Plastic beams at finite deflection under transverse load with variable end-constraints

0022-5096/81/050447-30 $02.00/O 0 1981. Pergamon Press Ltd.

J. Mech. Phys. Solids Vol. 29, No. 516, pp. 447-476, 1981. Printed in Great Britain.

PLASTIC BEAMS AT FINITE DEFLECTION UNDER TRANSVERSE LOAD WITH VARIABLE END-CONSTRAINTS AYFER G~~RK~K Department of Engineering Sciences, Middle East Technical University, Ankara, Turkey

and H. G. HOPKINS Department of Mathematics, University of Manchester Institute of Science and Technology, Manchester, M60 lQD, U.K.

(Received

8 January

1981)

ABSTRACT THIS paper is concerned with the influence of finite deflections on the behaviour of rigid perfectly-plastic beams with support conditions specified by an axial restraint factor and a rotational constraint factor. Attention is restricted to quasi-static deformations and to uniformly-distributed vertical loads. First, a complete solution (that is, within the limitation of moderate deflections and small strains) is presented for axially-restrained, rectangular beams with a certain degree of rotational end-fixity constraint. Then, the restriction of complete axial restraint is removed and a separate analysis is given in order to treat the effect of horizontal displacements which may occur at the supports. For an axially-restrained beam, it is found that, even at central transverse deflections of the order of beam thickness, the load-carrying capacity is considerably increased and also the membrane state is reached. It is also shown that, at moderate deflections, the magnitude of this increase is strongly dependent upon the degree of axial restraint provided at the supports.

1.

INTRODUCTION

DURING the past three decades, the behaviour of structures at finite deformations has been a subject of considerable interest since it is highly pertinent to engineering structural design. Theoretical and experimental studies so far reported indicate that the response of structures at loads greater than the limit load is strongly dependent on the geometry changes which may affect the equilibrium equations and hence also the load carried by the deforming structure. Consequently, when the influence of finite deformations is retained in the analysis, plastic flow progresses under increasing, constant or decreasing loads according to the circumstances. It is to be noted that neutral geometry changes, which justify the assumption of continued plastic flow under constant load, are exceptionally few (HILL, 1951). The compilation of a bibliography of published work in this field is not within the scope of this paper, but 447

448

AYFER GCIRKGKand

H. G. HOPKINS

the reader may note that a relatively comprehensive review is given in HODGE (1971), JONES (1972), GURK~K and HOPKINS (1973), and also in the textbook by MASSONNET and SAVE (1972). In the analysis of beams according to the theory of rigid perfectly-plastic solids, it is assumed that the beam remains completely undeformed below a certain level of the applied load. Once this critical load is reached, plastic hinges develop at yielding sections and the beam is said to become a rigid-plastic collapse mechanism. Although it is assumed that the beam can undergo indefinitely large deflections under that load, only infinitesimal departures from the undeformed state are considered and the equilibrium equations and geometrical relations are referred to the initial configuration of the beam. Therefore, the analysis of the post-yield response of beams within the framework of the simple plastic theory excludes the effects of both finite deformations and strain-hardening. Since the latter always strengthens the beam, its neglect leads to a certain reserve factor of safety. Further, HAYTHORNTHWAITE(1957a) has reported that the effect of strain-hardening may be disregarded unless the beam has a small span-to-thickness ratio. However, it is well known from the theoretical and experimental studies of the behaviour of axially-restrained beams that the deflections taking place subsequent to the formation of a rigid-plastic collapse mechanism have a stabilizing effect on the strength and thus the corresponding loaddeflection curves show a rising character (see HAYTHORNTHWAITE,1957a, 1961; CAMPBELLand CHARLTON, 1973 ; GILL, 1973 ; GURKOK and HOPKINS. 1973 and PANG and MILLAR. 1978). This is due to the favourable influence of induced axial forces. On the other hand, WASZCZYSZYN(1966), JONES (1973) and HODGE (1974) have shown that, if the support conditions permit horizontal displacements at the ends, then the limit load gives a quite reasonable estimate of the carrying capacity. Therefore, if the beam is restrained against axial support displacements, as would happen in an interior span of a continuous beam, and if the transverse deflections are other than infinitesimal, then the effect of geometry changes (i.e. axial forces) must be considered in the analysis. The object of this paper is twofold : first, to study the influence of finite deflections on the behaviour of axially-restrained, rectangular beams with various degrees of rotational freedom, and second, to analyse the same problem without the restriction of complete axial restraint in order to treat the effect of horizontal support displacements in the presence of finite deflections. The solution presented in the second part, therefore, describes the post-yield response of a rigid perfectly-plastic, rectangular beam with support conditions specified by an axial restraint factor and a rotational constraint factor, and it reduces to that obtained in the first part for the case of complete axial restraint. Attention is restricted to quasi-static deformations and to uniformly-distributed vertical loads. ONAT’S (1960) method of approach is followed, which requires the formulation of the proper rate problem. Furthermore, a small strain, moderate deflection assumption is used, effecting a considerable simplification in the analysis without a great loss in accuracy (see GURKUK and HOPKINS, 1973). The bending moment and axial force fields and deflections, up to the limiting transition to a wholly membrane (string) state, are traced through two consecutive phases of deformation, which show distinct features. A previous paper by the authors (GURKOK and HOPKINS, 1973) presented the

Plastic beams at finite deflection

449

fundamental equations for an exact theoretical analysis of the post-yield response of rectangular beams, and developed a complete solution (that is, within the limitations of small strain, moderate transverse deflection) for a simply-supported but axiallyrestrained beam (which may be called a pinned beam). Furthermore, the validity of small strain, moderate deflection assumptions was verified by attention to the correct condition of large strain for the primary phase of deformation. This analysis indicated that, at a central transverse deflection of magnitude equal to the beam thickness, the load-carrying capacity was increased fourfold and the beam became a plastic membrane or string of parabolic shape with no bending resistance. The solution presented in the first part of the present paper extends the previous analysis to a beam with similar geometry and loading except that the support conditions are now specified by a rotational constraint factor. It, therefore, includes the solutions for the particular cases of a pinned beam and a built-in beam and also a beam with support conditions between these two extreme cases. To the writers’ knowledge, studies in this field have so far been limited to obtaining solutions under the “purely vertical displacement” assumption. The case of a rigid perfectly-plastic, fully-constrained, rectangular beam carrying a centrally located concentrated load has been analysed by HAYTHORNTHWAITE (1957a), who also reported some experimental data for mild steel beams. His results indicate that the increase in the theoretical load-carrying capacity is proportional to the square of the central deflection until the bending moments vanish. HAYTHORNTHWAITE(1961) and GILL (1973) have studied the case of a built-in beam which is subjected to a uniformlydistributed transverse load. In their analyses, it is assumed that the beam material is rigid perfectly-plastic and that all material points on the beam centre-line move vertically downwards. HAYTHORNTHWAITE(1961) has considered ideal-sandwich and rectangular beams. His deformation modes for the former are similar to those described in the present work (and also by GILL, 1973); for the latter, however, only a single mode was considered, and it was assumed that as soon as the load was increased beyond the limit load, the central plastic zone started to spread outwards toward the supports. He also gave approximate solutions which were based either on the incipient velocity field or on the membrane velocity field. On the other hand, GILL (1973) has studied built-in and pinned beams of rectangular cross-section. Although his deformation modes are exactly the same as those described in the present work (and also in GURKOK and HOPKINS, 1973), his results for the built-in beam are almost identical with those of HAYTHORNTHWAITE(1961) who considered only one mode throughout the analysis. It must be emphasized that it is virtually impossible to satisfy precisely the zero displacement condition at the supports either in engineering practice or in the laboratory. It has been shown experimentally by WASZCZYSZYN(1966) that the rise in the carrying capacity is strongly dependent upon the degree of axial restraint provided at the supports. JONES (1973) has studied the effect of in-plane displacements at the boundaries on the behaviour of various laterally loaded, rigid perfectly-plastic beams and a square plate. He assumed that these end displacements were proportional to the square of the maximum transverse deflection. Later, HODGE (1974) presented an analysis of a simply-supported, rigid perfectly-plastic beam which was subjected to a concentrated load at mid-span. In this work, it was assumed that 23

450

AYFER G~RK~K

and H. G. HOPKINS

the axial support movement was proportional to the induced axial force. Although both these workers did not consider the possibility of travelling discontinuities with increase in deflection, their solutions clearly indicate the significant effect of support movements on the post-yield response of beams at finite deflections, In the second part of the present work, a complete solution is given in order to discuss the influence of such displacements which obviously prevent or delay the development of axial forces.

_. 7

STATEMENT OF THE PROBLEM

WITH SOME PR~LIMINARY~ONSIDERATI~~S

Consider a beam of span 2L and depth h, which is axially-restrained so that the distance between the supports remains constant. Assume that the beam is made of rigid perfectly-plastic material and subjected to a uniformly-distributed vertical load li. We shall be concerned with the quasi-static deflections of the beam in the presence of geometry changes. as this load increases beyond the limit load k,. Note that it is almost impossible in practice to provide complete rotational constraint at the supports of a built-in beam. If the supports lack rigidity, a certain amount of rotation occurs and consequently the plastic moment in the end sections cannot be achieved. In other words, the necessary rotations of the end sections at the yield-point state are provided by imperfections rather than the formation of plastic hinges in the beam itself. From another point of view, since actual structures are made of elastic--plastic material, the end constraints may undergo elastic deformations before the yield-point state is reached. ~eformabiiity of end connections may lead to the development of axial forces which obviously reduce the bending moment capacity of these sections and hence the load-carrying capacity at zero-defection (HAYTHORNTHWAII’E, 1957b). Since the present analysis is concerned with rigid perfectly-plastic beams, the axial force at zero-deflection is assumed to be zero, and the effect of elastic deformations prior to the development of the collapse mechanism is not treated. However, if M = -nMo (where 0 < n < 1) denotes the clamping moment at the end sections of an axially-restrained beam at the yield-point state, then the rotations which may occur at these sections due to elastic deformations and/or imperfections may be included in the analysis through the rigidity coefficient n. It is obvious that the value fi = 0 corresponds to complete rotational freedom at the supports (pinned beam), II = 1 to complete rotational constraint (built-in beam), while values ofrr which lie in the range 0 < PI< 1 represent intermediate degrees of rotational constraint. If II # 1, as will be discussed later. the rigid-plastic collapse mechanism involves only one plastic hinge which acts at the centre of the beam. The complete statement of the problem requires, first of all, the derivation of the equilibrium equations, yield condition and the strain-displacement relations in terms of the generalized variables. The next step is the determination of the yield-point state. Then the analysis of finite deflections requires the formulation of the proper rate problem (ONAT, 1960) which involves the rate equations, discontinuity relations and, finally, the proper boundary conditions. In 1960, ONAT formulated the proper rate problem together with the associated

Plastic

beams at finite deflection

451

criteria of uniqueness and stability for the case of plane curved beams (arches) made of rigid work-hardening material. In this formulation, the influence of axial forces on the mechanical behaviour of the beam was neglected, and consequently the extension of the centre-line was taken as zero. Later, BATTERMAN(1967) included additional terms in the equations in order to take into account the influence of axial forces. In the subsequent analysis, the Euler--Bernoulli hypothesis is assumed to hold: that is, plane material cross-sections originally perpendicular to the centre-line remain plane and perpendicular to it after deformation, and hence shear strains do not affect deformation. As a result of this last assumption, the shear force S, although necessarily retained in the equilibrium equations, does not influence the yielding of the material. Note that this assumption has been shown to be quite reasonable for beams with slenderness ratios which lie within the range of applicability of the beam theory (see, for example, HODGE, 1957). In the present analysis, a Lagrangian description will be followed: all dependent variables are regarded as functions of a Lagrangian material coordinate sO (arc length measured along the initial beam’s centre-line to a generic cross-section) and the time I. A fixed material cross-section s,, occupies at time t the geometrical position s where s is now arc length measured along the deformed beam’s centre-line at time t. As the beam deforms over a time at, plastic work Y&-p(s,, t) 6s, St is done on the beam material element 6s,, i.e. the rate of plastic work per unit length of the initial beam’s centre-line is i”p(sO, t). In the present viewpoint the only internal generalized forces that do work are the bending moment M and the axial force F. The shear force S is a (generalized) reaction (see PRAGER, 1953, 1955) because it corresponds to a strain-rate which has been taken to vanish. Now, we may write YieQ= Mk, + F& where M = M(s,, t), F = F(s,, t) and the generalized strain-rates kN, & are defined formally by k, = i?ti”p/3M, d, = (7f?‘P/?F. In fact, as will be proved in the Appendix, aN is the engineering strain-rate referred to the initial beam’s centre-line but ~~ = ?$/Cs, is the change of spin 4 with respect to the length measured along the undeformed beam’s centre-line and not the time rate-of-change of curvature of the deformed beam’s centre-line. The fundamental equations required for an exact analysis of the post-yield response of initially-curved and initially-straight, rectangular beams were given in GURKOK and HOPKINS (1973) (for the cases of large and small strains). Since the analysis here is concerned with initially-straight beams, the appropriate equations are rewritten below for ease of reference and then supplemented by the proper boundary conditions so that the effect of axial and/or rotational support movements can be included. Since symmetry is being assumed in both geometry and loading, attention is confined henceforth to the right-hand half of the beam. Further, the centre of the undeformed (straight) beam’s centre-line is taken as the coordinate origin with the I’axis directed vertically downwards. Non-dimensional

notation

All subsequent equations and the analysis are simplified through the use of the following non-dimensional variables which also put them in a form more appropriate

AYFERGORKGKand H. G. HOPKINS

452 for numerical

computations

:

M/M,,

R=

F = F/F,,

$= IS/MO,

so = s,/L,

s = s/L,

x0 = x,/L,

i-i = U/L,

Q = V/L,

a, = UJL,

rl = Lit.

[I=

3.

L2k/Mo,

f2.1)

u, = zr,/L,

ci = MJLF,.

FUNDAMENTAL EQUATIONS

3.1.I ~~ui~~~~~~~~~~f~~~~~s. In figure 1, a beam element is shown in its initial and positions with the stress resultants M, F and S acting over a cross-section

current

corresponding

to a generic

particle

:P with displacement

components

U(s,,

r) and

x

FIGURE 1. Geometry and equilibrium for an initially-straight beam. V(s,,

t). The positive senses of all quantities

are also indicated

in the figure. Note that

the load k (or the corresponding load factor 8) denotes the vertical load uniformlydistributed with respect to tength measured aIong the current centre-line of the beam. It may be shown that summing normal

directions

the internal

to the deformed

element

and external

forces in the tangential

and

PQ gives

and while moment

equilibrium

gives n;i,,+S=

0

provided that the inertia terms are disregarded and are used. In (3.1), j?, and Pn are the tangential and dimensional load j%.Here, and elsewhere, subscripted used to indicate partial differentiation with respect

(3.2) non~dimensional variables (2.1) normal components of the noncommas preceding variables are to these variables.

Plastic beams at finite deflection

Equations (3.1) and (3.2) may be referred beam through the use of the relation

to the undeformed

ds = (1 ++)d-YO which is valid for initially-straight

beams.

c&,

F, r, -

= ‘1dx,

453

configuration

of the

(3.3)

Thus,

x0+ UrlP, = 0,

LX4to + Q. jL,+ W/B” = 0, A, ,tO+ $

(3.4)

= 0. 1

The previous analysis by the authors (GURKC)K and HOPKINS, 1973) has shown that, for initially-straight beams, it is more convenient to use the vertical and horizontal equilibrium equations and also the corresponding displacement (or velocity) components. It is straightforward to show that these equations are

where sin 4 = - p,&,

cos$

= (1+~.,,)/rl,

and the shear force s has been eliminated by the use of the moment equilibrium equation (3.4),. The corresponding rate equations may easily be obtained by differentiating (3.4) and (3.5) with respect to time or a suitable time-like parameter. Since the analyses here will be restricted to conditions of moderate deflection and small strain, these rate equations are not given. 3.1.2 Struin-displacement load will obviously set up be associated strain- and For an initially-straight related to the horizontal manner :

relations.

The movement of the beam under the applied axial strain and curvature of the centre-line and there will curvature-rates. beam, the axial strain Ed and the curvature K = +., are and vertical displacement components in the following

EN= {(1+r7,,0)~+~~~0~~-l, f? = 4.s=

PJ,\-,x,I/x,-9~&(1

(3.6) +&JlV3.

(3.7)

It is now straightforward to obtain the corresponding strain-rate velocity relations from (3.6) and (3.7), but, for brevity, these results are not given here. Note that the generalized strain-rate corresponding to the bending moment M = M(s,, t) is R, = $,.*, and may be obtained from (3.7) with use of the relation (3.3). 3.1.3 effect force figure

Yield condition andpow rule. Since it is assumed that the shear force 3 has no on yielding of the material, the yield curve for the bending moment A? and axial F of a rectangular cross-section comprises two parabolic arcs, as shown in 2, with equations

+&?+F2

= 1.

(3.8)

AYFER G~RK~K and H. G. HOPKINS

454

FIGURE 2. Yield curve

for a beam with rectangular

cross-section.

In any region (section) of the beam where the yield condition is satisfied then plastic deformation may occur, and the stress/strain-rate relations for such regions (sections) are determined by the fundamental flow rule (or normality rule). Since all quantities are regarded as functions of the Lagrangian variable .YOand the time t, then the application of the flow rule (3.9) to the yield function

(3.8) gives (3.10)

In (3.9) and (3.10) & >, 0. It is obvious that the yield curve has discontinuities in slope at the points ti = 0, F = + 1, and hence the Aow rule represented by (3.9) needs modification at these singular points, in the usual way.

Yield-point

state

It may be found by the methods of limit analysis that, for an axially-restrained beam under consideration (see figure 3), the limit-load factor is & = 2(1 +n),

(3.11)

where II specifies the degree of rotational constraint provided at the supports and lies in the range 0 < n < 1. At p = /I,,, the bending moment M at .Co = 0 and X, = 1 reaches the values + 1 and -n, respectively, and the plastic hinges develop at plastically yielding sections (that is, where lJi? = 1). This instant is chosen as the time origin, that is, t = 0. It is obvious that, if n # 1, the rigid-plastic collapse mechanism involves only one hinge which acts at X, = 0, and the state of stress at this hinge is represented by point

Plastic

455

beams at finite deflection

c I i

Bdx,=2Moa+2nAiB

2J:x,(I-xo)

&=koL2/Mo=2(I+n)

FIGURE3. Collapse mechanism.

A on the yield curve shown in figure 2. In the case of a built-in beam (n = 1) there is stress-state is represented by another hinge acting at X, = 1 and the corresponding point B. Since the yield condition is only satisfied at these hinges, incipient plastic deformation occurs there, while the remaining parts of the beam tend to rotate rigidly. The generalized stress resultants I%?and Fat t= 0 may be obtained by integration of the equilibrium equations F, 2, = 0,

Rx,\-,+P0

Thus, from (3.12) with use of the F(&,, 0) = 0 and (3.1 l), it is found that F=

0,

= 0.

conditions

,Gf = 1 -(l

(3.12)

&#(O, 0) = + 1, M(l, 0) = -n,

+n)X&

(3.13)

If the effect of finite deflections is included in the analysis, the beam’s response following the formation of a rigid-plastic collapse mechanism has been shown (GURKOK and HOPKINS, 1973) to be described by two successive phases of deformation up to the final state of a plastic membrane. The primary phase begins when the load factor @ increases beyond the limit value PO given by (3.11). For p > PO, it is conjectured tentatively that the plastic hinge at X, = 0 will split into two laterally moving boundaries leaving behind them a descending curved rigid region. In the case of a built-in beam, the two outer hinges present at the supports are assumed to remain stationary. In view of these assumptions, the right-hand side of the beam, in this phase, is composed of two rigidly moving regions (inner curved and outer straight regions) separated by a plastically deforming section (see figure 4a) where the bending moment assumes its maximum value and the discontinuities in the velocity components and curvature are concentrated. This type of behaviour, which involves the occurrence of finite deformations at a plastically yielding section arises from the idealized nature of the material (that is, rigid perfectly-plastic material), and of course it is locally unrealistic. In reality, plastic deformations occur over plastically yielding “zones” rather than at isolated cross-sections. Here, the length of such zones 1, say, is assumed to be such that l/L@ 1. Certainly, I # 0 and we have in mind only situations l/L6 1 so that such local effects are discounted completely here.

456

AYFER G~RK~K and H. G. HOPKINS

Region

FEURE 4. Modes of deformation

I

(b)

: (a) primary phase, and (b) secondary phase.

Now, let &, = r(r) denote the distance to the right-hand boundary. Since F is a monotonically increasing function of t, it will be taken as the parametric time variable, and superposed dots used in the rate equations henceforth will denote differentiation with respect to r. The primary phase terminates. as will be shown in the course of the analysis, when the right-hand moving boundary has moved out to the cross-section So = c = i/3. On further loading, the secondary phase begins in which the two boundaries originating at .Yo= l/3 move in opposite directions, the one outwards toward the support (as before) and the other inwards toward the midspan. Now the plastic deformation completely extends over the entire region between these two boundaries. If the new boundary is denoted by 4 = c(g) and the original one (as before) by r, then the right-hand half of the beam in the secondary phase comprises two rigid segments joined by a plastically deforming segment whose length increases as the load increases (see figure 4b). Note that, if II = 1, then there is a stationary, extensible hinge acting at & = 1 (in addition to the regions considered above). The two phases of deformation will be reconsidered in detail in the course of the analysis. 3.1.4 ~~.~~~~?~~i~z~~f~ ~e~~~i#~s.In view of the previous discussion, the beam, in each phase of deformation, is divided into different regions whose number and position vary with time. The various mechanical quantities must satisfy certain continuity or discontinuity relations at the common boundary between any two such regions. For the primary phase, there are five relations (i)-.-(v) as follows. (i) equilibrium

requires that [A?] = [F] = [S] = 0

at X, = r.

(3.14)

In (3.14), and elsewhere, [ .] denotes the jump in the quantity enclosed in brackets.

Plastic beams at finite deflection

457

(ii) Since &? is continuous and assumes an extreme value at the boundary,

[i&=0

at .gO= Z.

then (3.15)

(iii) The cohesion of the beam material requires that [0] = [V] = 0 Across a moving boundary,

8,,,

at X0 = &

G, yn, and fi are discontinuous [c’.J+@] [&,]+[PJ

(3.16) but such that

= 0,

(3.17)

= 0.

(3.18)

(iv) Although the slope # is continuous, (I, (spin) may be discontinuous, So, the equilibrium rate equations on the two sides of the moving boundary should be linked by $I - +a#2 -t4Clll = 0, a[3 -I-F[& 4a/.I,[n] = 0. 1

(3.19)

(v) Since the strain-rates identically vanish on both sides of the boundary T0 = g, a continuity relation between them cannot be derived. However, a relation between the quantities ye and 4.z0 may easily be derived from geometrical considerations. Thus,

Clll= 2~~C#,.i;,lt

(3.20)

which, with use of the relation [$, /] + [$J = 0, may be rewritten as [n] + ZcG[fj] = 0.

(3.21)

The discontinuity relations given for the boundary &-,= rmust be satisfied across the two boundaries i, = (r, r) in the secondary phase, but such that, for any field quantity Q(&,, {) which is continuous across the boundary X,, = c, there holds [Q, &J f+ [& = 0.

(3.22)

3.1.5 Boundary conditions. For the axially-restrained beam under consideration, boundary conditions to be satisfied by fi, F, Y and IJ are P(1, f) = 0, -fi(l,

r)+nF*(l,

the

(3.23) c) = n,

(3.24)

@cE, f) + F’(E 0 = 1,

(3.25)

A, i&O, F) = 0,

(3.26)

is(0, %)= 0.

(3.27)

and, by symmetry,

Although the beam is completely fixed against axia1 movement, there is some amount of axial extension at the centre-line of the beam at X0 = 1. This may be best discussed as follows. In case of a built-in beam (where n = l), there is a stationary but extensible plastic hinge at X0 = 1 and this hinge produces a discrete extension & say, 24

AYFER GCRK~K and H. G. HOPKINS

458

at the half-depth of the beam there. As the load increases. the neutral axis at X, = 1 moves downwards, and since the fibres above the neutral axis are at the tensile yield limit, they stretch a certain amount, whereas the fibres below the neutral axis yield in compression and they are shortened. Thus, from figure 5, it follows that U(1. g, = -A, (3.28) -F. where Li = A(<) is the finite stretch at the half-depth and will be determined later in the course of the analysis. Note that this type of behaviour, as already discussed, arises from the idealized nature of the material.

X -Neutral

OXIS

i=t

FIGURE5. Condition at the support of an axially-restrained beam at some time t during deformation.

In order to take into account the effect of rotational imperfections, the support condition to be satisfied by the horizontal displacement field i?’ may be generalized as U(1. C) = -n& where 0 < M < 1. Note that, in beams with 0 < n < 1, the necessary rotation end sections and the resulting extensions may be provided by imperfections. 3.2 Small

(3.29) of the

struin ,formulution

So far the governing equations have been given without imposing any restriction on the magnitude of the displacements and strains. It has been shown by GURKOK and HOPKINS (1973) that the mathematical problem in the large strain solution is highly non-linear and complex. Thus, certain simplifying assumptions are required even for the primary phase of deformation. If the attention is restricted to moderate deflections and small strains, the righthand side of (3.6) may be expanded as a binomial series, under the assumptions that U,\O << 1

(3.30)

and (3.31 Then, (3.32 or (3.33

459

Plastic beams at finite deflection

If it is assumed that h/2L 6 1, I/ = O(h) and that IO/P1 6 1, then the series (3.32) may be truncated after the second term without introducing significant error. Next, in view of these assumptions, it follows from (3.33) that dS z d&. Furthermore, the terms D. 40x,p i0 and vi,, 0, _f0in (3.7) may be neglected in comparison with i;:x,x,. Hence, from (3.7) (3.34) The strain-rate equations now follow: 8, = i? = ~._~o+q,op,,

(3.35)

k, = i = - p p,x,’

(3.36)

The equations of horizontal and vertical equilibrium valid for small strains are F, X0= 0

and

(3.37)

FP _&J,+ crA, x0*G+ ap = 0, respectively, which on differentiation with respect to time or a time-like variable (e.g., r) give 8,

co

=

0,

(3.38) Fq ‘@,?, + FF i0x0+ aA, .fO,fO + lxj = 0. It must be emphasized that, within the small strain approximation, the load k becomes the uniformly-distributed load per unit length of the initial centre-line which is here taken as the x-axis. Thus, the distributed load is uniform per unit horizontal length. Next, the continuity or discontinuity relations (it(v) of Section 3.14 need to be reconsidered as follows (i) The relations (3.14), (3.15) and (3.16) remain valid. (ii) It follows from (3.34) that P has continuous first derivatives, i.e. [Pa,] = [VJ = 0

at 1, = (5,4).

(3.39)

U, x0 and 6 must still satisfy (3.17). (iii) The equilibrium rate equations (3.38) for both sides of the moving boundary are linked as follows .[A, i,] -&A]

= 0.

(iv) From (3.35), (3.36) and the flow rule (3.10) it may be found that -L [G] = -2aF[Q-J,

(3.40)

(3.41)

which is analogous to (3.20) or (3.21) applicable for large strains. 4.

ANALYSIS

In this section, a complete solution is given in order to discuss the influence of finite deflections on the post-yield response of axially-restrained, rectangular beams

460

AYFER G~~RK~K and H. G. HOPKINS

with a certain degree of rotational constraint. As already considered, the rotational rigidity of the support constraints is specified by a factor n which lies in the range Obndl. The limit load factor and the generalized stresses at c = 0 are given by (3.11) and (3.13) respectively. Since the velocity field which initiates the plastic deformation must satisfy the flow rule, it is obtained by integration of (3.36): t = 6°C1 - .Y()),

(4.1)

where v0 (> 0) is the mid-span deflection and the support condition (3.23) has been satisfied. It is obvious that the solution may be found only by assigning arbitrary values to VO,and hence, if the ensuing finite deflections are ignored, then the beam can undergo indefinitely large deflections according to the arbitrary virtual mode found. If, however, the effect of finite transverse deflections is to be taken into account, then it is necessary to solve a rate problem which involves stress-rates, velocities, and loadrates (ONAT, 1960). If the exact solution to the initial rate problem can be found uniquely, then the effect of subsequent deflections on the load-carrying capacity of the beam may be estimated. It is clear from (3.38) that p = 0 at t = 0 (i.e., at r = 0). Since the material shows no strain-rate sensitivity, unp monotonic increasing quantity associated with the progressive deformation of the beam can be used as a time. The mid-span deflection V, could be used as time (i.e., time-like parameter), so b = dfi/dPO = 0; that is, the initial slope of the load-central deflection curve is zero. Since the solution to the initial rate problem does not show whether the subsequent geometry changes are favourable or not, then it is necessary to obtain the solution to the rate problem at a later instant of time. HAYTHORNTHWAITE (1957a) has employed the incipient displacement field in order to study the post-yield response of axially- and rotationally-constrained beams in the presence of finite deflections. Similarly, JONES (1972) has analysed the influence of axial support movements on the behaviour of simply-supported and built-in beams under the assumption that the incipient displacement field remains valid for moderate the possibility of travelling hinges lateral deflections. In their analyses, (discontinuities) with increase in load is disregarded but instead it is assumed that there is a stationary yield hinge at the centre of the beam. In the present analysis, following the usual heuristic pattern of development, certain modes of deformation are assumed and then justified N posteriori when the solution is found.

4.1 Prinzary

pkuse

The primary phase of deformation begins when the load factor 11increases beyond the value PO = 2(1+ n) where the values of n range from zero for complete rotational freedom to one for full rotational constraint. For 11> 2( 1 +n), the axial restraints at the supports induce axial forces which act to reduce the value of the plastic moment at yielding cross-sections. It is then assumed that the central plastic hinge will split into two boundaries which move outwards towards the supports, so that the maximum moment now occurs at a location other than ,CO= 0. The mode of deformation is

plastic beams at finite deflection

461

shown in figure 4(a). In this phase, the right-hand half of the beam is now composed of two rigid regions separated by a plastically deforming section X0 = r. Region

I. In this region, the beam segment remains rigid and straight but it rotates about the support making an angle 4 with the horizontal. The state of stress at X0 = ris represented by the point P, on the upper parabolic part of the yield curve shown in figure 6, and as the load increases, the point P, moves toward the singular point D (where F = 1 and M = 0) due to the induced axial force. For a beam with rotational constraint factor n, the state of stress at X0 = 1 is represented by the point Pk on the lower parabolic part of the curve (see figure 6) represented by the equation -M+nnF2=n,

O
1.

FIGURE6. Curves representing equation (3.24).

It is clear from figure 6 that, if n = 0, then fi = 0 at X, = 1 and the point PR is on the positive F-axis and moves from the origin towards the singular point D as the deformation proceeds. If n = 1, then the point representing the stress-state at the stationary hinge at X0 = 1 is on the lower parabolic-- part of the yield curve. Note that this hinge causes some amount of extension A = A(4) of the beam’s centre-line there. Since the yield condition is not satisfied within the range e < X0 ,< 1 (the equality sign holds if n # 1) no deformation occurs. Consequently, the elements of the beam segment maintain their original length and have zero curvature, and use of the rate equations is unnecessary. It must be emphasized that, within the moderate deflection approximation, the axial force is constant along the beam, and then the stress-points corresponding to cross-sections i < X0 < 1 must lie on a straight line joining the points P, and PR (see figure 6).

462

AYFER GCJRK~K and H. G. HOPKINS

Since E

=

0,

IT

=

0,

(4.2)

integration of (3.32) and (3.34) with use of the conditions C
- l),

0 = tC2(?)(l

(3.23) and (3.29) gives, for

-x0)-nA(t).

-(4.3)

where

[4(T)l

C(F) = - tan

(4.4)

and A(g) is the total stretch at the centre-line of the beam at X0 = 1 up to time {, and may be determined as follows. Under the assumption that plane sections remain plane and normal, it is found from figure 7 (with use of the non-dimensional variables) that the unit extension at the half-depth is E = 2&j, which on integration

(4.5)

gives the total stretch up to time r as e --A(<) = 2cr W’)$(F’) d?. s0

+--? FIGURE 7. Stress and strain distributions

The generalized (3.37) as F=

(4.6)

1-41 in a rectangular cross-section moment and axial force.

stress fields F and fi are obtained {l-P(l-e)2;+

subject to a combination

from the equilibrium

forO
iz;i = ~((1-~)2-(1+n)(~o-~)2)

for e<

where B = fl/flo, and the integration constants have conditions (3.23) (3.24), (3.25) and M, :,(t, 5) = 0.

of bending

equations

1, (4.7)

X0 < 1, 1

been

determined

from

the

Region II. As the deformation proceeds, a typical element on the beam centre-line experiences extension and curvature instantaneously as the boundary passes through it, but after that instant it merely undergoes rigid-body motion. It follows that the state of stress within the central curved region is represented by a point inside the yield curve. Thus, B = 0,

i = 0,

(4.8)

463

Piastic beams at finite deRe&on

and the beam segment ~ufincreasing length) descends rigidly tu follow the rotation of the outer segment. The slope of the beam centre-line at any point 3,, = R, (0 ,< x0 G r) after the boundary has passed through it is, therefore, equal to that of the rigidly rotating outer segment (F d SO < 1) at that earlier time g = x0 when the boundary passed through that point. In view of these remarks, the slope of the bean centre-line over this region may be determined from (4.4) as 4 = -arctan

(C(.%,)),

(4.9

0 d _%,d c.

Next, from (3.36), (4.X) and (4.9) it follows that t & = 0,

0 < s* < z,

(4.10)

which with use of (3.271, (3.35) and (4.8) leads to a=

0,

(4.11)

u,< z.
Now, from integration of (4.10) and (4.11), appropriate quantities at X0 = <, it is found that

remembering

the

continuity

of

(4.12)

l7 = ~C’&)(l

-.Q+nE;{x*),

I

where A(.?,) may be ubtained by substituting c= X0 in (4.6). The bending moment field is determined frum the integration of the vertical equilibrium equation (3.373,. Thus, from (3.37), with use of the appropriate continuity conditions at SO = z and equation (3,26), it is found that P rJr = @(l -Q:,‘+(l

+n)(~“-.u;)j

+ ; [l -F(l-Q’;”

C&,) d.& I :<

Finally,

(4.13)

from (3.41), (4.3), (4.6), (4.10) and (4.11), it may be shown that Fz

whence,

.qa 6 L$

- U%)( 1 - $$w

+ n 1,

(4.14)

with use of (4.7),, p= --

j4EZ(1 +n)~-C2(~)~l-~f2’1~~401~(1+n)~(1-~)~)21*

(4.15)

Now that /? = /3(C, C{r)) is found, all the unknowns can be expressed in terms of r, and also C(r) is determined from the vertical equilibrium equation (3.37), as follows. Since g zO= 0, M, *a = 0 at .fO = 0, equation (3.37), when integrated with respect to Jy(jgives

W&+lxM&+cl&

= 0,

Now, on substitution

of (4.13), (4.14), (4.15) and

into (4.16) it is found

by continuity

at X0 = F that

0 d x, < r.

(4.16)

464

AYFER G~~RK~Kand

H. G.

HOPKINS

Since &# assumes an extreme value at the boundary .i-, = g, the relation (4.17) can also be obtained from (4.13) with use of (4.14) and (4.15). It is now straightforward to write the displacement and stress fields (4.3), (4.7) and (4.12) to (4.15) in terms of the Lagrangian variable & and the time-like parameter <: i7 = 2n(l +r@(l

-.Vo)/(l -F),

r7 = 2n2(1 -i-n)C$(l -s&o

-5)”

(4.20) From (4.19)r and (4.20), it may easily be shown that [d~/d(~(O, &Wje=O = t4&‘(1 +n)(I -r”)&

= 0.

which justifies the earlier remark made about the initial slope of the load deflection curve. the deformation of the beam for Equations (4.18)-(4.20) describe 0 < f z$ r* = l/3. More specifically, the primary phase continues until the boundary moves to a cross-section at a distance of one-third of the semi-span from the centre. If (4.19), is di~erentiated with respect to Z,, it is seen that the bending moment assumes its extreme value at .i;-,= 0, lu, = c and z() = (1 -?/2)-t

$(4r- 3?2)j.

(4.21)

It can easily be shown that R = ti;r(O,c) is a minimum for all times. However, KJ = Il;i(r3 %) is a maximum if F < I/3 and a minimum if F > l/3. Furthermore, for c > i/3, the yield condition is violated at X, = r- (that is, just behind the discontinuity). Note that the two solutions (4.21) are irrelevant since they lie outside the range 0 G .CO-C l/3 for i < l/3.

For c > l/3, the secondary phase of deformation begins with the single boundary x0 = e present during the primary phase being joined by a new boundary that separates from it to travel inwards. The plastic deformation now completely extends

Plastic beams at finite deflection

445

over the region between these two boundaries. Therefore, for any value ofn, the righthand half of the beam comprises two rigid segments joined by a plastically deforming segment. If n = 1, there is a stationary extensible hinge acting at the end section .+) = 1. If I, = c denotes the distance to the boundary moving towards the right-hand support, and if Z0 = r the distance to the new one moving towards the centre, then 06 r<

(4.22)

l/3 < 5< 1.

It is important to note that the stress and displacement fields must now satisfy the discontinuity relations across the two boundaries. Further, the quantities must be continuous at $= r= l/3. Region I. For the beam segment r 6 Z0 < 1, equations (4.2)-(4.7) remain valid in the second phase. Region

II. For the inner beam segment 0 ,< &, < c (which descends rigidly), the displacement fields may be obtained from (4.10) and (4.1 l), and the slope of the beam centre-line over this region may be written from (4.9) and (4.17) as C(.U,) = -2a(l

-+n)&(l-&),

0 6 x, < c

(4.23)

Next, from (4.16) with use of the continuity conditions (3.14) and (3.16) at f0 = c, it is found that (4.24) Region III. Within the moderate deflection and small strain approximations,

the axial force E is constant along the beam, and then the bending moment R must also be constant throughout the region %G i0 d e for continued plastic deformation. Thus, the state of stress at any cross-section in this region is represented by the same point, Pa (say). on the yield curve (see figure 6), and the stress/strain-rate relations are determined by the flow rule (3.10),. Thus, (3.1O)i with use of (3.35) and (3.36) gives ti. & + y _c* e & = - ‘a@ <,.S,’ r < x, < i. Since cr;i and F are constant throughout region III and continuous boundary .X0= r, it is found from (4.7) that lGl=p(f-~)~, F=

(1-fs(l-i’)2j~,

C&$&T, 06X,<

(4.25) across the (4.26)

1,

(4.27)

which satisfy the yield condition (3.8) for [< X0 d r. It is clear from (4.26) that the shear force S is zero throughout region III. Furthermore, constancy in values of A and F implies that the beam takes on a parabolic shape in this region. Now the vertical equilibrium equations for regions II and III, with use of the continuity of appropriate quantities across the two boundaries Z0 = (r, F), and the conditions

466

H. G. HOPKINS

AYFER G~~RK~Kand

and (3.26) give F = fiT3(1 -T)

(4.28)

and C(F) = -2a(l+11)~/~(1

-T)

or C(f) = - 2a(l +n)@/F, r <
(4.29)

fl=

([(1-~)4+4~(1-i’)2]i-(1-~)2)/2~(1-~)2

(4.30)

F=

([(1-~)4+4~(1-~)2]+-(1-~)2)/2~*(1-~).

(4.31)

and

Equations

(4.3) are then rewritten V = 2cr(l +n)Ql

as

-.Uo,@(l

-T).

a = 2~2(1+~~)2~2(1-.Y0)/~(l-~)2-nA(5),

r < x, < 1,

_-

(4.32)

where ? ({[(l -9)4+4Ql

A = 4x2(1 +n)

-Q)‘]*-(l

-W)

x

r -0 [2C(l-:)+Q3:-

1)?]/[4~(1-QJ])

(4.33)

dC.

Now, remembering that k. i-o= 0 for T< Y. ,< r and that [V] = [y \-,I = 0 across the boundary So = r, the vertical equilibrium equation (3.37), is integrated to give V=

a(l+n)(2~-92-.?;)/k(l-~),

r
g,

(4.34)

where the constants of integration have been determined from (3.16) and (4.32),. Knowing V(-Yo, f) for region III, the horizontal velocity component 0 is found from (4.25). Thus, integration of (4.25) with use of (4.34) and then (3.41) at .-U,= r gives f=Cr2(1+n)[(3~-1)/$(1-~)~]~[[(1-~)4+4~(1-~)~]*~(1-9)2]x .U,-2(1 +n)(+r3)/3;r, C-C x0 < r. (4.35) -It is now obvious that r = c(t) must be determined. The relation (3.41) at X0 = cwith appropriate quantities from (4.30) to (4.35) gives the non-linear, first-order, ordinary differential equation ---z? := -~(5,4)lh(5,4), where (4.36) -7 ~(~,~)=6~(1-~)~[(1-~)4+4~(l-~)2]f-1+~2), h(~,i)=(1-3~){2(~3-$)-3~[[(1-~)4+4~(1-~)2]*-1+~2]).

>

Finally, the vertical displacement field P(X,, <) for region II is found from (4.10) which, with use of (4.34) and the continuity of V and its first derivatives at X0 = r, gives ~=~((l+n){[(2~-~~-~)/~*(l-~)]++(xj&-+) +2ln[(l-,$)(l+c*)/(l+$)(l-r*)]), It is now necessary

to solve the non-linear,

first-order,

Of.?, ordinary

< C

differential

(4.37)

equation

Plastic beams at finite deflection

467

(4.36) subject to the initial condition

Equation (4.36) was integrated using the Runge-Merson method which also provides a technique for interval adjustment so that an estimate of the local truncation error at each step can be kept below a certain specified bound. It is not difficult to see that c = r = l/3 is a singular point of (4.36). In order to start the integration process, the initial values of gand rwere given with an increment 10e3 (i.e., e= c+ 10-3) which was also the initial step-length used for integration. The values of rand Tat the end of each step were used to compute /j, I/(0, %)/h and the other unknowns. The entire load-central deflection curves for axially-restrained, rectangular beams with three different rotational constraint factors are plotted in figure 8. Also shown in figure 8 are the experimental results of PANGand MILLAR(1978) for a built-in beam. It is obvious that equations (4.26), (4.28) and (4.34) for F = 1 and Ii?i= 0 give the membrane solution for axially-restrained, rectangular beams with rotational

/ -___ -8 /i

9.44 9

-Present 8

-

solution

-Membrane

:I,“.5 n=

solution

+ + + Experimental

I

/

equation

results

of

PANG and MILLAR(~~~~) 7

6

L/ 0

I

I 0.4

I

I 08

I

c I2

I

I 16

I

I 2.0

I

+, 2.4

I

I/(01/h

FIGURE 8. Load versus central deflection curves for axially-restrained rotational constraint at the supports.

beams with different degrees of

468

AYFER GCJRK~Kand H. G. HOPKINS

constraint factor II as 6 = ,s,, = [4j(l +n)][v(o)/%f_

(4.38)

Numerical results have shown that, for any value of n, the outer boundary x0 = g, reaches the end section when the inner one is at a distance .+?,,= r =: 0.05 from the centre. At that instant. e/(0, 1)/h * (1 +n)1.18, and the difference between the values of ppredicted by the present solution, fl, and the membrane solution, fim, is ~0*OOijl,. Since, within the restrictions of moderate deflection and small strain, the axial force is constant along the beam, then on substitution of 8= 1 into (4.7), it is seen that now F = 1 and hence &? = 0. It is therefore assumed that as soon as the outer boundary reaches the end section, the stress-points for the central zone of Iength z 0.05L tend to approach the yield curve at vertex D (see figure 6) almost instantaneously and hence the whole beam becomes a plastic string of parabolic shape. This discrepancy may result from the approximations of the small strain solution which justify the assumption of constant axial force along the beam span. Further analysis might require consideration of a third phase, but this is not considered here. In the large strain solution, the axial force will not be constant along the beam, and hence the stress-states at Z. = r and &, = 5 will be represented by the points P,; and P,,, say, on the yield curve shown in figure 6. The segment P,;P,; will correspdnd to the states of stress in the cross-sections along the region c< _YO < c. From figure 6, it is clear that the outer boundary Yc = % must reach the support before the inner one X, = Jreaches the centre. The load versus central deflection diagrams (figure 8) clearly indicate that axiallyrestrained beams can carry loads considerably greater than those determined by limit analysis. For the beam (with 0 6 II < 1) considered above, the load-carrying capacity is increased by a factor of ~4.72 when the membrane state is reached practically. The general conclusions are best drawn by considering the two particular bases of a built-in beam (i.e., a beam with n = 1) and a pinned beam (i.e., a beam with n = 0) only, since they are of much theoretical and practical interest. From figure 8, it is seen that the relative strengthening of the two beams is the same. Therefore, since the limit load of a built-in beam is twice that of a pinned beam, then, for the same value of c, the carrying capacity of the former is always twice that of the latter. On the other hand, if the load-central deflection curves are compared without making any reference to time (that is, the time-like parameter r), then the two curves plotted in figure 8 show that: (i) for 0 d 1/(0)/h < 0.9, &ui,t_in> flpjnned,and the ratio Bbuilt_in/Bpinned decreases from 2 to 1 as the deflection increases, (ii) for 0.9 < v(O)/~~< 2.36, J?bui\t_in < Bpinned,and the difference is quite small, and (iii) for v(OYh > 236, Pbuilt_in = fipinnd, and the two beams are now parabolicshaped strings (membranes) with constant F equal to 1. Note that the pinned beam has already reached the membrane state when V(0) z 1.181~.

Plastic beams at finite deflection 5.

469

SOLUTION WITH ALLOWANCE FOR END ROTATIONS

ANDAXIALSUPPORTMOVEMENTS The preceding theoretical analysis of the behaviour of axially-restrained beams clearly shows that the limit load of the simple plastic (bending) theory underestimates considerably the load which could be supported if deflections of the order of beam thickness or larger are permitted. WASZCZYSZYN (1966), experimentally, and JONES (1973), theoretically, have shown that the increase in the load-carrying capacity is strongly dependent upon the degree of axial restraint at the supports. In this section, a complete small strain soIution is presented in order to discuss the effects of axia1 (horizonta1) support movements on the post-yield response of the rectangular beam analysed in section 4. Since the influence of finite deflections is retained, the governing equations and most of the related discussions of section 3 will provide the basis for the following analysis. Assume that the support conditions of the rectangular beam analysed in section 4 are now specified in terms of the rotational constraint factor n (where 0 < n G 1, as defined earlier) and the axial restraint factor m (as will be discussed later). Now, if the (non-dimensional) axial displacement which may occur at each support of the beam is denoted by &, = a,(r), then the support condition (3.29) must be replaced by G( 1, g, = - nii(l!f) - A,&),

(5.1)

where a(T) is defined by (4.6). Note that a,(O) = 0, and that a,(r) > 0 under increasing load. It is obvious now that, if the supports lack rigidity, then the distance between them is not constant but a decreasing function of time. Since B,(O) = 0, the problem at t = 0 (or g = 0) reduces to that considered earlier and hence the yield-point state is defined by equations (3.1 l), (3.13) and (4.1). 5.1 Primary plme Now, it is assumed that, for fl> 1, the deformation modes of section 4 are still applicable. It should be emphasized that there is symmetry in geometry as well as in loading, that is, both the supports have the same amount of axial restraint. Thus, only the right-hand half of the beam will be considered. If there is some amount of axial restraint, for j?Zj > 1, then the central hinge will again split into two lateral boundaries which move out towards the supports. The distance to the right-hand moving boundary will be denoted by Z0 = c, as before. As will be seen later, if the beam is freely-supported, then the plastic hinge acting at S, = 0 at t = 0 will stay stationary for all time. In other words, the solution will reduce to that obtained by the simple plastic theory in which it is assumed that the beam can undergo indefinitely large deformations under p= 1 [i.e., fl = PO = 2(1 t-n)]. It must be emphasized that the experimental load-deflection curves even for freely-supported beams may have a rising portion after the yield-point load. But this stems primarily from the strain-hardening of the material, the effect of which is also completely disregarded in limit analysis, and secondarily from some friction at the supports which may cause the development of small axial forces.

470

AYFER GCJRK~Kand H. G. HOPKINS

In the primary phase, the right-hand half of the beam (with some amount of axial restraint) again consists of two rigid segments separated by a plastically deforming section X, = c. Following the procedures of section 4, it is found that the displacement fields for regions I and II (see figure 4) are

v = C(Q(i,

- l),

(5.2)

_-

s 5

v = C(Q(F-

l)-

C(.Xb) dx;, i-0

ii = +cz(zu,)( 1 - X0) - nh(x,)

(5.3)

0 d zu, 6 g,

- A,(.?,),

-where C(r) and A(5) are defined by (4.4) and (4.6) respectively, and A,(%) may be determined as follows. For the inner curved, rigid region 0 d X, < c, equation (3.32) with use of the appropriate quantities from (4.6) and (5.3) gives &(X0) = c, &$J[C(X())(

1 -20)

+ 2crnF(x,)]

- A,, #J.

(5.4)

1 - X0) + 2crnF(.?,)],

(5.5)

Now, if &. \-,(X0) = 2mC, i,(io)[C(X,)( then (5.4) becomes a(.fO) = (1 - 2m)C, ,,(.UO)[C(X,)(l -X,)-t

2crnP(.Y,)].

(5.6)

It is now obvious that m = 0 corresponds to complete axial restraint at the supports and m = l/2 to complete axial freedom, while the values of m in the range 0 < m < l/2 represent intermediate degrees of axial support restraint. Once the procedures used in section 4 have been understood and followed, then it is easy to obtain the (general) solution to the problem, provided that the appropriate support condition [that is, equations (5.1) and (5.5)] are used. The equations are obviously rather lengthy and complicated. Therefore, only the expressions which are regarded as essential in determining the load versus central deflection curves are given here. If To = p denotes the position of the right-hand moving boundary at the end of the primary phase, then 1 +n(l-2m) ‘=

(1 -g)[l

0
+n(l-2m)-2mR

(5.7)

and

iqo,r, =

2a[l +n(l-2m)]

+In

[l +n(l

*

(1 +n)(l-2m)f i[ 1 +n(l-2m)-2mF

(l-2m)

-2m)-2mZjf+[(1

1 +n)(l-2m)r]*

( [l+n(l-2m)-2m~]+-[(l+n)(1-2m)~]f -

31 +n)(l-2m)

[I

m

1

* arctan

r

1 f

2rnz

1 +n(l-2m)-%f

Ii

’

(5.8)

Plastic

beams at finite deflection

471

which is valid for 0 < m < l/2. When m = 0 and m = l/2, some of the terms have to be integrated by considering these specific values. Thus, it is found that P(O, r) -+CC as ttz -+1/2 and P(0. c) = 2a(l +I?) { In (S) for m = 0. Furthermore, can be obtained

-pi

the axial force F, which is taken as constant as F =

(1 +n)(l-2m)9 i 1 +n(l-2tn)-2m~

along the beam span,

1. *

(5.9)

It is seen that, for tn = 0, (5.7).-(5.9) reduce to the results already obtained in section 4 for axially-restrained beams. On the other hand, as m + l/2, V -+CC and F = 0 which implies that no membrane strengthening is obtained in beams with unrestrained supports, at least not according to small strain theory. Therefore, the beam cannot sustain loads greater than the limit load which corresponds to the case of inextensional deformation under constant load. It is also seen that, for any value of m, within the range 0 < m < l/2, the axial forces come into play just at the beginning of plastic deformation and increase the carrying capacity of the beam. It has been found that the yield condition (3.8) is violated just behind the discontinuity when e > z*. Furthermore, the results have shown that the length of the inner curved, rigid region at the end of the primary phase decreases as m increases from 0 to l/2, and as a limiting case, for m = l/2, the plastically deforming section practically stays at the mid-span, as expected. In other words, in freely-supported beams, the central plastic hinge of the rigid-plastic collapse mechanism stays at the centre under increasing deflection which is not accompanied by an increase in load. For c> r*, the secondary phase of deformation begins with two boundaries originating at the cross-section X0 = r (see figure 4) and moving in opposite directions as described before.

5.2 Secondary phase In this phase, the axial restraint factor plastically deforming stationary extensible expressions for fl and

deformed

shape of the right-hand half of the beam having any comprises two rigid segments joined by a segment id X0 d g. Note that, if n = 1, there is also a hinge acting at f, = 1. Following the procedures of section 4, the vertical deflection V at X, = 0 are found as follows: m (0 d m < l/2)

P = .fi (E 8,fi(5? where -.fi(5,i)= -7

4) =

i ), 1

{C1+~(l-2m)12(1-~)4+4(1+n)(l-2m)[1+n(l-2m)-2m~]x i”(1 -Q’)*

fi(5,

--

-[I

2(1 +n)(l-2m)[l

+n(l-2m)](l

-%)‘)[l

+n(l-2m)-2m~]~(l

(5.10)

+n(l-2m)], -c)2,

i

472

AYFER C%RK~K

and H. G. HOPKINS

(1 “i-iz)(l-2m) +(2e-$-f, ‘-. 1 2(1-Q il 11 +n(l-2m)-2mi]i (1 -202) “-------?7= 1 +ln [1+?2(1-221fl)-22m~]~+[(14~2)(1-2tn)~]~ ~-...____-___. ~+n)(1-2m)@ 1 i [l.tn(l-2m)-2tn~]t-[(l 4 2mT 2(1 +n)Cl-2m) $arctan _-------_L-_--_ - ~-----.-.O < m < l/2, m r 1 c i-tn(i--2m)-2mi_ I i

V(0, %)=

2a[l +n(l - 2m)]

P(O,n = 2i~(l+n)f(2~-~~-4~+3$)/2~~(1-~) -t-in [(l -t~~)/(l--~~)]j, i=

c(e)

m = 0.

must satisfy the first-order, non-linear, ordinary differentia1 equation ; = - .4fZ <)lh(T, Q).

where &,C)=

h(Z,r)=

6~~i-~)[f+n(l-2m)-2fn~~~~[l~~z(l-2m)]’(l-5)4 +4(l+~)(l-2~)[l~~(l-2~~)-2m~]~(l-~)z]~ -[1+n(l-2m)](1-~2)f4m~(l-~)~, ~(l-3~)[1+n(l-2m)-2m~]-2m~(l-~)~ x ~2(l+~)(l-2~){~3-.~)-3~~[[l+~~l-2m)]2~l-~)~ i-4~1Cn)(l-2m)[l+n(l-2m)-2r~i~]~(l-~)z]i -[i +n(l-2m)](l-~z)+4m~(l -fjjf .

1 (5.12) 1

~

Equation (5.12) was solved by using the computer subroutine which employs Merson’s form of the Runge-Kutta method. The integration was started with the initial value found for f = p from the moment expression as before. Note that the value of p depends on the axial restraint factor nt of the beam. At each step of integ~dtio~, the values of fs and ~~O)/~ were calculated with use of (5.10) and (5.11). The resulting load versus central deflection curves for some illustrative cases are shown in figure 9. It is evident that, at moderate deflections, the increase of load over the limit load of simple plastic (bending) theory is strongly dependent upon the degree of axial restraint at the supports. At large deflections, although of course these could well be quite unacceptable in structural engineering design practice, all beams with some axial restraint reach the membrane (string) state; hence, the effect of support movements becomes less important. It must be remembered, however, that the solutions presented here are valid only for deflections of the order of beam thickness. For larger deflections, the basic assumptions of the present analysis would cease to be sufficiently valid. naturally.

6.

CONC‘LUSION

The results of the present theoretical analysis of the effect of finite deflections on the post-yield response of rectangular beams under uniformly-distributed vertical load are presented in figures 8 and 9. In figure 8, the load/central deflection curves for axially-fixed beams with various degrees of rotational constraint are plotted. it is seen that deflections taking place subsequent to the formation of a rigid-plastic collapse

473

Plastic beams at finite deflection

I

III

0

0.4

08

I

I

I 1.2

I 1.6

I

I 2.0

I

Ii 2.4

V (0)/h FIGURE 9.

Load versus central deflection curves for pinned beams with different degrees ofaxial restraint at the supports.

mechanism have a stabilizing effect on the strength and the load is considerably increased beyond the so-called zero-deflection limit load. According to the results obtained, the membrane state is virtually reached at V(O) z 1.1811when the beam is rotationally-free (that is, n = 0) and at F(O) z 2.36h when it is rotationally-fixed (that is, n = 1); meanwhile, the Ioad-carrying capacity is increased by a factor of 4.72 for both cases. Thus, the analysis shows that the degree of rotational support rigidity is of no significance as far as the relative strengthening of the beam is concerned. If it is remembered that the factor n introduces the effect of imperfections and/or elastic deformations (which may occur at the supports of a built-in beam before the limit load and weaken the rotational stiffness of the support connections) then this result appears quite natural. The load/central deflection curves plotted in figure 9 for pinned beams with different degrees of axial restraint at the supports indicate that, for all beams with 0 G IN < l/2, there is a significant increase in the load-carrying capacity due to the induced tensile forces. But, at moderate deflections, the amount of this increase is closely related to the axial stiffness of the support connections. It is evident that even a small amount of support movement delays the development of axial forces which act

AYFERG~~RK& and H. G. HOPKINS

474

to reduce the plastic moment capacity of yielding sections. Therefore, the influence of such movements within the tolerable deflections is quite important and must be considered in design procedures. Otherwise, the finite deflection analysis ofbeams (as well as other engineering structures) under the assumption of complete axial restraint would be too optimistic. It is also seen from figure 9 that all beams with some amount of axial restraint reached the membrane state. Therefore, the effect of axial support movement becomes insignificant at very large deflections (the magnitudes of which are closely related to the amount of axial restraint). It should be remembered, however, that the physical model and also the basic assumptions of the preceding analysis are valid for deflections of the order of beam thickness and would become inadmissible for larger deflections. In the present work, the beam is assumed to be made of a rigid perfectIy-plastic material. Therefore, the effects of both elastic strains and strain-hardening are neglected. Since the latter always strengthens the structure, its neglect leads to a certain welcome reserve factor of safety. As regards elastic deformations, it is well known that the results obtained by employing the present model can be expected to provide a satisfactory approximation for the actual elastic-plastic solution if the final (total) strains are appreciably larger than the elastic strains. It should also be emphasized that the analyses of this type are usually poor in the neighbourhood of the limit load and such a discrepancy cannot be avoided unless a more realistic model is used. It is seen from the preceding sections that the analysis of structures in the

presence

of finite

deformations

is rather

complex,

and even

for the simple

case

(and also the simple model) treated here, consideration of successive phases of deformatjons is necessary. WASZCZYS~YN and ~;YCZKOWSKI (1963) and WASZC~YZYN (1967)

have studied

the finite elastic-plastic

cantilever beams subjected or point

loads).

to concentrated

deformations

external

CAMPBELL and CHARLTON (1973)

loadings

of axially-restrained

and

(that is, to end moments

have presented

an elastic-plastic

analysis

for a built-in beam of rectangular cross-section subjected to a single concentrated load at mid-span. They have assumed an instantaneous transition from the elastic to the plastic mode. To the present writers’ knowledge, published for beams having other types of loading conditions.

no work has been

REFERENCES BATTERMAN,S. C.

1967

CAMPBELL,T. 1. and CHARLTON,T. M. GILL, s. s. GJRK~K, A. and HOPKINS,H. G. HAYTHORNTHWAITE, R. M.

1973

Tram ASME 89, Ser. E, J. appt. Me&. 34, 500. ht. J. mech. Sci. 15,415.

1973 1973 1957a b

Ibid. 15, 465. S.I.A.M. J. appl. Math. 25, 500. Engineering 183, 110. Proc,. 9th Int. Gong. appl. Mech. (Brussels, 5-13

September Brussels.

1956), 8, 59. Free University

of

Developments in Mechanics (Proc. 7th Midwestern

Me&. Conf.) (Michigan State University. 6-8 September 1961) (edited by Lay, J. E. and Malvern, L. E.), Vol. 1, p. 203. NorthHolland, Amsterdam.

Plastic beams at finite deflection

Phil. Mug. 42, 868. Tram ASME 79, Ser. E, J. appl. Mech. 24, 453. Appl. Mech. Revs 24, no. 7, 741. Int. J. mech. Sci. 16, 385. Int. Shipbuilding Prog. 19, No. 218, 313.

1951 1957 1971 1974 1972

HILL, R. HODGE, P. G., JR.

JONES,N.

415

(AIAA/ASME/SAE 13th Structures, Structural Dynamics and Materials Conference, 1972, Paper No. 72-243.) 1973 MASSONNET, C. E. and SAVE,M. A. ONAT, E. T.

1972

1960

Int. J. mech. Sci. 15, 547. Plastic Analysis hnd Design of Plates, Shell.7 and Disks. North-Holland, Amsterdam. Plasticity (Proc. 2nd Symp. Naval Structural Mech.) (Brown University, 5-7 April 1960) (edited by Lee, E. H. and Symonds, P. S.), p. 225. Pergamon Press, Oxford.

PANG, P. L. R. and MILLAR,M. A. PRAGER,W.

1978

Int. J. mech. Sci. 20, 675.

1953

Proc. 8th Int. Cong. appl. Mech. (Istanbul,

1955

WASZCZUSZKN,Z.

1966 1967

WASZCZUSZUN,Z. and ~YCZKOWSK~. M.

1963

2&28 August 1952), Vol. 2, p. 65. University of Istanbul. Proc. 2nd U.S. Nat. Cong. appl. Mech. (University of Michigan, 1418 June 1954) (edited by Naghdi, P. M.), p. 21. American Society of Mechanical Engineers, New York. Bull. Acad. Polonaise Sci.. Ser. Sci. Tech. 14, 139. Acta Mechanica III/2, 219. Bull. Acad. Polonaise Sei., Ser. Sci. Tech. 11, 347.

APPENDIX NOTE ON GENERALIZED STRAIN-RATES FOR BEAMS The choice of the generalized variables for a rigid perfectly-plastic continuum undergoing finite deformations may be best discussed with use of the Principle of virtual velocities. For a given material body V, which occupies at some time t during deformation the volume V, the rates of internal and external energy dissipation in an equivalent form (that is, referred to the initial configuration) are given by

Q-,,, =

s s s v,

and

i,,, =

Q’il dl/,

T;u* dS 0 +

so

F&

v,

respectively. In equation (A.l), So denotes the initial surface area bounding V, ; Q = [Qi, . ., Q,,lT, the generalized stresses; 4 = [qi, . ., &IT, the rate of deformations (i.e., the generalized strain-rates); To = CT,,, ., To,]‘, the generalized surface traction per unit initial surface area; F, = [F,,, ., Fom]T, the generalized body force per unit initial volume; and u= [u,,. . .,u,,JT, and u* = [u:, .,u:]‘, the generalized interior and boundary velocities, respectively. If the generalized variables are both statically and kinematically

AYFER WRK~~K and H. G. HOPKINS

476

consistent, then the ~~~~z(.~~~~? of ~I~~~?~~~/ ~~~o~~r~~,~ states that G,;,, -- Yj,,,_

(it.‘)

This principle may be applied to a.beam element P,Q, which occupies at time f the position PQ (see figure 1) to show that R, = +,,, and d, are the generalized strain-rates corresponding to the generahzed stresses M(x,, t) and b’(w,, r), respectively. For this purpose, the equilibrium equations (which are referred to the initial coordinate x0) will be given in dimensional form. Thus. F~,O-S~,.~o+~~, = 0, S~,0+FF41,,o+& = 0, M, \‘n+ i$ = 11.1

(A.3)

Now these equations are muitipli~d in turn by t+, U, and d;. and then the resulting equations are added to give F. \“& + s. ,,U” + M. ,,i - sa.,,,Il, + F4, ,,I& + $4 + qk, U,+ sli,, II,, = 0.

(A.41

Note that I~,(_Y~, I) and z&0 t) are the tangential and normal velocity components of the material point PO at time t. Next (A.4) is integrated over the length P,,Qo and it is found by use of the equations k = k,t -+k,n, u = u,t+u,n,

and d; = (M”.,,-M,,,WL 6N = L~t,~o-~.~oUn~

that

Ql?

QO

(AS) f Fi, + MC&.J dx,. s PO I PO It is seen that the first term on the left-hand side of equation (AS) is the rate of energy dissipation due to the stress resultants at the boundaries of the beam material element; the second term is the work-rate of external loads, and the right-hand side would be, in view of (A.11 an,d (AJ), the rate of plastic work done on P,Q,. Therefore, it follows that the variables 8, and $,,, = ri-, respectively multiplying the generalized stresses F(u,, t) and M(.u,, t) are the generalized strain-rates. If (A.5) is transformed to an equation referred to a spatial coordinate s, then it is seen that the generalized strain-rates corresponding to the generalized stresses F(s, r) and M(s, t) are 8, (true strain-rate of the beam’s centre-line) and (&)., = R, (change of spin along the deformed beam’s centre-line), respectively. ~~~~+s~"+~~~~+

q(k+u)dx,

=