STUDY OF THE COLLAPSING PRESSUREOF THIN-WALLED CYLINDERS

HI LL IN I SUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

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UNIVERSITY OF ILLINOIS

BULLETINVol XXXIX November 11, 1941 No. 12

ENGINEERING EXPERIMENT STATIONBULLETIN SERIES No. 329

A STUDY OF THE COLLAPSING PRESSUREOF THIN-WALLED CYLINDERS

ROLLAND G. STURM

ROLLAND G. STURM

Price: $1.50

PUBLISHED BY THE UNIVERSITY OF ILLINOISURBANA

[Issued weekly. Entered as second-cleas matter December 11, 1912, at the post offile at Urbans, Illinois,under the Act of August 24, 1912. Acceptance for mailin at the special rate of postae provided for insection 1108, Act of October 3, 1917, authorised July 31, 19181

UNIVERSITY OF ILLINOISENGINEERING EXPERIMENT STATION

BULLETIN SERIES NO. 329

A STUDY OF THE COLLAPSING PRESSUREOF THIN-WALLED CYLINDERS

BY

ROLLAND GEORGE STURMRESEARCH ENGINEER PHYSICIST ALUMINUM

COMPANY OF AMERICA

(Formerly Graduate Student of University of Illinois)

PUBLISHED BY THE UNIVERSITY OF ILLINOIS

CONTENTSPAGE

I. INTRODUCTION . . . . . . . . .

1. R6sum6 of Previous Work . . . .2. Purpose and Scope of Investigation .3. Acknowledgment . . . . . . .

II. THEORETICAL ANALYSIS . . . . . . .4. Basic Arguments and Assumptions5. System of Coordinates6. Notation . . . . . .7. Deformed Element of Shell . . . .8. Internal Forces and Couples . . . .9. Internal Forces and Couples in Terms of

Displacement . . . . . . .10. Equations of Equilibrium . . . . .11. General Differential Equations .

. . . 7

788

99

101013

. . . 14

15. . . 16

. . . 17

III. SOLUTION OF BUCKLING EQUATIONS FOR ROUNDCYLINDERS, WITHOUT STIFFENERS . . . . . . 19

12. Uniform Pressure Applied to Sides Only . . . 19(a) Edges of Shell at Ends Simply Supported 20(b) Edges of Shell at Ends Fixed . . . . . 25(c) Edges of Shell at Ends Restrained . . . 30

13. Pressure Applied to Sides and Ends . . . . . 30(a) Edges of Shell at Ends Simply Supported . 31(b) Additional End Load . . . . . . . 32(c) Edges of Shell at Ends Fixed . . . . . 33(d) Edges of Shell at Ends Restrained . . . 33

14. Pressure on Ends Only . . . . . . . . 34

IV. ROUND CYLINDERS STIFFENED WITH RINGS .

15. Shell Round at Each Ring . . . .3636

4 CONTENTS (CONCLUDED)PAGE

V. CYLINDERS SLIGHTLY OUT-OF-ROUND . . . . . . 3716. Out-of-Round Unstiffened Cylinders . . . . 3717.. Out-of-Round Cylinders Stiffened With Rings . 40

VI. EXTENSION TO PLASTIC ACTION . . . . . . . . 4118. Cylinders in Which Stresses Are Beyond

Elastic Range . . . . . . . . . . 41

VII. EXPERIMENTAL WORK . . . . . . . . . . 4419. Effects Considered . . . . . . . . . . 44

A. Tests at the University of Illinois . . . . 4520. Object of Tests . . . . . . . . . . . 4521. Specimens . . . . . . . . . . . . 4522. Apparatus . . . . . . . . . . . . 4723. Discussion of Tests and Results . . . . . . 48

B. Tests at the Research Laboratories of TheAluminum Company of America . . . 52

24. Object of Tests . . . . . . . . . . . 5225. Specimens and Materials . . . . . . . . 5226. Apparatus . . . . . . . . . . . . 5527. Procedure . . . . . . . . . . . . 5528. Results and Discussion . . . . . . . . 57

VIII. SUMMARY AND CONCLUSIONS . . . . . . . . 6929. Summary of Analytical Results . . . . . . 6930. Comparison of Analytical With Experimental

Behavior of Aluminum Tubes . . . . . 7131. Conclusions . . . . . . . . . . . 72

BIBLIOGRAPHY . . . . 75

LIST OF FIGURESNO. PAGE

1. System of Coordinates . . . . . . .. . . . . . ... . 102. Element of Shell in Equilibrium . .... . . . . . . . 123. Sections of Deflected Cylinders . . . . . . . . . . . . . 214. Collapse-Coefficients; Round Cylinders With Pressures on Sides Only,

Edges Simply Supported; A = 0.30 ... . . . . . . . 245. Curves for Determining a-b Relation .... . . . . . . 266. Graphs for Determining a/b Ratios, Fixed Edges . . . . . . . . 287. Collapse-Coefficients; Round Cylinder With Pressure on Sides Only,

Fixed Edges; u = 0.30 ... . . . . . . . . . . . 318. Collapse-Coefficients; Round Cylinder With Pressure on Sides and Ends,

Edges Simply Supported; p = 0.30 . . . . . . . ... . 329. Collapse-Coefficients; Round Cylinder With Pressure on Sides and Ends,

Fixed Edges; I = 0.30 . . . . .. . . . . . . . . 3410. Typical Stress-Tangent Modulus Curves .. . . . . . . 4211. Thin-walled Pipe for Collapse Test . ... . . . . . . 4512. Section Through Iron-Pipe Bulkhead ... . . . . . . . 4613. View of 20-inch Steel Pipe After Collapse . . . . . . ... . 4614. View of Adjustable Bulkhead, Manometer, Air Ejection Pump,

20-inch Steel Pipe . . . . . .. . . . . . . . . . 4715. View of Collapsed 20-inch Steel Pipe and Calipers . . . . . . . . 4816. Radial Deflection Curves-20-inch Steel Pipe With Pressure on Sides Only;

A. = 0.22 Inches . . . . . . . . . . . . . . .. . 4917. Radial Deflection Curves-20-inch Steel Pipe With Pressure on Sides Only;

A. = 0.48 Inches . . . . . . . . . . . . . . .. . 5018. Radial Deflection Curves-18-inch Steel Pipe With Pressure on Sides Only 5119. View of 18-inch Steel Pipe After Collapse . . . . . . ... . 5220. Section Through Aluminum-Tube Bulkhead . . . . . . . . . 5321. Arrangement of Test Equipment and Specimen for Tubes Requiring Pres-

sures Less Than 14 lb. per sq. in. to Produce Collapse . . . . . . 5422. Arrangement of Test Equipment and Specimen for Tubes Requiring Pres-

sures Greater Than 14 lb. per sq. in. to Produce Collapse . . . . . 5623. Method of Measuring Diameter, 6-inch O.D. Tubing . . . . . . . 5724. Specimens 2, 3, 4, and 5 After Test, and Low Pressure Apparatus. . . 6025. Specimens 6, 7, 8, and 9 After Test . . . . . . . . ... . 6126. Specimens 11, 12, 13, 14, 15, and 16 After Test . . . . . . . . . 6227. Specimens 20, 21, and 22 After Test . . . . . . . . ... . 6328. Specimens 23, 24, 25, 26, 27, 28, and 29 After Test . . . . . . . . 6329. Radial Deflection and Maximum Stress Curves, Specimen No. 17,

Welded Aluminum Alloy Tube . . . . . . . . . . . . 64

6 LIST OF FIGURES (CONCLUDED)NO. PAGE

30. Radial Deflection and Maximum Stress Curves, Specimen No. 18,Welded Aluminum Alloy Tube . . . . . . . . . . . . 65

31. Radial Deflection and Maximum-Stress Curves, Specimen No. 1,Welded Aluminum Alloy Tube ... . . . . . . . . 66

32. Radial Deflection and Maximum Stress Curves, Specimen No. 10a,Extruded Aluminum Alloy Tube . . . . . . . . ... . 67

33. Radial Deflection and Maximum Stress Curves, Specimen No. 10b,Extruded Aluminum Alloy Tube . . . . . . . . ... . 68

LIST OF TABLESNO. PAGE

1. Tensile Properties of the Aluminum Alloys in the Tubes and Stiffeners . . 532. Collapsing Pressures of Thin-walled Aluminum Alloy Cylinders, Pressure

on Sides Only . . . . . . . . . . . . . . . . 583. Collapsing Pressures of Thin-walled Aluminum Alloy Cylinders, Pressure

on Sides and Ends . . . . . . . . . . . . ... . 59

A STUDY OF THE COLLAPSING PRESSURE OFTHIN-WALLED CYLINDERS

I. INTRODUCTION

1. Resumg of Previous Work.-The problem of determining theexternal pressure at which a thin-walled cylinder will collapse con-fronts the designers of boilers, penstocks, vacuum tanks, and similarunits of construction. In the design of hydro-electric or water sup-ply projects, problems of determining the collapsing pressure of thin-walled pipe and of evaluating the effect of stiffeners upon the strengthof the pipe are frequently encountered. Many industries using dis-tillation processes under partial vacuum are confronted with theproblem of designing tanks to withstand external pressure. Thedesign of submarines involves the same problem under complicatedconditions.

Many experiments have been made to determine the collapsingpressures of small pipe'* such as boiler tubes2 or similar tubes 3 andof heavy-walled lap-welded steel pipe4, but the conditions encoun-tered in large flumes, submarines, and tanks have not been studieduntil recently. Such tests have been made at the A. 0. Smith Cor-poration5, and at the U. S. Experimental Model Basin 6. Tests alsohave been made to determine the buckling strength of thin cylin-ders subjected to axial loads. Such tests were made for the purposeof obtaining a guide in estimating the strength of airplane fuselages8', 9,standpipe shells 10 , and to substantiate a new theory for the bucklingof thin cylinders under axial compression and bending."

Theoretical analyses of the behavior of cylinders under externalpressures have also been made by a number of investigators. Bryan12obtained (in 1888) the expression for the collapsing pressure of longthin tubes by means of the energy criterion for instability. Southwell13obtained (in 1913) an expression for the collapse of short tubes whichshowed that such tubes may buckle into more than two lobes. Un-fortunately, his expression contained an unknown parameter. Avalue for this parameter was determined (in 1914) by G. Cook 15 forthe case of hinged edges and lateral pressure only. R. von Mises de-rived (in 1914) an equation for the collapsing strength of short thintubes simply supported at the edges, and subjected to lateral pres-sure only e6 , which did not contain any undetermined constants.*Later (in 1929) he extended his work to include both lateral and end

*This and similar numbers refer to the bibliography at the end of the bulletin.

ILLINOIS ENGINEERING EXPERIMENT STATION

pressures"7. In 1920 Sanden and Gunther 18 used von Mises' formulasin studying the behavior of stiffened thin cylinders under uniformexternal pressure. In 1922 Westergaard 20 presented a general con-ception of the buckling of elastic structures which included generalequations for the gradual buckling of imperfect specimens or thosewith eccentric loading. In 1929 Tokugawa presented a paper 19giving a derivation of a formula for the collapsing pressure of cylin-ders which includes a "frame factor" applied to the length. Thisframe factor is determined experimentally. T'he paper also includeda study of the effect of stiffening rings upon the strength of the shell.

2. Purpose and Scope of Investigation.-The purposes of the studyherein presented are

(a) to analyze the elastic behavior of thin circular cylindricalshells subjected to uniform external pressure, and to determine thepressure at which such shells collapse for simply supported and forfixed edges; extensions in the analysis are made for plastic behaviorof the material, for "out-of-roundness" of the cylinder, and forstiffening effects of ring stiffeners;

(b) to study experimentally the behavior of thin-walled tubesunder uniform external pressures, for comparison with the results ofthe theoretical analysis.

3. Acknowledgment.-The investigation herein reported is a partof the thesis presented by the author in partial fulfillment of therequirements for the degree of Doctor of Philosophy in Engineeringat the University of Illinois in 1936.

The analytical work in the thesis and the experimental work onthe steel cylinders was done under the general guidance of H. M.WESTERGAARD who at that time was Professor of Theoretical andApplied Mechanics at the University of Illinois. The experimentalwork on aluminum alloy cylinders was carried out in the ResearchLaboratories of the Aluminum Company.

The author wishes to express his gratitude to Professor H. M.WESTERGAARD for his advice and guidance; to M. L. ENGER, deanof the College of Engineering, and to F. B. SEELY, head of the De-partment of Theoretical and Applied Mechanics, for their encourage-ment and assistance; to the Aluminum Company of America for the"use of its facilities; to Dr. F. C. FRARY, Director of Research andMr. R. L. TEMPLIN, Chief Engineer of Tests of the Aluminum Com-pany of America, for their cooperation; and to Mr. C. DUMONT,also of the Aluminum Company for his assistance with the tests.

COLLAPSING PRESSURE OF THIN-WALLED CYLINDERS

In editing the thesis for this bulletin of the Engineering Experi-ment Station, W. L. SCHWALBE, Associate Professor of Theoreticaland Applied Mechanics, extended the analysis of the problem toobtain the effect of certain terms that were neglected in the sim-plifying assumptions made in the thesis. The analysis herein pre-sented, therefore, contains some additions to that presented in thethesis, but the final results are substantially the same. The authorgreatly appreciates the careful analysis given the subject by Pro-fessor SCHWALBE.

II. THEORETICAL ANALYSIS

4. Basic Arguments and Assumptions.-The general procedurefollowed in this investigation is to derive expressions for the col-lapsing pressures of round cylindrical shells of elastic materials, andto treat deviations from these conditions as extensions of the firstderivation.

The argument used in determining the collapsing pressures is asfollows: Consider the cylinder deflected into some shape such thatthe differential equations of continuity and equilibrium combined,together with the boundary conditions, are satisfied. If the externalforces necessary to hold the shell in the deflected position are inde-pendent of the magnitude of the deflections as long as they are sosmall that they do not materially change the general shape of theshell, then the shell is in a state of neutral equilibrium. The lowestpressure at which neutral equilibrium may begin is the critical orcollapsing pressure of the cylinder. Below the critical pressure theequilibrium is stable, above the critical value the equilibrium isunstable.

The assumptions involved in setting up the general differentialequations are as follows:

(1) The shell is a round cylinder before buckling.(2) The shell is of uniform thickness throughout.(3) The material in the shell is homogeneous and isotropic, and

is elastic according to Hooke's law.(4) The thickness of the shell wall is small compared to the

diameter, so that the distribution of normal stress over the thicknessmay be assumed as linear.

(5) As a consequence of the preceding assumption, the radialstress, aZ, is negligible compared to the circumferential and longi-tudinal stresses, and the radial shearing detrusions are zero.

(6) Displacements are small compared to the thickness so that


FIG. 1. SYSTEM OF COORDINATES

certain small quantities may be neglected. Neglections are indicatedin the development of the analysis.

5. System of Coordinates.-The system of cylindrical coordinatesused to define and locate any particular element of shell is indicatedin Fig. 1. The whole system is based on a reference cylinder theradius of which is the average mean radius of the actual shell, andthe longitudinal axis of which coincides with that of the actual shell.The origin of coordinates is taken on the cylinder of reference, atthe point of maximum radial displacement of the deflected shell,and midway between the ends of the cylinder.

The y-axis is parallel to the axis of the shell, and lies on the cylin-der of reference, positive toward the reader.

The s-axis lies on the circumference of a right section of thecylinder of reference, positive in a clockwise direction. The coordi-nate is s= RO where R is the radius of the cylinder of reference,and 0 is the angle subtended by s in a right section of the cylinderof reference.

The z-axis coincides with the radius of the cylinder of reference,the radial displacement z being measured positive outward.

The meanings of all symbols used in the subsequent pages aregiven in the following section.

6. Notation.-In the discussion the following notation is used:A = constant (indeterminate in magnitude at collapse), repre-

sents the maximum value of deflectionA. = maximum initial departure from a round cylinder


a, b, c = arbitrary numerical constants determined from boundaryconditions and the general differential equation

B, C = constants determined by the differential equationsD = diameter of cylinder, inchesE = modulus of elasticity of the material in the shell or in the

stiffener, lb. per sq. in.E' = effective modulus of elasticity, lb. per sq. in.El = tangent modulus for the average stress, S 1, in the plastic

range, lb. per sq. in.EI, = flexural rigidity of combined stiffener and shell in length,

L,, lb.-in. 2t3

I = - = moment of inertia per unit of length of shell, in.12

I, = moment of inertia of combined stiffener and plate, in.4

K = numerical coefficient dependent upon the ratios of L/D and

D/t such that W, = KE (--D ,

K 1, K 2, K', K", Ki', K 2', Ki", K 2 " = numerical coefficients, depend-ent upon the ratios of LID and D/t used in evaluating K

L = length of cylinder, inchesLo = length of one wave under end load only, inchesL, = stiffener spacing, inches

M,e = bending moment in the stiffener, in.-lb.N = number of lobes into which the shell collapsesP = end load in addition to external pressure, lb. per linear inchPc = end load causing collapse, end load only, lb. per linear inchQ = the elastic limit of the material in the shell, lb. per sq. in.;

this value may be taken as 1.1 times the proportional limitas determined by Tuckerman 32

R. = radius from center of cylinder to the centroid of a right sec-tion of the stiffener and the plate effective with it, inches

r, r' = numerical ratios dependent on R/L and N introduced byloading conditions

S = maximum total stress in the shell (direct stress and bending),lb. per sq. in.

S = allowable total stress in an out-of-round shell, lb. per sq. in.S1 = average stress in the shell, lb. per sq. in.Sc = P,/t = unit longitudinal stress corresponding to Pc, lb. per

sq. in.Sp = unit longitudinal stress corresponding to an axial load

whether tension or compression, lb. per sq. in.

12 ILLINOIS ENGINEERING EXPERIMENT STATION

Notez - pos/itive downwvard:I -posit/ve toward reader.s - pos/'ive c/oc/w/kse.All// forces and coup/es -

pois//'-e as shown.

FIG. 2. ELEMENT OF SHELL IN EQUILIBRIUM


S. = modulus of failure of the material in the shell; the tensilestrength of the material may be used for S., lb. per sq. in.

S, = yield strength of the material, lb. per sq. in.S, = maximum circumferential bending stress in the deflected

shell, lb. per sq. in.t, = thickness of unstiffened shell having the same strength as

the stiffened shell, inchesW = the actual pressure applied, lb. per sq. in.W = allowable pressure of an out-of-round shell, lb. per sq. in.We = the collapsing pressure for the shell, lb. per sq. in.W, = external pressure at collapse of the stiffened shell, lb. per

sq. in.W. = ultimate collapsing pressure of an actual imperfect tube,

lb. per sq. in.Z, = radial deflection of the stiffener, inchesZo = initial departure from a round cylinder, inches(N 2L2a = 2--- + 1 = numerical ratio dependent on R/L and N

OE = slope of the tangent to the stress-strain curve at the yieldstrength, lb. per sq. in. (03 = 0.10 to 0.20 for materials notcold worked, and 3 = 0.05 to 0.10 for materials cold workedappreciably)

7,, = unit-detrusion in a tangential planee, = unit-strain in the circumferential directione, = unit-strain in the longitudinal direction

X =/ NL- + 1 = numerical ratio dependent on R/L and N

1 = Poisson's ratioe, = unit-strain in radial directionz = radial deflection of middle surfaceu = longitudinal displacement of a point in the middle surfacev = circumferential displacement of a point in the middle surface.

7. Deformed Element of Shell.-The element of shell considered isshown in its deformed state* in Fig. 2. If u, v, and z are the dis-placements in the directions y, s, and z, respectively, of a point inthe middle surface of the shell from the undeformed position, thenthe strains and the detrusion of the middle surface are defined as

au av v au OvS--= , f., = -- , "y, -- + . (1)

_y as R as Oy*A more complete description of a deformed element is given in Reference 34, p. 77.


By eliminating u and v, the following equation of compatibility 21, 22is obtained:

02e 02e 02a278 1 a2z+ - _ _ = 0. (2)

ay2 Os2 asay R ay2

56 is the infinitesimal angle at the center of curvature of thedeflected shell, subtended by the circumferential element of length.

It may be expressed as a function of 0; , = dO, and consists00

of the following parts:

(1) dO, original angle

1 32z(2) - -- dO, due to change in slope over length dsR 002

z(3) -- dO, due to radial deflectionR

(4) e,dO, due to circumferential strain.

HenceI# 1 02z z

-- = 1- ---- +e. (3)00 R 002 R

8. Internal Forces and Couples.-In Fig. 2 are shown the forcesand couples holding the element of shell in equilibrium.

External Force:W = external pressure, lb. per sq. in.

Normal Forces:(Positive when acting in the positive direction on the face of an

element facing the positive direction)P, = normal force acting on a unit of length of the face of the ele-

ment lying in a plane normal to the circumferential tangentto the shell, lb. per linear in.

P, = normal force acting on a unit of length of the face of the ele-ment lying in a plane normal to the longitudinal tangent tothe shell, lb. per linear in.


Shearing Forces:(Positive when acting in the positive direction on the face of an

element facing the positive direction)S,y = shearing force in the y direction on the unit of length of the

face of the element lying in a plane normal to the circum-ferential tangent to the shell, lb. per linear in.

Sy, = shearing force in the s direction on a unit of length of theface of the element lying in a plane normal to the longitudinaltangent to the shell, lb. per linear in.

S, = shearing force in the z direction on a unit of length of theface of the element lying in a plane normal to the circum-ferential tangent to the shell, lb. per linear in.

Sy, = shearing force in the z direction on a unit of length of theface of the element lying in a plane normal to the longitudinaltangent to the shell, lb. per linear in.

Couples:(Positive where the force of the couple farther away from the

center of the shell is positive)M, = bending couple or moment resulting from the distribution of

normal forces on a unit of length of the face of the elementlying in a plane normal to the circumferential tangent to theshell, in.-lb. per in.

M, = bending couple or moment resulting from the distribution ofnormal forces on a unit of length of the face of the elementlying in a plane normal to the longitudinal tangent to theshell, in.-lb. per in.

M,y = twisting couple or moment resulting from the distribution ofshearing forces on a unit of length of face of the elementlying in a plane normal to the circumferential tangent to theshell, in.-lb. per in.

M,. = twisting couple or moment resulting from the distribution ofshearing forces on a unit of length of the face of the elementlying in a plane normal to the longitudinal tangent to theshell, in.-lb. per in.

9. Internal Fortes and Couples in Terms of Displacement.-On thebasis of Hooke's law the normal and shearing forces may be expressedin terms of the strains and detrusion as follows:

21 22, 34

EtP" = (=, + .. ,)

1 -


EtP. = - (e, + py,) (5)

1 - p2

EtSa = ) y,,. (6)

The moments in terms of displacement, z, are:

El ( 2z p 02z z (My = -- - + - + - (7)1 - \ y2 R2 002 R

El 1 a2z z 02zM. = - - (-- + - + -y (8)1 - IA2 R 2 a62 R 2 ay2 )

EI 1 a2zM., = -- - (1 - () 9)

1 - A 2 R O9Qy

In the derivation of Equation (9) it is assumed that the dif-ference between M,y and My, is negligibly small. If this assumptionis not made, the left side of Equation (9) becomes Y (M, + My,).

The values for e,, ey, -y from Equations (4), (5), and (6) are sub-stituted into Equation (2). The result is

a2P, 02Pj 1 02Py 2 0aP,- + _

ay 2 ay 2 R2 02 R2 9022 (1 + I) a2Sa, Et 02z

- (10)R M06y R (y2

10. Equations of Equilibrium.-The conditions of equilibrium ap-plied to the force system shown in Fig. 2, neglecting differentials ofhigher order than the second, give the following equations:

OP. + Sy, a4- + + - = 0 (11)

as ay Os

aP.. OS... 02z- -.. S, -- =0

Oy as Oy2


OS., asy, 0a4' z az a2z+ - W +P--P P- -S P, Sy (13)

as ay as y 2 1say 3y)s

aMy 9M,.SW= - + -- (14)

ay as

aM, aMy,S az = - (15)

Os 9y

SM 8M2z-Sy - -Sy + M + My. - = 0. (16)

as ay 2

If products of forces and moments with the derivatives of dis-placements, which are small by comparison with unity, are neglected,and the shearing forces are eliminated from Equations (11) and (12)by means of Equations (14), (15), and (16), then, with M,y = My,,the following equation is obtained:

a2P, a2Py 1 + E, aM,. 2 (1 + e.) a 2M.,+- + +- = 0. (17)

Os2 ay 2 R as2 R OsOy

Equations (14) and (15) are combined with Equation (13); then

a2My a2M, 1 a,2M. 9

-- +2 + -= W+P,--ay 2 asay Os2 Os

a2z a2z 02z-P y -- Sy - Say-- . (18)

ay 2 OsOy OsOy

11. General Differential Equations.-The moments from Equa-tions (7), (8), and (9) are substituted into Equation (18); then

El 4z a 02z 2 8 4z++R

1 - Ap \ ay 4 R 2 y2 2 R a02 y 2

1 a4z 1 02z 1 Oi9+ - + = W + P-

R 4 a04 R 4 092 ) R 0692z 1 O2z 1 82z

- Pay -- - Sy. y Sy. R y (19)Oy2 R O99y R O00y


A second equation is obtained from Equations (17), (8), and (9):

1 021P, 2P, (1 + e,) El ( 1 04zR2 0'02 y2 R (1 - j2) R4 a64

1 02z (2 - M) )^z \+- 4 02 - R 2 0+2 Y = 0. (20)

A third equation is obtained from the equation of compatibility (10),a2z

combined with Equation (12), in which the small quantity Sy,--is neglected:

Oa2 , a2P, 1 O2P, y 02P. Et 02z-- + (2 + ) --- - - = - - (21)

Oy2 ay2 2 2 02 R Oy2

A fourth equation is obtained by differentiating the strain

1 1 /9v-, = (P -,P ) =- -+z

Et R (0a /

twice with respect to y, giving

02P , 02P, El ( 3v 2 (22)y2 A+ 9 (22)9y2 y R Vy2 80 y

The simultaneous Equations (19), (20), (21), and (22) representthe relations between the radial and circumferential deflections zand v, and the forces P,, Py, Sy., S,, and W, for a thin cylindricalshell which is not stressed beyond the elastic limit and which hassmall deflections.

To determine the external pressure W, the end load P,, theshearing stress S,y (or S,,), or any combination of these forces atwhich the equilibrium of the shell becomes indifferent, the shell isconsidered in the deflected shape which satisfies the foregoing dif-ferential equations and the boundary conditions. The function, z,representing this deflection must meet the requirement that itsmagnitude is indeterminate for some value of the external force.This value of the external force at which the condition of neutralequilibrium prevails is the critical load on the cylinder.

18


III. SOLUTION OF BUCKLING EQUATIONS FOR ROUNDCYLINDERS, WITHOUT STIFFENERS

12. Uniform Pressure Applied to Sides Only.-For this type ofloading the value of the circumferential force per unit length ofshell, P8, may be expressed as

P. = - WR + f (y, s).

WR is the average value of P, and f is a function of y and s whichexpresses the variation of P, from the average value. When thedeflection, z, of the shell is very small, then f (y, s) is also very small.

The longitudinal force, P,, has an average value of zero, for thisloading and the value of P, at any point departs from this averageby a small amount g (y, s), dependent upon the deflection z. Thisgives

P = 0 + g (y, s).

The shearing forces S,, and S,. have average values of zero forthis case of loading with variations from the average value byamounts h (y, s) and j (y, s), respectively:

S, = 0 + h (y, s) S,, = 0 + J (y,s).

The values of P,, Py, S,y, and Sy, are substituted into Equations(19), (20), (21), and (22), and products such as f with terms in9^ S02z 02z z

-- other than unity, g - , h --- , and j -- are neglected.90 9say s3y 9s3y

Then

El 4z a2-z 2 04z 1 ('z 04z++ R +

1 - 2 Ly 4 .2 R y, R2 Oay --Z 002 ( 0Z/J

1 1 a 2z z

R R - (23) R

1 a 2f 02g (1 + ,) El (1 04zR 2 2 y R3 (1 - 2) R2 004

1 02z a 4 Z+ - -- + (2 - ) - = 0

R2 a02t d'dy2 /


a2g 1 a'g a2f J a2f Et a2z

(2 + )-- +-- +- - -R =- (25)ay 2 R2 002 by 2 R2 002 R By2

02f 2ag Et ( v a0z )Y _ = + (26)

0y2 ay2 R 0 y20 0 by2 /(a) Edges of Shell at Ends Simply Supported.-The boundary

22z 02zconditions for z are z = 0, - = - = 0 for all values of 0 when

ay2 a62L fz

y = -. Because of symmetry - = 0 for all values of 0 when2 z 00

y = 0, and -- = 0 for all values of y at 0 = 0.90 v(v

For the circumferential displacement, - = 0 for all values of 0L 00

at y = --

These conditions suggest solutions of the form,

ryz = A cos NO cos --

L(27)

vyv = B sin NO cos --

L

Figure 3 shows cross-sections of a shell deflected in various waysso that N = 2, 3, and 4, giving respectively two, three, and fourlobes. The number of lobes has been found to depend upon theproportions of the shell and for any given shell will be that numberwhich will give the lowest pressure at neutral equilibrium.

When z and v are substituted into Equations (23), (24), (25),and (26) the following equations result:

El r 4 (2N2 - ) 7r2 N 2 N 4 -ry--- -- --- - A cos NO cos -

1 - /A L2 RR R 4 R 4 LW 2y 1

= -- (AN 2 + BN) cos NO cos - - f (y, s) (28)R L R


Edrges Simp/y Supported' Edges Fie'dSymmetri/ca/ About < Symmetrica/ About &

FIG. 3. SECTIONS OF DEFLECTED CYLINDERS

1 a2f 02g (1 - e,) EI [N 4 N2R2 002 Oy2 R3 (1 - 2) R .- R2

+ ( 2 - ) - A cos N cos - (29)L 2 J L

a2g 1 a2g yf(2 + ) - + - - + -

4y 2 R 2 002 Oy2

11 af Et r2 ry- - = -- - A cos NO cos -- (30)

R2 002 R L2 L

ayf 02g El 7r2-- - = -- -- (BN + A) cos NO cos-. (31)By2 ay 2 R L 2 L


From Equation (28) it follows that f (y, s) has the form

f (y, s) = C cos NO cos --L

and from Equations (29) and (30) that g (y, s) has a similar form,

g (y, s) = D cos NO cos --L

From Equation (31)R

BN + A =-- (C - pD).Et

The values of C and D are found to be

C EtA Roa

Et WR\ a+l+u- [N'{1+(X-1) (2-u)-1 ] 1

R3(1 -~) Et Xa

D EtA RXa

Et / WR\ l-M(a-l)+ [N2{1+(x-1) (2--)}-1] 1( E) - (-R3(1 -2) Et Xa

in which

N 2L 2

and N 2L 2

a =--+ 1.72R2


From Equation (28)

Et - pEI- (-.l-) [N21+(X-1)(2-)} -1] A cos NO cos--R2a' 2 R'(1 -_ ) ha LEl ry W ry

+ EI N2 [N22-(X -1) -] A cosNOcos- =--FA cosNOcos- (32)R4 (1 -,2) L R L

where1 .

F=N-1 N -1 +a

2 Xa

R-( 1 - [N2{1 +(X- 1)(2-A)} -1] 1-- [a(1-Ag)+ (1+A)2]+a+1+.R2(1-/.2)tXa1 E,)

Equation (32) indicates that solutions, different from zero, existonly if

Et El+-- N2-N2X -,(X-1)-1}

Ra 2 R 3 (1 -. 2)

- +l [N {1+(X- 1)(2--.)}-1 ]W = W,= F . (33)

As the uniform external pressure on a round cylinder increasesfrom zero to the value W, the cylinder remains round and in stableequilibrium until W reaches the critical value We. At that pressuresmall variations in the internal forces f, g, h, and j become possiblewith indeterminate deflections of the cylinder from the round. Theboundary values for displacements at the ends come into effectanalytically only at the critical load. Actually, on account of thebulkheads (Figs. 12 and 20) the values of z and v at the ends areapproximately zero during the whole range of values of W fromzero to We.


Values of LFIG. 4. COLLAPSE-COEFFICIENTS; ROUND CYLINDERS WITH PRESSURES

ON SIDES ONLY, EDGES SIMPLY SUPPORTED; / = 030

t3

Since I = -- an12

in which

D-= , Equation (33) may be written as

2t3 t

We = KiE- + K2 E--D 3 D

a+- 1 +--N2{N2W-,(X-1)- l- [N'{1+(X-1)(2-A) -1]

2 aX

F(1 -M 2)

2K 2 = --

afF

(34)

(35)

(36)


For the range of values D/t and L/R considered in this bulletin,F may be approximated by N 2 - 1. In order to plot values whichgive comparisons for various values of N, Equation (34) is written as

W. = (K +K 2 , E-- = KE( . (37)t2 D) D 3

This form of the equation is convenient to use, whether the valuesof K be plotted in the form of charts, or arranged in tabular form.

Equation (37) gives the uniform external pressure at which around cylinder may collapse into N lobes. The number of lobesgiving the minimum value of W has been found by plotting curvesfor K and L/R for various values of D/t and N. Figure 4 shows afamily of K-curves for A = 0.30. The curves are shown only in theregion of the minimum value of K.

(b) Edges of Shell at Ends Fixed.-Boundary conditions for thisaz av L

case are z = 0, - = 0, - = 0 for all values of 0 at y = +-ay 80 9 z 2

From symmetry and continuity, - = 0 for all values of 0 at y = 0Oz Oy

and - = 0 for all values of y when 0 = 0. A solution of Equa-a0

tions (23), (24), (25), and (26) satisfying these boundary conditions is

z = A cos NO cos -- + Ac cos NO cosh --aL aL

(38)7ry Try

v = B sin NO cos -- + Bc sin NO cosh --bL bL

From the boundary conditions and Equations (38) a relation betweena and b is found,

1 7r 1 r-tan- +-tanh -- = 0. (39)a 2a b 2b

1The curves in Fig. 5a show the relation of - tan - to a and

1 i a 2a- tanh- to b. Values of a and b have been chosen from these

b 2bcurves so that Equation (39) is satisfied. The corresponding rela-tion between a and b is shown in Fig. 5b.


tql0 (N 0N 0 N ( IM0 0O0OO

i 00 0 000I - - .. * -; -.- u -;- 0000

co 0 000 0co0o0o COO000 0 000 0000000 0000

00 - -

0-X C) p - 01 CO .C i


FiG. 24. SPECIMENS 2, 3, 4, AND 5 AFTER TEST, AND Low PRESSURE APPARATUS

Deflection measurements were made on Specimens No. 17, No.18, No. 1, No. 10a, and No. 10b. The results of these tests are dis-cussed individually in the order in which they were obtained.

Specimen No. 17. This specimen was a 12-in. diameter tubeformed from 21-gage (0.028 in. thick) sheet. The circumferentialseams were about 36 in. apart, giving a total length of about 144 in.for four sections. The net length (clear distance between bulkheads)was 143 in. Pressure was applied to both the sides and the ends,i.e., the end thrust from the bulkheads was carried by the specimen.Figure 29 shows the plotted values of deflection together with themaximum stresses. Apparently the edges were about half-way be-tween fixed and simply supported. The deflection was so gradualthat the tube could be made to breathe in and out, reproducing thesame deflection at given pressures whether the load was increasingor decreasing. This specimen was used for demonstration for sometime, and was deflected (A = about 0.30 in.) probably fifty times.

Specimen No. 18. This specimen was the former Specimen No. 17stiffened with two single angle (0.30 x 0.25 x 0.043 in.) stiffeningrings, one on each side at 18 in. from the center of the tube. Therings were made from 3S-3/4H material, and were attached to theshell with No. 2 machine screws spaced 1 in. center to center. These

COLLAPSING PRI -,URE OF THIN-WALLED CYLINDERS

FIG. 25. SPECIMENS 6, 7, 8, AND 9 AFTER TEST

rings were intended to give about the smallest size of stiffener whichwould develop the intermediate shell. Specimen No. 18 was testedunder the same conditions as Specimen No. 17.

Figure 30 shows the deflection and stress. Note the agreementbetween measured and computed values of radial deflection of thestiffeners. The deflections shown are the average values for the twostiffeners, each of which did not deviate more than 0.005 in. fromthe average. Until collapse, the deflection of the shell midway be-tween the stiffeners was about 20 per cent greater than the corre-sponding deflection at the stiffener. Figure 21 shows the specimenafter failure. The lower stiffener buckled and the lower end-sectioncollapsed before the center section collapsed. No measurements ofdiameter of the lower end section were obtained for this test. Sincefailure occurred by buckling of the outstanding flange of the stiffenerthere is no decisive evidence that the moment of inertia of the com-


FIG. 26. SPECIMENS 11, 12, 13, 14, 15, AND 16 AFTER TEST

bined stiffener and plate was adequate to develop the strength ofthe short lengths of the tube between stiffeners or between stiffenerand bulkheads.

Specimen No. 1. After Specimen No. 18 was tested the tube wasbrought back to a "nearly round" condition by hammering from theinside and by straightening the outstanding leg of the stiffener. Theoutstanding leg of the stiffener was then cut down to a height of1 in. to make Specimen No. 1. The same measurements were madeon this specimen as on Specimen No. 18. Pressure was applied tothe sides only, the end thrust being carried by an iron pipe insidethe specimen, as previously described.

Deflection and stress curves are shown in Fig. 31. The computed


FIG. 27. SPECIMENS 20, 21, AND 22 AFTER TEST

FIG. 28. SPECIMENS 23, 24, 25, 26, 27, 28, AND 29 AFTER TEST


~1

I,)iiK

N

I..

k~4J

/.0

0.8

06

0.4

0.2

u z.i

0 200

0.4 0.6 0.8 /. .2Rad/a/ Def/ect/on /Z /nches

0 4000 6000 8000 /0000 12000aw/K,/mum Stress /l /b. pver sq. 1'?.

FIG. 29. RADIAL DEFLECTION AND MAXIMUM STRESS CURVES, SPECIMENNo. 17, WELDED ALUMINUM ALLOY TUBE

collapsing pressure (Table 2) is determined as follows: assumingfailure as collapse of a short tube between stiffeners simply sup-ported at the stiffeners,

L 36 D 11.92- = - = 6.05, - = - = 425, K (from Fig. 4) = 17,R 5.96 t .028

We = KE = 17 X 107 X (425 = 2.20 lb. per sq. in.SD ) (425) 3

Between the stiffener and the end the clear length is 53.5 in. Forthis section one edge is fixed and the other simply supported.

The equivalent thickness t, for the member, considered as a long1

tube, is computed from I, = -- bt. 3 in which b = 36 in. and I,, the12

total moment of inertia, is 0.000098 in.4 For this computation a

I I I I I I I I I I I IUnder uniform exferna/ pressures on s/des and' ends

A/l ma/era/ 33S4H.Loni//ud'/na/ and carcumferen/'a/ jo/insf buff we/dea, hammered, and dressed'

Diameter //.90n. She// f/71ckness 0.082n1. Net len/th /43/n.=42 A, = 0.092 i-

Wc=083 lb. per sq in. - edges fi.\ed~c= .S7 lb. per sq. in. - edges simp/' supported

Ediges Fixeda'y,\ l _I f r(Measu.red PressureIr - at Co//apse/ -MeCasuered Deflecteon I

-- -- - Computed Oer/ec/on e_~ -- - - - Comiroed MaxI~'/?7tmF Stress

/t.

0.

0.

Ks

0.

2

0

8

6

4

2


--Unoer /2'form exfer'a/"/ pressures on s/des and' ends-Al// mater/,/ 3S71H.

- 17n9it/udaf/'na/ and c1rcunfe,--ren/2 7 o2/nsbu' wve/dea', haMmmerea, and dresse. I

- D/'a1meter /L.9V0. She// thi/cness 0.029 / Me /ent lelh /43/n. -Sf/fetners spaced 36/ . apCoar and S5.S /z from bu//lhead

- Stffener - S/g,/e ane 0.30"O..ZS"0.04Z. I

Measured__Vet/eat/n/iN

/~~~711

/ __/

___ //

-I-I ,'

/

T-7

.:onq ,'eg ourstanaI/n.

Co//apse5 b/ buckc//nq of outs. andn'g

S leg of Siffener

A., 0048 /t. for Sti ffener.S=/ / per sq "/ for S Iffened6 ebe.

I I __ I _ _ __ __ _

0 0.4 0.8 1.2 /.6 2.0 2.4Rad/ia/ Def/ect/on /,?'nches

0 2000 4000 6000 8000 /0000 /2000/ax/ri.t"am Stress /n StiffenTer / /&. per sq. /,.

FIG. 30. RADIAL DEFLECTION AND MAXIMUM STRESS CURVES, SPECIMENNO. 18, WELDED ALUMINUM ALLOY TUBE

length of 30t is assumed as acting with the stiffening angle. Thent, = 0.032 in., D/t, = 372, L/R = 24, K = 4.2, and W, = 0.815 lb.per sq. in. Since this value is less than 2.20, the tube is expectedto fail as a whole.

Stress in the stiffener flange is computed from Equation (63).The maximum moment is

3Ao WM. = El,

R,2 W, - W

The bending stress is M,c, I,. The average bending stress in thestiffener is computed by using c, = 0.057 in., the distance from


1.0

u2

04

.2

__ Under un/forrm externa/ pressure on sides o0ny.A/l mater/al 3S/H.

__ L Zong/fudi/'na/ and c/rcumferen/a/jointsI butt we/deao, 1hammerea, ard dresseO'

_ D/ameler 1//90 i1n. She/I th/'cAness 0026in. Met /et7,/ /43 in. _

__ ^ tr/' / ,e U' (JCi i/ i ./rlq.A,=O.OZ61 n. for sl ffener

W, =0.8/ lb. per s1 q h. for sriffened tube.tred Pressure at Co//apse

-- 9 i I i-^

r-Cmpfe/ 00Pcio

A

'~-A1easured /9ef/ec-t/oa

~eas~

0.4 0.8 1.2 1.6 2.0 2.4Rao'/a/ Def/ect/on n /Inches

0 O000 4000 6000 8000 /0O000 /000Mlaximum Stress in OutsAna'd/n' F/an~ge of Stffener

1 /b. per sq. i?.

FIc. 31. RADIAL DEFLECTION AND MAXIMUM STRESS CURVES, SPECIMENNo. 1, WELDED ALUMINUM ALLOY TUBE

the neutral axis of the stiffener section (angle and plate) to the mid-point of the outstanding flange. With Ao = 0.026 and R, = 5.98,

Wthe average stress is S = 1250

We - W

For W = 0.715,

For W = 0.74,

S = 8 950 lb. per sq. in.

S = 12 300 lb. per sq. in.

This stress is above the proportional limit, hence the resistance isreduced. By Equation (67)

E'l--- = 1E

1 /12300 - 9000 24 25 -- = 0.995.4 25 000 - 1000

I


Under uniform externa/ pressure on sides on/y.All mater/a/ 3S-H F

Diameter 6.00 I'. Shel//l thckness 0.042 i?. Ale/ /en/L7 11/4 n.L .,,., 0 7 -/'#J

- Wc = 8.5 lb. per sq. in. - edges fixed.We = 68 lb. per sq. in. - edges si/np/y supported.

E dges Simp/ly Supported

'0.0

8.0

6.0

4.0

2.0

06 0.8 1.0 .2Def/ect/on in /nches

6000 8000 /0000 . 12000Stress in /b. per sq. 17.

FIG. 32. RADIAL DEFLECTION AND MAXIMUM STRESS CURVES, SPECIMENNo. 10a, EXTRUDED ALUMINUM ALLOY TUBE

The corrected We is 0.995 X 0.815 = 0.810 lb. per sq. in.

0.74Then s = 1250 X - = 13 200 lb. per sq. in.

0.07

with a corrected value for E'/E = 0.993.

Hence We is very nearly 0.74 lb. per sq. in.

Specimen No. 10a. Specimen No. 10a was one of the 6 in. x 0.042in. extruded tubes, 10 ft. long over-all. The pressure was appliedon the sides only, while the end thrust was carried by an inner ironpipe which was 9 ft.-6 in. long, the effective length of the specimenthus becoming 9 ft.-6 in. External pressure was applied graduallyuntil the deflections started increasing rapidly. The pressure wasthen released and later a demonstration test (second test) was made

t~)

'N

viiI'

N

*1~k14.1

0.? 0.4Radi'a/l

2000 4000lvaaximumn

I-- Measurec uer/ection, /sr /0oaa/ny.-- .---- Measured Oef/ect/on, Znd Loadkng.--

--- - Computed Def/ection.

I A


'S

k.'3

'3

I',

'3I'

N

1..

4/

6

4

0

4 -1 //I -Ed es Sim vi Suppooree/

K/

-A-I

I

---- (.~?mputed ,'-Za'.4-,'mum ~5tressfor Edg~es F,~ed.

7 /-9- -- - - - - -/

-c-- A1e~zwred 'ef/ectAn.- - -- Comp~ded Def/ectk't.'.

0 0.2 0.4 0.6 0.8 1.0 1.2Racd/a/ Def/ecft/on ,7 /nches

0 800 /600 2400 3200 4000 4800/aximum Stress 7n lb. per sq. in.

FIG. 33. RADIAL DEFLECTION AND MAXIMUM STRESS CURVES, SPECIMENNo. 10b, EXTRUDED ALUMINUM ALLOY TUBE

in which the loading was continued until collapse occurred. Afterthe first test the central section of the tube retained a slight perma-nent set, not more than 0.010 in., which was practically removedby pressing this part of the tube by hand into an elliptical shapeso as to oppose the permanent set. The data are plotted in Fig. 32.

The bulkheads for sealing the ends of this specimen were madeand fitted for it. Therefore the edges at the ends of this tube wouldbe expected to be nearly fixed. The values computed for the con-dition of fixed edges agree with the test data much more closelythan do the values based on simply supported edges. The computedcollapsing pressure for fixed edges is 0.805 lb. per sq. in., which isclose to the actual pressure, (0.811 lb. per sq. in.).

- ^- - ,i'qea7sLred P-Mressvr- a7 L,,/t/jse

Under uniform externa pressure on s/des on.y.Al4/l maer/o/ 3S-H

D/ameter 6 00 In. She/ll fhicness 0042in. Net length 69 zn.- =23.0 -- =/43 Ao=0.008 //.

W-- -c=/2.0 /b. per sq. . - edges fred - -Wc = 8.8 lb. per sq in. - edges simp/y supported

daes :'d-, . ....

I

r

68

O


Specimen No. lOb. This specimen was 6 ft. long, cut from oneend of Specimen No. 10a, and was tested without end pressure. Theinner pipe was 5 ft.-9 in. long, making that the effective length.One end of this specimen fitted the bulkhead snugly, but the btherdid not, consequently the fixation of the edges was not complete.Nevertheless, the tube, which was slightly out-of-round, carriedmore than the computed value for a perfect tube with edges simplysupported.

The test data are plotted in Fig. 33 together with computedvalues. The values of deflection are appreciably smaller than thecomputed values based on the measured 0.008 in. initial radial de-viations from a round cylinder with fixed edges. Since the initialout-of-roundness was small, reasonable errors in measuring this valuewould be more than enough to account for the discrepancy. At apressure of about 8.9 lb. per sq. in., the measured deflections showa rapid increase which indicates a value of We less than that forfixed edges.

VIII. SUMMARY AND CONCLUSIONS

29. Summary of Analytical Results.-The collapsing pressure perunit of area, We, for thin-walled round cylinders under a uniformexternal pressure is equal to the product of the collapsing stiffness,KE, and the cube of the t/D ratio (thickness to diameter) of thecylinder:

W, = KE (t/D)3

E for linear elastic action (Hooke's law) is the modulus of elas-ticity of the material in the shell, and for non-linear or plastic be-havior a modified modulus denoted by E' is used.

Curves showing values of K for varying t/D and L/R (length toradius) ratios under several load and edge conditions are given in thefigures listed:

Load Edges at Ends FigureUniform pressure on sides only Simply supported 4Uniform pressure on sides only Fixed 6Uniform pressure on sides and ends Simply supported 8Uniform pressure on sides and ends Fixed 9

Axial End Load.-For linear elastic behavior the collapsing forcePc per unit of length of circumference, for a round cylinder under anaxial load distributed uniformly around the circumference, is given


by Equation (47). For small values of t/R (thickness to radius)and Poisson's ratio u = 0.30 Equation (47) reduces to

Sc = 0.6E-- (49)R

Pcwhere Sc = - is the stress per unit area in the shell at the col-

tlapsing load.

Stiffened Cylinders.-For round cylinders stiffened with rings, thecollapsing pressure is computed from We = KE(t/D)3 for a lengthof cylinder L,, between the rings, provided the rings are sufficientlystrong and stiff so that they do not fail. The flexural stiffness,EI,, of the stiffener which is necessary to withstand a pressure W,is given by Equation (53),

W,D 3L,El, = -- *

24

I, is the moment of inertia of the ring section and that part of theshell which has been assumed to act with the ring.

Cylinders "Out-of-Round".-The radial deflection of a cylinderslightly out-of-round proceeds gradually to the final failure. Undera uniform external pressure, W, the deflection A is expressed byEquation (55),

WAoA=

W, - W

Ao is the maximum initial deviation of the out-of-round cylinderfrom a round one. The initial radial deviations are of the samegeneral pattern as the finally deflected shell.

The maximum circumferential stress in the shell is given by Equa-tion (57), and the maximum pressure that causes a given stress isexpressed by Equation (61).

Non-linear or Plastic Action.-If the action is non-linear or plas-tic, a modified modulus of elasticity E' is proposed for use in thecollapsing formula for We. Two cases are of note: (1) when theaverage stress in the shell or stiffener is less than the proportional


limit Q* of the material the modulus is reduced according to Equa-tion (67) page 43.

E' = E 1 - 4 - $

in which S is the maximum stress (direct and bending) in the shell,S, is the modulus of rupture of the material, and S1 is the averagestress in the shell; (2) when the average stress is greater than theproportional limit, E' is estimated to be equal to the tangent modulusat a stress equal to the average stress. The use of the modifiedmodulus is illustrated in the computations for the collapsing pres-sure of an actual tube, specimen No. 1, on page 62.

For combined external pressure and longitudinal stress the effectof biaxial stress upon the inception and nature of plastic action mustbe considered. The stress causing a start of plastic action undercombined stress may be computed from Equation (69),

S2-S 1SSp + Sp2 = S 2.

30. Comparison of Analytical With Experimental Behavior of Alu-minum Tubes.-The values compared are the following ratios ob-tained from Tables 2 and 3.

S = average collapsing pressure by analysis, simply supportededges.

F = average collapsing pressure by analysis, fixed edges.

T = average collapsing pressure by test.

Number ofLoad Roundness Values T/S F/T F/S

Averaged

Sides only Round and out- 15 1.05 1.33 1.40of-round

Round 5 1.04 1.36 1.43

Out-of-round 10 1.16 1.17 1.36

Sides and Ends Round and out- 14 1.02 1.44 1.47of-round

Round 4 1.14 1.31 1.50

Out-of-round 10 1.16 1.20 1.40

No tests of shells with end pressure only were made as a part of this investigation. Tests re-ported in Bulletins 255 and 292 of the Engineering Experiment Station of the University of Illinoisresulted in a wrinkling stress considerably smaller than 8& given by Equation (49).

*See Section 6, Notation.


31. Conclusions.-From the analytical results, the test data, anda comparison thereof, the following conclusions are drawn:

A. Effect of Edge Restraint

(1) A fixed-edge condition as expressed analytically has a markedeffect on the collapsing pressure of a round thin-walled cylinderwith an L/R ratio between 1 and 80 and a D/t ratio between 20and 1000, for uniform side pressure only or uniform side and endpressure.

For a given collapsing pressure the maximum effect is equivalentto increasing the L/R ratio by 40 to 50 per cent.

(2) Analytical values of the collapsing pressure for both roundand out-of-round cylinders with edges simply supported are on theaverage less than the values obtained by test for both types of load,uniform side pressure only, and uniform side and end pressure.

(3) The analytical values of the collapsing pressure for both roundand out-of-round cylinders with fixed edges are considerably higheron the average than the test values obtained with both types of load.

(4) Test results indicate that some degree of restraint existed inmost of the tests but the degree of restraint could not be definitelydetermined.

B. Effect of Out-of-Roundness

(5) Analytically the effect of out-of-roundness varies with theratio of initial radial deflection Ao to the thickness t of the shell,the L/R (length-radius) ratio, the t/D (thickness-diameter) ratio,the elastic limit of the material, and the bending modulus of failureof the material. An expression of the effect is obtained for an initialout-of-roundness of the same pattern as the deflected shell at buckling.

C. Effect of Ring Stiffeners

(7) Analytically the collapsing pressure of thin-walled cylindersstiffened with circumferential rings may be computed as the col-lapsing pressure of the short lengths of cylinder between the stiffeningrings, provided the stiffeners have sufficient strength and stiffness.A rigorous determination demands that uncertainties as to bendingstiffness in the stiffened shell and as to kind of stress distributionin the stiffeners be removed. Tests on tubes with stiffeners are toofew in number to draw definite conclusions.


D. Effect of Inelastic Action

(8) The expressions for the collapsing pressure may be appliedto shells stressed beyond the elastic limit of the material if a modified"effective" modulus of elasticity is used.

The use of an empirical formula such as Equation (67) basedupon the behavior of straight members under compressive loads fortaking account of inelastic action in the curved shell needs furtherjustification but may serve for design purposes until a more satis-factory method is developed.

E. Final

(9) The results obtained point to the need for the study of rathersharply curved members under buckling loads when the maximumstress in the member exceeds the elastic limit of the material; forfurther study on the effect of stiffeners on the collapsing pressureof tubes; for tests on tubes with measured restraints as to displace-ments and rotations at the ends; for further analytical and experi-mental investigation of the out-of-round tubes; and for tests ontubes subjected to combined external pressure and axial loads,particularly tensile loads.

BIBLIOGRAPHY

No. YEAR

1848

June19061917

1906

Sept.-Dec.1931

Sept.-Dec.1931Nov.1934

1932

1933

Feb. 28,1933

Nov.1934

1888

1913

May1913Sept.1913Jan.1915

AUTHOR

William Fairbairn

A. P. Carman andM. L. Carr

A. P. Carman

R. T. Stewart

T. McLean Jasperand John W. W.Sullivan

H. E. Saunders andD. F. Windenburg

D. F. Windenburgand C. Trilling

E. E. Lundquist

E. E. Lundquist

W. M. Wilson andN. M. Newmark

L. H. Donnell

G. H. Bryan

R. V. Southwell

R. V. Southwell

TITLE AND REFERENCE

"The Resistance of Tubes to Collapse,"Philosophical Trans., Vol. 148, pp. 389-413.

"The Resistance of Tubes to Collapse,"Bull. No. 5, Eng. Exp. Sta., Univ. of Ill."The Collapse of Short Thin Tubes,"Bull. No. 99, Eng. Exp. Sta., Univ. of Ill."Collapsing Pressure of Bessemer SteelLap-Welded Tubes, Three to Ten Inchesin Diameter," Trans. A.S.M.E., Vol. 27,pp. 730-822.

"The Collapsing Strength of Steel Tubes,"Trans. A.S.M.E., Vol. 53, APM-53-17b,pp. 219-245."Strength of Thin Cylindrical Shells UnderExternal Pressure," Trans. A.S.M.E.,Vol. 53, APM-53-17a, pp. 207-218."Collapse by Instability of Thin Cylin-drical Shells Under External Pressure,"Trans. A.S.M.E., Vol. 56, APM-56-20,pp. 819-825."Strength Tests of Thin-Walled Dura-lumin Cylinders in Compression," Tech.Memo. No. 427, of the Natl. Advis. Com.Aero.

"Strength Tests of Thin-Walled Dura-lumin Cylinders in Compression," ReportNo. 473, of the Natl. Advis. Com. Aero."The Strength of Thin Cylindrical Shellsas Columns," Bull. No. 255, Eng. Exp.Sta., Univ. of Ill.

"A New Theory for the Buckling of ThinCylinders Under Axial Compression andBending," Trans. A.S.M.E., Vol. 56,AER-56-12, pp. 795-806."Application of the Energy Test to theCollapse of Long Thin Pipe Under Ex-ternal Pressure," Proc. Cambridge Phil.Soc., Vol. VI, pp. 287-292."On the General Theory of Elastic Sta-bility," Phil. Trans. Royal Soc. (London),Vol. 213, Series A, pp. 187-244."Collapse of Tubes," Phil. Mag.,pp. 687-698;

pp. 502-511;

pp. 67-77.


No.

15

16

17

18

19

20

21

22

23

24

25

26

27

28

YEAR

July1914

1914

1929

Aug.1933

1920

19291931

1922

May1928

July1933

April1935

May1935

1924

1889

18951890

1891

AUTHOR

Gilbert Cook

R. von Mises

R. von Mises

D. F. Windenburg

K. von Sanden andK. Giinther

T. Tokugawa

H. M. Westergaard

H. M. Westergaard

H. M. Westergaard

A. Foppl andL. F6ppl

E. E. Lundquist

E. E. Lundquist

T. Claxton Fidler

F. Engesser

A. Considere

TITLE AND REFERENCE

"The Collapse of Short Thin Tubes byExternal Pressure," Phil. Mag., pp. 51-56."Der kritische Aussendruck zylindrischerRohre," Vol. 58, pp. 750-755, V. D. I.Zeitschr."Der kritische Aussendruck fir allseitsbelastete zylindrische Rohre," A Stodola-Festschrift. Translated and annotatedby D. F. Windenburg, Report No. 366 ofthe -U. S. Exp. Model Basin, Navy Yard,Washington, D. C."Ueber das Festigkeitsproblem querver-steifter Hohlzylinder unter allseitig gleich-mitssigem Aussendruck," Werft UndReederei, Vol. 1, No. 8, pp. 163-168;Vol. 1, No. 9, pp. 189-198; Vol. 1, No. 10,pp. 216-221."Model Experiments on the Elastic Sta-bility of Closed and Cross-Stiffened Circu-lar Cylinders under Uniform ExternalPressure," Proc. World Eng. Congress,Tokyo, 1929, Vol. XXIX (1931), pp.219-279."Buckling of Elastic Structures," Trans.A.S.C.E., Vol. 85, pp. 576-654."Report on Arch Dam Investigation,"Part 3, Proc. A.S.C.E., pp. 231-266.

"Stress Functions for Shells," Tech.Memo. No. 351, U. S. Bur. Recl.

"Drang und Zwang," Vol. 1, Second Ed.,page 53, Eq. 59.

"Strength Tests of Thin-Walled Dura-lumin Cylinders in Combined TransverseShear and Bending," Natl. Advis. Com.Aero., Tech. Note No. 523.

"Strength Tests of Thin-Walled Dura-lumin Cylinders of Elliptic Section," Natl.Advis. Com. Aero., Tech. Note No. 527.

"A Practical Treatise on Bridge Construc-tion," Fifth Ed., page 162.Zeitschr. Architektur und Ingenieurwesen.(Architekten-und Ingenieur-verein, Hann-over.) Vol. 35, p. 455, Schweiz. Bau-zeitung, Vol. 26, page 24; p. 731, V. D. I.Zeitschr., Vol. 34."REsistance des Pieces comprimbes Con-gres International des Proced6s de Con-struction, Annexe a Comptes Rendus,"page 382.


No. YEAR

1910

March1935

1928

1929

1930

1932

193619251926

1931

1940

1939

1939

AUTHOR

Theodorvon Karman

William R. Osgood

H. M. Westergaardand W. R. Osgood

L. B. Tuckerman

Frary, Edwardsand Jeffries

W. Flugge

S. Timoshenko

W. Lode

A. Nadai

J. Marin andR. L. Stanley

J. M. Lessells andC. W. MacGregor

J. L. Holmquistand A. Nadai

TITLE AND REFERENCE

"Untersuchungen uiber Knickfestigkeit,"Mitteilungen uiber Forschungsarbeitenauf dem Gebiete des Ingenieurwesens, 81,Berlin."The Double-Modulus Theory of ColumnAction," Civil Engr. Vol. 5, No. 3, page173."Strength of Steel Columns," Trans.A.S.M.E., Vol. 50, Part I, paper No.APM-50-9, p. 65."Discussion of Paper by R. L. Templinentitled 'The Determination and Signifi-cance of the Proportional Limit in theTesting of Metals,'" Proc. A.S.T.M.,Vol. 29, Part II, page 538."Aluminum Industries." A chemicalanalysis of the aluminum alloy 3S, andthe significance of the cold work symbolsYH, H and 0, may be found on page 232,et. seq.

"Die Stabilitat der Kreiszylinderschale,"Ingenieur-Archiv., Vol. 3, p. 463."Theory of Elastic Stability."

"Berichte des Werkstoffauschuss," V. D.E. Diusseldorf; also Proc. 2nd Int. Congr.App. Mech., Zurich.

"Plasticity."

"Failure of Aluminum Subjected to Com-bined Stress," Am. Welding Soc. Jour.,Vol. 19, Part I, pp. 74s-80s.

"Certain Phases of the Combined StressProblem," Proc. 5th Int. Congr. App.Mech., Cambridge, Mass.

"A Theoretical and Experimental Ap-proach to the Problem of Collapse ofDeep-Well Casing," Am. Petroleum Inst.,Drilling and Production Practice, pp.392-420.

Documents

STUDY OF THE COLLAPSING PRESSUREOF THIN-WALLED CYLINDERS