1、A Reproduced CopyOFNASAReproduced for NASAby theScientific and Technical Information FacilityFFNo 672 Aug 65Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-“l r IProvided by IHSNot for ResaleNo reproduction or networking permitted without license fro
2、m IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.: . .Lu i,_ : . .:=_,qDESCRIPTION OF STRESS-STRAIN CURVES BY THREE PARAMETERSBy Walter Ramber
3、g and William R, 0sgoodSUMMARYA simple formula is suggested for describing thestress-strain curve in terms of three parameters: namely,Young_s modulus and two secant yield strengths. Dimension-less charts are derived from this formula for determiningthe stress-strain curve, the tangent modulus, and
4、thereduced modulus of a material for.which these three param-eters are given. Comoarison with the tensile and compres-sive data on aluminum-alloy, stainless-steel, and carbon-steel sheet in NACA Technical Note No. 840 indicates thatthe formula is adequate for most of these materials. Theformula does
5、 not describe the behavior of alclad sheet,which shows a marked change in slope at low stress. Itseems probable that more than three parameters will benecessary to represent such stress-strain, curves adequately.INTRODUCTIONAn assembly of the tensile and compressive stress-strain .curves for sheet m
6、aterials characteristic of air-craft construction is being obtained at the NationalBureau of Standards as the principal objective of a re-search project for the National Advisory Committee forAeronautics. Stress-strain, stress-devlation, secant-modulus, tangent-madulus, and reduced-modulus curves ha
7、ve been presented in reference 1. for various grades of sheetmaterials of aluminum alloy, carbon steel, and chromium-nickel steel. A second objecti_e of the same researchproject is a search for yield parameter,a that give a betterdescription of the stress-strain curve than those in useat present.The
8、 conventional descripti-on of the s.tress-straincurve of metals by the two parameters, Young_s modulus andyield strength, iS .inadequate for the efficient design ofmembers unless the material follows Hookes law up to. aProvided by IHSNot for ResaleNo reproduction or networking permitted without lice
9、nse from IHS-,-,-“_/ /.k- p._r_ i _2 NACA Technical Note No. 902yiel_ point at which it yields indefinitely under constantstress. This special behavior is approached, for example,by certain steels (fig. l) and by certain low-strengthmagnesium alloys, but it is not characteristic of manyhigh-strength
10、 alloys for aircraft.Examination of the stress-strain curves for aluminum-alloy sheet and chromium-nickel-steel sheet given in ref-erence 1 shows, particularly for the compressive stress-strain curves (figs. 2 and 3), a gradual transition fromthe elastic straight line for low loads toward the horizo
11、n-tal line characterizing plastic behavior. The type oftransition varies widely. Hence there is no hope of reduc-ing all stress-strain curves to a single typ_ of curve byuniform stretching, or affine transformation of coordinates.This rules out the possibility, which exists for affinelyrelated stres
12、s-strain curves (reference 2), of completedescription in terms of only two parameters, Youngts mod-ulus and secant yield strength. A minimum of three param-eters will be required to describe the changes in shapefor different materials.Several proposals have been made for describing thestress-strain
13、curve in terms of three or more parametersDonnell (reference 5) suggests as two yield parametersthe stresses sI, s_, at which the slope of the stress-strain curve is equal to 3/4 E and 1/4 E, where _ isYoung_s modulus. The stress-strain curve is then derivedfrom these two parameters on the assumptio
14、n that theslope varies linearly with the stress. This proceduregives a good descriotion of many tensile stress-straincurves of aluminum alloys, but it does not seem adequatefor the highly curved tangent-modulus curves found forthe compressive stress-strain properties in reference l,from which figure
15、 4 is taken. Furthermore there arepractical difficulties in determining the stresses corre-sponding to a tangent modulus of 8/4 E and 1/4 Equickly from the stress-strain curve.Esser and Ahrend (reference 4) noticed that thestress-strain curves for many materials may be approxi-mated by two straight
16、lines when they are clotted on log-log paper. They orooosed to define yield strength asthe stress at the intersection of these two lines.Description of the stress-strain curve above the yieldstrength _ould be obtained from the slope of the upperstraight line. The proposal is doubtless an advance ove
17、rthe description by an offset yield strength. It has the . . . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-s-ey- Sp-/ (i)wheree strains streameySpnstrain corresponding to yield strengthproportional limitconstan_SyIf the iogarithmof both sides is
18、taken in equation (_),it can be seen that equation (1) approaches Esser andAhrendls two straight lines as asymptotes for low and forhigh stress, respectively. The description of the transi-tion region is obtained by increasing the number of param-eters from three to four.In the study of plastic bend
19、ing, the second authorfound an analytical expression containing three parametersthat appeared to be well adapted for representing stress-strain curves. Further examination of the exoression inthe light of the data given in reference 1 confirmed thisview.Provided by IHSNot for ResaleNo reproduction o
20、r networking permitted without license from IHS-,-,-I4 NACA Technical Note No. 902: !ii iiANALYTI CAL EXPRE SSI ONStress-Strain CurveThe proposed analytical expression isE where K and n are constants.becomes the same as equation (2) if(2)Nadaits expression (1), r._Sp= 0ey _ = K(3)that is, if the pro
21、portional limit is taken as zero, andthe requirement is dropped that ey is the strain corre-spond_ng to a yield stress Sy.The expression (2) may be written in dimensionlessform in terms of the follo_ving variables (reference 6):S IS 1J(4)where sI is the secant yield strength, equal to theordinate of
22、 the intersection with the stress-strain curveof a line through the origin having a slope equal to m I E, 10) the experimentalvalues of tangent modulus for stresses below the secantyield strength sI differ less than _0.07 E from thevalues corresponding to equation (15). In the case of thevalues with
23、 the sharp knee (fig. 21) the maximum differ-ence was considerably greater. These differences do notdetract seriously from the usefulness of equation (2),however, since the region in which the agreement is notgood comprises a limited stress range. Consequently, inthis range the difference between th
24、e experimental andthe computed values of O corresponding to a given valueof tangent modulus are small.The comparison was confined to the materials in ref-erence l, which did not include alclad aluminum alloys.In the alclad aluminum alloys a change in slope at lowstress is observed which corresponds
25、to the yielding ofthe aluminum coating. It seems probable that inclusionof this effect will require the addition of at least onemore parameter to the three contained in equation (2).EXAMPLE FOR APPLICATION 0F THREE-PARAMETER METHOD/Computations based on elastic theory give a value ofSse = 87 l0 poun
26、ds per inch efor the critical compressive stress of a given specimen.The material of the specimen has the compressive stress-strain curve shown in figuAe 22. It is desired to deter-mine the stress:E rS r - SeEwhich is an estimate for the critical stress after takingaccount of the olastic yielding of
27、 the material.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12 NACA Technical Note No. 902, From figure 22 are obtained the two secant yieldstrengths|s_ = 43.0 X 103 pounds per inch sss = 38.0 x i03 poundsper inch sso thatS 1 - I. 132S_From figure
28、i0 this corresponds to a shaPe parametern= 8.15Entering figure 9 with this value of n and with th_ratioIgivess_1- _ 43.0 _ 0.494se 87.0_E _ 0.473Eso that the corrected critical stress is3 3“ 2s r = 0.473 87 lO = 41.2 X lO pounds per inchNational Bureau of Standards,Washington, D. C., April 8, 1943.P
29、rovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-j?L NACA Technical Note No. 902 13REFERENCE Si. Aitchison, C. S., and Miller, James A.: Tensile andPack Compressive Tests of Some Sheets of AluminumAlloy, 1025 Carbon Steel, and Chremium-Nickel Steel.T.N
30、. No. 840, NACA, 1942.2. 0sgood, W. R. : A Rational Definition of Yield Strength.A.S.M.E. Jour. App. Mech., vol. 7, no. 2, June 1940,pp. A61-A62.3. Donnell, L. H.: Suggested New Definitions for Propor-tional Limit and Yield Point. Mech. Engineering,vol. 60, no. ll, Nov. 1938, pp. 837-38._. Esser, Ha
31、ns, and Ahrend, H.: Kann die 0.2% Grenzedurch eine ubereinkommenfreie Dehngrenze ersetztwerden. Arch. f. Eisenhu_tenw., vol. 13, no. 10,1939-40, pp. 425-428.5. Holmquist, J. L., and N_dai, A.: A Theoretical andExoerimental Approach to the Problem of Collapseof Deep-Well Casing. Paper presented at 20
32、th AnnualMeeting, Am. Petroleum Inst., Chicago, Nov. 1939.6. 0sgood, W. R.: Column Curves and Stress-Strain Diagrams.Nat. Bur. of Standards Jour. Res., vol. 9, Oct. 1932,pp. 571-82.7o Timoshenko, S.: Theory of Elastic Stability. McGraw-Hill Book Co., Inc., New York, N. Y., 1936.8. von Karman, Th.: U
33、ntersuchungen uber Knickfestigkeit,_._iitteilungen uber Forschungsarbeiten. Ver. deutsch.Ing., Heft 81, 1910.rProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-“ . . L!i_i_/.NACATechnical Note No. 902ZOO, . :- -Fconsverse coM,oress/on Icompress/on_.50
34、_1Figs. 4,50 .50 ZOOS/8 _tdFigure 4.- Non-linear variation of tangent modulus withstress; chromium-nickel steel, full-hard,thickness 0.020 inch.5640_3z_ee16I/I/,/.S=_e i Il !IIl1/Jo .002 .004 .006 .008S_rainFigure 5.- Determination of secantyield strength./Provided by IHSNot for ResaleNo reproductio
35、n or networking permitted without license from IHS-,-,-. .- , ,!#.NAC A Tli,chnlcal#“8I_S/S = p0L0 ,F;q. 6_) +_ _o_D IIO) U) o ) ,-_ -_ _|,d(D%,-4X,7II“i I-,.-I 4-3or2I=I ,-44-3.- ,-4 0. Cc:)c_.r.-I“r _11Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-
36、,-: . .“ . . _ I INACA Technical“ ! “ : “: /, i -i/ “ “ :-iiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA Technical Note No. 90_ Fig. I0 i/5.010.5,9.08.04.025/.5 /.0 /./I , It I P J i I llqilll I F I I I I I I t I t I I I It II tllltl ILllll_l
37、 /IJllllll JlllillJl LLIIIII/.2 /.,3 /.4 /.5 /.6 /.7 /-8 /.9 at.O_,/e=I0.- Relation between n and Sl/S 2 .Figurei?log -Vn=l_-zogProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. !NACA Technical Note No. 902200150“/005OO,I0080_buf2O0Figs. 1,12I I I I
38、I I I I IFigure II.- Experimental and computed .,_,-,_“stress-strain curves_ -_-Cr-Ni steel, fUlll-hord, thicknessl_J _“ /-110.020 inch. X/r“ _ lI /X.; _ / l ._s. 7S .:“Tension, dred puledd Longitudinal AFrc/n$ verse. +Compression,Long/ltud/no/ oFf or?svef s8. xII2_00.E _ “ $z n_1000/b/s#7 in.2#,770
39、 18#. 7/473 4.8329,010 1800/32.7 zl.5826,/20 8Z3 5Z4 20829. 660 205. 6/62. 7 zl.81002 004 .006StrainMeos- Cor_-Fension, ured puledL ongi fudino/ a7tons verse. +CompressionL ongi fud/nol oTronsver-se. x.OO8 .010IE s, s, tt/000 Ib/sq in.IOEO0 52.2 5/.8 121.410,430 4_ ,#ll.6 10.93 _./06#.0 43.2 39./ 98
40、8I0, 710 48.0 45.6 13.43.012/.002A Jc2a I um Inumi r_ch.I I0O4. X._._;- _-_- - _ -:d:o _c- Experimental ancl computed-stress-strain curves;alloy 24S-T, thickness 0.064I006 .008 . OI 0 .012S#-oinProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. : _ .:
41、 :_: :i :, i !_._/: : :L : _:, : NACA Technical Nofe No. 902/008OZOO.8O.8OB“.40.c_f0Meas- Cam- E _ sz nb_ r Tensi-_, ured puled I000 Iblsq in. -_ Longitudinal _ - - 30,060 62S $38 ooTransverse. + - 3/,620 64/ GS,/ oo- Compression .Longitudinal o 30,780 b_.b 61.0 ooTransverse. x 32,660 G3.4 63.3 ooi
42、IiO01 002 .003 .004SfruinFigure t3.- Experimenfol end compufed sTress-sfroin curves, 1025corbon-sfeel, fhickness=O.054“.rlo 292qMu far / e I , /nchCr=/V/“ sfeel full-hard fh. 0.020 Iomq.,n =2oo3II2_00_QFigs. 13,140 .20 .40 .80 .80 LO0 lQO /,40G=S/8_Figure 14.- Experimenfal ond compufed fongenf modul
43、iin compression,2n3.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- , ,. LI IL,.!7i:- L:_:i!:NACA Technical Note No. 902LO0 _ _ ,3.10X_a_ _ 3.08 “.80 _.60-E-.40Figs. 15,16I I I I I I I I IMo fer / o I Incho 3.48 OczNi steel 1/4 hard fh. 0020 long.+
44、3.83 /12 -314 “ “ full “ “ _ “. .024 “I/4 “ “ .020 trans._0 Xo_ _ o.20 o t0CDODI00_Q0 .20 .40 ._0 .80 /.00 /.20 /.40d=S/S,Figure t5.- Experimenfa and comoufed fomc)enf moduli incompPessiom, 3n4.0498 Cr-/vi steel ful/-hord 0.0275 long.1.00 -_= A 410 1/2 “ .0 0 tr- s.+,.18 3i4 x,.si ; ,_ll ;t_._.ge .o
45、2.+.80 -,-,_ ,.E,“20v . 50 .20 .40 .80 .80 /.00 /20 140Figure 16.- ExpePimenfal and compufed fongen moduli incompnessioP, 4n5.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,. . =* #: L L :,. ;i-.Technical Note No. 902/.00 -.80.60 .in compression 6n
46、7%(.20n Mc/fe/-iol Incho 6.72 Or-N/Sfeel full-ho/-d fh. 0.02Figs. 17, 1875 tr-ons.LOOJ0II00_Q. _0 .20 .40 ._0 .80 ZOO 120d =sis,ZOO.80._0E-E-.40.2O71 Mafer/ala Z82 Al-ol/oy 24S-To 7.60 “ Inchfh. 0.08/ /on_7-“ .032 “I IFigure 18.-Experimental andco.mputed tangent moduin compresslon. 7n8. Irl=70 20 dO
47、 60 .80d=s/slzOO 120Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. .-. , r:iNACA Technical Note No. 902.80 _n AAofer/d/E o 8.59 AI-olloy 24S-RTth. long.7_- + 8.32 “x 8.018.20._0.2OIFigure 19-Experimenta end com-puted tangent moduli ncompression. _90ZOO I.806OE-E-4O.20 4O 60 .80#=s/s,ox 9.079.88+ 9.O7Mafer/al InchA/-o/oy/TS-T fh.0.032long 24S- T -084 “ “243-RT .20 -Figure 20-Experimenfo and com-puted tangent moduli ncompression. 9n/O0 20 .40 6o .8od-s/s,Figs. I9,20LOOdII00_Qv.-_ =8I
