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NATIONAU^lTVrSORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE

No. 1344

CRITICAL STRESS OF THIN-WALLED CYLINDERS IN TORSION

By S. B. Batdorf, Manuel Stein, and Murry Schildcrout

Langley Memorial Aeronautical Laboratory Langley Field,*ty&;>

i

Washington June 1947

~~SEPRöDUCED if . ,^'ONAL TECHNICAL .! INFORMATIONSERVICE il' lnA

yi fi

NOTICE

THIS DOCUMENT HAS BEEN REPRODUCED

FROM THE BEST COPY FURNISHED US BY

THE SPONSORING AGENCY. ALTHOUGH IT

IS RECOGNIZED THAT CERTAIN PORTIONS

ARE ILLEGIBLE, IT IS BEING RELEASED

IN THE INTEREST OF MAKING AVAILABLE

AS MUCH INFORMATION AS POSSIBLE.

: T I

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

^

TECHNICAL NOTE 110. 13^

CRITICAL STRESS OF THIN-WAULED CYLINDERS IN TORSION

By S. B. Batdorf, Manuel Stein, and Murry Schildcrout

SUMMARY

A theoretical solution is given for the critical stress of thin-walled cylinders loaded in torsion. The results are presented in terms of a few simple formulas and curves which are applicable to a wide range of cylinder dimensions from very short cylinders of large radius to long cylinders of small radius. Theoretical results are found to he in somewhat tetter agreement with experi- mental results than previous theoretical work for the Bame range of cylinder dimensions.

INTRODUCTION

For moBt practical purposes the solution to the problem of the buckling of cylinders in torsion was given by DonneTL in an important contribution to shell theory published in 1933 (reference 1). The present paper, which gives a solution to the same problem, has two main objectives: first, to present a theoretical solution of somewhat improved accuracy; second, to help complete a series of papers treating the buckling strength of curved sheet from a unified viewpoint based on a method of analysis essentially equivalent to that of Donnell but considerably simpler. (See, for example, references 2 and 3.)

The method of solution in the present paper,is that developed in reference 3« The steps in the theoretical computations of the critical stress are contained in the appendix. The results are given in the form of nondimensional curves and simple approximate formulas which follow these curves closely in the usual range of cylinder dimensions.

2 ' . NACA TTT No. 13H

' SYMBOLS

J,m,n integers

p arbitrary constant

r radius of cylinder

t thickness of cylinder vail

u axial component of displacement; positive in x-direction

v circumferential component of displacement; positive in y-direction

w radial component of displacement; positive outward .

x axial coordinate of cylinder »

y circumferential coordinate of cylinder

/ Et3 \ D flexural stiffness of plate per unit length / ]

E Young's modulus

L length of cylinder

Q mathematical operator defined in appendix

Z - curvature parameter ( ^~ Jl — \fi or [*) c yl —

an, fcn coefficients of deflection functions

kB critical shear-stress coefficient appearing in

u2

Mn = -

o formula Tcr w ks 2~£

L2t

(n2 + ß2)2 + ; XZZ2Pk ,

^(n2+ß2)2; 8ß

Vm,Wm deflection functions defined in appendix

MCA TN No 13IA 3

"•J X half wave length of buckles In circumferential direction

H Poisscn'.s ratio

T critical shear stress cr

V* = —T- + 2 + —f- bxk dx? öy2 0/ .

v" inverse of v , defined "by v~ *" v*w = w

'"'RESULTS AMD DISCUSSION

The critical shear stresses for cylinders are obtained from the equation

-n2D Tcr = ls:s-r-

L2t

The values'of 1:B for cylinders with either simply supported or clamped edges are given in the form of logarithmic plots in figure 1. The ördinate in this figure is the critical shear- stress coefficient ks The abscissa is a curvature parameter Z which Is Given directly by the theory and involves the dimensions of the cylinder pnd Poisson's ratio.

For very short cylinders the value of the shear-stress coef- ficient approaches the values for flat plates, 5-3^.when the edc;es are simply supported and C.98 when the edges are clamped. As Z increases ka also increases and the curves which defined ks are given approximately by straight lines. For simply supported cylinders, '• ' •

ks = O.85 Z3/k

For cylinders' with clamped edges,

ka = 0.93 Z3^

MCA TK No. I3IA

The range of validity of these formulae is approximately

100 < Z < 10 2- . t2

For the case of long cylinders the curves of figure 1 split into a series of curves depending upon the radius—thickness ratio. These curves, -which correspond to buckling of the cylinder into two circumferential waves (n = 2), depart from the straight lines

at approximately Z = 10^- or approximately - = 3jf • Because

the critical shear stress of a long cylinder is almost independent of end conditions, the curves for different values of r/t apply both to cylinders with simply supported edges and to cylinders with clamped edges. These curves are probably some- what inaccurate, however, because one of the requirements for the validity of the simplified equation of equilibrium used is that n2» 1. A calculation for long cylinders made by Schwerin and reported in reference 1 by Donnell suggests that all values corresponding to the curves given in the present paper for n = 2 are slightly high.

In figure 2 the results of the present paper are compered with those given by Donnell (reference 1) and Leggett (reference 4). The present solution agrees quite closely with that of Donnell . except in the transition region between the horizontal part and the sloping straight-line part of the curves. In this region the present results are appreciably less than those of Donnell (maximum deviation about 17 percent) but are in close agreement with Leggett1s results, which are limited to low values of Z.

In figure 3 the present solution and that of Donnell for the critical shear.stress of simply supported cylinders are compared on the basis of agreement with test results obtained by a number of investigators. (See referencesl, 5, 6, and 7.) The curves giving the present solution are appreciably closer to the test points. More than 80 percent of the test points are within 20 percent of the values corresponding to the theoretical curve for simply supported cylinders given in the present paper, and all points are within 35 percent of values corresponding to the curve.

In figure k the present solution for critical shear-stress coefficients of long cylinders which buckle into two half waves is given more fully than in figure 1 and is compared with test results of references 1 and 8.

NACA TN No Ijkh 5

The computed values from which the theoretical curves presented in this paper were drawn are given in tables 1 and 2.

CONCLUDING REMARKS

A theoretical solution is given for the buckling stress of thln-valled cylinders loaded in torsion. The results are applicable to a wide range of cylinder dimensions from very short cylinders of large radius to very long cylinders of small radius. The theoretical results are found to be in somewhat better agreement with experimental results than previous theoretical work for the same range of cylinder dimensions.

Langley Memorial Aeronautical Laboratory National Advisory Committee for Aeronautics

Langley Field, Va., March 20, 19V?

HACA TN No. 13IA

APPEEDIX

THEORETICAL SOLUTION

The critical shear stress at which buckling occurs in a cylindrical shell may be obtained by solving the equation of equilibrium.

Equation of equilibrium.- The equation of equilibrium for a slightly buckled cylindrical shell under shear is (reference 3)

_Ji Et„.li o^v _ . d?w . DV*w + —V ^ ^—j- + 2xcrt = 0 r2 ox* 3oc öy (1)

vhere x is the axial direction and y the circumferential direction. The following figure shows the coordinate system used in the analysis:

I I

HACA TN No. I3IA

Dividing through equation (l) by D gives

^*^$**4&-° <a> vhere the dimensionless parameters Z and kB are defined "by

and

k - TcrtL2 8" *2D

The equation of equilibrium may "be represented "by

Qw = 0 (3)

where Q is defined "by

I> ox4 L2 äx dy

Method of solution.- The equation of equilibrium may be solved by using the Galerkin method as outlined in reference 9» In applying this method, equation (3) is Bolved by expressing v in terms of an arbitrary number of functions (VQ, V±, . . . Vj, Wo, Wi,•. . ., Wj) that need not satisfy the equation but do satisfy the boundary conditions on vj thus let

J J w e 2L amvm + ]>" *mHn M m=Ö" m=0

MCA TN No. I3kk

The coefficient am and tfa are then determined by the equations

ax.ii

UO to VnQw dx dy = 0

? 2JI/1L

T^Qw äx dy = 0 0 I/O

(5)

where n = 0, 1, 2, . . ., j

The solutions given in the present paper satisfy the following conditions at the ends of the cylinder:

For cylinders of short and medium length with simply supported

edges w. = a* w = v = 0 and u is unrestrained. For cylinders of

short and medium length with clamped edges w - «jp = u = 0 and v is unresLrained. For long cylinders w = 0. (See references 2 and 3.)

Solution for Cylinders of Short and Medium Length

Simply supported edges.- A.deflection function for simply supported edges may "be taken as the infinite series

w = sin Ä C am sin SS + cos S V b* sin BS. "..*./-. ..... L • . \ Z_. .:• L.:- m=l ... nt=l ....

(6)

where \ is the half wave length of the buckles in the cir- cumferential direction. Equation (6) is equivalent to equation (k) if

MCA TIT No. I3W

Vn = ein AI sin S3S ° \ L

u = COg 2£ ein 2*£ A, L

(7)

Substitution of" expressions (6) and (7) into equations (5) and integration over the.limits indicated give

n <n2 +. ß2)2 + _JS!S!L_ *V + ß2)2

CO

/ in « Z__ n2 - m2

r-

l)a (n2 + ß2)2 + 12Z2n^

„4(n2 + ß2)2

m=l

CO >(8)

8knß \ mn + -f~; %- m=l

= 0

•where

"l n = 1; 2, 3, . • .

and m ± n is odd. Equations (8) have a solution if the following determinant vanishes:

10 MCA IN No. 13V*

n»l

n=2

n=3

n=5

n=6

•^8 .

2 a3

o e 0 £*fe °

° #3 ke

0

0 0

0 0

0 . 0

0

0

0

o ^k

ä5

0

0

0

0

0

0

0 ^

a6

0

0

0

0

0

0 tii*

Dl

0

.2 3

0

k_ "l5

0

.£. 35

2 3- 0

6 5 0

.10 21

0

b3 ^ *5 H •••

0 JL 0 i- ... 15 35

6 0 m 0 ... 5 21 0 12 0 2 «r • • •

7 3 L2 0 20 0 ... 7 9 o .20 0 22 ...

9 li 2 o .22 o ... 3 li

n =1 0 3

0 4 "15

0 6 ~35 •"

ri- =2 2 3

0 •I 0 2l o ...

ll: =3 0 6 5

0 12 7

0 2 'I •"

n= «If h

15 0 12

7 0 20

9 0 ...

n- --> 0 21

0 20 9

0 -22 ... li

x\- =6 6 35

0 2 3

0 22 li

o ...

^M1 0 0 0 0 0.

0 7^2 0 0 0 0.

0

0 0

0 ~i<U 0 0

*-s 0

0 .

0 T^-M^ 0 0 .

0 0 r±M* 0 .

oooo ^-M^ .

(9)

where

V _ Jt

8ß (n2 + ß2)2 + 12Z2nV

«Hr£ + ß2)2j

By rearranging rows and columns, the infinite determinant can be factored into the product of two infinite subdeterminants which are equivalent to each other. The critical stress may then "be obtained from the following equation:

NACA TIT No. I3IA 11

n=l

n=2

n=3

n=6

n=l

n=2

n=3

n=l*

n=5

n=6

al *2 a3

12

15

a5

0

0 .#

0

15 0

35

10 21

^ 0

0

0

0

0

0

0

0

0

0

0

0

0

12 7

1, ° f ^'T 20 9

*6

6 35

0

2 •5 *j

0

30 0 -f &5 If 2 3

JO 1, 11 ^s

0 & fi*

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 .

0 .

0 .

0 .

0 .

0 .

Dl

0

0

0

0

0

0

a2

0

0

0

0

0

0

D3

0

0

0

0.,

0

0

0

0

0

0

0

0

D5

0

0

0

0

0

0

0 .

0 .

0 .

0 .

0 .

0 .

&>• -f

0

'15 0

6_ "35

1Q 21

_2

0 -¥ ^ f ° 0 -1 .20

11 £"6

= 0

(10)

The firBt approximation, obtained from the second-order determinant, IB given by

•e; M3M2 (ID

•12 NACA TN No. l^kk

The second approximation^ obtained from the third-order determinant, is given by '

kD2 M1M2M3

-<$f**($h (12)

Th© third approximation, obtained from the fourth-order determinant, 1B given "by

*eVf + ^y _ ^(ufKl„2 + (|VMlMlt + (^)v3 • (§)V>

+ MiMgMoM^ = 0 (13)

Each of these equations shows that for a selected value of the curvature parameter Z the critical buckling stress of a cylinder depends on the wave length. Since a structure buckles at the lowest stress at which instability can occur, kB is minimized with respect to the wave length by substituting values of ß into the equation until the minimum value of kB can be obtained from a plot of ks against ß. This procedure is permissible when

itr that is,

when the cylinder buckles into more than two circumferential waves. For the limiting case of a cylinder buckling into two waves, see the section of the present appendix entitled "Solution for a Loflg Cylinder" which follows.

Figure 5(a) shows the convergence of the determinant for cylinders with Eimply supported edges.

Clamped edges.— A procedure similar to that used for cylinders with simply supported edges may be followed for cylinders with clamped edges. The deflection function used is the following series:

w = sin 2£ > am

5=0

mnx COB -= cos

+ cos -*£ \ bm

m=0

mnx (m + 2)nx cos —-- — cos J —-—

L L (l*)

II

Mf-GA TN No. I3H 13

Bach tern cf this «erlös satisfies the condition on v at the edges. The functions "Vjj end Wh are now derined ee follows:

Vn => ein £Z cos E2X _ L L

cos (° + 2)*X L

..

Wn = cos 2£ cos (n f 2)«X L

(15)

where

n = 0, 1, 2, . . .

When the Bame operations as those carried out for the case of simply supported edges are performed, the following simultaneous equations result:

For n = 0,

-go..

a0(2Mr) + M2) - a£M2 + ks ^L bm

m-1,3,5 L

m' 2 _ Jm+ 2)2_ m2 - if (m + 2)2 - 4

For n c 1,

a! (Mi + M3) - a^ + ks \ "bm

m=Q,2,4

m<= m' 2

m2 - 1 m2 - 9

- _lE±i?lL_ + (m + 2)g

(m + 2)2 - 1 (m + 2)2 - 9

For n = 2, 3, 4 . . .,

= 0

00

an(Mn + Mn+2) - an_2Mn - an+2Mn+2 + kB/ ha. m=0

sr m2 — n2

mc fm+ 2)

2 , faf 2)2

m2 - (n + 2)2 (m + 2)2 - n2 (m + 2)2 - (n + 2)2

lfc NACA TW No. !$&

vhere min is odd. <

For n = 0,

b0(2M0+ Mg) - "bgMg - ka ^ % »=1,3,5

m2 (m + 2)2

m2 - if (m + 2)2 - 4

For n = 1,

*l(Mi + M3) - "b3M3 - kB 2_ ^ m=0,2,4

m2 ng (m + 2)2

- - 1 m2 - 9 (m + 2)2 - 1

(m + 2)2 _ 9 _

For n = 2, 3, ^, . . .,

= 0

M^n + Mn+2) " bn-2Mn - *n+2Mn+2 - ks £__ % n=0

r ffi2 mc

nv ,2 _ n2 m2 - (n + 2);

(m + 2)2 + (m + 2)2

(m + 2)2-n2 (m _+ 2)2 - (n + 2)2 = 0 (16)

vhere m ± n is odd and

*-fi "(n2 + ß2)2 + ^-IgzSn^L—'

jt^(n2 + ß2)2

The infinite determinant formed "by these equations can be rearranged so es to factor into the product of two determinants which are equivalent to eech other. The vanishing of one of these determinants leads to the following equation (limited for convenience to the sixth order):

NACA. TN No. 1344 15

n=0

n=l

n=2

n=3

n=4

n=5

a,.

3£ 15

_il " 105

315

a2

-4^ 352

105

*3

ÜL 105

^W) - g| - ^M3

F'(Mp"rM4) 105 s

B

315

22 35

-g^

^3

^2 35

^ ^(M^m) - 315

4l6o

s

4l6o

693

315

1376 ,

1155

1287"

1155 V5 1287 kB^5+M7J

-^ -^^VMfi)

= 0 (17)

The first approximation, obtained from the second-order determinant, is given "by

- (^l)2 (2M0 + M2) (MX + M3) (18)

The second approximation, obtained from the third-order determinant, is given by

(Mi + M3) I (2M0 4 Mg) (Mg + Ml») - M22]

(19)

The third approximation, obtained from the fourth-order determinant, is given by

• 16 MCA TN No. 13kk

+ en £

i y-^

<n

#

%

OJ| ir-\

m

5 •HI

H

I

+

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CO •p

J&CA TU No. 13kh 17

Solution for e Long Cylinder

A long slender cylinder (Z>10$)

will "buckle into two

waves in the circumferential direction. If, in the previous cases of cylinders with simply supported or clamped edges, the half wave length in the circumferential direction X is taken as «r/2, it is possible to find the critical stress of a long slender cylinder with the corresponding edge conditions. This method of solution is laborious, however, because determinants of high order must be employed to obtain solutions of reasonable accuracy. The labor is greatly reduced by the use of the following deflection function:

v. = a^ < cos

I (Ks*a ¥)- cos (p -t- 2)itx

r 1 (21)

where p + 1 is the phase difference of the circumferential waves at the two ends of the cylinder measured in quarter-revolutions. This equation satisfies the single boundary condition w = 0. With this deflection function, the functions V and V all vanish except

Vl . cos^ + ^V cos (p + 2)?tx 2Z r

(22)

Use of equations (5), (21), and (22) and the relation 2X » rtr results in the following equation:

k„ m 3 J > + JL/LV* 12 + 12ZV 8L(P + l)|u «2W J n4[p2 + JL(^

h /L\2 <-J»?^G) 12Z2(p + 2)k

(P+.2)2 + 4 «2 mi > (23)

23 MCA TN No. 13^

This equation may te written

.' 8(p + 1) * r i/l - n£ if*

12Z2p^

^ + r?- /

zt Y

(P + 2)2 + * zt 1 2

j- *2r /TT^2 ' lgz2(p + -2)

(p + 2)2 + JL_Z± *2-/l-,2

(24)

For given values of Z and £ |/l — n2^ p is varied until a

minimum value of ks is obtained from a plot of p and corresponding values of k8. The critical Btress of a long slender cylinder is very insensitive to edge restraint; therefore, the solution applies with sufficient accuracy to cylinders with either simply supported or clamped edges. The shear-stress coefficient for long slender cylinders is plotted against the curvature parameter in figure k, and parts of these curves also^ appear in figure 1.

NACA TW No. 131A 19

REFERENCES

1. Donne11, L. H.: Stability of Thin-Walled Tubes under Torsion. MCA Rep. No. 1*79, 1933-

2. Batdorf, S. B.: A Simplified Method of Elastic-Stability Analysis for Thin Cylindrical Shells. I - Donnell's 'Equation. NACA TN No. 13M, I9V7.

3. Batdorf, S. B.: A Simplified Method of Elastic-Stability Analysis for Thin Cylindrical ShellB. II - Modified Equilibrium Equation". NACA TN No. 13^2, I9V7.

k. Leggett, D. M. A.: The Initial Buckling of Slightly Curved Panels under Combined Shear and Compression. R. & M. No. 1972, British A.R.C., I9U3.

5. Lundquist, Eugene E.: Strength Te3ts on Thin-Walled Duralumin Cylinders in Torsion. NACA TN No. k21, 1932.

6. Moore, R. L., and Wescoat, C: Torsion Tests of Stiffened Circular Cylinders. NACA ARR No. Iffi31, 19^.

7- Bridget, F. J., Jerome, C C, and Vosseller, A. B.: Some Naw Experiments on Buckling of Thin-Wall Construction. Trans. A.S.M.E., APM-56-6, vol. 56, no. 8, Aug. 193^, pp. 569-578.

8. Moore, R. L-, and Paul, D. A.: Torsional Stability of Aluminum Alloy Seamless Tubing. NACA TN No. 696, 1939-

9. Duncan, W. J.: The Principles of the Galerkin Method. R. & M. No. 181*8, British A.R.C, 1938-

IT!

NACA TN No. 1344 20

TABLE 1

THEORETICAL SHEAR-STRESS COEFFICIENTS AMD WAVE LENGTHS

OF BUCKLES FOR SHORT- AND ME3DIUM-LENGTH CYLINDERS

) r<

z

1 First approximation Second approximation Third app -oximation

ks ß k8 ß k8 ß

Cylinders with simply supported edges

0 1 5

10 30

100 300

1,000 10,000 100,000

5.60 5.69 6.68 8.36

14.93 34.09 76.80

189.5 107? 6050

0.770 .805

1.00 1.24 1.82 2.74 3.86 5.4o

10.0 17.9

5.34 5.42 6.22 7.55

12.69 27.86 62.47

153.0 871.2 4920

0.790 .860

1.015 I.265

1.875 2.91 4.18 5.95

11.2 20.1

5.41 O.865

7.545 1.27

* 61.47 4.32

851.9 4800

11.8 23.O

Cylinder s with clamped edges

0 1 5

10 30

100 1,000

10,000

9-55 9.57 9.90

10.79 16.13 35.40

206.3 6860

1.175 1.18 1.23 1.35 I.89 2.95 6.12

20.35

9.31 9.32 9.62

10.42 14.99 30.68

167.5 5449 '

1.205 1.21 1.27 I.38 1.97 3.14 6.70 23.2

9.09 1.205

10.19 I.38

1.

30.65 165.7

5310

3.12 7.00

24.8

COMM NATIONAL ADVI ITTEE FOR AEE

SORY ONAUTICS -

"I 11""

NACA .TN No 1344 21-

TABLE 2

THEORETICAL SHEAR-STRESS COEFFICIENTS

FOE LONG CHUffiES

ke f^l-l»2' Z

4 x 103 428

20 <

3 x 101*

10?

2,450

7,780

106 76,500

r 4 2.5 x 104 1,680

50 <

io5

-6

5,380

47,900

L IQ7 476,000

' 105 4,800

100 <

106 35,200

107 334,500

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

NACA TN No. 1344 Fig. 1

:~: --^ (V 4 -- -±±- •—

• -• ._. _ - — IT) o

O

—

ui .. *H

i

V

__ I Ü H

— • — - .... ._

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eda

es

>\ — £ to 2

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vs > — —

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impl

y s

up

dam

pe

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i

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o

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1

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a) Xi

— — o

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1 Ü _.. _. \ 1

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'l _o IMM Mil inn 1 1 1 II 1 L M mini i i i *o rO o O (\

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Fig. 2 JN.tt.C.tt. -LiN iNU. iOlt

k,-W

(a) Simply supported edges.

k •'CrtLf

z-J^FF NATION»!. WVISOBV

COMITTU FW UMMUTES

Figure 2.-

(b) Clamped edges.

Comparison of theoretical curves for critical stress of thln-walled cylinders in torsion.

43

TTI

NÄCA TN No. 1344 Fig. 3

_ w

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o Ni

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o

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(b) Clamped edges.

Figure 5.- Successive approximations of critical shear-stress coefficients for thin-walled cylinders in torsion. X

I TITLE: Critical Stress ol Thin-walled Cylinders In Torsion

AUTHOR(S): Batdorf, S. B.; Stein, Manuel; Schilderout, M. ORIGINATING AGENCY: National Advisory Committee for Aeronautics, Washington, D. C. PUBLISHED BY: Same

fiTD- 8023

none OWO. ACEKCY MO.

rUDUSHINO AQCMCY MO.

June '47 ooccun

Unclass. COUNTTT

U.S. Eng. FAoa

26 llUMTIATtOM

tables, graphs ABSTRACT:

A theoretical solution is given for critical stress of thin-walled cylinders loaded in torsion. Object is to present a solution of Improved accuracy and a method of analysis equivalent to Donnell's, but simpler. Critical shear- stress coefficient for simple supported cylinders is 0.85 times curvature parameter to the 3/4 power, and for cylinders with damped edges is 0.93. Results are presented in terms of simple formulas and curves which cover a wide range of dimensions.. Theoretical results agree with experimental results.

DISTRIBUTION: Request copies of this report only from Originating Agency DIVISION: ffimn Analvsl« nnrf Structures Cl\ I SUBJECT HEADINGS: Cvli DIVISION: stress Analysis and Structures (7) SECTION: Structural Theory and Analysis

Methods (2)

ATI SHEET NO.iR-7-2-8

SUBJECT HEADINGS: Cylinders - Stress analysis (285G5); Structural members - Stress analysis (90859)

Air Document! Oivlilon, Intalllganca Dapartmont Air Material Command

AIR lECHNICAL INDEX Wrf -ht-Pattorton Air Force BaM Dayton, Ohio