User:Eas4200c.f08.carbon.orear/HW6/MATLAB

Fri, 07 Nov 2008, 07:33:59 EST Dear class (and TAs):

As lectured in class on Mon, 3 Nov 08, part of the HW6 problem is on plate buckling as explained in my wiki page:

http://en.wikiversity.org/wiki/User:Eas4200c.f08/Plate_buckling

Read and report in HW6 the content of this wiki page (with proper reference to this page) and do the following plate buckling analysis on the NACA airfoil.

Refer to the conceptual figure of airfoil with the points B,E,H,F defined on the airfoil as done in my lecture p.22-1 and see also the figure in HW4 by Team Aero_(Eelman):

http://en.wikiversity.org/w/index.php?title=User:Eas4200c.f08.aero.e/Week_8&oldid=356604

Code Expansion
Add the following features to your matlab code.


 * compute the normal bending stress sigma_xx in the skin panels FB (upper) and EH (lower); identify which one is compressive and which one is tensile.

averageTop = -1.0401e+006 N/m^2

Top is compressive

averageBottom =  5.3318e+005 N/m^2


 * plot the stress sigma_xx as a function of the coordinate y of the airfoil for both skin panels FB and EH.








 * average the compressive stress sigma_xx to be compare with the compressive buckling stress to be computed below. Plot the average line of sigma_xx on the same plot of sigma_xx vs y for skin panels FB and EH.

averageTop = -1.0401e+006 N/m^2

averageBottom =  5.3318e+005 N/m^2

=Buckling Code=

Plate
Next, for your report on plate buckling as mentioned above and do the following:


 * replot the figure at the top of p.55 of the MIT lecture notes with clearly defined coordinate axes, plate dimensions, and simply-supported boundary conditions.


 * use matlab to plot the perspective view of the buckling shape of a simply-supported rectangular plate, taking the coefficient c_{mn}=1 in the formula for psi_{mn}. In addition, select a = 1.5 m, b = 1.0 m, and thus a/b = 1.5.


 * Plot two figures:




 * m = 1 and n = 1




 * m = 2 and n = 1


 * consider the factor with x as variable in the expression for u_z, i.e.,


 * $$ \sin ( \frac{m \pi x}{a} )$$


 * Find the period T of this function by considering the following equation:


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


 * Show that m is indeed the number of half wave-lengths.


 * use matlab to reproduce the figure of k_c versus a/b at the top of p.57 in the MIT lecture notes; use different line styles for different curves.



Airfoil

 * consider the NACA skin panel with compressive stress sigma_xx (either skin panel FB or skin panel EH) as a flat rectangular plate, with the width b along the y axis of the rectangular plate (as shown in the figure at the top of p.55 of the MIT lecture notes) being the length FB or EH of the skin panels (you can just add the length of the infinitesimal pieces of the skin in your matlab code to find the total skin panel width).


 * The thickness t_s of the skin is now the "height" h of the plate.


 * Consider a range of aspect ratio of the plate as represented by the ratio a/b, which is to vary between 0.5 and 2.


 * Find the values of critical buckling stress sigma_xx_critical for this range of aspect ratio, and plot sigma_xx_critical versus the aspect ratio a/b.




 * compare the compressive stress sigma_xx of the skin panel FB or EH with the critical buckling stress sigma_xx_critical found above for the same range of aspect ratio of the skin panel

The airfoil will not buckle for the values shown in this graph, and is safest beyond an a/b ~ 0.3.

=Clamped= Next, since the skin panel is supported by stringers and ribs (as explained in class), the boundary conditions can be neither purely simply-supported nor purely clamped; they are somewhere in between these two types of boundary conditions.

So for the rectangular plate with clamped boundary conditions, do the following:


 * use matlab to plot the perspective view of the buckling shape u_z of the clamped rectangular plate with aspect ratio a/b = 1.5 using the expression for u_z as shown in my wiki page (with reference to Timoshenko and Gere's book); use c=1.


 * use the excel file downloaded at the web address==


 * http://www.excelcalcs.com/repository/Strength/Buckling/Flat-plate-buckling.xls/
 * (Be sure to save this excel file right away, since you may not be able to download it again within one day.)


 * to compute the critical buckling stress for the same range of aspect ratio, i.e., from 0.5 to 2, and plot sigma_xx_critical versus a/b for clamped boundary conditions on the same plot as for sigma_xx_critical versus a/b for simply-supported boundary conditions.


 * now for each aspect ratio a/b, the true critical buckling stress for the aircraft skin panel would be bounded (below and above) by the critical stress for simply-supported plate and the critical stress for clamped plate.


 * Compare this range of critical buckling stress with the average compressive stress for the airfoil skin as obtained above, and draw some conclusion about the participation of the airfoil skin to resist bending.

Sincerely, Prof. Vu-Quoc

=Code=