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©(Q)[lJLJJ~~ ~m~~@lF[}=[] (Q)~

©(Q)~~m(LJJ©lY](Q)~~[L ~~W

by L. S. BEEDLET. V. GALAMBOSL. TALL

FRIlZ ENGINEERINGLABORATORY LIBRARY

FRITZ LABORATORY REPRINT NO. 177"Reprinted, courtesy of U. S. Steel Corp."

34

Dr. Lynn S. Beedle is presently Direc­tor of the Fritz Engineering Labora­tory, Lehigh University, Bethlehem,Pennsylvania. He received a B.S. de­gree in Civil Engineering at the Uni·versity of California at Berkeley in1941. He served in the United StatesNavy from 1941 to 1947 where hetaught in the Postgraduate School atthe United States Naval Academy andserved as Officer-in·Charge of Under'water Explosion Research at the Nor·folk Naval Shipyard. In July 1947 hejoined Lehigh University as a ResearchEngineer in the Fritz Engineering Lab·oratory. While at Lehigh, he receivedan M.S. degree in 1949 and a Ph.D.degree in 1952.Dr. Theodore V. Galambos received hisPh.D. degree at Lehigh in 1959, andis currently Research Assistant Profes·sor in Civil Engineering at Fritz Lab­oratory. He is currently engaged in re­search in the Structural Metals Divi­sion and is Project Director of UnitedStates Steel's sponsored project on"T·1" Steel columns.Lambert Tall is a Research Associatein the Fritz Laboratory, and has beenassociated with the research projecton residual stress and column insta·bility for five years. He received hisPh.D. degree in 1961.

ABSTRACT. It is shown in this paper that con­sistent explanation of column behavior can bemade for steels of different yield-stress levels andsubjected to different manufacturing and fabrica­tion processes. This is done by taking into ac­count the influence of fabrication proceduressuch as rolling, welding, and heat treatment. Thissummary is restricted to the behavior of centrallyloaded columns, but covers ASTM A7, ASTMA242, and USS "T-1" Steel. Wide-flange shapes,welded H- and box-shapes, and round bars werestudied in the research. Appropriate use of thetangent-modulus concept and a consideration ofinitial out-of-straightness makes it possible toarrive at a satisfactory explanation of column loadcarrying capacity.

INTRODUCTION. Recent technological de­velopments are now providing the structural en­gineer with a selection of steels having a widevariety of strength properties. This enables theengineer to provide more useful structures withgreater economy and beauty. His creativity ischallenged by his ability to choose not only mem­bers of varying sizes and shapes, but also mem­bers made up of different steels.

In the case of columns, the ready availabilityof steels of various yield stresses fortunately coin­cides with the development of rational techniquesfor predicting the strength of columns. These pro­cedures take into account numerous factors notrecognized or carefully studied heretofore. Al­though developed essentially for ASTM A7 steelcolumns, these procedures were necessary beforethe behavior of higher strength steel columnscould be understood.

It is the purpose of this paper to show howvariables in manufacture and heat-treatmentaffect the strength of columns. It will be shownthat a consistent explanation of column behaviorcan be made for steels that have different yieldstresses and that are subjected to different fabrica­tion processes.

One of the most important factors that in­fluence the strength of columns is the loading con­dition. Figure 1 shows a number of possibilities:central load and load plus moments of differentintensity at the ends. At one extreme is the cen­trally loaded column. At the other extreme is thecolumn bent by external moments in single curva­ture. The latter will support less axial load thanany other condition.

Although the scope of this paper is limited tocentrally loaded columns, the interaction curvesshown in Figure 1 indicate how significant theloading condition can be. The figure shows theaxial load-earrying capacity as influenced bybending moment, the information being presentedon a nondimensional basis for a typical A7 steelwide-flange column. When the moment is zero,the column will support its maximum axial thrust.On the other hand, if the moment were to reachthe full plastic value Mp, then the memberwould theoretically support no axial load what­ever. For intermediate cases, the axial load capac­ity is dependent upon the way the moments areintroduced. The curves (which have been con­firmed experimentally) show that the differenceis significant and pronounced. In fact, for thedouble curvature case, there is a certain rangeof moment for which the axial load capacity isdecreased only by a small amount due to thepresence of such a moment.

One might say that all columns have eitherreal or accidental end moments. This is true, butit is just as evident that it would unduly penalizea moment-earrying column to attempt to takeinto account accidental eccentricities in the limit­ing no-moment case. So there is real justificationin learning as much as possible about centrallyloaded columns. This summary of the state ofknowledge concerning such members is basedprimarily upon References 1, 2, and 3.

-I--ROIl ED 8_WE-3.l...

Figure I-Influence of loading condition on an H-shapedcolumn

(1)7T2EI

L 2

Even when the data is nondimensionalized totake into account the yield stress value, Figure 3,there is still more variation in column strengththan would appear to be acceptable.

The importance of the variation of the materialproperties as a factor influencing column strengthwas recognized by the Column Research Coun­cil when it was formed, and it assigned to its firstcommittee the task of determining the relation­ship between material properties and the strengthof columns. The first pronouncement of the Coun­cil was its Technical Memorandum No.1, "TheBasic Column Formula.,,4 This memorandumstates that the critical or ultimate failure load ofa straight, centrally loaded column is given bythe tangent-modulus concept. This concept re­quires a knowledge of the stress-strain relation­ship, namely, the "material properties."

Consider first an ideal coupon free from resid­ual stress as shown by the dotted line in Figure4a. Since E is constant up to the yield stress level,the Euler formula would apply up to that point.Application of the tangent - modulus conceptwould result in a horizontal line at the yield stresslevel because the tangent modulus E t is zero.

Early work has shown that rolled or fabricatedshapes do not behave according to predictionsbased on stress-strain curves from coupons.Among other things, such shapes contain residualstresses; also, the yield level may vary across thesection. As a result, the stress-strain curve ceasesto be linear above a certain point.

To construct a column curve for such a ma­terial, first the tangent modulus E t would be de­termined for various stress levels by using thestress-strain diagram in Figure 4a. Then a stress­vs-Et curve would be drawn as shown in Sketch b.E t = E until the proportional limit is reached,after which it decreases to zero at fYY ' Applyingthe tangent-modulus concept, the resulting "col­umn curve" of stress-vs-slendemess ratio wouldbe obtained as shown in Sketch c.

When failure occurs in the inelastic range, theinfluence of a nonlinear stress-strain relationshipis seen by comparing the solid line with thedashed line obtained for a member without initialstresses. In the elastic region residual stresseshave no influence.

Centrally Loaded Columns. The factors thatinfluence the strength of centrally loaded columnsare numerous and are pronounced as the scatterof points in Figure 2 suggests. The points plottedin Figure 2 represent tests of centrally loadedstructural columns. The average stress is plottedas a function of L/r ; the curve is the Euler hyper­bola for elastic column buckling and is a plot ofthe equation

Lll.=- =~

n •

a:A7 WELDED BOX

~........-==Rrv=

6=:..BE!ri·• ~-;'lI-nuNn~BAIi

+x:z.; .1l~llloltLBAR·

~

_:AI WELDED BOX

+=. RuONI! BAR

OXI. :w:r:. WElilC "A·XI5.

Cll.1!CL-:wr. ::rraOI'tG. AXIS

• x:r"Wl'". .DlN."EALEtl

AA-.2..4..2..~.::WE"AK AXJsOx::7. ""WFJ'cEc'"H

~lIC7 RlyE lEO H

41' (fjIUHFNI

E t

.. A

:'10

•

:.2.0

=.

oc:r.=. WEAKAXlS.@K:7. 'WF. ..5l1l0NG AXIS..

-=.;7ilE.::ANN EALED.

~ :wE. :.w:EAKAXlS.

[/=.=.H

oo

c

--- - - ....-- ---.--- - ---o .Ii·· ~

• • AI. ~ b

"ClD 0 A0. 0 %,_'.-.

D'

-:0 l..-_...L.._.....L__..L--_-'-_--L.

=: :.-0 ..,

=

I.o

t1 y

'::o'~_-",__""" ..,l,__"""__'--~....I.

r--- ~-:QE.:B.ES1D1JAl...SIREss.

--- MFMBFJrr:oNTA.lIlllNG::RESTDUAI:" STRESS.

Figure 4-lnfluence of residual stress and strain-hardeningupon the stress-strain, stress-modulus, and stress-slenderness

ratio diagrams

::rFigure 2-Results of column tests of A 7, A242. and "T·l" steel

Figure 3-Non-dimensional representation of results of columntests of A 7, A242 and "T-l" steel

=

pliO

AksL

:60

.ZO·

t1 Ir.&

t1y

.!l.A

I

36

-.

Another factor that theory has neglected untilrecently is strain hardening, an influence that isimportant for short columns. As shown in thestress-strain diagram of Figure 4a, structural car­bon steel strain-hardens after the plastic strain hasreached about 10 times the elastic limit value.This results in a D' versus E l relationship thatdiffers from the idealized case. Instead of a mod­ulus of zero at the yield stress, E l :;;:; 750 kilo­pounds per square inch (ksi), or about 1/40 thevalue of E. As a consequence, a column willcarry a greater average critical stress than the fullyield value when the slenderness ratio is less thanabout 15.

Thus, the strength of straight, centrally loadedcolumns is dependent on the stress-strain relation­ship of the complete cross section as a unit. Thelatter, in turn, is dependent on three importantfactors. These are

(a) The magnitude and distribution of residualstresses (which cause a lowering of the pro­portional limit and affect the shape of thecurve above crp) .

(b) The basic yield stress level of the material(which affects the practical upper limit ofcolumn strength).

(c) The strain-hardening modulus (which issignificant for very short columns) .

Columns With Initial Out-Oj-Straightness. Afinal group of factors, interrelated with the aboveto a certain extent, can be classed under theheading "Initial Out-of-Straightness." Unsym­metrical residual stresses have a similar influenceon the behavior of the member.

Straight, centrally loaded columns with sym­metrical residual stresses will start to bend at thetangent-modulus load as shown in the uppercurve of Figure 5 in which load is plotted againstlateral deflection. Straight columns with unsym­metrical residual stresses (such as those producedby cold bending) will remain straight until yield­ing commences. The bending begins at a load thatmay be lower or higher than the tangent-modulusload, depending on the magnitude of residualstress. As shown by the lower curve in Figure 5,columns with initial out-of-straightness will startto deflect laterally at the start of load, with a con­sequent lowering of column strength.

By taking appropriate theoretical account ofthe influence of yield stress, residual stress (bothsymmetrical and unsymmetrical), strain-harden­ing, heat-treatment process, cross-section shape,and initial out-of-straightness, it should be pos­sible to predict each of the test points shown inFigure 2 with sufficient accuracy.

1... _

t

,----------------iI

b+

-cENTER DE FLEe I IuN

Figure 5-Behavior at columns with cooling residual stresses,"cold-bending" stresses, and Initial out-at-straightness

Influence of Residual Stresses Due to CoolingAfter Rolling Structural Carbon Steel. Residualstresses are formed in a structural member as aresult of plastic deformations. In rolled structuralshapes, these deformations always occur duringthe process of cooling from the rolling tempera­ture to ambient air temperature; the plastic de­formations result from the fact that some partsof the shape cool much more rapidly than othersand cause inelastic deformations in the slowercooling portions. The situation is similar forwelded members and for heat-treated bars thatare other than furnace-cooled. Members that arecold-straightened contain residual stresses as aresult of the plastic bending deformations thatare applied. In general, it might be stated thatthere are no residual stresses if there is no plasticdeformation during the life of the material.Further, for elements under temperature gradient,the part to cool last will usually be in a state oftensile residual stress.

In a rolled wide flange (WF) shape, for ex­ample, as cooling continues from the rolling tem­perature, the tips of the flanges cool faster thanthe web and balance of the flanges. They becomehard and resist the contraction of the hotter ma­terial where the flange joins the web and causesplastic deformations there. As a result, it wouldbe expected that at ambient temperature theflange edges would be in a state of residual com­pression and the flange center in residual tension.

Of the many sets of residual measurements thathave been made, Figure 6 shows the results forthree WF structural carbon shapes of widely dif­fering size and geometry. Although the variationis considerable, the general pattern in the flangeis as expected. Insofar as columns are concerned,it will be seen later that the most important of thestresses are those at the flange tips. For A7 steel,the average stress measured there is about 13,000psi in compression.

To examine the influence of these stresses, con-

H

::T1f.W.r43 II,

'VER.i,E SIR_IN

38

-=)~_'AVERAGE"

.::wSTRIBUTION

+:4.7~~~'..z

Figure 6--Residual stress distribution In WF shapes CA-7 steel)

1b1

~rc+

Figure 7-lnfluence of residual stress on stress-strain curve

(2)

Figure 8--Testing a "stub column" on 5-million-pound maochine at Fritz Engineering Laboratory, Lehigh University.

Figure 9-"Stub column" stress·strain curve for as-deliveredmaterial in comparison with coupon test result

r1J-'-(

2.0

e1.0

/'",

II '----::::zs.-..:---o­I

II

II

I

sider Figure 7. The cutting of a coupon from theflange of a member relieves the residual stressespresent in the member prior to sectioning. Thusthe stress-strain diagram would be as shown bythe dashed line in Sketch b.

If, now, the load is considered as being appliedto the entire cross section containing its residualstresses, it is evident that the behavior will belinear until the applied stress becomes equal tothe difference between 0-y and O"rc. Then yield­ing will commence at the flange tips. A linear dis­tribution of residual stress has been indicated. Asshown in Sketch b, the stress-strain curve will re­main linear as long as the applied stress is lessthan tr p.

When more load is applied, the average stressand average strain are no longer proportional toone another because of yielding of the flange tips.Thus a nonlinear stress-strain relationship resultsfor the section as a whole. The circle in Sketch bcorresponds to distribution, d, the flange edgeshaving yielded.

After yielding has penetrated across the entiresection, the stress distribution is identical to thatof a shape containing no residual stresses. In ef­fect, they are "wiped out" and have no influenceon the yield stress level.

The curved portion of the stress-strain diagramin Sketch b, then, reflects the influence of residualstress. It causes a marked reduction in the pro­portional limit (to about 20 ksi for A7 steel)and a consequent reduction in the tangent-modu­lus values when this stress is exceeded.

The way to confirm this experimentally is totest an entire cross section, or "stub column."Such a test is shown in Figure 8. The memberis short enough to prevent column instability butlong enough to retain the residual stresses. The14WF426 shape is being compressed in a 5­million-pound capacity hydraulic testing ma­chine. The load at a yield stress level of 33 ksiwould be about 4 million pounds. Such a test notonly reflects the influence of residual stress, butalso automatically averages out the difference inyield stress for different parts of the cross section.

A typical stress-strain curve for such a stub­column test is shown in Figure 9. The behavior ofan isolated coupon is shown by the dashed line.The experimental points connected by the curveare from the test of the whole section or "stubcolumn." Very good correlation is obtainedwith the earlier predictions (Figure 7) for this18WFI05 member.

It is possible to examine the strength of col­umns containing "cooling" residual stressesthrough use of the tangent-modulus concept. Thetangent-modulus formula is:

7T2E t IL 2

39

I ra;- LX= 7f\lT r

(3)

and, as noted above,function of E t •

There is a difference in strength dependingon the flexure axis. When flexure occurs aboutthe weak (y-y) axis, the material most remotefrom the neutral axis does not contribute to themoment of inertia since it has yielded. The reduc­tion is not so drastic for a column bent about thestrong axis, and this tendency is shown in Figure10. The lower curves are for flexure about theweak axis of an H-section, and the upper curvesare for flexure about the strong axis.

The dashed lines in each case represent so­called "exact" solutions, solutions derived fromthe actual residual stress distributions. It will benoted that the curve for buckling about the strongaxis is approximately parabolic in shape; forbuckling in the weak direction, the curve may beapproximated by a straight line.

Figure 11 shows that the results of weak-axiscolumn tests are in good agreement with thestraight-line approximation previously men­tioned. The circles show the maximum load thecolumns carried. The coordinates are nondimen­sionalized in order that variation in E and cr y

could be eliminated. Also shown are the resultsof tests of stress-relief annealed columns (soliddots). Clearly their strength is well above that ofthe as-delivered members. Two tests at small L/r

values confirm the influence of strain hardening.Figure 12 similarly shows the results of columntests in which flexure was about the strong axisin comparison with the parabolic approximationof Figure 10. Again there is good correlation be­tween theory and tests.

Higher Strength Steels. The next question is,how do the material properties of higher strengthsteels compare with A7, and do they influencecolumn performance in a similar way?

Figure 13a shows comparative curves of con­structional alloy (USS "T-1"), high-strength lowalloy (ASlM A242), and structural carbon

in which Et is determined directly from thestress-strain curve. Equation 2 is "exact" only forthe special case of a rectangle bent about theweak axis. Without going into detail, the real keyto the solution lies in a consideration of the mo­ment of inertia of the yielded cross section. It hasbeen shown that this moment of inertia may beexpressed in terms of E t •

Consider the partially yielded cross sectionshown in Figure 10. E t = E for the elastic portion,but E t = 0 for the yielded tips. In effect, it is anew cross section of which the reduced momentof inertia, Ie' is that of the portion which remainselastic. Thus, the buckling equation is:

1T2EIe

L 2

Ie may be expressed as a

+4-I

=

H-- - -.- ---""\

\\

\\

\\

\

• 15TRE'Ss:BE I lEE ANN E ATTO=r:nTTIJINS

o AS·.DELI.'lERED=COL.UMNS:.

=~_!.-_!.-_.L-_"""'_..L-_""""_""""_--'

D:-

_ AEERUXlMAIE AOLllllDH..

..ro- - .E:X:XCT. SO.LOTIDN

::ro_

co l--_-L __----1. --'

:0:: =- :Ir.-a= = = _=0:-

X=JjuyJ..

?IT :F :r

Figure ll-Results of column tests of A 7 steel and straight·line column formula (weak axis)

DI--_-'-_---''--_-'-_---''--_~

:II" :n:L = = IlL =Figure 12-Results of column tests of A 7 steel and parabolic

column curve (strong axis)

=:I:-I

Figure 10-Column curves for WF shape with cooling residualstress

=

=

::LO

=

D:e

U ,D..:6

(I y 0 e:t1LllMJ5{:::r:ES1S..

40

Lr

=

.lCZ: .-__-~'---u"-----__,

:::D.:.-- --L-. L=

.tatFigure 13-Comparative stress-strain curves (a) and column

curves (b) for "T·1", A242, and A 7 steels

(ASTM A7) steels. They represent the auto­graphic record of typical tension coupon tests con­ducted at Fritz Laboratory. The similarities in thethree steels are as follows:

(a) The abrupt change at yield.(b) The relatively long plateau of strain dur­

ing which the stress remains constantwith increasing strain.

(c) Sensitivity to dynamic effects. The dipsrepresent points where the testing ma­chine was stopped; if the machine hadbeen operated at a faster strain rate, thecorresponding stress would have beenhigher.

Other than strength, differences in the threesteels are as follows:

(a) The absence of an upper yield point in"T-I" Steel. (In the final analysis, thisis of no consequence because the residualstress effect "wipes out" any upper yieldpoint that exists in the other steels.)

(b) The yield plateau in A242 and in A7 steelis usually horizontal up to strain harden­ing. The "T-I" plateau has a slight slope.

(c) Both A7 and A242 steel strain-hardenafter considerable strain and have (fromlimited tests on A242 steel) similar valuesof the strain-hardening modulus. (Et ~

750 ksi). "T-I" Steel on the other handhas a continuously rising curve with E l

~ 100 ksi. .Omitting for the moment the influence of resid­

ual stresses, the column curves for the threesteels are shown in Figure 13b. It reflects thesimilarities and the differences. The columncurves are constructed on the basis of the staticyield stress mentioned above: the sudden transi­tion from elastic to plastic region, the strain­hardening of A7 and A242 steel, and the gradualrise of "T-I" Steel column strength as L/r ap­proaches zero.

Of next interest are the magnitude and distri­bution of residual stresses due to air-cooling ofhigher strength steels. Measureme.pts have beenmade not only on A7 but also on A242 steels.

. They will shortly be made on "T-I" Steel, but inadvance of the measurements, calculations havebeen made on what the expected distribution andmagnitude might be. The results are shown inFigure 14 for 3 wide-flange shapes of differentgeometry and size (8WF31, 12WF50, and12WF65) . It shows that the residual stress magni­tude and distribution is about the same in the twomeasured steels. The theoretical examination for"T-i" Steel also suggests that the magnitude anddistribution of residual stress is not influenced

41

by the yield stress level. 0f JIiuchgreater impor­. tailceis the geometry of the cross section.

TIle yield stress level 0- y is also shown inFigure 14. As shown, (Tp = (ry - (T rc' Since

'. o-re d~snotvary, a comparison of the three. steels indicates thai the proportional liIriit ( (J'p= O"y -O'ro) mcre'ases as try increases.Thus, the influence of residual stress should 110tbe as pronounced ill the higher strength steels.

The corresponciing average stress-strain rela-

tionship! is shown in figllfe 15aon a nondimen­sional b?sis. The proportiQnaJUmit would be thelowest- for the A7steel. The "T-l" Steel shouldhave th~highest p~oporti~nallimit. the Jc6Iuriuicurve iru Figure 15h shows diagplriunatiCally .thatthe high~rthe yield istreSS level, the s~ronger Will.be thecolurnn (relatively speaking), Thus, wecan exp¢ct that in the higher .strength steels theinfluence of residJ.lal stress will be less Pl"O­nounce4·

;.

R))~o

....\

R

-" t-~,=~~~.;", .,.L. . __

Figure 14-Residual stresses ill rolled, shapes of differinggeometry and yield strength

:11:::.242

".

"

a: "

-----------......------.,--.,

.' 1.0L,,= 'lTV E. JL-.---"____...............__-------- ,

~;

Figure l~olumncurvesof·differentsteels as affected byresidual stress

..::a11--__..L-__....L.__-L.__..-L__--.JL.-

Figure 16 shows that these predictions are con­firmed by actual weak-axis column tests on A242steel, with the same shapes for which residualstresses shown in Figure 14 were measured. Theaverage curve for A7 steel affords a basis forcomparison. The per cent reduction in columnstrength due to this factor is less for the strongermaterial. -

Influence 0/ Weld;;zg Residual Stresses. Someof the data in Figure 2 were the results of tests onwelded shapes. Two questions arise with regardto such members: How do welding residualstresses compare with those set up because ofcooling after rolling? And how are these stressesinfluenced by the geometry of the shape?

Figure 17 shows a comparison of residualstresses in WF shapes, in universal mill plates,and in welded and riveted sections. The as-rolledplates contain significant residual stresses due tocooling after rolling. The welding process intro­duces high tensile residual stresses at the flange­web juncture (they frequently approach the yieldstress of the weld metal), and this gives rise tocompressive stresses that are higher than thoseencountered for the rolled shapes-at least forthese tests. In these tests, plates 9 inches by %inch and 9 inches by lh inch in size were joinedwith %6-inch fillet welds. The riveted shape hasonly those stresses that were present in the anglesand plates prior to fabrication, and these stressesare rather low.

Tests confinn the results that would be pre­dicted on the basis of measured residual stresses.Figure 18 indicates that the riveted columns withlow residual stresses exhibited relatively highercolumn strength. The welded columns with highercompressive residual stresses at flange tips gavelower strength. In fact, there is a greater reductionthan has been observed in the case of rolledshapes.

The magnitude and distribution of welding re­sidual stresses are markedly influenced by thegeometry. Further work was, therefore,requiredon members with cross-sectional shapes otherthan the H-section. It was expected, for example,that the use of welded H-shape columns wouldbe replaced more frequently by "box" sectionsand that these would undoubtedly show a higherstrength.

t:

--1

6 ksift-. AI

'+'"----=

:lr

-It-11

:1:2.

A - J._j (ly ~- 'IT ::E: -r::

::n::xJIlGE""R--l~"~..~ ~}+ =, -~

-:II"

=-

R~rVl

H :}~

~ ...J....__-L__--.J'---__.L-__..J.--C'

D- -.o:~

O;CDCLIMN IESIS CAZ4%)

\ ------ --""\ \

\ \\', \

" ,"... ,

= -

n::s

Figure 17-Residual stress patterns compared in rolled mem­bers. in plates. and in welded and rlveted members

(I

(ly

1:1:===-==I [U';" ~-L'IT~E .::T__

Figure 16-Column curves and test results for, A242 steelcompared with A 7' steel

Figure 18-Test results and column curves for welded andriveted members (weak axis)

43

typlC:AL -::T'XB"rl- - -

Figure 19--Resldual stresses In welded box shape comparedwith those In welded H-shape

:3Q

fI

bi:

•

~o"

fly =---::J:J:Iii

,\\\

\

wflnEIiH

fly =A3JW. (tEStS IN

~

______....' .__-4__.

::mJ

:::L

.(hI

44

Figure 2D--Results of tests on welded box and H-columns

This work is currently underway, but enoughhas been learned to show that the estimate wascorrect-in part, at least. Figure 19 shows theresidual stresses caused by welding of a boxshape. The stresses at the weld are above the yieldpoint of the base metal (because of the greaterstrength of the weld metal). To balance this, thecompressiv~ stresses are also fairly high (about30 ksi). For comparison, the residual stressesin a welded H-shape are also shown. The impor­tant point to note is that first yielding under com­pression load.does not take place at the comers(corresponding to the tips in an H-shape). Thus,in spite of the high compressive residual stresses,the elastic moment of inertia is more favorablefor column performance than in the case of awelded H-shape bent about the weak axis.

In Figure 20a are shown the results of a fewtests of welded box columns. The curve for rolledH-shapes is the upper one (average weak axis

-tests). The theoretical ultimate strength and ob­served test results for the welded H-shapes arethe lowest groups. The other points are for thebox shapes. As expected, they are stronger, infact, approaching the strength of the rolled mem­bers. The longest test column has the greatest ini­tial curvature. The short line near the test pointis the predicted value when this measured out-of­straightness is considered.

At a later date, box columns of higher strengthsteel will be tested in the Lehigh program. In themeanwhile, the only known tests on high-strengthwelded columns are from Japan. These arewelded H-shapes with results shown in Figure20b. As before, tests of stress-relief annealedmembers practically reached the· yield value.Also, as in the case of rolled members, the weld­ing residual stress effect appears to be less pro­nounced for the higher strength steel.

ROUND COLUMNS

"Cooling" Residual Stresses. The followinggroup of columns to be discussed are round col­umns 9f "T-l" Steel, a quenched and temperedconstructional alloy steel. These are membersthat are frequently used for the main legs of tele­vision towers. An example is shown in Figure 21.Round columns were studied in a program whichis being sponsored by U. S. Steel Corporation todetermine the effect of heat-treatment and coldbending on the residual stress and thus on thecolumn strength, This study.will also illustrate theeffects of initial out-of-straightness.

/I

/ ..

i ', I

../ !/:' ,/

.1 /

il /r t

j. /,I

/ //f

/ I

It

Figure 21-Round columns of USS "T-1", a quenched andtempered constructional alloy steel, are frequently used for

the main members of television towers.

45

·The yield stress at a zero strain rate.

The first illustration in this group, Figure 22,shows the types of residual stresses present inround bars due to heat-treatment. The stressesshown have been measured in a 2% -inch-diame­ter bar, which had been cooled from the temperingtemperature by water quenching. Because of this,the residual stresses are rather high (more thanhalf the yield stress in this case) .

Because the bar is solid, the residual stressdistribution can no longer be assumed to be uni­axial, as for thin-walled members (such as theWF sections), but it is triaxial. There are longi­tudinal, tangential, and radial stresses ( 0' z,0' e, and 0- r in the figure). The radial stresses

are small. They are the stresses that are necessaryto hold the tangential stresses in equilibrium. The"equivalent stress" shown in Figure 22 was usedin the analysis rather than the separate effects ofthe three different residual stresses.

Another variable in these steels is the methodof heat-treatment. Figure 23 shows what mightbe expected for different final treatment condi­tions. It compares bars that have been (a)quenched, tempered, and stress-relieved; (b)quenched, tempered, and air-cooled from thetempering temperature; or (c) quenched, tem­pered, and quenched from the tempering tem­perature. Of these, the quench, temper, and air­cool treatment is the normal production treat­ment. Figure 23 shows that the maximum com­pressive stress at the edges is from 10 to 20 ksiwhen the final treatment is stress-relieving or air­cooling. When the final treatment is quenching,the stresses are very high (up to 80 ksi for thesetests) .

The residual stresses seen in previous figuresinfluence the behavior of a stub column in muchthe same way as observed for stub columns ofwide-flange shapes. Figure 24 shows the resultsof such a test. The specimen was a 2% -inchround bar. (Similar information has been ob­tained for 71h-inch round bars.) The final treat­ment operation was quenching from the temper­ing temperature and should reflect the largest in­fluence that might be observed. The calculatedcurve was based on residual stress measurementsand the measurement of the static yield stress·.The points represent observations from the stub­column test, and the agreement is excellent. Be­cause of the relatively small area along the outersurface that is under high stress, the deviation inthe stress-strain relationship is less pronouncedthan was observed in the case of structural carbonsteel.

Of note is the fact that the stub-column curvedoes not quite reach the yield stress of a tension

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Figure 22-Residual stresses In round "T·!" steel bars(quenched, tempered and quenched from tempering

temperature)

Figure 23-lnfluence of final heal-trealment process onresidual stresses

-s:rmrm (in:fin.1Figure 24-Stub-column test of round bar containing residual

stress

46

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Figure 25--Column curves and test results for "T·l" steel withdifferent treatments

Figure 27-Effect of out·of·straightness on column strength

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coupon in the range of strain shown in Figure 24.The reason for this is the triaxial stress effect.It is clear that the influence is not great, and even­tuallywhen the stub column is sufficiently strained,the strength reaches the coupon value.

Figure 25 shows the column curves which re­sult from application of tangent-modulus theoryto the "T-1" stub-column curves for differentheat-treatment. (It has been shown that the fail-

ure load is proportional to ( ~t) 2 for round bars.)

In the upper curve, the final treatment after tem­pering is a stress-relieving. In the next curve, thefinal treatment is air-cooling. In the lower curve,the final treatment is quenching. The test resultsshow the correlation obtained.

The agreement is not too good in some cases,and the reason for this is initial out-of-straight­ness (which will be discussed later) and lack ofsymmetry of residual stresses.

Initial Out-Oj-Straightness. The final group ofcentrally loaded columns for which test resultsare shown in Figure 2 were those with some ofthe effects discussed earlier. This includes the col­umns that initially were not perfectly straight andthose which contain unsymmetrical residual stresspatterns. Both of these effects introduce eccen­tricity into the problem. The effect was showndiagrammatically in Figure 5.

In Figure 26, actual test data are correlatedwith theoretical predictions. Referring to Sketcha, a straight column with residual stresses willstart to deflect at the tangent-modulus load. Therewill be a small increase (depending on residualstresses and shape of cross section), and then theload drops. A column with initial out-of-straight­ness, Sketch a, will start to deflect from the verybeginning. The result is a lower strength.

Similar behavior is observed for the test of a7th-inch bar containing both initial out-of­straightness and an unsymmetric residual stress,Sketch b.

Sketch c shows a similar effect for a weldedbox shape. In this case, the residual stresses werealmost symmetrical. The column was centeredunder load, and thus, the effect of measured ini­tial out-of-straightness did not become evidentuntil substantial load had been applied to themember.

The effect of out-of-straightness in memberswith low residual stresses (stress-relieved bars)is shown in Figure 27a. The theoretical ultimate

80strength curves for ';:- = 0, 0.02, and 0.05 as-sume no residual stress. Evidently, the presenceof a small amount of out-of-straightness resultsin a substantial reduction in column strength. Testresults for a number of "T-l" Steel and structural

47

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48

Figure 28--Residual stresses in cold-bent bars

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THEORETICAL CURVE.

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Figure 29-Stub-column test of round bar after cold-bending

carbon steel columns are shown along with a shortsegment of the theoretical curve for the apparentout-of-straightness. The agreement is very good.

Since the effect of out-of-straightness is pro­nounced (as is also the residual stress effect), inorder to get a close prediction of the strength ofmost test columns, it will be necessary to considerthe influence of both of these factors.

Figure 27b shows first the influence of sym­metrical residual stresses for "T-l" Steel bars·quenched from the tempering temperature. It isthe lowest tangent-modulus curve given in Figure

25. For initial deflections of ~ = .02 and .05,r

the strength is lowered as shown in the corre­sponding· curves. The test points, shown in thesame way as before, indicate the good agreementthat was obtained with predictions. As expected,in these "T-l"Steel columns in which the residualstress effect is less pronounced than for A7 steelcolumns, the effect of initial out-of-straightnessbecomes more significant in the "knee" of thecolumn curve.

The remaining effect to be considered is thatof cold-bending. In the left portion of Figure 28are seen the stresses that are set up as a result ofstraightening. On one side of the member thestresses are compressive and on the other sidethey are tensile. These stresses may be predictedby a plastic analysis, and the comparison of themeasured residual stresses with the predicted dis­tribution shows that the agreement is very good.

In the right portion of Figure 28 is seen theeffect of adding an axial thrust sufficient to causeyielding. The addition of a compressive stresscauses yielding to occur on the "compressive"side. The shaded portion shows the correspond­ing yielded zone in the round bar. As a result ofthis yielding on one side, bending commences andwe have an eccentric column.

Figure 29 is a load-versus-strain curve obtainedfrom the stub-column test of a member whichhad been cold-straightened. The initial residualstress distribution is shown followed by variousstress patterns that correspond to the attainmentof certain limiting stress conditions in the crosssection. The corresponding points in the load­strain curve are indicated. Gradually, the sectionyields until finally the full cross section is yielded.Here, again, there is unusually good agreementbetween the theoretical prediction and the testmeasurements as shown by the points. .

Column tests have also been conducted oncold-straightened members, and the results con­firm the theoretical predictions.

SUMMARY. In summarizing, research on cen­trally-loaded constructional steel columns hasshown that

1. Residual .stresses due to cooling after rolling,due to welding, due to cold bending, and due tovarious heat-treatment processes have a pro­nounced influence on the stress-strain curve, Fig­ure 9, and generally decrease the load-carryingcapacity of columns, Figures 11, 20, and 25.

2. Residual stresses in rolled shapes do not ap­pear to be influenced by the level of yield stress,Figure 14. Thus, the higher strength steel col­umns show less reduction in strength because ofresidual stress, Figures 15 and 16.

3. Residual stresses due to full strength welds areusually greater than those set up due to coolingafter rolling, Figure 19. As a result, the strengthof welded H-shapes is less than that of rolled WFshapes, Figure 20a. However, since geometryplays a significant role, tests show that box shapesgive higher strength. Further research on the in­fluence of geometry and of weld size will shedfurther light on the problem.

4. The same analytical techniques used for A7and A242 steels may be applied to solid roundcolumns of "T-1" Steel for the predicting of "stub­column" strength, Figures 24 and 29, and the tan­gent-modulus load, Figure 25.

5. Surface residual stresses due to quenching ofround bars from the tempering temperature arehigh and may be predicted rather well, Figure 22.When the final treatment is air cooling from thetempering temperature or stress relieving, thestresses are markedly reduced, Figure 23,. as istheir influence, Figure 25.

4S

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Figure 31-Correlation of available theory and tests of "T-1".A242, and A 7 columns

a nondimensional basis to eliminate yield stressas a variable. The CRC's basic column curve isalso shown. In Figure 31 the same data are plot­ted, making use of available theory to take intoaccount (1) residual stresses due to cooling afterrolling, welding, and heat treating, (2) cold­bending residual stresses, (3) strain-hardening,(4) initial out-of-straightness, and (5) differentyield stresses. The agreement between the experi­mental values and the predicted values is verygood.

ACKNOWLEDGMENT. The research at Le­high University on round bars is currently beingsponsored by the United States Steel Corporation.The work on rolled and welded shapes is beingdone under sponsorship of the Column ResearchCouncil with a major portion of the financing bythe Pennsylvania Department of Highways andthe United States Department of Commerce, Bu­reau of Public Roads.

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6. Especially in the case of the solid round barsbut also in the welded A7 box shapes, it has beennecessary to take into account initial out-of­straightness to satisfactorily explain the observedload-carrying capacity, Figures 26 and 27. Forthe "T-1" steel members, the influence of residualstress is less pronounced, whereas that of initialcrookedness is increased. In part, these variationsoffset one another, and in part, it might be thebasis for using a somewhat higher design stressfor medium length columns.

On the basis of this research, we can now ac­curately predict the strength of steel columns.Figure 30 shows all of the available test data on

References

1. L. S. Beedle and L. Tall. "Basic Column Strength,"Proceedings ASCE, 86, ST7, July 1960.

2. L. Tall, "The Strength of Welded Built-Up Col­umns," Ph.D. Dissertation, Lehigh University,1961.

3. A. Nitta, "Ultimate Strength of High Strength SteelCircular Columns," Ph.D. Dissertation, Lehigh Uni­versity, 1960.

4. Memorandum, "The Basic Column Formula," Col­umn Research Council, Tech. Memorandum No. I,May 1952.

50