User:Eas4200c.f08.vqcrew.c/Homework 5/mememe

MATLAB Airfoil Problem Continued
The MATLAB airfoil problem, continued for HW 5, consisted of two parts. Both parts dealt with analysis of the effects of bidirectional bending on an arbitrary 4-digit NACA airfoil. Part I refers to an idealized case where only the axial effects of the stringers are considered. Part 2 considers the "full-blown" case - the bending effects of the airfoil skin, spar webs at 1/4 and 3/4 chord, and the stringers are considered. Below is a list of given and researched values used during the problem:


 * $$ A_B = 2 x 10^{-4}\;m^{2}\;$$ - Cross sectional area of the stringer at point B
 * $$ A_E = 2 x 10^{-4}\;m^{2}\;$$ - Cross sectional area of the stringer at point E
 * $$ A_F = 1 x 10^{-4}\;m^{2}\;$$ - Cross sectional area of the stringer at point F
 * $$ A_H = 1 x 10^{-4}\;m^{2}\;$$ - Cross sectional area of the stringer at point H
 * $$ t_s = .002\;m\;$$ - Airfoil skin thickness
 * $$ t_w = .003\;m\;$$ - Spar web thickness
 * $$ t_{str} = .005\;m\;$$ - Stringer flange thickness
 * $$ M_y = -1250\;N\cdot m\;$$ - y-direction bending moment (compressive)
 * $$ M_z = 500\;N\cdot m\;$$ - z-direction bending moment (tensile)
 * $$ \sigma_{u} = 1860\;N/m\;$$ - Ultimate tensile stress for steel 300M

Part I: Ideal Case


Figure ## shows a generic airfoil with spars located at 1/4 and 3/4 chord. The points B, E, H, and F, representing the intersection of the spars with the airfoil contour, are labeled. For the purposes of Part I, these 4 points represent the location of the idealized stringers. The stringer cross-section is not considered, and each stringer is represented as a "point area" during the calculations. Below is the full MATLAB code for this part followed by a sample output run.


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!MATLAB Code Part I - Bending Effects, Only Considering Stringers
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!MATLAB Code Part I Output >> vqcrew_4str(.5,2,4,15,1000);
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Figuree ## is the output plot from the MATLAB code for Part I. Several features are notable. First, the solid black circles located at the spar-airfoil intersections represent each stringer. The blue dashed lines represent the airfoil skin and spar webs. The red dash-dotted line running across the airfoil represents the neutral axis, and the black crosshair is the centroid of the stringers. The solid red circle represents the location of maximum normal stress on the airfoil.

A notable result from the output are the ultimate bending moments for the supplied ultimate stress. The calculated moments are of opposite sign to the original given moments. This is clearly explained by the result of the maximum bending stress analysis for the original loading condition. The determined stress was negative, indicating a compressive stress. The provided ultimate stress was given as a tensile stress, and since the neutral axis and thus the point of maximum normal stress do not change for this portion of the problem, the signs on the moments must switch to produce a tensile stress at that point. This also explains why much smaller magnitude moments produce a stress in excess of the ultimate stress by an order of magnitude - the ultimate stress provided is for tensile stresses, not compressive.

Part II: Full-Blown Case


For part II, the entirety of the airfoil was considered, including the cross-section of the stringers. In order to generate this cross-section, the assumption was made that the stringers were manufactured by stamping from a metal sheet of consistent thickness $$t_{str}\;$$. The following equation was used to determine the length of one flange of each stringer:


 * $$L_{brack} = .5*(A_{str}/2)/t_{str}\;$$

The stringers were assumed to be flush with the inner airfoil contour and the spar web wall. One stringer would consist of two "L-shaped" brackets on either side of the spar. Since the airfoil is assumed to have curvature, the brackets will not be identical, and the assumed manufacturing process means the angle between the two flanges will not be 90 degrees.

Figures ## and ## shows a generic depiction at both the 1/4 chord spar and the 3/4 chord spar. The diagram of the 1/4 chord spar is more representative of the general stringer shape. The stringers for the 3/4 chord spar are presented differently, however. The result of the above equation for the two stringers at the 3/4 spar was a bracket length of .005 m, which is identical to the provided stringer thickness. If one goes by the geometry provided in Figure ##, this essentially provides a quadrilateral cross-section, which is what is drawn in Figure ## and also used in the code for Part II, which is provided below. After the code is a sample output run.


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!MATLAB Code Part II - Full-Blown NACA Airfoil
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!MATLAB Code Part II Output
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Figure ## depicts the full output plot of the Part II code. The airfoil skin and spar webs were plotted as solid blue lines. The stringer cross-sections are plotted as small black circles, with each portion of the stringer wall represented with 5 points for ease of viewing. Zoomed in plots of the 1/4 spar and 3/4 spar are provided in Figures ## and ## respectively. The neutral axis is again plotted as a red dash-dotted line.

There are two centroids plotted on this airfoil; a zoomed in view of them is available in Figure ##. The green crosshair represents the centroid of the stringers when the stringer cross-sections are considered. The blue crosshair is the crosshair calculated in Part I. Also plotted is a red solid circle again representing the location of maximum bending stress on the airfoil, and a green solid circle represents the location of maximum bending stress in a stringer. Comparison and conclusions about these results follows in the next section.

Comparison and Conclusions
Below is a table showing selected data from the output of both cases. The stringer centroids, locations of maximum bending stresses, and neutral axis angle for both cases are more or less identical, which is not surprising given both are based off of the same airfoil with the same input data. The most apparent differences are in the data for ultimate bending moments and maximum normal bending stress. The higher maximum stress at lower supplied moment indicates the full-blown airfoil is slightly less efficient at carrying bending loads than the idealized airfoil. This is probably due primarily to the stringers being much, much better at carrying bending loads than either the spars or the airfoil skin. The only other notable point from the analysis is the stringer which carries the most stress in the full-blown case, which was found to be stringer B. This is not surprising, as this stringer is closest to the position of maximum thickness, which in combination with the positive slope of the neutral axis means it would be the farthest stringer from the neutral axis, and thus experience the highest stress.


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 * bgcolor="red"|
 * bgcolor="#99ccff" style="text-align: center;" |Ideal Case
 * bgcolor="#99ccff" style="text-align: center;" |Full-Blown Case
 * bgcolor="red"|Stringer centroid (meters,meters)
 * bgcolor="#99ccff" style="text-align: center;" | (0.2083 0.0080)
 * bgcolor="#99ccff" style="text-align: center;" | (0.2088 0.0080)
 * bgcolor="red"|Moment of Inertia Tensor Components (m^4,m^4,m^4)
 * bgcolor="#99ccff" style="text-align: center;" | (6.6875e-007,3.4375e-005,9.3322e-007)
 * bgcolor="#99ccff" style="text-align: center;" | (6.2884e-007,1.4533e-004,4.8673e-006)
 * bgcolor="red"|Neutral Axis Slope (radians)
 * bgcolor="#99ccff" style="text-align: center;" | 0 .0345
 * bgcolor="#99ccff" style="text-align: center;" | 0.0347
 * bgcolor="red"|Maximum normal bending stress (Pascals)
 * bgcolor="#99ccff" style="text-align: center;" | -8.1995e+007 Pascals
 * bgcolor="#99ccff" style="text-align: center;" | -1.1641e+008 Pascals
 * bgcolor="red"|Location of maximum normal bending stress (meters,meters)
 * bgcolor="#99ccff" style="text-align: center;" | (0.1436 0.0467)
 * bgcolor="#99ccff" style="text-align: center;" | (0.1434 0.0478)
 * bgcolor="red"|Ultimate Bending Moments (N-m, N-m)
 * bgcolor="#99ccff" style="text-align: center;" | (28355,-11342)
 * bgcolor="#99ccff" style="text-align: center;" | (20970,-8388)
 * }
 * bgcolor="#99ccff" style="text-align: center;" | (0.1434 0.0478)
 * bgcolor="red"|Ultimate Bending Moments (N-m, N-m)
 * bgcolor="#99ccff" style="text-align: center;" | (28355,-11342)
 * bgcolor="#99ccff" style="text-align: center;" | (20970,-8388)
 * }
 * }
 * }