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Computation of bounds on chosen measures of real plastic deformation for beams
Compurers & Srrurrures Vol. 61, No. I, pp. 171-182. 1996 Copyright 0 1996 Elsevier Science Ltd Printedin Great Britain. All rights reserved
PII: SOO45-7949(96)00021-l
0045.7949/96fl5.00+ 0.00
COMPUTATION OF BOUNDS ON CHOSEN MEASURES OF REAL PLASTIC DEFORMATION FOR BEAMS F. Giambancot and L. Palizzolo Dipartimento di Ingegneria Strutturale e Geotecnica, DISEG. University of Palermo, Viale delle Scienze, 901X-Palermo, Italy (Received 7 November 1994)
Abstract-A bounding technique for producing plastic deformation in elastic-plastic beam structures subjected to the contemporaneous action of a steady load and a cyclically variable load, such as the combined load results slightly above the shakedown limit, is studied. The technique is based on the proportionality between the kinematical part of the solution to the Euler-Lagrange equations relative to the shakedown cyclic load factor problem (for the structure subjected to a fixed steady load and to an amplified cyclic load), and the gradient, with respect to the cyclic load multiplier, of the kinematical part of the elastic-plastic steady-state response of the structure to loads at the shakedown limit. A suitable bounding principle allows us to evaluate a bound on the proportionality factor between the solution to the above mentioned problems for loads at the shakedown limit. Knowing such a bound and the kinematical part of the solution to the shakedown load factor problem, it is possible then to compute other bounds on any measure of real plastic deformation produced by loads slightly above the shakedown limit. A numerical example concludes the paper. Copyright 0 1996 Elsevier Science Ltd.
1. INTRODUCTION
study of the behaviour of elastic plastic structures subjected to a combination of a cyclic and a steady load has interested many researchers in recent years and a considerable effort has been devoted to this topic. The possibility of obtaining an approximate response with lower computational effort than the occurring one for a complete analysis, has also been studied. The result has been reached either having recourse to the approximated step-by-step analysis processes [ 11,or having recourse to suitable bounding techniques (see Refs [2-S]). A special bounding technique for plastic deformation, valid for elastic perfectly plastic structures subjected to the contemporaneous action of a steady mechanical load and a cyclic mechanical and/or kinematical load slightly above the shakedown limit, was studied by Giambanco, Palizzolo and Polizzotto [6]. The technique is based on the knowledge of the solution to the shakedown cyclic load factor problem (see e.g. Refs [2, 3, 7-111). Since the kinematical part of the solution to the Euler-Lagrange equations relative to such a problem (or, directly, the kinematical part of the solution to such a problem) is proportional to the gradient, with respect to the cyclic load multiplier, of the steady-state elastic-plastic response of the structure to the loads at the shakedown limit [ 121,determining first a bound on the proportionality factor between The
t To whom correspondence should be addressed.
the two solutions, it is possible then to compute other bounds on any chosen measure of real plastic deformation for loads slightly above the shakedown limit. The formulation proposed in the previous paper, although interesting from a theoretical point of view, needs to be specialized to the case of particular structure typologies in order to be applied. As a consequence, for computational purposes, an appropriate formulation for the bounding principle is desirable. A first step towards this direction was already undertaken by the present authors [ 131 in a paper devoted to the specialization of the bounding technique to the case of trusses. In the present paper the described bounding technique for the special case of beam structures is developed. Beam structure behaviour in steady-state is studied and the gradient, with respect to the cyclic load multiplier, of the elastic-plastic steady-state response to loads at the shakedown limit is obtained. The shakedown load factor problem for beam structures and the related Euler-Lagrange equations are studied. A suitable bounding theorem, which provides the way of computing a bound on the proportionality factor between the solution of the above mentioned problems, is proved. Knowing such a bound and the kinematical part of the solution to the shakedown load factor problem, it is possible then to compute other bounds on any measure of real plastic deformation produced by loads slightly above the shakedown limit. A numerical application confirms the reliability of the proposed bounding technique. 171
173
Real plastic deformation for beams elastic response to the cyclic load and a residual (self-equilibrated) bending moment, M,, i.e. M(.K, t) = MO (x) +
MC(.K,t) + Mr
(8)
cu.t) i.e.
= - Sg M’: - SC w:l + M,,
(5)
A4 = -AM’:) in (0, L). in which u’, = M’”(x) and n’E= u: (x, 1) are the elastic deflections produced by the loads P, and PC, respectively, and 5 < 5,” and z > 5: (&) are two selected load multipliers, determining the load PC = & P, + E PC. Denoting t as an instant subsequent to f0 and T = f - to as an auxiliary time variable within the range (0, At), the laws governing the steady-state phase can be written in the following form:
(9)
The solution to eqns (6a-i) or (6a-h) and (7~(9) is essentially unique [12]. In any case, the plastic dissipation density rate fi=
Mlip
(10)
turns out to be uniquely determined
in (0, L) and
VT E (0, At).
cp+= -S&w:-S&w:‘+M,-M,
,i+>O,
in (0, L),
VT E
(0, At),
cp-= +S%uS+SE,~l~-M,-M,~O, Lm cp- = 0
(6a)
A_ 20,
in (0, JC.),
VT E
(0, AZ), (6b)
in (0. L),
(SW:‘)” - P, = 0
kin. and mech. b.c.s. on PV,, (6~) in (0, L),
(Sc11,“)” - PC= 0
If in some region of the structure plastic accumulation Ak, # 0 exists, incremental collapse occurs; otherwise, if Ak, = 0 in the whole structure, low cycle fatigue of the material is probably induced.
4. THE STEADY-STATE RESPONSE TO CYCLIC THE SHAKEDOWN LIMIT
LOAD AT
When (still provided & > 5:) the multiplier &+{t+, the limit response is characterized by a plastic strain rate field and a residual stress rate field identically vanishing, namely
VT E (0, At),
lim Ic$= 0, <<_<‘i
lim tir = 0 :,+;;+
kin. and mech. b.c.s. on MJ~, (6d) in (0, L), M:’ = 0
in (0, L),
& = 1, - j._
mech. b.c.s on M,,
in (0, L),
VT E (0, At),
(60
VT E (0,At),
(6g)
k, = i Q, + kp in (0, L,) /& = -a;
in (0, L)
VT E (0,At),
VT E
(0, At),
(11)
(he)
kin. b.c.s on i,,
so that the residual stresses set up a time-independent self-stress field. i.e. li~li~+M, (x, t) = M,+(x) c
in (0, L)
(12)
Equations (6a-i), (7H9) become meaningless. However, by introducing the new quantities
(6h)
where i, = ti’,(x, t) and k, = f, (x, t) denote the residual displacement and strain rates, respectively. Equation (6i) implies that the plastic accumulation in the cycle
Ak, =
AI k, dr, s0
(13b)
(7)
is independent of to and compatible in (0, L) with the deflection increments in the cycle
(13c) (13d)
174
F. Giambanco and L. Palizzolo
equations (6a-i), (7~(9) take on the form =%
LP.,Aw:dx s0
~:=-S%W~-S~~W~+M~-M,~O, 8
3 0,
cpz i: = 0
in (0, L),
Vr
E
(0, At),
(18)
(14a)
,? 2 0,
q? 1: = 0
in (0, L),
Vt E (0, At),
By virtue of equations (14g, h), (15) and (17). and by virtue of the virtual work principle
(14b) (SW:)” - P, = 0
in (0, L),
km. and mech. b.c.s on w,, (SW$)” - PC = 0
in (0, L),
Vt
E
= 0
in (0, L),
I;,’ = 1: - I?
= %
(0, At),
kin. and mech. b.c.s on w. (Ml)”
(14c)
mech. b.c.s on M,i,
(14d) =,$I(-Sw:)(;i@)dxdt
(14e)
in (0, L),
Vt E (0, At),
(14f)
in (0, L),
V7
(0, At),
(14g)
f: = -(I+:)”
in (0, L),
E
” L JJ
=
in (0, L),
= &
(14i)
Ak,f =
bf I;,’ dr s0
in (0, L),
(15)
Aw: =
AI @: dr s0
in (0, L),
(16)
in (0, L),
(-SW;)@ dx dz
E
I II
Ak: = -(Aw:)”
dx dt
0
(14h)
d, &: dr = 0
(-Sw:)I;;
40
0
VT E (0, At), kin. b.c.s on ti$,
(-SW;)@ dx dr
II 0
+ f:=-@+~;
ArL JJ
=
’ (- Sw:)Ak,+ dx, s0
(19)
Ar L JJ
(- Sw:)k,? dx dz
5:
0
II
kin. b.c.s on Aw:. (17)
Equations (14a-i), (15)-(17) describe a deformation process for which the gradient with respect to the cyclic load multiplier of external work in the cycle, for loads at the shakedown limit, is given by:
ss AI
E’ = E,’ + E,’ = &
5:
=
P, ti: dx dt
0
ss Ar
L
0
A,
(-sw:‘)k;
0
0
dx dr.
(20)
0
L
JJ 0
L
M+I;+ P P dx dr =
L
M;Ak;
J0
dx = 0,
(21)
175
Real plastic deformation for beams
the plasticity conditions at any point and at any instant. Consequently, the problem of the shakedown cyclic load factor identifies itself with the following search problem:
equation (18) can be written as
5: = (-
Sw:))I;; dx dr
max , & subject to: I;,.;.,.*,,.“...H,I
(24a)
cp: = - sy, ,c: - S& ir:’ + l44; - M, < 0
d-(/i;) dx dr = D’ > 0,
=
(22)
where d+ = d+(f,+) is the gradient of the plastic dissipation density rate with respect to the cyclic load multiplier. The result of the latter is that the gradient with respect to the cyclic load multiplier of the external work in the cycle for loads at the shakedown limit, E', is equal to the gradient of the plastic dissipation in the cycle produced in the whole structure D’. The constants E,'and E: can be approximately calculted by the formulae
E,’
2
5:
ss Al
E - 5: o
I
0
( - SW:‘&, ds d7.
(23)
where k,, and Ak, are the real plastic deformations produced by the load PC = & P,+ 6 P,,being E slightly above the shakedown limit. The constants E,'and E,'from a qualitative ponit of view, characterize the steady-state plastic behaviour of the structure in the presence of a combination of a steady and a cyclic load slightly above the shakedown limit [I]: if E,'= 0 and E,'# 0 plastic shakedown occurs; if E,'# 0 and E: # 0 ratchetting occurs. 5. THE SHAKEDOWN LOAD FACTOR FOR CYCLIC LOADS
The necessary and sufficient conditions under which the relevant plane beam structure shakes down can be established through the statical theorem (Melan’s theorem). In the examined case, it states that shakedown occurs if a self-equilibrated time independent bending moment field W0 = M; (x) exists such that the total bending moments, obtained by summing up to the self stresses the moments produced by the fixed and cyclic loads, do not violate
in (0, L),
VT E
(0, At),
(24b)
in (0. L),
VT E (0. Al).
(24c)
(SW:)” - P,= 0 in (0, IY). kin. and mech. b.c.s on II’,. (24d) (SW:)” - P,= 0 in (0, L),
Vr E (0, AI),
kin. and mech. b.c.s on w’,, (24e)
(M;)” = 0
in (0, L),
mech. b.c.s on M;,,
e - 5<,< 0.
(240 (248)
Following the Lagrange multiplier method, denoting qc = (E: /<:) > 0 as a suitable constant, and I:; = ;: (x, r) 2 0, 1: = i: (x, I) > 0, Au, = AU, (.Y), 50 Aw; = 50 Aw; (x) and ‘I<> = (E, 150) > 0, re +: = ic ti: (x. f) as the Lagrange multipliers (r. and & in the last two terms playing the role of scaling factors not subjected to variation), the enlarged functional of problem (24) is
1(1= - & ‘I<+
dr 1. [( - SY, w6 -sy,H;I)+M;) ss0 0
x (;.: - A’ ) - M, (2+ + ;:‘)I dx ds
-
-
+
’ [(SW:)” - Po]l" Aw; dx s0 hr I ,, [(SW:)” - Pc]& ti; dx dr ss0
’ (M‘,)“Au, dx + (5 -
(25)
Taking the first variation of eqn (25) to all the variables, which are functions be as smooth as necessary, and since $ minimum with respect to the variables
with respect assumed to must have a of problem
176
F. Giambanco and L. Palizzdo
(24) and a maximum with respect to the Lagrange multipliers, the relevant Euler-Lagrange equations of problem (24a-f) can be obtained; they are: cst’-= -s<,
d - sy, d + nn;: - MY< 0, f: >, 0,
cp: k+ = 0
in (0, L),
Vr E (0, At)?
(26a)
rp’_= +.Szl W:+sq;w:-~@-M~~O, n’p_2 0,
cp’-if.. = 0
in (0, L),
Vt E (0, At), (26b)
% - C0< 0, (Swt)” - P, = 0
t/o (<. - Co) = 0, (26c)
V0> 0,
(26d)
n’r: = s& - 5)
in (0, JC,), mech. b.c.s on ML, in (0, L),
(@,
kin. and mech. b.c.s on tit:, (26g)
A& = -A&
in (0, t), dl
kin. b.c.s on Au,,
in (Q, L)$ tf? E (0, A?), (equilibrium)
(29c)
J^
(26h)
(30) it is possible
variable increment
(26i)
give of residual i.e.
L
-.Sw:~;dxdr=E,>OO,
G
II
(29b)
in which eqn (29a, b) is formally identical to eqn (14g, h), respectively. Since e and pP are cyclic, the moment field k must also vary cyclically, namely
@A&)” + ((SA@>“f” = 0 in (0, L), kin. and me&. b.c.s on Awit
(elasticity law)
homg. mech. b.c.s on &$
(260
‘dt E {O,At),
= 0
(29a)
in (0, L),
t/r E (0, Af)
Vr E (0, At),
in (0, L),
(S@” + (S(@)“)” = 0
in (0, L), VtE (0, At), kin. b.c.s on 6; (compatibility)
kin. and rnech. b.c.s on We, (26e) (Mi)” = 0
e = -(lit;)”
in (0, L),
kin. and mech. b.c.s on w,, (SW!)” - P, = 0
the role of residual ~sp~a~rn~~t rates. The eqns (26a, b, d, et(28) are formally identical to eqns (14a-d, f) and (15), respectively. Equation (26g) states that rC$is the residual displacement rate field produced by the piastic strain field ?$,, and that both vary cychcally with period At. This implies that G; and k; are not compatible, that k’; generates a self equilibrated residual bending moment rate field e in the structure, to which a residual curvature rate field 5 is associated, compatible with displacement +$ (and therefore also variable cyclically with period At). The eqn (26g) is thus equivalent to the equation system:
(26j)
Awr‘
c
where & = <: results, and where it has been set as !$, = ;t’: - ?_,
in (0, L),
Vr E (0, At),
AU,, harm,
(27) eqns
&r h$ dz Ak; = ,6
in (0, L).
(28)
work rates
ds
in
(31)
if definition (31) by virtue is verified too. (26i) that strain &k;, with we take that results to By virtue e) of work following same path left hand side in k) to represent by P, through mechanism by setting
The eqns (26a-k)-(28) describe some kind of plastic deformation process, associated with the cyclic loading P&!= 50 p0 + e: PC: in which !$ ptay the roIe of plastic strain rates, 2, and & play the rob of plastic multipliers, while @ pfay
taking into (32)
dimensional k)
that identical It is very to note that D+ introduced 4 meaning very
177
Real plastic deformation for beams value, while constant D‘ is a simple proportionality factor with an arbitrary value. Therefore the problems described by eqns (26a-k)(28) (32) and by eqns (14a-i), (15t( 18), (22), respectively, are the same except for the constants D+ and D”, while their solutions are equal, except for the kinematical variables, which differ for the ratios D‘/D+( =E‘/E+). If, in particular, E‘ = D‘ = 1 one has: ;;+t = E+&,
1:: = Efj”“’
&+ = E’k“P P
in (0, L), Ak; = E+Ak;,
Aw,! = E+A$
Vr E (0, At),
(33a)
in (0, L),
(33b)
If the solution to problem (26a-kt(28) provides &, > & (until now 5” = z,, has been supposed), the shakedown limit load multiplier <: shows insensitivity to the steady load variation, and this occurrence takes place when the point (&,,
6. THE BOUNDING PRINCIPLE
The steady-state elastic-plastic analysis of the structure subjected to the loads PC, = 5 P, + &PC, with & > 51, can be performed either through the solution to eqn (6a-i) or by a step-by-step full analysis of a convenient number of cycles until the steady state phase is reached. In any case, these approaches involve a considerable computational effort. However, for loads slightly above the shakedown limit (in practice E 15: = 1 + 1.2, approximately) it is possible to establish which collapse mode for the structure is to be expected, keeping the computational effort down. Equation (33b) allows us to establish which impeding collapse mode is going on at the shakedown limit, on the grounds of the solution to the shakedown load factor problem. If, for instance, Ak:, = 0 there is plastic shakedown, etc. If we know the constant E’, eqn (33), transforms into 2, z E+i:(&<:),
i_ s E+A’ (z-
k, z E+L$ (c - 5:)
in (0, L),
Aw, z E+A#
(& - 5:)
in (0, L),
and consequently the plastic part of the steady-state elastic-plastic response of structure to loads PC slightly above the shakedown limit, by utilizing the solution to the shakedown cyclic load factor problem could be calculated. Unfortunately, in order to compute E+, for instance by eqn (23) the plastic strain rate f, must be known, i.e. the solution to the analysis problem must be known and this makes further investigations needless. In order to sustain a computational effort, reduced with respect to the one necessary to solve the analysis problem, a suitable bounding technique may be employed. An upper bound E* on the constant E+ can be calculated, i.e. E’ < E*.
(35)
Then, utilizing eqns (34b) and (35) one can write IA& I d WA&
I(% - 53,
IAw, I < E*lAw: l(C - 5:)
in (0, L),
(36)
and similarly one could do it for the quantities present in eqn (34a). Relation (36) and other similar relations constitute bounds on the absolute value of the plastic part of the elastic-plastic response to the load Pt. The expression of E* can be obtained in the following way. Let BS denote the given continuous elastic-plastic beam structure, constituted by material for which associated plasticity laws hold [l l] and subjected to the load PC, whose behaviour is already described in Section 3. In addition to the compatibility and equilibrium equations, throughout (0, L) and for all instants r within the interval (0, At), the following relationships have to be satisfied:
k,=;M=
cp+ = +lV--IV,=
k = k, + k,,
(374
-%w,-Cw:+;Mr,
(37b)
-s&w: _ se WEll + M, - MY < 0,
cp_= -M-M,=
A+>O,
(37c)
+Scw: +SFw,’
t;),
1_ 20,
- Mr - My < 0,
(37d)
,&,=A+ -A_,
(37e)
t/r E (0, At), (34a)
(34b)
cp+ x, = cp_ 1_ = 0.
(370
178
Giambanco and
Palizzolo
-L Fig. 1. M and corresponding Drucker’s postulate I] can - M)&
0
in
structure: geometry
through applied
L),
t/r
(37c-f),
(0, Alt),
where &? is any bending moment fieId, provided that it is plastically admissible, i.e. so as to satisfy eqn (37c, d). Let BS* denote a fictitious structure. All the quantities related to BS* are marked by apex*. BS* is the same as BS, except for plastic behaviour: the complementarity constraints (37f) can be violated and the yield function is suitably perturbed. By symbols, throughout (0, L) and for all instants 7 within the interval (0, Ar), one has k* =k: +k,*,
(W
A eqn
like the defined, for but not (37fj hold, for which (40), but eqn (38) is called pseudoplastic. Stresses m* are admissible in BS*, but through eqn (37c, d) they also are admissible in BS, so that eqn (38) can be applied with R = m* (M* - M)& + a(-SC wl- .SC$ w:)k,< 0 in (0, L),
‘VTE (0, A.t). (41)
Stresses M are admissible in BS, but through eqn (39c, d) they also are admissible in BS*, so that eqn (40) can be applied with I@ = h4 - (M* - M)/c,* - a(-SZ w: - SL.$ WE)/+
By summing up eqns (41) and (42) one obtains
cpf=+m*-M,=+M* +a(-S&w:
load condition.
- sg Ml:')-My G 0, (39c)
(-S&
w: - St: WI’)& G (- SC0w: - sgt wc”)It$
lpp”=--m*-j&=-&f* _ a(-S&
n,* > 0,
w: - SC:WC)- My < 0
A? 3 0,
I$ = x,” - a?,
(39d)
in (0, L),
V7
E
(0,At),
(39e)
where the scalar CI> 0 is the multiplier of the perturbation, which is the elastic stress response f - sg UC - SC: WC)to load Pr:=&Po+t:Pc.
Because of the loosened form of eqn (39) with respect to eqn (37), the stress history M* in the cycle is not unique (as M is); let us suppose that we know such history. To m* and @, corresponding through (39c-e), the generalized Druker’s postulate [ 1I] can be applied
in (0, L),
VTE (0, Abt). (40)
g s> CL
Fie. 2. Cvclic load as a function of time.
(43)
179
Real plastic deformation for beams
Iktir
rletnrnts
with plastic hinges
Fig. 3. Discretization of the beam in eight elastic elements with plastic hinges and rigid nodes.
which, integrated throughout interval (0, At), becomes
(0, L) and within
the =
A1 L(M*-M)(f*-I;)dxdS ss0 0
-
ss
L “(M*-M);(ti*-ni)drdx=O, II II (46)
because by virtue of the virtual work principle it is
s
L(M*-M)(I;*-I;)dx=O, 0
(47)
and for the periodicity of (M* - M) is ” L (M* - M) (I$’ - &) dx dr. ss0 0
(44) “(M*-M);(ti*-ti)dr=O. s0
(48)
Taking into account that (f,* - f,) = (f* - I;) - ; (tii* - A&,
(45)
the third integral on the right-hand side in eqn (44) can be written as
L(M*
- M)
t-
(I;,* - f,) dx dz
L12
-
Equation (44), divided by (% - rs), taking into account eqns (46) (22) and (23), takes on the form
E+ = E,’ + E: z&
( - SW:‘)& dx d7
LIZ
V
Fig. 4. The bounded quantity: ratchet deflecton Aw, of the left span middle point.
180
F. Giambanco and L. Palizzolo
2.8
I‘-
2.8
2.6
:
2.4
2.4 2.2
l $
2.0
tk? +w
1.8
’
‘, I
I
\ \ \
2.0 1.6 1.2
1.6
0.R
1.4
0.4
1.2 1.0
I
I
I
I
0.2
0.4
0.6
0.8
0.8
w
IO
5”
0.6
Fig. 7. Curve of the quantity E+& (solid line) and its bound E*S&(dashed line) as functions of 5..
0.4 0.2 0.2
0.4
0.6
0.8
T E (0, At), of the quantities A,*, A!!, I;,*, M*, relative to the fictitious elastic-pseudoplastic process. These can be obtained by an elastic-pseudoplastic analysis, i.e. an incremental analysis with loose complementarity constraints (37f) [l]. The (not unique) solution to an elastic-pseudoplastic analysis problem can be obtained in various ways; an example will be shown hereafter in the application stage.
1.0
Fig. 5. Bree-diagram of the beam.
(- Sw;)Ak; dx
7. APPLICATION
(-SW:)@ dx dr
x dxdr
=
E,*+E: +E,*=E*,
(49)
which is the searched bounding principle (35). The computation of the right-hand side of eqn (49) implies the knowledge, in (0, L) and for all
In a previous paper, devoted to the optimal shakedown design of beams, Giambanco et al. [ 151 studied the case of a continuous beam constituted by two spans of length 15 (Fig. 1). Both spans were subjected to a uniformly distributed steady load P, and the span on the left was also subjected to a uniformly distributed load, cyclically variable between -P, and +P, (Fig. 2). The beam had a rectangular cross-section with dimensions B and H, and the width B was assumed as the design variable.
t 1.4 t 1.2 -
2.4 1.0 '4 '
0.8 -
+$ w
0.6 -
: :
i
0.4 -
2.0
b2.
1.6 -
?1
I.2 -
\ \ \ \ \ \
0.8 -
0.2 I 0.2
t 0.4
0.6
0.8
1.0
5"
Fig. 6. E,+6& (dashed line) and E:6& (solid line) diagrams as functions of r., for b& = O.l[:.
Fig. 8. Curves of the ratchet deflection Aw, (solid line) and its bound Aw,Z(dashed line) as functions of 5..
181
Real plastic deformation for beams The data
of the problem were: L = 400 cm; E = Young’s modulus = 21,000 kN cm-*; oy = yield stress = 40 kN cm-*; P, = 500 N cm-‘; P, = 500 N cm-‘. The beam was discretized in eight portions of equal length AL = L/4, with piecewise constant width B, (i = 1,2, . . ) 8), having purely elastic constitutive behaviour everywhere, except at the end cross-sections, which had rigid perfectly plastic constitutive behaviour (Fig. 3). The solution to the design problem provided the optimal widths: B, = 14.05 cm; BZ = Bi = 7.99 cm; B4 = Bs = 9.59 cm; B,, = B, = 5.20 cm; B8 = 5.03 cm. Afterwards the shakedown domain of the optimal beam was plotted and it was possible to verify that
SY, + bn > 0,
Yn 2 0,
(SY. + b.)TY, = 0,
H = 1Ocm;
min f YZSY, IY,,) SY, + bn 2 0,
(n = 1,2,.
.) (50)
where n is the index of nth step, SY, + b, = -cpn is the opposite of plastic potential vector of nth step, Y, is the plastic attivation increment vector of nth step, b, is the known term vector of nth step (taking into account the loads of nth step and the accumulated plastic strains at (n - 1)th step, as initial conditions of nth step), and S is a symmetric and positive semi-definite matrix. Rather than solving the sequence of problems (50), the equivalent sequence of quadratic programming problems has been solved
subjected to the conditions (n = 1, 2, (Yn 2 0)
the relevant structure found itself just at the shakedown limit. Due to this last beam feature, it is very interesting to analyze the structure behaviour for loads slightly above the shakedown limit and compute bounds on chosen measures of real plastic deformation. In the present application, the above referred optimal beam is considered and a bound on the ratchet deflection of the left span middle point (Fig. 4) is computed by means of the proposed technique. The following data are introduced: a = perturbation multiplier = 0.03; S& = cyclic load multiplier increment slightly above the shakedown limit = c - 5: = O.ltE. It is not necessary to specify
Y,* 3 0
the length of period Ar because, as noticed in Section 2, time only plays the role of an orderly parameter of the events. In Fig. 5 the Bree-diagram of the beam on plane (to, &) is represented. The trilateral line separating the zone E (elastic zone) and S (shakedown zone) to each other provides the elastic multiplier <: as a function of parameter &,, i.e. 5: =
(51)
i where the constraints Y. > 0, even if not explicitly imposed, are satisfied in the optimal conditions. In Fig. 6, the values of quantities E,+6& and E,+6<,, as functions of the parameter &, computed by eqn (23) for StC = 0.15: are represented. In the region relative to the alternating plasticity, E,’ is zero and E,’ is constant with respect to t0 ; in the region relative to the incremental collapse mode E,’ and E,’ vary as functions of &, vanishing at L = C. In order to obtain a fictitious plastic strain admissible history, to utilize in computing the bound E* on E+, a sequence of linear programming algebraical problems has been solved as
min aTY,* subjected to the conditions CYY, SY,* + b.* 2 0,
.),
(n = 1,2,
.)
(52)
1 where Y,* is the fictitious plastic activation increment vector of nth step, b,* is the known term vector of nth step (taking into account the loads and the perturbation at the nth step, as well as the accumulated fictitious plastic strains at (n - 1)th step, as initial conditions of nth step), SY,* + b,* = -cpz is the opposite of fictitious plastic potentials at nth step, while the objective function an is a suitable fixed vector. The solution Y.* of the sequence of problems (52) is a feasible solution of the sequence of problems SY.* + bt 2 0,
Yt > 0
(n = 1,2, . .)
(53)
similar to the sequence of problems (50), without the complementarity constraints. Usually, vector u. is chosen so that-at least foreseeably- the solution to eqn (52) is not much different than the one which would provide eqn (53), if the complementarity constraints (SYf + bX)TYd = 0 were present [5,63.
182
F. Giambanco and L. Palizzolo
In Fig. 7, the values of quantities E’S& and its bound E*@,, as a function of &,, computed by eqns (18), (23) and (49), are represented. In Fig. 8, the values of the ratchet deflection of the left span middle point Aw, = Aw, (L/2) and its bound, Aw,* = AwP (L/2), are represented. Near the first knee of the multiplier diagram t;s (&,), Aw, > Aw ,*, can result, even if only a little. This latter circumstance, absurd as it may appear, can be explained taking into account that the quantity Aw: appearing in eqn (34b) has been computed at the shakedown limit, while E+, E* and Aw, have been computed at 10% above such limit. The quadratic programming problems (5 1) and the linear programming problems (52) have been solved by means of a QPSOL FORTRAN routine [16] and by using a 3090J/VF IBM computer. 8. CONCLUSIONS The elastic-plastic response of beam structures subjected to a combination of cyclic and steady loads has been studied. The gradient, with respect to the cyclic load multiplier, of the elastic-plastic beam structure response to loads at the shakedown limit has been found. It has been shown that the relationship describing such a gradient is formally identical to the Euler-Lagrange equations relative to the shakedown load factor problem (for the structure subjected to a fixed steady load and to an amplified cyclic load), within a proportionality factor tying the kinematical part of the solutions to the two relevant problems. The kinematical part of the solution to the shakedown load factor problem for a beam structure can, therefore, provide some useful information on the steady-state elastic-plastic response of the structure subjected to a steady load and a cyclic load, such that the combined load condition be slightly above the shakedown limit. Simply by checking the increment in the cycle of the variables which play the role of plastic deformations, one is able to establish whether (in the real process) incremental collapse or alternating plasticity occurs; together with the computation of a proportionality factor, it consents to check, by verifying the identities (33a, b), the precision of the analysis problem solution slightly above the shakedown limit (identities (33a, b) tend to be exactly verified for &-+5:‘); executing an elastic-pseudoplastic analysis (which requires a computational effort lower than the required one for a full incremental elastic-plastic analysis) and computing a bound on the proportionality factor between the two responses, one is able to calculate other bounds on any chosen measure of real plastic deformation.
For the elastic-plastic beam model description, the simplifying hypothesis that the elastic and plastic strains are both linearly distributed in the beam height has been utilized. However, the authors believe that the described technique can also be easily performed in the case where the aforementioned hypothesis is removed. The effected numerical application allows us to confirm the reliability of the proposed bounding technique. REFERENCES
1. C. Polizzotto, Simplified methods of structural analysis for cyclic plasticity. Commission of the European Community, Nuclear Science and Technology, Final Report, contract no. RA 1-0116(N). 2. M. Capurso, L. Corradi and G. Maier. Bounds on deformations and displacements in shakedown theory. Matitriaux et Structures sous Chargement Cyclique, Ass. Amicale des Ingtnieurs A&ens Eltves de 1’E.N.P.C.. Paris. DD. 231-244 (1979). 3. J. A. Kanig, On u&er bounds ;o shakedown loads. 2. Angew. Math. 59, 349-354 (1979). 4. C. Polizzotto. A unified treatment of shakedown theorv and related dounding techniques. S.M. Arch. 7, 19-7i (1982). 5. F. Giambanco, M. Lo Bianco and L. Palizzolo, A
6.
I.
8.
9.
10.
11. 12. 13.
bilateral convergent bounding technique for plastic deformations. kfeccanica 25, i81-188 (i990). F. Giambanco. L. Palizzolo and C. Polizzotto. Delimitazione delle deformazioni plastiche per carichi oltre il limite di adattamento. In Atti X-Congress0 Nuzionale AIMETA. 1. vv. 159-164 (1990). D. A. Gokhfeld and’0. fi.&Cherniavsky, Limit Analysis of Structures at Thermal Cvcling. Sijthoff and Noordhoff, Alphen aan den dijn, The Netherlands (1980). W. T. Koiter, General theorems for elastic-plastic solids. In: Progress in Solid Mechanics (Edited by J. N. Sneddan and R. Hill), Vol. 1. pp. 167-221. North Holland, Amsterdam (1964). J. A. Kiinig, Shakedown of Elastic-Plastic Structures. PWN-Polish, Warsaw and Elsevier, Amsterdam (1987). J. A. Kiinig and G. Maier, Shakedown analysis of elastoplastic structures: a review of recent developments. Nucl. Engng Des. 66, 81-95 (1981). J. B. Martin, Plasticity: Fundamentals and General Results. MIT Press, Cambridge, MA (1975). T. Panzeca and C. Polizzotto, On shakedown of elastic plastic solids. Meccanica 23, 94101 (1988). F. Giambanco and L. Palizzolo, Bounds on plastic deformations of trusses. Int. J. Solids Struct. 31, 785-795 (1994).
14. C. Polizzotto, On the conditions to prevent plastic shakedown of structures; Part I: Theory, Part II: The plastic shakedown limit load. J. uppl. Mech. 60, IS-25 (1993).
15. F. Giambanco, L. Palizzolo and C. Polizzotto, Optimal shakedown design of beam structures. Struct. Optimization 8, 156167
(1994).
16. P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, QPSOL: a FORTRAN package for quadratic programming. Technical Report SOL 84-6, Department of Operations Research, Standford, CA (1984).
PII: SOO45-7949(96)00021-l
0045.7949/96fl5.00+ 0.00
COMPUTATION OF BOUNDS ON CHOSEN MEASURES OF REAL PLASTIC DEFORMATION FOR BEAMS F. Giambancot and L. Palizzolo Dipartimento di Ingegneria Strutturale e Geotecnica, DISEG. University of Palermo, Viale delle Scienze, 901X-Palermo, Italy (Received 7 November 1994)
Abstract-A bounding technique for producing plastic deformation in elastic-plastic beam structures subjected to the contemporaneous action of a steady load and a cyclically variable load, such as the combined load results slightly above the shakedown limit, is studied. The technique is based on the proportionality between the kinematical part of the solution to the Euler-Lagrange equations relative to the shakedown cyclic load factor problem (for the structure subjected to a fixed steady load and to an amplified cyclic load), and the gradient, with respect to the cyclic load multiplier, of the kinematical part of the elastic-plastic steady-state response of the structure to loads at the shakedown limit. A suitable bounding principle allows us to evaluate a bound on the proportionality factor between the solution to the above mentioned problems for loads at the shakedown limit. Knowing such a bound and the kinematical part of the solution to the shakedown load factor problem, it is possible then to compute other bounds on any measure of real plastic deformation produced by loads slightly above the shakedown limit. A numerical example concludes the paper. Copyright 0 1996 Elsevier Science Ltd.
1. INTRODUCTION
study of the behaviour of elastic plastic structures subjected to a combination of a cyclic and a steady load has interested many researchers in recent years and a considerable effort has been devoted to this topic. The possibility of obtaining an approximate response with lower computational effort than the occurring one for a complete analysis, has also been studied. The result has been reached either having recourse to the approximated step-by-step analysis processes [ 11,or having recourse to suitable bounding techniques (see Refs [2-S]). A special bounding technique for plastic deformation, valid for elastic perfectly plastic structures subjected to the contemporaneous action of a steady mechanical load and a cyclic mechanical and/or kinematical load slightly above the shakedown limit, was studied by Giambanco, Palizzolo and Polizzotto [6]. The technique is based on the knowledge of the solution to the shakedown cyclic load factor problem (see e.g. Refs [2, 3, 7-111). Since the kinematical part of the solution to the Euler-Lagrange equations relative to such a problem (or, directly, the kinematical part of the solution to such a problem) is proportional to the gradient, with respect to the cyclic load multiplier, of the steady-state elastic-plastic response of the structure to the loads at the shakedown limit [ 121,determining first a bound on the proportionality factor between The
t To whom correspondence should be addressed.
the two solutions, it is possible then to compute other bounds on any chosen measure of real plastic deformation for loads slightly above the shakedown limit. The formulation proposed in the previous paper, although interesting from a theoretical point of view, needs to be specialized to the case of particular structure typologies in order to be applied. As a consequence, for computational purposes, an appropriate formulation for the bounding principle is desirable. A first step towards this direction was already undertaken by the present authors [ 131 in a paper devoted to the specialization of the bounding technique to the case of trusses. In the present paper the described bounding technique for the special case of beam structures is developed. Beam structure behaviour in steady-state is studied and the gradient, with respect to the cyclic load multiplier, of the elastic-plastic steady-state response to loads at the shakedown limit is obtained. The shakedown load factor problem for beam structures and the related Euler-Lagrange equations are studied. A suitable bounding theorem, which provides the way of computing a bound on the proportionality factor between the solution of the above mentioned problems, is proved. Knowing such a bound and the kinematical part of the solution to the shakedown load factor problem, it is possible then to compute other bounds on any measure of real plastic deformation produced by loads slightly above the shakedown limit. A numerical application confirms the reliability of the proposed bounding technique. 171
173
Real plastic deformation for beams elastic response to the cyclic load and a residual (self-equilibrated) bending moment, M,, i.e. M(.K, t) = MO (x) +
MC(.K,t) + Mr
(8)
cu.t) i.e.
= - Sg M’: - SC w:l + M,,
(5)
A4 = -AM’:) in (0, L). in which u’, = M’”(x) and n’E= u: (x, 1) are the elastic deflections produced by the loads P, and PC, respectively, and 5 < 5,” and z > 5: (&) are two selected load multipliers, determining the load PC = & P, + E PC. Denoting t as an instant subsequent to f0 and T = f - to as an auxiliary time variable within the range (0, At), the laws governing the steady-state phase can be written in the following form:
(9)
The solution to eqns (6a-i) or (6a-h) and (7~(9) is essentially unique [12]. In any case, the plastic dissipation density rate fi=
Mlip
(10)
turns out to be uniquely determined
in (0, L) and
VT E (0, At).
cp+= -S&w:-S&w:‘+M,-M,
,i+>O,
in (0, L),
VT E
(0, At),
cp-= +S%uS+SE,~l~-M,-M,~O, Lm cp- = 0
(6a)
A_ 20,
in (0, JC.),
VT E
(0, AZ), (6b)
in (0. L),
(SW:‘)” - P, = 0
kin. and mech. b.c.s. on PV,, (6~) in (0, L),
(Sc11,“)” - PC= 0
If in some region of the structure plastic accumulation Ak, # 0 exists, incremental collapse occurs; otherwise, if Ak, = 0 in the whole structure, low cycle fatigue of the material is probably induced.
4. THE STEADY-STATE RESPONSE TO CYCLIC THE SHAKEDOWN LIMIT
LOAD AT
When (still provided & > 5:) the multiplier &+{t+, the limit response is characterized by a plastic strain rate field and a residual stress rate field identically vanishing, namely
VT E (0, At),
lim Ic$= 0, <<_<‘i
lim tir = 0 :,+;;+
kin. and mech. b.c.s. on MJ~, (6d) in (0, L), M:’ = 0
in (0, L),
& = 1, - j._
mech. b.c.s on M,,
in (0, L),
VT E (0, At),
(60
VT E (0,At),
(6g)
k, = i Q, + kp in (0, L,) /& = -a;
in (0, L)
VT E (0,At),
VT E
(0, At),
(11)
(he)
kin. b.c.s on i,,
so that the residual stresses set up a time-independent self-stress field. i.e. li~li~+M, (x, t) = M,+(x) c
in (0, L)
(12)
Equations (6a-i), (7H9) become meaningless. However, by introducing the new quantities
(6h)
where i, = ti’,(x, t) and k, = f, (x, t) denote the residual displacement and strain rates, respectively. Equation (6i) implies that the plastic accumulation in the cycle
Ak, =
AI k, dr, s0
(13b)
(7)
is independent of to and compatible in (0, L) with the deflection increments in the cycle
(13c) (13d)
174
F. Giambanco and L. Palizzolo
equations (6a-i), (7~(9) take on the form =%
LP.,Aw:dx s0
~:=-S%W~-S~~W~+M~-M,~O, 8
3 0,
cpz i: = 0
in (0, L),
Vr
E
(0, At),
(18)
(14a)
,? 2 0,
q? 1: = 0
in (0, L),
Vt E (0, At),
By virtue of equations (14g, h), (15) and (17). and by virtue of the virtual work principle
(14b) (SW:)” - P, = 0
in (0, L),
km. and mech. b.c.s on w,, (SW$)” - PC = 0
in (0, L),
Vt
E
= 0
in (0, L),
I;,’ = 1: - I?
= %
(0, At),
kin. and mech. b.c.s on w. (Ml)”
(14c)
mech. b.c.s on M,i,
(14d) =,$I(-Sw:)(;i@)dxdt
(14e)
in (0, L),
Vt E (0, At),
(14f)
in (0, L),
V7
(0, At),
(14g)
f: = -(I+:)”
in (0, L),
E
” L JJ
=
in (0, L),
= &
(14i)
Ak,f =
bf I;,’ dr s0
in (0, L),
(15)
Aw: =
AI @: dr s0
in (0, L),
(16)
in (0, L),
(-SW;)@ dx dz
E
I II
Ak: = -(Aw:)”
dx dt
0
(14h)
d, &: dr = 0
(-Sw:)I;;
40
0
VT E (0, At), kin. b.c.s on ti$,
(-SW;)@ dx dr
II 0
+ f:=-@+~;
ArL JJ
=
’ (- Sw:)Ak,+ dx, s0
(19)
Ar L JJ
(- Sw:)k,? dx dz
5:
0
II
kin. b.c.s on Aw:. (17)
Equations (14a-i), (15)-(17) describe a deformation process for which the gradient with respect to the cyclic load multiplier of external work in the cycle, for loads at the shakedown limit, is given by:
ss AI
E’ = E,’ + E,’ = &
5:
=
P, ti: dx dt
0
ss Ar
L
0
A,
(-sw:‘)k;
0
0
dx dr.
(20)
0
L
JJ 0
L
M+I;+ P P dx dr =
L
M;Ak;
J0
dx = 0,
(21)
175
Real plastic deformation for beams
the plasticity conditions at any point and at any instant. Consequently, the problem of the shakedown cyclic load factor identifies itself with the following search problem:
equation (18) can be written as
5: = (-
Sw:))I;; dx dr
max , & subject to: I;,.;.,.*,,.“...H,I
(24a)
cp: = - sy, ,c: - S& ir:’ + l44; - M, < 0
d-(/i;) dx dr = D’ > 0,
=
(22)
where d+ = d+(f,+) is the gradient of the plastic dissipation density rate with respect to the cyclic load multiplier. The result of the latter is that the gradient with respect to the cyclic load multiplier of the external work in the cycle for loads at the shakedown limit, E', is equal to the gradient of the plastic dissipation in the cycle produced in the whole structure D’. The constants E,'and E: can be approximately calculted by the formulae
E,’
2
5:
ss Al
E - 5: o
I
0
( - SW:‘&, ds d7.
(23)
where k,, and Ak, are the real plastic deformations produced by the load PC = & P,+ 6 P,,being E slightly above the shakedown limit. The constants E,'and E,'from a qualitative ponit of view, characterize the steady-state plastic behaviour of the structure in the presence of a combination of a steady and a cyclic load slightly above the shakedown limit [I]: if E,'= 0 and E,'# 0 plastic shakedown occurs; if E,'# 0 and E: # 0 ratchetting occurs. 5. THE SHAKEDOWN LOAD FACTOR FOR CYCLIC LOADS
The necessary and sufficient conditions under which the relevant plane beam structure shakes down can be established through the statical theorem (Melan’s theorem). In the examined case, it states that shakedown occurs if a self-equilibrated time independent bending moment field W0 = M; (x) exists such that the total bending moments, obtained by summing up to the self stresses the moments produced by the fixed and cyclic loads, do not violate
in (0, L),
VT E
(0, At),
(24b)
in (0. L),
VT E (0. Al).
(24c)
(SW:)” - P,= 0 in (0, IY). kin. and mech. b.c.s on II’,. (24d) (SW:)” - P,= 0 in (0, L),
Vr E (0, AI),
kin. and mech. b.c.s on w’,, (24e)
(M;)” = 0
in (0, L),
mech. b.c.s on M;,,
e - 5<,< 0.
(240 (248)
Following the Lagrange multiplier method, denoting qc = (E: /<:) > 0 as a suitable constant, and I:; = ;: (x, r) 2 0, 1: = i: (x, I) > 0, Au, = AU, (.Y), 50 Aw; = 50 Aw; (x) and ‘I<> = (E, 150) > 0, re +: = ic ti: (x. f) as the Lagrange multipliers (r. and & in the last two terms playing the role of scaling factors not subjected to variation), the enlarged functional of problem (24) is
1(1= - & ‘I<+
dr 1. [( - SY, w6 -sy,H;I)+M;) ss0 0
x (;.: - A’ ) - M, (2+ + ;:‘)I dx ds
-
-
+
’ [(SW:)” - Po]l" Aw; dx s0 hr I ,, [(SW:)” - Pc]& ti; dx dr ss0
’ (M‘,)“Au, dx + (5 -
(25)
Taking the first variation of eqn (25) to all the variables, which are functions be as smooth as necessary, and since $ minimum with respect to the variables
with respect assumed to must have a of problem
176
F. Giambanco and L. Palizzdo
(24) and a maximum with respect to the Lagrange multipliers, the relevant Euler-Lagrange equations of problem (24a-f) can be obtained; they are: cst’-= -s<,
d - sy, d + nn;: - MY< 0, f: >, 0,
cp: k+ = 0
in (0, L),
Vr E (0, At)?
(26a)
rp’_= +.Szl W:+sq;w:-~@-M~~O, n’p_2 0,
cp’-if.. = 0
in (0, L),
Vt E (0, At), (26b)
% - C0< 0, (Swt)” - P, = 0
t/o (<. - Co) = 0, (26c)
V0> 0,
(26d)
n’r: = s& - 5)
in (0, JC,), mech. b.c.s on ML, in (0, L),
(@,
kin. and mech. b.c.s on tit:, (26g)
A& = -A&
in (0, t), dl
kin. b.c.s on Au,,
in (Q, L)$ tf? E (0, A?), (equilibrium)
(29c)
J^
(26h)
(30) it is possible
variable increment
(26i)
give of residual i.e.
L
-.Sw:~;dxdr=E,>OO,
G
II
(29b)
in which eqn (29a, b) is formally identical to eqn (14g, h), respectively. Since e and pP are cyclic, the moment field k must also vary cyclically, namely
@A&)” + ((SA@>“f” = 0 in (0, L), kin. and me&. b.c.s on Awit
(elasticity law)
homg. mech. b.c.s on &$
(260
‘dt E {O,At),
= 0
(29a)
in (0, L),
t/r E (0, Af)
Vr E (0, At),
in (0, L),
(S@” + (S(@)“)” = 0
in (0, L), VtE (0, At), kin. b.c.s on 6; (compatibility)
kin. and rnech. b.c.s on We, (26e) (Mi)” = 0
e = -(lit;)”
in (0, L),
kin. and mech. b.c.s on w,, (SW!)” - P, = 0
the role of residual ~sp~a~rn~~t rates. The eqns (26a, b, d, et(28) are formally identical to eqns (14a-d, f) and (15), respectively. Equation (26g) states that rC$is the residual displacement rate field produced by the piastic strain field ?$,, and that both vary cychcally with period At. This implies that G; and k; are not compatible, that k’; generates a self equilibrated residual bending moment rate field e in the structure, to which a residual curvature rate field 5 is associated, compatible with displacement +$ (and therefore also variable cyclically with period At). The eqn (26g) is thus equivalent to the equation system:
(26j)
Awr‘
c
where & = <: results, and where it has been set as !$, = ;t’: - ?_,
in (0, L),
Vr E (0, At),
AU,, harm,
(27) eqns
&r h$ dz Ak; = ,6
in (0, L).
(28)
work rates
ds
in
(31)
if definition (31) by virtue is verified too. (26i) that strain &k;, with we take that results to By virtue e) of work following same path left hand side in k) to represent by P, through mechanism by setting
The eqns (26a-k)-(28) describe some kind of plastic deformation process, associated with the cyclic loading P&!= 50 p0 + e: PC: in which !$ ptay the roIe of plastic strain rates, 2, and & play the rob of plastic multipliers, while @ pfay
taking into (32)
dimensional k)
that identical It is very to note that D+ introduced 4 meaning very
177
Real plastic deformation for beams value, while constant D‘ is a simple proportionality factor with an arbitrary value. Therefore the problems described by eqns (26a-k)(28) (32) and by eqns (14a-i), (15t( 18), (22), respectively, are the same except for the constants D+ and D”, while their solutions are equal, except for the kinematical variables, which differ for the ratios D‘/D+( =E‘/E+). If, in particular, E‘ = D‘ = 1 one has: ;;+t = E+&,
1:: = Efj”“’
&+ = E’k“P P
in (0, L), Ak; = E+Ak;,
Aw,! = E+A$
Vr E (0, At),
(33a)
in (0, L),
(33b)
If the solution to problem (26a-kt(28) provides &, > & (until now 5” = z,, has been supposed), the shakedown limit load multiplier <: shows insensitivity to the steady load variation, and this occurrence takes place when the point (&,,
6. THE BOUNDING PRINCIPLE
The steady-state elastic-plastic analysis of the structure subjected to the loads PC, = 5 P, + &PC, with & > 51, can be performed either through the solution to eqn (6a-i) or by a step-by-step full analysis of a convenient number of cycles until the steady state phase is reached. In any case, these approaches involve a considerable computational effort. However, for loads slightly above the shakedown limit (in practice E 15: = 1 + 1.2, approximately) it is possible to establish which collapse mode for the structure is to be expected, keeping the computational effort down. Equation (33b) allows us to establish which impeding collapse mode is going on at the shakedown limit, on the grounds of the solution to the shakedown load factor problem. If, for instance, Ak:, = 0 there is plastic shakedown, etc. If we know the constant E’, eqn (33), transforms into 2, z E+i:(&<:),
i_ s E+A’ (z-
k, z E+L$ (c - 5:)
in (0, L),
Aw, z E+A#
(& - 5:)
in (0, L),
and consequently the plastic part of the steady-state elastic-plastic response of structure to loads PC slightly above the shakedown limit, by utilizing the solution to the shakedown cyclic load factor problem could be calculated. Unfortunately, in order to compute E+, for instance by eqn (23) the plastic strain rate f, must be known, i.e. the solution to the analysis problem must be known and this makes further investigations needless. In order to sustain a computational effort, reduced with respect to the one necessary to solve the analysis problem, a suitable bounding technique may be employed. An upper bound E* on the constant E+ can be calculated, i.e. E’ < E*.
(35)
Then, utilizing eqns (34b) and (35) one can write IA& I d WA&
I(% - 53,
IAw, I < E*lAw: l(C - 5:)
in (0, L),
(36)
and similarly one could do it for the quantities present in eqn (34a). Relation (36) and other similar relations constitute bounds on the absolute value of the plastic part of the elastic-plastic response to the load Pt. The expression of E* can be obtained in the following way. Let BS denote the given continuous elastic-plastic beam structure, constituted by material for which associated plasticity laws hold [l l] and subjected to the load PC, whose behaviour is already described in Section 3. In addition to the compatibility and equilibrium equations, throughout (0, L) and for all instants r within the interval (0, At), the following relationships have to be satisfied:
k,=;M=
cp+ = +lV--IV,=
k = k, + k,,
(374
-%w,-Cw:+;Mr,
(37b)
-s&w: _ se WEll + M, - MY < 0,
cp_= -M-M,=
A+>O,
(37c)
+Scw: +SFw,’
t;),
1_ 20,
- Mr - My < 0,
(37d)
,&,=A+ -A_,
(37e)
t/r E (0, At), (34a)
(34b)
cp+ x, = cp_ 1_ = 0.
(370
178
Giambanco and
Palizzolo
-L Fig. 1. M and corresponding Drucker’s postulate I] can - M)&
0
in
structure: geometry
through applied
L),
t/r
(37c-f),
(0, Alt),
where &? is any bending moment fieId, provided that it is plastically admissible, i.e. so as to satisfy eqn (37c, d). Let BS* denote a fictitious structure. All the quantities related to BS* are marked by apex*. BS* is the same as BS, except for plastic behaviour: the complementarity constraints (37f) can be violated and the yield function is suitably perturbed. By symbols, throughout (0, L) and for all instants 7 within the interval (0, Ar), one has k* =k: +k,*,
(W
A eqn
like the defined, for but not (37fj hold, for which (40), but eqn (38) is called pseudoplastic. Stresses m* are admissible in BS*, but through eqn (37c, d) they also are admissible in BS, so that eqn (38) can be applied with R = m* (M* - M)& + a(-SC wl- .SC$ w:)k,< 0 in (0, L),
‘VTE (0, A.t). (41)
Stresses M are admissible in BS, but through eqn (39c, d) they also are admissible in BS*, so that eqn (40) can be applied with I@ = h4 - (M* - M)/c,* - a(-SZ w: - SL.$ WE)/+
By summing up eqns (41) and (42) one obtains
cpf=+m*-M,=+M* +a(-S&w:
load condition.
- sg Ml:')-My G 0, (39c)
(-S&
w: - St: WI’)& G (- SC0w: - sgt wc”)It$
lpp”=--m*-j&=-&f* _ a(-S&
n,* > 0,
w: - SC:WC)- My < 0
A? 3 0,
I$ = x,” - a?,
(39d)
in (0, L),
V7
E
(0,At),
(39e)
where the scalar CI> 0 is the multiplier of the perturbation, which is the elastic stress response f - sg UC - SC: WC)to load Pr:=&Po+t:Pc.
Because of the loosened form of eqn (39) with respect to eqn (37), the stress history M* in the cycle is not unique (as M is); let us suppose that we know such history. To m* and @, corresponding through (39c-e), the generalized Druker’s postulate [ 1I] can be applied
in (0, L),
VTE (0, Abt). (40)
g s> CL
Fie. 2. Cvclic load as a function of time.
(43)
179
Real plastic deformation for beams
Iktir
rletnrnts
with plastic hinges
Fig. 3. Discretization of the beam in eight elastic elements with plastic hinges and rigid nodes.
which, integrated throughout interval (0, At), becomes
(0, L) and within
the =
A1 L(M*-M)(f*-I;)dxdS ss0 0
-
ss
L “(M*-M);(ti*-ni)drdx=O, II II (46)
because by virtue of the virtual work principle it is
s
L(M*-M)(I;*-I;)dx=O, 0
(47)
and for the periodicity of (M* - M) is ” L (M* - M) (I$’ - &) dx dr. ss0 0
(44) “(M*-M);(ti*-ti)dr=O. s0
(48)
Taking into account that (f,* - f,) = (f* - I;) - ; (tii* - A&,
(45)
the third integral on the right-hand side in eqn (44) can be written as
L(M*
- M)
t-
(I;,* - f,) dx dz
L12
-
Equation (44), divided by (% - rs), taking into account eqns (46) (22) and (23), takes on the form
E+ = E,’ + E: z&
( - SW:‘)& dx d7
LIZ
V
Fig. 4. The bounded quantity: ratchet deflecton Aw, of the left span middle point.
180
F. Giambanco and L. Palizzolo
2.8
I‘-
2.8
2.6
:
2.4
2.4 2.2
l $
2.0
tk? +w
1.8
’
‘, I
I
\ \ \
2.0 1.6 1.2
1.6
0.R
1.4
0.4
1.2 1.0
I
I
I
I
0.2
0.4
0.6
0.8
0.8
w
IO
5”
0.6
Fig. 7. Curve of the quantity E+& (solid line) and its bound E*S&(dashed line) as functions of 5..
0.4 0.2 0.2
0.4
0.6
0.8
T E (0, At), of the quantities A,*, A!!, I;,*, M*, relative to the fictitious elastic-pseudoplastic process. These can be obtained by an elastic-pseudoplastic analysis, i.e. an incremental analysis with loose complementarity constraints (37f) [l]. The (not unique) solution to an elastic-pseudoplastic analysis problem can be obtained in various ways; an example will be shown hereafter in the application stage.
1.0
Fig. 5. Bree-diagram of the beam.
(- Sw;)Ak; dx
7. APPLICATION
(-SW:)@ dx dr
x dxdr
=
E,*+E: +E,*=E*,
(49)
which is the searched bounding principle (35). The computation of the right-hand side of eqn (49) implies the knowledge, in (0, L) and for all
In a previous paper, devoted to the optimal shakedown design of beams, Giambanco et al. [ 151 studied the case of a continuous beam constituted by two spans of length 15 (Fig. 1). Both spans were subjected to a uniformly distributed steady load P, and the span on the left was also subjected to a uniformly distributed load, cyclically variable between -P, and +P, (Fig. 2). The beam had a rectangular cross-section with dimensions B and H, and the width B was assumed as the design variable.
t 1.4 t 1.2 -
2.4 1.0 '4 '
0.8 -
+$ w
0.6 -
: :
i
0.4 -
2.0
b2.
1.6 -
?1
I.2 -
\ \ \ \ \ \
0.8 -
0.2 I 0.2
t 0.4
0.6
0.8
1.0
5"
Fig. 6. E,+6& (dashed line) and E:6& (solid line) diagrams as functions of r., for b& = O.l[:.
Fig. 8. Curves of the ratchet deflection Aw, (solid line) and its bound Aw,Z(dashed line) as functions of 5..
181
Real plastic deformation for beams The data
of the problem were: L = 400 cm; E = Young’s modulus = 21,000 kN cm-*; oy = yield stress = 40 kN cm-*; P, = 500 N cm-‘; P, = 500 N cm-‘. The beam was discretized in eight portions of equal length AL = L/4, with piecewise constant width B, (i = 1,2, . . ) 8), having purely elastic constitutive behaviour everywhere, except at the end cross-sections, which had rigid perfectly plastic constitutive behaviour (Fig. 3). The solution to the design problem provided the optimal widths: B, = 14.05 cm; BZ = Bi = 7.99 cm; B4 = Bs = 9.59 cm; B,, = B, = 5.20 cm; B8 = 5.03 cm. Afterwards the shakedown domain of the optimal beam was plotted and it was possible to verify that
SY, + bn > 0,
Yn 2 0,
(SY. + b.)TY, = 0,
H = 1Ocm;
min f YZSY, IY,,) SY, + bn 2 0,
(n = 1,2,.
.) (50)
where n is the index of nth step, SY, + b, = -cpn is the opposite of plastic potential vector of nth step, Y, is the plastic attivation increment vector of nth step, b, is the known term vector of nth step (taking into account the loads of nth step and the accumulated plastic strains at (n - 1)th step, as initial conditions of nth step), and S is a symmetric and positive semi-definite matrix. Rather than solving the sequence of problems (50), the equivalent sequence of quadratic programming problems has been solved
subjected to the conditions (n = 1, 2, (Yn 2 0)
the relevant structure found itself just at the shakedown limit. Due to this last beam feature, it is very interesting to analyze the structure behaviour for loads slightly above the shakedown limit and compute bounds on chosen measures of real plastic deformation. In the present application, the above referred optimal beam is considered and a bound on the ratchet deflection of the left span middle point (Fig. 4) is computed by means of the proposed technique. The following data are introduced: a = perturbation multiplier = 0.03; S& = cyclic load multiplier increment slightly above the shakedown limit = c - 5: = O.ltE. It is not necessary to specify
Y,* 3 0
the length of period Ar because, as noticed in Section 2, time only plays the role of an orderly parameter of the events. In Fig. 5 the Bree-diagram of the beam on plane (to, &) is represented. The trilateral line separating the zone E (elastic zone) and S (shakedown zone) to each other provides the elastic multiplier <: as a function of parameter &,, i.e. 5: =
(51)
i where the constraints Y. > 0, even if not explicitly imposed, are satisfied in the optimal conditions. In Fig. 6, the values of quantities E,+6& and E,+6<,, as functions of the parameter &, computed by eqn (23) for StC = 0.15: are represented. In the region relative to the alternating plasticity, E,’ is zero and E,’ is constant with respect to t0 ; in the region relative to the incremental collapse mode E,’ and E,’ vary as functions of &, vanishing at L = C. In order to obtain a fictitious plastic strain admissible history, to utilize in computing the bound E* on E+, a sequence of linear programming algebraical problems has been solved as
min aTY,* subjected to the conditions CYY, SY,* + b.* 2 0,
.),
(n = 1,2,
.)
(52)
1 where Y,* is the fictitious plastic activation increment vector of nth step, b,* is the known term vector of nth step (taking into account the loads and the perturbation at the nth step, as well as the accumulated fictitious plastic strains at (n - 1)th step, as initial conditions of nth step), SY,* + b,* = -cpz is the opposite of fictitious plastic potentials at nth step, while the objective function an is a suitable fixed vector. The solution Y.* of the sequence of problems (52) is a feasible solution of the sequence of problems SY.* + bt 2 0,
Yt > 0
(n = 1,2, . .)
(53)
similar to the sequence of problems (50), without the complementarity constraints. Usually, vector u. is chosen so that-at least foreseeably- the solution to eqn (52) is not much different than the one which would provide eqn (53), if the complementarity constraints (SYf + bX)TYd = 0 were present [5,63.
182
F. Giambanco and L. Palizzolo
In Fig. 7, the values of quantities E’S& and its bound E*@,, as a function of &,, computed by eqns (18), (23) and (49), are represented. In Fig. 8, the values of the ratchet deflection of the left span middle point Aw, = Aw, (L/2) and its bound, Aw,* = AwP (L/2), are represented. Near the first knee of the multiplier diagram t;s (&,), Aw, > Aw ,*, can result, even if only a little. This latter circumstance, absurd as it may appear, can be explained taking into account that the quantity Aw: appearing in eqn (34b) has been computed at the shakedown limit, while E+, E* and Aw, have been computed at 10% above such limit. The quadratic programming problems (5 1) and the linear programming problems (52) have been solved by means of a QPSOL FORTRAN routine [16] and by using a 3090J/VF IBM computer. 8. CONCLUSIONS The elastic-plastic response of beam structures subjected to a combination of cyclic and steady loads has been studied. The gradient, with respect to the cyclic load multiplier, of the elastic-plastic beam structure response to loads at the shakedown limit has been found. It has been shown that the relationship describing such a gradient is formally identical to the Euler-Lagrange equations relative to the shakedown load factor problem (for the structure subjected to a fixed steady load and to an amplified cyclic load), within a proportionality factor tying the kinematical part of the solutions to the two relevant problems. The kinematical part of the solution to the shakedown load factor problem for a beam structure can, therefore, provide some useful information on the steady-state elastic-plastic response of the structure subjected to a steady load and a cyclic load, such that the combined load condition be slightly above the shakedown limit. Simply by checking the increment in the cycle of the variables which play the role of plastic deformations, one is able to establish whether (in the real process) incremental collapse or alternating plasticity occurs; together with the computation of a proportionality factor, it consents to check, by verifying the identities (33a, b), the precision of the analysis problem solution slightly above the shakedown limit (identities (33a, b) tend to be exactly verified for &-+5:‘); executing an elastic-pseudoplastic analysis (which requires a computational effort lower than the required one for a full incremental elastic-plastic analysis) and computing a bound on the proportionality factor between the two responses, one is able to calculate other bounds on any chosen measure of real plastic deformation.
For the elastic-plastic beam model description, the simplifying hypothesis that the elastic and plastic strains are both linearly distributed in the beam height has been utilized. However, the authors believe that the described technique can also be easily performed in the case where the aforementioned hypothesis is removed. The effected numerical application allows us to confirm the reliability of the proposed bounding technique. REFERENCES
1. C. Polizzotto, Simplified methods of structural analysis for cyclic plasticity. Commission of the European Community, Nuclear Science and Technology, Final Report, contract no. RA 1-0116(N). 2. M. Capurso, L. Corradi and G. Maier. Bounds on deformations and displacements in shakedown theory. Matitriaux et Structures sous Chargement Cyclique, Ass. Amicale des Ingtnieurs A&ens Eltves de 1’E.N.P.C.. Paris. DD. 231-244 (1979). 3. J. A. Kanig, On u&er bounds ;o shakedown loads. 2. Angew. Math. 59, 349-354 (1979). 4. C. Polizzotto. A unified treatment of shakedown theorv and related dounding techniques. S.M. Arch. 7, 19-7i (1982). 5. F. Giambanco, M. Lo Bianco and L. Palizzolo, A
6.
I.
8.
9.
10.
11. 12. 13.
bilateral convergent bounding technique for plastic deformations. kfeccanica 25, i81-188 (i990). F. Giambanco. L. Palizzolo and C. Polizzotto. Delimitazione delle deformazioni plastiche per carichi oltre il limite di adattamento. In Atti X-Congress0 Nuzionale AIMETA. 1. vv. 159-164 (1990). D. A. Gokhfeld and’0. fi.&Cherniavsky, Limit Analysis of Structures at Thermal Cvcling. Sijthoff and Noordhoff, Alphen aan den dijn, The Netherlands (1980). W. T. Koiter, General theorems for elastic-plastic solids. In: Progress in Solid Mechanics (Edited by J. N. Sneddan and R. Hill), Vol. 1. pp. 167-221. North Holland, Amsterdam (1964). J. A. Kiinig, Shakedown of Elastic-Plastic Structures. PWN-Polish, Warsaw and Elsevier, Amsterdam (1987). J. A. Kiinig and G. Maier, Shakedown analysis of elastoplastic structures: a review of recent developments. Nucl. Engng Des. 66, 81-95 (1981). J. B. Martin, Plasticity: Fundamentals and General Results. MIT Press, Cambridge, MA (1975). T. Panzeca and C. Polizzotto, On shakedown of elastic plastic solids. Meccanica 23, 94101 (1988). F. Giambanco and L. Palizzolo, Bounds on plastic deformations of trusses. Int. J. Solids Struct. 31, 785-795 (1994).
14. C. Polizzotto, On the conditions to prevent plastic shakedown of structures; Part I: Theory, Part II: The plastic shakedown limit load. J. uppl. Mech. 60, IS-25 (1993).
15. F. Giambanco, L. Palizzolo and C. Polizzotto, Optimal shakedown design of beam structures. Struct. Optimization 8, 156167
(1994).
16. P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, QPSOL: a FORTRAN package for quadratic programming. Technical Report SOL 84-6, Department of Operations Research, Standford, CA (1984).