L!189. ~LEHIGH UNJVERSITY PUBLICATIONS

Vol. XVI November, 1942,

THE INSTITUTE OF RESEARCH

No. 11

. ·Circular No. 179 Science and Technology, No. 152

STEEL COLUMNS OF ROLLED WIDE

FLANGE SECTION

PROGRESS REPORT No. 2

by

Bruce Johnston and Lloyd Cheney

Reprinted from a Report to the Committee on Technical Researchof the American Institute of Steel Construction

LEHIGH UNIVERSITY

BETHLEHEM, PENNSYLVANIA

JL-~ --'

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FRITZ ENmNEERfNG LABORATORYL~HIGH UN-lVERSITY

BETHLEHEM, PENNSYlVANIA

STEEL COLUMNS OF

ROLLED WIDE FLANGE SECTION

PROGRESS REPORT NO.2

BY BRUCE JOHNSTON AND LLOYD CHENEY

AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COLUMN RESEARCH AT LEHIGH UNIVERSITY

COMMITTEE ON TECHNICAL RESEARCH

AMERICAN I NSTITUTE OF STEEL CONSTRUCTION·

NO V.E M B E R I 942

CONTENTSPage

FOREWORD. 4

EULER FORMULA 5

"ECCENTRICITY" OR "SECANT" FORMULA. 7

FACTOR OF SAFETY 9

ALLOWANCE FOR ACCIDENTAL CROOKEDNESS AND END

ECCENTRICITY 10

REDUCED EQUIVALENT LENGTH ALLOWED BY FRAME ACTION. . 11

DESIGN FORMULAS FOR ECCENTRICALLY LOADED COLUMNS. 11

AXIALLY LOADED COLUMNS . i5

'l~EST RESULTS i5

DISCUSSION OF TEST RESULTS . 26

SUMMARY AND CONCLUSIONS .

[3 ]

34

FOREWORD

FOLLOWING Progress Report No.1, on the investigation of the localcompressive strength of wide flange columns, carried out at the Fritz

Engineering Laboratory, Lehigh University, the Committee on Tech­nical Research of the American Institute of Steel Construction presentsherewith Progress Report No.2, by Dr. Bruce Johnston, AssociateDirector; Fritz Engineering Laboratory, Lehigh University, and ?vlr.Lloyd Cheney, A. 1. S. C. Research Fellow, Lehigh University, wh'ichgives the results of tests of eccentrically loaded columns.

The investigation covered in, this Progress Report was completed' 111

June, 1942.

Early in the program, the Committee 011 Technical Research decidedto institute, as a part of the program, an investigation of the behavior.of columns as part of frames, and the behavior of stiffened plates incompression. These phases of the work are now in progress and, uponcompletion, will be covered by Progress Reports.

COMMITTEE ON TECHNICAL RESEARCH

AMERICAN INSTITUTE OF STEEL

CONSTRUCTION

1<'. H. FHANKLAND, Chuirll/(/'/1

C. A. ADAMS.

H. D. HUSSEY.

.JONA'l'HAN .JONES

,J. R. L.nIBEHT .

L. S. MOlSSEIFF .

\VAL'l'EH \VEISKOPF

NEW YORK, N. Y.NOVEi\IBER. 1942

American Institute of Steel Constrndion

. Consulting Engineer, Philadelphia, Pa.

American Bridge Company, New York, N. Y.

. Bet)llehem Steel Company, Bethlehem, ~a .

The Phoenix Bridge Company, Phoenixville, Pa.

Consulting Engineer, New York, N. Y.

Consulting Engineer. Nmi' York, N. Y.

f41

PROGRESS REPORT NO. 2

STEEL COLUMNS OF

ROLLED WIDE FLANGE SECTION

AMERICAN INSTITUTE OF STEEL CONSTRUCTION

COLUMN RESEARCH AT LEHIGH UNIVERSITY

BY BRUCE JOHNSTON * AND LLOYD CHENEyt

This progress report describes tests of eccentrically and axiallyloaded steel columns sponsored by the American Institute of SteelConstruction at Lehigh ·University. Progress Report No. 1 has pro­vided a general introduction to these tests and has presented testresults on the local compressive strength of column flanges. Thepresent report concerns tests wherein the column fails as a whole, bybuckling, bending, twisting, or combinations of such, as contrastedwith the local plastic buckling failures reported in Progress ReportNo. 1. The tests reported herein were covered in less detail in a pre-

. "ious memorandum 2, **.A brief statement of certain column formulas will be made. Deriva­

tion of the formulas will be found in the references. These formulasare used for predicting maximum loads, working loads, or stresses, andpresume that the failure of the column is integral and not due to localbuckling or crippling. Assumptions similar to those in the developmentof beam bending theory are made.

THE EULER FORMULA

In 1744 Euler presented his well-mown formula which gives thebuckling load for a pin-ended and axially loaded column. This formulais valid only so long as Hooke's Law holds and the proportional limitof the material is not exceeded.

* Associatc Director, Fritz Laboratory, and Associate 1'rofessor of Civil Engineering, LehighUniversity. (Absent on leave).

t Instructor of Applied Mechanics, Case School of Applied Science, Cleveland, Ohio,­Formerly A. I. S. C. Research Fellow, Lehigh University.

** Numerals refer to references listed at the end of Progress Report No. J.

[51 ..

6 . WIDF, ~'LANGE S'l'.EEL COLUMN'S

..-'EI ..-'EAp.=--,=---

, l' (+yp. = Euler critical load for column pinned at each endE = Young's. ModulusI = Least Moment of Inertia,

A ,= Cross-sectional Areal= Length between pin endsr = Radius of Gyration

(3)

If the column is not pinned at both end!? the Euler formula maybe modified.

..-'EIP' - --'­,,- (kl)' -

..-'EA P,k'

(4)

kl = the length between inflection points when the axial load is assmped to~hold the column in a slightly deflected or bent position '

The critical buckling load may be defined as the axial load whichwill just be sufficient to hold the column in a slightly bent position. ,

Equations (3) and (4) for critical buckling load may be modified totake account of loads which exceed the proportional limit. The modi­fication simply involves the substitution of a modified value of E whichwould be reduced from the elastic value of E toward a limit of zeroas a condition of pure plasticity is approached. There are two formsof the modification, one being known as the "tangent modulus'~ fOImulaand the other as the "double modulus" formula. '{he "tangent mod-'ulus" formula simply substitutes for E a value E r which is determinedby the slope of the tangent to the stress-strain curve obtained fOf thesame average compressive stress in a simple test on a short prism.Then,

(5)

The "double modulus" takes theoretical account of the fact that, asthe column starts to bend at the critical load, the elastic modulus appliesto the fibers which have relier from compression whereas the reducedtangent modulus is applied only to the fibers which have' increasedcompression. The tangent modulus is the simpler of the two modifica­tions, is on the conservative side, and has been shown in the case ofstructural aluminum alloys to agree well with tests20 • The results oftests would agree closely with the double modulus theory only if suchtests could be made with absolutely straight columns, perfectly homoge-

PROGRESS REPORT NUMBER TWO 7

neous material, zero eccentricity, and frictionless pin-end supports.Such conditions may be approached but never realized in the laboratory.In field practice the validity of such assumptions is even less and it iscommon procedure in design formulas'to make allowances only forinherent crookedness and eccentricity of loading. Nevertheless, in thecase of certain materials such as structural aluminum alloy or high-alloysteels the stress-strain diagram is markedly curved and it is necessaryto consider the reduced modulus in developing formulas. In the caseof ordinary carbon structural steel the yield point usually determinesthe strength of short columns not in the Euler range (since the E valueis nearly a constant below the yield point). Some investigators haveattributed to a reduced E the behavior of structural steel. columns inthe intermediate zone between the short column and Euler range (llrbetween 50 and 100). Although the behavior of structural steel columnsin this range may be satisfactorily approximated on this basis it isprobable that the more dominant factors in the case of carbon structuralsteel are: unavoidable end eccentricity, crookedness, non-homogeneity,ot residual internal stresses,

THE "ECCENTRICITY" OR "SECANT" FORMULA

The eccentricity formula gives the maximum stress in an eccen­trically loaded strut and includes consideration of the additionaldeflections due to bending. It is simply a ,"deflection theory" for strutsand according to Salmonll was first derived by H. ScheIDer in 1858.The eccentricity formula assumes that the eccentricity is known.Actually, most columns are framed, and the equivalent eccentricityintroduced by end moments in a framed column does not remain con­stant, but varies with changing column load. The eccentricitymay even,become zero and then reverse itself. Hence, laboratory tests with endeccentricity held constant do not bear a direct relation to the behaviorof framed columns.

In the case of short struts it is not necessary~ to take account ofadditional deflections due to bending and the maximum stress in theWF or I section loaded in one of the principal planes is:

P Me"rna. = A + -1- (6)

8 WIDE FLANGE STEEL COLUMNS

When a short strut is loaded with equal eccentricity "e" at each end.NI = Pe, and the maximum stress is

. P ( ec ) P ( e )(]'max = - 1 + - or = -- 1 + -

A r' A s

(ec eA e)

- = -,- = - = eccentricity ratio~ S s '

c = distance from neutral axis to extreme fibre[) = Section modulus 's = "core" or "kern" distance

(7)

(8)

If a factor of safety "n" with respect to yield-point stress is desired,Equation (7) may be rewritten to provide a direct design formula foraverage working stress

(]'./ntrw = 1 + cis

As a column becomes more slender the deflections due to bendingbecome"important and the maximum moment at the center is appre­ciably different from that at the ends:

l1fmax = Pe sec 1/.

where 11 may be expressed variously as

I ~/7' I ,.17' 7r /7'u = 2" EI = 2r " AE = 2 'z;:­

The maximum stress in the column then isP l1Im• x c P (' e )

(]'m.x ='-+ = -A 1 +-secuA Is.

- . P .The equatIOn for average stress (]'a = A IS

(9)

(10)

O'muz(]'a = ---~-=----- (11)

e I J-;;;-1 + -- sec --,,-

s 2r E

If a factor of safety "n" with respect to yield stress is desired, anEquation analogous to Equation (ll) is given by

(]'./n(]',. = --_---.:'------e I J ?I(],,.-

1 + - sec -,,--s 2r E

(12)

In Equation (12) it is assumed that the load "P" is applied in theplane of one of the two principal axes.

Prior to 1940 the A. R. E. A. Specifications for Steel Railway Bridgesprovided a similar formula for eccentrically loaded compressive membersas follows:

(]'./n(]'" = --------="-------

1 + (~+0.2.5) sec~.I n(]',.s' 21''' E

(13a)

PRO G RES S REP 0 R T N U 1\1 B E R TWO 9

, In the 1940 A. R. E. A. Specifications Equation (13a) was modifiedto take account of the fact that the load may be applied with simul­taneous eccentricity with respect to both principal axes. Consideringaxis 1 as the axis with the larger l/r and axis 2 that with the smaller l/r,the formula is then of the following type:

u.ln(TUI = ---------------. (e1 ) kl, Inuw e·, kl., Inuw

1 + - + 0.2.5 see --,-- +-- see ---,--81 2/'1 E 82 2/'2 E

(13b)

As is pointed out in a previous article3, this procedure neglects com­pressive str'ess due to torsion which in the case of torsionally weak Iand WF sections may amount to more than that added by the defiec- .tions due to bending.

In Equations (13) "k" is a reduction factor taken as 0.75 for rivetedends and 0.875 for pinned ends, but does not have the same real sig­nificance as "k" in Equation (4). The additional eccentricity ratio of0.2.5 is included to compensate for end load eccentricity, secondarymoments, and accidental crookedness. The significance and validityof both of these conti'adictory modifications of Equation: (12) tue'opento some question when applied to a framed column.5

There are three constants involving arbitrary selection in Equa­tions (13), i.e., (1) the factor of safety "'11", (2) the allowance for crooked­ness, secondary moment, and t~ccidental end eccentricity (0.25), and

, (:3) the reduction factor "k" which is applied to the l/r ratio. Thesewill be discussed in the order named.

FACTOR OF SAFETY

The selection of the proper factor of safety is an impor'tant anddiffi~ult problem. The real factor of safety in a structure is the "loadfactor", or ratio between load at the limit of structural usefulness andworking load. The factor of safety is sometimes thought of, however,as the ratio between yield-strength stress and allowable working stress.For structural steel with a minimum yield strength of 33 k. s. i. theA. R. E. A. uses an allowable working stress in tension of 18 k. s. i.Hence the so-called "factor of safety" in tension is given by:

. 33n = -- = 1.83

.IS

10 WIDE FLANGE STEEL COLUMNS

In compression, the A. R. E .. A. base stress for short columns· istaken as 15 k. s. i. and there is an implied eccentricity ratio of 0.25.Hence, by Equation (8), the base stress factor of safety with respectto yield stress is:

. u.n =--~--

33--,---,-,---,-,~c- = 1.7(j15 (1 + 0.2.5)

The A. 1. S. C. uses 20 k. s. i. as the tensile working stress and17 k. s. i. as the base column stress. Hence n = 33/20 = 1.65 in ten­sion. If the A.R. E. A. implied eccentricity ratio is carried over intothe A. 1. S. C. formula:

33/I = = 1.55

17 '(1 + 0.25)

if it is assumed that the same "n" is to 'be used for compression as fortension, a smaller eccentricity ratio is thereby implied. In this case,e/s = 0.1765, or roughly, 0.18.

The question now arises as to whether or not the implied eccen­tricity ratio of 0.18 is sufficiently high to take care of crooke.dness andaccidental end eccentricity in columns in building frames.

ALLOWANCE FOR ACCIDENTAL CROOKEDNESS ANDEND ECCENTRICITY

./;'revious investigations have demonstrated that the arbitrary' allow- .ance for crookedness and accidental end eccentricity may be lumpedinto one; either an assumed initial column curvature or an. assumedend eccentricity. The maximum stresses will be almost the same ineach case and it will be on the safe side to assume the combined effectas due to an equivalent end eccentricity.

It has been suggested that the A. 1. S. 1. rolling .and cutting toler­ances be taken as a basis for the equivalent eccentricity allowance.There are three types of tolerance which have a bearing on the selec­tion of the implied eccentricity ratio:

1. Tolerance for web displacement (±%-in.).2. Camber tolerance 1/960 (in.) with %-in. maximum.3. End milling tolerance (no A. 1. S. C. tolerance-fabricators as-

sume aboutW4-in. in 12 in.).On the basis of 1 and 2, arbitrarily assuming:(1) Minimum column section 6 WF 41.(2) Beams absorb 72 ecc~ntricity due to (1).

PROGRESS R~JPORT NUMBER TWO 11

Maximum';equivalent eccentricity ratio would be aboute llr

- = 0.18 + _.- (14)s 500

For llr = 0, Equatior (14) reduces to the present A. 1. S. C. allow­ance for cis, but results in an increasing allowance as the columnsbecome increasingly slender.

REDUCED EQUIVALENT LENGTH ALLOWED BYFRAME ACTION

An axially loaded column with· fixed, non-rotating, ends would havean llr reduction factor "k" equal to 0.5 and in the elastic bucklingrange would be four times as strong as the same column with frictionlesspinned ends. A tier building column which frames at regular intervalsinto beams will usually have a reduction factor "k" somewhere betweenOJ> and 1.0, when loaded axially.

It has been pointed out5 that in application to building frames withrelatively slender members the reduction factor "k" and the equivalentend eccentricity introduced b~' frame action may vary with variablecolumn load "P". In spite of the foregoing, the analysis of a simplifiedcase of frame action2 •5 indicated that the use of a "k" factor in the secaniformula may give good results for design purposes, at least for properlyspecified classes of columns. A more complete treatment of this veryimportant problem will he presented in a later progress report.

DESIGN FORMULAS FOR ECCENTRICALLY LOADEDCOLUMNS

The secant formula (Eq. 12 or 13) may be used for the design ofeccentrically loaded columns and Equation (13b) is so specified by theA. R. E. A. for cases where there is a known eccentricity. The practicaldrawback to Equation (12) or (13) is the fact that (J'w cannot be solveddirectly for any particular case but must be evaluated by cut and tryprocedure. This is not particularly difficult but will be avoided by thedesigning engineer if possible. The secant formula takes account ofadditional deflections due to bending but these are important only inthe case of relatively slender columns. When slender columns havingthe rolled I or H section shape are bent in their strong direction it isimportant to note that the secant formula (Eq. 12 or 13a) neglects thepossibility of lateral and twisting buckling such as is considered in the

12 WIDE FLANGE STEEL COLUMNS

(15)

(16)

design of steel"beams ,,'ith laterally unsupported compression flanges.In Equation (13b), however, the secant formula always assumes somelateral bendin(/ in the weak direCtion, hence compensates for the de­

ficiency of Equation (12) or Equaton (13a).

The A: LS','C. "Section 6" formula for eccentrically loaded columnsis given by:

fA +~=1FA F B

fA = <Tw = :LVcrage allowable axial stress in columnfB = maximum bending stress calculated by beam formula = MIS

FA = allowable working average column stress for un axially loadedcolumn

PB = allowable bending stress as a laterally unsupported beam forparticular III! ratio

If bending is in the weak direction only, the A. 1. S. C. allowablebeiittiiigstressF B = 20 k,sj, and the preceding expression 1.5 reduces to:

FA ,fA = (PA) e (k.s.!.)

1 + -.- -20 s

Equaiion (16) corresponds to a formula which has been called inGermaIly ,thq"Omega" formula and which presumably is used for bend­irig about both the weak and strong axes, It has recently been intro­duced into American literature:2l

,'A theoretical anaJysis3' 22 has been made to obtain the critical endload %r elastic lateral-twisting buckling of a column loaded with equalend eccentricities, in its strong plane. The condition for buckling isfound to be:

PM,'-+-- =1 (17), Per il'f2er

P = eolullln load at buckl ing under 'combined P and il'1.111 = Pe = moment at each end at buckling ,

Per = critical buckling load in weak direction for t,he column if axially loadedand with no end momentrr'EJ

Per = --- for pinned endsI'

. 4rr'EIPer = --- for fixed ends. I'

,11;r = critical moment for lateral buckling of the colullln loaded with equalend couples in the strong plane and with no axial load

Mer = ~~ (Ely) (GK),/7r'Q' + 1 (for pin ends) (18)I P

,-----': 2rr.l . / 4rr'a,2 '. .

Afcr'= -" (Ely) (GK),--,- + 1 (for ends fi;;:ed ~gamst bendmg m (19)~ I . . I~ the weak dIrectIOn)

l\, Q, = torSlo;'i f'onstants 2', 23

E = elastic tensile modulus = 29,000 k. s. i. for steelG ,= elastic shear modulus = 11,200 k. >. i. for steel

(Note ,lEG = 18,000 k. s. i.)

PROGHESS REPORT NU:\IBER TWO 13

The application of Equation (17) to a special case IS shown inFig. 24, adapted from a previous report3, and inclicating predictedaverage column stress at either yielding or elastic buckling in the caseof a 14 WF 30 loade61 eccentrically at. each end with ec/r2 = 1 in the

35

CLateral- orsionalBuckling for ~= ,

301-__-1--l-Y_-l- -h-:=-_E=.u=.:,IFr::...::B::....:u::..::c.:..::kl.:.:.cin-\t--r_---.0: - E in s rong

cr - ( .1)'2. Oir ctionrxx

<J.=33 k.s.i.·

251----1-+--+---+---j----t---\---jY

M1

P..-.r -=1Mcr PerOL-.-__.L-_~.L-~_J...-__J...-_-l.:---:-:-:

o 20 40. 60 80 100 120Ratio ~xx

Fig. 24.-lllustration of different types of column failure for 14WF 30 loaded

eccentrically in plane of web with eccentricity ratio ~ = I. (Note that for this sec­

tion the values on/ry" are about 4 times the indicated values of l/rxx ).

20quais

'Uj A 0 nt Formul.::i.

15c

g. --t5' --10h~

b5

plane of the web. If steel had a perfectly linear stress-strain diagramup to the yield point, and if all conditions were ideal, the secant formula(Eq. 12) would hold from A to 13 in Fig. 24 which indicates the averagestress at which the maximum stress reaches the yield point. From 13 to

14 WIDE FLANGE STEEL COLUMNS

C, and beyond, the condition of elastic buckling is the criterion ofstrength, as calculated by Equation (17). Actually, it might be expectedthat test results :"ould fall along line ADEC rather than ABC, assumingthat point E represents a state in which the maximlllm stress has reachedthe proportional limit of the material. Along the dashed line DE areduced effective modulus would govern and failure would probablyresult from inelastic lateral and twisting buckling. Such failure wasfrequently encountered in the eccentrically loaded column tests madein this program. -

It should be emphasized that in Fig. 24 the curve for the .secantformula is based on Equation (12). In this applicatIon of the secantformula, with the load applied eccentrically in the plane of the web,possible failure by lateral bending .or buckling has been neglected.Actually some accidental eccentricity in both directions will alwaysexist, which fact is considered in· the present A. R. E. A. formul~(Eq. 13b). The use of Equation (13b) in Fig. 24 would have resultedin a curve of critical load falling below the curve for lateral-tor~ional

buckling up to much higher values of ljr than indicated in Fig. 24.Elastic lateral-torsional buckling is importan~ as ~ limiting con-

, .dition of failure but usually is not a design problem when the'eccentricityin the plane of the web is relatively small and the slenderness ratio ofthe column is restricted to low values such as are commonly specifiedfor primary structural steel columns. The use of alloy steels wouldnecessitate re.-evaluation of this statement.

Equation (17) could be used to obtain working values for design,by dividing Per.and Mer by a suitable factor of safety, n. Equation (17)may be rewritten, translating to stress and to the notation used in theA. 1. S. C. "Section 6" formula, Equation (15).

fA + JIB)' = 1 (20)FA (FB )'

Equation (20) is similar to Equation (15) and gives slightly higherallowable stress values. Equation .(20) however, was derived on thebasis of elastic buckling whereas Equation (15) is simply an empiricalformula using specification formulas for FA and F B) and is intended toapply to short columns as well as to elastic buckling. In Equation (20)

"Pe; MereFA = -- and F B =--

An l.n·where n is the desired factor of safety.

If identical values of FA and FB are used in Equation (15) and Equa­tion (20) the A. 1. S. C. Section 6 formula, or Equation (15), results ina lower average allowable stress.than Equation (20).

~ROGRESS REPORT NpMBER TWO

NOTE ON AXIALLY LOADED COLUMNS

15

Design formulas for columns which are presumed to be' axiallyloaded, usually contain an allowance for eccentricity and crookedness,as previously discussed. As a result, the allowable stress in the shortcolumn range is less than the allowable tensile stress. In the longcolumn range (ljr > 100), current design formulas for working stressgive a factor of safety which is inadequate for a pin-ended' column(less than the ratio between tensile yield point and tensile workingstress)." Hence, it must be presumed that allowance is made for endrestraint in the long column range. This is a subject re"quiring furtherstudy. End restraint has no appreciable effect on strength in the shortcolumn range but may cause a 400 per cent increase in strength in thelong column range, over that of a pin-ended column. Whereas the pres­ent formulas may be unsafe in cases corresponding to frictionless pinnedends, it appears that further economy might be effected if columnscould be classified according to restraint' afforded by their particularframIng conditions. Some speculation along this line was presented ina previous memorandum2 but definite recommendations, if any, will bemade in a later progress report on framedeolumns.

, TEST RESULTS

Ninety-three te'sts were made on both axially and eccentricallyloaded columns, in addition to those reported in Progress Report No. 1.These tests were made with knife edge supports and constant eccen­tricity during test; hence they do not apply directly to framed columnsin which the equiv'alent end eccent~icity varies with the load. Eighty­nine of tlie tests were made with 3 I 5.7 sections, which have a'nominalflitnge width of 2.33 in., hence are similar in shape to a wide flangecolumn section. Seventy-six of the tests on the 3 I 5.7 sections wereplanned according to the schedule in Table IV, each letter, X repre­senting a test of the corresponding slenderness rati~, ljr, and e~centricityratio ecjr2 (or ejs). The same test program was carriecl'3-uti:n both theweak and strong' directions. The actual slenderness ratios were some­what larger than as indicated in Table IV, because of the bearing blocksbetween the ends of the column and the loading,knife edge.

The sixteen tests of 3 ~ 5.7 sections not fitting into the schedule inTable IV were pilot and miscellaneous tests preliminary to the mainprogram. An additional six tests were made on 6 WF 20 sections toprovide tests of a column section having different torsional and geomet­rieal eharacteristics.

16 WIDE FLANGE STEEL COLUMNS

It is recognized that a program involving :principally one size and"hape of section (and a small size at that) cannot be used as a basisfor any general conClllsions. It was outside the scope of the investiga­tion to enter into a more ambitious program. The essential purposeof the test program, therefore, was to give a correlated series of testsin which the primary variables were slenderness ratio and eccentricity.

TABLE IV

Slenderness Eccentricity Ratio ee/l"Ratio 1/1" 0 Y2 lY2 2 3 5 7

20 X X X X X X X X30 X X40 X X45 X X X X X X X X50 X X60 X X70 X X X X X X X XFlO X X

lOa X X120 X X

A typical test set-up for a 3 I 5.7 column test is shown in Fig. 25.Bearing blocks between knife edges and milled ends of columns wereabout three inches thick. For all loads within the kern area three-inchblocks machined smooth·on both' sides were used. In the case of theaxially loaded specimens, the load was centered by trial and error untilthe tensometers on the four corners (shown in Fig. 25) gave nearlythe same differences for load increments in the low load range. Eccentricload knife edge positions were located by measurement. ". In the case ofloads outside the kern, individual bearing blocks were welded to eachend of each column and the bearing blocks extended far enough to oneside to provide the loading arm which introduced the eccentric load.A correction24 in the effective l(T was made to compensate for the rigidlength of the bearing block. The correction was appreciable only inthe case of the very short specimens. The manner of loading intro-

.duced a degree of friction at the knife edge, since the hardened knifeedge was allowed to seat itself in the relatively soft steel of the bearingblock. The restraining moment introduced by such friction was calcu­lated to be relatively small and unimportant in the case of short columnsor columns eccentrically loaded. "In the case of the four tests madewith axial load, in the long column range (l(T = 100; 120), the frictionundoubtedly led to the higher than Euler loads that were recorded. Thestrength of an axially loaded slender column in the elastic bucklingrange is known to be very sensitive to slight variations in end restraint.

PROGREsS REPORT XC~B~R TWO 17

Fig. 25.-Test set-up for 3 I 5.7 column test.

5OOool--'-----,-t----t------::---t-----t-----+----+----t-----t --t

400001-----I------I------,,-+_--;P---I--6---_+_~---+_--oO::"::':~__t----_+_---___1

300001-----I------I---:--,?--+-----.~--I----f:f---+---,./"--+_---__t----_+_---___1

.ri.

~ 20000 1-:-.----+----7='-.:f------..i"7''''----_+_'----,---.;o-/-''-----j---,------j;J.---+---,---t--_-+- __t-----j,e.!;..~ ~

iii 0.0004 ;Yo10000 1-----!O-j.----d--+----~+--~~H----_+_----......------+----_+_---___1

oUNIT STRAIN

Fig. 26.-Typicaldress-strain diagrams in tension and compression for 3'1 5.7 column material.

Q

o

PROGRESS REPORT NUMBER TWO IH

The dj!l1~!lsionsof the specimens were measured carefully by microm­eter calipers. The moments of inertia in both the weak and strongdirection. ~v~.~ within three per cent of the nominal handbook values.The computed values of radii of gyration were within two per centof the handbook values. These constants of bending stiffness were alsochecked by l;~am tests in pure bending, and agreed closely with nominaland measured values of I and T for an ~ssumedmodulus E = 29;000,000p. s. i. Two different lots of steel were used in the case of the 3 I 5.7sections.·These had similar physical properties. The average yieldpoint weightjeld according to relative areas and based on tests of flangeand web material was 42.4 k. s. i. for the steel in Lot 1 and 40.8 k. s. k.in Lot 2. Fig. 26 shows typical stress-strain curves in tension andcompression for material taken from the junction between the web andflange. Th¢se tests were not used in computing the weighted average.The material in the 6 WF 20 sections had a weighted average yieldpoint of 39".8 k. s. i. The material met the A7-39 A. S. T. M. Specifi­cations in all respects.

The test results are presented in Table V and in Fig. 28 to 33.The second" column in Table V tabulates the length between knife. edgesupports. in the case of the short columns the lengths are corrected24

to compens;1te for the rigid bearing blocks which were assumed not tobend. The· correction was less than one per cent of the length evenin the shortest columns. Column 3 identifies the material designatedas Lot lor' Lot 2 for the 3 I 5.7 sections (see data as to physical prop­erties). Columns 4 and 5 indicate the orientation of the knife edgeand the load eccentricity. Column 6 gives the slenderness ratio, l/r,for the' direction of bending which is freely permitted by the rotationat the knife. edges. For tests C1 to C38, CPl1"and 6-1, 6-2, and 6-3,the free bending is in the weakest direction and the columns whentestedbeni·principally in this direction. Columns axially loaded as inTests No; 39 to 48, with free bending in the strong direction, failed inthe weak direction even though the ends were more or less fixed in thisdirection against rotation. In the case of the 3 I 5.7 sections the weakracpus of gyration is less than half the strong, hence such failure would <­

be expected'. In the case of the 6 WF 20 sections, the weak radius ofgyration is 'sliglitly more than half the strong. H~wever, previous tests,as well as those here reported, show that the length reduction factorfor a flat ~~d test is nearer 0.6 than 0.5. Column 7 reports 0.6X (theslenderness' ratio) in the direction normal to that in which free bendingis permitted". This is listed only when free bending is in the strong direc-

20 WIDE FLANGE STEEL COLUMNS

TABLE NO. V

25.7028.()02:3.8020.3023.2020 ..50

19.70H).GO19.4018.1017.3014.90

12.709.20

:30.0019.2015.301'l.l0

8 ..507.10

·24.0017.001:3.3010.00

7.005.39

20.4013.7011.359.18

6.8i>4.78

32.9033.4033.4034.40

31.0028.3023.3019.0015.201l.20

9.ll6.47

42.204.2.5042.2041.80

37.5033.8527.5522.1016.701l.33

30.20:3:3.202.5.7020.:3.5:31.5026.90

2.5.752.5.4025.1522.8020.7518.25

14.:;:31l.25:~2 .2028.0.523.7018.0.5

11.M8.9.5

29.GO22.3518.8014.31

9.097.H)

24.0516.6214.4111.58

30.2033.2025.7020.3;;29.7026.20

2.5.002'LOO24.2022.3020.liO17.80

14.4011.0030.802:3.4019.00lli.15

10.308.liO

28.60UL90Hi.4012.30

8.liO6.(i2

23.701li .401:3(;51l.10

8.30.5.81

42.2042.5042.2041.80

37.5033.8527.5522.1016.7011.33

27.941.755.762.6

li9.683.597 .•5

lll.0139.2167.0

48848.875.7n.873.873.8

73.873.822.632.642.747.1

.52.162.072.082.0

101.8121.6

105.7125.825 ..52:3.723.72:3.7

2:3.725 ..5.50.748.848.848.8

75.787.0

105.7125.825.93G.2

4li.350.75G.Glili.\l7.5.787.0

oooooooooo0.240.24

0.240.240.240.240.240.24

0.240.240.120.350.47o.n1.18I.G.50.120.3.50.470.71

1.171.650.120.350.470.71

1.18I.G5oooooooooo

Average Stress

WebWebWebWebWebWeb

Web'VebWebWebWebWeb

WebWebWebWeb\VebWeb

WebWebWebWebWebWeb'

WebWeb'VebWebWeb\Veh

WebWebWebWebWebWeb

WebWebFlangeFlangeFlangeFlange

FlangeFlangeFlangeFlangeFlangeFlange

222222

222222

222222

12211

121121

2222

\ 22

221112

13.50*18.82*24.1.';*21L 84*29.5034.81

40.1245.37~)lLOO

66.6213 ..50*18.82*

24.1.5*26.84*29 ..50:H.8140.124.5.37

.56.00Gli .G213 ..50*12.59*12 ..5\l*12 ..59*

12.59*13.50*2(i.84*2.5.8li2.5.8G25.8G

2.5.8G25.8li40.1239. ]239.1239.12

39.123.9.1227.6239.87.52.1958.37

64.5076.8189.12

101.37126.00150.62

C 31C :32C 33C34C :~.5

C 36

C 1C 2C 3C 4C 5C 6 I

C 7C8C 9ClOCllC 12

C 1:3C 14C 1.5CIGC 17C 18

C 19C 20C 21C 22C 23C 24

C 2.5C 2GC 27.C28C 29C :30

E '"0

I '" ~"'=' ~~"'=' -S"'='~ ~ ::;;: @~ ~.~ g ~ ~n~

c '0 ~- c:; H :.0-><~ oU~ ~ ~

---1----1--1----1--- ----------------2.5 .9 . . . . 43 .90 43 .90 34 .40:~6.2 . . . . 43.60 4:3. 60 34. :3046.3 .... 41. GO 41.60 33.2050 .7 . . . . 40 .4.5 40 .45 33 .40.5G .G . . . . 40.70 40.70 32.'90G6.9 .... 38.40 38.40 31.60

.c:nC :38C 39C 40C 41C 42

C 4:~

C44C 45C 46C 47C48

* These lengt~B were corrected to compensate for rigid bearing blocks.

P HOG It ];; S S REP 0 H. '1' N' U M HER TWO 21

TABLE NO. V-(Continued)

Average Stress

18.1017.6016.501.5.6016.1515.50

13.8012.7010.008.G4

23.4014.90

11.909.156.444.80

22.2013.:30

lO.GO8.45.5.904.49

17.751l.G2

12 .(i7. 10 .50n.02· 7.'15(i .53 5.404.81 :3.86

15.G211.868.1.56.29

27.201(i . :38

13 .:~O

11.107.41.5.G4

21.0.51:.L93

23 ..5022.8.520.45]\) .1020.0018.70

1G ..5014.0511.409 ..50

28.9019.00

12.679.02(i .•534.G9

12. 9:~

10.407.245 ..52

21.0:31:3.93

2:3.2022.6020.20JO .1020.0018.70

16 ..5014.9:)11.40

9 ..5028.0018.40

14.70113;~

7.9.5.5.94

27.10IG.30

22.322.322.322.347.147.1

72.072.072.072.0

47.147.147.147.172:072.0

22.6:~2 .G42.147.1.52.1G2.0

72.082.0

101.8121.622.322.3

2.02:~ .03.5.057.070.501 ..52

2.023.035.057.07

;"...,:§ 00

.... '"~~

'" '"~.-

~'"

1.011.011.011.011.011.01

1.011.011.011.010.501.52

2.023.03.5.057.070 ..501.:'12

FlangeFlangeFlangeFlange

FlangeFlangeFlangeFlangeFlangeFlange

FlangeFlangeFlangeFlangeFltLl1<TeFlange

FlangeFlangeFlangeFlangeFlangeFlange

FhngeFlangeFltLllgeFlangeFlangeFlange'

222222

2222

112222

222222

222222

27.6227.G227.G227.6258.:37.58.:37

.58.:3758.:37.58.:371)8 .:3780.1289.12

80.1280.1280.1289.12

27.G239.87.52.HI58.3764..5076.81

89.12101. 3712G.00150.6227.G227.(j2

ciZ

C49C50C 51C 52C 53C 54

C .'')5C 56C 57C 58C 59C 60

C 61C62C 63C64C 65C 66

C 67C68C69C 70C7l072

C 73C74C 75C'76

cr 1cr 2CP ;~

cr 4CP 5CP 6

CP 7cr 8cr 9CrlOCPll

1!).7.5'19.75'19.2918.8213 ..5018.82

24.1.520 ..5034.814.5.:3752. !\)

111111

11112

FlangeFlangeFlangeFlangeF1an<TeFlange

FlangeFlangeFlangeFllLngeWeb

ooooooooooo

1(). I1G.l1;3.81.5.411.015 A

19.724.128'.137.0

100.0

18.018.018.018.012.018.0

2:3.9:30.03G.048.0

4.0.042.84:~ .843.24:3.143.9

43.643.642.840.129 A

40.042.84:~.8

43.2.51.44.5.5

4:3.G43.642.840.129.4

:31.133 A34.233.633.834.2

34.034.033.6:31.126.5

6-1G-2G-36-4G-.5G-G

71.0071.007I .00

lt9.00119.00119 ..50

33333:3

WebWebWebFlangeFlangeFlange

o0.7G1.5:3o2.234.4.5

47.347.347.:346.746.746.9

4G.2

34.626.017.7::14.621.614.4

:34.627.920.0M.G2l.G14.4

29.221.914.929.229.0. ;12.1

* These lengths were corrected to cornpensnte for rigid bearing blocks.

22 -WIDE FLANGE STEEL COLUMNS

tion, in which case there is the possibility of lateral buckling. Column 8gives the average stress (load divided by cross-sectional area) at generalyielding of the column. General yielding was arbitrarily assumed inthese te~ts to have occurred when the average strain at the mid-lengthof the column, on the most stressed side, showed an offset of 0.001from the initial straight portion of the load-strain curve. These strainswere measured over a one-inch gauge length with tensometers. Theyield load, so defined, was in many cases identical with the ultimate,most commonly so when free bending was in the strong direction orwhen the columns were axially loaded.

Column 9 in Table V gives the average axial stress at the maximumtest load. Column 10 gives the average axial stress at general yiel,d,arbitrarily reduced to adjust the results to the minimum specificationyield point of 33 k. s. i. for structural steel. This reduction was not aconstant, but varied with liT so that in the elastic buckling range noreduction at all was applied. The variation in the reduction was madeas would be predicted by the secant or eccentricity formula.

The axially loaded columns. adjusted by trial to a uniform stresscondition at the mid-point, usually buckled suddenly at the maximumload. Theaxially loaded columns with knife edges parallel to the webbuckled freely in the weak direction. The axially loaded columnswith knife edge parallel to the flange also buckled in the weak direction,with a semi-fixed end condition which caused the results to correspondon the average to a length reduction factor .of about 0.6. .

The eccentrically loaded columns, with tile eccentricity causingbending in the weak direction, failed gradually and with a considerableamount of reserve strength after reaching the yield point. This wouldbe expected, as a result of analysis by the theory of plasticity. Afterinitial yielding, most of the bending was concentrated near the mid­height of the columns, particularly in the case of those loaded eccentric­ally to produce bending in the weak direction. The final shape of thesespecimens consisted of two fairly straight portions at each end and asharply bent portion at the center.

The columns loaded eccentrically to produce bending in the strongdirection usually. failed by plastic lateral-torsional buckling after initi­ally passing the yield point in the case of short specimens and somewhat'below the yield point in the long specimens. The tendency of thelonger specimens toward lateral-torsional buckling in the sub-yieldpoint and sub-elastic buckling range was discussed in connection withFig. 24. The condition of two of the shorter specimens after failure is

. illustrated in Fig. 27.

PROGRESS REPOHT NUMBEH TWO

.Fig. 27.-Tests Nos. C63 and CM after removal from testing machine.

24 WIDE FLANGE STEEL COLUMNS

The test results for the axially loaded columns and for those loaded. h . . . C cc I' F' d' dWlt . an eccentnclty ratIO -; = --:;::;- = 1 are s lawn III 19. 28, a Juste to

33 k. s. i. yield point in the short column range as previously discussed.The axially loaded columns with knife edge parallel to the flange wereplotted for llr = 0.6 ljr in the weak direction, the factor of 0.6 bringingthese results into line with those free to bend in the weak direction.For an eccentricity ratio of els' = 1, the average stress at which themaximum stress reaches the yield point, according to the secant formula,is shown by the dashed line. In the short column range there is a c'on­siderable reserve strength above the secant curve in the case of columnsin weak bending, and a slight reserve for those in strong bending. Thereserve for weak bending disappears at llr = 120, where the test resultsmerge with the secant curve. In the case of strong bending, the columnstrengths fall below those predicted by the secant formula for llr greater'than 80. The curve for elastic lateral-torsional buckling crosses thesecant curve at lh of about 115, hence these results appear to be inline with the discussion in connection with Fig. 24, indicating the transi­tion between yielding failure and elastic lateral-torsional buckling fail­ure. These longer specimens failed suddenly in this manner, therebysuppmting this explanation. This type of failure is associated usuallywith columns which are very slender in the weak direction.

Fig. 29, 30, and :31 show the test results for the three series in whichthe eccentricity ratio els was the variable, with the ljr rati.os of about23, 50, and n, respectively. Control points cross-referenced from thebest curves for els = °and els = 1 on Fig. 28 were given added weight

on Fig. 29, 30, and 31. The results are plotted in terms of 1 ~ cis as

the abscissa. This permits the entire range of eccentricity from axialload (els = 0) to pure bending (els = ex» to be plotted on one diagram.This manner of plotting also makes the short strut formula (Eq. 8without n) plot as a straight line.

Also shown on Fig..29, 30, and 31 are curves of load factor or truefactor of safety with respect to allowable working loads as computed bythe A. 1. S. C. "Section 6" formula (Eq. 15). In computing these loadfactors it was assumed that the laboratory test columns were straight,and an eccentricity of 0.1765, as implied by the A. 1. S. C. base columnstress, was added as a penalty on the test results.· For example, theallowable load by the A. 1. S. C. Section 6 formula would be calculatedfor an eccentricity ratio of els = 1.00. This value divided into theinterpolated test result for an eccentricity ratio of els = 1.1765 ,\Va.<;

2

6

20

36

Fig. 28.-Test results for axial andeccentrically loaded columns. Plotted

4 points give yield strength of columnadjusted to <Ty = 33 k.s.i.

8

24

28

32

6

•~~I

03"1 LOADED ~ OR ::¢• o 0 0

• • ..~r;;."33~ • 3"1S LOADfD)~ DR,~--~ --- \"" 06"1<1' LOADED DR I

2---.. ----1------

~I,'F .6"W' LOADED'] DRI~.

. \~ ... 1D"W LOADED -J OR "T.-AND fvlTH., ~ ..\-" FL NGES PLAI o BELOW I OMINAL

o \-;:;.~TH CKNESS

8

~\~"" (Nl TE- 3"1S L ADED-:f..-AI E PLOTTED\.;.. W TH '/r'Q6'" IN WEAK ENDING)

0 \

•1'\\4

~~.

0

~Elastic 00 .~

~.~"- 'a;'MI I"~ • • buckli g.

~~~r-.... •-r--!_.............~

~"

--1

--.J

~SECANT F RMULAl"'~-

~• (Eq.12) ~.................. "-

"-2

~ -:~~~i~

1

~. "

~ ........ .....,........... .....

4

8

3

3

20 40 60 80 100 120

SLENDERNESS RATIO llr

140 160 180o

200

26 WIDE FLANGE STEEL COLUMNS

considered to be the load factor for cis = 1.00. It should also benoted that for cis = 0, with knife edge parallel with flanges, the criticalllr is· in the weak direction and is larger than the other correspondingtest results. These are noted on Fig. 30 and 31 as based on an llr reduc-tion factor in the weak direction equal to 0.6. .

DISCUSSION OF TEST RESULTS

Assuming the laboratory test columns to be straight, Fig. 29, 30,and 31 make it possible to predict the column strength for any eccen­tricity ratio, cis, either applied or assumed as a design allowance. Theaxially loaded columns were adjusted in testing so as to be practicallyequivalent t6 straight columns. It should be noted that in Fig. 29 and30, corresponding to llr of about 23 and 48, respectively, the resultsindicate that for any eccentricity the short strut formula, neglectingdeflection and shown .by the dashed line, provides a satisfactory designcriterion.

Control points cross-referenced from the curves in Fig. 29, 30, and31 are shown in Fig. 32 and 33 to indicate the predicted strength ofcolumns of the type tested for cis =0 and cis = 1, with an addedallowance of cis = 0.1765 for accidental curvature and eccentricity.For the axially loaded case in Fig. 32, the secant formula is presumedto hold for llr ratios higher than those investigated. In Fig. 33, forcis = 1.1765, the predicted load factors are not carried beyond llr = 120.Although there are no control points for llr greater than 73 the curves

. between 73 and 140 are plotted on the basis of their similarity andnearness to the curves for cis = 1 in Fig. 28. In Fig. 33 there is alsoplotted the curve for elastic lateral-torsional buckling based on Equa­tion (17). The estimated curve for cis = 1, with knife edge parallelwith the flange, indicates that the transition between strength pre­dicted by the secant formula and that predicted by Equation (17) forelastic lateral-torsional buckling in the present case lies between llr = 80and llr = 140. The transition zone would vary with shapes of differenttorsional characteristics and steels with different proportional limits.The tests herein presented are very meager in this particular range andinadequate as a basis for conclusions; however, they do suggest thatthe secant formula IS not satisfactory in the region where there is atendency toward lateral-torsional buckling. The foregoing statementwould not necessarily apply to the type of secant formula specified bythe A. R. E. A., which makes allowance for accidental eccentricity inthe weak direction.

o;,;

Fig. 29.-Yield ·strength and loadfoetor for eccentrically loaded col.

umns with 1 = 23.r

.6

2.8

2.4~

~Cl...

2.03

(0:::- o • INDI IDUAL TEST

r---. 0 CON ROL POINT CROSS RE~ERENCED

"'-,,""-

~~ --- LOA FACTOR B A. I. S.C. Fa MULA

" ~W TH ADDEO II D 0.1755

'"~

:IT'"~i"" '"p.u> ..?~.> V·J ----"" K.~ - -- ---- --- -

"I"~ ~--- ---------- ~-~-

~--. I ,...----'-~~~--~---- ---..-___1._____--

- .__ .~ -f-- "

!55..L- f-- -

"" """ 0'\ 7.

"~ ~.r-s,~,

01'",

c;..",,, "-

~-<a"-

'1'%"-~...

~",~ r\.,

8

32

28

36

09 08 07 05 02 Ot a

28 WIDE FLANGE STEEL COLUMNS

The load factor curve in Fig. 32 indicates that even in the case offrictionless pin-supports the A. 1. S. C. parabolic design formula up tollr = 110 gives a factor of safety of 1.65 or better. This includes anallowance of els = 0<.1765 for accidental crookedness. It should be 0

remembered that this crookedness allowance is not as great as rollingtolerances ,,;ould imply. Above l/r = 110 the factor of safety fallsbelow 1.65. However, if end-restraint is considered to reduce the equiv­alent length by a factor of 0.85, the factor of safety does not fall below1.65 except for llr more than 200, in which case the formula IS notintended to apply.

If the equivalent end eccentl'icity which was adjudged to correspondto A. 1. S. 1. rolling tolerances (Eq. 14) is used as a design criterioninstead of the flat allowance of els = 0.1765, a more conservativedesign formula would be required in the long column range, to rilaintaina load factor of 1.65 with the assumption of a length reduction factorof 0.85. A formula of this type was discussed in the previous memo­randum2 . No new formula of this type will be proposed in this reportas any such recommendation should be coupled with appropriate allow­ance for end restraint, which depends on column framing conditions,to be studied as the next step in this program.

It is questionable whether a length reduction factor of 0.85 is per­missible in all cases, although a reduction of as much as 0.70 might bepermitted in certain cases of known end restraint. The general questionof-length reduction cannot be considered in this report. Columns mustbe tested as parts of frames, silllulating a variety.of known framingconditions, and the results coupled with theoretical analyses. If longcolumns could be classified according to dependable end restraint,further economy might be gained in some cases, but in other instancesthe present column formulas would seem to be on the unsafe side.

The load factors Of0 l' the case of eccentric loading are given in Fig. 33for the Secant, A. 1. S. C. and the "Omega" formulas. For bending inthe weak direction the A. 1. S. C. and "Omega" formulas are identical.For bending in the strong direction the following summarizes thedifferences between these formulas:

(1) The secant formula of Equation (12 or 13a) type allows for ad­ditional stress due to deflections, but neglects the tendency towardlateral-torsional buckling which may exist. When modified to Equation(13b) type, allowance is made for accidental bending, in .the weak direc­tion, thereby compensating for any neglect of the lateral budding'problem.

36 r--~---,-----r------r----,---....,.---...,....-----,~--........---r---....,

Fig. 30.-Yield strength and loadfoetor for eccentrically loaded col·

umns with +== 50.

0.. INOIVI UAL TESTS

I).. 0 CONT OL POINTS

f'~ f-....'..-YIELO STRENGTH32 r~"r-'''''',,..........---1f-------1--~---1----+----+----+---_L~OA~O~=AC=T=O~R~B=Y-+~J.~S~C.-F=O='R-+~U~LA----1

"~!:'-... WIT ADDED f 0.1765

~,""28t--+---r..............~,,--"""~.,---+---+----\------+----+---+----+_----l2.8 .~. 62.5 ~

'" ~DJUSTEO~ INIMUM) "~ _~.<!J ~ ~<l'd'V)24'1-c----t-----t----""~+_~0?~+_---+_---+_--'--1_---1_--___j---_12.4

e; ~ <"', :s~ ~~~ ~ b~ "'....<1' ..? '" /' ~_- -~-- ---- ---- ~

.pO~ I ~- Q~-=:201---_+---+----j-----"_~=""-~-=""-<£h----+---+----+---+-----12.c;t!i

~ I----f---- ----...:-- ~~ ....eo I- .---- ----V) I--~=I----;:=...----:=~--:--=-:--7---1~- --- l------'-- 1--165 7_1- .-

~ 16 ~"'" -- 1.6i -f-=--I ,,~~2 121----t----t----t----1-------I1------I11-......."'-----1-".------1---_+---____1

~ ~~" 8 \-----t----+---+---+_-----I---4---+-'~..__f\:--___iI_--__l

~~~'~

v;:::~,

4 1----t----t----I--------:I-------I------I----f----f--.....-"6:3.~~;_y'r__--____1

~~09 0.8 0.7 0.6 0.5

1·RATIO /. ~/s

0.4 Q3 0.2 0.1 o

30 WIDE FLANGE STEEL COLUMNS

(2) The A. I. S. C. Section 6 formula makes an arbitrary allowancefor the tendency toward lateral buckling, based on the lib reductionformula for beam benCling stress, but does not directly allowfordeflec­tion stresses.

.(3) The "Omega" formula neglects the tendency toward lateralbuckling and does not directly allow for the additional stress due todeflection.

Fig. 33 shows that the factor of safety falls below 1.65 for all three~ormulas in the range of llr between 60 and 80, and remains below 1.65 .up to llr = 120. The deflections of the small three-inch columns astested were probably more serious than in most practical.cases, whereasthe tendency toward lateral buckling is relatively less serious than insome of the deeper WF sections in the lightest rolling, the 14 WF 30for example. For llr of 120, the A. 1. S. C. Section 6 formula gave thehighest factor of safety.

In any special case, requiring accurate design, the most satisfactoryprocedure for bending in the strong direction would be to make calcu­lations both by the secant.formula (Eq. 11 or 12) and by the lateraltorsional buckling formula (Eq. 17). An allowance must also be madefor the transition between the two types of failure, depending on theshape of the stress-strain curve and the llr ratio of the column. In lieuof the foregoing, the A. I. S. C. Section 6 formula (Eq. 15) seems toprovide as good a design procedure as any, for general purposes. Sincethis formula depends on the lib reduction formula for allowable bendingstress it should be pointed out that for some sections the beam stressformula gives factors of safety as low as 1.1 for the condition of auniform load along the top ofa simply supported beam with pinned ends.Actually, of course, the ends are never frictionless pins and a higherfactor ofsafety than 1.1 would be realized.

The A. I. S. C,. Section 6 formula (Eq. 15) gives reasonably goodresults because the occasional low factor of safety in the bendingformula for F B is offset by the fact that (Eq. 15) gives lower allowableloads than (Eq. 20), when the same formulas for FA and FBare usedin each. Hence, it appears that (Eq. 15) should be continued in effectuntil such time as the bending stress formulas forlaterally 'unsupportedbeams are rationalized to take account of torsional· as well as lateralbending resistance. Such a step has been made in some specifications.21 •

22

--------r-------...T.....----

z

c;ti

...,Fig. 31.-Yield strength and load'foetor for eccentrically loaded col. :E

l Cumns with -= 73.

T

2.8

2.4 Q:

~

'"<tCl

2.0 ~

1.6

o • INDW IoUAL TfSTS

-, 0 CDNT OL POINTS, CROSS RfF RfNCfD,I- YIELD STRfNGTH

'~, --LOAD ACTOR BY .LS.C. FOR ULA

"', WiTH DDfD ~s- b.1765

8 "-

~,.J:s,

~~~ ~-1'....~ ,J'~

'\~~

""'a

[+-"<,,,0...:;~I~-1'-;.;

1/,.-97.5 ,~<.,..

jIt,(ADJUSTED ~~MINIMUM) ~7. --I~

.........~ J --- ....--- 1-=----- --+--------ts:~.~ ----- ---- --

~-~=~ I-- l,- " ..--= --:-::::... 1.6!...:L "---.

C-I -~~~. ,

~~; .,

~

~- I "

J6

4

2

8

32

a9 a8 0.7 0.6 a5

RATlO~, .. Is

0.4 0.2 0.1 ()

z[fl

Fig. 32.-Minimum yield strength andload faetor for axially loaded 3" Icolumns adjusted for crookedness al-

lowance and d. = 33 k.s.r.

2

2,4

20018016014080, 100 120SL£ND£RN£SS RA TIO ilr

604020

I--"a....

"""0 ± CONrRO POINrS CR

~~/£~~ i!i£D

~Wfi.~'/;;srs WJrH" AND- • ~- rAK£N AS 1765--..-

--- -- ........

~(NOr£ - MIS LOAD£ -I-AR£-- ADJUSr.. rOZ;~·a6~friNW£AK 8 NDING)

"'-.;: .

"l ,

f----- ----~------- '!!!!.H ~O~j.Q85~---_-: ------=--=- --,.1-----

r--___" ........-...--._'~

1--- 1--- 1----. 1----::..--- =--.;.~_~SZ-_ -_.-::-..,,--,~-~~o ~lfCrOH

7.

" .......................-!~_~N£~

" -------,LA FOR 1.

'<,S CANr FOR~

..... -Is' 0.176 AND

' ......~.3- ---,

~

36

o

28

8

32

'4

...'" 16

i:3... 12

'"~~...

Fig. 33.-Minimum yield strength andload foetor for eccentrically loadedcolumns (~= I) adjusted for crook­

edness allowance ( ~ = .1765) and

<T y = 33 k.s.i.

o

4

2.6

2.2"', I

')LOAD FACT R

-1-- SECA T FDRMUL 0.-33 -1.1765 , -1.65

" f----:~- f-- A.I S. . FORMULA

....., _0-;- HW" FORMULA

-r~....

"=-.,.........

8 - I.

~~~.......... L.F. 1.65 7

.1--- '---- '-- -">... . -.- ~>--.--- f----f---.................... r-...........

~................. I.

\-'_0_.-

~;LASTIC

0LATEI/AL- TORSIONAL

'" \'" GI.

~-I--- '"1----- ----~ """"'k'-~ 0} CONTROL POINTS CR SS REFERE f'!CED

~FROM T STS WITH lis THE VA! IABLE ANO--, [>(:r- • WITH~.r AI<EN AS 1765

~~~~ r ..... ---SECANT FORMULA ----

1. 7•176 r;;,'33K. I..4

8

36

2

32

20 40 60 80 100 120SLENDERNESS RATIO 7'r

140 160 180 200

34 WIDE FLANGE ~TEEL COLUMNS

SUMMARY AND' CONCLUSIONS

(1) A study of the present A. 1. S. C. column formulas for axiallyloaded columns indicates that for a load factor of safety of more than1:65 in the entire range up to llr = 200, a slenderness reduction factordue to end restraint is implied. This reduction factor varies from 1for llr = no to 0.85 for llr = 200.

(2) When columns are eccentrically loaded in the plane of the web,producing bending in the strong direction, the secant formula neglectsthe tendency toward lateral-torsional buckling which may exist, butmay be modified to compensate for this deficiency by assuming acci­dental end eccentricity in the weak direction.

(3) The A. 1. S. C. "Section 6" formula does not take direct accountof additional stress due to deflections, but makes an arbitrary allowancefor lateral~torsional buckling, which' will be conservative in all caseswherein the A. 1. S. C. stress reduction formula for beams in bendingis also conservative.

(4) The correct analysis of the lateral-torsional buckling of acolumn loaded eccentrically in the plane of the web requires use of thetorsion constants in the calculation.

(5) In cases where lateral-torsional buckling is not a criterion; theshort strut formula may be used for llr of 50 or less.

(6) Columns loaded eccentrically in the plane of the web maybuckle by plastic lateral-~orsional buckling at loads corresponding to nmaximum stress of less than the yield point.

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