DYNAMIC SHAKEDOWN BY MODAL ANALYSIS* Castrenze Polizzotto**

MECCANICA 19 (1984), 133-144

SOMMARIO. Si studia it problema dell'adattamento dina-

mico (shakedown) di una struttura discreta elasto-perfetta-

mente plastica e soggetta ad una storia di carichi prestabilita,

facendo uso a tale scopo delle caratteristiche dinamiche della

struttura fornite dalla analisi modale. Vengono presentati

svariati teoremi, sia di tipo statico che cinematico, tra cui

taluni teoremi di delimitazione superiore ed inferiore del

tempo minimo di adattamento. Nella formulazione dei

teoremi cinematici ha un ruolo eruciale la corretta defini-

zione di ((ciclo deformativo ammissibile)).

SUMMARY. Dynamic shakedown o f discrete elastic-perfectly

plastic structures under a specified load history is studied

using the dynamic characteristics o f the structure provided

by modal analysis. Several statical and kinematical theorems

are presented, including lower and upper bound theorems

for the minimum adaptation time o f the structure. In the

formulation o f the kinematical theorems a crucial role is

played by the appropriate definition o f (<admissible plastic

strain cycle)).

1. INTRODUCTION

Dynamic shakedown, initiated by Ceradini in 1969 [1],

has received much attention since then; statical and kinema-

tical theorems have been provided for various classes of

structural models (continuous, as well as discrete models,

workhardening models, second-order geometric effects, etc.). A few recent survey papers give a quite complete account of the most significant contributions and provide the state of the art in this topic [2-4].

In spite of the present rather satisfactory development

of the whole body of dynamic shakedown theory, there are

still a few aspects o f it which deserve further consideration,

in the author's opinion. For instance, one may remark that

as in quasi-static shakedown the so-called <<admissible cycles

of plastic strains>> follow up from the (optimality) require-

ment of being orthogonal to any time-independent self-stress distributions, so in dynamic shakedown such cycles of plastic

strains should result from the same requirement as above,

plus that of being orthogonal to any free-motion stress distri-

butions. The consequence is that in Dynamics the admissible

* This paper is part of a research project sponsored by the National (Italian) Research Council, C.N.R., Group of Structural Engineering, and by the National Department of Education (M.P.I.).

** Istituto di Scienza delle Costruzioni, Facolth di Ingegneria, Uni- versit~t di Palermo, Palermo, Italy.

cycles of plastic strains should possess some additional fea-

tures with respect to Statics, which however have never

been mentioned in the literature. Also, one may wonder to

what extent the use of a more appropriate definition of

admissible cycles of plastic strains would affect the formula-

tion of the kinematical theorems of dynamic shakedown [5,6].

The present paper is intended to study the above questions

together with other related to them. To this purpose, elasti-

perfectly plastic, discrete-type structural systems will be

considered. The results of the modal analysis will be suppos-

ed to be available in order to show that they may be of

considerable advantage in dynamic shakedown [7, 8] and

more generally in dynamic plasticity.

2. DEFINITIONS AND PRELIMINARY RESULTS

Let a discrete (or discretized) elastic-perfectly plastic structure be loaded by nodal imposed forces, F, and by

dement imposed strains, 0, both actions being specified

time functions. Once the initial conditions and the physical

properties of the system have been specified, finding the

response of it to the given actions in terms of nodal displa-

cements, u(t), and element stresses, o(t), constitutes a pro-

blem of dynamic plasticity which in general is very difficult

to solve. Making use of a piecewise linear description of the

plastic behaviour [9, 10], the governing equations can be summarized as follows:

Mii + Ku -- Cr D p = F(t) + CT D O(t) (2.1)

f = N r o -- k <~ 0 (2.2)

b = N},, ~./> 0 (2.3/

f r X = jsrX = 0 (2.4)

= D g u - -Dp --D0(t) (2.5)

wherep is the vector o f plastic strains, C is the matrix which

transforms nodal displacements, u, into (compatible) strains, 6, i.e.

e = Cu, (2.6)

D is the matrix of the element elastic coefficients, K = CTDC

is the stiffness matrix and M the mass matrix. To the set of

equations (2.1)-(2.5), which hold for every time t >/0, the following initial conditions must be appendea:

u =u0, h =~0, p =P0 , at t = 0 (2.7)

where u 0, E0 andP0 are specified vectors. When the response of the structure is such that plastic

19 (19841 133

deformation is produced only during a first phase of finite

duration whilst the whole subsequent phase is purely elastic,

the structure is said to shake down (to a purely elastic state) or to adapt (to the loads) and <<shakedown>> (or <<adapta-

tion>>) is the name given to this occurrence. Shakedown on

its own implies finiteness of the overall plastic deformation

produced, whose exact amount is actually unknown, just as

the adaptation time, i.e., the time at which the plastic phase

ends and the purely elastic one begins, is unknown.

The generalized Ceradini theorem is usually cast in the

following form [1, 11, 12]:

SHAKEDOWN THEOREM

A necessary and sufficient condition for dynamic shake- down is that there exist a finite time, r ~> 0, and some initial

conditions, (-rio' ~0' ~0 )' such that the purely elastic stress

response to the given load history with these initial condi-

tions, 0g(t), proves to be inside the yield surface at any sub-

sequent time, t >/r, i.e.

N T C ~ ( t ) - - k <,O, Vt >~r. (2.8)

Let the stress 0g(t) in Eq. (2.8) be given the form:

f ( t ) = i f ( t ) + oY(t) + p (2.9)

where o2r(t) is the purely elastic stress response of the struc-

ture to the loads F(t), .0(t), t f> 0, with arbitrary, but fixed,

initial conditions, i f ( t ) is the stress history associated with a free motion or natural vibration o f the structure considered

as purely elastic and P is a time-independent self-stress distri-

bution. With the change of the time variable

t = r + r , r>~O (2.10)

the inequality (2.8) can be rewritten as

NT[oE(r + 7") + ~OF(7") + p ] -- k ~< 0, Vr >/0 (2.11)

in which o~(r + 7"), r /> 0, is the elastic stress response trunca-

ted backward at time r, that is the elastic stress response to

the loads F(r + r), 0(r + 7"), 7" >~ 0, with arbitrary but fixed

initial conditions specified at r = 0, while aF(7") is a free-

motion stress associated with initial conditions specified at

r = 0. In the following, a finite time r >/0 for which the ine-

quality (2.11) holds with some oF(r) and P will be called

separation, or truncation, time.

An alternative form of the shakedown theorem is the fol-

lowing:

S H A K E D O W N T H E O R E M - ALTENATIVE FORM

A necessary and sufficient condition for dynamic shake- down is that there exists a separation time, or, in other

words, there exist a finite time, r >/0, a free-motion stress, oF(r), and a time-independent self-stress, p, such that the sum of these stresses with the elastic stress response to the

loads truncated backward at r, g/r(r + 7"), proves to be inside

the yield surface at any time r 1> 0. This alternative form of the shakedown theorem, which

will be referred to in the following, is completely equivalent to the previous one.

A few consequences can immediately be derived from

this theorem, maybe self-evident but nevertheless never

mentioned before, to the author's knowledge.

THEOREM 1 (Independence o f shakedown from the initial

conditions).

The capacity of a structure to shake down is independent of the initial conditions; in other words, if the structure

shakes down under the loads F(t) , O(t) with the initial

conditions {u 0, ~0' ~P0}' it shakes down also if the same loads

are associated with any other initial conditions, {u 0, ~'0' e'0 }"

Proof. If shakedown occurs under the loads if(t), 0(t)

and the initial conditions {u 0, ~t 0, P0}' there exists a separa-

tion time r >/ 0 and hence a ffF(f) and a p such that the

inequality (2.11) is satisfied; but the latter inequality is also

sufficient for shakedown under the same loads with any

other specified initial conditions. The theorem is so proved.

It is worth noting that while a change of the initial condi-

tions leaves unaltered the capacity of the structure to adapt,

on the contrary it has an influence on the amount of plastic

deformations produced and on the adaptation time.

THEOREM 2 (On the separation time).

If there exists a separation time, r, (and hence the structu- re shakes down), every time of the interval J(r) = {t : t >/r} is

a separation time.

Proof. If r is a separation time, then there exist a oY(r)

and a p such that the inequality (2.11) holds. Then, denoting

by r I = r + r 1, (r 1 >0), any time subsequent to r and setting r = ~1 + r ' , (r ' /> 0),from Eq. (2.11) there follows:

N r [ o g ( r + r l + r ' ) + o F ( r l + r ' ) + p ] - k ~ 0 , ~ ' r '>10

(2.12)

o r also,

N r [ o Z ( r l + r ' ) + o ~ ( r ' ) + o ] - - k < ~ O , Vr '~>0 (2.13)

where r 1 > r is another separation time and o/~(r ') is asso-

ciated with initial conditions specified at r ' = 0. Since r 1 is an arbitrary time of the interval J(r), the theorem

is proved. For a structure which is able to shake down it is meaning-

ful to determine the shortest separation time, r*. The latter

has the obvious property that no solution Qo~(7.), .p) to the

inequality (2.11) can be found for any r < r*. There imme-

diately follows

THEOREM 3 (On the lower bound to the adaptation time).

For a structure which shakes down, the shortest separation time, r*, is a lower bound to the adaptation time, t a, i.e.

r* ~<t . a

Proof. If shakedown occurs, the true post-shakedown

stress response can be set in the form (2.9) and satisfies the inequality (2.11) for any t >1 t a, so that t itself is a se- paration time and therefore belongs to the interval J(r*);

as a result t a >~ r*. If the initial conditions (2.7) are suitably adjusted, we

134 MECCANICA

can in principle obtain t a = r*, so r* can be viewed ad the

minimum adaptation time of the given structure.

THEOREM 4 (On the minimum adaptation time).

For a structure subjected to a specified load history, there exist some initial conditions which make the structure adapt within the minimum possible time and this minimum coincides with the shortest separation time.

Mathematical considerations for the characterization of r* will be given later on in this paper.

I f the inertia forces are negligible and thus quasi-static shakedown is involved, what has previously been said remains true provided the initial conditions are reduced to the initial plastic strain distribution, P0' the elastic stress response is

the relevant quasi-static response and the free-motion stress, o F, is dropped.

3. MODAL ANALYSIS

According to classical results (see, e.g., [13, 14]), the

structure considered as a purely elastic discrete model is dynamically characterized by its displacement mode shapes ~i ' its stress mode shapes, S i = DC ~i ' with the associated

natural frequencies, co i, (i = 1, 2 , . . . , N), N being the degree of freedom. The following orthogonality conditions hold:

1, f o r / = /

~ r M ~i = 8ij = (3.1)

0, for/=/=]

6oi 2 , for i = j

(3,2)

O, for i 4:j

Supposing that all these dynamic characteristics are known, the elastic answer of the structure to any excitation can be formulated in terms of them. For instance, the displa- cement associated with a natural vibration, u F ( t ) , h a s the representation

(3.3) N

uF(t) = y ~itai COS wit + b i sin % t l i= 1

where a i, b i a r e arbitrary constants related to the relevant initial conditions by

(3.4) N N

i = 1 i = 1

while the correspondig stress history is

(3.5) N

oF(t) = y Sita i cos w i t + b i sin w i t]. i= 1

Analogously, the elastic displacement response to the loads F( t ) and 0(t) can be cast in the form:

(3.6) N

ug(t) = Y ~i Y.(t) i - 1

where (Duhamel integral)

Y.(t) = {~/rF(t-) + S[O(~)} sin wi(t -- t) dt,

(3.7)

A more concise form can be given to Eq. (3.6) by means of the matrices

= [~1~2 ' " ~ 1 , s = [ s i s 2 . . . _ S ~ l (3.8)

= [-6Ol, 6o2 , . . . , CON_ 1 (3.9)

for which the following .identities hold [ 13, 141:

~ r M ~ = / , ~ T = M - 1 (3.10)

~ r K ~ = ~2 , cbg2-2ebr=IC- l=A (3.11)

Setting

• = 6o71 sin w i r, (i = 1, 2 , . . . , N) (3.12a)

x(r) = [-Xl(r), X2(r) . . . . , XN(T) I (3.12b)

and

N

H(T) = ~ X(r) ~ r = ~ ~i~T• ) (3.13) i = 1

Eq. (3.6) becomes

f t

ue(t) = H(t -- T) {F(?') + CrD 0(t)} dt.

J0

(3.14)

The matrix H(t -- t-) is a dynamic flexibility matrix which transforms the unit impulse applied at time 7 into the displa- cement response at a subsequent time t.

An alternative expression for utr(t) in Eq. (3.14) is obtain- ed by introducing the matrices:

7i(r) = 6o 72cos co i r, (i = 1,2 . . . . . ,N) (3.15a)

7(r) = [-71(r), 3,2(r) . . . . ,3~N(r)_ l (3.15b)

and

N

i= I

(3.16)

Since H(t -- }-) = d{G(t -- 7)}/dt and G(0) = A, integrating Eq. (3.14) by parts and supposing that F(0) = 0, 0(0) = 0 gives

f t

~ug(t) = ~g(t) - G(t - 7) {~(t-) + CrD ~(7)} dt (3.17) "0

where ~g is the statical displacement response, namely

~e(t) = A F(t) + L 0(t) (3.18)

where L = K-1 CrD. Eq. (3.17) says that the dynamic di- splacement response at time t can be considered as the sum of the statical displacement response to the load applied

at the same time, plus the (displacement) dynamic correction

19 (1984) 1 3 5

which accounts for the exact load evolution from t = 0 to

t. Moreover, Eq. (3.17) gives

f t

~g(t) = H(t -- }-) {~ (h + ~CTD 0(h} dt-.

"0

(3.19)

Further, introducing the matrices

Ad(r) = A -- G(r), La(r) = A#(r) CTQ (3.20)

enables the displacement response in Eq. (3 .17) to take on

the form

i t

u~(t) = {da(t--7)~(-i)+La(t--h~(t-'J}dT (3.21)

which shows that the matrices A d ( t -- t-) and La(t -- t)

transform, respectively, the unit force and the unit imposed

strain applied at time ? into the displacement response at a

subsequent time t and that they are the dynamic equivalent

of the matrices A and L, respectively, holding in the frame-

work of static loading processes. Analogous considerations hold true for the stress response

to the loads, i.e.

oR(t) = Q C uS(t) - -D 0(t) (3.22)

which, using Eq. (3.14), becomes

f t

olr(t) = D C H(t -- t-) F(t-') dt

"0

I t

+ D CH( t -h CrD O(h dT--D~O. ~0

(3.23)

Further, introducing the statical stress response, i.e.

f r ( t ) = L r F ( t ) + Z O(t), (3.24)

where Z = D C K - 1 C r D - D , and using Eqs. (3.17) and

(3.22) gives

f' f ( t ) = ~ ( t ) - z) e c ( t - - i ) P(?) dt + ~ 0 ~ ~ ~

(' - D C C ( t - 7) C r O 0(7t d? (3.25/

according to which the dynamic elastic stress response at time t is the sum of the statical stress response to the loads applied at the same time t, plus the (stress) dynamic correc-

tion accounting for the exact evolution of the load from

t = 0 t o t . Finally, introducing the matrix

Za( r) = D C A a (r} CrD -- D (3.26)

and recalling the matrix Ld(r) defined by Eq. (3.20), the stress response og(t) can be given the alternative form:

i t

~og(t) = { L T ( t - - h ~ ( t ) + Zd( t - - -D ~(7)} dt. (3.27)

The matrices L~( t -- t) and Zd( t -- t) transform, respecti- vely, the unit force and the unit imposed strain appliedat

time f into the stress response at a subsequent t ime t; they

are the dynamic equivalent of the matrices L T and Z, re- spectively, holding in the framework of statical loading pro-

cesses. Eqs. (3.14), (3.17), (3.21) and Eqs. (3.23), (3.25) and

(3.27) are of some interest in dynamic plasticity and in par-

ticular in shakedown analysis. For instance, Eq. (3.27)

enables one to extend to the dynamic field the classical con- cept, due to Colonnetti [15], that the elastic-plastic stress

response can be expressed as the sum of the purely elastic

stress response to the same loads, oE(t), plus the elastic

response to the plastic strains considered as imposed,

namely

f0' s = s + Za( t - 7) ~(7) dr. (3.28)

4. GLOBALLY QUASI-STATIC LOAD CYCLE

Using the definitions and the results of Section 3, we

can now investigate the characteristics to be possessed by

a load cycle, R~(7), 0 ~< 7 ~< t 1, such that the global virtual

work,

i t1

6L = {R~(7)}T6.0(7) dt, (4.1)

vanishes for any displacement 6v(7) chosen within the class

o f natural vibrations. Since 60 must have the form of Eq.

(3.3) with arbitrary constants a i, b i, i.e.

N 60([) = E ~i(ai cos 6oi'f Jr b i sin ~i7),

i= 1 (4.2)

substituting from the latter into Eq. (4.1) gives

N 6L = y {a s P/(.t 1) + biQi( t l ) }

i= 1 = 0 (4.3)

and thus the following Lagrangian equilibrium equations

must be met:

rf" pi( t l ) = d~ /~(t) cos 6oit dt = 0

~0

rf" Qi(tl) = �9 /~(t) sin ~ t d t = 0

(i = l, 2, . . ., N) (4.4)

Conversely, if the given load cycle satisfies the latter

conditions, the virtual work 6L in Eq. (4.1) vanishes for any

So(t) in the form of a natural vibration.

136 MECCANICA

In order to point out the implications of Eqs. (4.4) let

the first one be multiplied by cos coitl and the second one

by sin coitl; then, summing the two products gives

q bT_, /~(t) cos coi(tl--t) d{= 0, (i = 1, 2 , . . ., N) (4.5)

J0 ~

and this set o f identities, remembering Eqs. (3.15) and (3.16), implies

s q G( t l - - }) R~(7) dt-= (4.6) 0.

Further, multiplying the first of Eqs. (4.4) by sin wit 1 and the second one by cos coitl and then subtracting the

two products from one another yields

/~(7) sin coi(tl-- t) dt = O, (i = 1, 2 , . . . , N) (4.7)

and this set of identities, through Eqs. (3.12) and (3.13), is equivalent to

s t~H(t l - - 7) R~(t) dt = 0. (4.8)

According to Eqs. (3.17) and (3.19), Eqs. (4.6) and (4.8) state that the elastic displacement response, v(t), to the load cycle in question satisfies the following conditions:

O(t 1) = ~(tl), b(t 1) = 0. (4.9)

In other words, a load cycle satisfying Eqs. (4.4) produces

a motion which leads the structure from the initial motion- less configuration (V(0) = ~(0) = Q) to a final motionless

configuration (~(t 1) = 0) characterized by displacements E(tl) and stresses sQ1), which coincide with those that would be

produced by the same load cycle acting statically, i.e. (Eqs. (3.18) and (3.24)):

g(tl) = ~(t 1) = A{R(q) -- 8(0)} /

s(q) = ~(t 1) = Lr{R(tx) -- R(0)} ~ (4.10)

A load cycle satisfying Eqs. (4.4) will be termed <<globally

quasi-static>> in this paper. Analogously, if the load conside-

red is one equivalent to imposed strains, ~, namely it has

the form CrD ~, we will deal with a strain cycle which thus will be globally quasi-static when it satisfies the conditions (4.4). The latter now take on the more appropriate form of Lagrangian compatibility equations, i.e.

f tl

sT~t ~~(}-) cos c o j d t = 0

+0 (i = 1 , 2 , . . . , N ) (4.11)

f0" S ~(t) sin wit dt : 0

According to Eqs. (3.18) and (3.24), the final displace-

ment, o(tl), and the associated stresses, S(tl), produced by

a globally quasi-static strain cycle are

V(t 1) = ~(t 1) = L{~(t 1) -- ~(0)} /

s(t 1) = s-(t 1) = Z{8(q) - ~ ( 0 ) } J (4.12)

5. THE SAFETY FACTOR

F O R F I X E D MI N I MU M A D A P T I O N TIME

Coming back to Section 2 and to the positions there, let the given loads, F(t) and O(t), be multiplied by a positive

multiplier, ~, and let the following problem

max {~ INT[~ oZ(r + r) + ~F(T) q- p] - - k < 0, VT >.~ 0} (t,zr,s (5.1)

be solved for a fixed separation time, r. Here o~(r) is a free-

motion stress represented in the form

N ~~ = E S~i(ai cos r + b i sin coir) (5.2)

i=1

while p is a time-independent self-stress vector, thus satisfy- ing the condition c r p = 0. The maximum value of ~, ~* =

= ~*(r), determined by solving the problem (5.1), specifies the interval of the load multiplier, 0 ~< ~ ~< ~*, and thus the range of the amplified loads, for which the fixed r is a sepa-

ration time (and hence the structure shakes down); in parti- cular, r coincides with the minimum adaptation time of the structure subjected to the loads amplified by ~*(r). More precisely, the meaning of ~* can be stated as follows:

- f o r any ~ < ~*(r), the structure shakes down and its minimum adaptation time is not greater than r;

- f o r any $ > ~*(r) either the structure does not shake down, or it shakes down and its minimum adaptation time is greater than r;

- for ~ = ~*(r), the structure shakes down and its mini- mum adaptation time equals r.

In order to write out the appropriate optimality condi- tions for the problem (5.1), we consider the Lagrangian functional

= - a t + ~r{Nr[~o~(r + 7") + oF(r) + p] -- k} d r

~ 0 ~ ~

+ qT{-- OF(r) + E S i ( a i c o s c o i r + b i s i n c o i r ) } d r " 0 ~ i=1

- -vT cTo (5.3)

where /J /> 0, ~ and v are appropriate Lagrangian vector va- riables ~nd ~ is s o m e positive constant. Taking the first va- riation, we obtain

= r + + + el < q |

~ 0 , e T a = 0 , ~ = N ~ , r>~0 / (5.4)

f ' { o s + r)} T ~(r) dr = c~ (5.5)

19 (1984} 137

SrJo=q(r)( cos % r dr = 0

f

Sf t ~(r) sin coir dr = 0

( i = 1 , 2 , . . . , N ) (5.6)

= 21 (r) dr = Cv (5.7)

which are the desired optimality conditions.

Eqs. (5.4)-(5.7) describe a limit purely elastic process

associated with a sequence of instantaneous elastic-plastic states, each of which is holonomic in nature. In every instan-

taneous state, a plastic strain rate, c}, complying with the

plastic flow-rule, is associated with the allowable stress,

~ ( r ) = ~*o~(r + r) + oF(r) + p, which is unaffected by

the plastic strain previously produced. The entire process

can be viewed as one of simple production of plastic strain,

with no stress redistribution, which leads to a final collapse

mechanism described by the displacement vector o.

The plastic strain rate, c), constitutes a strain cycle which

proves to be globally quasi-static (Eqs. (5.6)) and globally

congruent (Eq. (5.7)). A strain cycle which complies with

Eqs. (5.6) and (5.7) will be called <<admissible strain cycle>> in the following.

A dual problem can be formulated, namely

f20 . . . . #r s {og(r+r)}TNtiCdr:a, OC>~O

(5.8) where the strain cycle ti c, qC = NtiC belongs to the class of

admissible strain cycles. The optimal objective values o f the

problems (5.1) and (5.8) equal one another, i.e.

f a~.*(r) = kr l tiC(r, r) dr (5.9)

]JO ~

where ~c(r, r) is the/an optimal solution to the dual pro-

blem. In the above developments, the problem (5.1) has been

treated as a true linear programming problem, for questions

which may arise in relation either to the infinite number

of constraints or to the consequent improper integrals

have been ignored. On the ground of this heuristic approach,

some properties of the function ~* = ~*(r) can be point- ed out.

The optimal value ~*, which will be called <<safety factor

for fixed minimum adaptation time>> in the following, turns out to be a nondecreasing function of r, for increasing r

is equivalent to reducing the number of constraints in the problem (5.1). This property is confirmed by the nonnega- tivity of the derivative d~*/dr, which can be derived from

the equality g'opt = -- a ~* and by differentiating it with respect to r, i.e.

d~* - a - l g * [ {6g(r + r)}r4(r) dr> 0 (5.10) dr J0 ~

where q = N~ belongs to the solution to Eqs. (5.4)-(5.7). Let us observe, in fact, that in consideration of Eqs. (5.4),

a component ~b h of ~ is nonpositive whenever the correspon-

dent component of ~ vanishes, i.e. ~o h = 0, hence whenever

li h ~> 0; as a result, ~h tih ~< 0 for any h, therefore

~T~ ={6a ( f ) } r~(r ) ={ ~ ( r + f) + s < 0

and the latter through an integration over the interval

(0, + oo) gives

= f0= s {6"(r)}rq(r) dr = {~( r + r)}T~(r) dr < 0, (5.12)

the strain cycle q(r) being a globally quasi-static one, hence

satisfying

f0 {~dY(r)}r~(r) dr = 0. (5.13)

Typical shapes of the function ~* = g*(r) are as shown

in Fig. 1. In Fig. l(a), ~*(r) increases monotonically and it

reaches a maximum, X, for r ~ + oo; this case - asymptotic case - corresponds to a nonperiodic loading which becomes

less heavy as time elapses. In Fig. l(b), ~* = ~*(r) is a straight

line parallel to the r-axis, ~* = X; this case - constant case - corresponds to a periodic loading, for the corresponding

backward truncated elastic stress response, ~E(r + r), assum-

ed to be the steady state response, remains unchanged when

r increases by a Ar equalling a multiple o f the period of the

loading so that ~* is the same at these values of r and hence

for all r's. An intermediate case of the two above is that of Fig. l(c) in which ~* increases first, then remains constant

as r increases; such a case may be produced by a loading

x -

(a)

(c)

r

( b ) r

I I I

r .~ I r=..~ ca)

Fig. 1. Typical shapes of the curve ~* = ~*(r) - safety factor (for fixed separation time, r) as funtion of r - and shakedown safety factor, X: (a) Asymptotic case, nonperiodic loading; (b) Constant case, periodic loading; (c) Intermediate case, nonperiodic loading, followed by a periodic one; (d) Unbounded case, loading of limited duration.

138 MECCANICA

which becomes periodic after some time has elapsed. Finally,

in Fig. 1 (c), [*(r) exists only for 0 ~< r < ~, while ~ * ~ + o~

when r --> F; if i is finite, this case -unbounded case - cor-

responds to a loading which acts only within the period

0 ~< t < [, as in fact, in such a case, an elastic stress response truncated backward at any r >/7, constitutes a free-motion stress and thus the dual problem (5.8) will possess no feasible

solutions and this implies unboundedness of the primal

problem (5.1). The shapes of the curves shown in Fig. 1, except the secon one relative to periodic loadings, are only

indicative and they are not meant to be examples of real shapes, nor to exhaust all the possibilities.

As concerns the uniqueness of the solution to Eqs. (5.4)-

(5.7), let us suppose there exist two distinct solutions,

(~*, ~~ ~Pl' ~1' ~Ul) and (~*, o2, P2' ~2' ~v2)' both associated with a single value of ~*. Applying Drucker's postulate, we have

U = [o~(r + r ) - - ff~(r + T)]T [q~l(r ) _~2(r ) ] t> 0, VT>~ 0, (5.14)

where o~ = ~ ~ + ~ ~ + Ph' (h = 1, 2). An integration over the time interval (0, + o~) gives

6. THE SHAKEDOWN SAFETY FACTOR

By definition, the (<shakedown safety factor>> of the struc- ture subjected to the given (nonamplified) loading history

is the maximum value, X, of the multiplier ~ such that for any ~ < X there exists some finite r for which the problem (5.1) has solutions. Since ~ *(r) is, for every finite r, the maxi-

mum value of ~ for which the problem mentioned has solu- tions and since ~*(r) is nondecreasing, it proves to be either

X = lira ~*(r) (6.1) r--+ -k ~

or, equivalently,

X = max ~*(r), (r)

(6.2)

provided X exists bounded. According to Eq. (5.10), since

a >0 , X is characterized by

lira i {6e(r + r)}rq(r) dr = 0 (6.3)

q- (p1--P2) f~(~tl--~t2) dr (5.15)

and in force of Eqs. (5,6) and (5.7),

f = U d T = 0 (5.16)

which implies that the inequality (5.14) holds with the equa-

lity sign at all times. Since this latter result applies also

componentwise, making reference to the k-th element, we

have that, if ~lk = ~2k = 0, the element stresses g~k and a~k are safe and may be different from one another; while

if ~lk -- ~2k q: 0, the stresses ~tc and a~k must belong to

one and the same (supporting) plane tangent to the element yield surface: during the intervals in which the supporting

plane touches the yield surface at a- corner, it must be o~k =

: a n d as a r e s u l t = a n d = hile in t h o s e

intervals in which the supporting plane coincides with a facet

of the yield surface, it must be true that ~lg = ~2k = ~k'

as well as Plk while ~/r and ~ k ' and hence o~11/c and o~2k, and

P2k' may differ from one another through a stress vector orthogonal to ~k" A complete uniqueness, i.e. d = ~aF22 '

~Pl = ~P2 and ~ql = ~2' would have resulted in case of smooth yield surface. If we view a piecewise yield surface as an approximation to a smooth one, in those zones o f the struc- ture where plastic activation takes place, uniqueness of solu-

tion to Eqs. (5.4)-(5.7) can be considered as proved in the general case, with exceptions for the plastic strain rate

at singular points and for the stresses o B and p at flat regions of the yield surface.

or also by

f0 ~ max {6g(r + T)}rq(r) dr = 0

(r.) ~ (6.4)

both corresponding to a tangent to the curve ~* = ~*(r)

parallel to the r-axis (see Fig. 1). Eq. (6.3), or (6.4), says

that when ~* ~ X the plastic strain cycle in Eqs. (5.4)-(5.7)

tends to comply with the elastic-recovery rule, for the

product ~br[J tends to vanish, (but the elastic-plastic process described by Eqs. (5.4)-(5.7) will not constitute a true in-

cremental elastic-plastic process since stress redistribution does not take place for any fixed r, nor for r -+ + ,~).

With reference to Fig. 1, let us observe that X is actually

an asymptotic value in case (a). In case (b), X can be eva-

luated by solving the problem (5.1) with an arbitrary r, hence

with r = 0; moreover assuming o ~ to be the steady state response to the periodic loading, with arguments like those

used in [16] one can show that X is also obtained by solving

(5.1) with oB(r) - 0. In case (c), again X can be obtained by solving the problem (5.1), but using an r not smaller

than the transition time between the two phases. Finally,

in case (d), for which X is unbounded and thus a = 0, Eq.

(5.10) shows that for r > f the derivative d~*/dr is indeter- minate.

7. THE MINIMUM ADAPTATION TIME

Let the load history be amplified by the scalar-~ as in Section 5, 0 < ~ < X, and let the following problem

min { r lNr[~ge ( r+r )+oF(r )+p] - - k< ,O , Vr~>0}(7.1) ( r , ~ ) . . . . .

be solved with a fixed load multiplier, ~. Comparing this

problem with the problem (5.1), we realize that the two va-

riables ~ and r play roles which are interchanged in the two

19 (zga4} 139

problems. Solving the problem (7.1) gives the smallest

value of the separation time, r*, such that for any r >1 r* there exist solutions to Eqs. (2.11). According to Theorem 4, r* coincides with the minimum adaptation time of the structure subjected to the specified loading history ampli- fied by the fixed ~.

Using the same procedure as for the problem (5.1), the Lagrangian functional to consider is now

xIJ 1 = 13r + s ~r{'Nr[~ ~ ( r + r) + s + 8] --k} dr +

+ ~r{--~ aF(r) + E Si(ai cos wir + bisin coir)} dr i= 1

-- v r Cr p (7.2)

where the appropriate Lagrangian variables are indicated by the same symbols as in Eq. (5.3) and 13 is some positive constant. Taking the first variation of 'I' 1, we obtain the following stationarity conditions:

= Nrt~ oE(r * + r) + aF(r) + el - k <~ O I

ti~>0, ~oTti=0, q = N t i , r~>0 / (7.3)

I

fo ~ {s * + r)}T~(r dr = --13 (7.4)

sT 4(r) cos % r dr = 0

S r ( | dr = 0

(i = 1,2 . . . . . N) (7.5)

s ~q(r) dr = Co. (7.6)

Since, according to Eqs. (7.5) and (7.6), the strain cycle ~(r) is admissible, from the equality ~~~ = 0 there follows:

fo fo {oe(r * + r ) } ~ ( r ) dr = .k r ~(f) dr. (7.7)

If the fight-hand member of the latter equation is positive, then the integral in the left-hand member of the same equa- tion is also positive, Eqs. (7.3)-(7.6) prove to be equivalent to Eqs. (5.4)-(5.7) and, as a result, the fixed multiplier coincides with the safety factor associated with r = r*, namely

= ~*(r*) (7.7)

The derivative dr"/d~, which can be obtained from the equa- lity [Jr* = ~lop t differentiated with respect to ~, proves to be positive, i.e.

_ 13-1~-1 {og(r * + r)}T~(r) dr >0 , (7.9)

e0

while comparing the latter with Eq. (5.10) and remembering Eq. (5.5) gives

dr* = I d a * ] - 1

In particular the derivative in Eq. (7.9) tends to diverge, i.e. (dr*/d~) -+ + ~ , when 13-+0 and thus when ~ - + X , according to Eqs. (7.10), (7.4), (6.3) and (6.4) (see cases (a), (b) and (c) in Fig. 1); on the contrary, it tends to vanish where, for 13 > 0, a -+ 0 (see case (d) in Fig. 1, with ~ -+ + ~ , such that r* = ~).

When the right-hand member of Eq. (7.7) vanishes, hence = 0 at all times r ~> 0, the strain cycle proves to be a trivial

one, therefore the first of Eq. (7.3) must be satisfied with s 0, p = 0, i.e. ~Nrs * + r ) - - k < O a t a l l t i m e s r > O . The latter condition can be satisfied only if ~ < ~*(0), so that r* = O.

In conclusion, the function r* = r*(~) is nondecreasing for 0 ~ ~ ~< X, while for ~ > X the problem (7.1) is not feasible.

According to Theorem 4, let us assume that the elastic stress response, o E* = ~ ( r * + r), (we now take ~ = 1 for simplicity), be associated with those particular initial condi- tions (optimal or extremal i.c.) which minimize the adapta- tion time, i.e. t a = r*. At times subsequent to r*, the real stress response, o* can be set in the form

o * = ~ ( r ~ + r ) + g F * ( r ) + p *, r > O , (7.11)

where the free-motion stress, s and the time-indepen- dent self-stress, p*, are the consequences of the elastic- plastic process prior to the adaptation time. Since ~* is an allowable stress, Drucker's postulate gives

(~a _ o,)T~ > 0 , r > 0 , (7.12)

where oa = s + o ~ + p and ~ are part of the solution to Eqs. (7.3)-(7.6). Through an integration over the time in- terval (0, + oo), Eq. (7.12) yields

i =(o a -- o*)r ~ dr =

- f - = (o F - o ~ * ) r 4 dr + (p -- p*) c~ dr (7.13)

which, remembering Eqs. (7.5) and (7.6), becomes

( ~(a" -- o*)T~ dr = 0 (7.14)

J0

so that the inequality (7.12) applies with the equality sign at all times. As a consequence the two stresses, ~k ~ and a~, at the k-th element belong to a single (supporting) plane

140 MECCANICA

tangent to the element yield surface at every time, provided

~x r Q" With the aid of arguments like those used to show

the uniqueness of the solution to Eqs. (5.4)-(5.7), and hence

to Eqs. (7~ the result is that g~ = a~, hence ~ * =

and p~ = Pk' provided that the yield surface has no flat

regions; while at a flat region, 9~k* and o~x, as well as 0~ and ~x' may differ from one another through stress vectors both

orthogonal to the corresponding ~k" In conclusion, the elastic-plastic stress-redistribution-free

process described by Eqs. (5.4)-(5.7), or by Eqs. (7.3)-(7.6), is <<plastically>> coincident with the real process which takes

place when the imposed initial conditions are the optimal

ones; in particular, when ~* ~ X, Eqs. (5.4)-(5.7) describe the <<limit strain cycle>> which takes place in the structure

at the shakedown limit.

It is worth noting that for periodic loadings (r* = O) Eq. (7.11) holds true with o ~* - 0Q (but in general p* 4: because plastic deformation may occur at time t = r = 0);

as a consequence, ~ in Eqs. (5.4)-(5.7) must vanish too and

this is equivalent to the statement that when the problem

(5.1) is to be solved for periodic loadings we can take ~ --- Q,

o ~* being the steady state response [ 16].

8. KINEMATICAL THEOREMS

According to a definition given previously, an admissible

strain cycle, ~(r) = N~(r), ~(r) >/ 0, is one which satisfies Eqs. (5.6) and (5.7). These are here rewritten, for the sake

of greater convenience, in terms o f ~ > 0 alone, i.e.

S~N J=~(r)[ cos coir dr = 0

~ g O

Sr.N ( l J ( r ) sin co.r dr = 0

(i = 1 , 2 , . . . , N ) (8.1)

N (r) dr = Cv.

JO (8.2)

In Statics, Eqs. (8.1) relax and the above definition will coin-

cide with that proposed by Koiter [17, 18] within the field

of quasi-static shakedown.

To the set of admissible strain cycles defined by Eqs. (8.1)

and (8.2) belong also strain cycles for which ~ may be dif-

ferent from zero only within a finite interval, r I ~ r ~ r 2, so that the conditions (8.1) and (8.2) take on the more restricti- ve forms:

I q'2

~STi N]~ "~, f4r)_ cos coir dr = 0

sT ~(r) sin coir dr = 0

( i = 1 , 2 , . . . , N ) (8.3)

N/r2(~(r ) dr =Co (8.4)

/ * T 1

However, it is worth noting that a nontrivial admissible strain cycle cannot be given the form of an instantaneous mechanism, i.e. the form

u(r) = ~ 'A(r - r ') , ( r 1 < r' < r 2) (8.5)

where p' > 0 specifies a time-independent compatible strain vector, i.e. q ' = Npf, and A(r -- r ' ) is the Dirac function. In fact, substituting from Eq. (8.5) into Eqs. (8.3) and

(8.4), or Eqs. (8.1) and (8.2), gives

T t t t S . q coswir = 0 , STq 'sincoir = 0 ( i = 1,2, . ,N) (8.6)

q' = Co. (8.7)

sm co.r cannot Since cos coir' and " ~ ' vanish contemporaneously

Eqs. (8.6) imply that

S T -, _ i q = 0 , (i = l, 2, . . ., N) (8.8)

or, in a more compact form,

Srq ' = 0. (8.9)

Sustituting from Eq. (8.7) and remembering that S = D C~,

Eq. (8.9) then becomes

dpTKv = 0 ( 8 . 1 0 )

which gives v = 0 and thus q ' = 0. So the assertion is proved and, as a consequence, in Eqs. (8.3) and (8.4) the interval

(r I , r 2) cannot be taken as infinitesimal.

We can equally show that a nontrivial admissible strain cycle cannot be a sequence of strain rate distributions diffe- ring from one another only within a multiplier, i.e. q(r) =

= q~ in which .q~ = Np ~ where p~ /> 0 is time inde-

pendent, and ~7 = r~(r) > 0 for r 1 ~< r ~< r 2, while ~7 = 0 for r < r I and for r > r 2. In fact, starting from Eqs. (8.1) and

(8.2) and following a reasoning path like the one used pre-

viously, we again easily arrive at Eq. (8.10) and this proves our statement.

As a result we can state that a nontrivial admissible strain

cycle actually needs to be a time sequence of strain rate di-

stributions not equal to one another at different times.

It is worth noticing that the latter two statements are typi-

cal of dynamic shakedown; within quasi-static shakedown,

none of them holds any more, except that plastic collapse is excluded.

The following theorem can now be proved:

THEOREM 5 (Kinematical theorem for later adaptation time)

A necessary and sufficient condition in order that a

structure subjected to a specified load history does not shake down or, if it shakes down, its adaptation time is

greater than an assigned time, r, is that there exists some admissible plastic strain cycle, ~(r) t> 0, which satisfies the inequality

! r {J(r + r)} r Nti(r) dr > k r /i(r) dr. (8.1 1)

J ~ " 0 ~

19 (1984) 14l

Proof. Either the structure does not shake down, or it

shakes down and its adaptation time is always (i.e., whatever the initial conditions may be) greater than r, in both cases

r is not a separation time and the problem (5.1), solved with

this r, must give ~* < 1. On the other hand, the dual problem (5.8) must have a solution tiC(r) constituting an admissible

strain cycle such that

fo ~* ~ aE(r + p)}T NIjC(r) dr = k r f~c(r) dr (8.12)

and therefore

fo f" {o2r(r + r)}TN~C(r) dr > k r ~C(T) dr

"0

(8.13)

which coincides with Eq. (8.11). So the necessity part of the Theorem is proved. Supposing, then, that there exists

an admissible strain cycle, ~ /> 0, which satisfies Eq. (8.11), since the left-hand member of t~he latter equation is positive,

this strain cycle proves to be feasible for the dual problem

(5.8) and thus we can write

o f~ ~* {og(r + r)}rNlJ(r) dr <. k r ~(r) dr (8.14)

where ~* is the safety factor (for fixed minimum adaptation

time) associated with the assigned r. Taking account of

Eq. (8.11) then yields ~* < 1 and this implies that r is not a separation time and therefore either the structure does

not shake down, or it has a minimum adaptation time greater

than r. So the sufficiency part of the Theorem is proved too.

Suitable limit considerations would show that the above

Theorem holds true also if the inequality (8.11) is considered

with the sign >/. From Theorem 5 there immediately follows

fled time, r, be a strict lower bound to the minimum adapta-

tion time of a structure which shakes down under a given

load history is that there exists some admissible plastic strain cycle, /J(r) ~> 0, which satisfies the inequality (8.11).

Proof. If r is a strict lower bound to the minimum adapta- tion time, r is not a separation time and so the proof of the

necessity part of Theorem 5 applies. On the contrary, if the inequality (8.11) holds, the sufficiency part of Theorem 5

can be applied to show that r is not a separation time.

THEOREM 8 (Kinematical Theorem for the upper bound to the minimum adaptation time)

A necessary and sufficient condition for a specified time,

r, to be an upper bound to the minimum adaptation time

of a structure which shakes down under a given load hystory

is that the inequality (8.15) is satisfied for every admissible

strain cycle, ~(r) ~> 0.

Proof. If r is an upper bound to the minimum adaptation

time, no admissible strain cycle can exist which satisfies

the inequality (8.11), otherwise r would be a strict lower

bound; so, every admissible strain cycle must violate Eq.

(8.11) and therefore satisfy Eq. (8.15). On the contrary,

if the inequality (8.15) is satisfied for every admissible strain

cycle, no such cycle exists which complies with Eq. (8.11)

and so r cannot be a strict lower bound.

The kinematical theorems tout-court are usually formulat-

ed to characterize inadaptation or adaptation of a structure

[5, 6]. They now take the following forms:

THEOREM 9 (Kinematical Theorem for inadaptation)

A necessary and sufficient condition in order that a struc-

ture subjected to a given load history does not shake down

is that there exists an admissible strain cycle, ~(r) 1> 0, which

satisfies the inequality:

THEOREM 6 (Kinematical theorem for earlier adaptation time )

A necessary and sufficient condition for a structure sub- jected to a speciefied load history to shake down and its

minimum adaptation time to be smaller than an assigned

time, r, is that the following inequality

J0 r { oE(r + r)}rNtJ(r) dr < k r ti(r) dr (8.15)

is satisfied for every admissible plastic strain cycle,/ i(r)/> 0.

( lim _ {oZ(r + r)}rN~(r) dr > k r ~(r) dr. (8.16)

r - * + ~ J 0 ~ J 0

The proof of this theorem is as for Theorem 5 applied

to an unlimited value of r. The Theorem holds good also

if the sign/> is substituted in the inequality (8.16).

THEOREM 10 (Kinematical Theorem for adaptation)

A necessary and sufficient condition for a structure

to shake down under a given load history is that the inequa-

lity

Proof.. This Theorem is proved by using Theorem 5, for

Eq. (8.15) is the negation of the inequality (8.1 1). Alternative forms of Theorem 5 and 6 are the following

o n e s :

THEOREM 7 (Kinematical Theorem for the lower bound to the minimum adoptation time)

A necessary and sufficient condition in order that a speci-

( f- lim _ {o~(r + r)}TNf~(r) dr < k r ft(r) dr

r - - * + ~ J 0 ~ ~ r

(8.17)

is satisfied for every admissible strain cycle, ~(r) >~ 0. Since Eq. (8.17) is the negation of Eq. (8.16), this theorem

can easily be proved by means of Theorem 9. It is worth noting that the integral

] 42 MECCANICA

W(r) = {ag(r + r)}rN~(r) dr, (8.18)

which is the work performed by the elastic stress response

truncated backward at r, 9qL~(r + r), through the admissible

strain cycle, can be given a different form in terms of the relevant load history. Since at every r 1> 0

N~ = C ~ - - D - I ~ , (8.19)

where ~, s are the elastic displacement rate and stress rate

responses to the strain cycle ~ = N~ considered as statically imposed strains, the work W(r) in Eq. (8.18), through the

virtual work principle, takes on the form:

/0 W(r) = {F(r + r) --m~E(r + r)}r} dr

+ {O(r + r)}r ~dr. (8.20)

In this way the work W(r) is expressed through the load

history truncated backward at r, including the inertia forces

associated with the elastic response to these loads. The

expression in the tight-hand member of Eq. (8.20) can be

substituted for the integral in Eq. (8.18) in all the above

kinematical theorems, so obtaining alternative forms of them. In the case of periodic loadings (Fig. l(b)), since W(r)

proves to be independent of r in this case, in the application of Theorems 9 and 10 the inequalities (8.16) and (8.17)

can be replaced by the inequalities (8.11) and (8.15), re-

spectively, using an arbitrary r, hence also r = 0. An analo- gous statement holds in the case of Fig. l(c), provided r is chosen sufficiently large. In the case of a loading of limited

duration (Fig. 1 (d)), it certainly proves to be

lim W(r) = 0, (8.21) r - + + ~

the inequality (8.16) cannot be satisfied and therefore

shakedown alway occurs.

To close this Section, we remark that Theorems 9 and 10

are expressed only in terms of the given loading history -

through the relevant elastic stress response, or even through

the load themselves - as well as of trial admissible strain

cycles, as one would expect a kinematical theorem to be.

On the contrary, in the body of the known kinematical theo-

rems [5, 6], there appear some additional ingredients (such

as the initial conditions) which seem to be spurious, for they

are competent to statical theorem formulations.

9. CONCLUSIONS

In the present paper, making use of a piecewise linear description of plastic behaviour, we have formulated statical,

as weU as kinematical theorems for dynamic shakedown of

discrete, elastic-perfectly plastic structures subjected to a specified load history. In doing this, a systematic use has

been made of the results of a modal analysis, showing that the dynamic characteristics of the structure (such as displa-

cement and stress mode shapes) may be helpful, at least

as a theoretical tool, in dynamic shakedown theory, as well

as in the wider field of dynamic plasticity. But this is a

point to further clarify, particularly in relation to continuous systems.

The generalized Ceradini theorem has been reformulated

in terms of the two concepts of separation time (i.e., the

lower extreme of the time interval which Ceradini's theorem

refers to) and of backward truncated elastic stress response (i.e., the elastic stress response at time subsequent to the separation time).

This has enabled us to derive some useful features of dy-

namic shakedown and in particular to introduce the concept

of minimum adaptation time, which seems to be novel. Also,

the concept of safety factor for fixed separation time has been introduced and two problems, which are dual of one

another, have been formulated showing that the usual shake-

down safety factor, if any, is the limit of the above safety

factor associated with the separation time tending to diverge.

We have also found that the so-called <<admissible strain

cycle>> of dynamic shakedown can be defined in a more ap-

propriate way. Through this definition, several kinematical theorems have been formulatea, among which lower and

upper bound theorems for the minimum adaptation time.

In particular, the two kinematical theorems for inadaptation

and for adaptation, which are the equivalent of those given

in the literature, seem to have a more appropriate form

since the former contain only the given load history and the

trial admissible strian cycle.

In the author's opinion, the results presented in this pa-

per, although obtained within some restrictive hypotheses

(such as piecewise linear plasticity, discrete models, etc.)

should hold also in wider structural contexts in which the

above restrictions are relaxed. Such generalizations, as well

as other aspects of the theory, will be presented in a later paper.

Received:February, 10, 1983.

19 (1984) 143

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144 MECCANICA