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Plastic Design Methods Advantages and Limitations I BY D R . D . C . DRUCKER, 2 VISITOR

INTRODUCTION

T h e necessity for considering plasticity in all structural design is clear. Assembly and welding stresses add to the stress imposed by the loading, and it is inevitable that stress concentrations will produce some local plastic flow in the best of designs. Furthermore most heavy metal structures will enter appreciably into the plastic range before reaching their loado carrying capacity. An exact plastic design would be a formidable task but neglecting w o r k - h a r d e n i n g provides a simple and yet a reasonably satisfactory approximation termed limit design. At the limit load the idealized structure collapses. For the overwhelming majority of structural problems, a design based upon a reasonable factor of safety against this plastic collapse provides a more appropriate structure than a design based upon elastic action. Also, it requires far ' less effort. If designers devote sufficient attention to application of the theory much repeated trial corrected by analysis will be replaced by a direct design method. Also, the actual continuity and the real details of the structure will not have to be oversimplified as at present. Although limit analysis is relevant, a more complete analysis is needed for plastic buckling, strengthening by secondary membrane stresses, and brittle fracture. Illustrations are provided for these and all major points of the paper.

1 T h e r e s u l t s p r e s e n t e d in this paper were o b t a i n e d in the course" of research sponsored by the Office of N a v a l Research u n d e r Cont r a c t N o n r 562(10) with B r o w n U n i v e r s i t y . C h a i r m a n , D i v i s i o n of Engineering, Brown U n i v e r s i t y , P r o v i dence, R. I. P r e s e n t e d a t t h e A n n u a l Meeting, New York, N. Y., N o v e m ber 14-15, 1957, of THE SOCIETY OF NAVAL ARCHITECTS AND MARINE ENGINEERS.

The designer of large steel structures often thinks in terms of the stress-strain diagram for steel obtained in routine tests, Fig. 1. An allowable or Working stress is chosen at some fraction 1/n of the lower yield point. The apparent factor of safety n, is selected carefully b y specification writers to take m a n y conditions into account and so will v a r y for different members and different structures. Stresses are computed from the given loading b y formulas of the strength-of-materials type based upon elastic considerations, Mc/I, P/A +Mc/I, and so on. These stresses are restricted to be less than the specified working stress. A safe design thus is evolved, not due to the adequacy of the formulas but to reliance upon the experience of decades or centuries. An actual structure is a very complex body with an extremely complicated state of stress. I t is pierced b y m a n y holes varying from hatch openings to rivet holes. Reinforcements of all kinds are present; doubler plates, rings, bulkheads, stiffeners. In its fabrication local stresses are produced by welding and mismatch, and there are over-all assembly stresses. Individual structural elements such as beams and thick plates come f r o m the mills with cooling residual stresses which are often over one half the yield stress. M a n y secondary stresses arise owing to continuity of the structure. For example bending and torsion m a y aet on what are supposed to be simple tension members, and axial force and torsion act upon beams. The combination of unknown initial stress and stress concentration and redistribution due to discontinuities of the structure defy calculation. The net result in terms of a load-deflection curve, Fig. 2, is quite different in appearance from Fig. 1 even for relatively simple laboratory models. On the first loading there is likely to be an almost immediate local yielding at some point. As loading proceeds there is a marked departure from the linear elastic relation on which the design sup-

172

PLASTIC DESIGN METHODS--ADVANTAGES

AND LIMITATION:s

173

Sy

/

, bJ ag

W

STRAIN

STRAIN

FIG. 1

/ .

--

f

TENSILE STRESS-STRAIN DIAGRAM

~ -

FAILURE LOAD

STRESS

REAONEO

ACCORDINGTO CALCULATION

"

ACCORO,NGTO CA~ULAT,ON

DEFLECTION

FIG. 2

FIRST LOADING OF AN ACTUAL STRUCTURE

posedly is based. The experimental load-deflection curves will v a r y with m a n y random factors b u t will show clear evidence of appreciable, although small, plastic flow long before the working load is reached. Furthermore, as indicated in Fig. 2, nothing distinguishes clearly the point at which first yield is calculated. This is partly because yielding actually occurs earlier b u t mainly because the structure is continuous and local yielding has little effect on over-all deflection. The factor of safety N on load will v a r y greatly from one structure to another for a given so-called factor of safety on stress n. If the structure is essentially stable as in Fig. 2 nothing catastrophic happens at the flattened portion of the load-deflection curve. Generally, however, deflections would be considered excessive above some arbit r a r y failure load as indicated. Certainly, except for problems of fatigue in which failure m a y occur under cyclic stresses in the elastic range, the factor of safety N based on load is far more significant than n which is based on a nominally elastic stress distribution. T h e calculation or estimation of N requires careful consideration of plastic action. If elastic stress formu-

FIG.

3

IDEAL OR

PERFECT, PLASTICITY FOLLOWING ELASTIC ACT'ION

l a s a r e thought of as giving a first approximation to the load-carrying capacity of real structures, a second and much better approximation would be given b y an idealized stress-strain curve, Fig. 3, which ignores work-hardening. As will be seen, this initially elastic and then ideally or perfectly plastic solid generally proyides a v e r y good approximation on the safe side for a structural metal. Better approximations to the actual stress-strain curve can be devised easily. Although our present ability to solve structural problems is limited and there are few areas in which answers can be or need be obtained with such better approximations, work is progressing in this direction. As demonstrated later, computation of the loadcarrying capacity for an elastic-perfectly plastic solid usually does not require a detailed analysis of the state of stress as the load is increased from zero. The theorems of limit analysis, which will be discussed, m a y be used instead. All the difficulties of. elastic stress analysis and the still more complicated elastic-plastic stress analysis can be omitted. T h e calculation of load-carrying capacity b y use.of the limit theorems is much easier than the calculation of stress in the elastic range. Answers obtained are not only more significant but fortunately they are also simpler. The limit analysis of complete structures or of complex groups of structural elements and their connections is feasible for configurations in which elastic analysis can hardly give the order of magnitude of the m a x i m u m stress. One of the i m p o r t a n t general results is t h a t initial or residual stresses or thermal stresses do not affect the limit load. T h e simplicity of limit analysis opens the way to limit design, to direct design as contrasted with the trial-and-error procedure which is normally followed. The first steps taken along these lines will

174

PLASTIC DESIGN METHODS--ADVANTAGES

AND LIMITATIONS q

Mot

LIMIT

MOMENTf?

z~ ql' 124

A ~

L

(o)

c

( b ) ELASTIC S bh z

;=

i,u o

h .&

Sy

( C ) PLASTIC HINGE AT A AND AT B

Mo 3_

CURVATURE

FIG. 4

PURE BENDING OF BEAM oF RECTANGULAR CROSS

SECTION b X h Mo

•

( d ) PLASTIC HINGE AT A, R AND C

t_ be outlined. Much more remains to be done but the start is encouraging. There are numerous problems for which limit analysis is not adequate. These include problems of compressive loading producing buckling before full plasticity is reached and, in general, a n y problem in which the weakening or strengthening effect of deformation prior to collapse is large. Brittle fracture is another extremely important area in which conventional elastic or plastic analysis and design are not suitable although limit design does have relevance. These more troublesome problems are discussed in some detail in the paper. TheY supplement the examples given to illustrate limit analysis and design. T H E P L A S T I C B E H A V I O R OF B E A M S

A s t u d y of the behavior of beams is very helpful because despite .the simplicity of analysis and of experiment most of the features of more general structures appear. Consider first a beam of reetangnlar cross section under pure bending, Fig. 4. Suppose the beam to be free of stress initially and further idealize the material as elastic-perfectly plastic, Fig. 3. The moment-curvature relation or the load-deflection curve then is linear until the extreme fiber stress reaches the yield stress, sv. Until the corresponding value of initial yield moment, svbM/6, is exceeded the stress distribution across the section is linear. As the load is increased, the stress distribution alters as shown in Fig. 4. The rate of deflection increases and the curve asymptotically approaches the limit m o m e n t value, M0 = sybh2/4. The approach is seen to be quite rapid even for a rectangular cross section. For a W F or an I-section, the moment-carrying capacity lies almost entirely in the flanges so t h a t the initial yield m o m e n t is a much greater fraction of the limit m o m e n t and the curve comes very close to the two limiting lines, the elastic inclined

FXG. 5

UNIFORM LOAD ON FIXED-ENDED BEAM

line, curvature = M/E1, and the horizontal limit m o m e n t line M = M0. I t is proper, therefore, in treating problems of bending to approximate the moment-curvature relation b y two lines in the same Way as the stress-strain diagram is simplified, Fig. 3 . The sandwich b e a m or plate in which the core takes shear and all the moment-carrying capacity is in thin top and b o t t o m metal sheets comes very close to the ideal picture. An actual b e a m will have high residual stresses distributed over the cross section. In the cooling process in manufacture, tension is produced on the inside and compression on the outside of a solid section. When a bending m o m e n t is applied initial yield in compression will come much earlier because at the extreme fiber on the compression side the applied stress adds to the residual stress. However, b y the time the curvature is b u t a few times the m a x i m u m elastic value only a v e r y small trace of the residual pattern remains. I t is obvious t h a t the limit m o m e n t itself cannot be affected no m a t t e r how large the residual stresses m a y be at the start. I t is also clear t h a t a n y distribution of thermal stress is washed out b y the subsequent plastic deformation. The small significance of first yield of a stable structure has been discussed previously and is apparent in Fig. 4. To explore this point further, consider next a fixed-ended b e a m under uniform load q per unit length, Fig. 5(a). I t is in such problems of static indeterminacy t h a t the distinction between elastic and plastic analysis becomes large and of great importance in structural design. As is well known, the end moments are twice the center m o m e n t in the elastic range. F r o m considerations of s y m m e t r y and statics alone the sum

PLASTIC DESIGN METHODS--ADVANTAGES

AND LIMITATIONS

175

\S.EE. v SEE FIG 5c

ELASTIC

8 FIG.

6

LOAD-DEFLECTION CURVE SANDWICH BEAM

FOR

IDEAL

of the end and center m o m e n t at all times m u s t be qL2/8. The elastic deflection 6 is qL4/384E[, Fig. 6. As the load is increased the extreme fiber at each end will reach yield first. Suppose now t h a t the b e a m is an ideal sandwich or the extreme case o f a W F - b e a m with all its moment-carrying capacity in the outer fibers. T h e limit m o m e n t M0 then is reached at each end at a load q given b y qL2/12 = Mo, Fig. 5(c). With further increase in q, the ends will rotate as plastic hinges with a constant resisting m o m e n t M0. The rest of the b e a m remains elastic and the new loaddeflection line therefore m u s t have the same slope as t h a t of a simply supported b e a m or five times the initial slope, Fig. 6. The increment of deflection A8 is related to the increment of load Aq b y 5 AqL4/E I A~ = ~-4

When the m o m e n t at the center reaches M0, a plastic hinge forms there. No additional load can be carried and the b e a m collapses. T h e limit load q* is determined b y the static equilibrium condition q'L2~8 = 2M0. Now it is clear for this problem t h a t q* can be calculated without paying the slightest attention to the intermediate stages of loading. A rectangular b e a m with the same limit-moment v a l u e M0 as the ideal sandwich b e a m has a far more complicated stress history. As for any beam of constant cross section, when load is increased, the yield point is reached first at the extreme fiber at each end. The rectangular cross section then requires 50 per cent more m o m e n t to form a plastic hinge. Therefore just as in Fig. 4, no break in the curve of load versus deflection will occur as load is increased. As shown in Fig. 7, the plastic region spreads at the ends and eventually a plastic region starts at the center. Finally the central region spreads until there are effectively plastic hinges at

FIG. 7

DEVELOPMENT OF PLASTIC ZONES IN FIXED BEAM OF RECTANGULAR CROSS SECTION

the center and at the ends. As in Fig. 4, the approach to the collapse condition is asymptotic but the limit load is reached for all practical purposes at a small multiple of the elastic compone~,t.of deflection. The limit load q* once again is independent of the details of the p a t h b y which it is reached and is still given b y q'L2~8 = 2Mo or

q,

16Mo 4s~bh 2 L2 L2 . . . . . . . . . .

[1]

as contrasted with the value at first yield qL~212 = svbh~/6 or

2s~bh2 . . . . . . . . . . . . . . q~ = _ L2

[2]

T h e ratio of the load-carrying capacity to the load at which calculations show the yield stress first is reached is two to one. Experimental load-deflection curves for fixed beams of rectangular cross section exhibit the influences of residual stress and of work-hardening which have been ignored in this discussion. T h e y demonstrate clearly, however, t h a t the limit load computed is a good value to use for load-carrying capacity of an ordinary b e a m if excessive deflection is to be avoided and the distance between the ends cannot be depended upon to remain completely fixed. If the supports A and B of the b e a m of Fig. 5(a) were not at the same level, and if the ends permitted some rotation, the m o m e n t diagrams of Figs. 5(b) and 5(c) would be distorted badly. However, no m a t t e r w h a t the initial conditions and the end details, if the connections of the b e a m to the support at A and B are strong enough to withstand the

176

PLASTIC DESIGN METHODS--ADVANTAGES

limit m o m e n t M0, the m o m e n t diagram of Fig. 5(d) still is valid at collapse. T h e moments continually adjust themselves b u t cannot exceed the limit moment. Collapse cannot occur until M0 is reached at all three of the cross sections A, B, and C. The limit load of this stable structure is seen to be independent of support deflections and rotations and of residual and assembly stresses. Much point has been made of the need for a stable structure if the limit loads as computed are to have meaning. The b e a m already considered indicates the reason. Suppose it to have end connections which are very flexible and so allow considerable rotation. In the first stage of loading the numerical value of the m o m e n t at the center of the span then might well be much larger than the end moments. Suppose further t h a t the b e a m is an I - b e a m without lateral supports. When the flange becomes plastic at the center of the span, lateral buckling becomes a serious threat even fox;short beams. The load-deflection curve for a b e a m with absolutely zero initial twist might look as shown in Fig. 8. After t~ae b e a m has reached the buckling load B, as the deflection increases the load drops off. The load m a y increase again after an excessive deflection has been reached. Perfectly straight beams are not possible, however. With very small initial crookedness and the usual high initial cooling stresses, the load-deflection curve will peak at a considerably" lower load than would be computed for buckling of a perfectly straight beam. A s t u d y b y Zickel and Drucker of eccentrically loaded steel columns (1) 3 indicated t h a t for beamcolumns which twist as they fail, failure was imminent as soon as the stress produced b y axial force and m o m e n t reached the yield stress in any part of the flange. Initial eccentricities and subsequent deflections Were included in the computations. Similar conclusions were drawn b y Hill and Clark for beam-column sections of aluminum (2). I t is necessary therefore to provide adequate lateral support for beams and to use sections which avoid local buckling and permit the needed plastic hinge rotations to take place at the limit moment. Studies at Lehigh show t h a t the ratios of thickness to width of flange of almost all I and WF-structural steel sections are safe i n t h i s respect (3). The general problem of plastic buckling is very troublesome both theoretically and in practical design (4, 5). Under compressive load the changes in the geometry or configuration of the structure tend to have a weakening effect. Whenever bracing or stiffening permits the structure to develop fully plastic action, the limit loads are a

/

AND LIMITATIONS

/~"',ACTU A L

MEMBER

DEFLECTION FIG. 8

BUCKLING IN PLASTIC RANGE

syA

'4

•

t Numbers in parentheses refer to the Bibliography at the end of the paper.

FIG. 9 STRENGTHENING EFFECT OF CONSTRAINT AGAINST INWARD MOTION OF A AND B, FIG. 5

suitable basis f o r design. -Whenever ,adequate. ' support cannot be provided, limit-analysis calculations m a y be misleading. T h e limit load can be approached b y prestressing or poststressing a steel structure to put enough initial tension in compression members. F o r t u n a t e l y there is a bright side to the picture as well. Changes in geometry under load can be helpful instead of harmful (6, 26). Again the beam of Fig. 5 can be used to illustrate the basic action. Suppose the end supports hold the b e a m fixed in horizontal position as well as in orientation. Suppose further t h a t the span-to-depth ratio is very large. The b e a m then has little bending strength or stiffness and acts more like a string than a beam. As soon as a small deflection is produced the load is carried primarily b y tension in the "string" provided the end connections can take the large force applied. As yielding proceeds the string sags

PLASTIC DESIGN METHODS--ADVANTAGES more and more, Fig. 9, and so is able to carry higher and higher loads in accordance with t h e suspension-cable formula, approximately (s~A)~ = qL2/S or

q =

8svA~/L

2 -- 8svbh~/L 2 .......

[3]

I n this formula, ~ is the sag or total deflection, A is the cross-sectional area of the beam-string. The result is approximate because the m a x i m u m tension acts at the cable support and not at the center. For small sag compared to span this difference is negligible. C o m p a r i n g Equation [3] with Equation [1], it is seen t h a t for a deflection of half the depth of beam, ~ = h / 2 , the string loadcarrying capacity equals the limit load. The deeper the b e a m the more i m p o r t a n t the bending resistance and very likely the smaller the allowable deflection as a fraction of the thickness and the more relevant the limit analysis answer. No m a t t e r how deep the beam, however, the string effect eventually becomes dominant as the load is increased, Fig. 9. Whether the limit load in bending or string load should be employed in design depends upon the situation. For design against a catastrophe of small probability a string load or combined bending and string load seems proper even for moderately deep beams. Again the proviso m u s t be added about the end connections, and intermediate splices as well, being strong enough. Riveted connections are likely to be more deficient in this respect than welded connections b u t their strength too m a y help in an emergency. Interior spans of rolled beams continuous over supports automatically have sufficient end and interior strength. As H a y t h o r n t h w a i t e (6) has suggested, it m a y well be worth while to take some care in detailing to be able to count on this added resistance to complete failure. As stated at the beginning of the section, the behavior of beams provides an excellent guide for the behavior of all structures. These more general aspects of structural design and analysis will be treated in the following sections. THE THEOREMS OF LIMIT ANALYSIS

A more general t r e a t m e n t of structures requires a s t a t e m e n t of the limit theorems of D r u c k e r , Greenberg, and Prager and some corollaries (7). T h e y are remarkably simple and in accord with intuition. First it can be shown t h a t for a structure composed of elastic-perfectly plastic material, Fig. 3, when changes in geometry are neglected, as in most elastic.solutions : Collapse occurs under constant load and at con-

AND LIMITATIONS

177

s t a n t stress; plastic strains only take place. The theorems are then I (Lower Bound). If an equilibrium distribution of stress can be found which balances the applied load and is everywhere below yield or at yield, the structure will not collapse or will just be at the point of collapse. I I (Upper Bound). T h e structure will collapse if there is any compatible pattern of plastic deformation for which the rate at which the external forces do work exceeds the rate of internal dissipation. Theorem I reaffirms our faith in the material to adjust itself to carry the applied load if at all possible. I t gives lower bounds on, or safe values of, the limit or collapse loading. The m a x i m u m lower bound is the limit load itself. Theorem I I is a formal statement of the fact t h a t if a p a t h of failure exists the structure will not stand up. I t gives upper bounds on, or tmsafe values of the limit or collapse loading. The minimum upper bound is the limit load itself. I t is rare t h a t exa.ct limit loads can be found for problems of practical importance with their complicated geometry of individual parts and complete structures. T h e two theorems then enable bracketing the answer closely enough for practical engineering purposes. As the lower bound theorem permits a n y distribution of stress satisfying equilibrium and the boundary conditions, it m u s t be true t h a t within the limitation of no effect of change in geometry : Residual, thermal, or initial stresses .or deflections do not influence the limit load. Also addition of (weightless) material cannot result in a lower limit load. And increasing the yield strength in any region cannot weaken the structure., M a n y other theorems and corollaries have been stated and most are listed in the references at the end of the Bibliography. T h e apparently inverted form of the statements as in the last two corollaries is essential. I t is not necessarily true, for example, t h a t increasing the yield strength in one region will strengthen the structure. The theorems will be illustrated first b y returning to the uniformly loaded fixed-ended b e a m of Fig. 5. The parabolic m o m e n t diagram of Fig. 5(d) satisfies equilibrium and is everywhere at or below yield. F r o m lower bound Theorem I, therefore the distributed load q* given b y Equation [1 ] is below or at most equal to the collapse or limit load. In this simple problem, there is no need to proceed further. Clearly, a higher load than q* cannot be balanced b y any distribution of moments which nowhere exceeds M0. The max-

178

PLASTIC DESIGN

METHODS--ADVANTAGES

Q~

~ h

AND

LIMITATIONS

3Mo .P/ L

2

.L

L -___.

"1- 2

(a)

Mo

h8 0..-

Qh~p/2

IP/2 (b)

(c)

(d)

-~o ,,q=p,

Mo

V

Mo~L.~p •

~P=SMo/h lip

t_~p3

(f) P=Q=SMo/h, L=4h FzG. 10 EQUILIBRIUM DIAGRAMS AND COLLAPSB MECHANISMS

i m u m lower bound has been found and is the limit load itself. This static or lower-bound approach is convenient and sufficient for beams and also for frames without sidesway. When sidesway is permitted the upper bound or deformation pattern or mechanism approach is more convenient. A convincing demonstration requires examples which are too elaborate for inclusion here. T h e reader is referred to the work of Symonds and Neal (8) and the books of Baker, H e y m a n , and H o m e and of Neal. T h e elementary and somewhat artificial problem of Fig. 10 does indicate the main points, however. One foot of the rigid frame is fixed the other is hinged and the loads P and Q are presumed to produce bending stresses which are large

compared with the axial stresses they induce so t h a t the interaction problem (1, 2) can be ignored. The limit m o m e n t for the columns of height h is d e s i g n a t e d as M0. The limit m o m e n t for the b e a m of length L is taken for illustration as 3M0. Suppose now t h a t P and Q are given. Will the structure collapse? Fig. 10(b) indicates the start of an equilibrium approach with the m o m e n t diagram superposed on the applied force and reaction picture. The b e a m is considered as simply supported at its ends and the left column is shown as a cantilever with an end force Q. F r o m limit Theorem I if _PL/4is less than 3M0 and Qh is less than M0 the structure will not collapse. I t does not follow, of course, t h a t the structure will col-

PLASTIC DESIGN METHODS--ADVANTAGES lapse should either of the maximum moments for this particular equilibrium solution represented by the moment diagrams exceed the limit moment. The moment diagrams as drawn ignore continuity at the junction of beam and column and certainly can be improved upon. On the other hand, if a mechanism picture of possible plastic deformation at collapse is sketched as in Fig. lO(c)"and the rate at which P and Q do work exceeds the rate of dissipation in the plastic hinges the structure will collapse. Hinges are shown in the column rather than the beam at the junction of the beam and each column because the weaker of the two obviously will govern at the joint. Equating external work in the small virtual displacement 0, QhO, to the internal dissipation in the columns

MoO + MoO + MoO = 3MoO shows that Q*, the limiting value of Q, cannot exceed 3Mo/h

Q* Sy ~ net

(b)

AN EXTERNAL NOTCH (b) Is PREFERABLE TO AN INTERNAL ONE (a)

haunts all designers of outside structures and is an especial bugaboo of ship designers. ' Ferritic" steels are prone to brittleness at low temperatures. M a n y ships develop Cracks of appreciable length and quite a number have broken in two with a loud snap. The thicker the plating the greater the danger with a given composition and treatment of steel. Other size effects also m a y be present and ships are getting bigger. Brittleness and ductility are opposites so t h a t it might seem t h a t limit analysis with its assumption of infinite ductility is irrelevant to the problem. To the extent however, t h a t savings can be made in material, in particular t h a t thinner plating can be used, limit design m a y enable the use of. better steel or at least better behaved steel plate. Plasticity in general certainly is relevant because some plastic action does accompany so-called brittle failure and roots of notches and similar trouble spots do go plastic prior to fracture. Limit considerations themselves are of value in a number of instances. One danger in particular must-be avoided. If. at a temperature below.the appropriate transition the limit load is reached an d a notch or.flaw is present brittle fracture is to be expected. Occasional overloads even of short duration can be catastrophic. If the structure is not aligned well during construction the relevant limit load m a y be quite low. At higher temperatures the geometry of the structure can be realigned b y the load and will then carry the overload properly. At low temperature it cannot always make the necessary adjustments and brittle fracture m a y result. Furthermore, there are situations where conventional specifications m a y permit the limit load to be reached for well-constructed structures which fall outside the range of experience on which the

AND LIMITATIONS

specifications are based. Fig. 14 illustrates one type which m a y arise in various guises. Also, large pressure vessels designed for low pressures have some pitfalls. However, most catastrophic brittle fractures in full-scale structures apparently cannot be ascribed to exceeding the limit load. On the other hand, practically all static laboratory tests on virgin material for the investigation of brittle fracture do exceed the limit load (22). The values of the load at which fracture occurs are above but close to the limit load. For example, symmetrical specimens under tension normally require reaching the yield stress on the net section. Specimens with bending and tension or bending alone are mote difficult to analyze but they too fail at or above the limiting combination of loads. I m p a c t tests such as the Charpy are beyond present methods of analysis but a reasonable guess would be t h a t they too exceed a limit moment. Localized impact added to a static stress field (23) does overcome the barrier to initiation of fracture at static fields below the limit load. Recently the barrier to initiation for static tests has been overcome b y prestrain (24) in a manner which seems to simulate the failure of ships. The appearance of the fracture including the very important lack of ductility (no thumbnail) at the root of the notch is quite similar. Failure loads less than 3~ the limit load have been recorded m a n y times and 1~ the limit load has been approached for project steel E which is very prone to brittle fracture. Plastic analysis is helpful not only in the interpretation of test results but also i n planning an appropriate specimen. I t can be seen from Fig. 20 t h a t the popular internally notched tensile specimen, shown schematically, which seems to resemble a hatch opening in a deck is not as suitable as the externally notched (24, 25). The limit load and plastic-deformation pattern in the plane for the internally notched specimen are influenced strongly b y the presence of the free outside edges which have no analog on the ship. The external notches require an out-of-plane action and a local pattern at the notch which is characteristic of the notch and so corresponds to the prototype action. Size effects are likely to be less m a r k e d in externally notched specimens. A discussion of these and other related aspects of laboratory" testing and comparison with ship failures is given in (22). CONCLUSION

This survey of the present status of plastic analysis and design presents but a, few facts and techniques of immediate use to the designer of ships. Emphasis is placed on the plastic and not the elastic behavior because failure is generally

PLASTIC DESIGN METHODS--ADVANTAGES AND L I M I T A T I O N S preceded by appreciable plastic deformation and limit design is far simpler than elastic design. The main objective of the paper is to convey the vision of a very bright future along with a most useful present. Principles already established analytically and verified experimentally point the way to direct design of structures as a whole. Close estimates of the load-carrying capacity of complicated assemblages are feasible at present. If enough effort is expended by both academic staff not versed in naval architecture and by ship designers, each learning from the other, much of the present design art and guesswork can be replaced by a scientific and straightforward procedure. As a consequence of a more solid foundation the art should then be able to make significant advances. BIBLIOGRAPHY 1 "Investigation of Interaction Formula fa/Fa q- fb/Fb__