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NAVY DEPART'MENT- THE DAVID W. TAYLOR MODEL BASIN

-WASHINGTON 7, D.C.

OBSERVATIONS OF STRESSES AND STRAINS NEAR INTERSECTIONS

OF CONIGAL AND CYLINDRICAL SHELLS

l mr.-lM A

Matthe' F. Borg

Ii

RESEARCH AND DEVELOPMENT REPORT

March 1956 Report 911

420i387-56

OBSERVATIONS OF STRESSES AND STRAINS NEAR INTERSECTIONS

OF CONICAL AND CYLINDRICAL SHELLS

by

Matthew F. Borg

March 1956 Report 911

NS 731-033

TABLE OF CONTENTS

P age

ABSTRACT..............................................................................................

INTRODUCTION .................................................. I................................... 1

DESCRTPTION OF MODELS .................................................. I.............. ...... 2

TEST APPARATUS AND PROCEDURC ......................................................... 5

EXPERIMENTAL RESULTS .. ................................................................... 8

DISCUSSION ..................................................................... I............... 40

Comparison of Experimental Results with Theory.......................................... 40

Features of the Theoretical Strains ........................................................... 40

The Experimental Deviations................................................................... 42Factors Affecting Strain Results..........................................................

The Experimental Precision ................................................................. 44

Theoretical Approximations ................................................................. 44Effects of Imperfections in the Models......................................... ........... 45

CONCLUSIONS ................................................. ......... I.......................... 46

ACKNOWLEDGMENTS ....................................................................... .... 46

APPENDIX A - MODEL IMPERFECTIONS .................................................... 47Out-of-Roundness ................................................................................. 47Generator Rectilinearity......................................................................... 48

Thickness Measurements........................................................................ 48

APPENDIX B- THEORETICAL CALCULATIONS .................................... I........ 60

Comparison of the Theories .................................................................... 60Stress Index Factors............................................................................. 60

REFERENCES.............................................................................. ... 64

This DocumentReProduced FromBest Avaijable Copy

LIST OF ILLUSTRATIONS - W AvaiableCop

Page

Figure 1 - Nomenclature and Schematic Drawing of Cone-Cylinder Models ............... 3

Figure 2 - Range of Parameters for the Cone-Cylinder Models ...................................... 3

Figure 3 - Completed Model Prior to Instrumentation and Testing ................................ 5

Figure 4 - Typical Layout of Strain G ages ...................................................................... 6

Figure 5 - Location of Strain Gages on Cone and Cylinder of Models CC-SA,CC-21, and CC-10 in the Second Test ............................................................ 7

Figure 6 - Schematic Assembly of Feed-Through Device .............................................. 7

Figure 7 - Typical Test Setup Showing Model CC-21 ..................................................... 7

Figure 8 - Apparatus for Measuring Circularity an4 Rectilinearity ............................. 7

Figure 9 - Typical Plots of Experimenial Pressure versus Strain (Model CC-15) ...... 9

Figure 10 - External Longitudinal Strains in Models CC-8 and CC-8A forInternal Hydrostatic Pressure of 1 psi ............................................................ 10

Figure 11 - Internal Longitudinal Strains in Model CC-8A for InternalH ydrostatic Pressure of 1 psi ....................................................................... . 11

Figure 12 - Internal and External Circumferential Strains in Models CC-8and CC-8A for Internal Hydrostatic Pressure of 1 psi ................................ 12

Figure 13 - External Longitudinal Strains in Model CC-9 for InternalH ydrostatic Pressure of 1 psi ......................................................................... 13

Figure 14 - External Circumferential Strains in Modei UC-9 for InternalH ydrostatic P ressure of 1 psi .......................................................................... 14

Figure 15 - External Longitudinal Strains in Model CC-10 for InternalH ydrostatic P ressure of 1 psi ....................................................................... 15

Figure 16 - Internal Longitudinal Strains in Model CC-10 for InternalHydrostatic Pressure of 1 psi (Sccond Test) ............................................... 16

Figure 17 - Internal and External Circumferential Strains in Model CC-10for Internal Hydrostatic Pressure of 1 psi ............................ 17

Figure 18 - External Longitudinal Strains in Model CC-11 for InternalH ydrostatic Pressure of 1 psi ......................................................................... 18

Figure 19- External Circumferential Strains in Model CC-11 for InternalH ydrostatic P ressure of I psi .......................................................................... 18

Figure 20 - External Longitudinal Strains in Model CC-12 for InternalH ydrostatic P ressure of 1 psi .......................................................................... 19

Figure 21 - External Circumferential Strains in Model CC-12 for !ntc:nalH ydrostatic P ressure of 1 psi ....................................................................... 19

Figure 22 - External Longitu-dinal Strains in Model CC-13 for InternalH ydrostatic P ressure of 1 psi ......................................................................... 2 0

Figure 23 - External Circumferential Strains in Model CC-13 for InternalH ydrostatic P ressure of 1 psi .......................................................................... 2 1

Figure 24 - External Longitudinal Strains in Model CC-14 for InternalHydrostatic Pressure of 1 psi ............................................. '22

iv

Figure 25 - External Circumferential Strains in Model CC-14 for InternalH ydrostatic P ressure of 1 psi .......................................................................... 22

Figure 26 - External Circumferential Strains in Model CC-15 for InternalHydrostatic Pressure of 1 psi .............................................. ......... 23

Figure 27 -External Longitudinal Strains in Model CC-15 for InternalH ydrostatic P ressure of 1 psi .......................................................................... 24

Figure 28 - External Longitudinal Strains in Model CC-16 for InternalHydrostatic Pressure of 1 psi........................................................ 25

Figure 29 - External Circumferential Strains in Model CC..16 for InternalH ydrostatic P ressure of 1 psi ......................................................................... . 25

Figure 30 - External Longitudinal, Strains in Model CC-20 for InternalHydrostatic Pressure of 1 psi........................................................ 26

Figure 31 - Internal Longitudinal Strains in Model CC-20 for InternalH ydrostatic P ressu"?e of I psi ........................................................................ . 27

Figure 32 - Internal and External Circumferential Strains in Model CC-20for Internal Hydrostatic Pressui.. of 1 psi (First Test) ...................... 28

Figure 33 - Internal and External Circumferential Strains in Model CC-20for Internal Hydrostatic Pressure of 1 psi (Second Test) ............................ 29

Figure 34 - External Longitudinal Strains in Model CC-21 for InternalHydrostatic Pressure of 1 psi ...................................................... 30

Figure 35 - External Circumferential Strains in Model CC-21 for InternalH ydrostatic P ressure of i psi ......................................................................... . 31

Figure 36 - Schematic Diagram of Purdue University Models 2 and 3 ............................ 32Figure 37 - Internal Longitudinal Stresses in Purdue University Model 3

for Internal Hydrostatic Pressure of 1 psi .................................................... 33

Figure 38 - External Longitudinal Stresses in Purdue University Model 3for Internal Hydrostatic Pressure of 1 psi........................................ 34

Figure 39 - Internal Circumferential Stresses in Purdue University Model 3fur Internal Hydrostatic Pressure of 1 psi .................................................... 35

Figure 40 - External Circumferential Stresses in Purdue University Model 3for Internal Hydrostatic Pressure of 1 psi........................................ 35

Figure 41 - Internal Longitudinal Stresses in Purdue University Model 2fot Internal Hydrostatic Pressure of 1 psi .................................................... 36

Figure 42- External Longitudinal Stresses in Purdue University Model 2?or internal Hydrostatic Pressure of 1 psi .................................................... 37

Figure 43 - Internal Circumferential ',tresses in Purdue University Model 2for Internal Hydrostatic Pressure of 1 psi .................................................... 38

Figure 44 - External Circumferential Stresses in Purdue University Model 2for internal Hydrostaotic Pressure of 1 psi .................................................... 39

Figure 45 - Circumferential Circularity and Thickness Measurements Comparedto Strain Sensitivities at Points 1/4 Inch from the Intersectionon the C one of M odel C C -8A ............................................................................ 49

Figure 46 - Circumferential Circularity and Thickness Measurements Comparedto Strain Sensitivities at Points 1/4 Inch from the Intersectionon the Cylinder o " Model CC-8A ................................................................ 50

V

Figure 47 - Circumferential Circularity and Thickness Measurements Comparedto Strain Sensitivities at Points 1/4 Inch from the Intersectionon the Cone of Model CC-10 (Second Test) ..................................................... 51

Figure 48 - Circumferential Circularity and Thickness Measurements Comparedto Strain Sensitivities at Points 1/4 Inch from the Intersectionon the Cylinder of Model CC-10 (Second Test) ............................................... 52

Figure 49 - Circumferential Circularity Measurements Compared to StrainSensitivities at Points 3/32 Inch from the Intersectionon the C one of M odel C C -21 .............................................................................. 53

Figure 50 - Circumferential Circularity Measurements Compared to StrainSensitivities at Points 3/32 Inch from the Intersectionon the C ylinder of M odel C C -21 ....................................................................... 54

Figure 51 - Longitudinal Profiles of Generators on Coneof M odels C C -8 and C C -8A ............................................................................... 55

Figure 52 - Longitudinal Profiles of Generators on Cone of Model CC-10 ..................... 55

Figure 53 -Longitudinal Profiles of Generators on Cylinder of Model CC-10 ............... 56

Figure 54 -Longitudinal Profiles of Generators on Coneof M odel C C -20 (Second T est) ......................................................................... 56

Figure 55 - Longitudinal Profiles of Generators on Cylinderof M odel C C -20 (Second T est) .................... .i ................................................... 57

Figre 56 - Thickness Measurements on Generators of Cone of Model CC-10 ................ 57

Figure 57 - Thickness Measurements on Generators of Cylinder of Model CC-10 .......... 58

Figure 58 - Thickness Measurements on Generators of Coneof M odel C C -20 (Second T est) .......................................................................... 58

Figure 59 - Thickness Measurements on Generators of Cylinderof M odel C C -20 (Second T est) ........................................................................... 59

Figure 60 - Thickness Measurements on 0-Degree Generatorof Cone and Cylinder of Model CC-8A ............................................................ 59

Figure 61 - Maximum Theoretical Circumforential Stress Indexes in Conicaland Cylindrical Shell at the Intersection (External Fiber) h/H = 1 ............ 62

Figure 62 - Maximum Theoretical Longitudinal Stress Indexes in Conical andCylindrical Shell at the Intersection (External Fiber) h/H - ................... 63

vi

LIST OF TABLES

P age

Table 1 - Geometric Characteristics of Project Cylicone Models ................................... 4

Table 2 - Maximum Theoretical External Stresses and Strains, for A/H = 1,per psi of Internal P ressure .............................. ................................................. 41

Table 3 - Maximum Theoretical External Stresses and Strr 's, tor k/H # 1,per psi of Internal P ressure .................................................................................. 41

Table 4 - Average Deviation of Experimental from Theoretical LongitudinalStrain Sensitivities Close by t-e Intersection .................................................. 43

Table 5 - Initial and Allowable Out-of-Roundness of Cone and Cylinderat the Intersectio n ................................................................................................. 4 7

Table 6 - Comparison of Maximum External Stresses in Cone at Intersectionof M odels C C -9 and C C -10 ................................................................................. 61

Table 7 - Maximum Theoretical External Strain Sensitivities in Coneof M odels C C -9 and C C -10 .............. .................................................................... 61

ABSTRACT

To test theory, models of the intersection of a conical and cylindrical

shell, having three different apex angles in the cone and three different ratios

of thickness to cylinder diameter, were subjected to internal hydrostatic pres-

sure, and the resulting strains at various points were measured. Careful studies

were made of the unavoidable imperfections in the models. The results are in

substantial agreement with the Wenk.Taylor approximate theory described in

TMB Report 826, and with the approximate theory of Geckeler. Comparisons

are also made with other theories.

INTRODUCTION

In the strength analysis of the USS NAUTILUS (SSN 571), of the USS ALBACORE

(AGSS 569), and of the 30-ft tes-ffank at the Portsmouth Naval Shipyard, the Bureau of Ships

encountered for the first time the task of computing stresses at the intersection of conical

and cylindrical components of pressure vesseis. In both cases, special strength analyses

were developed in collaboration with the David Taylor Model Basin. Experience with these

design features has been favorable so that continued omission of shear brackets at such inter-

sections and use of conical transition pieces betWeen cylindrical hull sections of different

diameters can be expected. Thus, rather than attack each future problem as a special case,

the development of more adequate and general solutions for the strength design of these coni-

cal structures was believed warranted. As a consequence, a research program was establish-

ed at the Taylor Model Basin, designated Project Cylicone,' to evaluate the available know-

ledge of the subject, to derive new theory or simplify design procedures where necessary,

and to validate the analysis by experiment.

The first step, reported in TMB Report 826,2 was to extract from basic elastic theorya family of edge coefficients for cones to provide a convenient method of computing strength

of pressure vessels which incorporate conical elements. Application was considered to both

stiffened and unstiffened intersections. In this report, the proposed theory is confronted by

data collected from a series of twelve model tests, which comprise that experimental phase of

Project Cylicone dealing with elastic behavior of unstiffened intersections. Additional tests

are planned to investigate stiffened intersections. Since it is known that rather high local

stresses may develop at intersections without precipitating failure of the structure, additional

theoretical and experimental investigations will yet be required to establish what level of

elastic stress may be accepted in design. The tests described herein were confined to the

elastic range and thus do not throw any light on this question.

1 References are listed on page 64.This DOcument

BstodUiad FrCoyBest Avaibable Copy

"¢" " . . . . .. . , T o-. ,-

The first undertaking in the study of elastic behavior of cones was to examine the

existing theory and experimental work. Several theoretical paPers were found, but further re-

finement was developed because of the unknown accuracy of approximate analyses and because

of the impracticability of rigorous ones; this is explained in TMB Reports 8262 and 981. 3 On

the other hand, collateral, experimental work on the ejuilibrium; of intersecting conical and

cylindrical shells previous to the last decade was alimost totally lacking, probably because

experilnental techniques were unavailable to measur'internal strains of internally loaded pres-

sure vessels. 4 Withilh the last few years, however, with the advent of wire-resistance electric

strain gages, progress has beeni rapid. An extensi'e experimental investigation of steel and

plastic pressure vessels having various shapes of heads was conducted at Purdue University

under the sponsorship of the Design Division of the Pressure Vessel Research Committee of

the Welding Research Council, 5 7 but the models with conical heads were in a limited range

of thickness-to-diameter ratios from approximately 0.013 to 0.04. The Taylor,!Model Basin se-

ries of tests described in this report utilized models in which the range of thifckness-to-diameter

ratio.. was extended downward to include values of greater interest to naval designers. The

results from these tests have shown substantial agreement with the approximate theoretical

solutions for the differential equations of equilibrium of conical shells developed by Wonk andI1 aylor. 2 T

The data obtained from two models were also compared with theoretical values calcu-

lated by the approximate method introduced by J. Geckelor 8 ' 9 and sometimes called the"equivalent cylinder" method. This method is compared with, the results obtained by the

theory of TMB Report 826 and by the more rigorous theory of Love-Moissnor. 10 It is concluded

that, for the range of geometries studied, there is little difference between the final results of

either TMB Report 826, Geckeler, or Love-Moissner, although the maximum stresses computed

by the three theories diverge as the apex angle of the conical shell increases. ,On the basis of agreement between these three methods of elastic analysis, and their

support by experiment, the Model Basin is preparing for early publication, a general evaluation

of theory and a simplified procedure by which stresses at cone-cylinder intersections may be

rapidly computed.

DESCRIPTION Of MODELS

Twelve unstiffoned models of cone-cylinder intersections were designed so that the

effoct. of three geometric parameters of interest to naval lesigners could be studied individu-

ally. The parameters are a, h/21, and h/l, where a is the half apex angle of the cone, h is

the thickness of the conical shell, H is the thickness of the cylindrical shell, and R is the

radius of the cylinder taken to the middle surface; see Figure 1.

Three nominal values of h/2R were selected, lying below the range tested at P.urduo

University, namely (based on available matorials and assumed radii), 0,00391, 0.00617, cnd

0.01042. For various reasons, the actual values employed were somewhat different.. For each

nominal value of 4/21?, thro models were constructed, with a equal to 5deg, 30deg, and 60dog,

/

3

Qn./inderPROnterGct~

~j I End'Plate

7 eh

End Plate

L IL Inch Pipe Tap for Inlet and Drain

Figure 1 - Nomenclature and Schematic Drawing of Cone-Cylinder Models

60 - _1 -I- - Purdue University -

Project Cylicone A; J~t I"

0 ~21 19- -v~__

_

N~y20 __ _ ls Model (Purdue)

10--- Steel Models (Purdue)1C-12 Li 40.....I (0

ao 001 0.02 h 0.03 0.04 0.05

2R

Figure 2 - Range of Parameters for the Cone-Cylinder Models

respectively. For all but two of the models hs - H; for Model CC-20, I/H = 1.5, whereas for

Model CC-21, h/H =0.67. The paramieters of these models are listed in Table 1, together with

the yield stress of, thi material; the' values of ae and h/21? (nominal) are also plotted in Figure 2,including data for four ofthe Purdue University models.

Two of the models (Models 00.10 and CC-20) were subequently reinetruinented and re-

tested, and two models (Models 00.8 and CC-BA) were made as nearly identical as possible

in order to determine the consistency and reproducibility of the experimental results.

As the cest of machined models was considered prohibitive, the models wero) fabricated

by rolling and welding t 'gether steel sheets. The average variation in thickness of a given

steel sheet was ±1.0 percent. This low tolerance for stool thickness was deemed essential,as assumptioxis in the stress distribution theory are based on uniform thickne.-S, However,

the fabrication procedure did lead to a variation greater than 1 percent near the intersection,

*1

4-

TABLE I

Geometric Characteristics of Project Cylicone Models

Average Average Radius HCone Cylinder Average Material Yield of Half Apex

Model Thickness Thickness Stress, oY, psi Cylirder Angle of Cone h/2R A/2R

in. in. Cone Cylinder in. e Actual Nominal

CC-8 0.060 0.060 29,400* 29,400* 8.0 30 0.00375 0.00391

CC-8A 0.060 0.060 31,350* 31,350* 8.0 30 0.00375• 0.00391CC-9 0.094 0.094 38,000* 38,000* 7.6 30 0.00618 0.00617CC-10 0.096 0.096 39,750 39,750 7.6 60 0.00632 0.00617CC-11 0. 1046 0.1046 32,600 32,600 6.0 30 0.00872 0.01042

CC-12 0.0834 0.0834 30,250 30,250 7.6 5 0.00548 0.00617CC-13 0.0607 0.0607 32,300 32,300 8.0 60 0.00379 0.00391CC-14 0.1034 0.1034 3,,750 36,750 6.0 5 0.00862 0.01042CC-15 0.1061 0.1061 38,400 38,400 6.0 60 0.00884 0.01042CC-16 0.0725 0.0725 32,200 32,200 8.0 5 0.00453 0.00391CC-20 0.09.74 0.0607 39,900 31,850 7.6 30 0.00641 0.00617

CC-21 0.0607 0.0974 31,850 39,900 5.7 30 0.00532

* *Compressive, others tensile.

but this inconsistency did inot materially prejudice the overall comparison of the theoretical

and experimental results.All models were fabricated in a similar way. The. cylindrical and ccnical shells were

machine rolled to the desired internal diameters and welded along their longitudinal seams to

give two complete individual shells. The longitudinal weld was hand-ground flush with the

shell surface, A steel straight edge was placed against each shell, and, if any generator was* juaged to have excessive peaks or valleys that were difficult to straighten out, the model was

rejected.

Thin stiffening plates were tack-welddd on the inside of each of the accepted shells

at both ends in ,order to prevent any large distortions at the intersection when the cone and

cylindler were welded together. These plates were removed after the two pieces had beenjoineO. The weld at the junction was hand-ground flush with the shell surface. The difficultyat this stage was to produce as constant a thickness around the intersection ams could be ob-tained by hand grinding. Since the shells were not absolutely circular, the excess weld ma-

torial could not be removed by machining. Guide lines scribed on both shells were used bythe machinist in grinding the intersection into a sharp And true line. Thickness measurements

Ii-i

Ii ,

made at the intersection after the models

were completed, as shown in Figures '56 to

60 in Appendix A, gave no observable trend

as to overabundance or undercutting of s'hl---

Models CC-20 and CC-21, which had

different shell thicknesses for the cone and

cylinder, were constructed so that the inside

diameters muLchod; consequently, in addition

to the excess weld material, the abutting

shell material at the intersection had to be

ground off.

End plates were welded on each end-of the models. The plate at the cone end

was 1 in. thick; that at the cylinder end,

2 in. thick.

A completed model is shown in

Figure 3.

NP21-4972

Figure 3 - Completed Model Prior toInstrumentation and Testing,

TEST APPARATUS AND PROCEDURE

The models were instrumented with electric wire-resistance strain gages. Models CC.8

through CC-16 (except for Model CC-8A) and Model CC-21 had gages only on the outer surfaceof the shells. Models CC-8A, .CC-10 for the retest, and CC-20 for both testswere instrumented

on both the outer and inner shell surfaces, with gages back to back, to provide data for pos-

sible correlation of strain results with variations in thickness and initial departures from cjr-

cularity at their point of location.

For the high strain gr dient near the intersection of the shells, Type A-8, 1/8-in. gage

length, Sl-4 electric strain gages were oriented as close together us possible in the longitudi-

nal direction. In the circumferential direction, SR-4 Type A-7 gages of 1/4-in. gage length

were used, since no variation in strain was expected in that direction. Gages were cemented

to the surfaces and dried as recommendod by the manufacturer. Gages found to 'iave a resist-

ance to ground of less than 100 megohms were replaced. The distance of the centerlino of

each gage from the cone-cylinder intersection was measured with a ruler to the nearest 1/32 ia.

I

6

00 1200 2400

_____ _ , L ngitudinal CircumferentialI_ _ Gage Gaqze

no

Junction

H i- - -- i,,iT hu I i,

_T 4'

m eicL.... * ... . L

J Figure 4 - Typical Layout of Strain Gages

The gages were placed in pairs, one circumferentially, the other .longitudinally, froma point 8/32 in. from the intersectioAl to as far as 6 in. away on the conical shell. The pages,,

were located along generator lines arbitrarily labeled 0, 120, and 240 deg* The 0-deg line!: -contained the largest number of gages, while the other two lines were used for spot-chepking

the data from these gages. A typical layout for the gages is shown in Figure 4. Models in- -

atrumented only on the outer surface carried approximately 75 gages. Twice that number wereused on models instrumented on both inner and outer surfaces.,i

Models CC-8A, CC-21, and CC-10 in the second test had a large number of additionalgages on the inside or outside of both the conical and cylindrical shells, arranged in bands3/32 or 1/4 in.;irom the intersection; see Figure 5. These were installed to investigate pos-

~~sible variability in strains suspected toa develop from initial out.of-roundness.The lead wires from the inside gages were brought out through the large end plate by

means of a TMB feed-through device, 'shown in Figure 6. This feed-through device had 100

•holes, for 50 paired leads from the gages. For the models instrumented on the inside, whe're• more than 50 internal gages were used, common ground l'eads were employed. A typical test

setup is shown in Figure 7.

Prior to the stain gage installation, circumferential and longitudinal circularity, and

thickness, were measured along sections and generators on which strain gages were located.

Ci

7

Line of iintersecnticn of Conica I

345* I5 nd~ Cylindrical 'hl325'

450 r~ Cp - Plunger

25660 8 inch Square

iddle Ke2660 ~ ~----~. Boy .inich Square

Longitudinal a- f 0both Conical and Cylin -0 Ring .drical Shells

245' P20 ssureEnd

2050160"DiskISO*

Figure 5 - Location of Strain Gages on Cone Figure 6 - Schematic Assembly of Feed-and Cylinder of Models CC-8A, CC-21, Through Device

aind CC-10 in the Second Test

7I

11

Firrre I 'yp~cfl eetSetu Shwin Fiure Aparau,4 or ewir.n

Modelf ....Cruart-AdHctlnert

Switihilg Dxiq nd trai inicaom re n te fce- he ppaatu inidetheshel, i fu (Acumcreti1

grudlr~ tro ppt~M odehe CC-21 c rcularity anl R~tin the-.yif O CJ

ung lonfituttinal proties.

8

Ames dial gages, mounted on th shafts of manual deflectometers were used for the circularityij

measurements; see Figure 8. Circularity and thickness were measured to givo (a) an indica-

tion'bf the initial out-of-roundness of the model, (b) an indication of generator rectilinoarity,

and (c) a possible correlation between initial circularities, measured thickness, and observed

strain sensitivities.

Prior to testing, the models were completely submerged in oil for 24 hours. This pre-

vented erratic thermal effects by equalizing the temperature in the model and the dummy blocks A

on which tomp,,orature-comp'ensating gages were mounted. In testing, internal pressure was

applied by oil in magnitudes such that.ihe maximum observed strain was about 50 percent be-

A low that at Which yielding of tho model material was first computed to occur.

The strain gage readings were plotted against the applied pressure, and the slope or

strain sensitivity in microinches per inch per psi of pressure was oLtained for comparison with

theoretical calculations. Strain sensitivity was employed in preference to stress because

both gages at a point would have to be operative in order to obtain the stress, whereas strain

sensitivity could be determined from readings of a single gage. Positive strain or stress in-

dicates tension, and negative strain or stress indicates compression. Strain sensitivities in

the cone and cylinder for the 12 models, in microinches per inch per psi, are plotted as a

function of the distance from the intersection along the generator,

EXPERIMENTAL RESULTS

The plot of experimental strain versus applied pressure was essentially linear.

Figure 9 is a sample plot. Two pressure runs-were usually made, and the slopes of the

pressure-strain plots almost a!ways were identical. In all but a small number of gages, the

zero shift, after removal of the pressure, was not more than ± 10 pin/in. Where a gage did not

return to its origin;U'IF;ading, it was assumed that local yielding had occurred in the shell,

probably due to the i'nitial stresses induced by welding at the intersec.ion and the longitudinal

seam. All gages had a resistance to ground of at least 100 megohms prior to the tests.

Figures 10 through 35 show the experimentally determined strain sensitivitie' for all

models tested. Theoretical curves are drawn in these figur'es according to the theory described

in TMB Report 826, with the addition of a few curves based on Geckeler's theory. A compari-

son of the theory with steel models tested at Purdue University 5 is also of interest. Modelsand 3, shown scheriatically in Figure 86, wore chosen for this comparison because they

were the only models with a sharp transition from cone to cylinder. The experimental results

for these models are shown in comparison with theoretical curves %in Figures 37 through 44.

The viariability of the experimental results was checked by performing the same test on

two identical models (CC-8 and CC-8A), and by retesting two models,(CC-10 and CC-20) with

gages on different generators.*

*Model CC-8, which had been destroyed at the completion of the teat, was duplicated b . auie of insufficient.t 'ain data. Model CC-10 was reistrumented because of its extreme cone angle, and in order to obtain additionalcomparisons with TMIB Report 826 and Geckeler approximate theories.

IN

9

Longi tudincil Gages40---___

Co,\ical Distance C aliO Cylindrical Distancefr Intersection-Iin/ // iDisrcnce=2 in. { in. {

20 --- v -- -- __-_

toSensitivityz es'ii Sensitivity-4.00/Ai in/in /f.0 - 11 1per psi 10.00__

RunI 0 -100 -~-200 0 100 0 -100 -200 -300 -400SRunr 0 -100 -200 0 t00 0 -100 -200 -300 -400

Strain in microinches per inch

Circumferential Gages

40 Conical Distande~ Conical Distance CWirc6ita -2 in. in.

30 - -_ ----

to - ---- s_ it- i I7

0 -- - ~ - s e n s i t i ; T i j~S e n s it iv it y n s i v t - I4,67/4 in/in/ psi + 1.33

Run 1 0 -100 -20 00 50 0 -100 -200Run.U 0 - 100 -200 0 50 0 -100 -200

Strain in microinches per inch

Figure 9 - Typical Plots of Experimental Pressure versus Strain,,(Model CC-15)

From a comparison of results -in Figures 10, 11, 12, 15, 28, and 30-33, consistency be-tween identical tests was considered very good.

Text continued on page 40.

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40

DISCUSSION

COMPARISON OF EXPERIMENTAL RESULTS WITH THEORY

From Figures 10 through 35 it is evident that the TMB tests are in good agreement with

the Wenk-Taylor theory. 2 They agree equally well with the Geckeler curves drawn in Figures

13 through 17, the difference between the two theoretical curves being small in these cases.

The agreement with the Wenk-Taylor theory is especially good for Model CC-16 (Figures 28and 29), for the external strains in Model qIC-10 (Figures 15 and 17) and Model OC-11 (Figures18 and 19), and for Models CC-20 and CC-5!1 (Figures 30 through 35) except over limited ranges.

The largest disagreements occurred in the circumferential strains; this may have been due to

the use of approximAte edge coefficients in the theoretical calculations (see page 60).

The Purdue results shown in Figures 371through 44 also agree about equally'well with

both Wenk-Taylor and the Geckeler approximate theories; the difference between the two thee.

retical results is again small.

FEATURES OF THE THEORETICAL ,STRAINS

The maximum theoretical external stresses and strains, as calculated from the Wenk-

Taylor theory, are collected in Table 2 for models with equally thick cone and cylinder and in

Table 3 for the models with unequal thicknesses.

From the numbers in the tables, or by inspection of the curves in Figures 10 through 35,

the following general conclusions can be inferred:

1. General Comparison. At the intersection of cone and cylinder, the external stresses

and strains are all compressive in models with a-- 30 deg or 60 deg, the longitudinal ones

being larger than the circumferential because of bending of the shell. When a is 5 deg, how-

ever, both stress and strain are tensile except for the longitudinal strain, which is slightly

compressive; the circumferential stresses and strains are larger than the longitudinal because

of membrane stretching.

At a sufficient distance from the intersection, on the other hand, the external stresses

and strains are necessarily all tensile.

2. Longitudinal Strains.

a. The maximum external longitudinal strain occurs at the intersection and in thecylinder, when a is 30 deg or 60 deg, except in Model CC-21 (Figure 34), where themaximum occurs In the thinner cone (k/H < 1). For a = 5 deg, there is little differencebetween cone and cylinder at the intersection, and numerically larger stresses andstrains occur el sewhere.

b. In Model CC-20, with a thicker c/one (h1/H > 1), the calculated internal longitu-dinal stresses and strains exceed the external at. the intersection in both cone andcylinder. At a distance, greater external stresses and strains occur in both com-ponents,

41

TABLE2

Maximum Theoretical External Stresses and Strains, for h/H = 1,per psi of Internal Pressure :1

Tensile stress is positive.

,/2 R 0.00391 h/2R L 0.00617 /h/2R =0.01042Location cc

S x e 0 e ax .15 C C U C

At Intersection 60 -919.2 -1163.4 -19.0 -29.6 -416.4 -b20.U -8.7 -13.2 -261.4 -330.0 -5.4- 8.5of 30 -285.5 - 418.1 - 5.3 -11.1 -109.8 -185.9 -1.8 - 5.1 - 42.2 - 75.9 -0.6 - 2.1

Cone 5 57.6 2,2 1.9 - 0.5 50.0 2.1 1.7 - 0.4 39.3 4.7 1.3 - 0.2At Intersection 60 -881.2 -1205,4 -17.4 -31.4 -385.1 -523.3 -7.6 -13.6 -220.8 -365.4 -3.8 -10.0

of 30 -275.9 - 418.3 - 5.0 -11.1 -109.7 -185.9 -1.8 - 5.1 - 39.0 - 75.9 -0.6 - 2.1

Cylinder 5 57.9 2.3 1.9 - 0.5 51.0 2.3 1.7 - 0.4 39.4 4.7 1.3 - 0.2

Away from 60 284.8 440.5 6.9 13.4 158.6 233.2 3.7 7.0 110.5 152.4 2.7 4.7Intersection 30 162.1 182.9 4.3 5.0 89.4 85.1 2.4 2.4 51.4 50.7 1.4 1.3in Cone 5 108.5 66.4 3.1 1.2 83.4 ;0.I 2.4 0.9 57.9 33.2 1.7 0.5

Away from 60 183.1 304.8 4.7 9.5 100.5 149.7 2.6 4.8 69.7 90.9 1.9 2.7

Intersection 30 152.1 163.9 4.1 4.5 88.0 85.0 2.4 2.2 50.7 44.1 1.4 1.2in Cylinder 5 108.9 66.4 3.1 1.2 86,0 50.3 2.5 0.9 58.8 33.2 1.7 0.5

TABIiE B

Maximum Thooretical External Stresses and Strains, for h/H 5 1,' per psi of Internal Pressure

Tensile stress is positive.

Model CC-20* Model CC-21h/2R = 0.00617 ,M2R = 0.00548

Location Shell h/H = 1.605 h/H = 0.623

-137.4 -143.9 -3.2 - 3.4Cone -138.6 -259.2 -2.0 -7.3

At the - 28.1 221.5 -3.2 7.7Intersection -. -212.8 --409.3 -3.2 -11.5

Cylinder 70.4 534.7 -3.2 17.1 -81.4 89.3 -1.8 -2.2

Cone 96.2 113.6 2.5 3.2 104.8 110.7 2.9 2.9Away from 82.3 39.7 2.5 - 1.3 _

the - _Intersection Cylinder 141.6 138.5 3.8 3.6 67.1 70.4 1.8 1.9

131.4 65.9 3.8 - 1.2

SLowr vaiues for CC-20 are for internal flbers, others are for external fibers.

Ii "

-.-. ..... I

42

c. Away from the intersection, the maximum stress and strain usually occur, inboth cylinder and cone, at a digtance of 0.16 R to 0.26 R from the intorsection (whereR is the radius of the cylinder), the maximum stress and maximum strain not necessar-ily occurring at the same point.

3. Circumferential Strains. External circumferential strains at the intersection were, of

course, equal in cone and cylinder (provided the exact edge coefficients are used); the stress-

es are also nearly equal when A = H, whereas greater circumferential stresses occur in the

thinner component when h and H are unequal.

Away from the intersection, the liaximumn external circumterential stress and strain are

found within 0.1 R to 0.5 R of the intersection, the two maxima not necessarily coinciding.

The maximum stress is found in the cone for a = 30 deg or 60 dog, but for a 5 dog the maxi-

mum in the cylinder slightly exceeds the maximum in the cone.

THE EXPERIMENTAL DEVIATIONS

The deviations between observed and calculated strain sensitivities show no marked

,,consistent bias. The experimental points lie about equally above and below the theoreticalcurves.

flIn order to see whether the deviations vary systematically with cone angle a or thick-

hness , deviations were collected for all models having I = H, taken at two selected points

on both cone and cylinder. The deviations were determined as close to the intersection of

,one and cylinder as experimental values were known, and also in the neighborhood of the

maximum at a distance from the intersection. Both longitudinal and circumferential external

strain sensitivities were included.

These deviations were arranged by values of cc and of h/2 R in four squares, of which

a typical one is shown in Table 4. The deviations near the intersection were'nearly all posi-

tive, indicating a slight tendency here for the experimental strains to be numerically smaller

than the theoretical. Furthermore, all plots showed a slight tendency for the deviations to

increase with increasing a or decreasing h. The deviations vary so much, however, that the

significance of the features noted remains dubious. This negative conclusion was supported

by a chi-test 1 t made as if all deviations had reference to independent observations of the

same quantity.

It may be remarked that, on retesting certain models after 9 to 18 months, no substan-

tial change in the strain data was observed. The major sources of error probably lay in thee-

retical approximations or in imperfections due to fabrication techniques; these wi i l now be

discussed.

:-i

43

TABLE 4

Average Deviation of ExperimentOi from Theoretical Longitudinal

Strain Sensitivities Cloile by the Intersection

Strai[n sensitivities are given in microinc.A per inch per psi internal pressure.

5 Deg 30 Deg 60 Deg ,

Model CC-16 Models CC-8, 8A Model CC-13 Average(averaRe)

_

1.8 2.4.4

0.003910.1 1.8 3.7 1.9

Model CC-12 Model CC-9 Model CC-10

02.3 2.7 1.7

0.00617 ___,

* LuuJ1.8. 3.71.Model CC-14 Model CC-11 Model CC-15

0.042 0.5ZL 0.5 2.311

1.51. -

IZ~II 1.5 2.5OverallAverage Average

0.2 1.2 1.3

Top Box - Cone

Bottom Box - Cylinder1- - -

/),

4 4,

FACTORS AFFECTING STRAIN RESULTS

The Experimental Precision

Pressures were read to 1 percent, strains to 5 pin/in. In calculating strain sensitivi-

ties, a pressure range of 50 lb was employed. The reading error in the sensitivities should

thus not exceed 5/50 or 0.1 at most. In addition, there might be an error of 2 percent due to

an error of this magnitude in the strains. A glance at the figures will show that instrumental

errors of this sort, will not suffice to explain the observed differences between experimental

and theoreLical results.

Theoretical Approximations

Approximations made in the derivation of Report 826 theory which affect stress magni-

tudes in the cone are:

1. Use of approximate differential equation;

2. Use of approximate stress-resultant equations;

3. Neglect of longitudinal displacement u when computing the displacement normal to

the axis 9 due to bending; and

4. Use of approximate edge coefficient. in calculating edge forces.

Items 1, 2, and 3 above are discussed in Report 826.2 The approximate expressions for the

edge coefficients a and d on page 10 of Reference 2 were employed here. Use of these simpli-

fied coefficients results in slightly less computation, but the theoretical circumferential strains

* in the cone and cylinder at the intersection are caused to differ, whereas in reality they must

be equal. This difference becomes more pronounced as the half-apex angle increases. The

difference with a = 5 dog is small (Figure 25); when a = 30 deg, it may be small (Figure 19)

or it may be appreciable. In Model CC-10, with ce = 60 deg (Figure 17), the calculated circum-

ferential strain was -8.7 pin/in. in the cone and -7.6 pin/in, in the cylinder, a difference of

about 13 percent. A later calculation with use of the exact coefficients a and d on page 10 of

Reference 2 gave a strain in both elements of -7.9. Thus, here the use of the approximate

edge coefficients caused a maximum error of 10 percent.

Estimates are given on page 9 of Reference 12 of the maximum total error resulting from

all the approximations made in the Wenk-Taylor theory. The numbers given there agree closely

with the following formula (if Poisson's ratio v 0.3):

Maximum error at the intersection - 105 sin a tan ct (in percent)

At other points on the cone, R is to be replaced by x sin ca where x is the distance from the

apex. Maximum errors at the intersection calculated from this formula for Models CC-8 to

CC-16 vary from 0.6 to 12 percent.

O.

45

Effects of Imperfections in the Models

Three major defects encountered in any one model are lack of

1. Circularity.

2. Rectilinearity of generators.

3. Constant thickness at or close to the intersection.

All three defects are attributable to the fabrication of rolled and welded shells into a compos-

ite structure.

That initial deviations around the circumference and axially along a cylindrical shell

increase the stresses under hydrostatic loading is well known. 1 2 , 1 Reference 13 states that,

in a thin circular ring, an initial deviation equal to 1/100 of the radius may increase the maxi-

mum fiber stress due to circumferential bending 4 to T times over the circumferential membrane

stress. The initial deviation of Model CC-21 presented in Table 5 in Appendix A was greater

than 1/100 of the radius. Such large initial deviations in a conical shell may very likely

cause a far larger increase in the maximum fiber stress duo to bonding.

This nonsymmetric bending of the shell can be studied in regions away from the inter-

sections where almost pure membrane action should result in stresses and strains equal on

both sides of the shell. In Figure 32, for example, the strains in the internal fibers follow

the trend of the theoretical curve but nearly all lie above it; the external strains, similarly,all lie below the theoretical curve in the region of membrane stress.* Averaging the internal

and external strains to eliminate circumferential bending places the membrane strains in close

proximity to the theoretical curve. Unfortunately, the original deviations on Model CC-20 are

unknown. The initial circumferential irregularities preceding the second to~t were not exces-

sive, being within the tolerances discussed in Appendix A. The longitudinal generi.tors, asmay be seen from Figures 54 and 55 in Appendix A, were almost straight.

The longitudinal strains in Model CC-20, on the other hand, both internal and external,

agree closely with the theoretical strains.

The original theoretical calculations were made on the basis of uniformi thickness,

taken as the average measured thickness of the sheet steel prior to rolling. The thickness

variation close by the intersection does not vary monotonically, however. In fact, due to the

irregularity of hand grinding, it may vary in a random manner, "ad the raadoness of .'ckness

would be very difficult to compare with the strain.

Strains recorded on different generators, at 0, 120,i1 and 240 deg, but at equal distance. A

from the intersection, differed in magnitude. This spread in values is presumably due to a

stiffening effect from an excess or lack of weld material, which increases or decreases the

local bending when the shell is pressurized.

*Departures from circularity were apparently in an oval mode. Ro-aults would not have shown such marked

effects of nonaycunetric behavior if the imperfections in shape were more localized.

i

46

Excess weld material acting as a stiffener may contribute to the low compressive strain

at the intersection. Conversely, a deficiency of material at the juncture may induce high com-

pressive strain. The resultant strain depends upon whether the bending or membrane action

is the major contributor to the strains.

An attempt was made actually to correlate initial out-of-roundness with the experimen-

tal strains,, but with only limited success. Figures 45 to 50 in Appendix A show that only the

circumferential strain followed (in many locations) the deviation pattern of the original circum-

ferential circularity. An effect of thickness variation upon strain is unrecognizable.

More consistent results could undoubtedly have been obtained with machined instead

of welded models. The cost would have been very much greater, however, aad it is believed

that the accuracy obtained with the welded models was sufficient, not only to validate the

TMB Report 821 theory, but also to indicate the locations of maximum stresses and strains.

The inconsistencies caused by the initial imperfections in this sories of models may them-

selves be useful in estimating variations in elastic strains in welded pressure vessels due to

normal imperfections in workmanship.

CONCLUSIONS

1. The test results indicate substantial agreement with the Wenk-Taylor elastic theory

described in TMB Report $26, in spite of irregularities in fabrication of the models. It maybe concluded that the theory is an adequate representation, within engineering accuracy, ofthe discontinuity stresses arising at the intersection of a conical and cylindrical shell.

2. As the half apex angle 1 of the cone increases, there is a pronounced divergence be-

tween both the Gockeler andl TMB approximate analyses and the accurate Love-Meissner

analysis, but for cones with smaller apex angles there is very little difference between the

theories.

ACKNOWLEDGMENTS

Experimental verification of the theory was part of the program of analysis on conical

shells under the supervision ( f Dr. E. Wenk, Jr., and Dr. C.E. Taylor, who planned the program

and made many valuable sugg stions. Experimental work was begun by Mr. J.L. Bigio and con-

tOnued by the author. Assistan)ce in the computations Was furnished by Miss T.M. Morrow.

After the author's departure from the Model Basin, while the report was still in draft

stage, considerable revision was made by members of the TMB staff; Dr. E.H. Kennard

undertook the preparation of the final draft.

i-A

47

APPENDIX A

MODEL IMPERFECTIONS

OUT-OF-ROUNDNESS

The initial out-of-roundness and the allowable out-of-roundness of the intersection of

the cone and cylinder are compared for five models in Table 5. The actual out-of-roundness

was determined from circularity measurements on the inside of the shells. Measurements on

the internal surface were chosen in preference to those on the external surface since measure-

ments on the outside would include also the thickness deviations due to the grinding at the

intersection. The actual deviation was taken as the difference in inches between the two ex-

treme eccentric points of shell deviations; Figures 45 through 50.

TABLE 5

Initial and Allowable Out-of-Roundness of Cone and Cylinder at the Intersection

Cone Cylinder _ _

Shell Actual Out- Allowable Shell Actual Out- AllowableModel a Thickness of-Roundness Deviation Thickness of-Roundness Deviation

in. in. ASMEin. ASME

C C-8 30 0.060 0.060 0.08 0.060 0.065 0.08CC-8A 30 0.0G0 0.054 0.08 0.060 0.043 0.08II,CC-10 60 0.096 0.050 0.076 0.096 0.033 0.076CC-20 30 '0.067 0.061 0.076 0.097 0.032 0.076

CC-21 30 0.097 0.076 0.057 0.067 0.047 0.057

*Allowable deviation is 1 percent of the radius (Section UG-80, Boiler Code, 1952).

* I,

The actual measured out-of-roundness is compared with two widely accepted criteria

for allowable out-of-roundness in Table 5. The first, commonly used in submarine fabrication,is that the allowable deviation from circularity is equal to the shell thickness. t All the models,

with the exception of the cylinder of Model CC-8, wore well within this tolerance. The second

criterion is used in the fabrication of commercial unfired pressure vessels and may be found in

the ASME Boiler Code, Section UG-8O, 1952; the allowable out-of-roundness is 1 percent of thedifference between minimum and maximum diameters divided by the nominal diameter at the

cross section. The cone of Model CC-21 was the only shell that did not fall within this latter

t In the fabrication of cylindrical models which are to be tested under external hydrostatic pressure, the deviation

allowed at the Model Basin has been one-half the shell thickneus. As the cylicone models are tested exclusively

under internal pressure, this allowable out-of-roundnesn seoms unduly rigid for these models.

48

limit. As all the cylicone models were fabricated by identical procedures, the out-of-roundnessof the five models shown probably indicates that the out-of-roundnesses of all the models were

within recognized tolerances.

In Figures 15 through 50 the strain sensitivities of Models CC-8A, CC-10, and CC-21

are shown for comparison with shell thickness and circumferential initial circularity at pointsequidistant from the intersection on both cone and cylinder. Initial circumferential deviations

of cone and cylinder at these points have a two-lobe formation, and the deviation curves of

cone and cylinder of any one model are very similar, Circumferential strain sensitivities of

all three models have directional inclines resembling the initial circularity of its shell.

GENERATOR RECTILINEARITY

Longitudinal profiles were measured along the 0-, 120-, and 240-deg generators on both

cone and cylinder and also on the seam, to determine generator rectilinearity. In Figures 51

through'55 are shown comparisons of longitudinal profiles of the cone on Models CC-8 and

CC-8A, two separate generatoks of the cone and cylinder of Model CC-10, and generators of

the second test of Model CC-20. In general, it was noted that thoeiinitial deviation of any one

generator was never larger than the shell thickness. The directions of the deviations indicated

a tendency of the models to bulge outward.

THICKNESS MEASUREMENTS

Thickness measurements along the generators of Models CC-!0, CC-20 (second test),and CC-8A are illustrated in Figures 56 through 60. Thickness measurements, at points 3/32

or 1/4 in. from the intersection on cone and cylinder for Models CC-8A and C(-10 are shown

in Figures 45 through 48.

Measurements of points close to the intersection of Model CC-10 (Figures 56 and 57)

show that on the 0- and 120-deg lines, the conical shell thicknesses along the generator were

greater for the first than"for the second test. On the 240-deg line, and also away from the

intersection on the 0- and 120-deg lines, the thickness at equidistant points varied at most,

by 3 percent. On the cylindrical shell generators, thicknesses at equidistant points werepractically equivalent in the first ?nd second tests, with variations of up to 3 pe'cent.

The thickness variations along the generators of Model CC-20 (second test), Figures

58 and 59, showed renarkable similarity due to the technique employed in grinding off the

excess weld material at the intersection. All generators on the cone were undercut near the

intersection; the corresponding generators on the cylinder were thicker near the intersection

but otherwise quite uniform,

Thicknesses on the 041deg line on both cone and cylinder of Model CC-8A are shown in

Figure 60. The material close to the intersection on both cone and cylinder is thicker than theoriginal shell, whereas beyond about 3/4 in. from the intersection the effect of grinding has

dimitlished so that essentially the true shell thickness exists.

Text continued ov, pat 60.

49

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50

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53

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55

Outword- -Eccentricity - -- -

InwardEccentricity -

0-4 -

2

o U

Scam Scam120,dogree generator 240-degree eneraWo 0-degree generator

Figure 51 - Longitudinal Profiles of Genorators on Cone of Models CC-8 and CC-8A

6Generor of GBeicrtor o~f Outard - .01

Fir st -Scn Test Eccntricity __inch

-4

0 degree gelivrator 120 degree generator 240 degree generator seam

Figure 52 - Lonigitudinal Profiles of Generators on Cone of Model CC-1C

7 -,,

-IN

4.

G I

3

SI nc

Outwa-wn

6h

E4

Va

0-Sgeam 20dg 240*0degreeSem-ge nerator generator generator

Figure 54 -, Longitudinal. Profiles of Generators

on oneof ode CC20,(Second Test)

57

Outward -4-- -*Inward

7 6

.E 4E

0

41

0 - -

Scam m2~Ige 20-degree 0-degree,1gen0 2,4.tor generator generator

Figure 55 - Longitudinal Profiles of Generatorson Cylinder of Model CC-20 (Second Test)

.5- - --- -- - _

94

c_

j4~~ ---- - _80LL

? 0.09 0.10 0.09 0.10 0,10 0.1 0fl9 0.10 0.09 010 0.09 0.10 009 0.10O5 -degree 120-degree 240-degree Seamgenerator generator generator

(hickness in inches

Figure 56 - Thickness Measurempents on Generators of Cone of Modrl CC-10

t'

58 t7/

Z 58_ -_ .---- l-.

*t

,- - --E i

-~0.09 0.10 0 9)09 OO0 "-.09 0.09

, O-dogree •120-dog re 240-degree Seamgenerator generator; generator 'Thickn I In inches.

S Figure 57 - Thickness Measurements on Generators of Cylinder of Model CC-10

* S

I !

S----< -r ,-,

- ----- F ---- -- ' . .

4~

0

0.09 0.1 Q0 .1-t.-- - ..

0.06 0.10 0.000910 0.0 0.0 0.06 0.100 ,erM 120"dgre 240-gre rel 0-degree

ge.arotar t nerator ge nrorThickness in inches

FFgure 58 - Thickness Masurements on Generators

of Cone of Model, C6-20 (Second Test) "

..... !t~ ~

59

5' ~ ~

4 ..o -

3

- 1* -

0

S rn 1 9.06 :o-ege [4 -ore 0dgegeeao geerto geeao

0

0.6 0.0 06 0.0i 0 .06 oe 00

Thickness in inches

Figure - Thickness Measureme on Dgo Genrator1~~. of Cn n Cylinder of Model CC-cndTet

60

APPENDIX B

THEORETICAL CALCULATIONS

Theoretical distributions of stress and strain were calculated for all models, including

the Purdue models,5 using the approximate Weak-Taylor theory2 for the cone and Timoshenko's

equations 14 for the cylinder. Young's modulus and Poisson's ratio Were taken as 30 x 106

and 0.3, respectively. The approximate edge coefficients a anOi d on page 10 of Reference 2

were employed except. in..a few special calculations tj be desocribed later. For comparison,

calculations were also made for two models by the approximate method of Gec koler, some-

times called the equivalent-cylinder method. Finally, maximum sresses t the intersection

of cone and cylinder in these models were also calculated front the general equations of

E. Meissnerl,° as extended by F. Dubois.1s*

COMPARISON OF THE THEORIES

of.theTwomodels with h H and nominal A = 0.00617 were selected for a close comparison

of the theories, namely, CC-9 with a = 30 deg and CC-10 with a = 60 dog. The exact bxpres- (Ksions for the edge coefficients as given on page 10 of Report 826 were emp , oyed in these

calculations. The resulting edge moments Mo in the cone at the intersection, as calculated

by TMBReport 826 and Geckeler, agreed within 3 percent, and the edge forces H 0 Within

1 percent.

Stresses and strains at the intersection of cone and cylinder, as calculated from three

different theories, are listed in Tables 6 and 7. It will be noted that, for the model with

a = 30 deg, stresses calculated by the method of TMB Report 826 differed only by 1 to 3 per-

cent from the values calculated by t ie exact Love-Meissner theory, the Geckeler values dif-

feting by 3 to 5 percent. For the model with a = 60 deg, however, G0ckeler's method comes

closer, agreeing within 2 percent, whereas the TMB Report 828 values are off 3 to 4 percent.

The strain sensitivities show differences, as between TMB Report 826 and Geckeler, with a

maximum of 9 percent between the largest values (13.8 versus 15.1). All these differences

are satisfactorily small.

STRESS INDEX FACTORS

The general nature of the calculated results can be exhibited graphi "illy by plotting

stress index factors, or the maximum stress in the cone at the intorsectio Ided by;'the

ordinary hoop stress in the cylinder, pR/(24). This is done for all TMB i js with ii= H-in Figures 61, and 62.

'Nuamerical data for th use of Dubolg" equationis were given by Watts rin furrows,1 6 i tts and Lang. 17

i ,I4~-

It will be noted that the stress in dX fEi~ctors become numerically large as~the cone

angle a becomes large, but decrease numc:rioalLy as. 4,'21 increases, except when ct is small.

The index factors should depend on~ A onl&through the.. ratio v/2RThe results show graphically thee. nail dif'ferences between results computed'b 'the

various available theories.

TABLE 6

Comp& j 'n of Maximum ExteEnal Stresses in Cone at Intersectionof Model;FCC-9 and CC-10

Stress in psi per psi

Model Stress Ri eport 826 Gecke ler Love-Maissner

ExactICC-9 O -185 -9 -189.1 -179.3

c30 deg -109.8 -113.1 -110.8

OC-10 -52D.0 -530.0 -541.6ce 60 deg ~-1B.4 -390.9 -389.9

1-2ABRLE 7

Maximum Theoretical Ex0iorn*a Sl, rain Sensitivities in Cone.

of Modoli-F 'and CC-10

Max. Circu, Sta in - Max. Lo',git. StrainModel 1iin/in/P i ________/i /n/psi '

deg Geckeler 19~ 11Re pert 826 Geckeler TDRpr 2

CC-9 30 +2.2* +2 .5* - 5.1 -5.1C-10 60 -7.4 -7,7 -15.1 --13.8

*Away from the intersection. Others 91 L~nleieection.

62

-1 _Lao 0dere

- -- - -- 7MBKpf Q~V- Mppro IeIn

003 0. 0 000 000 000 000 0. 0 0. 0 0f

Figur 6-Mxmu hoecaCicumeeta tesnee nCnca n yidiaShl-tte4trscin(xenl ie)hI

CIL a

00

2r-r 1111 a 3 degres - 11

L- 1- --

I1 6 0 degrees 1

S-4

CON

-15. TMO Rpt 826 -A 7 proximaite index

x Geckel a'r Approxirr~ote Index

0.003 0.004 O.GO5 0.006 0.007 0.D68 0.00 ,oq 0 0.011h/ 2 R

P igure 62 - Maximum Tl~eoretical Longitudinal Stress Indoxes in Conical and Cylindrical"Shell1 at the Intersection (External Fiber) I/l 1

64

REFERE ICES

1. Research and Development Project Card, NS 731-038, dated 31 Mar 1952.

2. Weak, E., Jr. aiid Taylor, C.E., "IAnalysis of Stresses at the Reinforced Intersectionof Conical and Cylindrical Shells," Dayid Taylor Model Basin Report 826 (Mar 1953).

3. Taylor, C.E. and Wenk, E., Jr., "Analysis of Stresses in the Conical Elements of Shell

Structures," Pa.per submitted to Second National Congress of Applied Mechanics, Ann Arbor,

Michigan (1954). Also David Taylor Model Basin Report 981 (in preparation).

4. Knoistra, IL.F. and IBlasor, R.U., "Experimental TVechnique in Pressure-Vessel Testing," ITransactions, ASME, Vol. 72, No. 5 (Ju 1950), pp. 579-589.

6. O'Brien, H.L., et al, "'Design Correlations for Cylindrical Pressure Vessels with

Conical or Toriconical Heads," The Welding Journal, Vol. 29, No. 7 (Jul 1950), pp. 336s-342s.

6. Design Division, Pressure Vessel Research Project, "Final Report," Purdue University

(Aug 31, 1952).

7. Smith, IL.W., "An Investigation of Discontinuity Stresses in Pressure Vessels by Means* of Three-Dimensional Photo-Elasticity," Doctoral Thesis, Purdue University (Aug 1952).

8. Geockeler, J., "Uber die Festigkeit achsensymmetrischer Schalen," Forsch. lag. Wes.,

* Berlin, Vol. 276, (1926), p. 21.

9. Wetterstrom, E., "Discontinuity Stresses in Pressure Vessols," Master's Thesis,

Purdue University (1947).

10. Moissner, E,., "Uber Elastizitat und Festigkoit dunner Sehalon," Viert. Natur. Gesell-

sohat, ol. 0 (-915, p. 234,711. Villars, D.S., "Statistical Design and Analysis of Experiments for Developmental

R~esearch," Wmn. C. Brown Company, Dubuque, Iowa (1951), pp). 226-229.

12. Sturm, RG., "A Stuly of the Collapsing Pressuro of Thin-Walled Cylinders," IUniver-

sity of Illinois Bulletin No. 1_2 (Nov 1941). For further inforniition see Bodnor, S.R. and 4

Berks, W., Pibal Report 210, Polytechnic institutu of Brookly' (Doc IKZ2).

13. Wu, T.S., et al, "Effect of Small Initial Irregul ari ties en the Stressos in Cylindrical

Shells," Structural Research Series No. 50, University of Illinois (Apr 1953).

14. Timoshonko, S., "Theory of Plates and Shells," McGraw-Hill Book (C ompany, rInc.,

New York (1940). pp. 389-395 and 409.

15. Dubois, F.. "Uber die Festigkoit dor JKegelschalu," Doctorate Dissortation, Zurich

(1917).

16. Watts, G.W. and Burrows, W.R., "The Basic Flastic Theory of Vessel Heads under

Internal Pressure," Journal of Applied Miechanics8, Vol. 16 (194.9). pp. 55-73.

17. Watts, C.W, and Lang, II.A., 'Strosses in it Pressure Vessel with a Conical Hlead,

Transactions, ASMF, Vol. 74, No. 3 (Jun 1951,), pp. 315 .3. . wui,ions u.) the above

letter ftorn authior,-, (lattd Mar 16, i 953.

I ..... .... -.. .

85

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