On the Torsion of Wings Citation Zahorski, Adam Teodor (1931) On the Torsion of Wings. Master's thesis, California Institute of Technology. doi:10.7907/E0T0-P929. https://resolver.caltech.edu/CaltechETD:etd-04172008-075520 Abstract Methods of investigation of pure torsion differ in many ways from the general method of procedure for other types of stresses. First it is very difficult to subject a body to a pure torsion without superposing bending moments; second the exact determination of twisting moment is in most cases very doubtful; third, a body subjected to torsional shearing stresses in general does not deform according to a simple law, but this deformation is usually a function of the geometry of the body. Item Type: Thesis (Master's thesis) Subject Keywords: Aeronautical Engineering Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Group: GALCIT Defense Date: 1 January 1931 Record Number: CaltechETD:etd-04172008-075520 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04172008-075520 DOI: 10.7907/E0T0-P929 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1406 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 23 Apr 2008 Last Modified: 03 Oct 2019 23:07 Thesis Files Preview PDF (Zahorski_at_1931.pdf) - Final Version See Usage Policy. 573kB Repository Staff Only: item control page

AT OMIT MATA or oepparpect Dy “ a a 4-4 Aj P, Zacoresi Saltiliment of the Reguireuents he te fad for the Decree of Easter of: Science ae) in Aeronautics <2 » io te Fy ae) Ff pa in} fad a) 9) oF ee . coh ~ ct ra) na 1931, FOREWORD Methods of investigation of pure torsion differ in many ways from the general method of procedure for other types of stresses, First it is very difficult to subject a bo- dy to a pure torsion without superposing bending moments; sec- ond the exact determination of twisting moment is in most cases very doubtful; third, a body subjected to torsional shearing stresses in general does not deform according to a simple law, out this deformation is usually a function of the geometry of the body. It is convenient to study the torsion of iso- tropic bodies under'‘three different types: a) Rodies of solid section where the stresses are distributed throughout the area in accordance with a certzin law. b) Ecdies of cvlindrical shell type where the stress- es are a function of the boundary only, or in other words the thickness of the shell is small in comparison with the other dimensions, so thet the assumption of uniform stress distribution over the thickness of the shell is reason:bly justified, c) Built-un sections where the individual parts / of boot) can be analised separately anu by tae lav suyerposition or the methou of least work the stresses and the deformations of the «hole bo- ad: can be determined. One of the probiems of designing is to determine the rigidity of the given structure and at what approximate load failure will occur. igxperiments have shown that in the type b) and ec) we have two distinct cases: First, it has been found that up to a certain loading the geometrical characteristics of these bodies are not changed and the deformations remain small, and when additional load is apvlied a sudden iner- ease of deformation is observed accompanied by buckling. This state is generally called the first stability limit, Second, the other limiting load can be defined as that at which permanent deformation remeins after the load has been removed, The above leads us directly to the problem of an elastic stability which has not yet been solved, Lack of the exact theory by which we can predict even the first stability limit prevents an engineer from predicting at whet load failure of the structure may occur, In the type a) it is safe to assume that the failure gene- rally will occur when the shearing stresses have reuched a certain value which is characteristic of the given matei- ial, but in the types b) and ¢c) stability is the primary cause of failure, The aim of this paper is to sive to the desigh- er semi-empirical formulae by which he can judge the rela- tive magnitude of the stresses and defornations of vari-~ Jus types of airfoils under the assumption that the tor~ sional stresses are taken by the skin alone and to show that this assumption is reasonably justified and that the order of magnitude of the computed values for deformation agrees.with the order of magnitudes of experimental data. Furthermore, it should be noted that this anelysis holds only up to the first stability linit, It has been shown by J.R.VWheatley ( “.A.C.A. Technical Note + 366) that the best method of analysis is Prandtl's Membrane Analogy which gives the formula: Str (1) Hurthermore the angle of twist in radians is given by: o= 2 E (2) where S is the unit shearing stress; t the thickness of the shell, M the torsional moment, Anthe total area of the cross section, 8 the angle of twist in radians per length L, G the shear mogulus and P the total perimeter of the section, It can be seen from equation (1) that the stress is a function of the area, and from equation (2) that 9 is a function of the area and the perimeter, In order to simplify engineering computations in the case of airfoils the two variables A and P can be expressed in terms of different parameters which can be more readily obtained. In order to do this 14 of the most commonly used airfoils have been investigated, namely: Clark Ye-21, Clark Y-18, Clark Y-15, Clark Y, G-d87, i-6 Mel2, USA-35-A, USA-35-B, USA-27, USA-5, RAF-15, '-9, and N-10, Sach section has been plotted, using 100 ineh chord, The areas were measured by the plenimeter and the peri- meters were scaled by dividers, the tabulated results being the mean of four readings, and are correct with one per cent. Investigations were carried out: a) for total cross section, ¢ to 100% chord, table I ana figure l, b) for the front part of the section up to 30% chord, table II and figure 2, c) for the middle part of the section from 30% to 65% chord, table III fig.3 ad) from 65% to 100% chord, table IV fig.4 TABLE I O-100% Chora Airfoil h A Ka A' %err. P. C-Y-21 all 1457 ,694 1470 -0.9 210.1 G-Y-18 18 1249 .694 1260 -0.9 208.4 CeY-15 15 1040 {694 1050 -0.9 206.9 c-Y 11.7. 810 .693 820 -1.2 204.8 G=387 15,15 1015 .668 1060 --4,2 207.9 M6 12 855 .712 841 *1.6 204.2 Ua12 11.85 837 .706 831 +0.7 204.1 USA-35-A 18,18 1226 .675 1274 °3.8 209.8 USA-35=B 11,58 779 .673 812 -4,1 205.5 USA-27 11,04 811 .734 774 44.8 205.0 USA=5 6.38 463 734 448 +3,3 202.6 RAP=15 6.38 475 .745 448 +6,0 202.4 N-9 8.46 586 .693 592 1.0 203.0 N-10 11,22 782 ,700 784 0,2 204,3 es, 701 "205.0 s t sf aor FHKE 7 714 EK (3) 8 = MD PWD 2.05 C =1,04 ML Gt (2a)*G t (2%. 701lxnxc)* Gitar xC ~ c | Figure l. h is the maximum ordinate, t is the thickness, A is the total area, k, is the coefficient of area and is defined : , ~—A_ A@k,Cxn . In our case ks Toor are plotted against h and the straight line is drown thru On figure 1A the areas all the points, the slope of which. come out to be .701, This value corresponds to the mean value of all the k,'S. A' is equal to k Ch or ,701 Cash. In each case the percentage of error in A' have been com- puted by the formla % err= 4F,4+y100, which is tubula- ted in the column 5, Py is the total perimeter, the mean value of which for all sections considered may be set at 2.05 C, Substituting the above values for area and the perimeter in the equations (1) and (2) we obtain the equ- tions (3) and (4). TABLE II 0-30% Chord Airfoil h A k! At Serr. R p! C-Y-21 21 513.815. 517 -0.8 88.5 1,735 CeY-18 18 441.816 443 ~0.5 684.4 1.757 Ce¥-15 15 365 .815 369 1.0 80.2 1.780 Ca¥ 11.7 286 .816 288 -~O.7 75.2 1,800 G=387 15,15 377 ,829 373 41.0 81.3 1.800 M-6 12 284 .790 295 -2.8 75.4 1.795 M-12_ 11.85 281 ,.790 292 -2,8 75.2 1.795 USA-35-A 16,18 452 .829 447 +1.4 85.5 1,775 USA=35-=B 11.58 287 .826 285 +0,7 75.8 1,820 USAs27 11,04 273 .825 272 +0,4 74,2 1.215 USA=5 6.38 162 .847 157 43,0 68.6 1.800 RAP-15 6.38 165 .863 157 +5,0 67,9 1,865 = 8,46 208 .819 208 0.0 70.9 1,840 N=10 11.22 274 .815 276 -0,7 74.4 1,806 k#, 820 271,805 Mw . Ml One bd ~* ol xb (5) = Jf. - Ste ogre 2x ,82 g = iL 1.805(h +b) 2 67 (2 +b) My GE Tex .820 nepyt °° a pe OFS (6) ce) ~ ~ ~ ~ fe ~ h ~ ~e ~ ~ ~ ~ meee ee eee He a. ef le. D .70 0 RPifure 2, Table II is computed 1n the similar way as the table I, except that k' is defined k's ss x b is used because it seem to be more logical parameter then G, P, is the total perimeter ineluding ordinate at aia 52% chord, and p' is defined by Fee p' (hp b), and p is the mean value of p', Again the above values are substituted in the equations (1} and (2) and we get equations (5) and (&) TABLE IIT 30%-60% Chord Airfoil hay Nes h A xt A’ “err. C-Y-21 92 14.8 17.9 662 1,056 659 +0,7 C-Y-18 18 12.67 15.33 570 1,061 564 41,0 C-Y-15 15 10.58 12.79 478 1,066 471 41,5 CH¥ 11.7 3,25 9.97 366 1,048 367 -0,2 G-387 15,18 99.84 12,51 457 1.043 471 -3,0 M-6 12 8.8 10.4 391 1,073 383 42,0 M-12 11.85 .8.8 10.3 379 1,051 379 0,0 USA-35-A 18,18 11.36 14,77 S51 1.065 544 -1,4 USA-35=-B 11.58 7.56 9,57 352 1,050 352 09,0 USA-27. «11,04 8,64 9,84 358 1.040 362 -1,0 USA-5 6.38 4.8 5,59 197 1,007 206 ~4,0 RAF=15 5.9 5,.0° 5,45 197 1,033 200 1,5 Neg 8,46 5.76 7,11 264 1,062 262 40.7 Ne-10 11.22 7.81 9,56 355 1,062 352 -0,7 1,051 S tisk? seams = -476mKp (7) Q= 7 ths - Mi 2,005 (h+b) = 453th tb) _ML t 2 = FE (ox 1.051 h b)t h* pt & -—-~ aw Ve o ? Lin ’ hz0 * ~s / t 1g5 ~ AL Me _ ___-{ } { ~~ ew eee ee ~ > * 65 0 ~<. C a Figure &, Just as the. tables I and II table III is compu- h ted except that h is taken as the mean value of hag and hgs The equations obtain are (7) and (8). TABL2 IV 68%=-100% Chord Airfoil nh" Aw xk! Atm Zerr, BP," C-Ye2l 14.8 282° .544 290 -2.7 36.7 C-Y-18 12.67 238 .536 249 -3,.6 84,1 C-¥~15 10.58 197 ,532 207 -4,.8 81.7 C-¥ 8,25 154 .,633 162 -+4.9 79.3 G-387 9.84 181 .528 193 -5,8 81,2 M-6 3.8 180 .584 173 +4.9 179.35 Mele 8.8 177.) 6575 173 +2.9 79.3 USA-35#A 11.36 223 .560 223 0.0 83.4 USA-35-B 7,56 140° ,522 148 -5,.6 78,6 SA-27 3.64 180 .594 170 45.9 79.6 USA=5 4,8 104 ,620 95 +9.5 75.2 RAF-15 5,0 113 .646 98 415.2 75.2 N-9 5.76 114 ,556 113 -0.9 76.2 N-10 7,81 153 .560 153 0,0 76.6 ie =, 560 st: us = ,s92 1. (9) 2A 2%,56 hxb hxb Im Pp M(2 beh)L wb je 1 O= rt (28)? & wet = 2514, (Broth | (10) aa i fk b t ee meee eee t be CO ES < bo — ee ———$ —— C Table IV.is obteined in previously described manner, But it is felt that percentage of error involve in At is high and in practieekne aree of a given section con be estimted better directly then paremetricly. 14. DAP SRIMENTAL CHSC OF PRANDTL'S FORMU bd Virst attempt to check the validity of Prandtl's formula has been made by using experimental cata published by S.von Yakla in the Luftfehrtforschung, vol.4, book l, page 39; Munchen, 1925, The wine tested was 150 em, long with two upars and 20 ribs evenly spaced, The other properties of the sec~ tion are given in figure 6. Curve (1) gives the angle in degrees of twist per 150 cm. of the model without the skin. The curve (2) gives the angle of twist with the paper co- ver over the ribs, The curve (3) gives the angle using 08 cm. plywood cover, Tt can be deduced from the character of the curves that thé spars and the ribs contribu‘e very little to the regidity of the model. If we take the following values; M = 10 em.-Ke, C =m 20 on, h-wl.? cm t = .08 em, G = 30000 Ke/em. Le= 1Ld0cn. and substitute in the equation (2) we get a” wm 1,04X10457,3% 150 — .642 derrees ~ $0000 x.08x (1.7)®X20 “—_ “his is too low, The curve sives about 3 degrees, but with the spars and ribs included, hence we should expect slightly more than 3 degrees, Hor the second set of experimentel dete the autor is ine debted to the Northrop Aircraft Co., Burbank, Cal. A full size wing of the Alpha plane hive been tested, The wing was of monocoque constiuction having 9 ribs and longitudinsel stiffeners. The .ian of the wins is shown by dotted line on ficure 5, At the root C-Y-16 sec- tion was.used with 100" chord, The wing Das gradually ta- per both in the 3 a lan and the thickness so that at the tip t the section was C-Y=11,7 with the chord of 66 inches. Sor the purpose of this analysis it has been assumed that the wing is built-up of hollow cylinders: .. which a constant cross section between adjacent ribs and equal to the section which corresponds to the stations #1, #2, + 3, ete. It is also assumed that the eileron has no effect on rigidity of the wins. Table Yowas computed in the following manner; Column 2} ©" is the chord in inches 3) h" is the maximum ordinate in inches, 4) Ais the total area of the cross section com puted by the formule A 7.694 Ch ( this coeff- icient corresponds to the Clark Y family of airfoils. &) is the perimeter in per cent of chord. Values are obtaiec from Dr. Klein's charts. 6) PP" is the perimeter in inches 7) At is the area of theaileron section. It has been found that theaileron is located at 83.9%: chord and the areas were computed on the assum- 8) Column 9) 11) 12) 13) 14) be Oo . ption of triangular cross section. is the effective area, and is equal to the dif- ference between column 4) end column 7). Table VA is the rib number is the thickness of the shell. Up to the rib 3 for 74.5% chord t euuals .040 in, and 125.5% chord is 0,032 in, or average t is .0Q35 in. From rib 3 to rib 5 the average t is .0357, and from rib 5 to the tip t is .C2&S1 inches. Twisting moments applied to the wing are shown on figure 5, Mis the total moment in in-lbs. at given section, B is the angle in radians and was obtsied by using equation (2) is the totsl angle of twist for a given section. It can be seen that the totel deformation at. the rib 8 is .005230 radians, Since there is no momemt between rib 8 and rib 9 the same deflection is assumed for the tip. The chord at the tip is 66 inches long, therefore the def- 1gfion should be .005230 X66 or .345 inches. The experi- mental value is .70 inches, i’. ‘¢ bry 100" 7 Rib! Sta#o | \ { Clark Y- 18 nn #L_-- LLL bl i | 2 2. I A760 in-lb t re 9 ___ee LL d an { 3 4 I t Tevatt gf ------ §---- - == | ” i . , 4 6 ; 35570,in-lb f | ; nn boa -2----- | a | Py y 5 7) | o | | ; yO ee: i ° a { ‘ } 6 10 | __1 26070 in-Jb t \ 1 ! | & ees | eee ; I } 7 12 | , | | M4 es 5 ee | in | f 8 \4 | ! 21040 in-lb | | 3 ! — pe nS: ; , 1 ! ! {9 1 State Loo 1 Sta. Q to o oO NY OO Oo fF WwW 10 il 12 13 14 15 16 2 a 100.00 97,88 95,75 95.62 91.50 89,37 87.25 85,12 85,90 80,87 &,75 76,62 74,50 72,57 70,25 88.12 66,00 1250 1183 1094 1022 952 886 824 769 708 653 602 555 465 424 536 353 TABLE V PwC 207.8 207.5 207.3 207.0 206.9 189.6 L79.1 178,8 178.4 178.1 177.7 177.3 L77,0 176.7 176.3 176,90 175.7 pt 207.8 203.1 198.4 193.7 189,2 160.6 156.2 152,0 148.0 144,0 139.8 135.7 131.8 127.8 123.9 119.9 115.9 18. 9 Rib L mo OO SF OD OF Ff & Bw 10 (2Ae )* 5598000 4178000 2849000 2144000 15638000 1201000 785000 il t" 2035 2035 20557 20357 -0281 20281 20281 TABLE; VA 12 L" LOed 13 amit, ff LI" = 7 130840 82680 82680 47110 47110 21040 21040 L4 L5 68 rad. Q, rad. 2000930 2000930 2000695 2901625 2000877 2002502 900629 2003131 001032 2004163 .000568 2004731 ~ 000499 2005230 -005230 19, —» 8 >| 7.5 17 it re j an | 30}. Ls gl | 200 . | 25 Curve {1) | | | 20 Curve (2) { 15 | | | 10_L | | | M om,ke. Figure 6 el. DISCUSSICN OF RESULTS All computed values of deflection are much smaller then those which have been found experimentaly. The author feels that data obteinec by Fakla are not re- liable, In his report there is a lack of description of the set up and the manner in which the experiments were performed, All that can be inferred is that the testing model was in a horizontal position, that the loads were applied at one edge of the free end and that the other edge was attached to the celing by means of a spring be~ lace, Such a set. up implies that the elastic center is at mid-point of the section. It is however knewn that this ie not generally the case for airfoils, There is then a posibility that a vertical component of force was present when the couple was applied. Farthermore the question of fixed end is not discussed properly, and it is not known haw rigidly the modekewas attached, And finally the dimen- sions of the model wowed seem to be out of proportion. A specimen which is 150 em. long and only 20 cm. wide and of 1.7 cm, maxinum thiexness and made out of wood, propab- ly is too flexible for this experiment. On this bases it is very doubtful Fakla's data can be used as an experimental check of Prandlt's Tormula,. eke In the case of tne Northrep wing the calcu- lated value is about 50% smaller than the observed value, Sut in Northrop report it is stated that due to the set up of the experiment it is possible that from 25% to 50% of the observed deflection is in the supporting jig and not in the wing, There is another possibility of experimental error, Two equal forces in opposite directio were applied at the leading and the trailing edge respectively. Hence the assum- ption that the elastic center is at the mid-point of the section is also made in this case. Due to this method of loading we can not be certain that the two forces are ex- actly equal, therefore a linear deflection may occur in ben- ding, which would greatly increase the apparent angulur twist, This would be further agcrivated by the fact that readings were taken at rib 9 in place of rib 8 for which computations were made.