Lind, N. C.

Schroff, D. K.

English

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

Missouri University of Science and Technology Scholars' Mine International Specialty Conference on Cold-Formed Steel Structures (1971) - 1st International Specialty Conference on Cold-Formed Steel Structures Aug 20th, 12:00 AM Utilization of Cold Work in Light Gage Steel N. C. Lind D. K. Schroff Follow this and additional works at: https://scholarsmine.mst.edu/isccss  Part of the Structural Engineering Commons Recommended Citation Lind, N. C. and Schroff, D. K., "Utilization of Cold Work in Light Gage Steel" (1971). International Specialty Conference on Cold-Formed Steel Structures. 2. https://scholarsmine.mst.edu/isccss/1iccfss/1iccfss-session1/2 This Article - Conference proceedings is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in International Specialty Conference on Cold-Formed Steel Structures by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu. \JTlLIZATION OF COLD WORK IN LIGIIl' GAGE Sl'El!L By N.C. Lind, Prof•••or of Civil Enaineerlna, University of Waterloo, ~nd D. K. Schroff, Associate Prograt~~~~.er, 1BM Canada L:United, 36 King Street East, Toronto 111, Canada Waterloo, Ontario~ Canada INTRODUCTION Cold forming of light sage steel structural sections can modtfy the mechanical propertiea of the material sianificantly, sener•lly increasing the yield atrenath and the ultimate strength ant! reduc:Ln& the ductility. The modification in the cold formed r•gLona. the "corner•"· is beneficial for the load carrying capacity and B1Rnlf1cant for the aection at a whole, as Karren (1] has demonat .. ratrd. H• has also developed an analysh to predict the increase in load carryina capacity that, with minor modifications, has been incorporated in the AISI (1968) Specification [2], in the Canadian Standard S•lJh (1970) and possibly others. This analysis is workable; but it is complicated, impoaaible to understand without recourse to ref. [1), and very difficult to execute manually without referring to the apec;:Lfication or standard for details and numerical constants. The theory also aasume1 von Hhes yield condition, and isotropic strain hardeninM following a power law. This complicates and specializes the analyais and ia not in good agreement with material behaviour; the tt·st results Ill show on the average a 5% difference between the yield strenf!;th in t('ns ion and compress Lon. lt is the purpose of this paper to present a similar theory bas«>d on fewer assumptions; this analysis is considerably simplified .,nd intuitively obvious. Specialized for a simple strain hardening law (linear) it leads to a design rule of extreme simplicity without the dr.twbacks mentioned above. In this rule, all aspects of material bf'haviour are represented through the hardening margin, fu ... fy' between Hltimatt> und vlt>ld strength, specific for the material, and a hardening constant determined empirically and taken to be the same for all moHt"r 1.1ls, Bec<'luse this conatant is determined directly from the corner tl!'sts. r.lther than from stress-strain curves, it is not surprising that th(' results a~~:ree much better with the test data than Karren's original thE-ory. Af; proposed in a draft of the CSA-Sl36 Standard, the rule for calculating the .strength of a beam or a fully effective (Q = 1) column is simply to replace the yield strength by the ultimate strength over St 2 in each 90° corner, For other corner angles the area is increased proportionally. The theory presented by Karren had auch of a semi·empirical appearance, and it was thought wise not to transgress the bounds of experimental evidence in the application, leading to the restriction in the range of corner angles (up to 120°) and the &ap between yield and ultimate strengths (no less than 207~ of yield), The present theory shows clearly that these restrictions can be dispensed with. MATERIAL BEHAVIOR The analysis requires few restrictive assumptions about the hehavior of the material. For example, it is not neceasary to asaUIII• that the material is isotropic or that it has equal properties in 10 tension and compression, It is assumed that the material is elastic· plastic; elastic strains are assumed negligible in comparison with plastic strains. Referring to Figure 1, consider a flat element of virgin material of length h, width L and thickness t, bent by a pure uniformly distributed moment into a cylindrical shape of mean radius R (inside radius a). Because of symmetry, the coordinate axes indicate the principal directions of deformation. The element is assumed to be a contiguous part of a larger sheet that remains elastic such that its length h is unchanged in the defonnation. A layer of thickness de: is therefore strained uniaxially in the process in tension or in com· pression depending on its position relative to the neutral surface. The stress O'zz normal to the layer is small and is neglected in the analysis. Assuming orthotroptc material behavior, the principal stress directions will also coincide with the co-ordinate axes shown, such that oyz' axz and ~ vanish. xy The yield condition can therefore, be described by its intersection with the two-dimensional subspace (crx' ~y) of stress space (see Figure 2). In the usual fashion, strain vectors are associated with points of the yield surface; during the forming of the corner a layer of thickness dz follows the stress paths OA or 08 in Fig. 2, characterized as the locus of stress points with vanishing strain in the y-direction, The engineering strain and engineering stress in this plastic deformation in the layer are denoted by e and cr respectively, related by the strain hardening law a = cr(e) (1) For a corner in service, on the other hand, the stresses in the plane of the cross·section are small and are neglected. The con-dition is one of uniaxial stress in the y-direction, represented by COO in Fig. 2. The effect of the plastic deformation is to raise the yield stress in tension and compression in the y·direction. For example, a layer compressed to point A in Fig. 2 will have yield stresses indicated by points E and C in tension and compression, respectively. When the moments that produce the plastic defonnation are released, elastic unloading takes place under plane strain conditions. If point A (Fig. 2) represents the inside corner surface (.z :c t/2 in Fig. 1), it will unload to a point such as F. The slope of AF is equal to Poisson's ratio; the precise location of F can be determined -by elastic curved beam theory modified for plane strain. Curve FG indicates the locus of residual stress points for the section as a whole. When, as in service, the corner is subsequently loaded in the axial direction, it will undergo strains of only elastic magnitude until the average stress 11 ao high that the yield stress 1s developed over the entire eroas section. We aim to calcula~e the average axial stress, which will be called the yield strength of the corner. For e)J:aaple, if the corn•r 11 in tenaton. the layer ortsinally repreaented by B and G will enter a •tate of restrained plastic flow when point 1 is reached i yielding of the ~onar ao a whole will not o~cur until the utarial at point F rea~hos poiDt H. Evidently, the pattorn of yieldilll caD be quite complicated, dependins OD detailo of the reoidual otreu field and the location of intermediate yield ourfaceo. !loot of the ~omplexitioo of the analyaia arile becauae the atreaa-atrain f'elatlonahip in Bq. is not ayaaetrical with reapect to the origin, as evidenced by Fig. 2 and illustrated in Fla. 3. For a aiven magnitude of strain, the atress value in com.preaaion 11 usually higher than the atreaa value in teuion. The difference tends to increa1e with the strain. A oonaequence of this lack of symmetry is the gradual ohift of the neutral axio toward• the compression side; the material in the middle of the plate that ia swept by the neutral axis auffers a strela reversal. The probleQt is also coml'li· cated by deformation of the crosa•aect:l.on; the distance • of a particle from the mid-surface is not conserved, as a rule. Radial expansion ancl contraction cause the particles of the aid-surface to move outw._,rds relative to the surfaces of the sheet which remain approxlmately a distance t apart. As a consequence, the linear variation of engineerill& strain e = e(z) with radial distance in the corner reflects a nonlinear var.iation of e with distance from the mid-surface. This effect could be significant when the strains are high but it is, of necessity, left out of consideration. INFWENCE OF STRESS-STRAIN ASYMMETRY It will firot be shown that the difference between the teno ion and compression aegments of the stress-atrain curve, Eq. 1, can normally be neglected in practice. For this purpose, let a • f(e) and a • g(e) denote the even and odd components, respectively, of the stresa-atrain curve as shown in Fig. 3. Let the magnitude of compressive atrain on the inside surface of the corner be denoted by c•h and let F denote the area under the compression stress block OC in Fig. 3. Let c denote the strain for which the area under the mean curve (Ctt) equals F. By a series expansion of the integral of function f + g representing the compression curve: c-h J (f+g)de = F • F - h(f + g) + ! h2 (f • + g •) + ••• 0 (2) similarly. let c + h + d denote the strain for which the area under the tension curve f - 1 equals F; then c+h+d J (f·g)de • F • F - G + h(f - g) + t h2 (f' - g') + • , • 0 in which F and G are &Teas as shown in Fig. 3: c F • J f(e) de, 0 c G • J g(e) de. 0 (3) (4) Assume that G is amall in compariaon with F; the areas in Fig. 3 then show that h and d are both small in comparilon with c. Moreover. the quadratic terms in Eqs. 2 and 3 are aeen to represent areaa negli&ible in comparison with G, when the alopea f' and 1' both are 1111811 (in comparleon with f/c). Then, nealectina appropriate tarma, theae equations can be aolved for the strain differences h aU d to yield h :. --lL. f +I d ~o-jLz f -I '(S) llov, tho true curvature k caD be calculated frca the otreiD differential over the thiakDaoe t: k •· (c + h +d) - !-(c - h)) 2c + d t • --r- (6) Thil ia the curvature correapondiag to a true otreu block of orea r. Now, if the •an curve ll uaad inataad of the true atr•••·•train cunrea. * a otreu block of area F would be uoociated with the curva~ure value k 11 given by (7) Hence, with aome rewritina * d * " 2 k • k (1 + z;;-> • k ! 1 + ("-l:f--rl IC f - I (8) The laat term in th• bracket in this equation ahova the relative error * in aubatitutina k for k. The error depends on the ahape of the stress .. st~ain curve and the value of the strain. Normally, G does not •xceed tw•nty percent of ac and 1 doea not exceed twenty percent of f. so * that the error in k for a value of F ia leas than one percent. Inveraely, the error in F for an aaau.ed curvature will also be leas than one percent, approxiiDately, 10 that aayaDetry in the plane strain va. stress characteriatic, ax • f(ex), can be neglected in practice: (9) Moreover, it ia aaaumed that there exiata an effective streaa function and an effective strain function having the property that the plane yield atresa va. pre1train relations, f~ • f;(ex) and f; • f;(ex) in the y .. direction in tension and compression respectively, are affine to the stress vs. plane strain relation, crx • f(ex), in the x·direction. Thuo, (10) (11) where A and B are constants. Fin.ally, it is assumed that the slope of the yield loci near the interaection with the ay •axis in Fla. 2 can be taken as constant over the range of residual stress a: and of stress "y of interest. These assumptions are satiafied by von Mises yield condition and isotropic work hardenina. •• an example. In Fig. 4, let a • "x(z) • f(ex(z)), in4icated by Curve 1-1, be the variation of stress in the x·direction (referred to original area of elementa) through the thickness of the corner just when the last incre~ ment of plastic work in foraains la parfot11ed. This stress is syametrical with reapect to the mid-ourface, by Eq, 9. The corresponding yield atrength variation~ through the thickness are indicated by the curves f;(E) and f;(a). The presence of residual atresaea after unloading should not be nealected.. Curve 2·2 incllcatea the elaattc atreaaes a~. induced by unloadina the -entl in Fig. 1, plotted ouch that the cliatance o-: fr- 1-1 to 2-2 ia the reoidual otraoo fiald in the x•direction. The atreaaea CJ: are diatributed accord in& t• plane 1train curved be• theory, ..odifiad to refer to original arua of el-nta. Finally. curve 3 .. 3 indicatao the reatdual atreaa in the y•diractton.c-oed of a plastic part (oea path AI ill ria. 2), and an elutic part ouociated with the unl.oadina and related to ": by tho Poiaaon affect. Eviclantly, J .,; (a) do • 0 (12) r cr: (a) •• • 0 (13) where the tntearation extend• fr011 aurface to aurfaca. If there vel'e no raaidual etr••• in the x-dtrection, the oial forco por unit ore lonath of comer to produce yioldinl in tonoion would bo oqual to tho oroo botvoon 3•3 ond 4·4. Fia. 2 ahCNo how tho ylold atr111 11 influenced by tran•v•r•• reatdual atraea (c.,ara D aDd FB). Tho dovioUon of tho yiold loci fi'OOI thoir tonaonto ot tho pointe of tntareection vith the ax ·u.il (euch u I, D. and C) for ablchaaa tn the rana• of reatdual ttr••• a: il,by aaaumption, nesttatbla. Abo, totrlanal•" 0550 1n Fla. 4 11 naalectad. Then, 'the corner yield atrenath in tenalon, by Eqa. 10, 12 .nd ll can be calculatect 11: 1 t/2 t t 0 0 - J If (e (z)) + (fy)' <1 • <1yldz t y X X •t/2 t/2 2 t/2R 2R • ! J f(o (z))dz • _A! J f(o )do • -, AF(t/2R) (14) t X t Q X X •t /2 in which (f~)' denotea ·the 1lope of the yield loci at the points on the ay ·uh in Fla. 2, aaa\DM:d to be a constant. yield atrenath in teneion il found u: Slmllorly, tho eornor f 0 • BF(t/2R)/(t/2R) (15) yc Thua. the corner yield atrensth ia eatily determined, to within a constant factor,from bend radiua R • t/2, material thickneas t, and area function F(e) under the etreea•atrain curve (averaae of tension and compreaaion). ln application, the dhtinction between tension and compression would nonully be tanored, and only tension data uaed. The compreaaion yield strength would then be taken equal to the tension value. Eq. 14 confiru intuition that the corner yield atrensth should be independent of the corner angle. LINEAII IIARD!NING MODEL The nine uterhh atudied by Karren { 11 ahowed considerable variation in aeneral in mechanical propertiea, but the ultimate atratn •u wa• eaeenttally conetant, equal to 1/3 (variation between 0.31 and 0.40 for a 2 inch 1111 lenath). Thia auaeete that the aateriala can be modelled by an ideal uterial hardenina in teneion fr011 f~ to f~ over a strain that b the lame fraction, here denoted by 3A/4a, of e '"' 1/3 for all the •teriala exuined. Chooein.s the rigid•plaatie u linearly hardening material e(ft • ft) " • fy + (1/~)(3A/4;) aivel the area function eo that, by Eqo. 14 ond 15 t" ye (16) (17) (18) (19) l{hen no cold work 11 done, that ia for 1/R • 0, the yield etreqth Al and Bft are, relpectively, equal to the virgin material yield y y etrenath in tMdion, and 19 sivo ll/A • ft and ccapreseion y t 0 It' and beco•: y y t 0 reapoetively. y Eqa. 18 (20) (21) Here, a il the only unknown quantity. Solving theee equation• for cr and iuerting the experimental values by Karren [ 1] shows that a rana•• from 2.10 to 5.03 in tension and from 2.02 to 7.51 in compression for all the tests by Karren (totalling about 200 cot"ners), averaging 3.34 in tension and 4.29 in compression. This linear model permits a very simple representation for deaigtl purpoeee. In tenaion, the increment in yield force for a corner 1a, by Eq. 20, ueing the average value for a, ~P • (9Rt) (f;0 (22) where, in the last reduction the superscript has been suppreased. We define a fictitious compressive ultimate strength as (23) Then, Eq. (22) also will apply to the increment in yield force in compression, by Eq. 21; with a equal to 4.29, and 7t instead of 5t in the last equality. In .Practice, compression data is rarely available, and it is reasonable to neglect the apparently higher hardening rate in compress ion. Using the value St together with tension data also in compression, Eq. (2n may be interpreted into the following simple design rule: Cold work of a 90° corner may be taken into account by replacing the yield strength by the ultimate strength over an arc lengtha,s, of the corner of five times the thickness. Por other corner angle• this arc length ahall be changed in proportion to the angle. Evidently, a difference in the value of a may be interpreted directly in tel'IDI of arc lenatha. Por example, if the ultimate strenath ia 402. higher than the yield atrensth, a SO'l error in a is equivalent to an error of t x t in affective area of flan1e or crosa-aection. The acatter in a 11 therefore not 10 significant as it might appear at 12 I I I I I I I I y )-x z I : . }----------JL-·-·-·-· L h t Fig, 1 first sight. There is some systematic variation, with a tend•ncy to overestimating the strength of sharp cornera. The variation could be t::llminated by using a parabolic strain hardening (for example) and lettina depend on material properties, etc. Fig. 2 CT MEAN TENSION ENGINEERING STRAIN Fig. 3 /- CT • t; ( z l = A f ( I ex I l CT COIICWSIONS On the baala of a aimple ID&terial .odel and experilllental data by Karren [ 11. it 11 concluded that (1) The raiae in the yield atrensth of a corn1r dependa aaainly on the corner curvature ratio a/t and the area function F(e) of the mean of the tension and com.preaaion atreaa•atrain curvea for the 1q.t;erial; it is independent of the corner angle. (2) A llnoar otrain-hardenll\11 1110del, pootulatlng that the material is rigid-plastic and hardening from yield strensth to ultimate strength over a strain of 33 l/3X, aceou.nts for all teat results avail-able with good accuracy. By this model, the raise in yield strength depends only on a/t and the difference, fu ... fy' of the ultimate strength and the yield strength. (3) The linear strain hardening model is convenient for design purposes. To take the cold work strengthening into account tt is merely necessary to replace the yield strength by the ultimate strength at each 90° corner over an arc length of five times the material thickness. For other corner angles the arc length should be modified in proportion to the angle of bending. ACKNCAJLEDGHENTS Thank• are due to H. Suminski and H. Sachdeva who assisted with the data a.nalysis and progranning, and to the National Research Council of Canada for financial support. 1. z. REFERENCES KarrEm, K.W., "Corner Properties of Cold Formed Steel Shapes", Proc. ASCE, V. 92, No. STl, Feb. 1967, pp. 401-432. Specification for the Design of Cold-Formed Steel Structural Members, Ametican Iron and Steel Institute, New York, N. Y, 1968. 

Utilization of Cold Work in Light Gage Steel

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Utilization of Cold Work in Light Gage Steel

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