Lehigh UniversityLehigh Preserve

Fritz Laboratory Reports Civil and Environmental Engineering

1-1-1957

Plastic design in steel (1957)Lynn Beedle

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Recommended CitationBeedle, Lynn, "Plastic design in steel (1957)" (1957). Fritz Laboratory Reports. Paper 553.http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/553

Third National Construction Industry Conference

/

ARMOUR RI:SI:ARCH

Preprint for

CREATIVE TRENDS IN STRUCTURAL DES'IGN

Sponsored by

Armour Research Foundationof Illinois Institute of, Technology

December 11 - 12, 1957

INSTITUTE OF TECHNOLOGY

20 .(00

PLASTIC DESIGN IN STEEL

by Lynn So Beedle

Fritz Engineering LaboratoryLehigh UniversityBethlehem, Penna$

It is the purpose of this paper to define and describe plasticdesign in structural steel. A brief discussion of its advantage is aimedat showing why the time of the engineer and architect is warranted instudying the method, Next some examples will be given to show howplastic design is applied to continuous beams and to rigid frames~

A considerable number of structures have a.lready been builtboth in Europe and in the United States according to designs based uponthe plastic method o Some examples will be shown, and possible futureapplications will be discussed.WHAT IS PLASTIC DESIGN?

Plastic design is a design based upon the maximum load thestructure will carryo This maxinlum load is determined f~rom an analysisof the strength of steel in the tlplastic U range ~e=- which explains theorigin of the termo

Conventional elastic design assumes that the limit of usefulnessof a structure is the first attainment of yield(:apoint stress., However"the final criterion of the usefulrless of a. frame is its ability to carryload. If its strength is not limited by brittle fracture, fatigue,instability, or excessive deflection, then the only logical remainingdesign criterion is the maximum load it will support~ The mere fact thatcomputed yield-point stress has been reached at some point: is of noimportance in itself; in fact, many present design assumptions rely uponductility of the material to provide a. safe structurs1'l Plastic designgoes one step further and makes ftconscious lt use of this same ductilityupon which the engineer has become accustomed to rely~

To what kind of a structure would this method be applied? Fromthe present state of knowledge, plastic design may be applied tostatioally-loaded structural steel frames with rigid joints, and tocontinuous or restrained beams. It is suitable for statically indeterminate

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structures which are stressed primarily (although not exclusively) in bending.It is not intended for simple beams, because present procedures for suchmembers afford sufficient ease of design and at the same time are notwasteful of material~

The concept of a safety factori~is, of course, retained o Whereasin elastic design a sectioll is selected on the basis of a maximum allowablestress at working load, in plastic design one computes fir~ the ultimateload and selects a member that will fail at that loade The ultimate loadis determined, in fact, by multiplying the expected or working load by afactor which assures that there will be the same margin of safety inrigid frames as is presently afforded in the conventional design of simplebeams.

A steel rigid frame attains its maximum strength through theformation of so-called Itplastic hinges tl 0 These hinges, in turn, formunder overload because of the ability of structural steel to deform'plastically after the yield point is reachedo This ductility of steel ischaracterized by a flat UplateauU in the stress-.str~ain d:lagram as shownby the typical curves in Figo 1. They were obtained from tension couponscut from a rolled WF shape. Notice that after the elastic limit is reached,elongations of about 1,5 times the elastic limit va.lue take place withoutany decrease in loado Afterwards some increase in strength is exhibitedas the material strain..".hardens o

Fig o 2 shows that these strains are really quite small. Atultimate load the maximum s train will not have exceeded about 1 0 5%elongation; and for ordinary structural steel, final failure by ruptureoccurs only after a specimen has stretched about 20 times this maximumstrain that is encountered in plastic design.

Based upon the idealized curve of Figo 2s there is shown in Figo 3the action of a beam under bendi'ng momento If it is assumed that all of thematerial in a WF shape is concentrated in the langes then (when the elasticlimit is exceeded) the compression flange shortens at constant load andthe tension flange lengthens at constant loado The resulting moment istherefore constant; the member acts just like a hinge except that deformationoccurs under constant moment~ The magnitude of this maximum moment is calledthe plastic moment,Mpo Since the member is entirely plastic,9 Mp is equ.al

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to the yield point stress multiplied by the combined statical moment of theareas above and below the neutral axis o

It should now be evident why the attainment of yield-point stressdoes not correspond to failureo At the point of peak moment a zone ofyielding develops, a hinge forms, and the structure can then call uponits less-heavily-stressed portions to carry further increase in load.Eventually, when enough plastic hinges form, the structure will reachits ultimate load and fail by continued deflection at that loado

A Hungarian, Gabor Kazinszy, first applied these concepts tothe design of some apartment~type buildings in 1914G Early tests in Germanywere made by Maier-Leibnitzo Van den Broeck, Baker, Roderick, Horne,Heyman, Prager, Symonds, Neal, and Johnston have all made importantcontributions to the plastic theory of structures9 The work at LehighUniversity has feature~ the verification of the plastic method throughappropriate tests on large structures, the systematizing of design operations,and the theoretical and experimental' stUdy of secondary design requirementsthat must be met. Plastic design is already a part of certain specificationsand engineers are now making use of ito

JUSTIFIOATION FOR PLASTIC DESIGN'Quite properly one might ask, UWhat can plastic design do for me?U

Plastic design gives promise of economy in the use of steel and of savingof time in the design office by virtue of its sinlplicity. Further it willprovide building frames more logically designed for greater over~all strength.

The plastic method is certainly not the solution to all problemsin structural steel design. However, for the type of struct'tl:re noted above,it is a powerful tool to assist the designero

Since more effective use is made of all of the material in astructure, then for the same working load, a plastically-designed framewill be lighter than 'its elastically-designed counterpart. Attentionis focused upon the entire structure in plastic design rather thanupon individual members o The saving in weight will depend on a number offactors; while the "average" saving might run from 1.5 to 20%, highervalues than this have been reportedo

The design technique is simpler because the essence of the methodis that it reduces an indeterminate structur'e to a determinate oneo This,

ARMOUR R~St;ARCH FOUNDATION OF ILLINOIS lNSTITUT~ OF TI:CHNOLOGY

in fact, is the role of the "plastic hinges U ; .they provide for rotationat constant moment and thus Udeterminet! the magnitude of the moment atcritical sections. Since it is much easier to design a determinate structurethan an indeterminate one, a simpler design procedure may properly beexpected when the plastic method is used. The advantage of continuity

~.s retained without the difficulty of elastic rigid frame analysis.ILLUSTRATIVE EXAMPIES

The first example is~a plastic analysis for ultimate loadrather than a design prooedure" .. The problem of the rigid cross-barsupported by three rods shown in Fig o 4 is admittedly impractical, but itshould serve to illustrate some important principles o The 'problem is todetermine the ultimate load this assembly will support.

According to elastic conoepts the 'maximum allowable load wouldbe the yield load, P 0 However, the equilibrium equation ( ~ V = 0 )ydoes not solve the problem, because the assembly is statically indeterminate.We only know that P = 2T1 + T2 To compute the yield load it wouldbe necessary to consider, in addition the relative elastic deformation, ofthe three bars (the "continuity" condit ion) 0

Notice how much simpler it is to determine the .,ultimate loadaccording to the principles of plastic analysis. After the center barreaches the yield stress, the partially plastic assembly deforms as ifit were a two-bar system except that a constant .force equal to yield-point stress times the area is supplied by Bar 2 (the member is in theplastic range). This situation continues until the load reaches theyield value in the two outer bars, after which the ~ssembly would continueto deform at the load Puo The ultima.te load is simply equal to the sumof the yield forces in each rod, or,

p = 3f .Au y

Quite evidently, yielding of the center bar reduced the indeterminate'assembly to one which could be ana.lysed by statics, and this simplifiedthe solution to a procedure that took one step 1Design of a Restrained Beam:

Suppose it is desired to support a total load of 200 kips on abeam in a building with several spans of 40..feet~ One of the interior -

ARMOUR R:SEARCH FOUNDATION OF ILLINOIS INSTITUTE OF TJ;C~NOLOGY

from which

spans is shown in Fig. 5. 'The first step in the USta.ticalU method ofplastic design wo~ld be to multiply the working load by the load factor.This gives an ultimate load of 370 kips, and the problem is to selecta section that will just fail at this factored load~

The next step would be to remove the restraining e~d momentsand draw the moment diagram considering the beam as simplY-$upported. Themaximum moment ordinate at the center, from statics, is W

uL/8.

Next, the r'edundant end moments MA and ~ are applied to the beamgiving the redundant moment diagram as shoW!lG>

Finally these two moment diagrams are combirled-: (theredunda.nt momentsare subtracted from the determinate moments) in such a way that plastichinges are formed a.t the center and at the two ends o If less than thesethree hinge~ were formed, then part of the beam 'would not be used tomaximum effectiveness.

From the graphical construction of the composite moment diagram,the center determinate moment (WuL!8) is equated to Mp at the end plusM at the center, orp

WUL-r == 2~M ... ~ ... 92~k4.'t.,

P 10As noted\a.bove, every beam has its own plastic moment of resistance

(M = yield stress times statical moment of the area above and below thepneutral axis)-*o It is found that a 30 WF 108 will be adequate, furnishinga plastic moment of 949 k-ft.

How does one know when a structure is being used to maximumeffectiveness? The answer: when it will carry no additional load4 .Atthe ultimate load the plastic hinges continue to rotate and thereforethe structure simply deflects at constant load. In this respect theaction is similar to a linkage or "mechanismu (~here is no change in loadwith deformation)o So this same'term (uMechanismff ) will be used todescribe the further deflection of a beam or frame after it reaches theultimate load. Such a mechanism is shown at the bottom of Fige 5, and thuswhat was actually done in dra.wing the composite moment diagram was to draw* - ,A table'- of these M values for rolled WF and I shapes used as beams hasbeen prepared by tEe AlSO, and it is included at the end of this paper.'

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it in such a way that a mechanism was formed.Design of a Rigid Fram~

As a third example, a single-span rigid frame will be designedwith haunched connections. Although the structure shown in Fig. 6 isquite different from that of the previous example, the steps in the designare exactly the same as before~

Step 1: The ultimate load is determined by multiplYiJig the expectedworking load (40-kips) by the load factoro Thus the ultimateload is 74 kipso

Step 2: The redunda.nt is selected as the force H at the column bases.This amounts to assuming that the right-hand column is restingon a roller support o

Step 3: Since the structure is now determinate, the moment dia.gramcan be drawn as shown by the heavy solid lineo (The separateconstruction of the determinate and the redundant moment diagramshas been e~iminated)0

Step 4: The next step is to draw the redundant moment diagram insuch a way that a nmechanismu is formed~ (in this wa.y thematerial is used most efficiently)oThe problem will first be solved neglecting the fact 'that there

are haunch'es. The redunda.nt moment diagram (line lc=a...b-c,-7) is drawn insuch a' way that the horizontal line a-b-c bisects the center ordinate of1110 k-ft. The required plastic moment is therefore 1110/2 =: 555 k..ft.From the 'tPlastic Moment Table"',it is fOtlnd that a 24 I 7909 -Will beadequate (Mp = 558 k-ft.)

Considering the frame with a haunch makes the problem no moredifficult~ The redundant moment diagram is dravm in such a way thatplastic hinges form in the beam at the ends of the haunches and at thecenter (sections 3, 4 and 5)0

Step 5: The magnitude of the .required plastic moment can eitherbe computed from the geometry of the moment qiagram or scaledfrom the sketch." It is found to be 407 k-ft. The lightestshape that is suitable is a 21 WF 68 (M = 439 k-fto)p

ARMOUR RESEARCH FOUNDATION OF ILLINOIS INSTIl'UTE OF TECt-iNOLOGY

At section 2 the moment is 491 k-ft or greater than theplastic moment value of 407 k-ft. A larger shape is neededthere, and turning to the table of M-values a 24 WF 76p(Mp = 550 k-ft) will be specified.A final step in all designs is to examine details such as

connections, bracing, colunm buckling, etc., to make sur,e that some secondarydesign factor does not limit the usefulness of the structure. Such designguides will be available in Ref. 10 and Ref. 11 which will, give the backgroundof the ory and experiment upon which they are based;lEXAMPLES OF STRUCTURES DESIGNED BY THE PLASTIC :METHOD

About two-h1llldred industrial (single-story) frames have beendesigned in England by the plastic method, -- ~lso a' school building anda five-story office building. Ref c> 7 coniain~ in its bibliography s'~venpapers describing actual plastic designs. Figs. 7 and 8 are two of .,:hese o

On" this continent the first building to be designed plasticallywas in Canada (Ref o 1). It was a two-story frame with, beams continuousover 6 spans.

At least four plastically-designed structures have been built inthis country. Figso 9 and 10 show one of these during erection and afterthe structure was completed o The .span of the frames of this warehouse inSouix Falls, South Dakota is 88-ft,c (Rafo 2.). They are spaced at 18'-6"centers and the structure is designed for a dead load of 12 psf and a snowload of 3'0 psf 0 It required a 24 WF 94 sh,ape, uniform throughouto A comparisonwith the 30 WF 108 required by conventional design shows a saving of about13% in structural steel.FUTURE TRENDS

Plastic design is suitable f,or the design of continuous beams,single-story industrial building frames and multistory buildings in lJ\ThiC'hhorizontal forces are resisted by wall supports (such as by cross-bracing)oThe metl10d should be applied to those structures that '"have been sufficientlystudied and tested to give confidence that the oalculated maximum strengthwill be realized.

Undoubtedly the scope of application will extend to other typesof construction when such studies have been completed~ Even now the

ARMOUR RE:SE:ARCH FOUNDATION OF ILLINOIS INSTITUTE OF T'ECI-fNOLOGY

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philosophy of plastic analysis is being used to proportion certain elementsof naval vessels, and current research is aimed at application in a moregeneral way in the design of main ship frames and the stiffened plates thatma~e up decks, etc o Studies have also been made of the application of plasticdesign to arches, Vierendeel girders, and to stiffening frames of cylindricalshells.

What will plastic design mean?" To the usidewalk superintendent"it may mean nothing. The structure will look just the same as a conventionally-designed rigid frame. To the engineer it will mean a more rapid method ofdesign. To the owner it will mean economy because plastic design requiresless steel. For the building authority 'it should mean more efficientoperations because designs could be checked faster o For all of us it meansthat better use has been made of the natural resources'with which AlmightyGod has so richly blessed us.

SELECTED REFERENCES1. PLASTIC DES'IGN IS USED SUCOESSFULLY

D.T. WrightEngineering News-Record, April 11, 1957, po 59

2. PLASTIC DESIGN OF WAREHOUSE SAVES STEELE.R. Estes, Jr.Civil Engineering, September, 1957, p 44

3. PLASTIC DESIGN IN STRUCTURAL STEEL (Lecture Notes)L.S. Beedle, B. Thurlimann, RoLe KetterLehigh'University, (Bethlehem, Pao) and AmericanInstitute of Steel Construction (New York, N.Y.)September, 1955

4. PLASTIC DESIGN OF STEEL STRUCTURES (Leoture Note's)J.W. Dolphin, H.M. Nelson, D.T. WrightRoyal Military College and Queen's University publicationMay 1956

5. THE STEEL SKELBTON, VOL. II: PLASTIC BEHAVIOR .AND DESIGNJ.F. Baker, M.R. Horne, Jo HeymanCambridge University Press, Cambridge, England, 1956

6. THE PLASTIC METHODS OF STRUCTURAL ANALYSISB.G. NealJohn Wiley and Sons, New York, 1957

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7 PLASTIC DESIGN OF PORTAL FRA:MESJ. HeymanCambridge University Press, Cambridge, England, 1957

8. PLASTIC DESIGN OF STEEL FRAMESL.5. Beedle(To be published by John Wiley and Sons, New York)

9. ANALYSIS AND DESIGN FOR ULTI:MATE STRENGTHR.L. Ketter and B. Thurlimann(To be published by McGraw-Hill Book Cc e , New York)

10. PLApTrC DESIGN IN STEEL(To be published as a Manual by American Institute of SteelConstruction)

11. PLASTIC DESIGN(To be published as a joint report by ASCE and WRC)

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PLASTIC MOMENT TABLEBased on f = 33 ksiy

If f r 33 ksi, multiply M by f/33Y P(courtesy, American Institute of Steel Construction)

Mp M MP Pft. ft. ft.

kips SHAPE kips SHAPE kips SHAPE..

3450 36 WF 300 1038 30 WF 116 550 24 "WF 763210 36 WF 280 lOIs' 24 WF 130 539 14 .W 11!2960 36 WF 260 982 21 WF 142 528 20 I 952770 36 WF 245 527 21 WF' 82

950 30 WF 108 513 12 WF 1202590 36 WF 230 943 27 WF 114 513 16 WF 962520' 33 WF 240 926 24 WF 120 488 18 WF 85

874 21 "WF 127 488 20 I 852300 33 "WF ',220 857 12 WF 190

846 24 WF'110 473, 21 WF 732110 36 WF 194 465 16 WF as2080 33 WF 200 837 27 WF 102 4!~9 12 WF 1062020 30 WF 210 820 241 120 441 18 WF 771972 36 WF 182 765 24 WF 100 439 21 WE' 68

765 21 WF 112 417 12 WF 991833 36 WF ,170 Ll7 20 I 751814 30 WF 190 764 27 WF 94 4

PLASTIC MOMENT TABLE

(continued)M M MP p' Pft o ft. ft.

kips SHAPE kips SHAPE kips SHAPE

:307 1,8 WF 55 149'.9 J.4WF 34 47.8 12 B 14297 12 WF 72 144.2 12 I 40..8 45 Go 8 I IlL.4292 16 WF 58 14104 '12 WF 36 43.9 10 B 15

1340 8 8WF 48 43.4 -8 WE 17285 18 I 54-.7 41 0 4 ' 6 WE 20282 14WF ~l 1290) 14 WF 30 40.0 6M 20

129 0 1 10 WF 39 39.5 7 I 20,2-7.7 '18 WF 50 122$0 12 I 35269 10 WF 77 12009 12 'WF 31 39 0 1 12 JR 110 .8

11403 12 I 31 0 8 ' 3704 8 B 15255 16 WF 5Q. 109.7 8WF 40 32 0 8 'T I 1503

249 10 WF 72 310 9 6 B 16240 14 WF 53 lo4~4 12 WF 27 31 0 2 8 B 13238 12 WF 58 '96~7 10 I 35 31.2 5 WF 18.5228 10 WF 66 ,9504 10 'WF 29 30.5 5 M 18.9

: 95'04 8WF 35 28 0 8 6 I 17 0 25226 16 WF 45 .26 0 4 5.WF 16216 14 WF 48 81,1 10 'WF 2,':215 ,12 WF ,3 2504 10 JFt 9210 15 I 50 80.7 12 B 22 23 0 0 6 I 1~o5207 10 WF 60 77.1 10 I 25:04 22 0 7 6 B 12

7405 8WF 28 20

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