AxAT^iir; 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 
 
 No. 1182 
 
 THE PROBLEM OF TORSION IN PRISMATIC MEMBERS OF 
 
 CIRCULAR SEGMENTAL CROSS SECTION 
 
 By A. Weigand 
 
 TRANSLATION 
 
 "Das Torsionsproblem fiir Stabe von kreisabschnittformigem 
 
 Querschnitt" 
 
 Luftfahrt-Forschung, Band 20, Lfg. 12, pp. 333-340 
 
 ag 
 
 UNIVERSITY OF FLORIDA 
 
 DOCUMENTS DEPARTMENT 
 
 1 20 MARSTON SCIENCE LIBfIaRY 
 
 RO. BOX 11 7011 
 
 GAINESVILLE. FL 32611-701* USA 
 
 Washington 
 September 1948 
 
1^1, rll Hn 
 
 ^l /6>^^ 
 
 MTIOHfiL ADVISORY COMMITTEE FOE AEEONAUTICS 
 
 TECHmCAL MEMOEAIJDUM NO. Il82 
 
 THE raOELEM OF TOESION IN. PEISM/>.TIC MliMBSES OF 
 CIRCUIAE SEGMEIWAL CROSS SECTION* 
 By A. Weigand 
 
 SIJ^MARy 
 
 The problem is solved "by approxlriia.tion. "by setting up a 
 ftmction complyiiig vith the differential equation of the stress 
 function, and deterroining the coefficients appociring in it in such 
 a way that the "boundary condition is fulfilled as nearly as possible. 
 
 For the semicircle, for vhich the solution is known, the 
 method yields very accurate val-aes; the approximated stress 
 distri^bution is in good agreement with the accurately computed 
 distrihution. Stress and strain neasuxements indicate that the 
 approximate solution is in sufficient!;/- exact agreement with reality 
 for segmental cross sections. 
 
 I . FUIOAMEKTAL EQUATION OF TORSION AJ3D ITS APPROXIMATE 
 SOLUTION BY THE I-ETHOI) OF LEAST SQUARES 
 
 The torsion proT^lem for the prismatic memher stressed "by 
 twisting moments at the ends is formulated as follows. Find a 
 function -fCy^ z) which in the cross -sectional plane satisfies 
 the partial differential eq^^ation ..-■_. 
 
 On 
 
 S"f S"f 
 
 -^-i- 7^:^-1 (1) 
 
 , oy c32 
 and at the houndary of the cross section the condition 
 
 f = (2) 
 
 ' '!Das TorsionEpro"blem fur Stabe- von Icreisa'bschnittfSrraigem 
 Querschnitt . " Luftfahrt-Forschung, Band 20, Lfg. 12, Feh. 8_, l<^kh, 
 PP- 333 -3to. 
 
2 NACA TM No. Il82 
 
 This function f (y, z) then gives the torsion constant J, of the 
 mranber accordlaig to 
 
 J^ = ^ f f f(y, z)eLy dz (3) 
 
 tho double integral to "be extended over the cross section. The 
 angle of trd-st \{/ of a length I is 
 
 M the applied torq.uo, G the modTtLus of rigidity of the inaterial. 
 The components of "Uie shearing stress follov from 
 
 ^d ^f ^d hf ,,, 
 
 ^y ^d 3z ^- Jd ay 
 
 Oirtng to the eq.uations(5) which satisfy identical]^ the equilihrium 
 condition 
 
 St St 
 
 Sj c5z 
 
 f (y, z) is called the stress function of the torsion prohlem. 
 
 The differential eqi^ation (l) with the "boundary condition 
 equation (2) follows from the conclderatlon of the state of strain 
 and the relation hetwcen stress ai^d strain, which is given "by 
 Hooke 'e law. 
 
 Occasionally, it is appropriate to Introduce the polar 
 coordinates r,<p instead of the rectangular coordinates (y, z) 
 (fig. l). The differential equation of the stress function together 
 with the hotindary condition then reads 
 
 S^f ^ 1 Sf ^ 1 S^f , .. . 
 
 — + + __ -__ c, -1 (la) 
 
 Sr2 r ar r2 Sq^ 
 
 f = (2a) 
 
MAC A TM No. 11 82 
 
 while the torsion constant follows from 
 
 J. i'%' j I f{T, cp)r dr df? (3a) 
 
 and the shearing stress components from 
 
 ^^5f 
 
 T = 
 
 \^ 
 
 d df 
 
 "^d cir 
 
 (5a) 
 
 Rigorous methods for solving the potential problem posed by 
 eq.viations (1) and (2) will not "bo discussed. 
 
 The approxiraate solution can "be effected in three ways. A 
 fvinction can "he assvimed that satisfies equation (l) hut not 
 equation (2). If the differential equation is replaced "by a 
 variation prohlem, it results in the conventional Eitz method; or 
 a function satisfying the differential equation can he assumed and 
 the houndsj'y condition met in individual points or "on the averagej " 
 an exact explanation of what is meant hy "on the average'' will be 
 given later- Lastly the differential equation can be replaced by a 
 difference equaxion and the linear equation system ensuing from the 
 boundary condition solved by iteration mth the aid of the Liebmann- 
 Wolf method. Only the second method is discussed in the present 
 report J after having been pointed out, among others^ by Trefftz 
 (reference l) and Gt. BerQiiiann (reference 2), 
 
 Since the torsion problem of the segment is to be treated^ we 
 proceed from the differentia], equation (la) . It has the particxLLar 
 solutions 
 
 ' ■'■ 2 
 f = - -^, f J- = r^ cos kcp, gj^ = r^ sin }sp (6) 
 
 from which the general solution 
 
 ^ = " f ■" 21 IVk -^ ^kSk] (7) 
 
k MCA TM No. 1182 
 
 can "be Tmilt up. Haw the determinatiaa of the coefficients ajj. 
 
 and "bij. is involved. The next thing is to so determine them 
 
 that egttation (2a) is ccanplied vlthin Individijal points . Among 
 others, prolDloms relating to plate tending have already "been 
 solved "by this method. 
 
 Another way is the folloving: Rather than specifying strict 
 compliance vitli the houndary condition at originally established 
 points it is reqxiired that hy choice of the coefficients the integral 
 of tlie squares of the botaadary values is least. In this instance 
 the Doundary condition is said to Tse fulfilled "on the average." 
 
 This method is hereafter called the method of least squares. 
 In the foiimula the requirement on the factors reads 
 
 J = f da = Min. ds = Boundary clement (8) 
 
 The inte?;ral is to Ve extended over the entire 'bovmd.ai'j', this is 
 indica:,6i I:'/ t}.i3 sign. ^ . Putting equation (f) in equation (8) gives 
 
 r f 
 
 ■P -n . _. ^ 
 
 f '!^r "^ % ^/-- IVk^'^' 9' -^ \^'k^^' ^^ } ds = Min. (9) 
 
 The coof.-'iic-.irTitfj fcllcw from tiie 3?equirement 
 
 _-: . ^ 0, -^~ = 0, -~ = 0, k = 1 . . . n (10) 
 
 H ^\ ^\ 
 
 From equation (10 ) follovs a linear equation system for the 2n + 1 
 iinlaaovn. a^, aj-, and lo-^. 
 
 The practical use of the metlicd depends upon whether sufficiently 
 exact resvJ-ts consistent with a moderate amount of paper work are 
 obtainable, especially for the stresses, or in other words without 
 having to solve a great niauber of linear equations. 
 
MCA TM Wo. 1182 
 
 H. APPLICATIOK OX) THE SEMICIRCEE AND TEE SEGMENT 
 1. Semicircle J Strict and Approximate Solution 
 "by the Method of Least Squares 
 
 The strict solution of the torsion proljlem for the sector was 
 giren "by St- Yenant (Handb- d. Physik Bd VI, pp 153--15l|) . The special 
 case of the semicircle is easily treated ae will "be shown. 
 
 To remain In agreement with the notation for the segment 
 (fig. 6) the coordinate system of figure 2 is shown for the semicircle. 
 
 On the straight "boundaiy AB, 9 = 5- and 3|j in the cross 
 secticaa, -i = 9 = ^. 
 
 The stress function is expressed "by 
 
 fCr, (p) = >" X, (r) cos k<p (11) 
 
 1,3... 
 
 It already fulfills the 'boundary condition on the straight houndary, 
 since k is an odd ntanber. The constant 1 in -the internal 
 
 —- = 9 = -g- is expanded in a Fourier series . 
 
 h ^- , » 2 cos k(p , ^ 
 
 Introducing eq.Ta&tions (11) and (12) in equation (la), the comparison" 
 of the coefficients of cos kq) on "both sides of the equation gives 
 
 y ,, 3^ , .]^_ .1. (-1) 2" 
 
 ^k +?k -^^k= -;(--Y~ ^^^' 
 
 The solution of this differential equation, finite for r = 0, reads 
 
 k+1 
 
 ^^-'^^''-t^'^' w 
 
NACA TM No. 1182 
 
 Since Xj5.(E) inust "be = 
 
 k+1 
 
 
 (15) 
 
 end, hence, 
 
 1 2 <» 
 
 k+1 
 
 f (r, (J,) = — / 
 
 (-1) 1 
 
 ^ 1,33".. k(^ -l£^) 
 
 :f J - ©^ 
 
 C03 k<p (l6) 
 
 is the solution of the torsion problem for the semicircle. The 
 torsion constant J, and tho shearing stress distribution are 
 
 computed from equation (l6) . From equation (3a) follows on three 
 places exactly 
 
 Ji k 
 J^ = 0.297 E = KR^ 
 
 and from (5a) 
 
 "r " «jr j,3 }_3 
 
 (X - I) 3ln r. -A-^ (J^ 
 
 ^] sin 3<P 
 E/ 
 
 h A 
 
 1_ (t_ r\, 
 
 sin 5<p + - . 
 
 (17) 
 
 T = — 
 
 8\ 
 
 Krt jj3 
 
 i (l - 2|) cos q) 
 
 1 /^ or\ 
 + — ^ 3-;; - 2- COS 3<P 
 
 3(3^ -^)\R^ ^/ 
 
 - 4) Ve^ i^/ 
 
 5(5'' - ^) 
 
 cos 59 + 
 
 • • « 
 
 (18) 
 
NACA TM No. Il82 
 
 The raaximm shearing stress occxirB at A (fig. 2), tiiat is, 
 
 for r = and 9 = ^. Her© 
 
 T =-L_i.2.85-^ (19) 
 
 The shearing stress at C (fig. 2) is 
 
 Folloving the rigorous sol-ution for the semicircle em approximate 
 
 solution "bj the -.net.'iod of least sqtiares shall "be derived. 
 
 Sinc'o the cress sect:' on is symmetrical ahout cp = n, and 
 following eq.uation (7) we vrrite: 
 
 ^ r^ V- k 
 
 f s - — + / a^^ cog ]s;q) (20) 
 
 Instead of coefficient a, the q-uantity x, is introduced hy 
 
 ak = f - -^- (21) 
 
 1, ^ic 
 
 So vith ^ = ^> formula (20) reads 
 
 T,2 / 2 ^ k \ 
 f = £- (-X + 2_ Xj^ cos kqj) (20a) 
 
 The "bovaadary values are 
 
 j^2 / 2 n jj. icrt\ 
 
 on AB (fig. 2) f^ =-^ ^-X + 1, x^ cos — ^ 
 
 On BC (fig. 2) f_ = — (-1 + 2_ X. cos kq>) 
 BG i^ \ / 
 
8 
 
 NACA TM No. 1182 
 
 The method of least squares yields as conditioiial equation for x^ 
 
 \2 
 
 rt 
 
 ■X" + 2._ Xjj-X'^^coa "TT ) dX + / ( -1 + > 
 
 2 ^- ,k ^' 
 
 > X, COS top) d<p = Min. (22) 
 rt ^ ^ ' 
 
 2 
 
 .) 
 
 Therefore 
 
 \^( Jl ^ ,k lotVl Ijr „ 
 
 ■'■ / ("-^ "^ Z_ ^k ^^'^ •'^j '^^^ 29 d9 = 
 
 For Xjj the linear equation system vith syaimetrical matrix 
 
 X" 
 
 IL. 
 
 
 
 (23) 
 
 is applioaljle, with 
 
 Aqo = 1 ^ f 
 
 COT-??'? 
 
 A =.f + ?-.k =1, . . . n 
 
 ^ ^ 2.1c + 1 
 
 (2Ua) 
 (2lrt)) 
 
 Icl 
 
 lor ijr 
 
 cos 2 *^os 2 "L 
 
 II 1,1 I ^ .^» 
 
 k + Z + 1 2 
 
 sin (k - I)| sin (k + \)~ 
 
 L k - I 
 
 k + Z 
 
 k # I (24c) 
 
 So =1*1' =1 
 
 COS 
 
 Jrt 
 
 sin 
 
 IJT 
 
 I + 3 
 
 X = 1, 
 
 n 
 
 (21^) 
 
 The ntanerical calculaticn was effected for n = 1, 2 ... 6. 
 For n = 6 the equation system reads 
 
KACA TM Wo. 1182 
 
 + 
 
 H I m 
 
 It 
 
 II 
 
 HJlfS 
 
 I 
 
 11 
 
 II 
 
 HJITN 
 i 
 
 11 
 
 HJ OS 
 I 
 II 
 
 ^ 
 
 I 
 
 CO 
 I 
 
 ^ 
 
 it- h|{C^ h!on 
 
 + 
 
 
 V3 
 
 HJlH tAJH 
 
 + 
 
 >r 
 
 H( lf\ 
 I 
 
 >r 
 
 ir\! 
 
 1 H 
 
 (M 
 
 
 H 
 
 if 
 
 H| iTv 
 + 
 
 en 
 H 
 H(ro 
 
 + 
 
 .1 ur\ 
 
 i 
 
 CO 
 
 I 
 
 I 
 
 + 
 
 
 
 I 
 
 >f 
 
 >f 
 
 ITS 0\ 
 
 -';;l 
 
 i 
 
 i 
 
 
 >r 
 
 
 i-i\a\ 
 
 
 + 
 
 I 
 
 1^ 
 H loo 
 I 
 
 + 
 
 
 H|^- 
 i 
 
 
 4^ 
 
 HJON 
 + 
 
 M^ 
 
 + 
 
 H I m 
 t 
 
 
 I 
 
 >^ >P ^ 
 
 «|CV1 
 
 + 
 
 M 
 
 ,o 
 
 H I CO H j m 
 
 I 
 
 «^ ^ ^P 
 
 HJin HJITN HJt- 
 
 I I 
 
10 NACA TM No. Il82 
 
 The coefficients were changed to decimal fractions and considered only 
 Tip to the fifth place after the decims.1 point. Six approxiciationB 
 were computed; for the first approximation Xg = x-, = ■ • • = xg = 
 
 vas used. The result is presented in tahle I. Insertion of 
 equation (20a) in eq.uation (3a-)/ gives the torsion constant J^ as 
 
 S" (25) 
 
 J, -^ 2Br (- ~ + ~ Xr, - - X, + -^ X. - -i- X. + - . . A = ivl 
 d V 8 1+ ^ 3 1 3x5 ^ 5X7 ^ J 
 
 and the following approximations for ^ computed exact to three 
 places: 
 
 K^^^ =OAllf K^^^ =0.326 K^^^ =0.300 
 
 K = 0.2c8 •«, . = 0.300 K = 0.2q8 
 
 (!+) ' (5) (6) 
 
 The third approxims.tion compvited from four linear equations already 
 gives a torsion constant value that differs "bj no more than 
 2/3 percent from the rigorously computed value. 
 
 For the stress calculation, equation (20a) is inserted in 
 equation (5a.), so that 
 
 M, n k-1 ' 
 
 T = - — ~ *) tacj^X sin kcp (26) 
 
 2kE-^ 1 
 
 % ^^- 
 
 T = ~_ /_^ (-2\ + kx X^"-^ cos }q)) (27) 
 
 2rE^ 1 ^ • 
 
 ^ - -3 
 
 The shearing stresses '^g) a^icL "^p'"^ at the straight houndary 
 are 
 
 m=^ M, y Pi; \ 
 
 T 2 ^ . _ /3j_^ _ 3^^^ + 5^,^ - + . . .j (28) 
 
 r 
 
 2k E-^ 
 
NACA TM No. 1182 11 
 
 <P=- ^A 
 
 *P 2kE3 
 
 (^\ - 2x^ + 4xi^\3 . 5xg^5 + .j (29) 
 
 For X = 1 the shearing stjeeeses are 
 
 X=l ^d / \ 
 
 T = - ^ — r /xj^ sin <p + 22^ sin 2<p + . . .j (30) 
 
 9tR \ 
 
 T J**"-^ = r(-2 + X;l cos (p + 2x2 °*^^ ^^ ■'■ * * V (31) 
 
 Ihe two DBXiimm shearine stresses are: 
 
 
 r max 2k ^3 
 
 T <P=rt,X=l _ 
 <P 
 
 V = -a^3(-^-^*^- 
 
 (32) 
 
 (33) 
 
 Of these expressions t ^ end t, ~ must at least^ 
 
 9 1 
 
 approxiaately disappear. 
 
 How for a check of the extent to \^ch these conditions are 
 
 3net for the different approxlDaaticais and also of the oxte^t of the 
 
 differences "between the approximated and the exact values of t^ 
 n max 
 
 and T . a!he results are represented in tahle II and figures 3 
 
 to 5. Tlie fotirth approximation already gives a serviceahle result, 
 which is soae'iaiat further iriproved "by the fifth and sixth a|4)roxi- 
 matlons. 
 
 IE 
 Figures 3 and ' h show the approximations for t J*^"^ 
 
 and T which really shoiad disappear. It indicates good 
 agreement in the fifth end sixth approximations. Figure 5 shows 
 
12 -ITACA TM No. 11 82 
 
 It 
 
 the sheELTing stress T . at the straight "boundary plotted 
 
 r 
 against X = — . The fourth, fifth, and sixth approximations differ 
 
 little from each other and from the accurate stress distritution 
 designated "by g. A marked departure occurs in the immediate 
 vicinity of the comer (point B in fig. 2). 
 
 2. The Segment 
 
 (a) AT3tiroximate solution by the method of least squares . - Since 
 the cross section is symmetrical to (p = (fig. 6) the 
 formula (20a) is applied to the stress function f . The boundary 
 values are given "by 
 
 AB k 
 
 ho£^.,. Y_ Xj, ?H£^ cos k9^ = 9 = « • • • (3^^) 
 \^ cos^cp ^ cos^ / 
 
 7 v^ ^ < < 
 
 I - 1 + 2_ ^ COS kcp] a = 9 = 
 
 f =-— (" 1 + ^ x^ cos kcpj a = 9 = rt (S^h) 
 
 If ds^ j.s an element of the straight boundary AB and ds^ 
 an element of the arc BC, 
 
 d<p 
 ds = E cos a ■ (35a) 
 
 cos (p 
 
 dSg = E d9 (35b) 
 
 The expression that is to be made a minimum by the choice of the 
 coefficients x^ reads 
 
 1'°'/ cos^a T^ cos^a \^ cos a 
 
 — I 4 t;uti U, -r-^ COS a \ C( 
 
 J = / - _ + ^ xjj — — COS kcp d® - 
 
 i/O \ COS 9 cos 9 / cos^9 
 
 K- 
 
 1 + ^ Xj^ cos k9 ) d9 (36) 
 
MCA TM No. 1182 I3 
 
 Prom — —- = follows a linear equation system with symmatriq^il 
 
 matrix for x^, which is to "be written in the form of equation (23) . 
 The coefficients A and the right sides are given hy 
 
 Aqq = « - a + sin a (379-) 
 
 2, 
 
 2k+l P^ °°^ ^^ rt - a 
 Aj^ = cos a / g^-.g d(p + — — - 
 Jo cos'^^^cp 2 
 
 sin 23sa , , _ /^r^\ 
 Ic = 1, 2, . . . n (37b) 
 
 Ic+i+i P^ cos Iq) cos l<p 1 
 Ak^ = cos aj^ _____d9 
 
 cos ^'"cp 2 
 
 sin (k - 1)9 
 . k - 1 
 
 sin (k + l)qi 
 
 + ' ■ 
 
 k + 1 
 
 k = 1, 2, , . . n 
 2 = 0. 1,, ... n 
 
 (37c) 
 
 Bq = « - a + sin a f cos a + -^ sin a) (37d) 
 
 1+3 /'°' cos l(? ^ Bin la , „ , . 
 
 B, = cos a / r-T— d(p I =: 1, 2, . . . n (37e) 
 
 The integrals appearing in equation (37) are of the fona 
 
 — ) they can he defined "by expressing cos p9 hy 005^9, 
 
 coe^p 
 
 ■D-2 
 cos-^ (pj etc. A reproduction of the somewhat elaborate formulas is 
 
 omitted. 
 
 The matrix Aj^^ including the right-hand sides B, of 
 equation (23) were computed to five places with the calculating 
 
 I 
 
Ik 
 
 NACA TM No. 11 62 
 
 machine as functions of a. To keep the paper work within tolerable 
 limits the process vas carried to k, 2 = 6. The result is shov/n in 
 figvire-C^. The imkno\'ms Xq, x-]_ . • . xg vere computed hy 
 
 equation (23) ^ hy the Gauss method. The result is given in table IV. 
 
 After Xq, x^^ . . . xg have teen determined the torsion 
 constant J, and the shearing stresses can "be computed. 
 
 By equation (3a) the torsion constant is 
 
 •^d "^ ^ I I f (^^ 9)^ "Jr dq) 
 
 ABC 
 
 The double Integral is to be extended over the area ABC (fig. 6) 
 
 cos a 
 r = E 
 
 cos q_ 
 
 no 
 
 MO 
 
 a . // ■ r 
 
 AOC CBC 
 
 d(p I f (r. cp)r dr 
 ^0 Jo 
 
 + / dcp I f (r, (p)r dr 
 Ja ,0 
 
 Insertion of the expression for the shearing function f from 
 equation (23) in this formula gives 
 
 J^ = B 
 
 n - a sin 2a 
 
 ('2 1 2\ 
 cos a + — sin a 
 3 / 
 
 (sin 2^ 
 
 + 2 
 
 n x^ / sin ka\ 
 
 --R^ 
 
 (38) 
 
NACA TM No. 11 82 
 
 15 
 
 with 
 
 k+2 r** cos kq) 
 J^ = cos a -^-^^ d(? 
 Jo cos cp 
 
 (39) 
 
 By equatloiis (5) and (5a) the shearing stresseB are 
 
 xy 2ke3 
 
 2X cos 9 - 2_ ^ ^k^ °°^' 0' " l)^ 
 
 (to) 
 
 M^ 
 
 T , = - 
 
 « - ,3 
 
 2r..f?- 
 
 f 
 
 ,k~l 
 
 2\ cos 9 + / k X, X sin (k - l)i|> 
 
 (41) 
 
 ^0 { ^ 
 
 = ---V 2\ - > 
 
 9 2Kf;-^ *'• 
 
 k Xj^X cos kro 
 
 ■) 
 
 (J^2) 
 
 J^, -^V k-1 
 
 T = - — _ \ k x-iA Bin k9 
 
 (1*3) 
 
 Particularly importeiit s;"e tlie fommlaB for the stresses at the 
 on AB . 
 
 AB ^■■•' i~ .^- /coo a.\l^--"L 
 T — '; 1 2 COS a " > k r, / \ cos (k - 1) <p 
 
 ^ 2i:?r' i_ 1 ^VoG c?y 
 
 (UOa) 
 
 AB 
 
 X2 
 
 M, 
 
 2i«I-i'^ 
 
 2 CC3 a tan co 
 
 s;^ ^ /cos af -1 
 1 \cos 9/ 
 
 sin (k - l) 9 
 
 (J^la) 
 
l6 NACA TM No. 11 82 
 
 on BC 
 
 BC % / — \ 
 
 ■^ (D = 5 2 - >_ k Xj^ C03 k(p) (i^2a) 
 
 ^ 2kR^ \ 1 / 
 
 2P M, .^ 
 T = - — -- ■> k x, sin kcp (l)-3a) 
 
 ^ 2KR^ T ^ 
 
 Of these equations (^la) and C+Sa-) must disappear (at least 
 approximately) . 
 
 Lastly there are the formulas for the shearing stresses in 
 A and C (cp = jt). 
 
 T = T = 
 
 xy max gicR 
 
 ^d /o V^ , k-1 \ 
 I 2 cos a - > k Xv cos a) 
 
 T^ = — - 
 
 2kE- 
 
 2-!-^(-l) kx^ 
 
 (UOt) 
 
 (1+213) 
 
 The numerical values for the torsion constant and the 
 particularly interesting shearing stresses j ^ and ^^ follow 
 
 xy 9 
 from equations (38)^ C+O'b). and {k2'h) . These are also included in 
 tahle rv and in figure 7 plotted against a. 
 
 (h) Solution formula "by Fourier series ■ - The torsion problem for 
 segmental cross section can also he solved "by means of the Foiu'ier 
 series. The method is hriefly explained- 
 
 To transform equation (la) we put 
 
 r2 
 f = - f^ + $(r, cp) (hk) 
 
 h. 
 
 $ must he a potential function irhich assvimes the values 
 
 $ = 2? (1+5) 
 
 4 
 
 at the section "boundary. 
 
NACA TM Wo. 1182 
 
 17 
 
 Therefore 
 
 
 K cos a 
 h cos^cp 
 
 < (p = a 
 
 
 a = <p = Jt 
 
 
 u 
 
 
 (i^6) 
 
 This eyen fimction of cp Is developed in a Fourier series in the 
 
 < < 
 
 interval -it = cp = + n 
 
 
 
 (^7) 
 
 with 
 
 K^ 1 / sin 2a\ 
 
 ° ll- ^ \ 2 / 
 
 a^ = cos a / ^ — a^ 
 
 i+ rt 
 
 i/O cos <p 
 
 n 
 
 (i^7a) 
 
 The potential fimction $(r, (p) is tiiilt up vith the aid of yet 
 
 to "be detercilnsd c^oofficiefi.tG from per bj.ciilar solutions. 
 
 CO 
 
 * N~ - n 
 
 $ = /__ Oj^r cos nq) 
 
 *0 
 
 (»^8) 
 
 At the "boundary ass-jisi'ss ths fol3.o-wing values 
 
 ^ < < -s- \" - n cos^a 
 
 = tp = a $ = ^_ n E cos no 
 
 ■^' cos^cp 
 
 a=(p = n ^ = /_ \R^ 
 
 (»^9) 
 
 cos ntp 
 
 y 
 
18 
 
 KACA TM Wo. 1182 
 
 This even fxmction of cp is also developed in interval 
 -rt = q)=+« ma FoiATier series 
 
 Bo = 2 
 
 
 
 
 
 
 Bj^ cos 
 
 nq) 
 
 1 
 
 2rt 
 
 f(? 
 
 10^ 
 
 V 
 
 cos a 
 cos CO 
 
 30S kq)j 
 
 dtp 
 
 »rt /_S2. 
 
 2jr 
 
 l/a \ 
 
 \ k \ 
 
 2_ IjR cos kcpj dcp 
 
 1 
 
 J 
 
 Bn = 2 
 
 1 na / CO. j^ ^^gk^^ V 
 
 '^ ''-^ ^ " cos^cp / 
 
 JO \ 
 
 if(i:v'=->^) 
 
 cos kcpj cos nq) d<p 
 
 -\ 
 
 / 
 
 (50) 
 
 (50a) 
 
 The Fourier series (equations (k'j) and (50)) obtained for the 
 
 "boundary values of ^ must "be identical, that is 
 
 n n 
 
 n = 0, 1, . 
 
 (51) 
 
 This is an infinite linear equation system for the looked -for 
 coefficients 'bj^. With 
 
 ^-^-^^■n 
 
 n 
 
 k 
 
 (52) 
 
NACA TM No. Il82 
 
 19 
 
 the system reads 
 
 o + X a^=^ = « - a + 
 
 sin 2a 
 
 1 
 
 ''a 
 
 ■^ 2 /"* cos k© -„ sin Jca /-_» 
 > aT^x„ = cos'^a / ^ dcp ;:— (53) 
 
 1 (10 cos'^cp 
 
 The coefficients a and aj^^ are given "by 
 
 n 
 a„« = cos a 
 on 
 
 '^°' cos n(p 
 
 Jo cos (p 
 
 - dqp 
 
 sin ng 
 n 
 
 n = 1^ 2j 
 
 I • • 
 
 ia 
 
 a^„ = cos a / — ~- 
 
 ^ Jo cc-/^ 
 
 cos^9 jt - a sin 2na 
 
 dq) + — — - 
 
 l^n 
 
 n P<^ cog k® cos ri4) sin(n - k)a sin(n -f k)a 
 = cos a / ::: dq) 
 
 n 
 cos q) 
 
 2(n - k) 2(n + k) 
 
 (5M 
 
 / 
 
 Ohviously &^ ^ a^, that is, the Djatrix (ajav') is not symmetrical. 
 
 To solve for given a the torsion problem hy this process the 
 
 Pourior series must "be limited to finitely many terms j in other 
 
 words, the system (53) must be approximated by the section method. 
 
 2 
 For example, going as far as x- inclvisive means that 7 = i(-9 factors 
 
 aw have to "be computed. The numerical calculation thus becomes 
 very tedious and is therefore omitted. 
 
 lU. CHECK OF THEOKETICAL BESUI/T BY TEST 
 
 With the setup described in reference 3 "the torsion constant J, 
 
 of a member of segmental section was optically determined; while the 
 maximum shearing stress t^^^ (point A in fig. 6) was determined by 
 
 max 
 
 means of stress measxirements . The shaft sketched in reference 3^ 
 
20 NACA TM No- 11 82 
 
 figure 1 Uas machined to d = 70 mlUlineters aiid a flat siarface 
 milled out which gave the desired section. The milled surfaces 
 corresponded to the angles a = 20°, 1*0°, 6o°, and 8o°. The 
 comparison is illustrated in figure 7- The agreement is plainly 
 sufficient. 
 
 Translated "by J. Vanier 
 National Advisory Committee 
 for Aeronautics 
 
 REEERENCES 
 
 1. Trefftz, E.: Ein Gegenstiick zum Eitzschen Verfahren. 
 
 Verhandlungen des 2 . interna tionalen Kongresses ftir 
 technische Mechanik, Zijrich I926, p. I31. 
 
 2. Bergmann, St.: Ein Naherungsverfahren zur LSsung gewisser 
 
 partieller, linearer Differentialgeeichungen. Z. angaw. 
 Math. u. Mech. Bd. 11 (I931) , p. 323. 
 
 3. Weigand, A.: Ermlttlung der Formziffer der auf Yerdrehxmg 
 
 beanspruchten ahgesetzten Welle mlt Hilfe von FeindehnungS" 
 messungen. Luftf-Forschg. Bd. 20 (19^3), Lfg. ", p. 217 . 
 (Also availahle as NACA TM No- 1179 •) 
 
NACA TM No. 1182 
 
 21 
 
 2ABIE I 
 
 THE AITK)XIMATIOKS FOE TEE T3HKN0W1I IK THE EQUATION 
 SYSTEM (23) APPLICABIE TO THE SEKECIROIE 
 
 ^0 
 
 ^1 
 
 ^ 
 
 "3 
 
 \ 
 
 X 
 
 5 
 
 ^6 
 
 First approximation 
 
 0.486 
 
 -0.654 
 
 • 
 
 
 
 
 
 Second approximation 
 
 0.1135 
 
 -1.399 
 
 -0.638 
 
 
 
 
 
 Third approximation 
 
 .0136 
 
 -1.6553 
 
 -0.9412 
 
 0.3004 
 
 
 
 
 Fonrth approximation 
 
 -0.0086 
 
 -1.7364 
 
 -1.0859 
 
 -0 .4566 
 
 -0 .10095 
 
 
 
 Fifth approximation 
 
 -0 .0068 
 
 -1.73K) 
 
 -1.0711 
 
 -0.43465 
 
 -0.0835 
 
 -0 .00855 
 
 
 Sixth approxiinfl,tl on 
 
 -0 .00022 
 
 -1.6978 
 
 -0 .9967 
 
 -0.3251 
 
 0.0308 
 
 0.0900 
 
 0.0359 
 
22 
 
 NACA TM No. 1182 
 
 o ©. 
 
 
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 Q 
 
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 8 
 
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 CM 
 
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NACA TM No. 1182 
 
 23 
 
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24 
 
 NACATMNo. 1182 
 
 \ 
 
 
 
 
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 ^"x,. 
 
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 Figure 1.- The coordinates of the section points and the shearing 
 
 stress components. 
 
 Figure 2.- Semicircular section with coordinate system. 
 
NACA TM No. 1182 
 
 25 
 
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26 
 
 NACA TM No. 1182 
 
 '9 .- - ^ 
 
 tli " 
 
 2.8 
 
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 Figure 5.- The approximations for t at the boundary AB of the 
 
 semicircular section, 
 
 1:1. Approximation 4:4. Approximation 
 
 2:2. Approximation 5:5. Approximation 
 
 3:3. Approximation 6:6. Approximation 
 
 g: exact solution 
 
 Figure 6.- Notation at segment. 
 
NACA TM No. 1182 
 
 27 
 
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UNIVERSITY OF FLORIDA 
 
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 DOCUMENTS DEPARTMENT 
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