column str~tgth of tubes elastically restrained … · strains were measured with a ewing...

REPORT No. 615

COLUMN STR~TGTH OF TUBES ELASTICALLY RESTRAINED AGAINST ROTATIONAT THE ENDS

By V71LLLAM R. OEGOOD

SUMM.ARY

A study UMSmade of the e$ects oj known end restrainton commercially amiluble round and dreamiim tubingof chromium-molybdenum steel, duralumin, stainless steel,and heat-treated chromium-molybdenum ~teel; and a moreaccurate method than any prem”ously arailable, but SW apractical method, was dmelipedjor designing compressionmembers in m-refed or welded shwcture8, paMarlyaircraft.

Two hundred 8pecimen8 were tested as short, mediunt-[ength, and long columns w“th freely suppoded ends orelastically restrained ends. The test specimens werecentered under load on Lnife edges held in ca.rn”ers,andthe jree lengths were computed by a rational method notheretofore u8ed. Tensile and compresm”rete8ts were madeon each piece of ori~”nal tubing jrom which column8pecimens were cut. Ilhe column data mere reduced withthe aid oj these test8, and formulas were constructed torepresent the column strengths in terms of spem~ed tenm”leyield strengths ~“thfour materials used.

It wjawnd possible to extend work done by Bleich onthe &sign of elastically restrained compression membersin bridge-s and to pr~eni a method that shoufd be suitable

for designing such members in aircrafi. Z4.e design iafacilitated greatly by the we of tables and a nomographicchartl both included in this paper. A numerical exampleis also yire n.

lNTRODUCTIO?J

Compression members, particularly in riveted orwelded structures like bridges and fuselages of airplanesare columns elastically restrained against rotation atthe ends, and the strengtha of such members lie betweenthe strengths they would have if perfectly free to rotateat the ends, on the one hand, and if perfectly lked atthe ends, on the other hand. The interest of theBureau of Aeronautics, Naw Department, and of theh’ationaI Advisory Committee for Aeronautics in thestrengths of compression members of the kind men-tioned Ied to the transfer of funda to the hTationaIBureau of Standards in July 1928 and thereafter for aninvestigation of the subject-.

The column strengths of round and of streamlinetubuh.r specimens of chromium-molybdenum steeI, ofduratumin, of stainkss steel, and of heat-treated

chrominm-molybdenum steeI have been studied. Thedkuneters of the round tubes ranged from 1 inch to 2inches and the basic diametem of the streamline tubesfrom 1.%to 2X inches. The thicknesses of both sets oftubes ranged from 0.035 to 0.033 inch. Tests weremade which included comprehensive tensile and com-pressive tests of the material used, 186 tests of speci-mens as initid.ly centrally loaded cohnnns with freelywpported enda and with -mrious known eqmd elasticrestraints at both ends, and 14 such tests with one endfreely supportad and the other end reslmtined elasti-caHy in a known manner. k this paper these testsare described, and the cohmm tests, further, are inter-preted in the I&ht of the doubIe-moduhs theory of:ohunn action and are reIated to the mechanical prop-~ties of the materials of the tubea. The question ofksigning elastically restrained compression membemk also considered.

The author is particuldy indebted to Dr. L. B.I’uckrman for advice and suggestiona during theprogr~ of the instigation, and to se-reral otherraembers of the Engineering Mechanics Section of thehTationaIBureau of Standards for ideas and heIpfulwgestio~ in smoothhg ditlicukies encountered ~mtime to time in the Laboratory work. It is a pIeasureto acknowledge the assistance recei~ed in this way.Mr. E. E. Lundquist of the National Advisory Com-mittee for Aeronautics was much interested in theprobIem of designing compression members elasticallyrestrained against rotation at the ends and offeredvaluabIe criticism of the section on the design of suchmembers. Bis comments resulted in a great improve-ment in this section.

The author wishes also to express appreciation forthe tubing donated by the .Akc.inum Co. of America,by the SummeriH Tubing Co., and by the Na~ De-partment, and for the heat treatment of the heat-treated ohromium-molybdenum-steel tubing by Metal-lurgical Laboratories, Inc.

A considerable part of this paper was submitted as athesis in partiaI fulfillment of the requirements for thedegree of Doctor of Philosophy in engineering in thegraduate schooI of the University of HIinois, 1933.Acknowledgment is here made for permiasion grantedby the graduate school of the University of Illinois touse this m@u-iaI.

Lo1

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102 REPORT NO. 615-NATIONAL ADVIHORY COti~fITTEE FOR AERONAUTICS

MATERIAL AND MATERIAL TESTS

THE MATERIAL AND ITS PREPARATION FOR TEST

The round chromium-molybdenum+iteeI tubing hadbeen bought under U. S.’ Army Specification No,57–180–2A now covered by U. S. Army SpecificationNo. 57–18&2C, the equivalent of Navy DepartmentSpecification 44T18c, Feb. 1, 1937: Tubing, Steel(Chrome-molybdenum) Round, Seamless (Aircraft Use).In the latter specification:

The specified minimum yield strength was raised from 60,000to 75,000 lb. per sq. in, for the sizes of tubing that were used,and a more precise method of determining the yield strength wasspecified.

The specified mtnirnum elongation in 2 in. was raicd from 10to 12 percent for sfzes 0.036 to 0.188 in. thick.

The round duralumin tubing, donated by the Ahnni-num Co. of America, Pittsburgh, Pa,, w-assupplied toconform with Navy Department Specification 44A2jnow covered by N,avy Department Specification44T21b, May 1, 1937: Tubing, Aluminum-alloy (AJum-inum-copper-magnesium-manganese), Roundj seamless,Condition ‘T”, heat treated. In the latter specification:

The specified minimum yield strength was raised from 30,000”to 40,000 lb. per sq. in., and a more preoise method of determiningthe yield strength was specified.

The specified percentage elongation in 2 in. was lowered by 2for all sizes covered in the edier specification (Z to ilj$ in.,inoluaive, nominal outside diameter).

The round stainless-steel tubii was bought underProposed Navy Department Specification M-55b,temper C!tubing, now covered by Bureau of AeronauticsSpecification 44T27 (INT) 22 ApriI 1937 for Tubing,SteeI, Corrosionmxisting (18 percent Chromium and8 percent Nickel) %smless, Drawn, Round, Structural% H-CoId-Drawn, Pickled. III the Iatter specification:

The speoified yield strength was lowered from 150,000 lb. persq. in. to a minimum of 135,000 lb. per sq. in. and at the sametime the method of determining the yield strength was changedto conform in tiect to the definition given in Navy DepartmentSpecification 44T27, “The yield strength is the load per squareinoh of original oross eaction at which the materieJ exhibits anextension under load of 0.002 inch per inch in excem of thatwhich would be computed from Young’s modulus of elasticityof 25,000,000 pounds per equare inch and the usual forrmda:Unit stress= Young’s moduluex unit deformation.”

The spechied elongation in 2 in. was lowered from 8 percentfor a fufI-tube speoimen to a minimum of 6 percent for materialover 0.02 in. thick and not over 0.049 in. thick and to a minimumof 6 percent for material over 0.049 in. thick.

Flattening requirements were added.

All the chromium-molybdenum-steel tubing that washeat treated was donatid by the Summefi TubingCo., Bridgeport, Pa., and was beat treated free ofcharge by Metakgical Laboratories, Inc., Philadel-phia., Pa., to a requested temile strength of 175,000lb, per sq, in,, no specification applying.

The streamline chromium-molybdenum-steel tubingdonated by the Summerill Tubing Co., Bridgeport,Pa., was stipplied to conform with Navy DepartmentSpecification 44T17a, now coyered by Navy Depart-

ment Specification 44T17b, Dec. 1, 1936: Tubing, Steel(C?hromeMolybdenum), Structural, Streamline CrossSection, Seamless (Aircraft Use). In the latter speci-fication:

The specified minimum yieId strength was raised 10,000 lb.per sq. in., but the method of determining the yield strength waochanged so that most material passing the old specification wouldprobably also pass the new one.

The streamline duralumin tubing, donated by thoAluminum Co. of America, Pittsburgh, Pa,, was sup-plied to conform with Navy Department Specification44TZ2;now covered by Navy Department Spccifhtion44T22i, Feb. 1, 1937: Tubing, Ahuninum-aIIoy (AhI-minum-copper-magnesium-manganese), StrcamIino (Air-craft Use). In the latter specification:

Maximum ohemical contents of iron, silicon, chromium, andother elements (including zinc) were specified,

The specified minimum yield strength was raizcd from 30,000to 32,000 lb. per sq. in.

The streamline stairdess-steel tubing WWSsuppliedby the Navy Department, which bought it under NavyDepartment Tentative Specification M-55C, eitherGrade 1 or 2 tubing with physical properties specifiedfor temper “B” and with cross+ectional dimensions toconform to those shown in table 2 of Navy DepartmentSpecification 44’J!22,now covered by Navy DopartmeutSpecification 49T11, May 1, 1937: Tubing, StCcl,Corrosion-Resisting (18 percent Chromium and Spercenti Nickel), Seamless-Drawn, Streamline-Cross-Section (Aircraft Use), K H-Colddrawn, Pickled. Inthe liWer specitlcation:

The specified yield strength was lowered from 125,000 to11O,OOIIlb. per sq. in., and the method of determining the yieldstrength was changed to conform in effect to the definition notedunder the description of the round stainless-steel tubing.

This specified percentage elongation in 2 in. was lowered fromM for a strip specimen h 13 for materiaI over 0.049 in. thick andta 8 for material not over 0.049 in. thick.

Since for the purpose of the present invcatigat.ion thelatest specifications are, with the ~xceptions noted,essentially the same w those under w~lch the tubingwas supplied, reference to specifications hereimfter willbe confhd to the latest specifications.

The nominal dimensions of the tubes used twogivenin table I. One tensile specimen and one comprcssivospecimen were taken from each tube, usually fromopposite ends, most of the remainder of the tube beingavailable as column specimens. AI1 them specimenswere weighed, their lengths and the outside diametersof the round specimens were measured, and representa-tive determinations of density were made by the Divi-sion of Weights and Measures of tho Bureau. TIMcross+ectional area of each specimen was cornput.cdfrom the weight, the length, and the density, Tholeastradii of gyration of the cross-scctiomd areas of thoround specimens were determined from tho actualdimensions and of the streamline specimens by sub-jecting representative specimens ta pure bending,

COLUhfN STRENGTH OF TUBES 103

measuring deflections, and computing the radii of gyra-tion from the results so obt~ined and the values of themoduli of elasticity found for the compressive speci-mens.

The initiaI deflections of alI the column specimensexcept a few short ones were measured.

TEWILE AND COMPRESSIVE T~TS

The tensile tests were made in a 100,000-lb. (50,0bo-Iig) Amsler machine having scale ranges from O to101000, 20,000, 50,000 and 100,000 lb. Most of thecomprtive tests were made in a 50,000-lb. compound-lever machine, having scale ranges from O to 5,000and 50,000 lb., in which the movable head couId bemaintained very rigid. Most of the compressive speci-mens were tested with the lower end against a flatblock and the upper end against a hemispherical bearing.

Strains were measured with a Ewing extensometer,when possible, on a 2-inch gage length; otherwiseTuckerman strain gages were used for determinationsof moddi of elasticity, and Huggenberger extenso-meters for determinations of yield strengths. Themodu.li of elasticity were obtained from stress-straindata by means of difference curves (referance 1) drawnfor each of the tensile and compressive specimens.The stress-strain data for det- the modulus ofelasticity were usually taken after the specimen hadbeen strained to about 0.002 and the strain released.This procedure “ironed out” some of the initiaI stressesin the material and in many cases, particuhwly that ofthe stainlees steeI, made the determination of themoduhui of elasticity much more definite than it wouIdhave been if determined from redngs taken duringthe first ~oftding. In these cases, owing to the imme-diate curvature of the stress-strain diagram at lowstresses, the modulus of elasticity would have had tohave been determined as the tangent modulus at zerostress, a very unsatisfactory determination.

In most cases the tensile yield strengths were ob-tained from stress-strain diagrams, according to thedetitions in the latest Navy Department specificationsapplying to the type of tubing tested.1 The compres-sive yield strength was obtained from a stress-straindiagram as the stress corresponding to the intersectionwith the stress-strain curve of a line drawn throughthe origin with a slope 13E,where 0<~<1 and E is themodulus of elast.icity. This method is discussed later.The values of # used were: for the chromium-molyb-denum steeI, 5/9; for duralumin, 2/3; for stainksssteel, 5/8; and for heat-treated chromiurn-molybdenumSteal, 5/7.

~icaI stress+train diagrams are shown in figures 1,2, 3, and 4.

1In the = of the Lreet-treeted ehrominm-rndybdenrurded the tenslk yfdd-~-~eneetiexattbefnkeetfcm wfththe~strdnmrmdahe with a sIope equel to tbnt of the modnlns line and at a stmfn O.(IOZfrem it.

COLUMN TESTS

The primary series of cohmm test9 were tests withfreely supported ends. & can readiIy be shown theo-retica.IIy,the buckling strength of an ide~ column hav-ing known eIasticrestraintsat the ends is the same as thebuckIing strength of a cohunn with freely supported endshaving a Iength equal to the distance between the suc-cessive points of inflection of the center line of thecolumn with restrained ends. Local, crhkling failuresare not considered here. The length of the equivalentfieely supported column is calkd the “free length” of thecohnnn. Although the double-modnks theory ofCOIUDIRaction (reference 2) 2 furnishes a method ofdetermining the free length of an ideal cohmm, as willbe shown later, the imperfections of real cohmms madean experimental check seem advisable, and at Dr. L. B.‘l?nckerman’ssuggestion a considerable number of speci-mens were tested with known ekwt.icrestrainb. Therestrained ends also sirmdate more nearly the practicalcondition under which actuaI cohunns are used.

APPARATCS FOR PROCUSINQ RESTRAINTS AT ENDS

The diagrammatic sketch (@. 5), shows the lower fix-ture used for procuring an elastic restraint at the lowerend of a cohmm. Except for unimportant details, theupper Mum is the same as the lower. Each fixturecomists essentially of a carrier with a knife edge whichbears on a seat on a stationary support c.Iampedto thevveighing@ble of the testing machine. Means are pro-vided for holding the end of the test specimen in positionon the carrier and moving it horizontality under lowloads in a direction perpendicular to the Imife edge.Rotation of the carrier about the knife edge is restrainedby the helical springs shown. The degree of r~t.rtitmay be varied by dumging the active lengths of thesprings, provision for which is furnished. By means ofa dial gage, not shown in the figure, it is possible tomeasure the rotation of the carrier about the knife edge.Wing nuts on the ends of the rods through the springsmake it possible to compress the latter so that rotationof the carrier mill not cause one spring ta go out ofaction (the springs cannot be uwd in tension).

PEOCEDUFtE FOE MAKING A T~T

The same -rerticaI testing machine was used for thecolumn tests as for most of the compressive teds pre-tioudy mentioned. Before a series of columns withektstically restrained ends was tested, it vm necessaryto adjust the active Iengths of the springs to procurethe degree of restraint against rotation desired and then,after. compressing the springs by means of the wingnuts, to deternine the restraint accurately. Thisdetermination was made by hanging a series of lmown

~& IndfeetedfnW WW.thetheory- dewlopedby mntrfbntfom of semralmem It cennot jastly be nemed a!tersmy one of them, nor even two, without denyinguedtt dne elrewhero. Dr. L. B. Trmkermen snggestd the neme ‘WubkrmduIus-.”

-...—

1(}4 REPORT NO. 615—NATION.4L ADVISORY COMMITTEE FOR AERONAUTICS

/io,ooo

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FIGURE 1.—TypIcd sfress-strstn dlogmms of chrrp&r.molybdenum steeI.

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‘hid mcdnlus=lO,Ol),OM Ib. per sw In.

COLUMN STRENGTH OF TUBES 105

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FIGCEX 3.—T~icaI stres%rsh d!sgmms of sta!nIess steal.

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Dffterence of straln-Straln--Odul

Tfil moduIns-WXsl,033 lb. permq. km

.. .

106 ‘ REPORT N(). 615-NATIONAL ADVISORY CO~fM17YEE FOR AERONAUTICS

weights on one of the hangers at the ends of the carrier(fig. 5), thus causing the carrier to rotate about theknife edge, and noting the corresponding readings ofthe dial gage indicating rotation (not shown in thefigure). From data obtained in this way the momenton the carrier could be plotted against the rotation ofthe carrier, and the moment per unit angular rotation,which wiIl be called the “restraint”, m, could bedetermined as the sIope of the resulting diagram,The restraints used are given in table I.

The procedure proper, for making a column testconsisted in placing the specimen between the upperand lower carriem, centaring under load S with thesprings out of action, and then bringing the springa

k-rip mW$fnq nut

—.

FIGUEE 6.-D@aroruatfc sketob showing appsmtus for proourfng eleetic restrrdntat endofcalumn.

into action if the test was to .be made with restrainedends, and loading to faihwe.

RESULTS

DETERMINATION OF THE FREE LENGTH

The results of the coInmn tests were plotted in termsof the ratio of slenderness, based on the free length,and the average stress at failure, and also in terms ofnondimensional variables that took into account theproperties of the material. The iirst question thatpresented itsdf in preparing such a graphict-d repre-sentation was the determination of the free hmgth.The rigid carriera at the ends of the columns compli-cated the situation somewhat. Engeeser (refarence 5)had shown in the case of straight ehtstic columns howsuch rigid portions of equal lengths at freely supportedends could be treated rationally, The problem of thestraight elastic column of uniform crose section elasti-cally restrained against rotation at the ends had beentreated by Nater and others (references 6, 7, and 8),and B1eich (reference 9) considered the case of thestraight inelastic column so restrained. IrI the presentinvestigation there were many slightly curved, inehwticcolumns with rigid portions and both with and withoutelastic restraints at the ends. AU these columns werecentered under load with the expectation that such

I Chrlstle (reference8) wm probably the dret to compmsate for imperfections in srractnel column by shiftfrrgthe ende of the cnfur.m refncive to the eupports. Conefd&re(referenoo 4) swms to ham been the tit to employ centerfng under Irad egsteruatf-celly in a S8rke of eolnmn teete,

centering would idealize the elastically restrainedcolumns as well as the freely supported columns,+ Arational method of determining the lengths, and thusthe ratios of slenderness, of equally strong columnsfreely supported at the ends was then sought. Thctstrength of the difkent cohmms could then bc studiedas a function of the ratio of slenderness bttscd on thefree lengths so found.

If the double-modulus theory (referenco 2) of columnaction is adequate to describe the be.hvior of columnsunder load, as is being questioned loss and less today,and if the departures from straightn~ of th columnsare not too great, then it should be possible to dctcr-miue the free length of any test column on the basis ofthis theory.J If, when the free lengths of the testcolumns have been computed, plots of aver~~e stressat failure against ratio of slenderness based on thocomputed free leng~or modifications of them quantitiesthat take into account the variations in tho propertiesof the material, lie on a reasonably smooth curve, thisfact may be regarded as proof that small init.id curva-tures under condition~ of proper centering do not affectthe strengths of even inelastic columns freely supportedor eI&kally restrained at the ends.

The equations determining the freo length of anaxiaIIy loaded straight column of uniform cross sectionwith rigid portions and elastic restraints at the endsmay be obtained as outlined in the following sections.The procedure is to write down the differentitdequationfor the deflection of the center line, to integrate it, tointroduce the boundary conditions, and to dctmminounder what conditions the displacement of the centerbe becomes indeterminate. It will be found that thisdisplacement becomes indeterminab at a dcflnite load,the buckling load.

The notation that will be used is:

A, the cross-sectional area of the columni, the leastmadiusof inertia or radius of gyration of

the cross section of the cohmm measuredparallel b the plnne of bending.

I=Aiz1, the length of the cohunn spwimcn.

so, the length of the rigid portion between the upperend of the specimen and the upper knife edge.

81,the length of the rigid portion between the lowerend of the specimen and the lower knife edge.

~Welfbown teate by von K6rm&n(refereneeIO) show tlrntslfghtimperfm~[onsuoh as small fn[tfrdcurvahms do not al?et tha sfrengths of frrdy support.sd columr.uVbWlcanteredrmder load. Zimmerme on (reference11) k ShOwlltheoretically thathe strangth ef elastk coIurnns Is not atktd by Wght fnltfsl curvature when thehrrme ore wntered properly,snd Rein (reference L!4),fn* emeful mrfa of testsj hodmluded some definitely kmt whmmr of struetorrd steel, eonfirmhg Zirnmernwn’sealllta.

~In this theory it k emuroed tlmt the columns ere strafght, that the uMerfat IsIO~OU& that the I@sd fs smiled in tie with theda of tho column, and tlm[eformstions due to shear are negligible. Sluce tho behavlcwofs rnetmtal W&wadmyond the propxtlonel Ilrnft In &mlumn depende upon its previoun stra[n blstory,he strength of columns In which the matarfsl fs so strmmd depmxfs on the lcadfIwdetory. A etsudard loadiog hfetory muet therefore al.w be esmmed in Ore thmy.‘t is emumed thet the compreas[on Is unlftxrn over sll the erosa .vuxkm until thewelding bed is rmebed. and not until then does krding teke place.


P, the load on the oolumn at failure by buckling.F, the transverse force acting at each knife edge on

the ends of the cohmm.~, the double modulus.

mo, the elastic restraint resisting rotation at the topend of the oohunn.

ml, the elastic r=traint resisting rotation at thebottom end of the column.

#0, the rotation at the top end of the cohmm,positive clockwise.

41, the rotationat the bottom end of the cohun.n,positi~e cloolm-is.e.

fi~o=m~o, the restraining moment at the top end ofthe column.

ilil= –ml#,, the reshining moment at the bottomend of the column.

Kith the coordinate itxes taken as in figure 6 thedifTerentiaIequation of the deflectd center line of thecolumn is for smaIl deflections a,

(1)

where M is the bending moment at my section,

.JI=.MO+F(SO+X) –Py

Integration of equation (1) and substitution of the

boundmy conditions, x=O:dy dy&=#o, Y=80!JO; Z=l: ~=

k, Y= –%vh, fieldsfourhomogeneous linear equationsin #o, #1, and two constants of integration. Ths de-flection y becomes indeterminate when the determi-nant of the coefhients in these equations is equal tozero, and hence the buckling load is ‘defied by theequation

A=o

Upon e-iahmtion of the determinant, there is found interms of the nondimensional variables

‘=~’(’~+?)nndp’=p’(’;+?)’

-( )1 (90’1)+9 1+~+$ ,uN, C’os 4+2 1+~+~ Pd.@=o (2)

~It nmy be noted that equation (I) applks to eny coInron that rematns strdght upto the Ir&emt of faUrMOee In the standard Ioeding history. IO tbk me themmegestrm Is conste.ot and hence ~ k Independent of r.

If the length of a freely supported column having thesame strength as the given cohunn is denoted by l.,then

(3)

(the original double-modulus equation), and thisequation makes it.possible to write

(4)

Equations (2) and (4) determine @ and & when theother quantities are known.

It should be noted that the determination of the free[ength & does not require a kuowkdge of the due of

~ the freeIengths 1, so,

Equation

—

iII

-.x—

FLGGPJ 6.—Deliected center line of eohmnn.

length bein det+mined solely by%and S1and t e variables ~ and pl.

(2) is aimpli&l cmsiderablyspecial cases. Thus, when So=sl=s andintroduce the nondimensional variablea

()~= 1+2+ r;=~l-;

the

.

108 REPORT NO. 615—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

and

, “-H++?)“-’+?m “)

and by factoring the left-hand side obtain

(l:hw: ‘$W)[(1+2V’) (1—cm @)—P’@ sin +]=0 (7)

It maybe noted that if #=2nm, where n is an integer,the left-hand side of this equation becomes indetermi-nate. Substitution of this value of @ into the originalequation, (2), shows that d =2nr is not a solution, Ifthe first of the factors on the left-hsmd side of equation(7) is equated to zero, the solution obtained correspondsto the case #0= —#1 (fig. 6) and, if the second factor isequated to zero, the solution obtu.ined corresponds totlhecase #0=#1, The first solution yields the smallestvalue of 4 and is the only one of practical interest.There is obtained then in a convenient form

cot $+V4=0 (8)

Equation (8) may also be written in terms of the trigono-metric functions

s ‘#=-–l and t=l–&4:sm ~ (9)

t+s=-; (8a)

W7here tables of t+s (reference 13) are available,equation (8a) will be the most convenient form for use.

If, when sO=q =s and k=O, there is introduced thenondimensional variable

equation (2) for the determination of ~ reduces to

f )s 18 1

t=–+; l+~+p,,. ~ ~“——_— (11)

which may be solved by triaI with the aid of table VII.Finally, if 80=sI=0, equation (2) may be written in

the form/klPl(t2-s2)+(/JO+ /JJt+l=o (12)

Zimmerman (reference 13) gives this equation andPrager (reference 14) gives it in a modi6ed form, butthey assume it to apply for da.stic buckling ordy. Ithas also been presented in a paper by the author(reference 15). The equation may be solved by trialwith the aid of table VII, or it may be solved directly

by means of the nomogram (fig, 7) the idea for which isdue to L. B. Tuclwrman?

In order to use the nomogram, a straight line is runthrough the points of the circle determined by t.hovaluesof ~ and Y1read on the circular scale. This line willintersect the spiral curve in at least onc point, Thewdue of 4/r corresponding to this point, or tho lowervalue if there are two intersections,aread on tho scaleof the spiral curve will be the lowe9t value for whichbuckling can occur,

The necessary constants for the column spccimcnswith equal (or no) restraints at opposito ends rind forthe column specimens freely supported at tho top andrestrained at t-he bottom being computed, cquutions(8), or (8a), and (11) could be solved for@ rind iho freelengt~ determined from equation (4). The ratios ofslenderness, k/i could now bo found; and tho corre-sponding average stressesat fuilure,PIA, wcro obtnincdfrom the mtt.simumloads and tho cross-sectional areas.The vahes of m and ml that were used ranged fromOto about 450,000 lb.-in. per rrtdian. (See t.abloI.)

COLUMN DATA

The values of ratio of slenderness, 10/i, and corre-sponding average stress, P/A, are given in trtbleH andare plotted for the chromium-molybdenum-steel speci-mens in figure 8, for the duralumin specimensin figure 9,for the round stainless-steel specimens in figuro 10,and for the heat-treated chro]r]ium-~llolyb(lmul+kclspecimens in &are 11. One of the main causes, prob-ably the main cause, of the scatter in the points inthese diagrams is the unavoiclablo variation in thoproperties of the mate~al from tube to tube ancl alongthe kmgth of any one tube from which specimens wcrocub.-+his variability can be corrected for to theextent to which the compressive stress-strain relationsremain invariable along the length of any on? givenspecimen and are a5ely related from specimen tospecimen.

Suppose, for example, t.lmtthere is dctcmnined fromthe compressive stress-strain diagram of the nmtcrirdin each specimen a certain stress, S, as the intersectionwith the stress-straincurve of a line through tho originhawing a slope 13.E,where p is a constant for a gbrcnmatarial and 0</3<1, and E is the modulus of clM-ticity of the material, Let therenow be constructed insach case a reduced compressive stress-strain dingrmninwhich there is plotted, instead of stressagainst strain,3tress divided by S agninst strain divided by S/E.*

7Equetion (U!) Is In rme of Clerk’s mnorrfcel forms of eqontlone of nomtmreph[o~der 4. (See refemnm 16.)

I In the ease of two @rrte of fnkrseetlon the hfglrer vehre of ~lr uxreaponds to usmsteble cmdftion of equilIbrlurm

t The reduced strem-stmln dtegram bes been used by IIohensrnser (mkrcnm 17),rho does not, however, determine S s@tleeJly by the metbnd hem med.


)/(2. I f.60

/s0

o[.40

/20-0-/

1.20

‘.? L/0

/ \$? o

Q.A

o

Q 1 I I I I I

2 w pm

s o

FIGGEX7.—N-omcgram for determhdng dfr when rwo mlues of p are given.

III$ II

I1

lo@ooI

\,o •~io

90,000 0 Om*OI}

f“ 5 o.035”_

3

c10, \ e Ir&=o, m#oO* \ nm=O

nm*O

’04! ; “\J

1.%6% 0.049”❑ ~.O,q#O

80,000● m=O

\* aeo \

o m#O1

I.W x 0.058” —

&o e w=o, q*o\2 o!m=o

1

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70,000 \ \~\<

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60,000 E \

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0 20 40 60 80 100 120 140Rotti of slend~ess, 1,/t

Ftatm 8.-TIuJ$ ~-di8gmni h dwamhIIu-mdYMenuuwiwl tab.. . .

+

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@ ‘“”X3

114 REPORT NO. 616-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

If the various origid stress-strain curves are ffielyrelated,l” then they all reduce to one and the samecurve, in particular, that part of the reduced diagramwithin the elastic range is identicaI for all materials,Just as there is corresponding to any given stress-strain curve, a definite double-modulus column curve,P/A= T21?/(l#Jg,so there is corresponding to our reducedstress-strain curve, a defhite reduced double-moduluscolumn curve in which P/A and 3 in the double-moduIus formula me replaced by P/(AS) md J?/E,respectively, and in which 10 must be replaced bylJ~.’l

If now there are introduced the nondimenaiomd wmi-ables lx

(13)

the reduced column curve

E‘U=~E ---- (14)

is identically the same for all columns having geometri-ca~y similar cross sections (E depends among otherthings on the shape of the cross section) and made ofmaterials having compressive stress-strain diagramsthat are a.5eIy related.

Although it is too much to expect nll the compressivestress-strain curves of a material to be aflinely related,nevertldess they are to some degree of approximationso related; n.nd experience has shown that, where theproperties of the materkd difl’er widely, the correctionproposed is a reaI correction (reference ‘18). Accord-ingly, the values of XOCand a,, equations (13), werecomputed for each column specimen. For E in eachcase was used the modulus of elasticity of the com-pressive specimen for the tube from which the cohunnspecimen was cut. For the determination of S, whichmay be called the “compressive YieId strength”, in thecase of the chromium-molybdenum steel and the dural-umin tubes, p was so chosen that were the tensile yieldstrength to be determined for a material just passingspecifications, then by the PE method exactly the sameyield strength would be obtained as by the methodspecfied for round tubing. This consideration gave

P=: for the chromium-molybdenum steel and f?=: for

the duralumin. The value of I?determined in the same

way for the %-H stainless std gave fl=~; this vahe

was arbitrarily reduced to ~, which was the value used

for all the stainless-steel specimens. For the heat-

treated chromium-molybdenum steeI f?=$ was used.

This value would correspond to a specified yie~dstrength of 150,000 lb. per sq. in., determined as indi-

11By~rl~~~]~~~~~t the~]~~~t~~~~ bet= theWV=r(~f)-oand IV&, @-u where p and c are mnetants.

II The q~antfty ~~~ h tie ]~th Ofan e~fO ~l~mn ~h~ wO~d b~~ atthe average stress IS.

l! sim!lw variablm were used by Tuckwrnan, Petrenko, and Johnson (reference 1S).

cated in footnote 1. For curves that are strictly af-flnely rdated the value of #?is immaterird, within t.holimits O<f?<l -withinwhich it ha my rnefming. Forcurves that me approximately dllnely related tho bestvalue of p is that value which most nearly brings thcmall to the same reduced stress-strain curve, but prrtc-tictd.ly the best value may be regarded as the vnlucwhich is most effective in reducing the scatter in the&Ii, P/Adiagrams. In order to make sure that thovalues of B adopted were reasonably good, other vrducswere tried: ~ for chromium-molybclenurn steel andduralumin, and ~~~and ~ for stainless std. Therewas little difference in the results, which were not mttdosignific~tly better nor worse. This result was to boexpect~d, for in order that the method have any valueat all, the stress-strain curves must show some sem-blance of &e relationship, and if they do, the VCIIUOof ~ most convenient to use w-illbe practically as goodas any other.

The &,,u,-diagrams thnt result from the proccdurcjust outlined are shown in figure 12, 13, 14, ml 16.

Compmisons of the chromium-molybdenum-steel dia-grams (figs, 8 and 12) and the durnhunin dingrnms (figs,9 and 13) show some improvement; but the improw-ment is most marked in the stainlesssteel and the hcat-treated chromium-molybdenum-sted diagrams (figs. 10and 14 and figs, 11 and 15). That tho improvement isnot greater in the chromium-molybdenum-steel rcaulkmay be exphined by the fact that the rnechanicnlproperties of the materinl vary considerably along thelength of a given tube. The vnlue of the comprcasivoyield strength for a cohmn specimen mny differ appre-ciably from the value of the compressive -yieldstrengthas actualIy determined on the compressive specimen,Moreover, the compressive stress-strain curves of thechromium-molybdenum-steel showed comparativelypoor af%nerelationship. No great improvement in thoduralumin resultswould bo expected becaum of t-hegen-eral uniformity of all the material used.

The next most probable cause of scatter in the LJ i,P/A- and ~O,,urdiagrams after variations in thoproperties of the materials, is the unccrtninty of theconditions at the ends of the test cohunns. The bestmeasure of the success with which a column has beencentered is the subsequent loaddeficction curve ol.)-tained on testing the column; or better, for compnrativopurposes, a plotmf load, P, divided by maximum lon(i,P-, againstdeflection within the free length dividedby the free length (reference 19). Tho shnrpcr the“knee” of such a diagram, that is, tha smaller the small-est-radius of curvature of the reduced load deflectioncurve, the better centered or the better adjusted thomds of the column may be assumed to lmvo been.Representative diagrams of this kind are shown in6gures 16 and 17. The deflection, 5., in the freelength was obtained from the observed deflections onthe assumption that equation (1) represented thedeflected center line.


I I I

~o.ool~ 6JG ~ 0.00IL 8./2,

(a) ohromium-mdybdemm steel. (b) Dnre.lundn.

FIGURE 16.—RepresentetIm reducd Ioaddefleetfon curws for cbromlmn-molyb-denmnweel and for durnlumin tubes.

.—

116 REPORT NO. 616—NATIONAL ADVISORY CO~MITTEE FOR AERONAUTICS

Comparison of the low points on the h,,,u,-diagramswith the reduced Ioaddeflection diagrams for thecorresponding specimens indicated that one chromium-molybdenum-steel, three duralumin, four stainless-steel,and two heut-treated chromium-molybdenum-steelcolumns were probably tested with unsatisfmctcn~endconditions, since the reduced loaddeflection curvesshowed large radii of cwwature at the knees.

The error in the free length, ~, due to an error in therestraint, m, was estimated to be not. greater than 1percent in any case.

The teste on freely supported round columns areregarded as the primary data, the tests with restrainedends and the tests of the stremnline sections being re-garded as check data. The degree to which the checkdata agree with the p~imary data is an indication of theaccuracy of the method of computing free length bymeans of equations (4) and (8) or (11). It will be notedthat; iri general, in figures 12, 13, 14, and 15 the pointsrepresenting the check drttrtfall in approximately equalnumbem above and below the point representing theptiary data.

FORM~ASAND CURVES FOR DESIGNfNG

The data were adapted for use in designing by thefollowing procedure. Empilicrd fornnh were de-. . .veloped to gme a good apprommatlon to the AOS,u*-values for each of the materials, curvw representingthese formulas being shown in figures 12, 13, 14, and 15.Use of these formulas or curves requires a knowledge ofthe compressive yield strength of the materirdj and thisinformation is not usualIy available to the designer,The specified property most nearly related to it is thetensile yield strength. Therefore, the a~erage ratios ofcompressive yield strength to tensileyield strength weredetermined for the several materials, and the values ofthe specified tensileyield strength in tho se~eral currentspecitlcations were introduced into the empirical equa-tions to give the column strength as a function of ratio ofslenderness for material just paasing the specitlcation.Curves representing these results are shown in figures 8,9, 10, and 11. The detaik of this procedure will now betaken up.

The results of the tests on the freely supported roundchromium-molybdenum-steel, columns can be repre-sented in the &u~-diagram by a curve of the farm I*’

and the reduced Euler curve

“’=6‘orz%~“’>0 (15b)

11~This type of curve, in terms of the rstlo of slenderness ~d the RVS2WeSk=,wss prowed by Krefsgw (referfmceX0. -

where K. and K1 are empiricrdconstants. T1iesocurves

me shown in figure 12 for KO=5.6 itnclKl=l:

5.6 – ho: 560 10

“8=5.29+h O? ‘or F@ “~E (ltla)

(m)u,=+for ;Z u,>O.

It may be noted that for A“I=O, equation (15a) rmluccsto the Johnson parabolic formuln.

The results of the twts on the freely supporhxlround duralumin columns can be rcprcscnted in thok.,,u,-diagram by the straight lino and the rcduccdEuler curve

u,=l.175— O.575~&l for 1.176ZC,20;6

or u#=l.175—o.445A.# for 1.175au,20.6 (17ri)

1and US=—2h for 0.6 au,>O (17b)

0s

which are shown in &ure 13.The results of the tests on the freely supported round

stainless-steelcolumns can be represented in t.hoX@,,u,-diagram by a modification of a curvo proposed byAarflot’8

1a4=KJO?+(l-KJ~m , K,S l,K& O (18)

for all values of ~.,, & and K8being empirical constants.The curve is shown in figure 14 for &=0.08, &=8:

1~’=0.68&;+0.32~&~+8 (10

It maj be noted that for K*=1 or .&=O equation (18)reduces to the Euler formula.

The redts of the tests on the freely supportwl roundheat-treated chromium-molybdenurn-steel columns canbe represented in the A,,, a,-diagram by tho sevcnth-degree parabol~ and the Euler curve,

r,= O.943(1—0.0751YW7)for 0.9432 r,z 0,733 (20fL)

1“=z

for 0.7332 u,>O (20b)

which are shown in figure 15.The ~elations between &/i and PIA, usually dmircd,

may be obtained by substituting the values of ha, andus from (13) in (16a) to (20b), solving for P/A, dintroducing the numerical values of tho quantitics Sand E; but tliis procedure -would msdt in equationsapplicable only ta the particular material tcskl.What is wanted are equations in IQfiand PfA which

give safe results when applied to any material passingspecifications. The specified property most cIoscIyrelated to the compressive yield strength of a materialis the tensile yield strength, and if the ratio of the.aotwo strengths is known for a particular material, a

IIAermt(referem uj pr-1

u-;~+~~

COLUiIN STRENGTH OF TUBES 117

column formula mitten in terms of the comprewiveyield strength may be given in terms of the speci6edyield strength. The a~erage ratios of the compr=iveyield strength to the tensile yield strength for the roundtubes of the four materials used in this instigationwere found to be 1.000,0.908,0.827, and 1.120for cbromi-um-molybclenum steeI, duralumin, stainks steel, andheat-treated Chromium-molybdenum steel, respectively.

Navy Department Speci&ation 44T18c for cbro-mium-molybdenum-st*l tubing requires a tensile yieldstrength not less than 75,000 lb. per sq. in. for tubingup to 0.188 inch thick. The compressive yield strengthof tubing just passing this spechication may be esrpectedto be S=1.000 X 75000=751000 lb. per sq. in. Theaverage -ralue found for E for the round tubes -was29,800,000 lb. per sq. in. Consequently, the columnstrength in pounds per square inch of tubing up to0.188 inch thick -dich passes the specification justmentioned and for which the ratio of diameter d tothickness f does not exceed about 50 (value obtainedfrom unpublished tests on short specimems) may berepresented by the forndas

la ‘P~=75000 ()22000– ; for 79400Z~a 32600,

()20700+ 10a

(ala)=%

p 294000000 ‘ -“—=A q’

()

for 326002$>0. (21b)

z

Curves corresponding to thwe forndas are shown iufigure 8. They represent the strength that may beexpected from tubw which just pass the specification.

ATm-y Department Specification 44T21b for heat-treated durahunin tubing requires a tende yieldstrength not I- than 40,000 lb. per sq. m. Thecompressi~e yield strength of tubing just passing thisspecification may be expected to be S’=0.908X40000=36,320 lb. per sq. in. The average value found for Efor the round tubes was 10,590,000 lb. per sq. in. Conse-quently, the cohmm strength in pounds per square inchof tubing which pass- the specification just mentionedand for which the ratio of diameter d to thickness t doesnot exceed 55 (due obtained from unpublished testson short specimens) maybe represented by the formulas.

P 104500000_—A _.’

(’)t

P‘or 218002 z~ 0.

(22b)

Curves corresponding to these formulas are shown mfigure 9. They represent the strength that may beexpected from tubes which just pass the specification.

NTavy Department Specification 44T27 (IhTT) forstainkss-steel tubing, 3/4 H—Cold drawn, requires atensileyield strength not Iess than 135,000 lb. per sq. in.The compressive yield strength of tubing just pas~~

this specMcation may be expected to be S=0.827X135000=111,600 lb. per sq. in. The average valuefound for E for the round tubes was 26,300,0W lb. persq. in. Consequently, the cdmm strength in poundsper square inch of tubing which passes the specificationjust mentioned and for which the ratio of diameter d to

thickness f does not exceed about 35 (see table I) may berepresented by the formula

The curve represented by this formula is shown infigure 10. It represents t-he strength that may beexpected from tubes which just pass the specitkation.

As stated prewioudy, no specifications apply to theheat-treated chromium-molybdenum-steeI tub~, butalI the tensile specimens showed a yield strength above150,000 lb. per sq. in. determined as indicated in foot-note 1. The variation in the ratios of compressiveyield strength to tensile yield strength was so great,more than 20 percent, and the number of tubes, five,from which specimens were cut was so smdf that insteadof using the average ratio for this material, it seemsde&able from considerations of safe@ to use the leastratio found of compressive yield strength to tensfieyieId strength, namely, 0.99, in obtaining a rdationbetween P/A and lo/i based on a specified tensile yieldstrength. If a spetied minimum tensileyield stremgthof 150,000 lb. per sq. in. is assumed, the oompr~veyield strength of tubing just passing the specificationmay be expected to be JS=O.99X 150000=148,500 Ib.per Sq. in. The average value found for E for theround tubes was 30,000,000 lb. per sq. in. Conse-quently, the cohmm strength in pounds per square inchof hea&treated chromium-molybdenum-steed tubingsimilar to that tested, ham a specified mjnimurn ten-siIe yield strength of 150,000 lb. per sq. in., for which theratio of diameter d to thickness t do= not e--teed about35 lZ (see table I) may be represented by the formulas

~=140000[1 –(0.01547~i)7] for 140000 s~a108900

(24a)P_296 100000

~– ~for 1089002~>0 (24b)

Curves represented by these formulas are shown in figure11. They represent the strendh that may be ~ectedfrom tu& which just pass ‘a specification requiringa minimum tensileyield strength of 150,000lb. per sq. in.

L{u f,g~ thsto~ eOlUMIIepecbneuWWIw.sti.i’tiedby crtnklhg at one endbut tbts spwtmen WMthe shmteet one tesbrd with so Mgh a YSIUSofd/t.Fire otherIonser speobnens with the smue vabm of #t, two of them cut from the same tdm esthe speofmen b qnestio% MIed by prfmery bncklbrg e.t higher awsgs stxesws. Itfs extrem@ dLOkdtto renters short spedmen beeenswthe dedeetions under thecenter@ Ioad ere m smelL Thts eondiffon fe not of greet lmrmrtenee for suohepeci-menswhen feUnreoccnm by@mezybncklin& bntlt!s like&to aEeetar)mectsblythe aremse stress at fellure when the CeIInrek by mfnklbg. It seems probable,theWorq that thes~ menttonalwasIraded eccentr&sl@, end thet the strengthdcas no+ represent the streneth wblch wonld have been obtdned Mder a centcdly.sKwIledIcm&

.—. .-.—

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I,.

1

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zo

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m for Naw standard durahunln tubes.

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Y. S.-l86~lb. WIKI. In.E-?6,8uI,&XI lb. w W. in.

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I

126 REPORT NO.- 615—NATIONAL ADVISORY CO-MMITTEE FOR AERONAUTICS

Figures 18,19,20, and 21 represent curves of bucklingload P, plotted against free length, &,for Navy Depar&ment standard sizes of tubing up to and inclu&ng 3inches in diameter. These curves were obtained fromequations (21a) and (24b) by inserting the appropriatevalues of A and i and solving for P in terms of ?0.

In the analysis of trusses continuous at the jointsit is necessary to use the ratio r=Z/E, which is a func-tion of the average stress on the cross. section of acohnnn at the instant of buckling, since Z is a func-tion of this stress, The relation between ~ and theaverage stress may be obtained by eliminating A08from equation (14) and the empirical equationapplying for the particular material being used, andthen substituting for u,- its value in terms of PIA.Thus, for chromium-molybdenum steel, by ehninatingX,, from equations (16a) and (14) and substituting

P(

P.—m lb. per sq. in.

“= 1.000.75000A A )

1–0.000 01260:r= O.00007467$ for 79400z~2 32600

1+0.0000133$ (25a)

and from (16b) ~= 1, for 32600a~>0. (25b)

Similarly for duraIumin, equations (17a) and (14) giveP

(P.

on substituting u,=)0.908. 40000A ~ m lb” ‘er ‘q” ‘“

Pr= O.0001920

< )p 2for 42700a $221800,~ 1–0.00002343Z

(26a)

and from (17b) r= 1, for 2180Q”~.~>0 (26b)

And for stainless steel, equations (19) and (14) give onP.

(P.

substituting uS= —m lb. per sq. in.0.827,135000A A )

Finally, for heat-treated chromium-molybdenumsteel, equations (20rL)and (14) give, on substituting

P P.(“’=0.99 .150000A Xm lb” ‘er ‘q” ‘“ )

P(

P‘=0’000014112 1‘140000A )

? for 1400002~a 108900

(28a)and from (20b) n

7=1 for 108900a~>0, (28b)

T~bles III, IV, V, and VI give “valuesof 7 for differentvalues of P/A in equations (25a) to (28b). The quan-

tity ~J

T~r is alsolisted in these tables as it willbe found

convenient to have it.

DISCUSSION

$bme of the materialused fortestdid not pass speci-fications in all particulars, and in the evrduation of theresults of the tests, this matter should bo cm.aidcrcd.One failure to meet specifications which might be con- “sidered significant in the present investigation is thefailure to reach the specified tensile yield strength, butthis fadure can be adequately corrected for by using the& a,-method of plotting, provided that enough speci-mens which do pass the specification for yieki strengthare also tested as checks. Two round column speci-mens did not pass the specification for straightness(maximum allowable departure from ~traigbtncsa: ratioof initial deflection to length 1 to 600), but t-hewspeci-mens showed no lower strength than other compnralhspecimens; nor was the effect of initial deflection on thostrength apparent in any other case.

Occasional high points in the lJi,P/A- and h,,

u,-diagrams,asinfigures8 and 9, and figurm 12 and 13,were due to friction at the knife edges and have nopractical sif#cance. This friction was minimizcciin the later twte by vibrating the specimen slightly bymeans of a light buzzer attached ta We middle of thespecimen during test. Three conspicuously low pointsin figures 11 and 15 were due b failure by local bucklingat the ends of the specimens where they bore on thepIatm of the carrie~. Two of these specimens werofrom round tubes with ratios of diameter to thickness of35.7 and 50.0, respectively, and tho third was from astreamline tube with ratio of basic round diameter tothiCkUSSSof 32.1.

It seemssafe now to conclude that round tubs hrtvfi]gratios of dimmeter ta thickness not greater than wasmentioned previously and conforming to Navy Depmt-ment Specifications 44T18c, 44T21b for hcd-treatedtubing or 44T27 (INT) for ?i II-cold drawn tubing, inparticular tubes having departures from straightnessnot much greater than allowed by tho specifications,may be designed as columns with elastically rcstrnincdends on the basis of the double modulus and oquntions(21a), (21b) for chromium-molybdenum steel; (22a),(22b) for dumlurnin; and (23) for stainlesssteel. Sinti-larly, round heabtreated chromium-molylxlcnum-stwltubes with ratios of diameter to thickness not muchgreater wan. 35 and with tmsile yield strengths not lessthan 150,000 lb. per sq. in. may be designed with theaid of equations (!Ma) and (Mb),

As far as the shape of the cross section is wnccrncd,theoretically the streamline tubes should be slightlyweaker than the round tubes; but this difference wou~dbe so small as to be masked by other considerations in aseries of tests made under necessarily practicnl condi-tions. If the difference did show itself, it should doso independently of the material; but no such con-sistent difference appears in the test rcwdts.


The points representing the chromium-molybdenum-steel streamline tubing in the Aa,,u,-diagram (~. 12)are W low except one, but these low values are believedto be due to the appreciably flatter knee of the com-prwsi-re stress-strain diagram (reference 22) of thematmiaI of the streamline tubes. (See ICB-C in@g. 1.) If this particularly flat knee may also beexpected in round chromium-molybdenum-steel tubing,further tests with round tubing having this character-istic wouId be desirable. The results available indicatethat for streamline chromium-molybdenum4. eeI tubingnot over 0.186 inch thick, passing hTavy DepartmentSpecification 44T17b, for which the ratio of basicround diameter d to thickness f do= not apprec.iablyexceed 35 (see table 1), formuIa (21a) may be replacedby

12()

—g‘=82400- for 87200Z ~Z 35800A.

()18900+ 4%

For TW.Iuesof P/A less than 35,800 lb. per sq. in.,equation (21b) applies.

The points representing the duraIumin streamlinetubing in the h~,ut-diagram (fig. 13) all except one highone closely folIow the points for the round tubing.

The points representing the sthkss-steel streadinetubing in the ~o,,u,diagram (fig. 14) are the most erraticof any for the four materials tested, but this fact is notaltogether surprising since the matmial was not thesame m that of the round stai&.ss&eel tub=. Thetrend of the points for the streamline tubing relativeto those for the round tubing is consistent with thediHerences show by the compressive stress-straindiagrams. The stress-strain curves of the materird ofthe streamline tubing had sharper knees than those ofthe material of the round tubes (see 2SC?-Cin fig. 3)and the streamline-cohmm specimens show high valuesof r, for high values of & and low values of u, for lowdues of Ao,, which would be expected (reference22).

The points representing the heat-treated chromium-molybdenum-sted tubing (Q 15) cIosdy follow thepoints for the round tubing, except two which me themuhs of teste on specimens with restrained ends.The restmint was heavy, 440,000 lb.-in. per radian andthe reduced loaddefkxtion curves for these specimenswere not smooth at low loads and showed very bhmt“knees.” It is probable that, as load was applied, themovable head of the testing machine did not moveparaIIel to itseIf, thus producing a rotation of the topend of the specimen and bending, which resulted in avery nonuniform distribution of stress. Since the kneeof the stress-strain curve of this material is so sharp,premature bending of the specimen would lomr thecohmm strength more pronouncedly than would be theease with a mater.id with a blunt knee.

Iwmm—m

NOTE ON ‘THE DESIGN OF COMPEESS1ON MEMBERSELASTICALLY RESTRAINED AT’ THE ENDS

In a truss or a framework (Slafmetz) continuous atthe joints the members are interdependent, and in par-tictiar the buckling strength of a member depends onthe restraining moments (positive or negative) producedat its ends by the other members meeting there. Thesemoments, moreover, depend on the geometrkd andmaterial properties of alI the membe~ of the buss orframework. It is not possiile, therefore, to considerthe buckling of a comprtion member by itself but oDlyas part of the structure as a whole. Theoretically, it ispossible to determine with giveD conditions of lodingthe mm.mum load to produce failure of any truss, butpractically, the solution of the trtmscendentd equationsinvolved is out of the quwlion except for the simplestcases. Approximate solutions based on simplifyingassumptions at present seem to offer the onIy way out.

Only planar trusses with the joints assumed to beimmovable d be cmsiderecl here. The case of mov-able joints has been treated by B1eich (reference 9),Prager (reference 23), and othera; the case of spacestructures has been treated notably by Friedrich andHamsBleich (reference 24).

In order to get anywhere it is necessary ta assumethat the trues can be broken up into sufEciently simplegroups of membem to enable the stability of each groupto be investigated separately. Two such groups =e (a)an individual compression member and the membe~meeting it at its ends, the far ends of the latter membersbeing considered as freely supported (fig. 22); and (b)three members forming a triangle (fig. 23). As WW beshown presently by examples, the first group can betreated by means of equation (12), and the stability ofthe second group can be investigated by means of thecandition o~buc-kling of a trian~-e. -

t’*,+t’jk S’jk SfII

s’ jk t’jk+t’~[ S’kl

s’ lj s’ki t’ki+t’ L*

= O(25)

where

rmd the subscripts refer ta the members as shown in&me 23. This equation is given as applying to theelastic case by Borkmann (reference 25) and may bederived by applying the equation of four moments,Bleich (reference 9), successively to the members ij,jk; jk, ki; and ki, ij.

In the groups of members the stabtity of which is tobe considered, P for each member represents the loadin the member, positive for compression and negativefor tension. The quantity ~, -which occurs in theexpressions fors and t, is always given by

r-(26)

.-.


but when P is negative (tension), s and t become [ Similarly for any member meeting ij at j

~ –I and t=l–&4‘=sinhj

If a member carries no load, or is assumedno load, an assumption which may be made

-- (9a)

to carryfor sim-

—~Jk — 1

‘Jk—~jk~Jk tJk

Now since

ml+rnu+mu+.., +rnm+.., +mh=O

r

\

1, k ~j

mFICIURE22.-A member of a truss, touether with the members meet~ It at Its ends. FIoua.E 23.-Three meurbera of a tu formInKa trfangla

plicity and with safety for most practical tension andmembers (see the appendix) then s’ and t’ or s/(P.0 and ~/+?njl+~J2+. .+~jk+... +!?tjn=())

t/(Pi) become indeterminate. These quantities may beevaluated by the usual methods, and one finds there can be obtained by simple substitution IS

1 1(

J- y+y+...+p+)G=m and ~=3EI

(9b) fil=P*J*,

( 1

(28)~ y+p++...+y)Equation (12) applied to the group of members shown ~j=P1jJ1j

in figure 22 may be writtenThe method of treating such groups of members as

#l#J(tll*—sl;)+ (#l+ Pj)tIj+l=o (27) thdesignated (a)and (b)depend..on whether itis

36. 0“ 29.4”~A J94+”2 .f061”z II

.?9.4” 27.911.lml=~ a

\\\\\\ 29.5”\ .l**=

‘t

.o;~;+B . 0::242;4 J) 01659”4 F .

4.33W

Iw O“=/n, o“~=Sq.h.. 04. j~.4

FIGCRE ‘Z4.-Portlon of a fowlnge.

where pf and pl are the values of y at i and j, respec-tively, and it is necessary tu%etermine Pt and p, interms of the loads and the dimensions of the membersof the group. The quantities pi and p, depend on therestraintsoffered by the members meeting ij at i and atj and may be found easily by applying equation (12)to.these members. Thus if the member hi is consideredand the value of ~ at h is put equal to p~f=0, since themember is. supposed freely supported at h, there isobtained on solving for the value of p at i for thismember

desired merely to check the stability of existing groupsunder given loads or whether tito groups arc to bcdesigned to carry specific loads. Tho actual proceduremay be much the same in the two crises. In the ihlcase the left-hand side of equation (27) or (25) iscomputed, whichever one applies, and is comparedwith fie righ~hnnd side, O. A70simple crit.erincan begiven for assuring stability by menns of such a com-parison, however, and it is recommended that fur agroup like that shown in figure 22 the nomogram,figure 7, be used in conjunction with equation (27) to

I u The expression for M and YI were dfflerently determined In reference 18 andweretherecalIedTfandTj.


determine whether a member is stable. It should be Example: Figure 24 shows in outline a portion of anoted that no value of @/T>l .4303 must. be aIIowed loaded cbrcnnium-molybdenum-steeI trms. Posted onto occur in any member ih or jk, because djr= 1.4303 each member is its length 1, cross-sectionnl area A,

-.

represents the condition of one end freely supported cross-sectiomd moment of inertia 1, and load P. By aand the other end fixed. For a triangular group (fig consideration of the group of members FE, FC, I?D,23), care must be ~ercised in using equation (25) FH, HJ, HG, and HE, it has been found that thisthat no values of @/T occur larger than those which group just reaches a neutral state of equilibriumcorrespond to the lowest critical load@ In the sec- when IT= 1,450 pounds. It is desired to check theond case, that of deaignhg membem for specific loads, other compression members in order to determineequation (27) or (25) must be solved for one unknown, whether they are in stable equilibrium and possibly to~, if the other membe~ are assumed to be known, or redesign some of them. Ody t~o groups will be con-if more than one member is unknom, the equation sidered here, the group (a) JI, JG, JL, LIi, LI and thethat applies must be sol~ed by trial for such a combi- group (b) BC, CD, and DB. It is convenient tonation of @’s as will satisfy it. A numerical example arrame- at least part of the computation in some suchwilI help to clarify the procedure. tabuhw form as the follo-ming:

The vahes of P in the table me obtuined from figure24 with W= 1,450pounds; the values in cohunn 7 areobtained from trebleIII; those in column 9 are 1 one-thousandth of the product of those in Columns 2, 7,and 8; those in columns 10, 11, and 12 are obtainedfrom table WI except for member CD; and the sourceof the other entries is obvious. It was assumed forsimplicity, and to be on the safe side, that membersJG md LI carried no stress.

If equation (27) is now appfied to the group (a),there is first obtained from equation (2)

~J=0.01900 (6.510+13.24)=0.3752jf==0.01900 (23.67+15.69)=0.7478

Now substitute into equation (27) and find0.3752XO.7477X3.829+ (0.3752+0.7478) X2.6 H+1>0The group is safe. The desigg, however, may beuneconomical and the possibility of redesiggg themember JL is investigated. Any of the other membersor any two or more members might have been con-sidered. Substitute p. and p. as just found intoequation (27) and adv’e it for +JL by t.rirdtith the aidof table WI or, more simply, determine @m directlyfrom the nomogram of figure 7:

k= 1.784a-

This is the due of @/r required for t-hemember J1in order to bring tbe group (a) into neutral eqfibrium.The value of @/r in the original design is only 0.701,so that a considerably smaller member JL would beadequate. Assuming tubular construction, such a

.-.—

member may be picked from tbe chart of figure 18 by -entering it at the load Pm=2,349 lb. and the freelength, computed from equation (4)

~ _la 22.4&-iTli@2”56 ‘“r

It is found that a X-in. by 0.035-in. tube would besatisfactory. Whether this tube would represent apractical possibility is, of course, another matter.

Lf some member of the group other than the memberJL -were to be redesigned, equntion (27) mtiththe ex-pressions for P from equation (28) could be solveddirectly for the value oft required for the member. Therequired value of #/r could then be looked up in tableWI in the case of a compression member, or computed

from equation (9a) in the case of a tension member. Anew compression member could then be picked fromfigure 18 as just explained. A new tension memberwould have to be chosen by satisfying equation (26)by trial. If more than one member of the group wereto be redesigned, a relation connecting the variousdir’s could be obtained from equations (27) and (28)and this relation satisfied by trial. H vrouId probablybe at least as expeditious, however, to modify onemember nt n time rather than two or more simul-taneously.

Let the second group of membem now be eonsidered—the triangle composed of the members BC, CD, DB.Proceed in the same general way as with the first group,that is, compare the Ieft-hand side of equation (25)tith zero. If the determinant-is expanded, the valuesof s’ and t’ in terms of 23’,P, 1,s, and t me substituted,

130 REPORT NO. ‘(H5-NATIONAL ADVISORY COMM-ITTEE FOR AERONAUTICS

and the result is divided through by E8, there may bewritten

(h ~ tCDxtCD j tDB

)(

tDB ; hC

PBCIB~ PcD~CD )= PDB@d l’DBIDB p=

(

tCD ~ tDB

)+ 2PBA~~ZZ~DB~DB *’ Pc.lm P=tin=

(tDB:tBO

~ Pms~2 PDBk~B -m )SDB2

( )

ho :s.0‘pDB~~B2 &&~ Podm (29)

Substitution from the table gives0.004375–0.0007816–0.01239 +0.00001433+0.003766=–0.005017<0,The group is safe. The design may be uneconomical,however, and the possibility of redesigning the memberEC iEinvestigated. Any of the other members or anytwo of them might have been considered. For thepresent-purpose it is more converiient to write equation(29) in the form

1(

~+p:&;DBPBo2tBo2PCDLCD )— (tso’-%’)

[( )(tDB ‘_

‘ + ‘:’ , &30‘&m k+pr+ ~h:;:&2Jt~ lDB)1—

‘ PDBalDB2Pm~~+2 PCD1:?22PB4B0(tCDs-SCD’)hB=o

‘Pm21m2P~B~DB

Substitution. from the table of all vahws except SBC_andt~o gives after simplification~o’–sBo2+5.366t,c+ l.645~o+6.~94=()

Solution of this equation by trial with the aid of tableVII gives for the value of 41Trequired to cause buckling,

4W=I.799T

Since this value is greater than the actual value, 1.043,a smalk member may be picked from figure 18. If, asbefore, the free length is determined

and the chart (fig,

~m 23.46 in.10=—=~

r

18) entered with this length and t-helo&dP~c=2,218 lb., a x-in. by 0.035-in. tube is found tobe adequate.

It may be noted that when for any member #/Tis less than unity, the member has reserve strengthconaidere.d as a freely supported column and can actto restrain the membe~ meeting at its ends. In theprevious axample, the values of ~/T for all the mcmlxmwere less.than unity except for the member BC, Novalue of #/T greater than 2 is possibla if the structurcis to remain stable.

It may happen that the most severe condition ofloading for a given member is not the ono that producesthe greatgst average compressive stress in the member.Strictly, then, every member should be invdgatmi foreach condition of loadhg which produces compressionin it. Practically, however, this procedure will not bonecessary, and if a member in compression is satisfac-tory for the condition of loading producing tho greatcstaverage compressive stress in it, it will usually bo pos-sible to tell by inspection whether other conditions ofloading should alEobe .consi.dered. —

NATIONAL BUREAU OF STANDARDS,

H’ashington, D. C., September9, 1997.

APPENDIX

The simplifying assumption that tension members in they will be. In case of doubt, for csample, extremclya group carry no load is safe so long as the quantities high values of ]P1/A,the simplifying assumption should

~~ and %1 which occur for these members satisfy the not be made.REFERENCES

conditions

—1smh rpand

Pl –lPll ~ 3EI—.–-l-

t 1–*tanh tj5

(a)

(b)

respectively. In order to investigate whether theseconditions are satisfied in any given case, it is necessaryto know the value of 7 for computing .@. Since noinformation concerning T will usuaIly be available fortension members, it is suggested that the values givenfor the same stress in compression be used. It is notto be expected that the discrepancy will be large. Forr= 1 conditions (a) and (b) W always be satisfied,and for any practical member it is almost certain that

1. Tuckerman, L. B.: Dimussion of paper by R. L. ‘l?emplh,

The Determination and Sigd%xume of the ProportionalLimit in the Test~mgof Metals. Proo. Am. Sec. TestingMats., vol. 29, pt. II, 1929, pp. 538-646.

2. Osgood, William R.: The DoubIe-Modulus Theory cfColumn Action. Civil Engineering, vol. 5, no. 3, March19a5, pp. 173-175.

3. ChrhXIe, James: Experiments on the Strength of Wrought-Iron Struts. Trans. Am. Sot. Civ. Eng., vol. 13, .4priiE34, p. 85.

4. Consid&e, A.: R&istance des Pibeea Comprim&s, CongrtiIntamational dea Prcc&Ms de Construction, 1891, p. 371.

S. Engemer, Fr.: ~ber die Knickfestigkeit von Stibcn verilnder-licheu Trtigheitsmomentes. Zeit. d. ~sterreich. Ing.- u.Aroh.-Vereines, vol. 61, 1909, p. 544.

6. Nater, H.: Knickung eIastisch eingeapanntcr Stabc.Sohweizerieohe Bauzeitung, Bd. 71, 18. Mai 191S, S. 215.

7. Usinger, P.: Beitriige zur Knfckihaorie. Dcr Eisenbau,Bd. 9, Aug. 1918, S. 169.

COLUMNSTRENGTH OF TUBES

S. Leduc, R.: Contribution h I’l%ude des Poutres Prfsmatiqum.Services Teohrdques de l’A&onautique, Bulletin Tech-nique ATO.@ Jurn 1929, p. 43.

9. Bleiah, Fr.: Theorie und Bereohmmg der eisernen Ihticken.Julius Springer (Berlin), 1924; also Die ~ckfestigkeitelasticher Stabverbindungen. Der Eisenimu, Bd. 10, Feb.1919, s. 27.

10. von IMrn&, Th.: Untersuchungen fiber ICnickfestigkeit,Mitteihmgen fiber Forschungmrbeiten. Ver. deutsch.Ing., Heft S1, 1910.

11. Zmermann, H.: Die IMckfestigkeit von Stiiben mifnic!ht gerader Adse. Sitzungsberichte der PreussischenAkademie der ‘7i%menschdten, 18. Oktaber 1923, S. 262;also Lehre vom Knicken auf neuer Grundla~ WilhelmErnst und Sohn (Berlin), 1930.

12. Rein, W.: Versuche zur Ermittlung der Knickspannungenftir ~erschiedene Bmmtlihle. Julius Springer (Berlfn),1930.

13. Zimroermann, H.: Wickfestigkeit der Stab~erbindungen.‘Whelm Ernst und Mm (Berlin), 1925.

14. Prager, W.: The Buckling of an ElastioalIy Enctired Strut.Jour. R. A. S,, VOL49, no. 311, Nov. 1936, pp. s33-s34

15. Osgood, Wm. R.: Contribution to the Design of CompressionMembers in Aircraft. Bur. Standards Jour. Res., VOL13,no. 1, July 1934, pp. 157–160.

16. Soreau, Rodolphe: Nomographic ou Trait.4 da Abaques.Tome I, Etienne Chiron (park+, 1921, p. 172.

17. Hohenernser, K.: Flieszversuche an Rohren aus Stahl beikombinierter !ifug- und Torsionbeanapruahung. Z. A.M. M., Bd. 11, Feb. 1931, S. 15.

IS. Tuckexman, L. B., Petrenko, S. N., and Johnson, C. D.:Strength of Tubing under Combined Axial and TransvemeLoading. T. N. No. 307,N. A. C.A.,1929.

19.Domke, O.: Die Auebiegung eines Dru&stabes bei Uber-schreitung der Kr&klast. Die Bauteohnik, Bd. ~ 26.NOV. 1926, S. 747.

20. Kretiger, H.: Bkirag tiLi friigen om SWVOH kn~ckning.Teknisk Tidskrift (Sweden), Wig- och Vattenbyggnads-konst., VOL45, Sept. 15, 1915, p. 10L

21. Aarfiot, AI. G.: A New ~lm Fo~~. B~. Am. Soc.Swedish Eng., VOL26, Feb. 1931, p. 7.

22. Osgood, W’iIIiem R.: Column Curves and Stress-StrainDiagrams. Resemch Paper No. 492, Bur. StaudardsJour. Res., VOL!3,no. 4, Out. 1932, pp. 571-5S2.

23. Prager, W.: EIaatic Stability of Plane Frameworks. Jour.Aero. Sci., VOL 3, no. 11, Sept. 1936, p. 3SS.

24. Bleic~ Friedrich, und Bleich, Hans: Die Stabilitiit riium-Iicher Stabverbindungen. Zeit.. d. &terreich. hg.- u.Arch.-?’ereines, Bd. SO, 1928, S. 345.

25. Borkmann, IL: Charts for Checking the StabiLity of Cam-pnsxsion Members in Trusses. T. hi. h’o. S00, N. A. C. A.,1936.

TABLE L-NOMINAL DIMENSIONS OF TUBESRESTRAINTS USED

I I 1 I 1

ROUND TUBES

t 6teinless steal

I 1 , b 1

I I t t I

Heat-treated cbromIum-moIyMenom steeI

L---- 0.042

E

:= am A1 -..–-. . ms E! .4297 0I . . . . . . . .085 lm.oI -.-----

.m A.049 z&7 : B% . KM A II 1 1 I I

STREMfLINE TUBES

I I

I Heat-treated chromium-molybdenum steeI I

131

AND .

---

132 REPORT NO. 615-NATIONAL ADVISORY COMMITTEE FuR AERONAUTICS

TABLE 11.—RESULTS OF. TESTS ON COLUMNS

ROUND TUBING

BOTH ENDS FREELY SUPPOETED CJx RESTRAINED

CHROMIUM-MOLYBDENUM STEEL

1 in. dlomekm by O.(BI in. thick. Nominal dI’4-32.0

C&:.: . . . . . . ..-.-. !a3 I 22.27I 90!910 0.402- . . . . . . . . . . . . . . m 42. O& 7’0,720 . m7

u 917 IICA-b-.. - 69,3s 00#090 :cg.805 CA-2. _~.::::::: 46: 73.02 44091 .

0.667.443

1% [n. diometer by 0.049 In. thick. Nominal d/4=23.O

DA*------------------ 222 4L 24 0.723 0.846 DA-2. --_- . . . . . ..-%%

4b: 70.20 6ab30 1.237DA-5 .. . . . . . . . . . . . . . . . m 49.73 .876 .793

:%

DA-rzc . . . . . . . . . . . . . . . . o 07.79DA-4.. -—..-.-.. 89.35

bbr730 1.193 .004w! mo L 292

lM In. diemeter by 0.0SSin. tldck. Nombial W-22.9

IAD-2__i_____N-L- “-—-.---.----A-

C-2. -.- . . . . . . . . . . .C&--------

------------------A_F-2.- . . . .._.. -----AB-2 . . . .._.. --.._.N-iL...-.G-..=- . . . . .

%::::5:::::::::~L:---------

. ---------------------— ---------

I-3 . . ..-.. ._.._._.v-2___________yin-----------

- -----------------3-L____________I-2-------------------F-2. . . . . . . . . . . . . . . . . . . .

g:-:::-_:.:;-:-:Em.-:.:._.-_:_M-3 . . . . . . --------H-:.:. __-,. ______

- —. -... . ----P-2____________K-2.. _.__. _._ . . ..-E+ . . . . . . . . . . . . . . . . .M-2-. .-.. --.- . . .

M. 6264307L M6412O&O&72. WI7&29;; g

ri 597b.cd7a. 2377. w

EE842287. 2J37.77

;;

91:7010L 8Im 4102, b1027l!M3120.21&3.oK&2

p:.-:- ...._.. _- 461- . . . . . . . . . . . . . . . . 3ao

A-b. . . . . . . . . . . . . . . . . . 431D-2----------------- 82110+.-. . . . . . . . . . . . . . . . . . .21MAK-2. ..- . . .._.. ---- mA-3 . . . . . . . . . . ..-...--. 132B+.. --.. -..- . . .._..-.T-2.. .-.. _________ $AJ-2 . . . .._. __._...-o-2. .-... _________ 832D-2-_ . . . . --------- 230M--------------- 0P+--------------------v-3--------------- AK-2----------------- 230A-2. -_.. _.._. . . . . .0-6------------------ %B-2-------------u-3--------------- %R-3-------------- sloJ+... . . . . . . . . . . . . . . . . . WIP4-------------- 0w... . . . . . . . . . . . . . . . . .D-6-_ . . . ..-.. -------- 26:A-.3.._ . . . .._... --- 250AI-2--. - . . . . . ..-.-. 461AII-2..-...-..- . . .._. 461II-a. . . . . . . . . . . . . . . . . . . 1320+ . . . . . . . . . . . . . . . . . . . 182

a 3a4.322.329.243.262.276.409.472.629. &as.665.602.m. b79.607.618.733. 74Q.801.633.824.325.000. W&

:%

L 031L ma1. m

o. WL au.m.926. W.’331.920.926

:E

:E

;%

.W2

.s!74

J :=

.379

.W2

.79)

.324

.798

. mo

.720

.702

.716

.&as

0.644

s.632. 6i4. 04a. b24.541

:%.614. b17.49[. bl~.454.442.410.422. 3s7;KnJ

.261

.823

.221

.a17

.Z5b

.22a

.219

.132

. 14b

DURALUMIN

12n. diameter by O.~in. tb20k. Nomirmldlt-23.t9

EA-4.- . . . . . . . . . . . ------ m 2L M 4&tno o:Ag L 041 EA-bb.__ . . . . .._..–EA-2 . . . . . ..- . . ----- 283 4L 95 23, mEA-ba.-..._..__——_—

EA-2------—--0 59.74 27, WO L 192 :M EA-6d .____ .._ . . ..-

EA-b . . . . . . . --------- 0 24.84 27, w L 194 .002 23A-5e._:_— -----EA-bo.-- . . . . .._.._-. .0 06.22 %$20 L MU . 5%3 _

.? ‘: T ‘E

sHtn. dlameterby0.0.5Ski.t hkk. NomlnaldE-3M

T..

BC-O.i.-=________BA-6--77_..._-_BC4.--. . . . . . . . . . . . . .BAG-------------------BB~...-:---------_.--BB-!L-.:---------------BC-3-—-----------BA-2--___—---BC-2-----7

BW2b -----------------BA-6.._ . . . . ..-. -_-_––BB-7._.._.---- . . ..-BB-5 . . .._— -----BC-6--------------:g:&.: . . . .._.. -------

. . . . . . . . . . . . . . ..-BA-4.- . . . . .._. _-._.BB-4..-..._._-__--BB-o._..——-

a w.203. W3.3s7.349.82.3.S2J.740.755.747

0. 4i3.697.698.662.690.72JI.7W.916.473.9X

4a 94: 74.320 74. wo 29.33

: $1~o0 147.70 147.9

al,4201$ mI&m1074011,3707, 6M7, m

tk%

1. OM1. 4c12L 610L 991L 0022.3619334X 94b1PM

f ytJ

.444

.!459

.276

.182

. lib

.117

.m

I

2imdiameterby0.lWIn.thick. Nomin8id/t-34.6

FAD-------------------- aa 4L as a%990 0.819 O.am FA-6_...-__- . . ----FA+..._...--_-._

49. n233 46.36

29,em 0.029 0:7&3Q340 .393 .749 FA-2 . . .. A---------- 4; 03.02 26j9M L 148

STAINLESS STEEL

-

lln. diameter by 0.0351m thick NoWd/t-rn.6

CIA-7. . . . . ..–-...-.--– 4bl 25.22 124!4M o. Ml L 084 clB-o ______________ oGAB ------------------ o

6a 43 59, mo L 29236.62 87,130

a 201.762 .724 CIA-6-.-___ .__—__ o

GB-10---------------- 3h04 93,250 .762 .720oa31 b&4co 1.2J4 . 4b!4

CiA-6. . . . . . . . . ..——: 62.25

0 73.70&:---------

4&3NI 1.432 . X4b69,180 1.077 . b73 QA-3_________ 0 %W ; g2 .218

. . . . . . . . . .-. .— -- ban b%410 L 077 .627 QA-2. ..---- . . . . . . . .45!

o&f

C4B-2-...-...-. .__.-I& 470 .127

w. 12 69.610 L 204 .493 .:

COLUMN STRENGTH OF TUBES

TABLE 11.—RESULTS OF TESTS ON COLUMN&Continuedt t, 1 t

E&y R&gf A;h= Ell&x.=_ m>. Ratio of

Epeebnen Ill-% 13pecbnen IO-% -==-@## “~ ~~ : h% r (4b&) @

STAINLESS BTEEI.-Contlnned

1 Im diameter by 0.0S3 In. tbkk. Nomfrd dK=12.O

133

HB-7..—-–-—--- 233 W. 02 Ill g 0.623 a!2T4 Hs—6—..— 63. 2S IL429HB%._-..-——_— 37.87 .779 .777 Ecs-6—— ----- 8 77.46 2ZHE-4---.---–-.-—- 43! 43.84

% .343mm LIXS ECS-S-------------

:% HE-L --------------0 1o24 ~g 2.136~m

HB-a . . ..-. ---..-_– 0 lEs2 6~ 6ZI LI02 o 132.a :?%

lM in. d!.ameter by 0J123 k tbfek Nomtnal Oi’t43.7

IO-IO---. _..-_-— 2Q.61 ll&300 0.721 0.79s IG-9--- 40.33&

oICHL----------

97, w : El a 64939.25 07, m .955 .649 IG-S-..----—--- 0 46.02 am .M4

lx h diameter by 0.M3 h tbICk. NOMbl!dd/t-192

m.- .....-..-----------451 1*Soo 0.333 L 811n-lo.-....-----––––

n-s–————.w S&m .S12

I

o 7S.16 3&652o .7’16

O-USJB–6._—____—.— 0 1026 % 760 k%l . !21Zl

lx hi. diemeter by 0.049 in. tbiok. N“muInald/t=2al.6

K~=___________ 223 l% 1# Q3E9 g L173 KS-7 __ —-—___E: ,-

0 7552 % 310 L 822 0.315— o .S14 .72s

HEAT-TREATED CHROMIUM-MOLYBDENUM STEEL

1 In. diameter by 0.042 hL thick Now dlt=XLS

7Ez—2---__---–-—— 0 2L a l@s!m o. a14 awn lHz-L_.—_—_—_——— 0lHz-3s . . ..-–-_––_ 0

4Q. 10 lam yg26.61

a 602M&100 .Wa LEz-7___---–––.. –––

W.m Mmo s0 3Q.16 3!## .477

IHz-I ___________ 0 .917 lH.z-4_.__.-..- . . . ..- o 64.74mz-3_——_— 0 39.95 U300 .W2 .MO LHz—3------ 0 9L 76 2%s60 kz :?%

m1% k die.meta by 0.086 In. Udok. Nomtnal a7t-Xi.7

IW,3CQ U& LU2il sEY-7__________%fl E&w

0 829s.W6 LHY–E__-__-_—_

n

Ilm,m L322 a590o % 762 .621

XI.93 ~u~m .s24 .819 LHY-3-–---——_ o EE 7L230 2%m+.. --.-... -.–.--–

.=.SS

2E 2% %o 79. &? 47,230 .254

LHY-6_.——:E

0 E/9. 6 17,740 kz .W547.37 LT. 500 LIS6

1% in. diameter by 0.033 in. tldek Nomfnel dK=30-O

lHx-l...___-.--——— 0 l% 37 ‘w~ . .. -—-—- ---— =-2- . . . ..- . . . ..--– 0 I m. 8 IM7x-zl ----------- o 4&23 IQ m .—-—--- --.—— VsInea of S canld mt be o tabmd

17,62Q --------------------

I I I I 1. II

1% IL dlmneterby 0J349 In. tbfck. Nomfnd d/t-86.7

lH’iV-l. .._.. _...-__ o acn 17&m o. Us awo LHsv-2___________ o 79.73 46. Slo LW7 a236IHw-35..—__—— 0 Mm Mm .W LHvi+L--.._._.._.--

157, Mm iEo %77 3L MO 2.403 .173

mm--w. -.-.-—.— o 42 Is .m

LOWEREm RESTE~

OHROMIUM-MOLYBDENUM STEEL

1 in diameter by 0.U33In. tkdek. Nom5naI eYt=2S.6

CM-x!.-..---.-–..——— 450 46.’W qm) a386 o. m cA-sd ----------- Z33 4acm 77, ml am aiw

M Lo.diameter by 0.022 In- tldek. Nominal d/t=.Z3.O

DA*------------------ 430 h26a 63. WO o. 92i aw3 DA*------------------- 3a3 Mm m 910 a0i9 a728

lx h. diameter by 0.0SSh. tidck. NombJ a7t-23.9

AA-z. -- . . . . ..-.- .---– 450 59.86 M, lM 0.W4 a r= AA-2a ____________ mI

6&07 6Z950 L071I

a 737

DURALU?WN

1,4 in. dlametec by 0.05S in. tldck. NooAual d/f-2&9

BD-8..-.-.-----—---[ MI I 3S.07 ! 2LW ! 0.746 I 0.S6L II BD+- –1 Z3[W4[8WOI0-7671W2

134 REPORT NO. 615—NATIONAL ADVISORY CQ~hfITTEE FOR AERONAUTICS

TABLE 11.—RESULTS OF TESTS ON COLUhfhfS-&mtiw&l

STAINLESS STEEL

E&d~; Ratio of A;h~A.= c)-

ELl%ln; Ratio of A~v&we

Spedmen lc+n aiendar-

J

A.- cb-6pe2brren

slender.

!% ~$ ~ m

lo+md

x

‘b&h%r!y W (lb.p# ~ My ++ E m

1 fn, diameter by 0.M5 [n. thick. h’ombrd d/b2S.6

GA-IO. . . . . . . . . . . . . . . . . . . 451 2L 170B-6- . . . . . . . . . . . . . . . .

182.SW ‘ 0.464283 m. 18

L lW GB-4.–-.._..._. _.._..*326 L072 .575

450 86.92 ~m 1.410 0.443

1 in. rlhmetez by 0.M3 in. thick. N’omfnrdd/t= 12.O

HB-10____________ W 24.07 119,Oxl --------------------- ------------ --------- ---------- ---------- . . . . . . .._

l%ln. dle.met8r by0.C&5in. thlok. Nonr4nafd/t-M.7

rcM3..- . . . . . . ..- . . .._-.. m M. 26 71PS50 L 820 0.478 ----------------- .—-— . . . . . . . . . . ---. _...._ . . . . . . . . . . . . . . . ..-

l%in. dimuater by 0.0051n. tbfck. Nombm14t=19.2

JB-5-_ . . . . . . . . . . . . . . . . . 238 76.84 39,040 L 591 0.349 ------------------------- -------- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

STREAMLINE TUB~Gb

BOTH ENDS FREELY SUPPORTED OE RESTRAINED

CHROMKfM-MOLYBDENUMTSTEEL

1.f385by 0.714 by0.035 in. h’omfnald/i-M,7

lcB-4a ___________ o 35.05 77040

. 1.

o:;% 0:8A lCB-2____________ICB-l. ..__. ..__ -_... 88.61 g:

o 5s. 37 L 049 0.661IcR-3____________ .0

4;%%

lCB4b._-___...__. 4A 91 .794 .7737Q.51 L 405 .430

. L. .—

3.872 by L429by0.0S31n. Nomfnaid/t-30.I

lCA-l . . .._ . . . . ..__.. - o 53.30 65070 0.988 0.700 lCA-8..:_- . . . ______ o1.

78.74 4W0 1.340. 0.512

DUEALUMIN

L6S5byo,714 by0.035in. [email protected]

lDB-l . . . . . ..--. ._._- 39. wlDB-4n . . . . . . . . . . . . . . . . . 44:

285s0 o. Ta o. w40.25 27W0

lDB-2__________ o.S86 .7W LDB-3-J

saw ;7~o 79.08

a w—.--—--—— ;% .40h

&372 by L4!Wby0.065in. Nomlnnld/t=3S,5.—

lDC-2------------- 0 89.24 28550 0.706 0.344lDC-L . . . . .._-. . . . . . .

lDC-3___________o 59. L4

os30io L fw .744

79.70 17W 1.434 0.497

STAINLESS STEEL

1.005 by0.714by0.035 in, Nomfnal d/t-36.7.- —--- ----

lSB-3a . . . . .._. ______ 440 42.40 86100 0. 8eo o. TWn

IsB-l___________ 0 m. 05 27610 Z.rlu o. !46a

2,023 by0.SS7by0.04’J1n. Namfndd/t-Wl.6

lSD-3 . . . .._–--.__ lswl..:fi_” ______ o W 16 41360 L 767 0. w.

2.687 byl.148by0.05$ in. h-ominald/t=34.5

lSF-2a_.._.._.___.- 0 XL 65 110300 0.493 0.337

3.372 by L429by0.065in, Nominrdd/t-W.5---

2sc-l ___________ o 69.32 64460-—

[ 1.1681 aw[l-.:.; I

HEAT-TREATED CHROMIUM-MOLYBDENUM STEEL

1.517 by0.043by!M12.6in. Nominrdd/t-32.l

TlHO-SL . . . . _______ 440 35. lx! . M82m 0.34 0.865lHC+3K.._._._.._ 40.43

IHo-l..__.-.–—-– 01474@)

03.09 757en l.bld a 439

44:.071 .855

lHO-5_ . . . ..__ . . . . . . . .lEG-2-._i..._.. ._._-

41.43 141mo .$?35o

.3Z?79.24 477X 1. m .27

fI.035by L2SOby0.05Sfn. Nomfnald/t=38.8

lHE-8___-_—-—_ 440 47.55 l159mI 1.181 0.002

lHE-L.. -._—___ o 49. i’7lHE-2_________ o

1251w1

1.134 .716 :. I “43I F-~Lxnlfa!lnm b Sb.ea are given by major diameter, m4nor dIarneter, rmdtbfckne%. knlnal dlt la given for baaic round dinmemr.


<TABLE 111.—VALUES OF r AND 1~ & FOR CHROMIUM-MOLYBDENUM STEEL

E=29,WJ300 lb. pm W. in.

135

Pit (l&.p ~4

Ioa IA

P/&(i~lper ~4

101 1A A Pl~(J~)per , ~

f

1(JJ 1

~b~A

. .A

O“@=). .

G b%

— .

0 to32@o 1 llo35a 49W as- o. Om 65,000

0 0 la.471

7O.mm

aa,mm!22

LLMI .0383 60,033 .WQ28

.M40 66#oo1

.4420

.m19 8

34$00 .W3 .05s330

61#m .s1081

8 1.Wta ai,mo

19.M2 .0909

835JMI .996 .05s4

2452JXKJ .791

35.0633

4 I5s&i0

%3.aw .0944

.9a6mO0 .W2 .0395

Maa#m .m

41

5 2.0664 Ca,wo .651

22.09s5

a7#m .!337IO

.053732 47

54JXHI .749 .06746

70Qm2

.m22

. loa2

WMO . 9s110 32

.0m9. ~ 56@o .7Zao

.00E4 71,01m .s7

. 10s

39,CUI22

.97410

. owl32

5&m .m .MQ453

8 272@o .256 .LIM

40,0m .%a 12

.mm33

57030 .E32 .OW63

10 8ialw .222 .I!2a9

41,0m .95a24 Ia

.Oafbsa4

S&ow .am .o’i19lo6

IO74,000

8.Isa .Ia4a

em .9-M25

.0m9L4 w

60,m .= .0733Ml

n75,mo

4.la4 . 14s6

43,(03 .csam M

.(W334

6Wam -m . oHa 76@o201

la 4.lm .=

44,0m .922’211 17

.0m735

61,WI . ml .0%5W

lan,mo

5 al.033 . !UU5

45,0m .fwl . @312la aa

62S01 .M5 .07s3 m,am626

14 5 2a.049 .mdl

46020 .m5 . m17m

caJ3m .62736

.m2441

16m,ooo . Ola .W2

54i@10 .am .(W2

a 22 13WoOO

.

16.49Q Ows

53,400 0 .

4%M0 .W2a

.Om25

17 6

TABLE lV.-VA.LUES OF r AND ~ /

.—

~~ ~ FOR DURALIIMIN

I?=I0,690.IJX!lb. per W. in.

P/$@k.)w ~

f

mlA A ll~(lb.lpw ,

. .z b%

~ ~g & pl:f~y . A %’+ A(in b.Vrl

— .

Oto21#am 1 O.c%a 29, m

RO.an a 1294

436,000 0.169 o.2aa

aom . WI32

.0m2a7 m

lqxkl .Mo17

m. ml at m

14.147 .ma

qam .ma32 41

.09932L

aQm .6M21

26 M.18i2 37, m .126 .X6

awl .m931 45

. Iolom

aqm .G25

2a. MD a7, Wo

16.lm .aol

~mo .911z?

.102549 la

aom .44530

29 16.IM6 a%ooo .Ws .ml

24, Om .awxl

.1o41!W 17

U500 .4L637

29 la.L5M aa.wo .Oil .253

WW3 .m8L 50

.105915

Q(I3O .’J0447

30.Ia7a 39, m

w.OM .41a

aom .m29 6

.107a Q5m .35614 62

ao.lc-i 29, m

m.042 .47i

25.W3 .79ax! 7

. 109sE?

lqooo .ama6

al.171 40, m

21.mo .503

am .762!2a a

.I.lm 33.WO10

.Za7 . lii126

3140, m

24.020 .5s8

2ejmo .73127 9 a

.I144 Umo .%0 . I&xm

K2 264LOO0

27.012 .ss3

27, m .6m10 6

.1170 &m .24aWo

81 27.19a am .W6 L25S

27, m .mam 12 4

.llOi mm .217 .210917

az 30amo .m x 176

mom .m24 Ia 2

.lmi m.xn .19a .22a6129

a2 azqwl .mo &204

am .60424 M

.12590 .

32 35gm o .

s.. .

136 REPORT NO. 616-NATIONAL ADVISORY CObf~ITTEE FOR AERONAUTICS

TABLE V.–VALUES OF ~ AND ~ijr FORSTAINLESSS’I’EE.L

E=26,3WX10 lb. par W. In.

0.624

.819

.304

.799

.773

. 7s

.720

.722

.704

.0S9

.008

.649

.630

. 6U

. &al

. b?l

. Ml

.621

.610

.434

.468

—

A

—

16

lb

16

16

17

17

17

18

Ia

13

19

19

19

20

m

a

xl

21

21

21

!zl—

—

A

—

o

0

1

1

I

1

2

2

2

2

3

3

3

4

4

4

4

b

6

6

e—

-

A

—

0

6

7

7

8

8

9

9

10

10

11

11

12

13

14

lb

16

17

13

al

n—

—

A

—

o

2

2

3

3

6

b

6

T

7

8

9

10

10

U

19

12

12

13

14

lb—

A

m

24

27

w

33

87

4s

47

88

01

n

35

104

lxl

171

137

3$2

640

MM

m

---(I@ 1

(K/l ,%-r1P &

(L/lb.%)-

0.02s0

.06M

.0b#2

.0299

. me

.0714

.0722

.0731

.0740

. 07KJ

. “m

. G771

.0732

. O?W

.Wa7

.0621

.WM

.Ctsb2

. (t309

.!3%97

.om7

r A

—

0.44721

.42621

.40822

.2S322

.28192

.33922

.81722

. 2Jb23

.27223

. ‘m23

.22623

.X023

.193a

.18723

.13423

.111n

ass24

.00424

.04024

.01610

0

PIA (lb.sq. In. F

0.0923 IoLow

4000

& 000

arm

10,Om

12,000

14,000

16,020

I&030

2), 000

!QOm

24#000

28,WI

%Wl

2a,m

3&WI

34 cm

3&WI

3&C02

40,Cmo

I

1. O@)

.Ea

.902

.W2

.’aw

.Wd

.Ea

.974

.367

.900

.962

.943

.m

.923

.912

.W

.3s8

.876

.803

.349

aooz

.0621

.0221

.0’322

. Ma

.0$24

.0928

.0227

.0220

. M31

. Mm”

.0996

. M39

.0942

.0046

. Olwl

.0684

.00.58

.0663

.0653

.0074

.0951

. mm

. Im

.1023

,1026

. llm

.11.14

, llQ1

.1244

.1365

.1377

.1462

. MM

.1020

.1307

,2104

.2464

.allo

.43u3

m

TABLE VI.—VALUES OF_r AND ;J

& FOR HEAT-TREATED CHROMIUM-MOLYBDENUN1 STEIH,.—

E-KvXMKW3 lb. P sq. IL

-f

l@ 1A 9-E

(ii b.Vr)

.“-5

.Q5m6

.09306

.ns927

. a5948

. 050e9

. anl’a‘a

.cW?11

.m12

.061014

.0014u

—

—

A

—

2

2

2

!2

2

3

a

4

4

4

6

—

—

A

—

8

7

9

10

la

16

19

97

40

a

m

—

r A

— .

10

LKUIo

1. Wo1

.9991

.6$%a

. ‘am!4

.0942

. W33

.WJ3

.9304

-f

I@ 1.75 A

(in./ b.lP)—

0. 0s61o

.05310

.MS1o

. 0S91I

.05320

. Om

. 06s3 x

.asw :

.0tia41

. 05s51

IllA

f77X

(tn. b.q—

.061917

. Wlx19

.003222

. m4126

.005128

.0M334

.mi340

. !xG750

.072466

,070401

,Cgion

m

P/A (lb.Sq. in.r

o to

106, WI

109, Ow

110,000

111,m

112,m

113,m

114,m

lI& O@)

ll& Ooo

117,m

P[A (lb.eq. in.rr

rr

m.ml

130#cQo

1?s, 020

13g m

13&am

l% m

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137,mo

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140,m

.330

.363

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. 70!3

.736

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.645

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.476

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m,m

119,ml

lx), m

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W-MO

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127,030

1% m

.Qa2

.im

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. SI06

.053

.080

.941

.932

. r121

.m

.895


TABLE VH.-VALUES OF s, t, AND W–N

137

*‘-drop

2f–1 ~ 4*tnterpalat[on for & t, and &-# when 0.99<4~LOI fe not @Iei end it wfIl ke @TicfentIg emrrrste to takes +,t —a , emdv–~- +. gtily, I*&~fstf~~

#/r

o.01.02

:ti.G5.m.07.03.00.10.11.12.La.14.16.L6.17.La.19:g

.22

.23

.24

.36

.=

.27

.%

.m

.20

.81

.32

.?3

.64

.35

.86

.m

.2a

::

:%.4a.44.45.46.47.4a.49.&l.51.62.62

::.66.67.5a.69.a.61

::

:%.68.67.Ca.69.70.n.72.72.74.76.76.77.is.79.m.81.s2.=.s4

.:

1–= I-‘r r

fors and t when L99<$<2.LHI 8snot PUSSLbIqend then with suffkfent2 2

eccmecys--sndt= —.*2 2A--r r

1s

0. mo16.OcoM.00149.03264.m-4L2. m695:fi

. OL248

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. 03im

. 64?39

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. LOL42

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. L21?51

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. Mle4

. 152crz

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. 2LS9L

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. 249E3mom:=

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. 2.4X31

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.33422

.40728

.48L24

.45662

. 4mLa

.51095

.54014

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.mmo

.6MM:=

.74941

.79102

. m&90

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. W4118

. BIBL IX478L094201. L5’m7L 22Z1OL 281sL 866L8L44MIL 53016L 6.2)90

1%%L S3564:=

Z 232L6z 4679z 6679ZS442&0493a.anss2.62722.s0774 L224L 4ii&mmh2455

t V4

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. moll

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.00646

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. mom’

. 0L352

.01651

. 0L997

.02297

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. m379

. m976

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.E%2

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.12247

. 12S59

.=7

.17686

.19746

. 22Lm

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.27611

. Smm

.23964

.2i6a2

. 4162S

.45970

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. 53S19

. 612s4

.67419

. 72W

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.8706L@306L L6SlL!M87L27S0L 5018L maaL 7SL0

k%%2 ma9!L4947z 7L2Sa.’alwla mla4m22, MB

:~4

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kgmomLZ!M4IaamLhI15L6.274

+4/.

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.92

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.94

.95

.96

.97

.m-99

ML02L~L04L~L06L07L@SL09L IOLLILIZLL8L 14L 15L 16L 17LL9LL9LmL 21L22

RL25L!ML27L%L29L 30L S1LS2L aa:.L%L 87L23L.39L40L4L1.42L4aL44

?$L47L4.9L49L60L 61L62L62L64L65LiML57L69L59

:%

H

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L67L6SL69L70L i’L1.72

s

hssm&mmr. 2642&Mm9. 24n

LO.6722

M%

EEaL 3s1g%

-L%.%-52024-2&m4–Z 099-22 Ix-7—La 772-m4Lo-14542-L& 274-La Lm–LLm6-L& am-9.93S7-9. 4L14-s. mi9-&W-8.2XB–7. 9LE4–7. ml–7. 4L2S-7.202L—7.0L29–&.S4Sa-6. mm’-6.6636-6.4201-e 3L2Q-6. 2LSQ-&m-&M-6.9759-6. 9LM–5. S542–b m40–5 moo–5. 722046S27–L 6229–6. 5W?–6. 624a–5. mm-6.6WS-&W-&E4 m21-5. m–5. M67-6. Mm–5. m-6. n24-6.7461-g g;

-6. Si65-&mm-6. mm-&05441-a L24?-6.m1742ss2-E S7m–K472S–Km-6. m4L-6. S177-a 95L2–7. O%ia–7. 2510–7. 498–7.6X6–7. 7W2-s. 012?3

t

7. amo7. mms.im29.7019

LLLwLzm714.onL6.481LaSIaZ&S74m.!u349.985m.w

–m. mi+9. ma-23.2s2-24 ma-19. m–I& 457-M. m–L2 215-Ill 787

–9. 6257-am-7. mm—7.ma-&6UM-6. Lmm46X9-&!m62–4 S4L446010-41sm-zKS164633042$80–a. 14s4-2. 92io-2 n72–2 m7a–23m7–214s6-L W78–L 7969-L 6317–L 4TI1-L 3148–L 16L0-L 0L05-aS6m–m7165-a572L-a 42914.2s69+ 1454-al

a2is5O.#all0.66s5a707a:s529

L 1491LaL4644L 6L12

;Eam22745z 45L2Zama2. S7410a. Maoa. amo6.4244a64L2aswo4L0274&tM460s24mL06. L6ioh 4702

t:+

Ha%%HE4392mm

E?%LL9.95

%:

-&mw-214. m–147. 70–LLL54-wm-SL 547–?% 196–65. m-m. E474690-62 ML-49. an4ss7-4&Sn-4%Lm*6674?s2-29. X2-.212al-67. 4L7–a% am–2s am-ah m-2AS$4-842s2-m.904-m-2L%L-s2W-2Z7LI-22488-S2.!E3-22109-aL 959–2L 829–3L i20-2L629-2L 564–8L 496–SL 462-2L ML-6L402-m. 297-2L 4CQ-6L4LS-8L 448-8L 476-3L 6X-2L 672-8L 62L-2L6W–3L m-2L Sal=* g

+zg

-22.239-SI.452-s2 570–a 69a-22s19-22950

=!%+245* 6L0--m. am-66. SLl-w-a4L26-84 2s4

L 76L 74 w%L ?b L 46%’9L 761.77 u?%L78 h 6W3L79 b. 62b4b

b. 66497M h6waL82 6.71770

b. 74911i; h 78054L 8!5 5.8119bL80 b.84336

NO. 616-NATIONAL ADVISORY FOR AERONAUTICS

TABLE .VIL-VALUES OF s, t, AND t~–+Continued

9

-& 2466-aKm–& ?7Eu–k 0772–L40M–Q i7!23

-16.176–la 821–lL 110–lL 671-Lz !2Q4

, –n. Wa~;; .yO

t

-16.792-17. M4–la m-m. ala-22. x%-lb. Mb-B. 796–w. 624-4a 161–5a 129-66.764

-100.07-m 03

-Q I

column str~tgth of tubes elastically restrained … · strains were measured with a ewing...

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