1960Artigo_Gvozdev_The Determination of the Value of the Collapse Load for Statically Indeterminate Systems Undergoing Plastic Deformation

IJtt. d. Mech. Sci. Pergamon Press Ltd. 1960. Vol. 1, pp. 322 335. Pri,,ted in Great I~ritaill

THE DETERMINATION OF THE VALUE OF THE COLLAPSE LOAD FOR STATICALLY INDETERMINATE

SYSTEMS UNDERGOING PLASTIC DEFORMATION*f

A. A. GVOZDEV

T R A N S L A T O R ' S P R E F A C E

THE article by Professor A. A. Gvozdev which is the subject of this translation c o n t a i n s w h a t a p p e a r s to be t h e e a r l i e s t p r o o f o f t h e t h e o r e m s o f l i m i t a n a l y s i s

for p r o p o r t i o n a l l o a d i n g . I n t h e course o f his p r e s e n t a t i o n , G v o z d e v i n t r o d u c e s

m a n y c o n c e p t s n o w f a m i l i a r in p l a s t i c i t y t h e o r y , n o t a b l y g e n e r a l i z e d forces a n d d i s p l a c e m e n t s , t h e p r i n c i p l e o f m a x i m u m p l a s t i c w o r k a n d t h e e q u i v a l e n t n o r m a l i t y c o n d i t i o n for s t r a i n r a t e s a t y ie ld .

T h e a r t i c l e a p p e a r e d in a s l im v o l u m e e n t i t l e d Proceedings o f the Conference

on Plas t ic Deformat ions , December 1936, w h i c h was i s sued b y t h e A c a d e m y of

Sc iences o f t h e U . S . S . R . in 1938, u n d e r t h e g e n e r a l e d i t o r s h i p o f A c a d e m i c i a n

B. G. G a l e r k i n . $ O n l y f i f teen h u n d r e d copies were m a d e in t h e f i rs t p r i n t i n g

a n d t h e d o c u m e n t a p p e a r s to h a v e r e m a i n e d u n n o t i c e d o u t s i d e t h e U . S . S . R . , a l t h o u g h f r e q u e n t l y r e f e r r e d to b y R u s s i a n a u t h o r s . §

G v o z d e v ' s p a p e r is e s p e c i a l l y n o t a b l e b e c a u s e o f his use o f g e n e r a l i z e d forces a n d d i s p l a c e m e n t s . Th i s c o n c e p t was e m p l o y e d in 1952 b y P r a g e r s w h e n

e s t a b l i s h i n g a g e n e r a l t h e o r y o f l i m i t des ign , a n d H e d g e 9 h a s g i v e n i t a c e n t r a l

p l a c e in his r e c e n t t e x t on t h e p l a s t i c a n a l y s i s o f s t r u c t u r e s . T h u s t h e p a p e r

is e s s e n t i a l l y m o d e r n in a p p r o a c h , d e s p i t e i t s age , a n d th i s , t o g e t h e r w i t h t h e s i m p l e a n d d i r e c t m a n n e r o f p r e s e n t a t i o n o f t h e m a t e r i a l , is c o n s i d e r e d to

m a k e a ful l t r a n s l a t i o n w o r t h w h i l e .

P r o f e s s o r G v o z d e v has w r i t t e n m a i n l y on a s p e c t s o f r e i n f o r c e d c o n c r e t e

c o n s t r u c t i o n . T h e A d d i t i o n a l B i b l i o g r a p h y (B) c o n t a i n s a l l a r t i c l e s b y h i m

w h i c h t h e t r a n s l a t o r has seen m e n t i o n e d in t h e R u s s i a n l i t e r a t u r e .

* Proceedings of the Conference on Plastic Deformations, December 1936, p. 19. Akademiia Nauk SSSR, Moscow-Leningrad (1938).

Translated from the Russian by Professor R. M. HAYTHOI~TH'vVAITE, Brown University, Providence, R.I. (Now at University of Michigan, Ann Arbor, Michigan.)

See Additional Bibliography (A).

§ In the case of Gvozdev's paper, recognition appears to have come rather late even in Russia and in fact it coincided with a great outpouring of work on similar lines in Britain and the U.S.A. When reviewing developments in plasticity in the U.S.S.R, for the period 1917 47, Ishlinskii 1 dismisses the entire subject with the single sentence (p. 244) "The general idea of the carrying capacity of structures and the method of computation at the limit load was given by A. A. Gvozdev". Meanwhile neither Sokolovsky ~ nor Leibenson s had made any mention of his work in their books on the theory of plasticity. Nor did Kachanov a or Markov 5 recognize the limit theorems as derivable from the variational principles when specialized to proportional loading. Later on, writers interested in structural applications, such as Bezukhov 6 and Rzhanitsyn 7, have given considerable prominence to this and other papers by Gvozdev.

322

Statically indeterminate systems undergoing plastic deformation 323

1. G E N E R A L A P P R O A C H

One problem of s t ruc tura l const ruct ion which must occupy a central posi t ion is the computa t ion of the carrying capaci ty, i.e. the de te rmina t ion of the largest live load which a s t ruc ture can carry. Only in the case of bri t t le mater ials which obey Hooke ' s law almost to f rac ture can this problem be solved complete ly by the methods of the t heo ry of s t ructures and of elasticity. Of grea ter pract ical impor tance is the de te rmina t ion of the carrying capaci ty of s t ruc tures composed of elements which yield on reaching a certain stress state.

I t is more convenient to formula te the plastic relat ions in terms of the elements f rom which the par ts of the s t ruc ture are composed ra ther t h an for the mater ia l itself, i.e. for an e lement of length of a beam, for an element of a plate obta ined b y cut t ing th rough the ent ire thickness, for an e lement of a body, for a beam of a f ramework, etc. Thanks to this approach, the field of appl icat ion of the theory under considerat ion is broadened appreciably.

The approach becomes par t icu lar ly clear when applied to a segment of a reinforced concrete beam in bending. I t would be incorrect to say t h a t the mater ia l of such a beam can y ie ld- -y ie ld ing is possible only in the reinforcement . However , the relat ionship between the value of the bending m o m en t M and the angle of re la t ive ro ta t ion A¢ of the ends of the section (Fig. 1) shows that, when the moment reaches a certain value M = -/]i, the magnitude of the

deformation increases almost without change in the loading. In the graph there is a typical yield zone ab. In reality, when the moment reaches the

value M = 217, the reinforcement s tar ts to yield, bu t in the concrete a crack opens, progressively diminishing the height of the compression zone. The m om e n t M can be de te rmined by a formula due to A. F. Loleit. The equal i ty , M = ~]7, is suggested as the condit ion of yield for the reinforced concrete beam in bending.

A

/ \

Fro. 1.

In wha t follows it will be necessary to define what is mean t by the s ta te of collapse of the s t ructure . I f for known condit ions the elements of a s t ruc ture are able to yield, t hen the s t ruc ture can be considered as disabled as soon as the displacements of the system can increase solely on account of the plastic deformation of its elements. This s ta te will be referred to below as the s ta te of collapse, and the corresponding in tens i ty of the live loading as the collapse load.

2. CHARACTERISTICS OF THE ELEMENTS

The methods of calculation of the value of the collapse load must be adapted to a w~de class of structural forms, so the relationships which are to be written for an element of a structure should be formulated in a sufficiently

324 A.A. GVOZDEV

general manner . On the other hand, idealization of the propert ies of' the material is unavoidable in order t ha t our calculations should not be excessively complex. This leads, for example, to ignoring strain hardening and also to el iminating ent i re ly the factor of t ime f rom the calculations.

For materials such as reinforced concrete, the relat ion between the deformations and the t ract ions up to the s ta te of yield is exceedingly complex; therefore a new simplification is in order and for the plastic s ta te we will in t roduce the most e lementary relations in the spirit of Sain t -Venant and yon Mises.

The following characterist ics will be a t t r ibu ted to the elements of the s t ruc ture :

(a) For an element under the influence of" constant forces, percept ible deformat ion is impossible except in the s ta te of yield. In reali ty, reinforced concrete s t ructures can deform cont inuously under the action of a cons tant load, due to the creep of the concrete ; however, the magni tude of such deforma- t ion is of quite ano ther order f rom the magni tude of the deformat ion at collapse, as defined above.

(b) In the s ta te of yield, the generalized displacement (deformation) of an element can increase indefinitely for a constant value of the corresponding generalized force.

(c) For each possible mode of plastic deformat ion, the corresponding generalized force has a specified constant value. The set of generalized forces corresponding to all possible modes of deformat ion determines the Held condition, i.e. the relat ionship satisfied by the forces acting on the elements when in a s ta te of yield.

The characterist ics (a) and (b) can be considered as defining an ideally plastic body or element.

In the example of the reinforced concrete beam bent in one plane, which has been considered above, the value of the momen t producing plastic deforma- t ion was found to be M = 21), where for definiteness we shall set. M > 0.

I f the beam has double reinforcement , then for bending in the opposite sense the corresponding value of the momen t producing yield is found to be M = M where _M < 0. Consequently, possible values of the bending moment in the beam must sat isfy the condit ion _M ~< M ~< ~IT/. Such conditions, indi- cating tha t by vir tue of the plastic characterist ics of an element the load act ing can change only within set limits, will be referred to as limit conditions.

If, as in the example considered, the stressed s ta te of the element is determined by only one generalized force, the yield conditions (and hence the limit eondit, ions) are par t icu lar ly simple. Often, however, the stressed state of the e lement is character ized by values of several l inearly unre la ted generalized forces, for example, a longitudinal tbrce N and bending moments M~ and M~ in a beam, or six stresses (ax, sy, ~ , T~y, rye, %x) on the sides of an e lementa ry cube with sides perpendicular to the co-ordinate axes x, y, z, or, on the other hand, three principal stresses (~1,~2,~a) on the sides of a correspondingly or ienta ted e lementa ry cube. For such eases the yield condit ion must be studied in more detail.

Sta t i ca l ly i n d e t e r m i n a t e sys t ems unde rgo ing p las t ic d e f o r m a t i o n 325

The deformat ion of the e lement can be described with the aid of the generalized displacements corresponding to the generalized forces which have been selected. For the sake of convenience, we select as generalized forces values of ident ical dimension, for example in the ease of a beam, the moments abou t three axes which do not intersect a t a point bu t which lie in the plane of the section. For this the generalized displacements will have identical dimensions.

We shall now obtain in a convenient fashion a geometrical in te rpre ta t ion which is often used for the s tudy of s t rength theory . In a space of n dimensions, where n is the number of generalized forces necessary to define the stressed s ta te of the element., we set out along the axes of one set of co- ordinates the values of the generalized forces Sl, S2, Sa . . . . ,s,~, and along the axes of another , the values of the corresponding generalized displacements, q , %, e 3 . . . . , %. The corresponding forces and displacements are set out parallel and in one and the same direct ion (Fig. 2).

/

F~(~'. 2.

Any stressed s ta te of an element is then represented by the vector s with components s i (i = l, 2 , . . . , n), and the plastic deformat ion of the e lement by the vector e with the components e~ (i = 1, 2 . . . . . n). Since, however, the deformat ion for a plastic s ta te can increase indefinitely, the length (magnitude) of the vector e is indeterminate . This vec tor determines only the direction in space, whereas the vector s determines a point. Therefore, in place of the vector e it is more convenient to obta in the corresponding uni t vec tor e', i.e. the vec tor the length (magnitude) of which equals uni ty .

The p r ope r ty of the ideally plastic e lement given under (e) above can now be s ta ted as: for each possible deformat ion, represented by the uni t vec tor e', where

e r = e e2, i

there corresponds an expendi ture of work

T = e ' s = f ( e ' ) .

All the vectors s the ends of which lie on a linear ( n - 1)-dimensional manifold ( " p l a n e " ) or thogonal to e' and d is tant T = f(e ' ) f rom the origin o f co-ordinates sat isfy this condit ion (Fig. 2).

326 A . A . GVOZDEV

The equation of this linear manifold has the analytic form*

E e' i s i = f.

The set of such (n - 1)-dimensional manifolds for all possible modes of deformation (of vectors e') produces a convex body in the n-dimensional space. The points of the surface (of the boundary) of this body satisfy the yield condition ; all points of the body satisfy the limit condition.

If only discrete modes of deformation are possible,$ the yield condition determines a polygon, the boundaries of which are orthogonal to the corresponding vectors e'. I f the possible vectors e' form an analytic manifold, then at the yield conditions they satisfy the points of an ( n - 1)-dimensional manifold (of a "surface") skirting a family of linear ( n - 1)-dimensional manifolds ("planes") corresponding to all possible vectors e'. Thus in this ease the deformation corresponding to the stress state represented by any point A of the boundary of the convex body, i.e. the plastic deformation, is represented by the vector e~, parallel to the normal to the surface of the body at the point A (Fig. 3).

FEe. 3.

The origin of co-ordinates (the unstressed state) always satisfies the yield condition. Since the work being expended in the plastic deformation is always positive, the vector e~ is directed along the outwards normal of the yield surface at the point A. Thus the following results are deduced from the definition of an ideally plastic element given above:

Result I. In the space (sl, s2, ...,sn), the yield condition gives an ( n - 1 ) - dimensional manifold ("surface") which is a limiting convex body, referred to as body B.

Result II. For each stressed state represented by a point A lying on the boundary of the body B (i.e. a state of yield) there corresponds a deformation represented by a vector e~ directed along the outwards normal to the boundary of the body B at the point A.

* Trans l a to r ' s n o t e - - T h e or ig inal is E ei8 i = f , which appears to be a mispr in t .

t T rans l a to r ' s n o t e - - T h i s s t a t e m e n t is unnecessar i ly res t r ic t ive . I f any two modes of deforma- t ion occur a t the same s t ress s ta te , a po lygona l form will resu l t ; howeve r , ce r ta in l inear eombina- t ions of the modes are then also poss ib l e ) °


I f F -- 0 represents the yield condition, then the second result is written in the analytical form

e i=C~F/~s~ ( i = 1,2, . . . ,n) ,

where C is a positive quant i ty depending on the co-ordinates and on time but independent of the indices i.

This shows that the function F represents the potential of the flow, as introduced in the theory of plasticity by yon Mises in 1928. I t is necessary, however, to remark that von Mises's result possesses an entirely formal character. Von Mises wrote the function F as a quadratic which did not change with variations of the hydrostatic pressure and he sought to obtain an expression for the deformation in the form of linear functions of the stresses which again were to be independent of the hydrostatic pressure. Here, on the other hand, the role of the function F as a potential of the flow is derived directly from the definition of the ideally plastic material and is unrelated to the analytic expression for the condition of yielding and displacement.

From the definition of the ideally plastic material at the yield condition, the points lie on a convex body; hence it is easy to deduce the following result.

Result I I I (Fig. 3). The work T = eAs4 being expended on the plastic deformation of an element is not less than the virtual work T' = e A s D which would be performed for the same deformations by combinations of any load- ings (stresses) on the element which are permitted by the limit conditions.

We shall now clarify by means of examples what we understand by possible modes of plastic deformation. Suppose that for an element of some isotropic material only plastic deformation representing slip is possible, i.e. pure shear in the direction of one of the three principal shearing directions. In all there will be six possible modes of deformation (counting the sign of the deformation). All six of the possible vectors e lie in the plane e 1+ e 2 + e 3 = 0. In view of the similarity of the axes sl, s 2 and s 3, the yield conditions define in this case a regular hexagonal prism (maximum shear stress hypothesis).

I f not only pure shearing but also all deformation which does not change the volume of the element is possible, and if the work T = e 's is constant, i.e. does not vary with the direction of e', then the yield conditions define a cylinder (Hencky-von Mises theory).

For concrete and rock, the condition of plasticity defines the boundary of a body which is broadening in the direction of increase in the hydrostatic pressure (Fig. 3). Evident ly in this case the vector e is always projected in the positive direction of the axis* le~ + ]e£ + le£, i.e. the plastic deformation must be accompanied by an increase in the volume of the element.

This theoretical prediction is confirmed by the results of tests. In the University of Illinois, Richart, Brandtzaeg and Brown t subjected concrete cylinders to longitudinal compression and simultaneous hydrostatic pressure on the lateral surfaces. At the start of the test, the volume of the concrete

* T r a n s l a t o r ' s n o t e - - S e e Fig. 3. The oc tahedra l axis is m e a n t .

t T r a n s l a t o r ' s n o t e - - T h e a u t h o r p robab l y ha s in m i n d F. E. R icha r t , A. B r a n d t z a e g a n d R. L. Brown, A Study of the Failure of Concrete under Combined Compressive Stress, Bull. 185,. U n i v e r s i t y of Il l inois Eng inee r ing E x p e r i m e n t a l S ta t ion (1928).

328 A . A . GVOZDEV

decreased owing to the smallness of Poisson's ratio. With increasing load, the reduction in volume slowed down and when the concrete started to flow it increased in volume; however, in many samples the original size remained the largest even up to the conclusion of the test.

Often the expressions for the limit conditions prove to be rather compli- cated. The following two examples show how they can be simplified for the purpose of practical calculations.

For an element of length of a metal I-beam subjected to the action of a longitudinal force and bending in the plane of the web, it is easy to construct the limit conditions for any position of the neutral axis, recalling that all that, part of the section lying on one side of this axis is stressed to the yield limit in compression and the remaining part of the section is stressed to the yield limit in extension. The lens-shaped limit curve [Fig. 4(a)] is constructed from two parabolas having common tangents at the points C1, (~2, Ca and ('~.

( m (c :

FIG. 4,

7 b • " 2 ,

As is evident from the graph in the figure, it can be approximated closely with the aid of a hexagon (or allowing rather large error, by means of a rhombus). This corresponds to the substitution of a fictitious effective cross- section for which only discrete positions of the nentral axis are possible, as shown in Fig. 4(b) for the case of the hexagon and in Fig. 4(e) for the case of the rhombus.

For the reinforced concrete section with double reinforcement, as shown in Fig. 5, the limit moments for the upper and lower reinforcement are com- puted in accordance with the draft, of technical specifications and standards for reinforced concrete construction (1936). In the case of eccentric extension, of pure bending and of eccentric compression with large eccentricity, the yield curve is composed of two line segments.* Only in the range of small eeeentrici- ties, i.e. at the approach to brittle fracture, does the yield curve depart from

* Trans l a to r ' s note M o and M o are the bend ing m o m e n t s associa ted wi th the ax ia l forces when considered ac t ing a t O and O" respect ive ly . The energy d iss ipa t ion is a lways g iven by D = O M o + O ' M o, where 0 and 0' are the ro ta t ions t a k e n abou t whichever bar is not e x t e n d i n g : hence M 0 and =¢I~ are t rue genera l ized stresses. Note t ha t , if bo th bars ex tend , 0 and 0' are bo th non-zero, the va lues being ob ta ined by a l lowing opening of the jo in t first abou t one ba r and then a b o u t the other. Nega t ive ro ta t ions , i.e. ro ta t ions opposi te in sense to tbe cor responding momen t s , are not i)ermissible.


these lines. I f we restrict ourselves to cases far from brittle fracture, the limit conditions can be given approximately in the form M'~< ~ ; M/> M. Two modes of plastic deformations correspond to this: (1) The lower reinforcement yields, the element opens and its parts rotate about an instantaneous centre lying at the level of the upper reinforcement. (2) The upper reinforcement yields, and opening of the element proceeds, by the turning of its parts about an instantaneous centre lying at the level of the lower reinforcement.

Thus in these two examples it appears to be sufficient for practical purposes to consider only two or three modes of deformation instead of the infinite number of modes which actually exists.

Brittle fracture

' t

Brittle ~ ture

/ //

/ / . o ~ ,¢,*:--~ M; .~r.,

:FIG. 5.

3. LIMITATIONS ON THE APPLICATION OF THE THEORY

Before passing on to an explanation of the principles and methods of calculation, it is necessary to make several reservations.

In all that follows, it is presupposed that, up to the instant of collapse, the deformations and the displacements remain sufficiently small so that one can ignore the change in the geometric quantities (angles, lever arms, spans, etc.) entering the equations of equilibrium of the elements and also the kine- matical constructions and calculations. Such a restriction is usually made in the calculation of elastic systems, but in our case it is evidently more trouble- some because the restriction is carried to very late stages of loading of the structure. None the less, for structures which are sufficiently rigid, as will usually be the case in construction, the restriction which has been referred to seems to be appropriate.

A danger which appears to be a more serious limitation on the application of the theory is that the structure might lose stability in whole or in part before reaching what is being designated here as the collapse load. This limitation is important in the case of metal construction,* but it plays hardly any role for reinforced concrete. However, for the latter material it is necessary to consider the possibility of brittle fracture prior to collapse due to yielding.

* Author's note--The question of loss of stability has decisive significance for compressed beams in metal frameworks, even when their flexibility is not very large. The author has, inci- dentally, given a diagram for the approach to failure in such beamsfl I

22

330 A.A. GVOEDEV

The possibility of prior collapse by instabi l i ty or by bri t t le f rac ture requires individual verification and, for design, a s t rengthening of the elements involved.

In the following presenta t ion of the theory, these questions are not touched upon; it will be assumed tha t the possibility of collapse by instabi l i ty or by bri t t le f racture has been el iminated with the aid of appropr ia te constructional i n e a s u r e s .

4. T H E P R O B L E M OF T H E D E T E R M I N A T I O N OF T H E C O L L A P S E LOAD. L I M I T I N G E Q U I L I B R I U M

The state of collapse, as it has been defined above, is the s ta te of limiting equil ibrium of a system under the action of (1) a dead load of given in tens i ty and (2) a live load of in tens i ty P.

Our problem becomes the de terminat ion of the value of the pa ramete r P. For this purpose, it is assumed tha t the dead load is insufficient to cause the collapse of the system. The directions of the forces comprising the live loading are given; therefore only positive wdues of the pa ramete r P are of interest .

For the state of collapse, two groups of conditions must be tulfilled: Group I. The fbrces in all e lements of the system nmst satisfy: (a) the

conditions of equil ibrium; (b) the limit conditions. Group I I . The configuration of the yield zones and the mode of possible

deformat ions of the elements belonging to these zones must allow displacements of the points of the system. In this connection, the elements which are not in a s ta te of yield are considered as not being deformed.

E v e r y s ta te of the system which meets the conditions of Group i will be called state I and the corresponding in tens i ty of the live loading will be denoted by PI. A set of states I exists to which a set II~ of the values P~ corresponds. For example, those states th rough which the system passes when loaded from P = 0 up to collapse belong to this set. Ev iden t ly the set lI~ is bounded from above; we shall denote its upper limit by P~.

E v e r y state t ha t meets the conditions of Group I I will be called s ta te II . I t is possible to construct (mental ly at least) the set of states for which the conditions of Group I I are satisfied.

Each state I I can be brought into correspondence with some in tens i ty of the live loading PJI, as deternfined in the following manner : we give the system I I an indefinitely small displacement which permits deformat ions of its plastic zones and we denote by T~,I ~ the work being expended in the elements of the system for the deformat ion corresponding to this displacement. We will compute also the work ~ done by the dead loading and the work Tj, done by a uni t live loading on the same displacement. We determine the value Pn from the equal i ty

E , ' which meets the condit ion


Evidently fulfilment of this condition is necessary for the equilibrium of a system in state II. Since we are interested in the states for which Pn > 0, then*

Tp>0.

This follows from the assumption that the dead load is insufficient for collapse of the system and hence,

Tn, ii - Tg :> 0.

This analysis will be needed below. We will consider the set 1] n of positive values Pn. The lower bound of

this set will be denoted by -Pn. Since in the state of collapse both condition I and condition II must be fulfilled, the intensity of the collapse load P lies simultaneously in both sets H I and II n (their intersection). Hence

P I >>. P >~ Pn.

We shall now show that P~ is not larger than -Pn- Suppose that this is

not so and Pi > - e I I ;

then it would be possible to take values P; and Pn such that

Pi > r ;I"

The state I' is a state of equilibrium, therefore the work done by the forces of this state on any possible displacements must be equal to zero., From the possible displacements we select the displacements of the state II ' , with the aid of which the load P'II is determined. Denoting by TI, ii the work done by the forces acting on the elements of the system in state I ' for the deformations of the elements in state II ' , we write

P'~ ~, + T a = T~, n.

The condition determining P'II has the form

P'n Tj, + 5 = Tn. II '

TI[ 1 i - T I i I a n d s o v ; = ,

In accordance with Result I I I deduced from the definition of an ideally plastic element, the numerator of the right-hand side is not less than zero; and sinee

T, > 0, then r'II ~> P;"

We have arrived at a contradiction. Hence it follows that PI is not larger than -Pu and so the collapse load P is the largest of the loads for which the conditions of Group I can be satisfied; and at the same time is the smallest of the loads for which the conditions of Group II are satisfied

P I . . . . = e = PII lniil"

* T rans l a to r ' s n o t e - - T h i s is equ iva len t to res t r ie t ing s t a t e s I I to d i sp lacement s in which t h e live loading does posi t ive work on t he s t ruc tu re .

T rans l a to r ' s n o t e - - T h i s is t he principle of v i r tua l work for rigid bodies.

332 A.A. GVOZDEV

Thus the establishment of the collapse load may be approached either by a determination of P = P~ max or by a determination of P = Pn m~" A combina- tion of these two methods is also possible. Now it is easy to show that factors producing internal stresses in the system (for example, removal of supports, residual deformations caused by previous cycles of loading, change of tempera- ture, etc.) do not influence the value of the collapse load, provided, however, tha t the corresponding displacements are small and that it is possible to ignore changes in the mechanical properties of the material brought about by these factors. Actually, neither the conditions of equilibrium nor the limit conditions change, unless the loads involve the induced internal stresses as factors.

The fundamentals of the method of determining the collapse load will be illustrated briefly by two examples.

5. DETERMINATION OF THE COLLAPSE LOAD _P =PImax FOR STATICALLY INDETERMINATE BEAM SYSTEMS WITH

n UNKNOWNS

Making use of the conditions of equilibrium, it is possible to represent any force s~ in a statically indeterminate system as a linear function of the load P~ and the statically indeterminate parameters X 1, X2, ..., X~

i=n sk = sk~ + P~ skr, + ~2 ski X i . ( 1 )

i = 1

Here Skg is the force system in the primary structure due to the dead loading; Sky is the force system in the primary structure due to unit live loading, and ski the force system in the primary structure due to the forces X i = 1.

The forces in the statically indeterminate system must, moreover, satisfy the limit conditions. Therefore the conditions which the load intensity Pt must satisfy can be obtained by substituting the expression (1) for the force % into the yield conditions.

As was shown above, the limit conditions can be represented in the ibrm of the union of expressions of the form

i=m Z e;. 8j <f, (2)

i=1

where s i (j = 1, 2 .. . . , n) are the forces characterizing the stressed state of the element and m the number of these forces.

After substitution of the expression for sj from (l), the condition (2) takes the form

P~ Eel,. sk,~ + Ee'k sk, + Exi E% ski -~ f

or, using the notation Ee'l: Skp : ~p, Z e k 8kg : ~g, Ze~ Ski : 3i,

P~ ~p + ~, + ZX~ ~ < ,f. (3)

The condition (3) has a simple mechanical meaning: it is easy to see that the coefficients 8~), $g and 3, represent displacements (generalized), which arise in the basic structure in the direction of the corresponding live loading, the dead loading and the unknowns X i when the element for which this condition arises undergoes unit deformation e'.


Thus the condition (3) shows that the work done by the external forces applied to the basic structure on the displacements corresponding to the deformations e' of the element being considered does not exceed the work expended in the deformation e' of the element. I t is indeed possible to consider the coefficient ~ as an influence coefficient.

Using this interpretation, the conditions (3) can be written immediately without resource to equation (1) or condition (2). Each possible deformation of each of the elements of the system corresponds to a new form (3) and the problem can be solved by a consideration of the union of these conditions.

The basic structure as a whole can now be considered as an element. To each of the possible deformations of the element there corresponds a condition of the form (3). The union of these conditions determines the relation between the forces Pr, X1, X2, ..., X , acting on the basic structure.

In an ( n - 1)-dimensional space with co-ordinates X1, Xe, ..., Xn, PI, the union of conditions (3) determines a convex body B. To the internal points of the body correspond the states of the system for which the yield condition for the single element is not met. Each point A of the boundary of the convex body B lies on at least one of the n-dimensional manifolds determined by the condition (3). Therefore, for the state of the system represented by the point A of the boundary, there exists an element or elements the stressed state of which allows plastic deformation of the type defined.

If, however, point A corresponds to P~ < PI ...... then deformations of the systems, as has been shown already, cannot result from the consideration of the individual plastic deformations.

I f for conciseness of language it is agreed that the positive direction of the axis P~ is directed upwards, then the determination of the collapse load is reduced to finding the segment of the upper supporting plane of the convex body (3) which cuts the axis ~ .

6. DETERMINATION OF THE COLLAPSE LOAD PIInfin FOR A REINFORCED CONCRETE PLATE

We will consider as an example an annular reinforced concrete plate,

supported along the inside and outside edges, with radially symmetric reinforce-

ment above and below and loaded with a radially symmetric vertical load [Fig. 6(a)].

I t is easy to explain the manner of its collapse. For possible plastic deformations of the elements, as understood above, one can take opening of cracks above and below, accompanied by rotation of the parts of the elements with respect to the horizontal axes, at the level of the upper and lower reinforcement. Owing to the axial symmetry of the plate and the loading, the cracks must be circumferential and radial. I t is easy to convince oneself that collapse without circumferential cracks is impossible. However, a circumferential crack must be accompanied by radial cracks extending from the circumferential crack to the external boundary of the plate. The cracks which have been enumerated are formed on one (the lower) side of the plate [Fig. 6(b)]. I f there were no other cracks, then the plate would rise at the external edge

334 A .A . GvozDEv

and t h a t is clearly impossible. Therefore radial cracks also appea r on the uppe r surface f rom the in ternal to the external edge of the pla te [Fig. 6(c)]. Only the combina t ion of these two deformat ions satisfies the condit ions of the specification, and for this it is not difficult to a r range a re la t ionship be tween these two modes of de format ion [Fig. 6(d)]. The collapse scheme is set t led wi th precision to wi th in the p a r a m e t e r de te rmin ing the posi t ion of the circumferential crack. After this it is easy to fo rm the work equa t ion

+ =

I

~NzA I i I

!

i '--~

FIG. 6.

F r o m this equa t ion the value of Pn is de te rmined as a funct ion of the posi t ion of the c i rcumferent ia l crack. The usual m i n i m u m prob lem remains to be solved.

A c i rcumstance mer i t ing ~ t ten t ion is t h a t bo th the uppe r and lower circumferential re inforcement mus t yield f rom the c i rcumferent ia l crack to the externa l edge. Thus the outer pa r t of the p la te is subjec t to extens ion in the c i rcumferent ia l direction. Hence the in ternal p a r t of an annu la r p la te behaves like a c lamped s t ruc tu re up to the m o m e n t of collapse. This resul t is pe rhaps somewha t unexpec ted and is, i t seems to us, of considerable interest f rom the point of view of developing sui table designs for re inforcement .

R E F E R E N C E S

1. A. Iv. ISHLINSKII, Mechanics in the U.S.S.R. during the Thirty Years 1917-1947, p. 244. GITTL, Moscow (1950).

2. V. V. SOKOLOVSKY, Theory of Plasticity. Akademiia Nauk SSSR, Moscow (1946). 3. L. S. LEIBE~SON, Elements of the ~V1at]~ematical Theory of Plasticity. Moscow (1943).

(Translation: Note RMB19 of the Graduate Division of Applied Mathematics, Brown University, 1947.)

4. L. M. K n c g ~ o v , Prikl. ~lat. ~)iekh. 6, 187 (1942). (Translation: Note RMB7 of the Applied Mathematics Group, Brown University, 1946.)

5. A. A. MARKOV, Prikl. Mat. Mekh. 11, 339 (1947). (Translation: Note All-T3 of the Graduate Division of Applied Mathematics, Brown University, 1948.)

6. N. I. BEZUKIIOV, The Theory of Elasticity and Plasticity. GITTL, Moscow (1953). See also his article in: Structural Mechanics in the U.S.S.R., 1917-1957 (edited by I. M. RABINOVICH) p. 233. Stroiizdat, Moscow (1957).

Sta t i ca l ly i n d e t e r m i n a t e sys t ems unde rgo ing p las t ic d e f o r m a t i o n 335

7. A. R. RZgANrTSY~, Calculations qf Structures taking account of the Plastic Properties of Materials (2nd Ed.) . S t ro i izdat , Moscow (1954).

8. W. PRAGER, Sec t ional Addres s a t t he Eighth International Congress of Theoretical and Applied Mechanics, Istanbul. 1952 (1955).

9. P. G. HODGE, The Plastic Analysis of Structures. McGraw-Hi l l , New Y o r k (1959). 10. D. C. DRUCKER, A .~lore Fundamental Approach to Plastic Stress-Strain Relatio~ts.

A.S.M.E. (1952). P roceed ings of t he F i r s t U.S. N a t i o n a l Congress of App l i ed Mechanics , 1951, p. 487.

] 1. A. A. (]VOZDEV, Proekt i stand. (1934).

A D D I T I O N A L B I B L I O G R A P H Y

(A). Proceedings of the Conference on Plastic Deformations, December, 1936 (edi ted b y B, G. GALERKI~). A k a d e m i i a N a u k SSSR, M o s c o w - L e n i n g r a d (1938). C o n t e n t s :

A. A. IL'IUSHI~', On the Question of Visco-Plastic Flow of Material. A. A. GVOZDEV. This paper . S. A. BEa~SHTEI~, The Computation of Statically Indeterminate Beams in the Plastic Range. B. N. GORBU~OV, On the Questio~t of the Factor of Safety for Computations by the Theory of Plastic Deformation. S. A. BERNSH~EIN a n d V. S. TURK~, Experimental-Theoretical Investigations of the Elasto-Plastic Action of Continuous Steel Beams. K. S. ZAVRIEV, The Computation of Metal Structures for Critical Conditions, taking account of Plasticity. A. G. NAZAROV, The Application of the Concept of an "Ideal Profile" to the Analysis of the Carrying Capacity of Statically Indeterminate Systems.* N. M. BEHAEV, Theories of Plastic Deformation.~

(B). A. A. GVOZDEV, Stroiternaia promyshlennost' Vol. 10, No. 1 (1932); Vol. 11, No. 1 (1933); Vol. 12, Nos. 5, 6 (1934); Vol. 17, No. 3 (1939); Vol. 21, Nos. 1, 2 (1943); Vest. V I A No. 30 (1940); Izv. Akacl. Naulc SSSR, O T N 9, 19 (1943); 4 (1953); Prestressed Concrete Structures (edi ted b y V. E. ]:~ATT8). Stro i izda t , Moscow (1947); Inzh. sborn. 5 (1948); 5 (1949); Computation of the Carrying Capacity of Structures by the Method of Limiting Equilibrium. Stro i izda t , Moscow (1949); Questions of Con- temporary Reinforced Concrete Construction (edi ted b y A. A. GVOZDEV). S t ro i izda t , Moscow (1952). A. A. GVOZDEV a n d M. S. BORISHANSKII, Proekt i stand. No. 6 (1934). A. A. GVOZDEV a n d V. I . MURASHEV, Directions for the Computation of Reinforced Concrete Bents and Frames. Stro i izda t , Moscow (1932). A. A. GVOZDEV, V. I . MURASHEV, V. N. GORNOV a n d V. Z. VLASOV, Directions for the Design and Computation of ,~lonolitt~ic Thin-walled Roofs and Floors. T s N I P S , Moscow (1937).

* Nazarov is referring to an ideal sandwich section. He uses a piecewise linear m o m e n t - curvature relation.

f Beliaev's paper also appeared in Izv. Akad. Naulc SSSR, OTN 1 (1937), and a translation of the lat ter appeared as Note RMB-14 of the Applied Mathematics Group, Brown University, 1946.

Documents

1960Artigo_Gvozdev_The Determination of the Value of the Collapse Load for Statically Indeterminate Systems Undergoing Plastic Deformation