1、t. / JNATIONAL-AD_rSORY COMMITTEEiFOR AERONAUTICS-L .TECHNICAL NOTENo. 1344CRITICAL STRESS OF THIN-WALLED CYLINDERS IN TORSIONBy S. B. Batdorf, Manuel SLein, and Murry SchildcroutLangley Memorial AeronauticalLangley Field, _.?JLaboratory .=/i!t/fProvided by IHSNot for ResaleNo reproduction or networ
2、king permitted without license from IHS-,-,-I |!Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NOTICEIS BEINGOF MAKINGHAS BEENCOPY FURNISHED USAGENCY. ALTHOUGHCERTAIN PORTIONSRELEASEDAVAILABLEREPRODUCEDBYITTHIS DOCUMENTFROM THE BESTTHE SPONSORINGIS
3、RECOGNIZED THATARE ILLEGIBLE, ITIN THE INTERESTAS MUCH INFORMATION AS POSSIBLE.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-T IfProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIOHAL ADVISORY COM
4、MITTEE FOR AERONAUTICSTECIE_CAL NOi_ _0. 134_CRITICAL STRESS OF THIN-WALleD CYLINDERS IN TORSIONBy S. B. Batdorf, Manuel Stein, and Murry SchlldcroutA theoretical solution is given for the critical stress ofthin-walled cylinders loaded in torsion. The results are presentedin terms of a few simple fo
5、rmulas and curves which are applicableto a wide range of cylinder dimensions from very short cylinders oflarge radius to long cylinders of small radius. Theoreticalresults are found to be in somewhat better agreement with experi-mental results than previous theoretical work for the same rangeof cyli
6、nder dimensions.INTRODUCTIONFor most practical purposes the solution to the problem ofthe buckling of cylinders in torsion was given by Do_ne_ in animp6rtant contribution to shell theory published in 1933 (reference i).The present paper, which gives a solution to the same problem,has two maln object
7、ives: first, to present a theoretical solutionof somewhat improved accuracy; second, to helpcomplete a seriesof papers treating the buckling strength of curved sheet from aunified viewpoint based on a method of analysis essentiallyequivalent to that of Donnell but considerably simpler. (See,for exam
8、ple, references 2 and 3.)The method of solution in the present paper,is that developedin reference 3. The steps in the theoretical computations of thecritical stress are contained In the appendix. The results aregiven in the form of nondim_nsional curves and simple approximateformulas which follow t
9、hese curves closely in the usual range ofcylinder dimensions. !Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2SYMBOLSHACA TN No. 1344J,m,n integersP arbitrary constantr radius of cyl_ndertU:.thickness of cylinder wallaxial component of d_splacement
10、; positive in x-directionv circumferential component of displacement;positive iny-directionw radial component of displacement; positive outwardXYDELaxial coordinate of cylindercircumferential coordinste of cylinder“flexural stiffness f plate per unit length _12(_Pt!2)“_Youngs moduluslength of cylind
11、erQ mathem_tlcal operatordefined in appendixZ _anJ I_nkscurvature parameter 1. A calculation for long cylinders made by Schwerinand reported in reference 1 by Donnell suggests that all valuescorresponding to the curves given in the present paper for n = 2are slightly high.In figure 2 theresults of t
12、he present paper are compared withthose given by Donnell (reference l) and Leggett (reference 4).The present solution agrees quite closely with that of Donnellexcept in the transition region between the horizontal part andthe sloping stralght-line part of the curves. In this region thepresent result
13、s are appreciably less _han those of Donnell(maximum deviation about 17 percent) but are in close agreement withLeggetts results, which are limited to low values of Z.In figure 3 the present solution and that of Donnell for thecritical shear stress of simply supported cylinders are comparedon the ba
14、sis of agreement with test results obtained by a numberof investigators. (See referencesl, 9, 6, and 7.) The curvesgiving the present solution are appreciably closer to the testpoints. Morethan 80 percent of the test points are within 20 percentof the values corresponding to the theoretical curve fo
15、x simplysupported cylinders given in the present paper, and all pointsare wlthin 35 percent of values corresponding to thecurve.In figure 4 the present solution for critical shear-stresscoefficients of long cylinders which buckle into two half wavesis given more fully than in figure i and is compare
16、d with testresults of references i and 8.i! i|iProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-yNAOA _ No 1344 5The computed values from which the theoretico_ curves presentedin this paper were drawn are given in tables 1 and 2.CONCLUDING RE_Jd_KSA t
17、heoretical solution is given for the buckling stress ofthln-walled cylinders loaded in torsion. The results are applicableto a wide range of cylinder dimensions from very short cylindersof large radius to very long cylinders of small radius. Thet_eoretical results are found to be in somewhat better
18、agreement withexperimental results than previous theoretical work for the samerange of cylinder dimensions. Langley Memorial Aeronautical LaboratbryNational Advisory Committee for AeronauticsLangley Field, Va., March 20, 1947iiIProvided by IHSNot for ResaleNo reproduction or networking permitted wit
19、hout license from IHS-,-,-6 NACA TN No. 1344THEORETICAL SOL_iIONThe critical shear stress at which buckling occtu,s in acyl_nlrical shell may be obtained by solving the equation ofequilibrium.Equation of equilibrium The equation of quillbrium fora slightly buckled cylindrical shell under _shear is (
20、reference 3)_x_ + 27cr t - = 0_x _y (l)where x is the axial direction and y the circ,nnferent_ald_rection. The following figure shows the coordinate systemused in the analysis: iJiiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NAOA TN No. 1344Divid
21、ing through equation (i) by D gives7-_- _-g+_s =o (2)where the dimensionless parametersandZ andZ - -rtks are defined bywwhereiThe equation of equilibrium may be represented by-o (3)Q is defined byloZ2v_ _ 34 _2 82o.=v _ +_ _ + 2_=,L2_xtMethod of solution.- The equation of equilibrium may be solvedby
22、 using the Galerkin method as outlined in reference 9. Inapplying this method, equation (3) is solved by expressing w interms of an arbitrary number of functions (Vo, V1, . . . Vj, WO,Wl, . , Wj) that need not satisfy the equation but do satisfythe boundary conditions on w; thus letm=0 m=0Provided b
23、y IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NAOA TN No. 13_+4The coefficient am and bm are then determined by the equationsj-wnQ dx -0UO Uowheren = O, i, 2, . ., JThe solutions given in the present paper satisfy the followingconditions at the ends of the
24、 cylinder:For cylinders of short and medium length with simply supported_wedges w. = _ = v = 0 and u is unrestrained. For cylinders of8wsho1_t and medium length wlth clamped edges w = _x = u = 0 and vunrestrained. For long cyliz_lers w = O. (Sge references 2 and 3.)o . . -“Solution for Cylinders of
25、Short and Medium LengthSimply supported edges., A deflection function for simplysupported edges may be taken asthe infinlte seriesiswherecumferential direction.if-w =: L“sin-_Z :;am sin _i-:$ COS “ m“i_sinT_y T :k L. “ k _. “ ; : _=1_,._ m=l_ik is the half wave length of the buckies in the ciP-.Equa
26、tion (6) is equivalent to equation (4) .Z(6)I i,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN No. 1344Vn = s_n _Z sin n_xX :LCWn = cos _y sin n_xxk LSubstitution of_expresslons (6) and (7) into equations (5) andintegration over the limits in
27、dicated give9(7)an I (n2 +. 62)2 + 12Z2n4 -_ bm mn =-: m=lI.bnl (n2 + #2)2 + l_Z2n4 . 8_- nun + “ ) am .=4fn2 _e_2 _ _ n2 - m2-: m=l J(8)whereand m ndetermin_nt vanishes:.on = I, 2, 3,is odd. Equations (8) have a solution if the followingProvided by IHSNot for ResaleNo reproduction or networking per
28、mitted without license from IHS-,-,-I0n=ln=2n=3n=_n=5n=6n=ln=2n=3n=4n=5n=6a2 a_ a4 a5 a6 . bIa1ks_l 0 Q0 00 0D .00 0 0 . 00 0 0 . -_o o o . o 65o kD_M4 o o . 4 o“ -1-5o o _ o . o .1o_6 o0 0 0 “ 35_0w .oNACA TNNo. 1344b 3 b_ b5 b6 .o /L 0 %- .15 35o 12 o .5 21o z._2 o e7 3 “-_- o 207 7 o .o -_ o 30 .
29、9 ii._ o .3o o .3 ii,w . o -2 o -! o 63 -5 39 “ ks“Z2 0 _ 0 _ 0 o“- - .6 _ 1._2 0 2 0o 5 o 7 “ “120 o .20 0 . 015 7 9o !o o _o o _39. o21 9 ii “6 0 2 _ 0 . 03-5 5 o _z , ,0 0 0 0 0 .0 0 0 0 .o _s_3 0 0 0 .0 0 kls-_4 0 o .0 0 0 .1-M 0ks 5 “0 0 0 0 _ . ,J , (9)Ywhere_-l-(n2 + _2) 2 +By rearranging row
30、s and columns, the infinite determinant can be factoredinto the product of two infinite subdeterminants which are equivalentto each other. The critical stress may then be obtained from thefollowing equation:Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS
31、-,-,-NAOA TN No. 13_4 iin=l:I=2n=3n=4n=5n=6en=ln=2n=3n=4n=9n=6aI b2 a3 b4 aT b6 . b1o _ o T_ “ oo o. o6 1 12 2 0o “_ k%-M3 T o _ ._ 20o z2 I_%._ o . o19 7 _s 9o lo o .P.O. _ 3o_“2“f 9 _ . o6 2 303“5 0 _ 0 -ii . 0a2000000b3 at b5 a6 .0 0 0 0 o o o o .0 0 0 0 . oll,0, 0 0 0 .0 0 0 0 .0 0 0 0 . Q 0 00
32、00 00 0O. 00 00000000 00 00 00 0O 0 0 0I “e - 1 .e_ o 4o ._ 3 -J.7o . -_ _% -_ o zo3 5 210 .:. 0 6 _ 125 3“T oo o-“17 k.0 . 0 i0210 6 039o 6w_ eo35gee2J I 0 0 e3 e0 -i_“ “g 3 _63 ii “ The first approximation, obtained from the second-order determinant,is given by=0(zo)(.z)Provided by IHSNot for Resa
33、leNo reproduction or networking permitted without license from IHS-,-,-“12 Kerations as those carried out for the caseof simply supported edges are performed, the following simultaneousequations result:For n = O,ao(2M 0 + M2) - a_242 + ks i5 m2 (m+ 2)2 7 =Z bm m-2-4 + (m“+_-2)2-A 0m=1,3,5%For n = I,
34、al(M1 + M3) - a3M3 + ks_ _(m. +Fcr n = 2, 3, 4 ,(m + 2)2 _ = 0(m + 2i 2 - 9J(m + 2) 2 - n 2 (m+ 2)2 != o(_+ 2)_ - (=+ 2)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 NACA _ No. 1344!where m + n is odd. ,For n = 0,bo(_o+M2)-_2 -ks T_m=l, 3,5+ = 0
35、a_ m2-_ (m+ 2)2-For n = i_Oobl(Ml + M3) - b3M 3 - ks _-m=O,2,_r.2_2 .=2 (m+ 2)2,_Lm2-1 ,_2-9 (m+2)2_“= 0For n : 2, 3, 4, . . ., . ,.,- m2 m2bn(lv*n + _:in+2) bil-_Mn, -bn+2Mn+2-ks h:_=r, am m2 - n2 -m 9- - (n -I: 2) 2w= Cm + 2)2 + _ (m + 2) I = 0 (16)(m + 2)2:-n 2 (m + 2)2 - (n + 2)2where m n is odd
36、 andMn = _ n2 + _2)e +_ _4(n2 + #2)2|The infinite determinant formed by these equations can be rearrangedso es to factor into the product of two determinants which areequivalent to eech other. The vanishing of one of these determinantsleads to the following equation (limited for convenience to theSi
37、xth order):F:! |Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN No. 1344 15n=On=ln=2n=3n=4n=5ao bl a2 _3 a4 B53_9 _s(MI+M3) 352-5 1o91 352-k-_ _(MeM4)105105 - ks 3 315o 32 _ !_435- 32 o _137.6315 1155-_- o _ 3_2.1o5 3.5147._2 - ks“1-_4 1376I1
38、55315693416o is(M4+M6) 94_q693 12871 94_0 1 (Ms+M7)-k-_ 1287 ks:o (_7)The first approximation, obtained from the second-orderdeterminant, is given by_s2 = i0 r2 will buckle into twot2rwaves in the circumferential direction, If, in the previouscages of cylinders with simply supported or clamped edges
39、, thehalf wave length in the circumferential direction _ is takenas _r/2, it is possible to find the critical stress of a longslender cylinder with the ccrrespondlng edge conditions. Thismethod of solution is laboriousj however, because determinants ofhigh order must be employed to obtain solutions
40、of reasonableaccuracy. The labor is reatly reduced by the use of the followingdeflection function: “(2l)whereat the two ends of the cylinder measured in quarter-revolutions.This equation _atisfies the single boundary condition w = O.With this deflection function, the functions V and W allvanish exce
41、ptp + 1 is the phese difference of the circumferential waves(22)Use of equations (5), (21), and (22) and the relationresults in the following equation:+ (p + + + (23)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-18 NACA TN No. 1344This equation may
42、 be writtenks .-8(;+ i) p2, 4 zt _2 + . _2z_4( 4 zt _2Zt “_2_2 rjl -_ 2_;_. (P + 2) 2 +12Z2(ip + .2) 4 tZt 2_2 _/ : : _2r 1- .(2_). :. -:; For given values of Z and -_Jl- #2 p is varleduntil_ aminimum value of ks is obtained from a #lot of p and correspondingvalues of ks. The critical stress Of a lo
43、ng slender cylinder isvery insensitive to edge restraint; therefore, the solution applieswith sufficient accuracy to cylinders with either simply supportedor clamped edges. The shear,stress coefficient for long slendercylinders is plotted against the culwature parameter in figure _,and parts of thes
44、e curves also appear in figure i.IillilProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I%_CATN No. 1344 19REFERENCESI. Donnell, L. H. : Stability of Thin-Walled Tubes under Torsion.NACA Rep. No. 479, 1933.2. Batdorf, S. B. : A Simplified Method of El
45、astic-StabilityAnalysis for Thin Cylindrical Shells. I - DonnellsEquation. NACA TN No. 1341, 1947.3. Batdorf, S. B.: A Simplified Method of Elastic-StabilityAnalysis for Thin C_lindrical Shells. II - ModifiedEquilibrium Equatio_. NACA TN No. 1342, 1947.4, Leggett, D. M. A. : The Initial Buckling of
46、Slightly CurvedPanels under Combined Shear and Compression. R. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- NACA TN No. 1244 Fig. 1it! ,$o o _o _jl_u)O%OO_D!o_.4oO_o_._i0_“4r_!Provided by IHSNot for ResaleNo reproduction or networking permitted w
47、ithout license from IHS-,-,-i0 _: _-I0NACA . _ _ _Present _lution - .,_o Leggetts solu_I I I IIIII t0I I I llll 1 I I 11111 t I I II11102 10 104(O) Simply SUpported edges.I 1 1 I111110 5 i0 _I0Donnells solution, jPresent sTlution _,x_o Leggetts solution1 I I I1111 I I III1,cONATIONAL I_VISORY(OIOIINH _ /_AOII_*TKSI i I I itlll I I I Illil 1 I 1111i0 I I0 102 1031_2Z - r-t-,i2“_(b) Clamped ed
