Prandtl function with respect to y and z.

Recall that the gradient of a scalar function f(x,y,z) is simply the partial differential of f in each direction (x, y, and z) or (i, j, and k).

$$\vec{\bigtriangledown} f(x,y,z)=\frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial y}\hat{j}+\frac{\partial f}{\partial z}\hat{k} $$

In lecture 27, the class found that the sum of the partial derivative with respect to y of stress in the yx direction and the partial derivative with respect to z of the stress in the zx direction results in zero. Now we can replace the stresses with the Prandtl stress function equivalents.

$$\frac{\partial \sigma _{yx}}{\partial y}+\frac{\partial \sigma _{xz}}{\partial z}=0$$

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

$$\frac{\partial ^{2}\phi }{\partial y\partial z}-\frac{\partial ^{2}\phi }{\partial z\partial y}=0$$

Since $$\phi$$ is continuous and smooth then the second mixed derivative is interchangable.

Now the Prandtl stress function $$\phi (x,y)$$ can be introduced in terms of the shear stress as in equation 3.15 (Sun, 70).

$$\tau _{xz}=\frac{\partial \phi }{\partial y}$$ $$\tau _{yz}=-\frac{\partial \phi }{\partial x}$$

It follows that:

$$\gamma _{xz}=\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}=\frac{\partial w}{\partial x}-\theta y$$

$$\gamma _{yz}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}=\frac{\partial w}{\partial y}+\theta x$$

$$\frac{\partial \gamma _{yz}}{\partial x}-\frac{\partial \gamma _{xz}}{\partial y}=2\theta $$

$$\gamma _{yz}=\frac{1}{G}\tau _{yz}$$

$$\gamma _{xz}=\frac{1}{G}\tau _{xz}$$

$$\frac{\partial \tau _{yz}}{\partial x}-\frac{\partial \tau _{xz}}{\partial y}=2G\theta $$

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

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

This equation for the Prandtl Equation is equivalent to step F of the Torsional analysis Roadmap.

To find a function for the Prandtl Function, a two dimensional case is first looked at:



In the above image, a cross section of a non-uniform bar is shown with a force (t) pu.



This image represents the stress vector (t) on an infinitesimal surface of a non-uniform beam.