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Uniqueness criterion for incremental variation of steady state and symmetry limit
J. Mrch. Phn. Solidv Vol. 37. No. 4. pp. 49>514,
0022-5096Gl9 53.00 + 0.00 c 1989 Pergamon Press plc
1989.
Primed 10Great Britain.
UNIQUENESS CRITERION FOR INCREMENTAL VARIATION OF STEADY STATE AND SYMMETRY LIMIT KOJI UETANI Department of Architecture. Kyoto University, Sakyo. Kyoto. Japan (Received
30 June
1988)
ABSTRACT A SUFFICIENT condition for uniqueness of the incremental variation of steady state is established for a cantilever beam-column subjected to completely reversed tip-deflection cyciing. The critical steady state. at which the sufficient condition for uniqueness is first broken, is found on the sequence of symmetric steady slates generated under a conlinuously increasing tip-deletion amplitude. In a previous paper (UETANI,K. and NAKAMJRA.T., J. Me&. P~.Ys.Soli& 31,449, 1983). a symmetry limit has been defined as the critical steady state, at which transition from a symmetric steady state to an asymmetric steady state occurs first, and its theoretical solution has been obtained. The uniqueness limit turns out lo coincide with this symmetry limit. Though the previous theory for symmetry limits was constructed on the basis of five hypotheses introduced without proof. the presenttheory involves no such hypothesis.
1. INTRODUCTION WHEN AN initially straight cantilever beam-column is subjected to an idealized completely reversed tip-deflection cycling program with continuously increasing amplitude under a constant axial compressive load, the beam-column exhibits a sequence of symmetric steady states, in each of which a pair of extremely deflected configurations are symmetric with respect to its initial member axis as shown in Fig. l(a), during some range of smaller tip-deflection amplitudes. But when the tip-deflection amplitude reaches a certain limit, transition from a symmetric steady state to an asymmetric steady state occurs and subsequently the beam-column exhibits a sequence of asymmetric steady states as shown in Fig. l(b) with increasing amplitude of an antisymmetric deflection mode. UETANI and NAKAMURA (1983) have called the critical state, at which the transition occurs, a symmetry limit and developed a theory for predicting the symmetry limit without tracing the entire hysteretic response of the beam-column. The essential ideas of the previous theory are stated as follows :
(a)
(b)
A sequence of steady states generated under a tip-deflection cycling program with continuously increasing amplitude may be regarded as a path in a special space, each point in Which represents a steady state. Such a path has been called a steady-state path. A symmetry limit is regarded as the bifurcation point on a steady state path 495
496
K. t
C’ETAh.1
Time COIDA program denseiY disposed constan+-amp)tude CYCum Proces8es Of COntnUOUSly mcreashg ampCtude
a pa, Ofextremely deflected shapes at 8” asymmetru steady state
FIG
(a) (b) I. A cantilever beam-column subjected lo 3 up-deflection cyclmg program axial
(cl
comprewve
COIDA
under a constant
load.
representing a sequence of symmetric steady states. from which another branching path representing a sequence of asymmetric steady states emanates. The symmetry limit may be found as the symmetric steady state with the smallest tip-deflection amplitude among those satisfying the condition that “to the problem of finding possible rates of steady-state variables with respect to the tipdeflection amplitude. there exists a solution involving non-vanishing amplitude of an anti-symmetric deflection mode”.
The previous theory. however. was developed on the basis of five hypotheses introduced without proof. Because of the lack of proof, a numerical analysis of the hysteretic response had to be performed to verify the hypotheses and the prediction results. It was shown by the response analysis that transition to an asymmetric steady state actually occurs at the theoretically predicted symmetry limits. In the present paper, a new theory for predicting symmetry limits is developed on a more clear and more certain basis than that of the previous theory in the sense that all the uncertain assumptions are abandoned. First a sufficient condition is established for uniqueness of the incremental variation of steady state caused by a given increment of tip-deflection amplitude. The critical symmetric steady state. at which the sufficient condition for uniqueness is first broken. is found along the sequence of symmetric steady states and shown to be identical with the symmetry limit obtained in the previous paper.
2.
ANALYTICAL
MODEL
AND
BASIC
EQUATIONS
The present theory is concerned with the behavior of an initially straight cantilever beam-column subjected to an idealized completely reversed tip-deflection cycling program ofcontinuously increasing amplitude Y. referred to as COIDA, under a constant
Uniqueness criterion and symmetry x=Hef
limit
491
IN=AEn
FIG. 2. Analytical model. axial compressive dead load N as shown in Fig. 1. In a COIDA program, infinitely many infinitesimal increments are consecutively applied in such a way that after every infinitesimal increment of the amplitude, the deflection cycling at that increased amplitude is repeated as many times as is necessary for the beam-column to attain a steady state. Infinitely many COIDA programs with different micro-compositions can be conceived. The analytical model used here is the same as employed in UETANIand NAKAMLJRA (1983), which is a model beam-column with a uniform idealized sandwich cross section as shown in Fig. 2. Normal stresses are transmitted only through the two thin flanges of cross-sectional area A/2 placed at distance 2H. and shear stresses only by the core, which makes no shear deformation. The material of the flanges obeys the bi-linear hysteretic stress-strain relation of Fig. 3, specified by elastic modulus E, strainnotations are introhardening modulus E,, and yield stress tiv. Non-dimensionalized duced for the quantities shown in Fig. 2 as follows: 5 = x/H for axial coordinate. I = L/H for length of the beam-column, n = N/EA for constant axial compression. oL = iiL/E for compressive stress in the left flange, o,, = i?JE for compressive stress in the right flange, u = U/H for displacement component in the direction of the initial member axis, 1: = V/H for displacement component in the direction perpendicular to the initial member axis. r = E,/E(O < cx < 1) for strain-hardening ratio. On the assumption of small but finite deflection conventionally employed in the post-buckling theory of beam-columns, the equations of equilibrium are written as
FIG. 3. Bi-linear stress-strain
relation of the flanges.
K. UETANI
FIG. 4. Three stead!-state
lines and a steady-srate
loop
_!(a,+uL ) = I?. !(a, where a subscript i following boundary conditions are
U[
a comma
I,;, +
171‘,;;
denotes
0
=
(1)
differentiation
with respect to <. The
u(0) = r(O) = r.,,(O) = 0. O,(/)_UL(/)
= 0.
z.(I) = 2.1.
(2)
where 1’Tis a given value of tip-deflection. Compressive strains I:, and I:~ of the left and right flanges are related to lateral displacement 1. and compressive axial strain P of the central fiber by EL = r-l’,i:
= -u,:-
!(r..J2-z’,!i.
&k = I?+ I‘,;; = - II._- 4(I’,; ) 7+ I‘,;, The displacement
component
u can be calculated
zd(<) = -
(3)
from LJand 1’by
,; [c-t (r.,:)’ 21 d:. s
The stress-strain
relations
of flange elements
are
fJ=I:
for virgin elastic response.
u = W+ (1 - ~)a).
for compressive
c = rc-(1
for tensile strain-hardening
0 = C--E’
-r)a,,
for unloading
strain-hardening
response.
response,
response. (4)
where .sr denotes the residual plastic strain. The cyclic response of a flange element in a steady state may be depicted by a closed locus of the state point in the stress-strain plane and classified into four different types shown in Fig. 4. A steady-state line E represents a purely elastic response of a flange element. Steady-state lines C and U represent shakedown states attained after some histories of plastic deformation of flange elements. A steady-state line C starts from and returns to the strain-hardening branch. whereas a steady-state line U does not.
499
Uniqueness criterion and symmetry limit
There is only one type of steady-state hysteresis loop denoted by P. It represents a cyclic behavior of a flange element called alternating plasticity where tensile and compressive plastic strains are alternated. 3.
A SUFFICIENT CONDITIOK FOR UNIQUENESS OF INCREMENTAL VARIATIOX OF STEADY STATE
A beam-column is supposed to be in a steady state. where the same passage of deformation is repeated for every cycle of tip-deflection with a fixed amplitude I(/. It is assumed that the steady state satisfies the following condition (H3’). Condirion (H3’). The strain direction of each element in the right flange is reversed from compressive to tensile only at an extremely deflected state r’ of r(l) = ti and from tensile to compressive only at I-” of c(l) = -I//. The strain direction of each element in the left flange is reversed from tensile to compressive only at I-’ and from compressive to tensile only at F’.
This is a little more specific statement than that of the hypothesis (H3) in the previous theory. When there is no bifurcation point on the closed equilibrium path representing a steady-state cyclic behavior, the whole of the steady-state behavior may be specified only by a set of reversal-point configurations u’ = {e’(t), ~‘(5)) and u” = (e”(r). L.“(C)). The quantities belonging to I-’ and to Y” are indicated by the superscripts I and II respectively. The relations between reversal-point stresses and reversal-point strains are written for each type of flange-element cyclic behavior as type E :
a; = CL,
typeU:
a; = EL--t&.
typec:
ak = rck + ( - 4ay,
ai = $+(a-l)(&-ady),
aif = LY$’+ ( - r)av.
a: = $+(z-
type P :
a: = E:,
a:’ = c:,
ai = El:-&.
a; = EL, al =&P--C;,
EL-&E;.
I)($-ay),
ok = r&+(
- z)ay .
01 = a&--(l-a)a,,
aLI' =
- %)a\.
a: = r&--(1
TE:’ + (
a: =
-a)aY.
(5)
When an infinitesimal incremi el It dl// of tip-deflection is applied to the beam-column in the known steady state under a tip-deflection amplitude $, the beam-column will move to an adjacent steady state (II’ +I? dl//. u”+ti” d$), where 6’ and I?’ denote the rates of change of reversal-point configurations u’ and u”, respectively, with respect to II/. Suppose that there could be two distinct possible rate solutions (i~‘.ti”) and (ti”.i~“‘). The former solution (a’. a”) satisfies the following basic rate equations derived from the equations (l)-(3). Field equations of equilibrium :
_!((ik+ti:, i(iil: Strain4isplacement
= 0.
ltir:‘) =
relations
:
0.
h((il, -ir:),:;
+d:: .. . = 0,
i(irl: -c$$:
+n2;: = 0.
(6)
UETAM
K.
500 $
=
p’ __$
.I’ El
_ -
.I’ :I’ -_I :t, c .5.
dk
.::7
=
.‘I = CR
2
+
I&
pll +
p, .., ’
(7)
where p” = _ r;‘! _ ‘.I!‘:‘! . _, .. .
p’ = - lj’: - r.‘$‘: ., .,* Geometrical
boundary
:
conditions
P’(l) = 1.
r:‘(O) = d(O) = 0, P(0) Mechanical
boundary
= r:!:‘(O) = 0,
1:“(I) = - 1.
(8)
:
conditions c&(f) -6;(l)
b;(l)
= 0.
-rif(l)
= 0.
(9)
A beam-column in a steady state may be divided into some number of regions. in each of which the left and right flanges of every element exhibit the same pair of types of cyclic behavior, e.g. [E, E], [C, U] etc. since ti, ti,<. bK-tiL and (hR --r!~,),~ -nnl:,: should always be continuous throughout the beam-column, then at every inter-region boundary I”(q-0)
= F’(q+O),
P(‘l-0)
= P(q+O).
c$(q-0)
= &(t]+O),
1?(q-0)
= Z(q+O).
@k-ti:)I;=,
0 = (d:,-4_)l:r.,+,.
(&ti:‘,];=,_,,
The relations given by
between
= (+c+:‘)]~=,_,,,
(&tit_),J~=q_o
= (c&&).:I~=,+o.
(&-&‘,.iI;_,_,,
= (fY;-6~‘.:]t=11+o.
reversal-point
(10)
(11)
stress rates and reversal-point
tik = ,&k.
61: = V&k’+ (pLR- VK)&.
ti:’ = ~“2:‘.
ci( = \.tc: + (FL - v,)c’:‘,
strain
rates are
(12)
where the coefficients p and v are to be selected in accordance with Table I. It should be noted that these coefficients depend not only upon the current type of steady-state TABLE 1. Coe#icients.for
Type of steady-state behavior
real and.fictitious
Direction of reversal-point strain rate
materials
Uniqueness
criterion
and symmetry
501
limit
response of the flange element but also upon the direction of its reversal-point strain rates. The rate equations (6) of equilibrium in the field and the rate equations (9) and (I 1) of mechanical boundary conditions and inter-region boundary conditions are equivalently given by the variational equations
(13) where dsL = 6e - &q,:;, for all the continuous
virtual
displacement
&
= de+ 6’1.;:
fields &I( = {ae, SC}) satisfying
&I(O) = SC.:(O) = &I(l) = 0. The subtracted field ed( = ir’ - 8” = (PI-@‘I, ti’ - t”}) following equations from (13), (7), (8). (9) and (12). O’[:(b&_
+&&}
-nti;‘&T.<] dt = 0.
(14)
may be shown
to satisfy
the
g”R= ed+LC,
Et = $‘-&,
s tid(0) = z’?(O) = 0, ci; =
Cd(l) = 2, .d VREK,
cf"L =
f$(/)-C?:(I)
= 0,
VLdf.
(15)
The other solution (ir”. I?“) also satisfies a set of equations given by replacing all the unstarred quantities in the equations (6)-( 13). (15) by the corresponding starred quantities. From the unstarred and starred equations of (15). it can be shown that the difference fields (Air’, Ati”) defined by Ai,’ = i,” _$,
Ai,” = $1’ -8”
(16)
satisfy the equation I [j {v,AE~&’ + v,A&r)~~} -nA?:6r:.;]
d{ = 0.
(17)
I0 Since Aztd(0) = AZ?(O) = APd(I) = 0. the equation
(I 7) should
still hold with 6u = Aird :
~[~{vL(AEd,)‘+vR(A$)2j-n(Ati$)2]d~
= 0.
i A sufficient
condition
for uniqueness
of the subtracted
field is therefore
that
SO'
K. UETASI
nyti, =
> J‘[l(vl.(i:,)~+\.R(~R)2J-n(l:.i)~]d: 0.
(18)
0
where 6,
for all non-vanishing
continuous
=
i-r:
::.
.. .
ER = (i_tr:
fields ti subject
:. ..
to the geometrical
constraints
1:(O) = r:,,(o) = t:(I) = 0.
(10)
It should bc noted that this condition guarantees uniqueness of the subtracted held it’--ti”, but not uniqueness of the steady-state rate solution (I?.$‘). Under the condition (18). two distinct solutions (it’. 8”) and (ti”. II”‘) are connected by the relations AC;’ = A$‘_
At:’ = A,:“.
Aii; = A$‘.
Ai’, = A&.
(20)
Subtraction of the virtual-work equations in (13) respectively equations for the starred quantities yields
from the corresponding
Since AI? and Ati” are subject to the same geometrical constraints should still hold after replacing 6u by A6’ or 66” respectively.
These equations the steady-state
suggest the introduction rate solution as follows
:
of a sufficient
condition
as (14) on &I, (21)
for uniqueness
of
+(ir:!‘_~l’)(ril”_-tll!)+(~~ -n{(2:;; -I:!: )z+ -cf!)‘}] (I y
for all sets of it’, ti”, it”, I?” satisfying
(8)
di > 0
(22)
Uniqueness
criterion
and symmetry
503
lrmit
The condition (22) is hard to apply to uniqueness limit analysis of the incremental variation of steady state. because the material coefficients ,u’s and Y’Sin (12) depend upon the signs of strain rates. in other words. because A&:.k, AC& are not related to Ail,R. A$!. k in the sense of (12). By analogy with the fundamental inequality introduced by HILL (1958) for deriving a relaxed sufficient condition (over-sufficient condition) for uniqueness of equilib~um path. a fictitious material obeying the following constitutive relations is envisaged here. (ik = &dk,
ci’l: = sgbf + (pi - s;)t$.
tit’ = p:s:‘.
6; = tf$ + (p[ - st’)d:‘.
(23)
The coefficients $‘s and \-r’s are given in Table 1 and depend only upon the current type of steady-state response of the flange elements. Under the condition (18) the following inequalities and equalities hold : (~k’-~~)(~~-~~)+(~~*-~~)(~~~_~~)
FAi; + (& - v;)AE;‘lAi; >, j&A& fdi:, + \’\‘R (61’ - (iL)($ -&) + (6;’ - $)($’
= Z&(A&)‘,
-#$)
> {$A& + (p: - \~~~A~~!~A~~ + fpt:‘A.$’)A$ = 2$(A&)2.
(24a.b)
These can be proved in Appendix I. In view of (24) it can be shown that I-I(ir’,ir” ; 8”. i”‘)
3 I-IF(Air’)
(25)
where
From (22) and (25). we obtain a more relaxed condition incremental variation of steady state as II’(C) > 0
for uniqueness of the
(26)
for all non-vanishing continuous displacement rate fields I satisfying the geometrical constraints (19). The inequality v 2 pr yields that I-P(C) 2 I-I’(il). This means that the condition (1X) is necessarily satisfied whenever the condition (26) holds. In consequence it may be concluded that (26) provides a sufficient condition for uniqueness of the incremental steady-state variation from a known steady state satisfying the premised condition (H3’).
4.
SEQUEISCE
OF
SYMMETRIC
STEADY
STATES
In the present and next sections the symmetry limits of cantilever beam-columns subjected to a tip-deflection cycling program COIDA under constant axial compression will be obtained by applying the uniqueness condition derived in the previous section. Existence of a sequence of symmetric steady states is obvious from the fact that both the initial configuration of the beam-column and the loading system are symmetric with respect to the initial member axis. In the plane of Fig. S. whose coordinates are tip-deflection amplitude ci, and amplitude rh of anti-symmetric deflection mode, this sequence of symmetric steady states is represented by the straight lint C>SF along the vertical axis emanating from the origin. which has been called a fundamental steady-state path. The steady-state point definitely moves up along this fundamental symmetric steady-state path as the tip-deflection amplitude is increased. at least as far as the sulhcient condition (26) for uniqueness of the incremental steady-state variation holds. In the present section. analytical solutions to the sequence of symmetric steady states are obtained. The solutions will first be derived presuming that the premised condition (H3’) holds and thereafter the validity limit of the solutions. within which (H3’) holds, will be clarified. 4. I. V~riu~ion f.$sraw wgions Consider an incremental variation from a generic symmetric steady state to an adjacent symmetric steady state caused by a given increment of tip-deflection amplitude. The rates of change of the steady-state variables during this process arc connected by the symmetry relations _
+I_
.I
.II
CL = &R
q : TIP-DEFLECTION AMPLITUDE
steady
states
ANTI-SYMMETRIC DEFLECTION AMPLITUDE
: Vb
FIG.5. A seqwnceof symmetric
steady siatcs and a paw of sequences of asymmetric
steady states
Uniqueness criterion
symmetry limit
and
Since both the current steady state and the incremental consideration are symmetric,
Using (27) the constitutive
relation
(12) is reduced
of (28). (7) into the second equation
.I VC.;:~;
under
to
for
(28)
of (6) and the first equation
of (9)
(29)
= 0.
When the current steady state involves no P-region, exhibiting a steady-state loop P, from Table 1
In this case, the solution
variation
= 0,
+nff::
A(I)
v= 1
steady-state
6: = v&+(/i-v)&.
6; = pck, Substitution yields
505
(30) where every flange element
is
0 < r
of (29) under the boundary
conditions
sink,,(i-I)-k,,rcosk,,l+sink,,l ---.-~ sin k,,lkEEl cos kEEI
cl = -
for
(8). (30) is given by
OG<
where
The corresponding
stress rates are given by .I
-
CJu -
-6:
=
-nsink,,(<-I) sin k~El-k,,icos
It is shown by (3 1) that 6; is monotonically the condition that
lip;7 decreasing
for
0 < r < 1.
(31)
from the base to the top under
7[--
” -=c 4/’ The right-hand-side of (32) is identical with column with a rigid constraint of tip deflection. is sketched in Fig. 6(a). It may be concluded cases of (32) the variation of state regions in steady-state path has the following properties
(32)
the Euler buckling load of a cantilever The distribution of irk for n = n2/(41*) from the above discussion that in the the beam-column along the symmetric :
506
(b)
(a) FIG. 6. Distrih~~tions
of c?k : (a) for n = a’;(4/‘);
(b) at ,I~.~ = &
(3 At every tip-deflection amplitude level below a certain Iimit. the whole beamcolumn is occupied by an E-region only as shown in Fig. 7(a). when n < cl.. (33) (ii) A C-region appears from the base and spreads monotonically toward the tip as shown in Fig. 7(b). (iii) Any U-region does not appear at least until the appearance of a P-region. (iv) A P-region appears from the base as shown in Fig. 7(c).
Our subsequent discussion is restricted to the cases of (32) and (33). After appearance of a P-region, such an arrangement pattern E-C-P of state regions as shown in Fig. 7(c) is to be maintained as far as Irk(T) > 0 over the range 0 < 4 d 1. This condition is to be broken when the P-region spreads to such an extent that &k(O) = 0. or equivalently that 8;;(O) = 0. It is shown in Appendix II that a non-vanishing reversal-point deflection rate I:’ satisfying tiftc(0) = 0 can exist only when the inter-region boundary < = Q.,, between the C-region and the P-region is such that
(a)
(b) FIG.7.
Vmalmn
of arrangement patterns al state regions.
62)
Uniqueness k,,
criterion
and symmetry
507
limit
cos kppr7cp sin A.rr(~c.r- I) -kEE sin kpprjcp cos kEE(qCP -I)
= 0,
(34)
where
Denoting the smallest positive root of (34) by q &. the condition over the range 0 d 5 d I is given by R-P
<
that irk(<) > 0 all
(35)
rl?P.
Figure 6(b) illustrates the distribution of irk at qcp = $JJp. The condition (35) guarantees that the P-region never diminishes and then that the inter-region boundaries qcp and qEc move monotonically toward the tip as II/ is increased. This implies that the arrangement pattern of state regions varies in the order of E, E-C and E-C-P and that the pattern E-C-P is preserved at least as far as (35) holds. 4.2. Symmetric
steady-state
solutions
The symmetric steady-state solution may be derived for each arrangement pattern of state regions. At any symmetric steady state the state variables of F’ and F” satisfy the conditions cr = _ r’tr)
e’ = e”, The constitutive E-region
:
C-region
:
equations
&k = &:l,
-Z)by,
a: = E: -(I
(36)
-Z)(&‘K-(Ty).
CT; = a$-(1
-r)av.
The field equations governing 1.’obtained general solutions are written as follows :
-Z)dy.
by substituting
t:f:::: +nrflc = 0,
u: = J: sin kEt, + K1, cos k,,(
+ B:< + 0;.
: eI
=!+(.!-z) a -. (t’!:; -cry). z
rItt;: +nz!!:: = 0,
r$ = J~sink,,i+k~cosk,,5+~:.5+~~, e;=p.
n-(1
-~)a~ .Lx
-
(37)
(37), (3) into (I) and their
: e’ = n,
C-region
f.7; = D:‘.
: ak = rek+(l
E-region
a: = a;,
for each state region are given as follows :
a; = r&+(1 P-region
E: =&;,
n(J:- sin kEEr + KL cos kEEr),
508
P-region
K. t.kTAI.;l :
rb = Jb sin A,,< + lub cos X-r& + Bb{ + 0;. The inter-region
boundaries
.Z = vtc and f = rjcp are characterized
(38) by
ak@kc) = cY. ~:,(Vc,)-o:(rlcr) These conditions
may be transformed r:;.::(k)
= &.
with (37). (3) and (38) to = fly -n[
G.:: 07CP) = fly [=
= L.:,::(&c)].
(39)
&:hdI.
(40)
The constants of integration and the inter-region boundary locations may be determined from the boundary conditions (3) and an appropriate set of inter-region boundary conditions, i.e. the continuity conditions for t’, c!;, (crk -a:), (r& -a:).; and characteristic conditions (39) and/or (40). The resulting solution of symmetric steady states is given for each arrangement pattern of state regions by (E-solution) J;
=L:
6 _.
__.___._.
tank,,/-&,I‘
z-l: = -k,,Jk. (EC solution)
The equations
tank,,I,
0: = -lg..
(41)
of (41) and
,:- = .I:. 41/sin k&Fjrc
K: = -&
lu:. = K:, -I)
B& = &,
D: = D:..
=
(ECP solution) g/ =
‘f n
_____. t? i tank,,(Fjcr,--1)
+ r(k,,Jsin
&p~cos kppqCp - sin kppvCP)
kppkp +cos kpp~cp) + 1 - 2 , I
Uniqueness criterion
and symmctv
WY
ltmlt
Once any one of the quantities +, qEc. qcP is specified. all other unknowns together with the arrangement pattern are determined directly from (41). (42) or (43). 4.3.
Validity range oj’lhe condition (H3’)
Though the symmetric steady-state solutions have been obtained on the assumption that (H3’) holds, nothing has yet been discussed about (H3’). Besides, it has been implicitly assumed that no bifurcation path of equilibrium emanates from the closed equilibrium path representing the cyclic response of a symmetric steady state. In this sub-section these assumptions are examined and the validity range of the symmetric steady-state solutions is clarified. The condition (H3’) can be assured by showing that at every configuration in a symmetric steady state
and
aK(5) b 0.
o;(t)
< 0
over
0 < << I
when
z.‘(j) > 0
gK(U G 0,
a;(t)
3 0
over
0 d r < I
when
r’(I) < 0,
where ( )’ denotes differentiation with respect to an equilibrium-path parameter I. The governing equations for the incremental change in the state of equilibrium caused by a given tip-deflection velocity may be written as follows : The equilibrium boundary conditions and inter-region equations, straindisplacement relations. boundary conditions may be obtained from (6)-( I 1) by replacing {( ‘)‘}‘s by {( )‘}‘s. Constitutive equations are given by
where I
for or
P= z
for
Iu-zrel for
< (I-~)a,
IG--OLCI= (I-r)ay
la-a&l
= (I-a)aY
and and
(a-as)&
(a--~)&‘>
< 0, 0.
It should be noted that a steady-state cyclic response of a flange element of type C involves no plastic deformation because the stress-strain state point is reversed immediately from the arrival point on the compressive strain-hardening branch as shown in Fig. 4. It follows that at every configuration in a steady state represented by the E or EC solution,
p’_= pa Following
the same procedure
= I
over
as in deriving
0 < < < I. (3l), it can be obtained
that
K.
510
UETASI
It is shown by (31) that under the conditions (32) and (33). (H3’) holds in all symmetric steady states given by the E or EC solution. For a symmetric steady state of ECP. distribution of the values of pt. and pR in the P-region differs at different configurations. Examination of stress-rate distribution in general provides that the most critical with respect to (H3’) is the configuration with the largest yielding portion in the P-region. i.e. the reversal-point configurations I-’ and F”. where p, = /jK = I
for
ticp d 5 d 1.
[‘L = /)H = 2
for
0 d < < qcp.
It can be shown by repeating similar discussions to those leading to (35) that (H3’) remains to hold at the symmetric steady states represented by the ECP solution as far as
R-P Q rl?P.
As to the second assumption. application of the criterion by HILI. (1958) for uniqueness of equilibrium paths yields the conclusion that the closed equilibrium path representing a symmetric steady state should be unique, in other words that there is no branching point on this closed path. if fIcq(ti) =
~:[ji~~(EI)‘+~~(~K)~)
-n(zL)‘]dr
> 0.
s where PR = {‘r = I /)K = !I.
p, = I
/+ = 11, = r for all non-vanishing continuous Since p, z ,$ and pK 3 &.
displacement
for E-region, for C-region, for P-region, rate fields ir satisfying
rYJ(il) 2 l-V(o).
(19).
(44)
The equality in (44) holds only for purely elastic symmetric steady states. The inequality (44) guarantees that there is no bifurcation point on the closed equilibrium path ofevery symmetric steady state satisfying the sufficient condition (26) for uniqueness of the incremental variation of steady state.
5.
UNIQUENESSLIMIT AND SYMMETRYLIMIT FOR CANTILEVERBEAM-COLUMNS
In the previous section it has been shown for the cases of n < dY and n < 7r’/(41’) that there exists a unique sequence of symmetric steady states for a continuously increasing tip-deflection amplitude. When a beam-column is subjected to a COIDA program of tip-deflection cycling, it should actually follow the sequence of symmetric
511
Uniquenesscriterionand symmetry limit
steady states, which is represented by the line OS in Fig. 5, as far as the sufficient condition (26) for uniqueness of the incremental variation of steady state holds at each symmetric steady state. In this range. therefore, there is no possibility that an\ transition from a symmetric steady state to an asymmetric steady state can occur. Any transition to an asymmetric steady state necessarily requires the uniqueness condition (26) to be broken. In the present section, the critical symmetric steady state, at which the uniqueness condition (26) is first broken, will be found. From the critical symmetric steady state of uniqueness limit, transition to an asymmetric steady state may occur as shown b> SB or SB’ in Fig. 5 or may not occur. Then the resulting solutions of uniqueness limit will be compared with the symmetry limit solutions in UETANI and NAKAMURA (1983). The sufficient condition (26) for uniqueness may be restated equivalently that min (IIF(ti)). l&s where S denotes the set consisting satisfying the geometrical constraints Q(i) o
-
> 0.
(45)
of all the continuous (19) and a normalizing
displacement condition
rate fields
’ [P’ + (Cc;)‘] dc = o. I0
being an arbitrary small positive constant. The field ti minimizing IIF under R = w necessarily I-I:, (ti, 6u) +iR,,
(46)
satisfies the variational
(Ii, h)
equation
= 0
(47)
with a special value of the Lagrange multiplier i., for all continuous variational displacement fields 6u satisfying (14). I-I:, and R, , in (47) are the bi-linear functionals derived from the quadratic functionals II’ and R respectively. It should be noted that the trivial field ti = 0 satisfies (47) but not (46). The field differential equations and the mechanical boundary conditions derived from (47). together with the geometrical boundary conditions, constitute an eigenvalue problem. Let the ith eigenvalue be denoted by %(‘)and the associated eigenmode by iP. which is normalized so that R(G) Since the equation n:,($“, From
this equation
= 0.
(47) is satisfied when i. = I”’ and ti = 6u = ti’“, we obtain i”‘)+
).(‘jQ,
,(fi(", 8"') =
2{fl'(fi"')+
j.'"Q(i+")} =
that
0.
and (46), nF(hC”) = _jI”o .
.
This implies that the minimum value of II’ is given by the largest eigenvalue I.,,, and that uniqueness of the incremental variation of steady state can be assured by the condition that i.,,, Since the condition
(48) is to be broken
< 0. with i.,,,
(48) = 0, the critical
symmetric
steady
K. UETASI
512
state, at which the sufficient condition (26) for uniqueness is first broken. is characterized by the existence of non-vanishing displacement fields satisfying the equation rI:,(iL6u)
= 3
’ [p’(&+ i I,
I:,;:&.,~~) - nr:,:6z..;] d; = 0
(49)
for all continuous variational displacement fields du satisfying (II). Operating on (49) by integration by parts. the following set of equations may be derived as necessary conditions for (49). :
Field equations
/AL;;
P = 0. Boundary
+ nz:,;<= 0.
:
conditions
(50)
C(O) = C,:(O) = 0. Inter-regional boundary conditions $‘:,::, pFc,i:; should be continuous. For purely elastic symmetric the non-vanishing solution q;, where J is an arbitrary
= 0.
r:(l) = L:,i<(l) = 0. (at < = qE(.. qcp) : the quantities
steady
l:(t) = J
constant.
states,
the boundary-value
sinl\-.EE(C:-I) ._~ cos Ii,,1 i
-kEF(r-I)
1:. I:...
problem
I
I (50) has
,
if and only if tan (X-,,l) - kEEl = 0.
(51)
The cases of (32). however. never meet (51). For the symmetric steady states represented by the EC or ECP solution, boundary-value problem (50) has the non-vanishing solution
the
e(r) = 0. JIR,(sin(A-,,~)-k,,~}+Rzjcos(k,,~)-
I}]
for
0 d c < rItc..
for
rlEC < : G 1.
r:(r) = i J[sinkFF(~-/)-RIX.PP(r-1)] where R, = sin A_,,+,.sin
A.tE(qtC - I) + (krE/kpp)coskpp~cc
R2 = COSA.~~~~~~ sinA,,.(r],,
-I)-(kEelkPP)
cosk,,(q,,.
-I).
sink,,tl,,cosk,,(rl,,.-I),
if and only if
-/G, ..,.cosk,,(~,.~-I){sinli,,~,c-kpplcoskpp~,:(-)
= 0.
(52)
It can be shown that the lowest critical solution is given by the smallest positive root r$c of (52). Then by substituting ?,& = & into the EC solution (42) or the ECP solution (43), the critical symmetric steady state, at which the uniqueness condition
Uniqueness
criterion
and symmetry
513
limit
(26) is first broken, can be obtained. It could be shown by some numerical examination that in almost all cases the condition (35) holds at least up to this uniqueness limit. The uniqueness limit conditions (51) and (52) coincide with the symmetry limit conditions (79) and (75) in UETAIGIand NAKAMURA (1983). It may be concluded that the symmetry limit can be found as the uniqueness limit of the incremental variation of steady state for the present problem of cantilever beam-columns.
REFERENCES
1958
HILL,R. UETANI,K. and NAKAMURA.T.
1983
J. Mech. Phys. Solid.7 6,236. J. Mech. Phys. Solids 31,449
APPENDIX I For a flange element exhibiting any type of steady-state behavior except type C, the of (24a) or (24b) obviously hold because the constitutive equations (23) are identical for this element. For a right flange element exhibiting a steady-state line C. the sign of the difference left side from the right side of the inequality in (24a) is estimated in view of the last of (20) as follows : when
CL > 0
when
8’,’ > 0
and
D = -2(1 when
$
< 0
D of the equation
(AI-l) & < 0
-a)d:,(dk’-&) and
with (12)
CL > 0
D = 0, and
equalities
> 0.
(Al-2)
L)1< 0
D = 2(1 -z)($-&)’
> 0.
The equality in (24a) is evident from the last equation of (20). The inequality in (24b) can be proved in the same manner as shown above.
(Al-3) and the equality
APPENDIX I I A general solution
of (29) may be written
for each region by
L& = ti:.,;: = SE sin fk,, + T, cos k,,<. .I L’p.:: = SP sin kpp< + T, cos k,,,<. where k,, = xl’njcx. By using (28) with Table 1 and (7), the 1st and 3rd conditions 10 &:hr) G..:::(k,) From (30) and (AII-I)
through
(AII-4).
of (11) at r = +-rare
(AH-l) (AIL2) reduced
= 4.::0kF.L
(AH-3)
= GJ.&kP).
(AII-4)
K.
514 Tf
=
-
UETANI Sr
tank,,l.
(All-5)
(All-6)
(All-71 Substitution
of (AlI-2).
(All-6)
and (AII-7)
into Cb.iz(0) = 0 yields (34).
0022-5096Gl9 53.00 + 0.00 c 1989 Pergamon Press plc
1989.
Primed 10Great Britain.
UNIQUENESS CRITERION FOR INCREMENTAL VARIATION OF STEADY STATE AND SYMMETRY LIMIT KOJI UETANI Department of Architecture. Kyoto University, Sakyo. Kyoto. Japan (Received
30 June
1988)
ABSTRACT A SUFFICIENT condition for uniqueness of the incremental variation of steady state is established for a cantilever beam-column subjected to completely reversed tip-deflection cyciing. The critical steady state. at which the sufficient condition for uniqueness is first broken, is found on the sequence of symmetric steady slates generated under a conlinuously increasing tip-deletion amplitude. In a previous paper (UETANI,K. and NAKAMJRA.T., J. Me&. P~.Ys.Soli& 31,449, 1983). a symmetry limit has been defined as the critical steady state, at which transition from a symmetric steady state to an asymmetric steady state occurs first, and its theoretical solution has been obtained. The uniqueness limit turns out lo coincide with this symmetry limit. Though the previous theory for symmetry limits was constructed on the basis of five hypotheses introduced without proof. the presenttheory involves no such hypothesis.
1. INTRODUCTION WHEN AN initially straight cantilever beam-column is subjected to an idealized completely reversed tip-deflection cycling program with continuously increasing amplitude under a constant axial compressive load, the beam-column exhibits a sequence of symmetric steady states, in each of which a pair of extremely deflected configurations are symmetric with respect to its initial member axis as shown in Fig. l(a), during some range of smaller tip-deflection amplitudes. But when the tip-deflection amplitude reaches a certain limit, transition from a symmetric steady state to an asymmetric steady state occurs and subsequently the beam-column exhibits a sequence of asymmetric steady states as shown in Fig. l(b) with increasing amplitude of an antisymmetric deflection mode. UETANI and NAKAMURA (1983) have called the critical state, at which the transition occurs, a symmetry limit and developed a theory for predicting the symmetry limit without tracing the entire hysteretic response of the beam-column. The essential ideas of the previous theory are stated as follows :
(a)
(b)
A sequence of steady states generated under a tip-deflection cycling program with continuously increasing amplitude may be regarded as a path in a special space, each point in Which represents a steady state. Such a path has been called a steady-state path. A symmetry limit is regarded as the bifurcation point on a steady state path 495
496
K. t
C’ETAh.1
Time COIDA program denseiY disposed constan+-amp)tude CYCum Proces8es Of COntnUOUSly mcreashg ampCtude
a pa, Ofextremely deflected shapes at 8” asymmetru steady state
FIG
(a) (b) I. A cantilever beam-column subjected lo 3 up-deflection cyclmg program axial
(cl
comprewve
COIDA
under a constant
load.
representing a sequence of symmetric steady states. from which another branching path representing a sequence of asymmetric steady states emanates. The symmetry limit may be found as the symmetric steady state with the smallest tip-deflection amplitude among those satisfying the condition that “to the problem of finding possible rates of steady-state variables with respect to the tipdeflection amplitude. there exists a solution involving non-vanishing amplitude of an anti-symmetric deflection mode”.
The previous theory. however. was developed on the basis of five hypotheses introduced without proof. Because of the lack of proof, a numerical analysis of the hysteretic response had to be performed to verify the hypotheses and the prediction results. It was shown by the response analysis that transition to an asymmetric steady state actually occurs at the theoretically predicted symmetry limits. In the present paper, a new theory for predicting symmetry limits is developed on a more clear and more certain basis than that of the previous theory in the sense that all the uncertain assumptions are abandoned. First a sufficient condition is established for uniqueness of the incremental variation of steady state caused by a given increment of tip-deflection amplitude. The critical symmetric steady state. at which the sufficient condition for uniqueness is first broken. is found along the sequence of symmetric steady states and shown to be identical with the symmetry limit obtained in the previous paper.
2.
ANALYTICAL
MODEL
AND
BASIC
EQUATIONS
The present theory is concerned with the behavior of an initially straight cantilever beam-column subjected to an idealized completely reversed tip-deflection cycling program ofcontinuously increasing amplitude Y. referred to as COIDA, under a constant
Uniqueness criterion and symmetry x=Hef
limit
491
IN=AEn
FIG. 2. Analytical model. axial compressive dead load N as shown in Fig. 1. In a COIDA program, infinitely many infinitesimal increments are consecutively applied in such a way that after every infinitesimal increment of the amplitude, the deflection cycling at that increased amplitude is repeated as many times as is necessary for the beam-column to attain a steady state. Infinitely many COIDA programs with different micro-compositions can be conceived. The analytical model used here is the same as employed in UETANIand NAKAMLJRA (1983), which is a model beam-column with a uniform idealized sandwich cross section as shown in Fig. 2. Normal stresses are transmitted only through the two thin flanges of cross-sectional area A/2 placed at distance 2H. and shear stresses only by the core, which makes no shear deformation. The material of the flanges obeys the bi-linear hysteretic stress-strain relation of Fig. 3, specified by elastic modulus E, strainnotations are introhardening modulus E,, and yield stress tiv. Non-dimensionalized duced for the quantities shown in Fig. 2 as follows: 5 = x/H for axial coordinate. I = L/H for length of the beam-column, n = N/EA for constant axial compression. oL = iiL/E for compressive stress in the left flange, o,, = i?JE for compressive stress in the right flange, u = U/H for displacement component in the direction of the initial member axis, 1: = V/H for displacement component in the direction perpendicular to the initial member axis. r = E,/E(O < cx < 1) for strain-hardening ratio. On the assumption of small but finite deflection conventionally employed in the post-buckling theory of beam-columns, the equations of equilibrium are written as
FIG. 3. Bi-linear stress-strain
relation of the flanges.
K. UETANI
FIG. 4. Three stead!-state
lines and a steady-srate
loop
_!(a,+uL ) = I?. !(a, where a subscript i following boundary conditions are
U[
a comma
I,;, +
171‘,;;
denotes
0
=
(1)
differentiation
with respect to <. The
u(0) = r(O) = r.,,(O) = 0. O,(/)_UL(/)
= 0.
z.(I) = 2.1.
(2)
where 1’Tis a given value of tip-deflection. Compressive strains I:, and I:~ of the left and right flanges are related to lateral displacement 1. and compressive axial strain P of the central fiber by EL = r-l’,i:
= -u,:-
!(r..J2-z’,!i.
&k = I?+ I‘,;; = - II._- 4(I’,; ) 7+ I‘,;, The displacement
component
u can be calculated
zd(<) = -
(3)
from LJand 1’by
,; [c-t (r.,:)’ 21 d:. s
The stress-strain
relations
of flange elements
are
fJ=I:
for virgin elastic response.
u = W+ (1 - ~)a).
for compressive
c = rc-(1
for tensile strain-hardening
0 = C--E’
-r)a,,
for unloading
strain-hardening
response.
response,
response. (4)
where .sr denotes the residual plastic strain. The cyclic response of a flange element in a steady state may be depicted by a closed locus of the state point in the stress-strain plane and classified into four different types shown in Fig. 4. A steady-state line E represents a purely elastic response of a flange element. Steady-state lines C and U represent shakedown states attained after some histories of plastic deformation of flange elements. A steady-state line C starts from and returns to the strain-hardening branch. whereas a steady-state line U does not.
499
Uniqueness criterion and symmetry limit
There is only one type of steady-state hysteresis loop denoted by P. It represents a cyclic behavior of a flange element called alternating plasticity where tensile and compressive plastic strains are alternated. 3.
A SUFFICIENT CONDITIOK FOR UNIQUENESS OF INCREMENTAL VARIATIOX OF STEADY STATE
A beam-column is supposed to be in a steady state. where the same passage of deformation is repeated for every cycle of tip-deflection with a fixed amplitude I(/. It is assumed that the steady state satisfies the following condition (H3’). Condirion (H3’). The strain direction of each element in the right flange is reversed from compressive to tensile only at an extremely deflected state r’ of r(l) = ti and from tensile to compressive only at I-” of c(l) = -I//. The strain direction of each element in the left flange is reversed from tensile to compressive only at I-’ and from compressive to tensile only at F’.
This is a little more specific statement than that of the hypothesis (H3) in the previous theory. When there is no bifurcation point on the closed equilibrium path representing a steady-state cyclic behavior, the whole of the steady-state behavior may be specified only by a set of reversal-point configurations u’ = {e’(t), ~‘(5)) and u” = (e”(r). L.“(C)). The quantities belonging to I-’ and to Y” are indicated by the superscripts I and II respectively. The relations between reversal-point stresses and reversal-point strains are written for each type of flange-element cyclic behavior as type E :
a; = CL,
typeU:
a; = EL--t&.
typec:
ak = rck + ( - 4ay,
ai = $+(a-l)(&-ady),
aif = LY$’+ ( - r)av.
a: = $+(z-
type P :
a: = E:,
a:’ = c:,
ai = El:-&.
a; = EL, al =&P--C;,
EL-&E;.
I)($-ay),
ok = r&+(
- z)ay .
01 = a&--(l-a)a,,
aLI' =
- %)a\.
a: = r&--(1
TE:’ + (
a: =
-a)aY.
(5)
When an infinitesimal incremi el It dl// of tip-deflection is applied to the beam-column in the known steady state under a tip-deflection amplitude $, the beam-column will move to an adjacent steady state (II’ +I? dl//. u”+ti” d$), where 6’ and I?’ denote the rates of change of reversal-point configurations u’ and u”, respectively, with respect to II/. Suppose that there could be two distinct possible rate solutions (i~‘.ti”) and (ti”.i~“‘). The former solution (a’. a”) satisfies the following basic rate equations derived from the equations (l)-(3). Field equations of equilibrium :
_!((ik+ti:, i(iil: Strain4isplacement
= 0.
ltir:‘) =
relations
:
0.
h((il, -ir:),:;
+d:: .. . = 0,
i(irl: -c$$:
+n2;: = 0.
(6)
UETAM
K.
500 $
=
p’ __$
.I’ El
_ -
.I’ :I’ -_I :t, c .5.
dk
.::7
=
.‘I = CR
2
+
I&
pll +
p, .., ’
(7)
where p” = _ r;‘! _ ‘.I!‘:‘! . _, .. .
p’ = - lj’: - r.‘$‘: ., .,* Geometrical
boundary
:
conditions
P’(l) = 1.
r:‘(O) = d(O) = 0, P(0) Mechanical
boundary
= r:!:‘(O) = 0,
1:“(I) = - 1.
(8)
:
conditions c&(f) -6;(l)
b;(l)
= 0.
-rif(l)
= 0.
(9)
A beam-column in a steady state may be divided into some number of regions. in each of which the left and right flanges of every element exhibit the same pair of types of cyclic behavior, e.g. [E, E], [C, U] etc. since ti, ti,<. bK-tiL and (hR --r!~,),~ -nnl:,: should always be continuous throughout the beam-column, then at every inter-region boundary I”(q-0)
= F’(q+O),
P(‘l-0)
= P(q+O).
c$(q-0)
= &(t]+O),
1?(q-0)
= Z(q+O).
@k-ti:)I;=,
0 = (d:,-4_)l:r.,+,.
(&ti:‘,];=,_,,
The relations given by
between
= (+c+:‘)]~=,_,,,
(&tit_),J~=q_o
= (c&&).:I~=,+o.
(&-&‘,.iI;_,_,,
= (fY;-6~‘.:]t=11+o.
reversal-point
(10)
(11)
stress rates and reversal-point
tik = ,&k.
61: = V&k’+ (pLR- VK)&.
ti:’ = ~“2:‘.
ci( = \.tc: + (FL - v,)c’:‘,
strain
rates are
(12)
where the coefficients p and v are to be selected in accordance with Table I. It should be noted that these coefficients depend not only upon the current type of steady-state TABLE 1. Coe#icients.for
Type of steady-state behavior
real and.fictitious
Direction of reversal-point strain rate
materials
Uniqueness
criterion
and symmetry
501
limit
response of the flange element but also upon the direction of its reversal-point strain rates. The rate equations (6) of equilibrium in the field and the rate equations (9) and (I 1) of mechanical boundary conditions and inter-region boundary conditions are equivalently given by the variational equations
(13) where dsL = 6e - &q,:;, for all the continuous
virtual
displacement
&
= de+ 6’1.;:
fields &I( = {ae, SC}) satisfying
&I(O) = SC.:(O) = &I(l) = 0. The subtracted field ed( = ir’ - 8” = (PI-@‘I, ti’ - t”}) following equations from (13), (7), (8). (9) and (12). O’[:(b&_
+&&}
-nti;‘&T.<] dt = 0.
(14)
may be shown
to satisfy
the
g”R= ed+LC,
Et = $‘-&,
s tid(0) = z’?(O) = 0, ci; =
Cd(l) = 2, .d VREK,
cf"L =
f$(/)-C?:(I)
= 0,
VLdf.
(15)
The other solution (ir”. I?“) also satisfies a set of equations given by replacing all the unstarred quantities in the equations (6)-( 13). (15) by the corresponding starred quantities. From the unstarred and starred equations of (15). it can be shown that the difference fields (Air’, Ati”) defined by Ai,’ = i,” _$,
Ai,” = $1’ -8”
(16)
satisfy the equation I [j {v,AE~&’ + v,A&r)~~} -nA?:6r:.;]
d{ = 0.
(17)
I0 Since Aztd(0) = AZ?(O) = APd(I) = 0. the equation
(I 7) should
still hold with 6u = Aird :
~[~{vL(AEd,)‘+vR(A$)2j-n(Ati$)2]d~
= 0.
i A sufficient
condition
for uniqueness
of the subtracted
field is therefore
that
SO'
K. UETASI
nyti, =
> J‘[l(vl.(i:,)~+\.R(~R)2J-n(l:.i)~]d: 0.
(18)
0
where 6,
for all non-vanishing
continuous
=
i-r:
::.
.. .
ER = (i_tr:
fields ti subject
:. ..
to the geometrical
constraints
1:(O) = r:,,(o) = t:(I) = 0.
(10)
It should bc noted that this condition guarantees uniqueness of the subtracted held it’--ti”, but not uniqueness of the steady-state rate solution (I?.$‘). Under the condition (18). two distinct solutions (it’. 8”) and (ti”. II”‘) are connected by the relations AC;’ = A$‘_
At:’ = A,:“.
Aii; = A$‘.
Ai’, = A&.
(20)
Subtraction of the virtual-work equations in (13) respectively equations for the starred quantities yields
from the corresponding
Since AI? and Ati” are subject to the same geometrical constraints should still hold after replacing 6u by A6’ or 66” respectively.
These equations the steady-state
suggest the introduction rate solution as follows
:
of a sufficient
condition
as (14) on &I, (21)
for uniqueness
of
+(ir:!‘_~l’)(ril”_-tll!)+(~~ -n{(2:;; -I:!: )z+ -cf!)‘}] (I y
for all sets of it’, ti”, it”, I?” satisfying
(8)
di > 0
(22)
Uniqueness
criterion
and symmetry
503
lrmit
The condition (22) is hard to apply to uniqueness limit analysis of the incremental variation of steady state. because the material coefficients ,u’s and Y’Sin (12) depend upon the signs of strain rates. in other words. because A&:.k, AC& are not related to Ail,R. A$!. k in the sense of (12). By analogy with the fundamental inequality introduced by HILL (1958) for deriving a relaxed sufficient condition (over-sufficient condition) for uniqueness of equilib~um path. a fictitious material obeying the following constitutive relations is envisaged here. (ik = &dk,
ci’l: = sgbf + (pi - s;)t$.
tit’ = p:s:‘.
6; = tf$ + (p[ - st’)d:‘.
(23)
The coefficients $‘s and \-r’s are given in Table 1 and depend only upon the current type of steady-state response of the flange elements. Under the condition (18) the following inequalities and equalities hold : (~k’-~~)(~~-~~)+(~~*-~~)(~~~_~~)
FAi; + (& - v;)AE;‘lAi; >, j&A& fdi:, + \’\‘R (61’ - (iL)($ -&) + (6;’ - $)($’
= Z&(A&)‘,
-#$)
> {$A& + (p: - \~~~A~~!~A~~ + fpt:‘A.$’)A$ = 2$(A&)2.
(24a.b)
These can be proved in Appendix I. In view of (24) it can be shown that I-I(ir’,ir” ; 8”. i”‘)
3 I-IF(Air’)
(25)
where
From (22) and (25). we obtain a more relaxed condition incremental variation of steady state as II’(C) > 0
for uniqueness of the
(26)
for all non-vanishing continuous displacement rate fields I satisfying the geometrical constraints (19). The inequality v 2 pr yields that I-P(C) 2 I-I’(il). This means that the condition (1X) is necessarily satisfied whenever the condition (26) holds. In consequence it may be concluded that (26) provides a sufficient condition for uniqueness of the incremental steady-state variation from a known steady state satisfying the premised condition (H3’).
4.
SEQUEISCE
OF
SYMMETRIC
STEADY
STATES
In the present and next sections the symmetry limits of cantilever beam-columns subjected to a tip-deflection cycling program COIDA under constant axial compression will be obtained by applying the uniqueness condition derived in the previous section. Existence of a sequence of symmetric steady states is obvious from the fact that both the initial configuration of the beam-column and the loading system are symmetric with respect to the initial member axis. In the plane of Fig. S. whose coordinates are tip-deflection amplitude ci, and amplitude rh of anti-symmetric deflection mode, this sequence of symmetric steady states is represented by the straight lint C>SF along the vertical axis emanating from the origin. which has been called a fundamental steady-state path. The steady-state point definitely moves up along this fundamental symmetric steady-state path as the tip-deflection amplitude is increased. at least as far as the sulhcient condition (26) for uniqueness of the incremental steady-state variation holds. In the present section. analytical solutions to the sequence of symmetric steady states are obtained. The solutions will first be derived presuming that the premised condition (H3’) holds and thereafter the validity limit of the solutions. within which (H3’) holds, will be clarified. 4. I. V~riu~ion f.$sraw wgions Consider an incremental variation from a generic symmetric steady state to an adjacent symmetric steady state caused by a given increment of tip-deflection amplitude. The rates of change of the steady-state variables during this process arc connected by the symmetry relations _
+I_
.I
.II
CL = &R
q : TIP-DEFLECTION AMPLITUDE
steady
states
ANTI-SYMMETRIC DEFLECTION AMPLITUDE
: Vb
FIG.5. A seqwnceof symmetric
steady siatcs and a paw of sequences of asymmetric
steady states
Uniqueness criterion
symmetry limit
and
Since both the current steady state and the incremental consideration are symmetric,
Using (27) the constitutive
relation
(12) is reduced
of (28). (7) into the second equation
.I VC.;:~;
under
to
for
(28)
of (6) and the first equation
of (9)
(29)
= 0.
When the current steady state involves no P-region, exhibiting a steady-state loop P, from Table 1
In this case, the solution
variation
= 0,
+nff::
A(I)
v= 1
steady-state
6: = v&+(/i-v)&.
6; = pck, Substitution yields
505
(30) where every flange element
is
0 < r
of (29) under the boundary
conditions
sink,,(i-I)-k,,rcosk,,l+sink,,l ---.-~ sin k,,lkEEl cos kEEI
cl = -
for
(8). (30) is given by
OG<
where
The corresponding
stress rates are given by .I
-
CJu -
-6:
=
-nsink,,(<-I) sin k~El-k,,icos
It is shown by (3 1) that 6; is monotonically the condition that
lip;7 decreasing
for
0 < r < 1.
(31)
from the base to the top under
7[--
” -=c 4/’ The right-hand-side of (32) is identical with column with a rigid constraint of tip deflection. is sketched in Fig. 6(a). It may be concluded cases of (32) the variation of state regions in steady-state path has the following properties
(32)
the Euler buckling load of a cantilever The distribution of irk for n = n2/(41*) from the above discussion that in the the beam-column along the symmetric :
506
(b)
(a) FIG. 6. Distrih~~tions
of c?k : (a) for n = a’;(4/‘);
(b) at ,I~.~ = &
(3 At every tip-deflection amplitude level below a certain Iimit. the whole beamcolumn is occupied by an E-region only as shown in Fig. 7(a). when n < cl.. (33) (ii) A C-region appears from the base and spreads monotonically toward the tip as shown in Fig. 7(b). (iii) Any U-region does not appear at least until the appearance of a P-region. (iv) A P-region appears from the base as shown in Fig. 7(c).
Our subsequent discussion is restricted to the cases of (32) and (33). After appearance of a P-region, such an arrangement pattern E-C-P of state regions as shown in Fig. 7(c) is to be maintained as far as Irk(T) > 0 over the range 0 < 4 d 1. This condition is to be broken when the P-region spreads to such an extent that &k(O) = 0. or equivalently that 8;;(O) = 0. It is shown in Appendix II that a non-vanishing reversal-point deflection rate I:’ satisfying tiftc(0) = 0 can exist only when the inter-region boundary < = Q.,, between the C-region and the P-region is such that
(a)
(b) FIG.7.
Vmalmn
of arrangement patterns al state regions.
62)
Uniqueness k,,
criterion
and symmetry
507
limit
cos kppr7cp sin A.rr(~c.r- I) -kEE sin kpprjcp cos kEE(qCP -I)
= 0,
(34)
where
Denoting the smallest positive root of (34) by q &. the condition over the range 0 d 5 d I is given by R-P
<
that irk(<) > 0 all
(35)
rl?P.
Figure 6(b) illustrates the distribution of irk at qcp = $JJp. The condition (35) guarantees that the P-region never diminishes and then that the inter-region boundaries qcp and qEc move monotonically toward the tip as II/ is increased. This implies that the arrangement pattern of state regions varies in the order of E, E-C and E-C-P and that the pattern E-C-P is preserved at least as far as (35) holds. 4.2. Symmetric
steady-state
solutions
The symmetric steady-state solution may be derived for each arrangement pattern of state regions. At any symmetric steady state the state variables of F’ and F” satisfy the conditions cr = _ r’tr)
e’ = e”, The constitutive E-region
:
C-region
:
equations
&k = &:l,
-Z)by,
a: = E: -(I
(36)
-Z)(&‘K-(Ty).
CT; = a$-(1
-r)av.
The field equations governing 1.’obtained general solutions are written as follows :
-Z)dy.
by substituting
t:f:::: +nrflc = 0,
u: = J: sin kEt, + K1, cos k,,(
+ B:< + 0;.
: eI
=!+(.!-z) a -. (t’!:; -cry). z
rItt;: +nz!!:: = 0,
r$ = J~sink,,i+k~cosk,,5+~:.5+~~, e;=p.
n-(1
-~)a~ .Lx
-
(37)
(37), (3) into (I) and their
: e’ = n,
C-region
f.7; = D:‘.
: ak = rek+(l
E-region
a: = a;,
for each state region are given as follows :
a; = r&+(1 P-region
E: =&;,
n(J:- sin kEEr + KL cos kEEr),
508
P-region
K. t.kTAI.;l :
rb = Jb sin A,,< + lub cos X-r& + Bb{ + 0;. The inter-region
boundaries
.Z = vtc and f = rjcp are characterized
(38) by
ak@kc) = cY. ~:,(Vc,)-o:(rlcr) These conditions
may be transformed r:;.::(k)
= &.
with (37). (3) and (38) to = fly -n[
G.:: 07CP) = fly [=
= L.:,::(&c)].
(39)
&:hdI.
(40)
The constants of integration and the inter-region boundary locations may be determined from the boundary conditions (3) and an appropriate set of inter-region boundary conditions, i.e. the continuity conditions for t’, c!;, (crk -a:), (r& -a:).; and characteristic conditions (39) and/or (40). The resulting solution of symmetric steady states is given for each arrangement pattern of state regions by (E-solution) J;
=L:
6 _.
__.___._.
tank,,/-&,I‘
z-l: = -k,,Jk. (EC solution)
The equations
tank,,I,
0: = -lg..
(41)
of (41) and
,:- = .I:. 41/sin k&Fjrc
K: = -&
lu:. = K:, -I)
B& = &,
D: = D:..
=
(ECP solution) g/ =
‘f n
_____. t? i tank,,(Fjcr,--1)
+ r(k,,Jsin
&p~cos kppqCp - sin kppvCP)
kppkp +cos kpp~cp) + 1 - 2 , I
Uniqueness criterion
and symmctv
WY
ltmlt
Once any one of the quantities +, qEc. qcP is specified. all other unknowns together with the arrangement pattern are determined directly from (41). (42) or (43). 4.3.
Validity range oj’lhe condition (H3’)
Though the symmetric steady-state solutions have been obtained on the assumption that (H3’) holds, nothing has yet been discussed about (H3’). Besides, it has been implicitly assumed that no bifurcation path of equilibrium emanates from the closed equilibrium path representing the cyclic response of a symmetric steady state. In this sub-section these assumptions are examined and the validity range of the symmetric steady-state solutions is clarified. The condition (H3’) can be assured by showing that at every configuration in a symmetric steady state
and
aK(5) b 0.
o;(t)
< 0
over
0 < << I
when
z.‘(j) > 0
gK(U G 0,
a;(t)
3 0
over
0 d r < I
when
r’(I) < 0,
where ( )’ denotes differentiation with respect to an equilibrium-path parameter I. The governing equations for the incremental change in the state of equilibrium caused by a given tip-deflection velocity may be written as follows : The equilibrium boundary conditions and inter-region equations, straindisplacement relations. boundary conditions may be obtained from (6)-( I 1) by replacing {( ‘)‘}‘s by {( )‘}‘s. Constitutive equations are given by
where I
for or
P= z
for
Iu-zrel for
< (I-~)a,
IG--OLCI= (I-r)ay
la-a&l
= (I-a)aY
and and
(a-as)&
(a--~)&‘>
< 0, 0.
It should be noted that a steady-state cyclic response of a flange element of type C involves no plastic deformation because the stress-strain state point is reversed immediately from the arrival point on the compressive strain-hardening branch as shown in Fig. 4. It follows that at every configuration in a steady state represented by the E or EC solution,
p’_= pa Following
the same procedure
= I
over
as in deriving
0 < < < I. (3l), it can be obtained
that
K.
510
UETASI
It is shown by (31) that under the conditions (32) and (33). (H3’) holds in all symmetric steady states given by the E or EC solution. For a symmetric steady state of ECP. distribution of the values of pt. and pR in the P-region differs at different configurations. Examination of stress-rate distribution in general provides that the most critical with respect to (H3’) is the configuration with the largest yielding portion in the P-region. i.e. the reversal-point configurations I-’ and F”. where p, = /jK = I
for
ticp d 5 d 1.
[‘L = /)H = 2
for
0 d < < qcp.
It can be shown by repeating similar discussions to those leading to (35) that (H3’) remains to hold at the symmetric steady states represented by the ECP solution as far as
R-P Q rl?P.
As to the second assumption. application of the criterion by HILI. (1958) for uniqueness of equilibrium paths yields the conclusion that the closed equilibrium path representing a symmetric steady state should be unique, in other words that there is no branching point on this closed path. if fIcq(ti) =
~:[ji~~(EI)‘+~~(~K)~)
-n(zL)‘]dr
> 0.
s where PR = {‘r = I /)K = !I.
p, = I
/+ = 11, = r for all non-vanishing continuous Since p, z ,$ and pK 3 &.
displacement
for E-region, for C-region, for P-region, rate fields ir satisfying
rYJ(il) 2 l-V(o).
(19).
(44)
The equality in (44) holds only for purely elastic symmetric steady states. The inequality (44) guarantees that there is no bifurcation point on the closed equilibrium path ofevery symmetric steady state satisfying the sufficient condition (26) for uniqueness of the incremental variation of steady state.
5.
UNIQUENESSLIMIT AND SYMMETRYLIMIT FOR CANTILEVERBEAM-COLUMNS
In the previous section it has been shown for the cases of n < dY and n < 7r’/(41’) that there exists a unique sequence of symmetric steady states for a continuously increasing tip-deflection amplitude. When a beam-column is subjected to a COIDA program of tip-deflection cycling, it should actually follow the sequence of symmetric
511
Uniquenesscriterionand symmetry limit
steady states, which is represented by the line OS in Fig. 5, as far as the sufficient condition (26) for uniqueness of the incremental variation of steady state holds at each symmetric steady state. In this range. therefore, there is no possibility that an\ transition from a symmetric steady state to an asymmetric steady state can occur. Any transition to an asymmetric steady state necessarily requires the uniqueness condition (26) to be broken. In the present section, the critical symmetric steady state, at which the uniqueness condition (26) is first broken, will be found. From the critical symmetric steady state of uniqueness limit, transition to an asymmetric steady state may occur as shown b> SB or SB’ in Fig. 5 or may not occur. Then the resulting solutions of uniqueness limit will be compared with the symmetry limit solutions in UETANI and NAKAMURA (1983). The sufficient condition (26) for uniqueness may be restated equivalently that min (IIF(ti)). l&s where S denotes the set consisting satisfying the geometrical constraints Q(i) o
-
> 0.
(45)
of all the continuous (19) and a normalizing
displacement condition
rate fields
’ [P’ + (Cc;)‘] dc = o. I0
being an arbitrary small positive constant. The field ti minimizing IIF under R = w necessarily I-I:, (ti, 6u) +iR,,
(46)
satisfies the variational
(Ii, h)
equation
= 0
(47)
with a special value of the Lagrange multiplier i., for all continuous variational displacement fields 6u satisfying (14). I-I:, and R, , in (47) are the bi-linear functionals derived from the quadratic functionals II’ and R respectively. It should be noted that the trivial field ti = 0 satisfies (47) but not (46). The field differential equations and the mechanical boundary conditions derived from (47). together with the geometrical boundary conditions, constitute an eigenvalue problem. Let the ith eigenvalue be denoted by %(‘)and the associated eigenmode by iP. which is normalized so that R(G) Since the equation n:,($“, From
this equation
= 0.
(47) is satisfied when i. = I”’ and ti = 6u = ti’“, we obtain i”‘)+
).(‘jQ,
,(fi(", 8"') =
2{fl'(fi"')+
j.'"Q(i+")} =
that
0.
and (46), nF(hC”) = _jI”o .
.
This implies that the minimum value of II’ is given by the largest eigenvalue I.,,, and that uniqueness of the incremental variation of steady state can be assured by the condition that i.,,, Since the condition
(48) is to be broken
< 0. with i.,,,
(48) = 0, the critical
symmetric
steady
K. UETASI
512
state, at which the sufficient condition (26) for uniqueness is first broken. is characterized by the existence of non-vanishing displacement fields satisfying the equation rI:,(iL6u)
= 3
’ [p’(&+ i I,
I:,;:&.,~~) - nr:,:6z..;] d; = 0
(49)
for all continuous variational displacement fields du satisfying (II). Operating on (49) by integration by parts. the following set of equations may be derived as necessary conditions for (49). :
Field equations
/AL;;
P = 0. Boundary
+ nz:,;<= 0.
:
conditions
(50)
C(O) = C,:(O) = 0. Inter-regional boundary conditions $‘:,::, pFc,i:; should be continuous. For purely elastic symmetric the non-vanishing solution q;, where J is an arbitrary
= 0.
r:(l) = L:,i<(l) = 0. (at < = qE(.. qcp) : the quantities
steady
l:(t) = J
constant.
states,
the boundary-value
sinl\-.EE(C:-I) ._~ cos Ii,,1 i
-kEF(r-I)
1:. I:...
problem
I
I (50) has
,
if and only if tan (X-,,l) - kEEl = 0.
(51)
The cases of (32). however. never meet (51). For the symmetric steady states represented by the EC or ECP solution, boundary-value problem (50) has the non-vanishing solution
the
e(r) = 0. JIR,(sin(A-,,~)-k,,~}+Rzjcos(k,,~)-
I}]
for
0 d c < rItc..
for
rlEC < : G 1.
r:(r) = i J[sinkFF(~-/)-RIX.PP(r-1)] where R, = sin A_,,+,.sin
A.tE(qtC - I) + (krE/kpp)coskpp~cc
R2 = COSA.~~~~~~ sinA,,.(r],,
-I)-(kEelkPP)
cosk,,(q,,.
-I).
sink,,tl,,cosk,,(rl,,.-I),
if and only if
-/G, ..,.cosk,,(~,.~-I){sinli,,~,c-kpplcoskpp~,:(-)
= 0.
(52)
It can be shown that the lowest critical solution is given by the smallest positive root r$c of (52). Then by substituting ?,& = & into the EC solution (42) or the ECP solution (43), the critical symmetric steady state, at which the uniqueness condition
Uniqueness
criterion
and symmetry
513
limit
(26) is first broken, can be obtained. It could be shown by some numerical examination that in almost all cases the condition (35) holds at least up to this uniqueness limit. The uniqueness limit conditions (51) and (52) coincide with the symmetry limit conditions (79) and (75) in UETAIGIand NAKAMURA (1983). It may be concluded that the symmetry limit can be found as the uniqueness limit of the incremental variation of steady state for the present problem of cantilever beam-columns.
REFERENCES
1958
HILL,R. UETANI,K. and NAKAMURA.T.
1983
J. Mech. Phys. Solid.7 6,236. J. Mech. Phys. Solids 31,449
APPENDIX I For a flange element exhibiting any type of steady-state behavior except type C, the of (24a) or (24b) obviously hold because the constitutive equations (23) are identical for this element. For a right flange element exhibiting a steady-state line C. the sign of the difference left side from the right side of the inequality in (24a) is estimated in view of the last of (20) as follows : when
CL > 0
when
8’,’ > 0
and
D = -2(1 when
$
< 0
D of the equation
(AI-l) & < 0
-a)d:,(dk’-&) and
with (12)
CL > 0
D = 0, and
equalities
> 0.
(Al-2)
L)1< 0
D = 2(1 -z)($-&)’
> 0.
The equality in (24a) is evident from the last equation of (20). The inequality in (24b) can be proved in the same manner as shown above.
(Al-3) and the equality
APPENDIX I I A general solution
of (29) may be written
for each region by
L& = ti:.,;: = SE sin fk,, + T, cos k,,<. .I L’p.:: = SP sin kpp< + T, cos k,,,<. where k,, = xl’njcx. By using (28) with Table 1 and (7), the 1st and 3rd conditions 10 &:hr) G..:::(k,) From (30) and (AII-I)
through
(AII-4).
of (11) at r = +-rare
(AH-l) (AIL2) reduced
= 4.::0kF.L
(AH-3)
= GJ.&kP).
(AII-4)
K.
514 Tf
=
-
UETANI Sr
tank,,l.
(All-5)
(All-6)
(All-71 Substitution
of (AlI-2).
(All-6)
and (AII-7)
into Cb.iz(0) = 0 yields (34).