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Large displacement analysis of elastic-plastic trusses with unstable bars
Large displacement analysis of elastic-plastic trusses with unstable bars Czeslaw C i c h o n Cracow Technical University, Cracow, Poland
Leone C o r r a d i Department of Structural Engineering, Politecnico, Milano, Italy (Received April 1980; revised November 1980}
Large displacement analysis up to the collapse of trusses is considered. Bars are elastic-plastic and possibly unstable post-critical behaviour is accounted for. The collapse situation may correspond either to a limit point (snap-through buckling) or to the occurrence of bifurcation in Shanley's sense. Attention is mainly focused on the global stability test, which permits the collapse situation to be identified. It is shown that the test can be reduced to a check on the sign of a single number, which can be easily obtained for any given structural configuration. It is also shown how this test can be introduced into a large displacement elastic-plastic analysis code with little additional computational effort.
Introduction In this paper truss type structures will be considered. Bars exhibit an elastic-plastic behaviour and the possibility of local instability is accounted for in the force-elongation diagram of the bar which may present a negative slope (softening) after a given force level is reached (as is typical of unstable post-critical behaviour on the compression side of the diagram). The analysis up to the point of collapse of such trusses is considered. Usually, collapse corresponds to a limit point on the loading path and it is often due to global instability of the partially yielded structure. However, as Shanley 1 pointed out stable bifurcation may occur prior to the structural load carrying capacity being exhausted; this phenomenon is also considered as critical for the structure. A truss configuration will be defined as critical when it corresponds to a limit point along the loading path or if stable bifurcation may occur for some load increments. Conditions to detect a critical configuration have been proved for the case of associated flow laws and conservative external forces. 2, 3 Under widely general hypotheses, they can be reduced to a test on the sign of a single number, which can be easily defined for any known configuration. Part of this paper was presented at the 20th Polish Solid Mechanics Conference, Porabka-Kozubnik, September 1978
210
Eng.Struct., 1981, Vol. 3, October
This test can be introduced into an analysis algorithm and its practical effectiveness has been demonstrated 4 within the framework of first-order theory, with softening bars. In this paper arbitrarily large displacements are accounted for, even if strains are supposed to remain small. Previously established results are reformulated in a way suitable for truss problems and a general numerical procedure is described which is able to analyse elastic-plastic trusses up to the occurrence of collapse or bifurcation. The fundamental steps of the procedure are discussed and demonstrated by means of a simple but significant example. Elastic truss r e l a t i o n s Consider a plane truss. Figure 1 shows the genericjth bar (j = l, ... , n); its axial force is denoted by oj, while ei indicates its change in length. These quantities are proportional, respectively through the initial cross sectional area Aoj and the initial length 10/of the bar, to the usual engineering stress and strain definitions and will be referred to as generalized stress and strain. The crj-ej relation will be assumed to be piecewiselinear, as indicated in Figure 2. The total strain e/is conceived as the sum of an elastic (recoverable) part ej and of a plastic (irreversible) contribution p/: e~ = ei + Pi
(la)
0141-0296/81/040210-09/$02.00 O 1981IPC BusinessPress
Elastic-plastic trusses w i t h unstable bars: C. C i c h o n a n d L. C o r r a d i 0
EA
By making use of equation (2), splitting Q/in the same way as q], expressing ~] as a function of d] through equation (3a) and by imposing the .condition that the resulting relation holds for all q 11, di, the following equilibrium equation is reached:
(~
!---'Z23
,,
I
Figure I
Assumed generalized stress and strain measures
oi c; oj
(5a)
Cj(dj) = [3e/Od x 3e/Ody]/= [cos0 i sin0/]
(55)
=
/ /
/ ./A2d
AIN
with
D i = - Q , ] : Q ~ ] or Q i : B ; D i (6a, b) Since Ci is to be evaluated in the final configuration, the
H2A2Ny H 1A
equilibrium equation (5a) is nonlinear. If equations (3) are combined with equations (1), the following relation is obtained: D; = [ c M i ) ] ' s i ( ~ M i ) - ,~.) (7) It is often convenient to assume that the final configuration is reached through a sequence of steps. The relations governing the truss transition from a known configuration Z N to Z = ~lv + AY. can be obtained from previous equations by imposing the condition that they hold in Y'N and by expressing the final values of quantities as the sum of their known values in ~N and of their variations over A~.. These equations can then be linearized for an incremental (infinitesimal) configuration change ~ = ~St from ~N. In particular, from equation (7) one obtains:
E
5) / Figure 2
Di = C~qSj(6I -- I~j) + C'tONj
L
Piecewiselinear elastic-plastic constitutive
law
(8a)
The last addend in equation (8a) can be written as follows:
(Sb)
(:~ioNj = o~j[~C/~al.dj = o N i O . d i where: The elastic part is linearly related to oi:
o] =S/e/
1
S/= (EAo/lo)]
(Ghk)j = [aCh/adk]i = ~ (~h* --
(lb, c)
The relation between oi and Pi will be discussed later in the paper. oi and ei are defined in a local reference system. They are connected, respectively through equilibrium and geometry, to nodal forces Qi and nodal displacements qi, referred to a global, fixed frame. Figure 3 shows barj in its initial and deformed positions. The final configuration is defined by the four components of vector q], taken in the following order:
GG)i (h,k-l,2)
(8c)
and ~hk is the Kronecker symbol. Moreover, for infinitesimal increments one can write:
(8d)
~i = [aeJ/adilN di = CNiai
From equations(8) one obtains: lji = , Nidli -- C~viSipi
(9a)
with:
qi = {q~ qt}} = {q Ix q ly q2x q2y}~
KNi =
SiC~viC~vi +
(9b)
ONiGNi
q2i can be considered as the vectorial sum of a rigid translation q~i and of a differential displacement di, accounting for both rotation and stretching:
d/=q2j-qlj=B]q]
B~ =
[_10 :] 0 --1
x
V
0
(2a, b)
i~
xo
,j
- q ~ _ ~ Y o
Simple geometrical considerations (see Figure 3) provide the following relation between ei and d/:
el(di) = ti(di) - Zoi l](di) = ((Xo + dx) 2 + (Yo + dy)2)] ,2
~2
(3a)
(3b)
I:
The equilibrium conditions, relating the components of vector Q / t o ei, can be written by enforcing the virtual velocity equation with respect to the final configuration. One obtains:
Q~qi =
°iei
(4)
--I
Figure 3
Initial and deformed bar geometry
Eng.Struct., 1981,Vol. 3, October 211
Elastic-plastic trusses w i t h u n s t a b l e bars:
C.
C i c h o n a n d L. C o r r a d i
KN/is the tangent stiffness matrix of bar]
in configuration t EN- It is composed of an elastic portion SjC]v/CNj and of a geometric contribution ONiGNj.Note that the elastic part, as defined above, depends on current geometry (through CNi), in contrast to some widely used alternative definitions in which the 'elastic' portion of the tangent stiffness is taken as constant. 5 The above relations can be regarded as governing the elastic behaviour of all n (unassembled) bars if indices are dropped and symbols are given the following meaning: q ={qt . . . . . qtn}t
C = diag (C/)
(10a, b)
All vectors are defined by relations of the type of equation (10a), with obvious symbol substitutions. Analogously, all matrices are defined according to equation (10b). Let u and F denote the m-component vectors of nodal displacements and forces for the assembled truss. The assemblage relations read:
q =Lu
F= LtQ
( l l a , b)
where L is a Boolean matrix. Equations (2), (6), (7), (10) and (11) provide the set of relations governing the large displacement, elastic response of the truss subjected to external loads F and to plastic strains p, conceived as given inelastic strain distributions. The corresponding incremental relations read:
d = BLEt
(lZa)
P = LtBt19
(12b) (12c)
= KNd -- CfvSlJ
Plastic law The actual value o f p is to be determined on the basis of the bar constitutive law, illustrated in Figure 2. The plastic strain p/in bar[ can be expressed as a linear combination of Y~ non-negative plastic multipliers Aft, i = 1 . . . . . Y/, each pertaining to one of the Y/segments (yield modes) characterizing the (supposed piecewiselinear) plastic behaviour of the bar:
lJ/= ni~/ X/J> 0
(14b)
4)] = n;6! - h/X/ <- 0
(14c)
$~Xj = 0
114d)
(14a)
In equations (14) lower-case symbols denote subvectors or submatrices obtained by considering in the corresponding upper-case quantities only the entries relevant to the yield modes that are activable in Z N. Equations (14a, b) immediately follow from equations (13a, c)written in incremental form; equation (14c) is a consequence of the requirement, (13d), that ~ / b e non-positive in the incremented configuration ~2v + 6 .~; equation (14d) prevents the simultaneous activation (X/i > 0 ) and unloading (@ < 0) for the same yield mode i. As in the previous section for elastic relations, equations (14) can be assumed to govern the plastic behaviour of all n bars if indices are dropped and sy.mbols are interpreted according to equations (10) (e.g., X = {X] . . . . . Xt~}t, n = diag (hi), etc.). The incremental p r o b l e m
Governing relations Meaningful information on the truss behaviour in a given configuration EN is provided by the set of relations (12), (8) and (14) governing the incremental problem. In what follows, in order to simplify the algebra, subscript N will be dropped; however, all variable quantities (already introduced or to be defined) have to be considered as evaluated in Z N. Note first that using equations (1), (8d) and (14a), equations (12) can be replaced by the following relations: P = Xuu tJ - KupnX
(15a)
d
(15b)
=
Kpu il
-
KppnX
where:
p~ =N/A/
(13a)
Kuu = LtBtgBL
(16a)
A/= { A / I , . . . , A/y/}
(13b)
Kup = K;u = LtBtcts
(168)
A//> 0
(13c)
Kpp = S
(16c)
A yield mode is said to be activable if it may contribute to plastic flow. Let us define a Yj-vector ¢bj of plastic potentials, depending linearly on oj and Aj, whose components cannot assume positive values. If yield mode i is activable, the corresponding component ~P/iof @j vanishes, while it is strictly negative otherwise. The linear expression for ff~jis written as follows: 6 _
t
*j - N)o/ - H/A/ -
R i <~ 0
(13d)
The Yj x Yj-matrix H i of hardening coefficients accounts for the modifications of plastic potentials due to yielding; H i will be assumed to be symmetric, a hypothesis which permits the description of several hardening laws of practical interest.6 Rj is a Yj-vector of positive constants (plastic capacities). For instance, for the plot of Figure 2, Yi = 3,N/= [1 1 - 1 ] , R / = { o 1 02 o03}t a n d H / = diag (Hi, H2, H3)j. Suppose now that in configuration ZN, Yi out of the Y/ yield modes of bar j are activable, i.e. that a Yi'subvect°r $! of Oj is null. Then, the incremental plastic response of
212
the bar is governed by the following relations:
Eng. Struct., 1981, Vol. 3, October
If (15b) is used to eliminate d from (14c), the set of relations (14b-d) assumes the form:
¢ = n % . u -,,,X< 0
(17a)
X~>0
(17b)
$'X -- o
(17c)
with:
Oo = h + ntSn
(18)
The solution of equations (15) and (17) provides the complete incremental truss response to imposed nodal displacements u. X must first be obtained from equations (17); then, P and d are provided by equations (15). The set of relations (17) is known in mathematical programming theory as a Linear Complementarity (LC) problem. If matrix Kuu is non-singular, equation (15a) can be solved for ti; thus, from equations (15) one obtains:
t~ = Duu lk + Duo nX
(19a)
d = D p u F + Dpp n~,
(19b)
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi
where: (20a)
Duu = (Kuu) -1 _
t
Dup - Dpu = DuuKpu
(20b)
Dpp = Kpu Duu Kup - Kpp
(20c)
If d is eliminated from (14c) by means of (19b), the set of equations (14b-d) reduces to the following LC problem: = n%.r
-
(21a)
0
~,~>o
(21b) (21c)
= 0
where:
a =h -
n
tOpp n
(22)
Equations (19)and (21)govern the incremental truss response to given external load increments b~; first X is evaluated by solving equations (21); then, zi and d are obtained from equations (19). As it appears from (19c), matrix Dpp (also indicated with Z in analogous formulations 6) provides the plastic strain rate contribution to the incremental stress response. E q u i v a l e n t elastic s t r u c t u r e Consider a configuration ~N for the truss in whichy yield modes are activable. Suppose that all of them will be actually activated during the incremental process, i.e. that ~ = 0. In this case, if oa is non-singular, equation (17a) can be solved for ~(, to obtain:
= 6~-lntKpuft
(23)
If the above expression is introduced in equation (15), one obtains:
[: = KPuuil
(24a)
6 = r ~ u fi
(24b)
where:
KPhk=Khk--Khpnt~-lnKph
(h,k=u,p)
(24c)
Equation (24c) defines the stiffness properties of an elastic truss equivalent to the actual elastic-plastic one in configuration ~N, in the sense that the elastic stiffness of the activable bars is replaced by the current elastic-plastic stiffness. Obviously, equations (24) furnish the actual incremental response of the system only for those ti, if any, for which all the yield modes activable in Y'N are actually activated. Analog.ous results can be obtained starting from equation (2 la). If q~- 0 and a is non-singular, the equation can be solved for )~, to give:
= a-lntDpuP
(25)
By introducing equation (25) into equations (19) one obtains: ti = D~uuP
(26a)
d =/9~puti
(26b)
with:
D~hk= Dhg + Dhpnta-lnDpk
( h , k = u,p)
(26c)
As before for stiffnesses, the matrices defined by equations (26c) represent the flexibility properties of the equivalent elastic truss in configuration ~N- They are related to the
matrices defined by equations (24) by means of equations formally identical to equations (20). Equations (26) represent the actual truss solution only for those F, if any, for which ~ = 0.
Critical configurations A truss configuration Y'N will be called critical if one of the following events occurs: (i), the second-order work /+tti is non-positive for some zi, or (ii) an incremental response to some/~ does not exist or it is not unique. In order to clarify the meaning of the above definition, note first that condition ~'tli > 0 for all ti ensures stability in Drucker's sense 7 for the truss in configuration EN. Even if stability in the elastic-plastic range is a rather controversial argument, it is generally agreed that Drucker's condition represents a valid stability definition in the present context (conservative forces, associated flow laws). However, as first pointed out by Shanley and generally recognized since,a lack of uniqueness of response might occur prior to actual loss of stability (according to Drucker's definition); possible occurrence of this event, also considered as critical, is accounted for by condition (ii) above. Thus, a configuration Y'N which is non-critical is stable in a rather strong sense, since stable bifurcation phenomena are also ruled out. The following results were proved in references 6 and 3: (1) An incremental response to all imposed displacement increments fi exists and is unique if and only if matrix 60, equation (18), is positive definite. (2) An incremental response to all imposed force increments F exists and is unique if and only if matrix a, equation (22), is positive definite. The above results are based on a theorem in mathematical programming theory 8 which ensures the existence and uniqueness of solution for the LC problems equations (17) or (2 I) if and only if the relevant symmetric matrices oa or a are positive definite. The following statement has also been proved. (3) Configuration ~N is non-critical (in the sense defined previously) if and only if one of the following (fully equivalent) conditions hold: (3.1) both matrices a and Ku~ are positive definite. (3.2) both matrices oa and Kuu are positive definite. Note that the positive definiteness of a alone is no evidence of a non-critical behaviour. In this case, only the existence of a unique incremental response to all F is ensured, which may be the case also for clearly unstable configurations. For trusses, lack of positive definiteness of oa may occur only if the stress-strain diagram of some bars drops vertically after the activation of a new yield mode (critical or sub-critical softening6); ~ ceases to be positive definite when one of these modes becomes activable during the structural evolution. If no bar exhibits a behaviour of this kind, w will remain positive definite along the whole loading history. In this context, the following statement was also proved in reference 3. (4) If 60 is positive definite along the whole loading history, a critical configuration is first reached when a ceases to be positive definite. At this point, also KPuu ceases to be positive definite, while Kuu may remain so. By virtue of this property, a test on the positive def'mitehess of a is sufficient to detect when a critical situation is
Eng. Struct., 1981, Vol. 3, October
213
Elastic-plastic trusses with unstable bars. C. Cichon and L. Corradi reached for the first time. This situation corresponds either to a bifurcation or to a limit point along the loading path. When the positive definiteness of ~o cannot be ensured a priori, the test on a must be supplemented by a test on oa.2 Illustration The above results will be demonstrated with reference to the simple example of Figure 4a. In the equivalent scheme of Figure 4b unbarred symbols refer to dimensionless quantities, obtained by dividing forces by EAo (e.g., F = F/EAo, a = O/EA o) and displacements b y / 2 (e.g., u = a/17, e = g/I-]). The bar's behaviour is shown in Figure 4c, in terms of dimensionless quantities. Yielding in compression occurs for o = -Oo(Oo > 0). The behaviour of the yielded bar is governed by the dimensionless hardening parameter h. As long as no unloading from a plastic state occurs, the value of the axial force o in the bar is: o = (l - lo)/lo
if o ~> - Oo
(27a)
o = ( h ( / - l o ) - o0)/(I +hlo)
if o ~< - Oo
(27b)
where: l(u) = (1 + (u - H o ) 2 ) '/2
(28a) (28b)
lo = (1 +11o) x/2
Making use of the above equations, the elastic matrices defined by equations (16) and (20) (which are scalars in the present case) can be expressed as functions of u only. In particular, one has:
a =h
Dpp
( .'~0~t)
co = h + 1/1o
( 30b )
with Dpp defined by equations (29b) and (27b). It wilt be assumed that h >--1/lo, which ensures ca > 0. The load F will be increased monotonically until a critical situation (in this case, collapse due to snap-through buckling) is reached. The collapse configuration will be denoted by a superposed cap. Depending on the value of Oo and on the initial geometry, the following alternatives are possible. Buckling in the elastic range. This occurs if Kuu, equation (29a), vanishes prior to yielding. This is the case when: O0 ~
(31 )
1 - - l o 2/3
The equation Kuu = 0 identifies the collapse configuration, which is characterized by the following quantities: = lg/3
(32a)
t)a = Ho - ([2a - 1)1/2 k . = (([a
lo)/lo)((•a-
(32b)
Ho)/[a)
(32c)
Buckling in the plastic range. If equation (31) is not verified, yielding will precede buckling. The elastic limit configuration (index E ) is characterized by the following quantities: IE = 10(1 -- Oo)
(33a)
Kuu = (u -Ho)2/(lo 12) + o/13
(29a)
UE=Ho
( 1 2 - 1 ) 1/2
(33b)
Dpp = - o/(l(u - Ho) 2 + olo)
(29b)
FE = o o ( H o - UE)/IE
(33C)
Moreover, the equilibrium equation reads (see equations (5), (6) and (1 lb)) F = (o/1) (u - H o )
(29c)
When the bar yields the 1 x 1-matrices a, equation (22), and ¢o, equation (18), can be defined. Their expressions are:
Moreover, in this configuration one can define: a E = h - Dpp E
(34a)
DppE = ao/(lo(l~(1 - 00) 3 - 1))
(348)
Two alternatives are possible, depending on the sign o f a E . Collapse at first yielding. When aE <~ 0 (h <~DppE). The collapse configuration coincides with the elastic limit configuration, equations (33). Collapse after plastic flow. When aE > 0. In this case collapse is reached when a, equation (30a), becomes equal to zero. The collapse configuration is determined by the condition a = 0, associated with equations (29b) and (27b). After some algebra, one obtains:
a
b
ib 2 = ((Oo + hlo )/h) 1/3
(35a)
ab2 = H o - (i~,2 - 1 ) '/2
(35b)
[;82 = h ( i b 2 - lo) -
] +hlo
S~=~o h=_1 h = O ~
C
• E
-%
h > O ~ / / /// h=oo
Example1. Unbarredsymbolsdenotedimensionless quantities,definedby dividingforcesby E l l o and lengthsby Figure 4
214
Eng. Struct., 1981, Vol. 3, October
Oo G 2
-
ib2
Ho (35c)
The diagrams o f t / a n d ~ b as functions of the hardening parameter h are illustrated in Figure 5 for Ho = 0.5, Oo = 0.05; for these data, yielding precedes collapse. Snap through buckling occurs at first yielding when h <~Dpp E = 0.6236 (t~ = UE = 0.1421 , F = FE = 0.0169). For h >Dpp E both t~ and F increase and approach, for h ~ the elastic collapse values (t~ ~ t~a = 0.2221, F-+ F a = 0.0192) For the same data and h = 2, the complete forcedisplacement curve is plotted in Figure 6, up to u = Ho. The values of Kuu, equation (29a), and a, equation (30a), are also plotted as a function of u. Clearly, a is not defined
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi 0022
026
!
h ~<063261
0 021
Snao at y~ldlr~
Elastic behaviour
h ~oo
i
024'~
fmstJ I
-
I uo=0.2221
'1<.0 0 2 0
0.22 ~ o
8 o o19
~
0 20_~
0018
50
018~
i=o 0169 ~
017
_UE=01421
y
016
~
~
0 016
014 5
I h =- Dpp E = 0 6236 I
0 015 01
I ill
05
I
I
I II
I
I
I II
I
I
I II
10 5 0 10 50 100 Hardening c o e f f i o e n t , h
012
500
Figure 5 Collapse load and collapse displacement for truss of Figure 4, as function of hardening parameter h; H o = 0.5, a o =
Analysis up to collapse of trusses
0.05
® I
0 020i----
I
~ 0 20
\
J
\
0 015
015
o \
o
0 010
010
-0006
i1)
h=20
I
\~
I
005
~ c~
m
\1
×
. I
'\\
ooo5
-
0
_--
I 0125
Ill_ II
-0.06
\
F -o oloi
o
,ioi 1_
Q 250 Displacement u
I 0 375
""
(C) If the analysis is continued after the first critical situation, a may return positive. Property (2) ensures that in this case a unique incremental response to all F exists. In Figure 7 the structural behaviour in the configurations corresponding to u = 0.2 ( a < 0) and u = 0.4 (a > 0 ) is depicted. In the second case the incremental response always exists and is unique; the structure deforms plastically (tJ = KPuuF)i f / ~ < 0 and unloads elastically (tJ = Kuu/z) if F > 0. Note that uniqueness of the incremental response is by no means an indication of the uniqueness of response; as well known, alternative solutions, involving large displacement changes Au exist whatever the sign of F.
The above results have been incorporated into a large displacement, elastic-plastic code so as to produce a truss analysis tool able to detect a critical situation when first reached. In this section, the characteristic features of the procedure are outlined. Further details can be found elsewhere .9 The loading history is conceived as a sequence of finite steps. Step N leads from configuration Z ( N - I ) to configuration EN- At the beginning of each step it is assumed that it is known which o f the activable yield modes will be actually activated and which will unload. The truss behaviour during the step is assumed as holonomic; thus, due to the piecewiselinear nature of the constitutive law, the step analysis concerns a linear elastic structure, with the properties of the equivalent elastic truss defined earlier in the paper. However, the step analysis is nonlinear because of finite geometry changes. The step ends when an assigned load increment A F is reached or when a new mode is predicted to activate for aAF, a < 1. Two main operations are to be performed during each step, namely: (i), the iterative solution of the geometrically nonlinear governing equations; and (ii), the step size adjustment when the activation of a new yield mode is predicted to occur during the step. Operation (i) is performed as follows. At the beginning o f the step the flexibility matrix D~uu for the equivalent elastic structure in configuration E(N-1) is known. A first
-,0.10 0 500
P Figure 6 Load-displacement curve and Kuu, Kuu,a diagrams for example of Figure 4, with h = 2.0; H o = 0.5, a o = 0.05. Zone 1 : elastic; Kuu = KPuu> 0 (stable; unique incremental response). Zone 2: plastic; Kuu > 0 , a > 0, Ku"u > 0 (stable; unique incremental response). Zone 3: plastic; Kuu > 0, a < 0, K~u < 0 (unstable; incremental response does not exist or is not unique). Zone 4: plastic; Kuu < 0, a > 0, KPu < 0 (unstable; unique incremental response)
as long as u < ue. For completeness, the value of the equivalent stiffness K~u is also illustrated. Some of the results quoted in the section on critical configurations are illustrated by the diagrams in Figure 6. In particular, the following points are worth underlining. (A) Since Kuu > 0 as long as the truss is in the elastic range and since co > 0, property (4) ensures that a critical situation is reached when a is no longer positive. At this point K~u also ceases to be positive. (B) By virtue of property (3), all configurations such that t~ ~< u ~< 0.5 are critical. In fact, in this interval either a or Kuu are negative and KPuuis always negative.
I
F= 0.0176
-- --
-
-
]L F< 0
a
u=O.2
F= 0 . 0 0 8 5 - -
b
,iX:
-- - - - ~ - - - - ~ 7
o =-03966<0
-- ~ - - -
.
.
.
.
.
.
I u=04
a= 2 9 9 9 8 > 0
Figure 7 Incremental behaviour on post-collapse portion of curve of Figure 6. (a) u = 0.2 (zone 3): F > 0, no incremental response; < 0, alternative incremental responses (elastic unloading or progressive yielding). (b) u = 0.4 (zone 4): te > 0, unique incremental response (elastic unloading); ,E < 0, unique incremental response (progressive yielding)
Eng. Struct., 1981, Vol. 3, October
215
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi
trial value of the displacement increment vector is obtained on this basis (2xu N ---D~uuSXF); once zSu N is estimated, it is possible to evaluate, by means of the exact nonlinear relations, the corresponding stress increments A ON and the consequent unbalanced forces, on which basis the solution has to be corrected. The correction is performed by means of a standard Newton-Raphson technique, which requires the matrix KPuu to be re-evaluated at each iteration and a different system of linear equations to be solved; in spite of this, it appears preferable to other existing techniques which operate without updating matrices, since geometrical nonlinearities may cause either an increase or a decrease of the truss stiffness during each step; the adopted tangent correctier: method is likely to reduce convergence problems which might arise in this case. If no new yield modes activate, the correct step solution is obtained when the unbalanced force vector is found to be null. Otherwise, the step size adjustment procedure (it) is applied at each iteration within the step. This operation simply amounts to scaling the estimated displacement increment vector Au N to an amplitude such that the associated stress increments 2XON are exactly those needed to make a new mode activable. In other words, the reduced step size is found under the hypothesis that A u N changes amplitude but not the ratio between its components. If this were actually the case, the correct step solution would be found at the end of the step size adjustment and the unbalanced force vector would be null. In general, further iterations are needed; however, even if they modify the relative values of the components of AUN, the step size will change only little, so that a fairly rapid convergence is to be expected. When the step ends with the activation of a new mode, the scalar a is evaluated by means of the relation:
a = h - ntD~pp n
(36)
where the scalar h and the vector n refer to the new activable mode. Note that for trusses, vector n has a single nonvanishing (unit) entry, so that the matrix product ntD~pvn simply singles out a particular diagonal element of matrix At each iteration within the step, when matrix KPuuis re-evaluated, a test on its positive definiteness is performed. If the test fails, a critical configuration is predicted to be reached during the step, prior to the activation of a new mode. In this case the collapse load is determined by means of a procedure identical to the one used for elastic structures, r° but based on the equivalent elastic truss. This computation requires a comparatively lengthy sequence
Y= o / EAo2
~$I
_~f a
Ii
,4
-05
b
Example 2. Dimensionless quantities dividing forces by EAo~and lengths by L~ Figure 8
216
= ~3(2 0)--i~ -
Eng.Struct.,1981,Vol.3,October
are defined
by
of operations. However, in the majority of cases, the first critical configuration reached corresponds to the activation of a new mode; thus, a test on the sign of a, equation (36), is usually sufficient. I f a is found to be negative, the collapse load has been attained and the computation stops. If it is positive, the next step can be started on the basis of a new equivalent structure, whose properties are evaluated from the relations given in the section on equivalent elastic structure. Since the value of a can be easily computed on the basis of the step solution and of the knowledge of the new activable mode, the truss analysis is fairly rapid. The computational efficiency is not jeopardized by the test on the positive definiteness of KPuu, which is performed in any case whenever the matrix is evaluated;the adopted principal pivoting procedure 11 requires only a limited computing time even for comparatively large matrices. At the beginning of each step a search for the activated modes is to be performed. The solution of the LC problem equation (21) (which can be achieved as described elsewhere 12) tells which modes will be activated in an incremental step from Z(N-1). Consistent with the assumed stepwise holonomic behaviour, activated yield modes are supposed to remain active during the whole step. Note, however, that under proportionally increasing loads the occurrence of unloading of plastic bars is rarely experienced. In this case, the preliminary LC problem solution is avoided, with a further consistent increase in computational efficiency. The possible occurrence of unloading is checked (in an approximate way) at the end of the step. If unloading is detected, the step is repeated (if necessary, under reduced load amplitudes) by incorporating the preliminary search for active modes. This operation is performed in any case whenever the load increment vector changes shape. The procedure is illustrated by means of the simple example given in Figure 8a. Dimensionless quantities are used, as defined earlier. The bar behaviour is elasticperfectly plastic and it is depicted in Figure 8b. The parameter/3 defines the ratio between the cross-sectional areas of the upper to the lower bar (3 ---AoffAo2); for 3 = 0 tile structure of Figure 8a reduces to that of Figure 4a, with Ho = 0.5. Even if simple and quite unrealistic, this example is significant because it incorpor:,tes a variety of possible actual truss behaviours for different values of 3. Solid lines in Figure 9 show the force-displacement curves for some typical values of 3; black dots indicate critical situations (always corresponding to snap-through buckling). For low values of 3 (3 < 0.0980) snapping occurs at the first yielding, which indicates that destabilizing geometrical effects bring the structure to collapse even if the upper bar is still elastic. For high values of 3 (3 >0.2676) collapse occurs when the lower bar has yielded in compression and the upper one in tension. This situation corresponds to that causing plastic collapse in first-order theory and, in fact, the limit analysis collapse load FL = 0.0931 is approached as/3 increases and geometric nonlinearities become less influential; for/3 >1.3623 the collapse load is F = 0.0920, with only a marginal decrease with respect to F r ; however, geometric effects give the post-collapse curve a negative slope, which indicates their destabilizing nature. In the internal 0.0980
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi
The dashed curve in Figure 9 represents a sort of limit curve which is always reached for/3 > 1.3623. Collapse occurs when this curve is attained on a point with negative slope (i.e., for u < 0.6550); otherwise, the curve can be followed up to the asymptotic limit value. For completeness, the structural response in the hypothesis of a purely elastic bar behaviour is also indicated (by thin lines) in Figure 9. For 13< 0.1968 snap-through buckling occurs and the collapse load is marked by white dots. Plastic results were obtained with the proposed numerical procedure. A maximum load step size of AF = 0.1 was adopted. Results for typical values of ~ are shown in Table 1. No step ended because the assigned maximum load ~crement was reached, rather because of yielding (in tension or compression) of either bar or because of elastic unloading of previously yielded bars. The event causing the step to end (ESC) is indicated for each step, together with the results of the step solution. For simplicity, the equivalent flexibility matrix (a single number in the present case) is indicated by Duu; superscript P is used to indicate its updated value at the end of the step, which becomes the first iteration Duu value for the subsequent step. If no new modes become activable at the end of the step,D~uu = Duu. Up to the figures indicated, the results coincide with the closed form solution of the problem. For illustrative purposes, a rather unrealistic example was chosen. In fact, not only large displacements but also large strains are required to cause yielding. The example permitted a wide variety of behaviours to be explored by simply changing the value of 43. However, the presence of large strains is inconsistent with the formulation proposed, based on engineering stress and strain definitions, which become questionable as strains become large. The main novelty of the proposed procedure rests on the test on the sign of a to detect whether or not a critical
Table 1
Duu Step 1
ESC F u o~ o2
Duu a
P Duu
Step 2
ESC F u oI o5
Duu a
P Duu
Step 3
1.0
1.1287
1.8783
0.3 3.5088
0.2
0.05
4.0064
5.0865
lyldten 2yldcmp2yldcmp2yldcmp2yldcmp 0.0873 0.0721 0.0334 0.0279 0.0196 0.0977 0.1421 0.1421 0.1421 0.1421 0.1 0.0734 0.0220 0.0147 0.0037 -0.0359 -0.05 --0.05 -0.05 --0.05 1.1262 2.0786 5.3706 6.9396 12.3609 0.1835 0.7056 0.4062 0.2636 - 0 . 2 2 9 7 8.6812 2.6344 11.8064 23.4742 Collapse 2 y l d c m p l y l d t e n 2unld 2unld 0.0920 0.0913 0.0686 0.0457 0.1421 0.1916 0.5000 0.5000 0.1 0.1 0.0824 0.0549 -0.05 --0.05 --0.05 --0.05 11.3122 2.5164 9.0171 17.4520 --0.1333 --0.0302 --Collapse Collapse 9.0171 17.4520
ESC F u oI o5
Duu a
D:u Step 4
2.0
ESC F u oI o2
Duu a
P Duu
1 yldten 0.0803 0.5986 0.1 -0.0457 7.6394 --0.0552 Collapse
l y l d ten 0.091 5 0.8708 0.1 0.0095 4.2301 0.0756 7.9114 2 yld ten 0.1419 1.1 029 0.1 0.1 3.2248 0.2067 13.9470*
* For ~ = 0.2, structure can carry additional load with both bars y i e l d e d in t e n s i o n u p t o a s y m p t o t i c v a l u e o f F = 0 . 2 .
0.2C
F'=O2
01E
t~ 13 o O 0.1C FL =O O931
situation is reached when the step ends with the activation of a new mode. As already mentioned, this event is the most frequently experienced and the simplicity of the test permits its extremely easy implementation in a truss analysis code. On the basis of the computational experience gained so far, storage requirements rather than computer time proved to be the critical factor. Realistic plane truss problems can usually be analysed by exploiting core facilities only. The extension to space trusses, even if conceptually straightforward, requires some care in order to preserve the computational efficiency in the presence of a substantially larger number of variables.
W
Conclusions ODE
o
O
0,5 lO 15 Displocement, u Figure 9 Load-displacement curves for example of Figure 8 and various cross-sectional ratios/3. F ' = asymptotic limit toad value; F L = first order theory collapse load
A procedure for large displacement analysis of elasticplastic trusses is proposed. The following assumptions are made: (i), strains are small even if displacements can be arbitrarily large; (ii) local instability effects are accounted for in the force-elongation (stress-strain) diagram of the bar, supposed to be piecewiselinear; (iii) the plastic behaviour of bars can exhibit both hardening and/or softening and is governed by a matrix H which is supposed to be symmetric. Within the framework of the above assumptions, the analysis problem is formulated and previously established
Eng. Struct., 1981, Vol. 3, October
217
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi
results, which characterize a critical situation, are specialized. The formulation is implemented into a numerical procedure able to follow up to collapse the truss evolution. The term 'collapse' is intended in a broad sense since it refers not only to the exhaustion of structural load carrying capacity but also to the possible occurrence of bifurcation type instability in Stanley's sense. Because of the assumed piecewiselinear nature of the bar constitutive law, the structural evolution can be regarded as a sequence of steps in which the truss behaves elastically, provided that the current elastic-plastic stiffness is used to replace the original elastic bar stiffness. Relations permitting the generation of the properties of this equivalent elastic truss are provided. In spite of this, the step analysis is nonlinear because of geometrical effects. Attention is focused mainly on the case in which collapse corresponds to the activation of a new mode, i.e. it is caused by an abrupt change in the current stiffness of the bar, which makes the truss unable to carry additional load. It should be noted that this occurs in the majority of practical truss analysis problems. In this case it is shown that, in order to detect whether or not a collapse situation has been reached, a test on the sign of a single number is sufficient; this number can easily be obtained on the basis of the information provided by the step solution, with little additional computational effort. This test represents the main novelty of this paper and its introduction is the characteristic feature of the numerical procedure proposed. For the numerical solution of the geometrically nonlinear step equations and the treatment of other events which might occur during the truss evolution (collapse not corresponding to a new mode activation, elastic unloading of previously yielded bars) already established techniques are used and reference is made to the existing literature. The aim of this paper being simply to present the procedure and to indicate its potential, only very simple illustrative examples are presented. Ackn owledgemen t This research was sponsored by C.N.R. (Italian Research Council).
References 1 2 3 4 5 6 7
218
Sewell, M. 'Plastic buckling'. In 'Stability' (Leipholz, H. H. E., ed.), SM Study No. 6, University of Waterloo Press, 1972 Corradi, L. 'On a stability condition for elastic plastic structures',Meccanica 1977, 12, 24 Corradi, L. 'Stability of discrete elastic plastic structures with associated flow laws', SMArchives 1978, 3,201 Corradi, L., De Donato, O. 'Collapse analysis of elastic plastic trusses with unstable bars',Mech. Res. Comm. 1977, 4 , 4 1 7 Mallet, R. H., Marcal, P. V. 'Finite element analysis of nonlinear structures', Proc. ASCE, J. Struct. Div. 1968, 94, 2081 Maier, G. 'Incremental plastic analysis in the presence of large displacements and physical instabilizing effects', Int. J. Solids Struct. 1971, 7,345 Drucker, D. C. 'On the postulate of stability of materials in the mechanics of continua', J. Mdcanique 1964, 3,235
Eng. Struct., 1981, Vol. 3, October
8 9 10 ll
12
Samelson, tt. eta!. 'A partition theorem tot I'uclidean :Vspaces'. Prec. A mer. Math. Sac. 1958, 9,805 Cichon, C. and Corradi, L., 'Large displacenlcnt and stability analysis of elastic-plastic trusses', ISTC Tech. Rep. 4-81. Politecnico Milan(), 1981 Riks, E. 'An incremental approach to the solution of snapping and buckling problems', Int. J. Solids Street. 1979, 15,254 Cattle, R. W. 'The principal pivoting method of quadratic programming'. In 'Mathematics of decision sciences' (l)antzig, G. B. and Veinott, A. t:., eds), part 1, American Mathematical Society, 1968 Franchi, A. and Cohn, M. Z., 'Computer analysis of elasticplastic structures', lnt. J. Camp. Meth. Appl. Mech. 1980, 21 (3), 271
Notation a
matrix governing stability under load control, equation (22) bar cross-section A rigid translation matrix, equations (2) B geometric matrix, equation (5b) C differential displacement vector for bar and d,D corresponding force vector Duu , Dup , Dpp structural flexibility matrices, equations (20) e elastic portion of bar elongation Young's modulus E nodal force vector for assembled truss F hardening matrix and corresponding subO,h matrix for activable modes K.., I%, Kpp structural stiffness matrices, equations (16) bar length l connectivity matrix L outward normal matrix and corresponding N,n submatrix for activable modes plastic portion of bar elongation P nodal displacement and force vectors for q,Q individual bars vector of plastic capacities R elastic stiffness of bar, equation (lc) S nodal displacement vector for assembled truss U bar elongation (generalized strain) C bar slope 0 bar stiffness matrix, equation (9) K plastic multiplier vector and corresponding A,X subvector for activable modes axial force in bar (generalized stress) 0 Z configuration plastic potential vector and corresponding subvector for activable modes matrix governing stability under displace~0 ment control, equation (l 8) subscript referring to bar j J subscript denoting initial values 0 subscript denoting values in configuration Z N N superscript marking updated matrices for P equivalent elastic truss
Leone C o r r a d i Department of Structural Engineering, Politecnico, Milano, Italy (Received April 1980; revised November 1980}
Large displacement analysis up to the collapse of trusses is considered. Bars are elastic-plastic and possibly unstable post-critical behaviour is accounted for. The collapse situation may correspond either to a limit point (snap-through buckling) or to the occurrence of bifurcation in Shanley's sense. Attention is mainly focused on the global stability test, which permits the collapse situation to be identified. It is shown that the test can be reduced to a check on the sign of a single number, which can be easily obtained for any given structural configuration. It is also shown how this test can be introduced into a large displacement elastic-plastic analysis code with little additional computational effort.
Introduction In this paper truss type structures will be considered. Bars exhibit an elastic-plastic behaviour and the possibility of local instability is accounted for in the force-elongation diagram of the bar which may present a negative slope (softening) after a given force level is reached (as is typical of unstable post-critical behaviour on the compression side of the diagram). The analysis up to the point of collapse of such trusses is considered. Usually, collapse corresponds to a limit point on the loading path and it is often due to global instability of the partially yielded structure. However, as Shanley 1 pointed out stable bifurcation may occur prior to the structural load carrying capacity being exhausted; this phenomenon is also considered as critical for the structure. A truss configuration will be defined as critical when it corresponds to a limit point along the loading path or if stable bifurcation may occur for some load increments. Conditions to detect a critical configuration have been proved for the case of associated flow laws and conservative external forces. 2, 3 Under widely general hypotheses, they can be reduced to a test on the sign of a single number, which can be easily defined for any known configuration. Part of this paper was presented at the 20th Polish Solid Mechanics Conference, Porabka-Kozubnik, September 1978
210
Eng.Struct., 1981, Vol. 3, October
This test can be introduced into an analysis algorithm and its practical effectiveness has been demonstrated 4 within the framework of first-order theory, with softening bars. In this paper arbitrarily large displacements are accounted for, even if strains are supposed to remain small. Previously established results are reformulated in a way suitable for truss problems and a general numerical procedure is described which is able to analyse elastic-plastic trusses up to the occurrence of collapse or bifurcation. The fundamental steps of the procedure are discussed and demonstrated by means of a simple but significant example. Elastic truss r e l a t i o n s Consider a plane truss. Figure 1 shows the genericjth bar (j = l, ... , n); its axial force is denoted by oj, while ei indicates its change in length. These quantities are proportional, respectively through the initial cross sectional area Aoj and the initial length 10/of the bar, to the usual engineering stress and strain definitions and will be referred to as generalized stress and strain. The crj-ej relation will be assumed to be piecewiselinear, as indicated in Figure 2. The total strain e/is conceived as the sum of an elastic (recoverable) part ej and of a plastic (irreversible) contribution p/: e~ = ei + Pi
(la)
0141-0296/81/040210-09/$02.00 O 1981IPC BusinessPress
Elastic-plastic trusses w i t h unstable bars: C. C i c h o n a n d L. C o r r a d i 0
EA
By making use of equation (2), splitting Q/in the same way as q], expressing ~] as a function of d] through equation (3a) and by imposing the .condition that the resulting relation holds for all q 11, di, the following equilibrium equation is reached:
(~
!---'Z23
,,
I
Figure I
Assumed generalized stress and strain measures
oi c; oj
(5a)
Cj(dj) = [3e/Od x 3e/Ody]/= [cos0 i sin0/]
(55)
=
/ /
/ ./A2d
AIN
with
D i = - Q , ] : Q ~ ] or Q i : B ; D i (6a, b) Since Ci is to be evaluated in the final configuration, the
H2A2Ny H 1A
equilibrium equation (5a) is nonlinear. If equations (3) are combined with equations (1), the following relation is obtained: D; = [ c M i ) ] ' s i ( ~ M i ) - ,~.) (7) It is often convenient to assume that the final configuration is reached through a sequence of steps. The relations governing the truss transition from a known configuration Z N to Z = ~lv + AY. can be obtained from previous equations by imposing the condition that they hold in Y'N and by expressing the final values of quantities as the sum of their known values in ~N and of their variations over A~.. These equations can then be linearized for an incremental (infinitesimal) configuration change ~ = ~St from ~N. In particular, from equation (7) one obtains:
E
5) / Figure 2
Di = C~qSj(6I -- I~j) + C'tONj
L
Piecewiselinear elastic-plastic constitutive
law
(8a)
The last addend in equation (8a) can be written as follows:
(Sb)
(:~ioNj = o~j[~C/~al.dj = o N i O . d i where: The elastic part is linearly related to oi:
o] =S/e/
1
S/= (EAo/lo)]
(Ghk)j = [aCh/adk]i = ~ (~h* --
(lb, c)
The relation between oi and Pi will be discussed later in the paper. oi and ei are defined in a local reference system. They are connected, respectively through equilibrium and geometry, to nodal forces Qi and nodal displacements qi, referred to a global, fixed frame. Figure 3 shows barj in its initial and deformed positions. The final configuration is defined by the four components of vector q], taken in the following order:
GG)i (h,k-l,2)
(8c)
and ~hk is the Kronecker symbol. Moreover, for infinitesimal increments one can write:
(8d)
~i = [aeJ/adilN di = CNiai
From equations(8) one obtains: lji = , Nidli -- C~viSipi
(9a)
with:
qi = {q~ qt}} = {q Ix q ly q2x q2y}~
KNi =
SiC~viC~vi +
(9b)
ONiGNi
q2i can be considered as the vectorial sum of a rigid translation q~i and of a differential displacement di, accounting for both rotation and stretching:
d/=q2j-qlj=B]q]
B~ =
[_10 :] 0 --1
x
V
0
(2a, b)
i~
xo
,j
- q ~ _ ~ Y o
Simple geometrical considerations (see Figure 3) provide the following relation between ei and d/:
el(di) = ti(di) - Zoi l](di) = ((Xo + dx) 2 + (Yo + dy)2)] ,2
~2
(3a)
(3b)
I:
The equilibrium conditions, relating the components of vector Q / t o ei, can be written by enforcing the virtual velocity equation with respect to the final configuration. One obtains:
Q~qi =
°iei
(4)
--I
Figure 3
Initial and deformed bar geometry
Eng.Struct., 1981,Vol. 3, October 211
Elastic-plastic trusses w i t h u n s t a b l e bars:
C.
C i c h o n a n d L. C o r r a d i
KN/is the tangent stiffness matrix of bar]
in configuration t EN- It is composed of an elastic portion SjC]v/CNj and of a geometric contribution ONiGNj.Note that the elastic part, as defined above, depends on current geometry (through CNi), in contrast to some widely used alternative definitions in which the 'elastic' portion of the tangent stiffness is taken as constant. 5 The above relations can be regarded as governing the elastic behaviour of all n (unassembled) bars if indices are dropped and symbols are given the following meaning: q ={qt . . . . . qtn}t
C = diag (C/)
(10a, b)
All vectors are defined by relations of the type of equation (10a), with obvious symbol substitutions. Analogously, all matrices are defined according to equation (10b). Let u and F denote the m-component vectors of nodal displacements and forces for the assembled truss. The assemblage relations read:
q =Lu
F= LtQ
( l l a , b)
where L is a Boolean matrix. Equations (2), (6), (7), (10) and (11) provide the set of relations governing the large displacement, elastic response of the truss subjected to external loads F and to plastic strains p, conceived as given inelastic strain distributions. The corresponding incremental relations read:
d = BLEt
(lZa)
P = LtBt19
(12b) (12c)
= KNd -- CfvSlJ
Plastic law The actual value o f p is to be determined on the basis of the bar constitutive law, illustrated in Figure 2. The plastic strain p/in bar[ can be expressed as a linear combination of Y~ non-negative plastic multipliers Aft, i = 1 . . . . . Y/, each pertaining to one of the Y/segments (yield modes) characterizing the (supposed piecewiselinear) plastic behaviour of the bar:
lJ/= ni~/ X/J> 0
(14b)
4)] = n;6! - h/X/ <- 0
(14c)
$~Xj = 0
114d)
(14a)
In equations (14) lower-case symbols denote subvectors or submatrices obtained by considering in the corresponding upper-case quantities only the entries relevant to the yield modes that are activable in Z N. Equations (14a, b) immediately follow from equations (13a, c)written in incremental form; equation (14c) is a consequence of the requirement, (13d), that ~ / b e non-positive in the incremented configuration ~2v + 6 .~; equation (14d) prevents the simultaneous activation (X/i > 0 ) and unloading (@ < 0) for the same yield mode i. As in the previous section for elastic relations, equations (14) can be assumed to govern the plastic behaviour of all n bars if indices are dropped and sy.mbols are interpreted according to equations (10) (e.g., X = {X] . . . . . Xt~}t, n = diag (hi), etc.). The incremental p r o b l e m
Governing relations Meaningful information on the truss behaviour in a given configuration EN is provided by the set of relations (12), (8) and (14) governing the incremental problem. In what follows, in order to simplify the algebra, subscript N will be dropped; however, all variable quantities (already introduced or to be defined) have to be considered as evaluated in Z N. Note first that using equations (1), (8d) and (14a), equations (12) can be replaced by the following relations: P = Xuu tJ - KupnX
(15a)
d
(15b)
=
Kpu il
-
KppnX
where:
p~ =N/A/
(13a)
Kuu = LtBtgBL
(16a)
A/= { A / I , . . . , A/y/}
(13b)
Kup = K;u = LtBtcts
(168)
A//> 0
(13c)
Kpp = S
(16c)
A yield mode is said to be activable if it may contribute to plastic flow. Let us define a Yj-vector ¢bj of plastic potentials, depending linearly on oj and Aj, whose components cannot assume positive values. If yield mode i is activable, the corresponding component ~P/iof @j vanishes, while it is strictly negative otherwise. The linear expression for ff~jis written as follows: 6 _
t
*j - N)o/ - H/A/ -
R i <~ 0
(13d)
The Yj x Yj-matrix H i of hardening coefficients accounts for the modifications of plastic potentials due to yielding; H i will be assumed to be symmetric, a hypothesis which permits the description of several hardening laws of practical interest.6 Rj is a Yj-vector of positive constants (plastic capacities). For instance, for the plot of Figure 2, Yi = 3,N/= [1 1 - 1 ] , R / = { o 1 02 o03}t a n d H / = diag (Hi, H2, H3)j. Suppose now that in configuration ZN, Yi out of the Y/ yield modes of bar j are activable, i.e. that a Yi'subvect°r $! of Oj is null. Then, the incremental plastic response of
212
the bar is governed by the following relations:
Eng. Struct., 1981, Vol. 3, October
If (15b) is used to eliminate d from (14c), the set of relations (14b-d) assumes the form:
¢ = n % . u -,,,X< 0
(17a)
X~>0
(17b)
$'X -- o
(17c)
with:
Oo = h + ntSn
(18)
The solution of equations (15) and (17) provides the complete incremental truss response to imposed nodal displacements u. X must first be obtained from equations (17); then, P and d are provided by equations (15). The set of relations (17) is known in mathematical programming theory as a Linear Complementarity (LC) problem. If matrix Kuu is non-singular, equation (15a) can be solved for ti; thus, from equations (15) one obtains:
t~ = Duu lk + Duo nX
(19a)
d = D p u F + Dpp n~,
(19b)
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi
where: (20a)
Duu = (Kuu) -1 _
t
Dup - Dpu = DuuKpu
(20b)
Dpp = Kpu Duu Kup - Kpp
(20c)
If d is eliminated from (14c) by means of (19b), the set of equations (14b-d) reduces to the following LC problem: = n%.r
-
(21a)
0
~,~>o
(21b) (21c)
= 0
where:
a =h -
n
tOpp n
(22)
Equations (19)and (21)govern the incremental truss response to given external load increments b~; first X is evaluated by solving equations (21); then, zi and d are obtained from equations (19). As it appears from (19c), matrix Dpp (also indicated with Z in analogous formulations 6) provides the plastic strain rate contribution to the incremental stress response. E q u i v a l e n t elastic s t r u c t u r e Consider a configuration ~N for the truss in whichy yield modes are activable. Suppose that all of them will be actually activated during the incremental process, i.e. that ~ = 0. In this case, if oa is non-singular, equation (17a) can be solved for ~(, to obtain:
= 6~-lntKpuft
(23)
If the above expression is introduced in equation (15), one obtains:
[: = KPuuil
(24a)
6 = r ~ u fi
(24b)
where:
KPhk=Khk--Khpnt~-lnKph
(h,k=u,p)
(24c)
Equation (24c) defines the stiffness properties of an elastic truss equivalent to the actual elastic-plastic one in configuration ~N, in the sense that the elastic stiffness of the activable bars is replaced by the current elastic-plastic stiffness. Obviously, equations (24) furnish the actual incremental response of the system only for those ti, if any, for which all the yield modes activable in Y'N are actually activated. Analog.ous results can be obtained starting from equation (2 la). If q~- 0 and a is non-singular, the equation can be solved for )~, to give:
= a-lntDpuP
(25)
By introducing equation (25) into equations (19) one obtains: ti = D~uuP
(26a)
d =/9~puti
(26b)
with:
D~hk= Dhg + Dhpnta-lnDpk
( h , k = u,p)
(26c)
As before for stiffnesses, the matrices defined by equations (26c) represent the flexibility properties of the equivalent elastic truss in configuration ~N- They are related to the
matrices defined by equations (24) by means of equations formally identical to equations (20). Equations (26) represent the actual truss solution only for those F, if any, for which ~ = 0.
Critical configurations A truss configuration Y'N will be called critical if one of the following events occurs: (i), the second-order work /+tti is non-positive for some zi, or (ii) an incremental response to some/~ does not exist or it is not unique. In order to clarify the meaning of the above definition, note first that condition ~'tli > 0 for all ti ensures stability in Drucker's sense 7 for the truss in configuration EN. Even if stability in the elastic-plastic range is a rather controversial argument, it is generally agreed that Drucker's condition represents a valid stability definition in the present context (conservative forces, associated flow laws). However, as first pointed out by Shanley and generally recognized since,a lack of uniqueness of response might occur prior to actual loss of stability (according to Drucker's definition); possible occurrence of this event, also considered as critical, is accounted for by condition (ii) above. Thus, a configuration Y'N which is non-critical is stable in a rather strong sense, since stable bifurcation phenomena are also ruled out. The following results were proved in references 6 and 3: (1) An incremental response to all imposed displacement increments fi exists and is unique if and only if matrix 60, equation (18), is positive definite. (2) An incremental response to all imposed force increments F exists and is unique if and only if matrix a, equation (22), is positive definite. The above results are based on a theorem in mathematical programming theory 8 which ensures the existence and uniqueness of solution for the LC problems equations (17) or (2 I) if and only if the relevant symmetric matrices oa or a are positive definite. The following statement has also been proved. (3) Configuration ~N is non-critical (in the sense defined previously) if and only if one of the following (fully equivalent) conditions hold: (3.1) both matrices a and Ku~ are positive definite. (3.2) both matrices oa and Kuu are positive definite. Note that the positive definiteness of a alone is no evidence of a non-critical behaviour. In this case, only the existence of a unique incremental response to all F is ensured, which may be the case also for clearly unstable configurations. For trusses, lack of positive definiteness of oa may occur only if the stress-strain diagram of some bars drops vertically after the activation of a new yield mode (critical or sub-critical softening6); ~ ceases to be positive definite when one of these modes becomes activable during the structural evolution. If no bar exhibits a behaviour of this kind, w will remain positive definite along the whole loading history. In this context, the following statement was also proved in reference 3. (4) If 60 is positive definite along the whole loading history, a critical configuration is first reached when a ceases to be positive definite. At this point, also KPuu ceases to be positive definite, while Kuu may remain so. By virtue of this property, a test on the positive def'mitehess of a is sufficient to detect when a critical situation is
Eng. Struct., 1981, Vol. 3, October
213
Elastic-plastic trusses with unstable bars. C. Cichon and L. Corradi reached for the first time. This situation corresponds either to a bifurcation or to a limit point along the loading path. When the positive definiteness of ~o cannot be ensured a priori, the test on a must be supplemented by a test on oa.2 Illustration The above results will be demonstrated with reference to the simple example of Figure 4a. In the equivalent scheme of Figure 4b unbarred symbols refer to dimensionless quantities, obtained by dividing forces by EAo (e.g., F = F/EAo, a = O/EA o) and displacements b y / 2 (e.g., u = a/17, e = g/I-]). The bar's behaviour is shown in Figure 4c, in terms of dimensionless quantities. Yielding in compression occurs for o = -Oo(Oo > 0). The behaviour of the yielded bar is governed by the dimensionless hardening parameter h. As long as no unloading from a plastic state occurs, the value of the axial force o in the bar is: o = (l - lo)/lo
if o ~> - Oo
(27a)
o = ( h ( / - l o ) - o0)/(I +hlo)
if o ~< - Oo
(27b)
where: l(u) = (1 + (u - H o ) 2 ) '/2
(28a) (28b)
lo = (1 +11o) x/2
Making use of the above equations, the elastic matrices defined by equations (16) and (20) (which are scalars in the present case) can be expressed as functions of u only. In particular, one has:
a =h
Dpp
( .'~0~t)
co = h + 1/1o
( 30b )
with Dpp defined by equations (29b) and (27b). It wilt be assumed that h >--1/lo, which ensures ca > 0. The load F will be increased monotonically until a critical situation (in this case, collapse due to snap-through buckling) is reached. The collapse configuration will be denoted by a superposed cap. Depending on the value of Oo and on the initial geometry, the following alternatives are possible. Buckling in the elastic range. This occurs if Kuu, equation (29a), vanishes prior to yielding. This is the case when: O0 ~
(31 )
1 - - l o 2/3
The equation Kuu = 0 identifies the collapse configuration, which is characterized by the following quantities: = lg/3
(32a)
t)a = Ho - ([2a - 1)1/2 k . = (([a
lo)/lo)((•a-
(32b)
Ho)/[a)
(32c)
Buckling in the plastic range. If equation (31) is not verified, yielding will precede buckling. The elastic limit configuration (index E ) is characterized by the following quantities: IE = 10(1 -- Oo)
(33a)
Kuu = (u -Ho)2/(lo 12) + o/13
(29a)
UE=Ho
( 1 2 - 1 ) 1/2
(33b)
Dpp = - o/(l(u - Ho) 2 + olo)
(29b)
FE = o o ( H o - UE)/IE
(33C)
Moreover, the equilibrium equation reads (see equations (5), (6) and (1 lb)) F = (o/1) (u - H o )
(29c)
When the bar yields the 1 x 1-matrices a, equation (22), and ¢o, equation (18), can be defined. Their expressions are:
Moreover, in this configuration one can define: a E = h - Dpp E
(34a)
DppE = ao/(lo(l~(1 - 00) 3 - 1))
(348)
Two alternatives are possible, depending on the sign o f a E . Collapse at first yielding. When aE <~ 0 (h <~DppE). The collapse configuration coincides with the elastic limit configuration, equations (33). Collapse after plastic flow. When aE > 0. In this case collapse is reached when a, equation (30a), becomes equal to zero. The collapse configuration is determined by the condition a = 0, associated with equations (29b) and (27b). After some algebra, one obtains:
a
b
ib 2 = ((Oo + hlo )/h) 1/3
(35a)
ab2 = H o - (i~,2 - 1 ) '/2
(35b)
[;82 = h ( i b 2 - lo) -
] +hlo
S~=~o h=_1 h = O ~
C
• E
-%
h > O ~ / / /// h=oo
Example1. Unbarredsymbolsdenotedimensionless quantities,definedby dividingforcesby E l l o and lengthsby Figure 4
214
Eng. Struct., 1981, Vol. 3, October
Oo G 2
-
ib2
Ho (35c)
The diagrams o f t / a n d ~ b as functions of the hardening parameter h are illustrated in Figure 5 for Ho = 0.5, Oo = 0.05; for these data, yielding precedes collapse. Snap through buckling occurs at first yielding when h <~Dpp E = 0.6236 (t~ = UE = 0.1421 , F = FE = 0.0169). For h >Dpp E both t~ and F increase and approach, for h ~ the elastic collapse values (t~ ~ t~a = 0.2221, F-+ F a = 0.0192) For the same data and h = 2, the complete forcedisplacement curve is plotted in Figure 6, up to u = Ho. The values of Kuu, equation (29a), and a, equation (30a), are also plotted as a function of u. Clearly, a is not defined
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi 0022
026
!
h ~<063261
0 021
Snao at y~ldlr~
Elastic behaviour
h ~oo
i
024'~
fmstJ I
-
I uo=0.2221
'1<.0 0 2 0
0.22 ~ o
8 o o19
~
0 20_~
0018
50
018~
i=o 0169 ~
017
_UE=01421
y
016
~
~
0 016
014 5
I h =- Dpp E = 0 6236 I
0 015 01
I ill
05
I
I
I II
I
I
I II
I
I
I II
10 5 0 10 50 100 Hardening c o e f f i o e n t , h
012
500
Figure 5 Collapse load and collapse displacement for truss of Figure 4, as function of hardening parameter h; H o = 0.5, a o =
Analysis up to collapse of trusses
0.05
® I
0 020i----
I
~ 0 20
\
J
\
0 015
015
o \
o
0 010
010
-0006
i1)
h=20
I
\~
I
005
~ c~
m
\1
×
. I
'\\
ooo5
-
0
_--
I 0125
Ill_ II
-0.06
\
F -o oloi
o
,ioi 1_
Q 250 Displacement u
I 0 375
""
(C) If the analysis is continued after the first critical situation, a may return positive. Property (2) ensures that in this case a unique incremental response to all F exists. In Figure 7 the structural behaviour in the configurations corresponding to u = 0.2 ( a < 0) and u = 0.4 (a > 0 ) is depicted. In the second case the incremental response always exists and is unique; the structure deforms plastically (tJ = KPuuF)i f / ~ < 0 and unloads elastically (tJ = Kuu/z) if F > 0. Note that uniqueness of the incremental response is by no means an indication of the uniqueness of response; as well known, alternative solutions, involving large displacement changes Au exist whatever the sign of F.
The above results have been incorporated into a large displacement, elastic-plastic code so as to produce a truss analysis tool able to detect a critical situation when first reached. In this section, the characteristic features of the procedure are outlined. Further details can be found elsewhere .9 The loading history is conceived as a sequence of finite steps. Step N leads from configuration Z ( N - I ) to configuration EN- At the beginning of each step it is assumed that it is known which o f the activable yield modes will be actually activated and which will unload. The truss behaviour during the step is assumed as holonomic; thus, due to the piecewiselinear nature of the constitutive law, the step analysis concerns a linear elastic structure, with the properties of the equivalent elastic truss defined earlier in the paper. However, the step analysis is nonlinear because of finite geometry changes. The step ends when an assigned load increment A F is reached or when a new mode is predicted to activate for aAF, a < 1. Two main operations are to be performed during each step, namely: (i), the iterative solution of the geometrically nonlinear governing equations; and (ii), the step size adjustment when the activation of a new yield mode is predicted to occur during the step. Operation (i) is performed as follows. At the beginning o f the step the flexibility matrix D~uu for the equivalent elastic structure in configuration E(N-1) is known. A first
-,0.10 0 500
P Figure 6 Load-displacement curve and Kuu, Kuu,a diagrams for example of Figure 4, with h = 2.0; H o = 0.5, a o = 0.05. Zone 1 : elastic; Kuu = KPuu> 0 (stable; unique incremental response). Zone 2: plastic; Kuu > 0 , a > 0, Ku"u > 0 (stable; unique incremental response). Zone 3: plastic; Kuu > 0, a < 0, K~u < 0 (unstable; incremental response does not exist or is not unique). Zone 4: plastic; Kuu < 0, a > 0, KPu < 0 (unstable; unique incremental response)
as long as u < ue. For completeness, the value of the equivalent stiffness K~u is also illustrated. Some of the results quoted in the section on critical configurations are illustrated by the diagrams in Figure 6. In particular, the following points are worth underlining. (A) Since Kuu > 0 as long as the truss is in the elastic range and since co > 0, property (4) ensures that a critical situation is reached when a is no longer positive. At this point K~u also ceases to be positive. (B) By virtue of property (3), all configurations such that t~ ~< u ~< 0.5 are critical. In fact, in this interval either a or Kuu are negative and KPuuis always negative.
I
F= 0.0176
-- --
-
-
]L F< 0
a
u=O.2
F= 0 . 0 0 8 5 - -
b
,iX:
-- - - - ~ - - - - ~ 7
o =-03966<0
-- ~ - - -
.
.
.
.
.
.
I u=04
a= 2 9 9 9 8 > 0
Figure 7 Incremental behaviour on post-collapse portion of curve of Figure 6. (a) u = 0.2 (zone 3): F > 0, no incremental response; < 0, alternative incremental responses (elastic unloading or progressive yielding). (b) u = 0.4 (zone 4): te > 0, unique incremental response (elastic unloading); ,E < 0, unique incremental response (progressive yielding)
Eng. Struct., 1981, Vol. 3, October
215
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi
trial value of the displacement increment vector is obtained on this basis (2xu N ---D~uuSXF); once zSu N is estimated, it is possible to evaluate, by means of the exact nonlinear relations, the corresponding stress increments A ON and the consequent unbalanced forces, on which basis the solution has to be corrected. The correction is performed by means of a standard Newton-Raphson technique, which requires the matrix KPuu to be re-evaluated at each iteration and a different system of linear equations to be solved; in spite of this, it appears preferable to other existing techniques which operate without updating matrices, since geometrical nonlinearities may cause either an increase or a decrease of the truss stiffness during each step; the adopted tangent correctier: method is likely to reduce convergence problems which might arise in this case. If no new yield modes activate, the correct step solution is obtained when the unbalanced force vector is found to be null. Otherwise, the step size adjustment procedure (it) is applied at each iteration within the step. This operation simply amounts to scaling the estimated displacement increment vector Au N to an amplitude such that the associated stress increments 2XON are exactly those needed to make a new mode activable. In other words, the reduced step size is found under the hypothesis that A u N changes amplitude but not the ratio between its components. If this were actually the case, the correct step solution would be found at the end of the step size adjustment and the unbalanced force vector would be null. In general, further iterations are needed; however, even if they modify the relative values of the components of AUN, the step size will change only little, so that a fairly rapid convergence is to be expected. When the step ends with the activation of a new mode, the scalar a is evaluated by means of the relation:
a = h - ntD~pp n
(36)
where the scalar h and the vector n refer to the new activable mode. Note that for trusses, vector n has a single nonvanishing (unit) entry, so that the matrix product ntD~pvn simply singles out a particular diagonal element of matrix At each iteration within the step, when matrix KPuuis re-evaluated, a test on its positive definiteness is performed. If the test fails, a critical configuration is predicted to be reached during the step, prior to the activation of a new mode. In this case the collapse load is determined by means of a procedure identical to the one used for elastic structures, r° but based on the equivalent elastic truss. This computation requires a comparatively lengthy sequence
Y= o / EAo2
~$I
_~f a
Ii
,4
-05
b
Example 2. Dimensionless quantities dividing forces by EAo~and lengths by L~ Figure 8
216
= ~3(2 0)--i~ -
Eng.Struct.,1981,Vol.3,October
are defined
by
of operations. However, in the majority of cases, the first critical configuration reached corresponds to the activation of a new mode; thus, a test on the sign of a, equation (36), is usually sufficient. I f a is found to be negative, the collapse load has been attained and the computation stops. If it is positive, the next step can be started on the basis of a new equivalent structure, whose properties are evaluated from the relations given in the section on equivalent elastic structure. Since the value of a can be easily computed on the basis of the step solution and of the knowledge of the new activable mode, the truss analysis is fairly rapid. The computational efficiency is not jeopardized by the test on the positive definiteness of KPuu, which is performed in any case whenever the matrix is evaluated;the adopted principal pivoting procedure 11 requires only a limited computing time even for comparatively large matrices. At the beginning of each step a search for the activated modes is to be performed. The solution of the LC problem equation (21) (which can be achieved as described elsewhere 12) tells which modes will be activated in an incremental step from Z(N-1). Consistent with the assumed stepwise holonomic behaviour, activated yield modes are supposed to remain active during the whole step. Note, however, that under proportionally increasing loads the occurrence of unloading of plastic bars is rarely experienced. In this case, the preliminary LC problem solution is avoided, with a further consistent increase in computational efficiency. The possible occurrence of unloading is checked (in an approximate way) at the end of the step. If unloading is detected, the step is repeated (if necessary, under reduced load amplitudes) by incorporating the preliminary search for active modes. This operation is performed in any case whenever the load increment vector changes shape. The procedure is illustrated by means of the simple example given in Figure 8a. Dimensionless quantities are used, as defined earlier. The bar behaviour is elasticperfectly plastic and it is depicted in Figure 8b. The parameter/3 defines the ratio between the cross-sectional areas of the upper to the lower bar (3 ---AoffAo2); for 3 = 0 tile structure of Figure 8a reduces to that of Figure 4a, with Ho = 0.5. Even if simple and quite unrealistic, this example is significant because it incorpor:,tes a variety of possible actual truss behaviours for different values of 3. Solid lines in Figure 9 show the force-displacement curves for some typical values of 3; black dots indicate critical situations (always corresponding to snap-through buckling). For low values of 3 (3 < 0.0980) snapping occurs at the first yielding, which indicates that destabilizing geometrical effects bring the structure to collapse even if the upper bar is still elastic. For high values of 3 (3 >0.2676) collapse occurs when the lower bar has yielded in compression and the upper one in tension. This situation corresponds to that causing plastic collapse in first-order theory and, in fact, the limit analysis collapse load FL = 0.0931 is approached as/3 increases and geometric nonlinearities become less influential; for/3 >1.3623 the collapse load is F = 0.0920, with only a marginal decrease with respect to F r ; however, geometric effects give the post-collapse curve a negative slope, which indicates their destabilizing nature. In the internal 0.0980
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi
The dashed curve in Figure 9 represents a sort of limit curve which is always reached for/3 > 1.3623. Collapse occurs when this curve is attained on a point with negative slope (i.e., for u < 0.6550); otherwise, the curve can be followed up to the asymptotic limit value. For completeness, the structural response in the hypothesis of a purely elastic bar behaviour is also indicated (by thin lines) in Figure 9. For 13< 0.1968 snap-through buckling occurs and the collapse load is marked by white dots. Plastic results were obtained with the proposed numerical procedure. A maximum load step size of AF = 0.1 was adopted. Results for typical values of ~ are shown in Table 1. No step ended because the assigned maximum load ~crement was reached, rather because of yielding (in tension or compression) of either bar or because of elastic unloading of previously yielded bars. The event causing the step to end (ESC) is indicated for each step, together with the results of the step solution. For simplicity, the equivalent flexibility matrix (a single number in the present case) is indicated by Duu; superscript P is used to indicate its updated value at the end of the step, which becomes the first iteration Duu value for the subsequent step. If no new modes become activable at the end of the step,D~uu = Duu. Up to the figures indicated, the results coincide with the closed form solution of the problem. For illustrative purposes, a rather unrealistic example was chosen. In fact, not only large displacements but also large strains are required to cause yielding. The example permitted a wide variety of behaviours to be explored by simply changing the value of 43. However, the presence of large strains is inconsistent with the formulation proposed, based on engineering stress and strain definitions, which become questionable as strains become large. The main novelty of the proposed procedure rests on the test on the sign of a to detect whether or not a critical
Table 1
Duu Step 1
ESC F u o~ o2
Duu a
P Duu
Step 2
ESC F u oI o5
Duu a
P Duu
Step 3
1.0
1.1287
1.8783
0.3 3.5088
0.2
0.05
4.0064
5.0865
lyldten 2yldcmp2yldcmp2yldcmp2yldcmp 0.0873 0.0721 0.0334 0.0279 0.0196 0.0977 0.1421 0.1421 0.1421 0.1421 0.1 0.0734 0.0220 0.0147 0.0037 -0.0359 -0.05 --0.05 -0.05 --0.05 1.1262 2.0786 5.3706 6.9396 12.3609 0.1835 0.7056 0.4062 0.2636 - 0 . 2 2 9 7 8.6812 2.6344 11.8064 23.4742 Collapse 2 y l d c m p l y l d t e n 2unld 2unld 0.0920 0.0913 0.0686 0.0457 0.1421 0.1916 0.5000 0.5000 0.1 0.1 0.0824 0.0549 -0.05 --0.05 --0.05 --0.05 11.3122 2.5164 9.0171 17.4520 --0.1333 --0.0302 --Collapse Collapse 9.0171 17.4520
ESC F u oI o5
Duu a
D:u Step 4
2.0
ESC F u oI o2
Duu a
P Duu
1 yldten 0.0803 0.5986 0.1 -0.0457 7.6394 --0.0552 Collapse
l y l d ten 0.091 5 0.8708 0.1 0.0095 4.2301 0.0756 7.9114 2 yld ten 0.1419 1.1 029 0.1 0.1 3.2248 0.2067 13.9470*
* For ~ = 0.2, structure can carry additional load with both bars y i e l d e d in t e n s i o n u p t o a s y m p t o t i c v a l u e o f F = 0 . 2 .
0.2C
F'=O2
01E
t~ 13 o O 0.1C FL =O O931
situation is reached when the step ends with the activation of a new mode. As already mentioned, this event is the most frequently experienced and the simplicity of the test permits its extremely easy implementation in a truss analysis code. On the basis of the computational experience gained so far, storage requirements rather than computer time proved to be the critical factor. Realistic plane truss problems can usually be analysed by exploiting core facilities only. The extension to space trusses, even if conceptually straightforward, requires some care in order to preserve the computational efficiency in the presence of a substantially larger number of variables.
W
Conclusions ODE
o
O
0,5 lO 15 Displocement, u Figure 9 Load-displacement curves for example of Figure 8 and various cross-sectional ratios/3. F ' = asymptotic limit toad value; F L = first order theory collapse load
A procedure for large displacement analysis of elasticplastic trusses is proposed. The following assumptions are made: (i), strains are small even if displacements can be arbitrarily large; (ii) local instability effects are accounted for in the force-elongation (stress-strain) diagram of the bar, supposed to be piecewiselinear; (iii) the plastic behaviour of bars can exhibit both hardening and/or softening and is governed by a matrix H which is supposed to be symmetric. Within the framework of the above assumptions, the analysis problem is formulated and previously established
Eng. Struct., 1981, Vol. 3, October
217
Elastic-plastic trusses with unstable bars: C. Cichon and L. Corradi
results, which characterize a critical situation, are specialized. The formulation is implemented into a numerical procedure able to follow up to collapse the truss evolution. The term 'collapse' is intended in a broad sense since it refers not only to the exhaustion of structural load carrying capacity but also to the possible occurrence of bifurcation type instability in Stanley's sense. Because of the assumed piecewiselinear nature of the bar constitutive law, the structural evolution can be regarded as a sequence of steps in which the truss behaves elastically, provided that the current elastic-plastic stiffness is used to replace the original elastic bar stiffness. Relations permitting the generation of the properties of this equivalent elastic truss are provided. In spite of this, the step analysis is nonlinear because of geometrical effects. Attention is focused mainly on the case in which collapse corresponds to the activation of a new mode, i.e. it is caused by an abrupt change in the current stiffness of the bar, which makes the truss unable to carry additional load. It should be noted that this occurs in the majority of practical truss analysis problems. In this case it is shown that, in order to detect whether or not a collapse situation has been reached, a test on the sign of a single number is sufficient; this number can easily be obtained on the basis of the information provided by the step solution, with little additional computational effort. This test represents the main novelty of this paper and its introduction is the characteristic feature of the numerical procedure proposed. For the numerical solution of the geometrically nonlinear step equations and the treatment of other events which might occur during the truss evolution (collapse not corresponding to a new mode activation, elastic unloading of previously yielded bars) already established techniques are used and reference is made to the existing literature. The aim of this paper being simply to present the procedure and to indicate its potential, only very simple illustrative examples are presented. Ackn owledgemen t This research was sponsored by C.N.R. (Italian Research Council).
References 1 2 3 4 5 6 7
218
Sewell, M. 'Plastic buckling'. In 'Stability' (Leipholz, H. H. E., ed.), SM Study No. 6, University of Waterloo Press, 1972 Corradi, L. 'On a stability condition for elastic plastic structures',Meccanica 1977, 12, 24 Corradi, L. 'Stability of discrete elastic plastic structures with associated flow laws', SMArchives 1978, 3,201 Corradi, L., De Donato, O. 'Collapse analysis of elastic plastic trusses with unstable bars',Mech. Res. Comm. 1977, 4 , 4 1 7 Mallet, R. H., Marcal, P. V. 'Finite element analysis of nonlinear structures', Proc. ASCE, J. Struct. Div. 1968, 94, 2081 Maier, G. 'Incremental plastic analysis in the presence of large displacements and physical instabilizing effects', Int. J. Solids Struct. 1971, 7,345 Drucker, D. C. 'On the postulate of stability of materials in the mechanics of continua', J. Mdcanique 1964, 3,235
Eng. Struct., 1981, Vol. 3, October
8 9 10 ll
12
Samelson, tt. eta!. 'A partition theorem tot I'uclidean :Vspaces'. Prec. A mer. Math. Sac. 1958, 9,805 Cichon, C. and Corradi, L., 'Large displacenlcnt and stability analysis of elastic-plastic trusses', ISTC Tech. Rep. 4-81. Politecnico Milan(), 1981 Riks, E. 'An incremental approach to the solution of snapping and buckling problems', Int. J. Solids Street. 1979, 15,254 Cattle, R. W. 'The principal pivoting method of quadratic programming'. In 'Mathematics of decision sciences' (l)antzig, G. B. and Veinott, A. t:., eds), part 1, American Mathematical Society, 1968 Franchi, A. and Cohn, M. Z., 'Computer analysis of elasticplastic structures', lnt. J. Camp. Meth. Appl. Mech. 1980, 21 (3), 271
Notation a
matrix governing stability under load control, equation (22) bar cross-section A rigid translation matrix, equations (2) B geometric matrix, equation (5b) C differential displacement vector for bar and d,D corresponding force vector Duu , Dup , Dpp structural flexibility matrices, equations (20) e elastic portion of bar elongation Young's modulus E nodal force vector for assembled truss F hardening matrix and corresponding subO,h matrix for activable modes K.., I%, Kpp structural stiffness matrices, equations (16) bar length l connectivity matrix L outward normal matrix and corresponding N,n submatrix for activable modes plastic portion of bar elongation P nodal displacement and force vectors for q,Q individual bars vector of plastic capacities R elastic stiffness of bar, equation (lc) S nodal displacement vector for assembled truss U bar elongation (generalized strain) C bar slope 0 bar stiffness matrix, equation (9) K plastic multiplier vector and corresponding A,X subvector for activable modes axial force in bar (generalized stress) 0 Z configuration plastic potential vector and corresponding subvector for activable modes matrix governing stability under displace~0 ment control, equation (l 8) subscript referring to bar j J subscript denoting initial values 0 subscript denoting values in configuration Z N N superscript marking updated matrices for P equivalent elastic truss