User:Eas4200c.f08.vqcrew.c/Homework 6/my

other problem

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!HW Problem - Proof of Zero Warping for Solid Circular Bar Consider the following two relations for the shear strains $$\gamma_{yx}\;$$ and $$\gamma_{zx}\;$$:
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 * $$\gamma_{yx} = \frac{\sigma_{yx}}{G} = \frac{\partial u_x(x,y)}{\partial y} - \theta_z$$


 * $$\gamma_{zx} = \frac{\sigma_{zx}}{G} = \frac{\partial u_x(x,y)}{\partial z} + \theta_y$$

Note also the following two relations for the shear stresses $$\sigma_{yx}\;$$ and $$\sigma_{zx}\;$$:


 * $$\sigma_{yx} = -G \theta_z\;$$


 * $$\sigma_{zx} = G \theta_y\;$$

Substituting the shear stress equations into the shear strain equations yields the following relationships:


 * $$\frac{-G \theta_z}{G} = -\theta_z = \frac{\partial u_x(x,y)}{\partial y} - \theta_z$$


 * $$\frac{G \theta_y}{G} = \theta_y = \frac{\partial u_x(x,y)}{\partial z} + \theta_y$$

By moving the $$\theta\;$$ from the left hand side to the right hand side of the above relations, the following conditions are immediately apparent:


 * $$\frac{\partial u_x(x,y)}{\partial y} = 0$$


 * $$\frac{\partial u_x(x,y)}{\partial z} = 0$$

The combination of these two conditions results in the overall condition $$u_x(y,z) = c\;$$, where $$c\;$$ is a constant that is not necessarily 0. In other words, the displacement in the x plane is not dependent on y or z; thus, there is no warping for a uniform bar with a solid circular cross-section.
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Read and Report - Plate Buckling Page
The plate buckling analysis page is split into two main sections: one for simply-supported boundary conditions and one for clamped boundary conditions.

The first portion of the simply-supported section considers the buckling mode shape relation $$\psi(x,y)\;$$, followed by a a relation for the out-of-plane displacement due to buckling. These relations describe the physical response of a "flat plate" to buckling stresses. The section further describes the critical buckling load $$(P_x)_{cr}\;$$, followed by a discussion of the dimensions of the quantities involved. This information can be used to determine the critical buckling stress $$(\sigma_{xx})_{cr}\;$$, which can be used to characterize the plate's resistance to buckling loads.

The second portion consists of essentially the salient equations from the simply-supported section, but adjusted for the different boundary conditions.

MATLAB Coding Portion

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!Full MATLAB Code - Buckling Analysis
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!Command Window Output for MATLAB code
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The above tables contain the full MATLAB code for buckling analysis as well as the command window output for a run of the code using the parameters listed in | Homework 5. The first part of the question requested a graph of of normal bending stress in the panels BF and EH (as described in Figure ##), with another curve showing the average compressive stress in the appropriate panel. This graph is shown in Figure ##. The red curve represents the stress in panel BF, the blue curve panel EH, and the average stress by the green curve. It is readily apparent from this plot that the compressive stress is experienced by panel BF.

Next, the problem requests a replotting of a figure from the MIT OpenCourseWare notes relating to the loading and support conditions on a flat plat; this is provided in Figure ##. The buckling mode shapes for the cases of $$m\;=\;1\;,\;n\;=\;1$$ and $$m\;=\;2\;,\;n\;=\;1$$ are plotted in Figures ## and ## respectively. The next portion of the question refers to determining the period of a portion the buckling mode shape function; that analysis is provided below.

Determining Period of Buckling Mode Shape Function
Consider the term of the $$\psi (x,y)\;$$ function represented as $$\;\sin ( \frac{m \pi x}{a} )$$. In order to find the period of this function, the following relationship is useful:


 * $$\sin ( \frac{m \pi (x + T)}{a} ) = \sin ( \frac{m \pi x}{a} )\;$$

Rudimentary algebra makes it immediately obvious that the following is also true:


 * $$\sin ( \frac{m \pi x }{a} + \frac{m \pi T}{a} ) = \sin ( \frac{m \pi x}{a} )\;$$

Noting the trigonometric relationship that $$\sin(\theta + 2\pi) = \sin(\theta) \;$$, the following can be written:


 * $$\frac{m\pi T}{a} = 2\pi\;$$

Thus, an equation for the period of this function is immediately apparent:


 * $$T = \frac{2a}{m}\;$$

For $$m\;=\;1\;$$, the period of this function is $$2\;a$$, indicating one half-wavelength present in the plate. For $$m\;=\;2\;$$, the period is $$a\;$$, indicating the presence of two half-wavelengths, or one full wavelength. Finally, for $$m\;=\;3\;$$, the period is $$\frac{2a}{3}$$, indicating the presence of 3 half-wavelengths, or 1.5 full wavelengths. Thus, it is apparent that $$\;m$$ is the number of half-wavelengths.

MATLAB Coding Portion Continued
The problem further requests the replotting of another figure from the MIT notes; this time, the $$k_c\;$$ vs. $$a/b\;$$ curves for various values of m. This plot is provided in Figure ##. Each curve refers to a different value of m; $$m\;=\;1$$ is plotted as a solid blue line, $$m\;=\;2$$ as a red dotted line, $$m\;=\;3$$ as a green dash-dotted line, and $$m\;=\;4$$ as a magenta dashed line.

The final portion of the question referred to an analysis of the airfoil skin panel as a flat plate, such as the type shown in Figure ##. The height of this idealized plate will be equivalent to the airfoil's skin thickness, and the length $$b\;$$ equivalent to the length of the airfoil skin panel. The skin material was considered to be Aluminum 2024-T3, as enumerated in a separate email about the problem. Using this idealized plate, the critical buckling stress $$(\sigma_{xx})_{cr}\;$$ for both simply supported and clamped boundary conditions was calculated for a range of aspect ratio such that $$(a/b) \in [.5,2]\;$$, and plotted in Figure ##. The red curve is for the clamped conditions, and the blue curve for the simply supported. Also, the buckling mode shape for the clamped case was also determined, and is provided in Figure ##.

One important note about the determination of critical buckling stress for the simply-supported conditions relates to the determination of $$m\;$$ for the calculations. There was no condition enumerated in any communication on the problem about this issue; thus, the assumption of using the $$m\;$$ corresponding to the minimum $$k_c\;$$ at a given aspect ratio would be used. Thus, for aspect ratios less than $$1.414\;$$, $$m\;=\;1$$ was used; otherwise, $$m\;=\;2$$ was used.

When comparing the previously calculated average compressive stress to these two critical buckling stress curves, when noting that the actual support condition is somewhere "inbetween" the two conditions considered, it is apparent that the airfoil skin is not particularly suited for bending resistance. The average compressive stress was approximately $$5.9x10^7\;$$ Pascals. From the critical stress graph, for any aspect ratio greater than approximately .75, this would be a higher stress than could be handled by the simply supported condition, which indicates the airfoil skin would experience buckling failure under those conditions. Only very small aspect ratio panels would be able to handle that kind of loading. Thus, it is apparent that the airfoil skin is not useful from the standpoint of bending resistance.