User:Eas4200c.f08.carbon.orear/HW7/MATLAB

=MATLAB Code=

Mon, 24 Nov 2008, 07:21:38 EST Dear class (and TAs):

Again, the goal of HW7 is the NACA 2415 airfoil. This time, you will analyze the NACA airfoil to find the shear flows in its panels (skin and spar webs) when the airfoil is subjected to transverse shear forces in both y and z directions, as a result of the drag and the lift forces on the airfoil.

See for example a figure of the 3-cell NACA airfoil with all defined points by Team Aero_Eelman at:

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

Now, consider the following shear forces:


 * horizontal shear force $$V_y = 10000 N$$, applied at point L (leading edge)


 * transverse shear force $$V_z = 5000 N$$, applied at point B

=Part I: Single-cell NACA airfoil=

all components of the moment of inertia tensor: I_22, I_33, I_23
 * I_22 = 6.3913e-007
 * I_33 = 8.3333e-006
 * I_23 = -7.6055e-008

Stringer Centroid = (0, 0.2083 , 0.0081) m


 * Area_B = 0.0002 m2
 * Area_E = 0.0002 m2
 * Area_F = 0.0001 m2
 * Area_H = 0.0001 m2

Stringer locations
 * B =    0    0.1250    0.0458
 * F =    0    0.3750    0.0263
 * H =    0    0.3750   -0.0132
 * E =    0    0.1250   -0.0285

In your shear flow analysis, using the method explained in class(see also the examples in the book), solve problem P2 by using the following single cut:


 * cut off the skin at the trailing edge point T

The path "s" to use in your solution of problem P2 is the same path of the shear flow as shown in red in the above figure, starting from the trailing edge point T, following the upper skin to the leading edge point L, then wrap around the airfoil to the lower skin to end at the trailing edge.

For the moment equation, take the moment of all shear forces and shear flows about point D on the spar web BDE.

Find the final true shear flow in each skin panel.



$$q'_{TF} = 0$$

$$q'_{FB} = q'_{TF} + q'_{F} = 0 + (V_z * Q_fy)/(I_y) + (V_y * Q_fz)/(I_z) = -(10000N * 0.0001 m2 *(0.2083-0.1250))/(6.3913e-007) - (5000N *  0.0001 m2 * (0.0458 - 0.0081))/( 8.3333e-006) = -1.3260e+005$$

$$q'_{FB} = -1.3260e+005$$

$$q'_{BL} = q'_{FB} + q'_{B} = -1.3260e+005 + (V_z * Q_by)/(I_y) + (V_y * Q_bz)/(I_z) = -(10000N * 0.0002 m2 *(0.3750-0.2083))/(6.3913e-007) - (5000N *  0.0002 m2 * (0.0263 - 0.0081))/( 8.3333e-006) = -1.3260e+005 +  -5.2383e+005 =  -6.5643e+005$$

$$q'_{LE} = q'_{BL} = -6.5643e+005$$

$$q'_{EH} = q'_{LE} + q'_{E} = -6.5643e+005 + (V_z * Q_by)/(I_y) + (V_y * Q_bz)/(I_z) = -(10000N * 0.0002 m2 *(0.3750-0.2083))/(6.3913e-007) - (5000N *  0.0002 m2 * (0.0081 + 0.0132))/( 8.3333e-006) = -6.5643e+005 + + 5.2420e+005 = -1.3223e+005$$

$$q'_{HT} = q'_{EH} + q'_{H} = -1.3223e+005 + (V_z * Q_by)/(I_y) + (V_y * Q_bz)/(I_z) = -(10000N * 0.0001 m2 *(0.2083-0.1250))/(6.3913e-007) - (5000N *  0.0001 m2 * (0.0081 + 0.0285))/( 8.3333e-006) = -1.3223e+005 + 1.3253e+005 = 0.0030+005$$

We see that this result will be the same if we insert more cells but make cuts in those cells! The only change will occur when those hypothetical cuts are replaced by the complete skin.

=Part II: 3-cell NACA airfoil=

In your shear flow analysis, using the method explained in class (see also the examples in the book), solve problem P2 by using the following triple cuts:


 * cut off the skin at the trailing edge point T


 * cut off the spar web FH


 * cut off the spar web BE

The path "s" to use in your solution of problem P2 is the same path of the shear flow as shown in red in the above figure, starting from the trailing edge point T, following the upper skin to the leading edge point L, then wrap around the airfoil to the lower skin to end at the trailing edge.

For the moment equation, take the moment of all shear forces and shear flows about point D on the spar web BDE.

Find the final true shear flow in each skin panel and spar web.



$$q'_{TF} = 0$$

$$q'_{FB} = q'_{TF} + q'_{F} = 0 + (V_z * Q_fy)/(I_y) + (V_y * Q_fz)/(I_z) = -(10000N * 0.0001 m2 *(0.2083-0.1250))/(6.3913e-007) - (5000N *  0.0001 m2 * (0.0458 - 0.0081))/( 8.3333e-006) = -1.3260e+005$$

$$q'_{FB} = -1.3260e+005$$

$$q'_{BL} = q'_{FB} + q'_{B} = -1.3260e+005 + (V_z * Q_by)/(I_y) + (V_y * Q_bz)/(I_z) = -(10000N * 0.0002 m2 *(0.3750-0.2083))/(6.3913e-007) - (5000N *  0.0002 m2 * (0.0263 - 0.0081))/( 8.3333e-006) = -1.3260e+005 +  -5.2383e+005 =  -6.5643e+005$$

$$q'_{LE} = q'_{BL} = -6.5643e+005$$

$$q'_{EH} = q'_{LE} + q'_{E} = -6.5643e+005 + (V_z * Q_by)/(I_y) + (V_y * Q_bz)/(I_z) = -(10000N * 0.0002 m2 *(0.3750-0.2083))/(6.3913e-007) - (5000N *  0.0002 m2 * (0.0081 + 0.0132))/( 8.3333e-006) = -6.5643e+005 + + 5.2420e+005 = -1.3223e+005$$

$$q'_{HT} = q'_{EH} + q'_{H} = -1.3223e+005 + (V_z * Q_by)/(I_y) + (V_y * Q_bz)/(I_z) = -(10000N * 0.0001 m2 *(0.2083-0.1250))/(6.3913e-007) - (5000N *  0.0001 m2 * (0.0081 + 0.0285))/( 8.3333e-006) = -1.3223e+005 + 1.3253e+005 = 0.0030+005$$

Plot the perspective figure of the shear buckling shape u_z using 5 buckling mode shapes for the plate aspect ratio a/b = 1.5 as explained in class on Fri, 5 Dec 08. The derivation of the transverse buckling shape u_z and the perspective plot form an important part of HW7.

=Buckling analysis= Buckling analysis of skin panels FB and EH due to shear flows.


 * refer to my plate buckling wiki page at http://en.wikiversity.org/wiki/User:Eas4200c.f08/Plate_buckling, particularly the section on shear loading. Do the following:


 * consider Eq.(27); take the determinant of matrix K (2x2), and verify Eq.(28) for lambda


 * compute the lowest critical buckling stress $$(\sigma_{xy})_{cr}^{2,\star}$$ shown in Eq.(29) for a range of aspect ratio a/b from 0.5 to 2.


 * compute the more accurate critical buckling stress $$(\sigma_{xy})_{cr}^{5,\star}$$ shown in Eq.(31) for the same range of aspect ratio a/b from 0.5 to 2.


 * Plot the values of $$(\sigma_{xy})_{cr}^{2,\star}$$ and $$(\sigma_{xy})_{cr}^{5,\star}$$ versus the aspect ratio a/b from 0.5 to 2.


 * the above is for simply-supported plate; for clamped plate, report on the same plot above the 3 points corresponding to K = 1, 2, infinity. For K=infinity, also report the experimental data point with K=4.1.

For each skin panel (FB or EH), do the following:


 * find the curvilinear length of the skin panel by adding the infinitesimal length as you integrate along the skin panel (you may have done this computation in HW6, but verify to weed out possible theoretical or coding bugs)


 * compute the shear stress from the shear flow obtained for the skin panel, and compare the skin panel shear stress to the theoretical buckling shear stress $$(\sigma_{xy})_{cr}^{5,\star}$$ and draw some conclusion.