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Unformatted text preview: AAE 352
Summer 2009 Test 2 Name: DA Marni//" Hours spent preparing for this test (separate from normal reading or'homewOrk):
THE NUMBER OF POINTS ADDS UP TO 77. YOU HAVE 75 MINUTES (~ 1 point/minute). l. (3 pts) List three conditions that, when met, guarantee a unique structural solution
a. ﬁg {4/ b. wM/Qaf/ 5/ 0 (QM/Méﬁ wig/(2’
2. (2 pts) Which condition of the above do our stress functions (like Prandtl) automatically satisfy? Ezm 'A'v‘am 3. (3 pts) Given an arbitrary body with a boundary condition of 1),; = 0 , write that in terms of the torsion stress function phi (Prandtl). ﬂ 5 0
: —§Q ' i  I 22: ‘ dz _
4. (15 pts) Given the following cross sections; ﬁnd the torsion constant for each (J). Wewill'
assume that the ends are not constrained (warping is allowed to develop). Work on the 
following sheets, but copy your answers here. ALL WALLS HAVE UNIFORM THICKNESS . 0F 0.01 m. I a. Circle: X/a ‘3/‘44—  t
333 x/cf? 45* \/ (“3'05" 57; ‘f/ 0. Half circle: 2.5%? /0 I“! ’5}; 233 b. Ibeam:
12/
174" . V gr H’
d. Two half circles (=circle with cross bar): 575 ﬂy? M :,
F: 0.6m
d : £3723 d 5 (::Z:L:Z¢0Km*ii:ﬂ{ (92, ‘ zé/thi; (7 mm) + QM?) 31M]
Qzazé 5. (5 pts) Pretend that the best cross section was the I—beam (not true), J= 1E6 m4 (not true),
material is aluminum with G=27GPa (true). Build a torsion member 200 cm long, and apply a
w far will the bar rotate (degrees)? r>< :2! [Xv—12465. j, a. (3 pts) The answer above assumes that warping is not constrained. If it was, would the total rotation be more or less?
A 55 3 6. (5 pts) Primary warping is zero at point A in an Ibeam. Find the warping (in meters) at B. Use
the numbers I gave you in the preceding problems for any values needed. By symmetry, we
know that the center of twist is at the centroid. 0.2 m  . (3 pts) Draw a case where shear lag could be a problem. ,
We 5 flag Mm real/é? ﬂaw? WM m” / ﬁfe/W“ I
4/ 59M ﬁrm/ml?”
é? £91m»: fry/Mal
fir/“afar 7% mar/var Quart? 00""
. (3 pts) Draw or describe a beam which you could NOT analyze with the Euler assumptions for
a simple beam (remember, the simple beam includes both the sophomore and the bi—directional
variants). p / /J' T
% ZLA [413 nm» {<1
X . (5 pts) Given the following beam, ﬁnd the shear and moment at point C. Complete
shear/moment diagrams are not required. MD=75 N*m pz=5N/m 10. (5 pts) Given the following beam cross section and moment diagram, what are the maximum
tensile and compressive stresses? M = 300N*m M = —200N*m 20cm FM! 20 2 Z; A;
_ 214‘ a I
l :. OWN mama“) + ( 2 7* WM“, )6 3w...)
(zacdzw) + (#szem) " [Sam foam fie Ame29m_ 1 I] L IL, F'D’LO/ f _ 7 13 . '3
4/ /“3 5 (waif r (ZQMIZMN l§~’/(¢)L+ (T +(#cm){3(gw)(:9mmf'
— 3 ,., Jr ; 13,3 1' 5?60 7D $60 7“ ﬁZO — /é,¥xm cmlf(ﬂ%c_;) E 3 /6 7‘ Arm—5’4” :: ll. (3 pts) Write the con‘eet equation for stress for each of the following cases. There are no applied moments. 1A!”
a. Loading in 2 only, symmetric about the z axis. . .
M 3
G _: _£L_._. 3
I 5
1:). Loading in y only, nonsymmetric
Mg 20/ s 0 M
— I3 ? e I M;
. . I5"! 'r‘ «‘3 a “33%
c. Loadlng in y and z, symmetric about the z axis
I 15% :. (9 _
M .5 M
U. 3 3—4— 4" 5' {j :1 £13,? .f— % [j
"" II R 3 I2— I T"—
J J—g 12. (10 pts) The Boeing 787 is being designed. Let’s assume that the wing has uniform cross
section along its length (not true). The loading in ﬂight is shown below, along with the shear and moment diagrams. Find the equation for the deﬂection of the wing (w). Leave in terms of
E, I, and X. O p,=26,666 N/m L=30 m M=(p/2)(x2—2xL+L2) (Space for # 12) n"; can.. is 13. (7 pts) Assume that your answer is w(x) = L[—— +1] (not true). One designer wants the wing to have the angle at the tip of the wing be 30 degrees when loaded so that it will look
really cool. What Iy would you need to use to get that? Assume an isotropic composite with E=100 GPa. 30 deg a. (2) Assume that your answer from the Boeing problem above is I),=4X10'3 m4 (not true).
Assume that the cross section is symmetric about y and that the total height of the cross
section is 0.3 m. What is the maximum compressive stress? Is it on top or bottom of the wing? 5Q %/Of€ C9; 300 a7“ (/4? r}: 73 545122244, 55 ggoagcé/uj 1/ C,er 'z‘rfﬁ/I ﬂf’ 2’;Z /’ _______ HWMMM—H
)fo ". 46L,
4%?“ a. i 91:“; MM 3a“ : @352? 14. (5 pts) Given the shape below, ﬁnd 1y, 12, and Iyz. Use the webstringer assumption. Box your l i BIISWGIS . ((7 : 356?: __, x440) Hz, (on/173%.)” J
3/4 www.mwmmmﬂumm m (5 7' 7' 1‘ Awake; ; A: ( 436%...)3 Ab/wégzmﬁf 4M~ZMIL ; [567 MmbL :1 f] 7':
,4] 62,4?)(2, 34) + All (12,6?)0014) 7‘— 4; {mmww
ﬂ é Z'SW'mAL 7J— g.gm«wé “25,3 #1,”,‘1” ...
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 Fall '08
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