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INTRODUCTION TOCOMPUTATIONAL PLASTICITY
Raaele CasciaroDipartimento di Strutture, Universita` della Calabria
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Contents
1 Plasticity for absolute beginners 31.1 Initial remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Basics of plasticity theory . . . . . . . . . . . . . . . . . . . . 51.3 Some more . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 The Drucker postulate (1951) . . . . . . . . . . . . . . . . . . 71.5 Limit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 The static theorem of limit analysis . . . . . . . . . . . . . . 101.8 The kinematic theorem of limit analysis . . . . . . . . . . . 111.9 Additional comments . . . . . . . . . . . . . . . . . . . . . . . 121.10 Plastic adaptation (shakedown) . . . . . . . . . . . . . . . . 131.11 The Melans theorem (1936) . . . . . . . . . . . . . . . . . . 141.12 Relations with admissibletensions criteria . . . . . . . . . 15
2 Finitestep incremental analysis 162.1 Holonomic plasticity . . . . . . . . . . . . . . . . . . . . . . . . 172.2 The HaarKarman principle (1908) . . . . . . . . . . . . . . 182.3 Reformulation in terms of elastic prediction . . . . . . . . . 192.4 The PonterMartin extremal paths theory . . . . . . . . . . 202.5 A deep insight . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Convexity of the extremalpath potential . . . . . . . . . . 222.7 Consequences of convexity . . . . . . . . . . . . . . . . . . . . 232.8 Principle of Minimum for Druckers materials . . . . . . . 242.9 Principle of Minimum for elastic perfectlyplastic materials 25
3 Computational strategies 263.1 Numerical algorithms for plastic analysis . . . . . . . . . . . 273.2 Incremental elastoplastic analysis . . . . . . . . . . . . . . . 293.3 Solution of subproblem P1 . . . . . . . . . . . . . . . . . . . . 303.4 Example 1: constant tension plane elements . . . . . . . . . 313.5 Some comments . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6 Example 2: framed structures . . . . . . . . . . . . . . . . . . 333.7 Solution of subproblem P2 . . . . . . . . . . . . . . . . . . . . 343.8 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.9 The arclength method . . . . . . . . . . . . . . . . . . . . . . 363.10 Riks iteration scheme . . . . . . . . . . . . . . . . . . . . . . 373.11 Convergence of Riks scheme . . . . . . . . . . . . . . . . . . 38
1
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3.12 Implementation of the Riks scheme . . . . . . . . . . . . . . 393.13 Adaptive solution strategy . . . . . . . . . . . . . . . . . . . . 403.14 Some implementation details . . . . . . . . . . . . . . . . . . 41
4 Finite element discretization 424.1 Some initial remarks . . . . . . . . . . . . . . . . . . . . . . . 434.2 Further comments . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Simplex mixed elements . . . . . . . . . . . . . . . . . . . . . 454.4 Fluxlaw elements . . . . . . . . . . . . . . . . . . . . . . . . . 464.5 HC elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6 Flat elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.7 Superconvergence, Multigrid solutions and much more... 49
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Chapter 1
Plasticity for absolute beginners
This section intends to provide motivations and basic hy-
potheses of elastoplastic theory and presents some of the
more classical results of the theory.
In particular, we present the fundamental theorems of limit
analysis (the static and kinematic one) and the Melan theo-
rem for plastic adaptation (shakedown analysis).
The discussion claries the real meaning of traditional pro-
cedures based on elastic analysis and admissible tension cri-
teria.
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1.1 Initial remarks
The structural behaviour is essentially nonlinear. Concepts like Failure or Structural safety can not beframed within a linear relationship between causes and
eects.
Traditional analysis procedures, based on elastic solu-tions and admissible tension criteria, have only a con-
ventional basis.
The great computational power of new lowcost comput-ers allows, now and even more in the future, a generalized
use of more sophisticated procedures, based on nonlinear
analysis algorithms.
The new European norms on buildings (Eurocodes) arebased on a nonlinear analysis philosophy.
Therefore, the concepts and procedures of nonlinear analy-
sis will have an increasing importance in the future.
Remarks:
The nonlinear behaviour of structures descends both fromphysical aspects (nonlinear constitutive laws: Plastic-
ity) and geometrical aspects (nonlinear displacements
deformations laws: Instability).
Only the rst theme is treated here.
4
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1.2 Basics of plasticity theory
The elastoplastic structural behaviour can be framed within
a well consolidated theory (Incremental elastoplasticity).
The theory is based on the following primitive concepts:
Structural materials have limited strength.That is, tensions must be limited.
This is formalized assuming that, for each point of the
analyzed body, the tension := {xx, xy, , zz} lieswithin a domain of the tension space, called elastic do-
main of the material
De := { : f [] 1}(plastic admissibility condition).
The function f [] depends, in general, on state para-
meters taking into account the dierences in the behav-
iour which are related to mechanical or thermal processes
(hardening, fatigue, ...).
Irreversible deformations are produced.This means that in a loadingunloading cycle the struc-
ture does not completely recover the initial conguration.
This is formalized by decomposing the deformation in-
crement in two components:
= e + p
e is the elastic part and p is the plastic part. Only
the rst one is related to the increment of the tension:
= Ee
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1.3 Some more
At low tension levels the behaviour is nearlyelastic.
This is formalized by assuming that the plastic compo-
nent of the incremental deformation, p, is dierent from
zero only if the tension belongs to the boundary of De(yield surface):
p = 0 only if f [] = 1For smaller values of (smaller in the sense just speci-
ed), the incremental behaviour is purely elastic.
Remarks:
The problem is governed by equations (equilibrium, kine-matic compatibility, elasticity) and inequalities (plastic
admissibility).
The elastoplastic behaviour is essentially nonholono-mic. The presence of residual deformations implies that
the tension and deformation state due to the application
of a load depends not only on the nal load attained but
also on the loading process (pathdependence)
The constitutive laws are expressed in incremental form ;this allows an easier treatment of the dierent behaviours
during the loading and unloading phase.
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1.4 The Drucker postulate (1951)
The elastoplastic behaviour can be more precisely described
once it is accepted the following postulate, proposed by Drucker:
Let the structure be subjected to a loadingunloadingcycle due to the application of some additional forces.
We have:
1) during the loading phase, the work done by the
additional forces is non-negative;
2) during the entire cycle, the work done by the ad-
ditional forces is nonnegative.
Considering a loadingunloading cycle which takes the ten-
sion from an initial admissible state a (that is, belonging to
De) to a point y lying on the yield surface and then returnsto a, the postulate gives:
(y a)T p 0Since the condition must be true for each a De and
each y De, the postulate leads to the following results: The elastic domain De is convex. The plastic deformation p belongs to the subgradientof De, that is to say, it is normal to the yield sur-face.
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1.5 Limit analysis
Using the few results obtained is now possible to give an
answer to the following problem:
Let the structure be subjected to a proportional load-ing p, evaluate the maximum value for the amplify-
ing factor .
The technical relevance of this problem is evident: if p cor-
responds to the nominal load, the evaluation of max implies
an estimate of the failure safety of the structure.
Remarks:
The problem makes sense only if the function f [] isconstant with the time (elasticperfectly plastic behav-
iour) or, at least, if it presents a bounded limit surface.
The presence of isolated plastic zones, embedded in anelastic matrix limiting their deformation, does not rep-
resent necessarily a cause of structural danger.
The situation is dierent when the plastic deformations,not surrounded by an elastic ring, form a kinematically
admissible mechanism.
p = up
This last occurrence, to which we will refer using theindex c, corresponds to the collapse of the structure.
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1.6 An example
The plasticization of the central bar alone, does not causea plastic mechanism, which is prevented by the lateral
bars behaving elastically.
Figure 1.1: 3bars simple structure
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1.7 The static theorem of limit analysis
Consider an elasticperfectly plastic structure and let
c , c , uc , c
be the tension, the incremental plastic deformation, the in-
cremental plastic displacement (c and uc are assumed to
be kinematically compatible) and the load multiplying factor
at collapse. Further, let q and f be the body and surface
external nominal loads.
With regard to a generic statically admissible tension eld
a (that is to say, satisfying equilibrium and plastic admis-
sibility conditions), the virtual work equation provides:BTc
c dv = c{BqT uc dv +
BfT uc ds}
BTa
c dv = a{BqT uc dv +
BfT uc ds}
where a is the load multiplier associated to a.
Subtracting these equation, we obtain:B(c a)T c dv = (c a){
BqT uc dv +
BfT uc ds}
The rst member is nonnegative, and the integral within
brackets in the second member is positive (as a consequence
of the Drucker postulate, if it is assumed that is internal
to the elastic domain). As a consequence we can derive the
following inequality:
c awhich corresponds to the static theorem of limit analy-
sis):
The collapse multiplier is the maximum of all the sta-tic multipliers, that is to say, those multipliers asso-
ciated with statically admissible tension elds.
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1.8 The kinematic theorem of limit analysis
Let {up, p} be a generic plastic, kinematically admissiblemechanism.
In each point of the body for which p = 0, we can asso-ciate the tensionp to the known plastic deformation, dened
by the normality law.
Then, dening the associated kinematical multiplier pthrough the energy balance:
BTp
p dv = p{BqT up dv +
BfT up ds}
and recalling the equilibrium equationBTc
p dv = c{BqT up dv +
BfT up ds}
we obtain, by dierence:B(p c)T p dv = (p c){
BqT up dv +
BfT up ds}
where the rst member is nonnegative and the bracketed
integrals are positive. As a consequence we derive the in-
equality:
p cwhich corresponds to the kinematic theorem of limit
analysis):
The collapse multiplier is the minimum of all kine-matic multipliers, that is to say, those multipliers as-
sociated with admissible plastic mechanisms.
11
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1.9 Additional comments
Both theorems provide neither the failure tension eldnor the failure mechanism, but only the collapse multi-
plier.
The failure load does not depend on the initial conditionsand on the loading process.
Thanks to the static theorem, the traditional analysisbased on nominal elastic solutions and admissible ten-
sion criteria, gets a rational meaning: in fact, as a con-
sequence of the use of an equilibrated and plastic admis-
sible tension eld, it provides a limit elastic multiplier
e, actually representing a lower bound for the failure
multiplier.
Possible errors related to the loss of information aboutthe initial tension state (which is generally partially avail-
able or not available at all) are irrelevant to this scope.
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1.10 Plastic adaptation (shakedown)
When the structure is subjected to repeated loadingunloading cycles, the plastic failure is not the principal
cause of structural danger.
Even if the failure condition is never attained during theloading process, new plastic deformations can nucleate
at each loading cycle.
for repeated cycles, the total plastic deformation can con-sequently grow indenitely or, anyway, (if successive de-
formations balance each other) can bring to fatigue dam-
age.
In both cases, the loading process implies the structuralruin.
Therefore, it is necessary that the plastic process endsrapidly, that is to say, after few cycles (the runningin
period) the structure gets again a purely elastic behav-
iour.
When this occurs, we say that the structure gets plasticadaptation .
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1.11 The Melans theorem (1936)
Lets consider the loading process p[t] and let
[t] = E [t] + [t][t] = E [t] + [t] + p[t]
be the out coming elastoplastic solution, expressed in terms
of the elastic solutionE[t] e E[t] and of the dierence [t]
(corresponding to an autotension eld).
We consider (if it does exist) a nominal elastic solution,
always internal to the elastic domain:
[t] = E [t] + 0 , f [[t]] < 1
this solution corresponds to E[t], except for the tension 0,
which represents the possible dierence in the evaluation of
the initial stress state (obviously, this dierence will be an
autotension eld).
Introducing the quantity:
[t] :=B( )TE1( ) dv 0
we obtain:
[t] = B( )T p dv 0 ( < 0 if = 0)
Hence [t] is both nonnegative (bounded from below) and
always decreasing during the plastic process. This implies
that the plastic process will surely stop.
As a consequence, we derive the following statement:
If a nominal elastic solution (internal to the elas-tic domain), does exist, the structure will get plastic
adaptation.
14
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1.12 Relations with admissibletensions criteria
The previous results clarify the real meaning of procedure
based on admissible tensions criteria.
The traditional analysis, based on (nominal) elastic so-lutions and admissible tensions criteria, provides a lower
bound for both the plastic collapse and the plastic shake
down.
An erroneous evaluation of the initial tension state isirrelevant.
The use of procedure based on limit analysis conceptsalone is not safe with respect to shakedown problems.
The plastic adaptation theory provides a synthetic ap-proach to the analysis, which does not require a com-
plete information about the time evolution of the loading
process, but only requires the knowledge of the maximum
value attained by the tensions.
Anyway, the Melans theorem does not give any detailedinformation about the extension that the plastic phase
reaches before the structure gets adaptation.
A complete information can be only provided by a realelastoplastic analysis based on pathfollowing solution
algorithms.
15
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Chapter 2
Finitestep incremental analysis
In this part some theoretical basis of the incremental elasto
plastic analysis will be introduced.
In particular it will be presented an analysis procedure ori-
ented to furnish the complete time evolution of the structural
response due to a given loading process.
The solution, represented by a path in the load/displacement
space, is obtained through the computation of a sequence of
equilibrium points {uk,pk} ne enough to reconstruct aninterpolating curve.
Since the earlier 60s, the interest for these topics has been
strongly increasing, thanks to the availability of powerful
computers and to the development of specic numerical al-
gorithms.
16
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2.1 Holonomic plasticity
In order to obtain incremental elastoplastic solutions (using
small but necessarily nite loading steps), it is necessary
to solve the following problem:
Let {0, 0} be a known initial state and (p1 p0)a given load increment, determine the corresponding
stepend solution {, }.
Remarks:
Because of the irreversibility of the plastic behaviour andthe pathdependence of the nal state, the set of data
which denes the problem is not complete.
Referring to the gure below, bothu1 andu2 are possiblesolutions for the load increment (p1 p0).
17
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2.2 The HaarKarman principle (1908)
A possible way for getting an holonomic law in the step is to
express directly the equation of the incremental elastoplastic
theory in terms of nite increments. In this way, the step
end solution can be characterized by the following extremal
condition (HaarKarman):
[] :=1
2
BTE1 dv +
BTp0 dv +
B(N)T u ds
under the following constraints:
: satisfying equilibrium conditions, f [] 1In fact, recalling the relations:
= e + p
= Ee
( a)Tp 0where p = p p0, we obtain:
=BT (e + p0) dv
B(N)Tu ds
=BTT dv
B(N)Tu ds
BTp dv
=B( a)Tp dv 0
The HaarKarman principle can be expressed in the following
way:
The elastoplastic solution minimizes the total com-plementary energy of the system under the equilib-
rium and plastic admissibility constraints.
18
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2.3 Reformulation in terms of elastic prediction
Denoting with E the elastic solution obtained from the
initial (stepbeginning) state and the assigned load incre-
ment, the HaarKarman principle can be rewritten as:
[] :=1
2
B( E)TE1( E) dv = minimum
under the constraints:
: satisfying equilibrium conditions, f [] 1It is worth observing that the functional [] represents, in
an energy norm, the square of the distance between and
E. Hence the principle can be stated as:
Among the stress states verifying equilibrium and plas-tic admissibility conditions, the stepend elastoplastic
solution has the minimum distance (in an energy
norm) from the elastic solution E of the same prob-
lem.
Remarks:
This formulation is particularly convenient from both thenumerical and the computational point of view.
The solution can be easily characterized. If the point belongs to the elastic domain, we have = E. Other-
wise, is found as the tangential contact point between
two convex surfaces: the yield surface and a level line of
the elastic strain energy.
Since the latter surface is strictly convex, the uniquenessof the solution is proved.
19
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2.4 The PonterMartin extremal paths theory
The extremal paths theory, formulated by Ponter & Martin
in 1972, furnishes a convenient link between incremental and
holonomic plasticity. The following results hold:
Among all the incremental elastoplastic paths start-ing from the initial state {0,0}, there are some ex-tremal paths that realize both the minimum of deforma-
tion work (for xed stepend strain 1) and the maxi-
mum complementary work (for xed stepend stress 1)
The use of extremal paths denes a holonomic law inthe step that, for Druckers materials, satises the con-
ditions:
0 (2 1)T (2 1) (2 1)TE(2 1)
For elastic perfectlyplastic materials, the extremal pathsolution corresponds to the one dened by the Haar
Karman principle.
Figure 2.1: Extremal solution ({1, 1}
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2.5 A deep insight
Let [t] be a path between 0 and 1 in the stress space and
let [t] be the corresponding path in the strain space, image
of [t] through the constitutive law.
The complementary work on [t] is dened as
U [[t]] :=[t]
T dt
We dene extremal the paths characterized by the following
(extremal) condition:
U [1] := U [[t]] U [[t]]In order to complete the denition domain of U , we assume
that
U [1] := +when no admissible path between 0 and 1 does exist. So,
U [1] has been characterized as a function dened in all the
space, and will be called extremalpath potential .
From the decomposition of the total strain in the elastic
and plastic components ( = e+p), it is possible to derive
the decomposition of the complementary work in the elastic
part Ue[] and the plastic part Up[]. Only the latter is
pathdependent, and we can express it in this way:
Up[[t]] :=[t]
Tp dt
Then, by dierence, we obtain the extremalpath plastic
potential
U p[1] = U [1] Ue[1]
21
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2.6 Convexity of the extremalpath potential
Lets consider an extremal path between 0 and 1, and a
dierent path made up of an extremal part between 0 and
2 and a linear part between 2 and 1:
L[t] = 2 + tL , L = (1 2)We have, as a consequence of the previous denition:
U [1] U [2] + 10(1 2)T (2 + L[t]) dt
= U [2] + (1 2)T 2 + 10(1 2)TL[t] dt
The last expression can be rewritten 10(1 2)TL[t] dt =
10{ t0TLL d} dt
For Druckers materials (which satisfy the condition T 0) this expression is nonnegative.
We can derive, therefore, the following fundamental in-
equality:
U [1] U [2] T2 (1 2) 0As a consequence, using the standard results of convex analy-
sis, we obtain some important results:
1. The functional U [1] is convex.
2. The strain belongs to the subdierential U of U :
[1] U [1] := { : U [2]U [1]T (21) 0 , 2}In the same way it is possible to prove the convexity of the
extremalpath plastic potential U p[1] and the normality law
for the plastic strain p
p[1] Up[1]22
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2.7 Consequences of convexity
Let 1 and 2 be two stepend stresses; using the property
of convexity of the extremal potential, we obtain
U [2] U [1] T1 (2 1) 0U [1] U [2] T2 (1 2) 0
then, adding up the two members of the equations:
(2 1)T (2 1) 0In the same way, using the property of convexity of the
extremalpath plastic potential, we obtain:
(2 1)T (p2 p1) 0This relation, applying the strain decomposition law:
= e + p , = Ee
which implies that
(2 1)TE(2 1) = (2 1)T (2 1)+(p2 p1)T (2 1) + (p2 p1)TE(p2 p1),
provides the condition:
(2 1)TE(2 1) (2 1)T (2 1)The two previous inequalities can be rearranged in the fol-
lowing form:
0 (2 1)T (2 1) (2 1)TE(2 1)Later on, this inequality will have a great importance.
23
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2.8 Principle of Minimum for Druckers materi-
als
Let be the elastoplastic extremalpath endstep solution,
and eq a generic stress eld, in equilibrium under the same
endstep loads. The following extremal condition must hold:
U [eq] U [] []T (eq ) 0Moreover, since eq is an autotension eld, we have:
B[]T (eq ) dv
B(N(eq )Tu ds = 0
Therefore, by integrating the inequality over the domain B,
we obtain:BU [] dv
B(N)Tu ds
BU [eq] dv
B(Neq)
Tu ds
This result can be expressed with the following statement:
Among all the equilibrated stress elds, the elastoplastic extremalpath solution minimizes the extremal
path potential.
The statment represents a generalization of the principle of
minimum of the total complementary energy, valid for elasto
plastic, stable materials which satisfy the normality law
p[] Up[]
24
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2.9 Principle of Minimum for elastic perfectly
plastic materials
With respect to elastic perfectly-plastic materials, the follow-
ing conditions hold:
any admissible stress can be reached through a purelyelastic path.
let and a be two admissible stresses and p the in-cremental plastic strain associated to , we have:
(a )T 0
Moreover, the pathdepended part of the complementary
work can be written as: T0(p[t] p[0])T [t]dt =
T0
{ t0p[ ]Td
}[t]dt
= T0p[t]T
{ Tt[ ]d
}dt =
T0p[t]T (a [t])dt 0
and this proves that the complementary work is maximized
when p = 0.
As a consequence, the extremal paths can be characterized
as purely elastic paths. Then we can write:
U [] =
Ue se f [] 1+ se f [] > 1
and the principle of minimum becomes equivalent to the
Haar-Karman principle:
Among all the equilibrated and admissible stress elds,the elastoplastic extremalpath solution minimizes
the complementary energy.
25
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Chapter 3
Computational strategies
This part intends to discuss some numerical strategies suit-
able for the nite element elasto-plastic analysis of complex
structures.
In particular, the discussion will refer to the incremental
elastoplastic problem, using the so called initial stress strat-
egy combined with a pathfollowing incremental method.
26
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3.1 Numerical algorithms for plastic analysis
1. Limit analysis:
Linear programming algorithms (based on limit analy-sis theorems and on the statickinematic duality).
Nonlinear programming (algorithms based on limitanalysis theorems).
Alternative variational formulations combined withthe use of specialized solution algorithms.
This class of algorithms has been very popular (at
least within the Italian academy) during the 60s
70s. At the present time, it is not used at all, ex-
cept for rare cases, because it is considered much less
ecient then the incremental approach.
2. Plastic adaptation (shakedown):
The situation is the same as limit analysis.In a certain sense, the problem is similar to that
of limit analysis, and therefore similar solution al-
gorithms could be used. However, the methods re-
cently proposed are poorly ecient, even if there is
the possibility of providing synthetic results for com-
plex loading combinations, circumstance that makes
this approach particularly interesting.
Because of the technical relevance of the topic, it is
desirable an increasing development in the research.
27
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3. Incremental analysis:
Heuristic algorithms (Euler extrapolation ...). Quadratic programming. Initial stress approach: Secant matrix iteration
NewtonRaphson method
Modied NewtonRaphson
Riks arclength method
....
Explicit algorithms based on pseudodynamic ap-proach.
The algorithms belonging to this class are widely used,
thanks to their good eciency.
Some implementations of the initial stress method,
combined with an incremental Riks strategy, are stan-
dard features of the commercial FEM codes.
Pseudodynamical algorithms usually require very high
computational power.
28
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3.2 Incremental elastoplastic analysis
Let the structure be modeled by nite elements an let p[]
be the assigned loading path; we have to solve the following
problem:
Determine a sequence of equilibrium points (uk, k)ne enough to obtain the path by interpolation.
The problem has a recurrent structure and can be decom-
posed in two subproblems:
P1: Knowing the initial state (at the beginning of the step)
and the nal nodal displacement vector u (at the end of
the step), determine the corresponding vector of internal
nodal forces, s[u].
P2: At the end of the step the vector of nodal loads p is
given; determine u such that the following equilibrium
equation is satised:
s[u] = p
Remarks:
Only the subproblem P1 requires the description of theelastoplastic response of the structure (i.e., contains the
physics of the problem).
The subproblem P2 actually corresponds to the solutionof a implicit nonlinear vectorial equation.
29
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3.3 Solution of subproblem P1
The theory of extremal paths provides a good theoretical
frame for the subproblem P1.
Making use of extremal paths, in fact, the load step is per-
formed through a real incremental elastoplastic path, which
in addition presents some theoretical advantages (as shown
by Ponter-Martin, a little shift in the trajectory leads to neg-
ligible dierences in the attained nal state).
By integrating the conditions written for the material on
the overall structure, we obtain for s[u] a representation char-
acterized by the following inequalities:
0 (s[u2] s[u1])T (u2 u1) (u2 u1)TKE(u2 u1)where u1 and u2 are two possible nal displacement vectors
and KE is the standard elastic stiness matrix of the struc-
ture.
Practically, the nodal forces s[u] can be obtained through
a standard elementbyelement assemblage of the single ele-
ment contributions. That is to say:
1. For each element we determine the elastic solution due
to the initial stress state 0 and to the assigned displace-
ment increment (u u0) (elastic prediction):E = 0 + E , := D(u u0)
2. The nodal displacements in the element are assigned.
Therefore there is no nodal equilibrium condition to be
satised, and no interaction among elements. Conse-
quently, the HaarKarman minimization condition only
implies local, independent conditions dened for each el-
ement.
30
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3.4 Example 1: constant tension plane elements
Let and ij be the cubic and deviatoric components of
the nal stress, and let E, Eij be the corresponding elastic
predictions.
The elastic strain energy [] and the plastic admissi-bility condition (Mises) f [] 1 are expressed by:
[] :=1
2E
(1 + )
ij2ij + 3(1 2)2
f [] :=1
2y
ij2ij < 1
Figure 3.1: Haar-Karman solution
In the deviatoric space, the level curves of [] and f []have both spherical shape. The HaarKarman solution
[] = min. , f [] 1which corresponds to the tangent point, is then obtained
as:
ij :=Eij
max(f [E]1/2, 1), := E
31
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3.5 Some comments
The solution process described can be employed, moregenerally, in the case of compatible elements based on
numerical integration; in this case it will be applied to
each Gauss point.
In presence of a discontinuity between elastic and plasticzones, it isnt possible to localize with accuracy the in-
terface between the dierent behaviours in the element.
Then, the discretization error is always proportional to
the dimension h of the element, even using complex ele-
ments, that in elasticity present a better error estimate
(proportional to hn, being n 24). The error usually appears in the form of locking and canbe relevant.
The use of mixed elements is a way to avoid lockingphenomena.
Because of the particular structure of the HaarKarmanprinciple, hybrid elements, dened by compatible bound-
ary displacements and equilibrated internal stress elds,
are particularly convenient.
32
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3.6 Example 2: framed structures
We use a hybrid beam element, dened by the endsections
displacements and rotations and by internally equilibrated
elds for the bending moment and the axial strength. The
element behaviour depends on 6 kinematic and 3 static vari-
ables.
Let N be the axial strength, Mi and Mj the bending mo-
ments (in the two edge sections of the element), and NE ,
MEi, MEj the elastic predictions corresponding to the as-
signed nodal displacements.
The complementary strain energy is dened by:
[] :=1
2
&EA
N2 +3&
EJ(M 2i +M
2j 2cMiMj)
where
c :=1 /22 +
:=12EJ
GA&2
The plastic admissibility condition (expressed only forthe end sections) provides:
My Mi My , My Mj My The Haar-Karman solutions is directly obtained from thefollowing 3steps algorithm:
1) Mi := max{My,min{MEi,My}}2) Mj := max{My,min{MEjc(MEiMi),My}}3) Mi := max{My,min{MEic(MEjMj),My}}
33
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3.7 Solution of subproblem P2
The equilibrium at the end of the step is expressed by the
condition:
s[u] = p
which corresponds to a nonlinear implicit equation in the
unknown u.
The equation can be iteratively solved using the modied
NewtonRaphson method:
rj := p s[u]uj+1 := uj + K
1rj
where K is a suitable approximation for the Hessian matrix
Kt[u] :=ds[u]
du
The convergence of the MNR iteration scheme can be dis-
cussed introducing the secant matrixKj, dened in the jth
iteration step by the following equivalence
Kj(uj+1 uj) = s[uj+1] s[uj]The iteration scheme implies
rj+1 =[IKjK1
]rj
Therefore, we derive the following (sucient) convergence
condition:
[IKjK1
]< 1 , j
where [] is the spectral radius of the matrix.The convergence condition can be rearranged in a more
useful form:
0 < Kj < 2K
34
-
3.8 Some remarks
The inequality0 (s[uj+1]s[uj])T (uj+1uj) (uj+1uj)TKE(uj+1uj)which is true for solutions obtained through extremal
paths, starting from u0, can be written in the following
matrix form:
0 Kj KEHence, the second part of the convergence condition (i.e.,
Kj < 2K) is trivially satised once it is assumed (as in
the initial stress method)
K := KE
(or a reasonable approximation of KE).
the rst part of the convergence condition (Kj > 0)is more critical because, even if we have Kj 0, theiteration scheme fails to converge near to the collapse,
where:
Kt[u] 0 For this reason, the MNRbased incremental process isnot able to describe the collapse state of the structure.
Actually, when the plasticization process goes on, theconvergence of the iterative process rapidly decreases,
and this generally means that the process itself will pre-
maturely stop.
35
-
3.9 The arclength method
The convergence problems arising near the limit pointof the equilibrium path are related to the parametric
representation adopted to describe this curve. In fact,
the representation has been assumed in the form u =
u[], even if the curve we are looking for is not analytical
in .
Figure 3.2: Loadcontrolled analysis and arclength method
These diculties are easily avoided if an analytical rep-resentation is used. A proper description is obtained as-
suming a curvilinear arclength abscissa as description
parameter, in the {u, } space. The arclength method has been proposed by Riks in1974 for the pathfollowing analysis of nonlinear elastic
problems. At the present time, it can be considered as
the de facto standard method for nonlinear analysis.
36
-
3.10 Riks iteration scheme
The iteration scheme due to Riks represents the rst, and jet
more ecient, implementation of the arclength method.
Its main idea is to introduce explicitely the load multiplier
as further unknown. At the same time a further constraint
is needed, and it is given in the following way:
uTMu + = 0
where M and are appropriate metric factors. This condi-
tion implies the orthogonality (in the enlarged space {u, })between the iterative correction
u = uj+1 uj = j+1 j
and the total step incrementu = uj u0 = j 0
If the iteration starts from a suitable extrapolation {u1, 1}that realizes the required distance from {u0, 0}, the con-straint represents an approximate, but computationally e-
cient way to x the arclength.
With this choice, the iteration scheme can be rearranged:Ku p = rjuTMu + = 0
where
p :=dp[]
d
37
-
3.11 Convergence of Riks scheme
The use of the Riks iteration scheme leads to the relation:
rj+1 =[IKjK1
][I jBj] rj
the matrix Bj and the factor j are dened by:
Bj :=pdTjpTdj
, j :=pTdj
j + pTdj
Where we have called:
dj := K1Muj
The following sucient condition for convergence is de-
rived:
[[IKjK1][I jBj]
]< 1 , j
Remarks:
The better performance of the Riks scheme is strictlyrelated to the lter eect produced by the matrix
[I jBj]
The lter does not aect the components of rj orthogonalto dj, and reduces the parallel components of a (1j)factor.
For 0 and, in any case, near to a limit point (where 0), we have j 0 and then the lter correspondsto an orthogonal projection on the direction dj.
38
-
3.12 Implementation of the Riks scheme
The Riks scheme is a very powerful tool, but its eciency
partially depends on the appropriate choice of the metric
factors ed M.
A convenient choice can be to assume = 0 and M such
that
dj = u := K1p
(These choice is equivalent to the assumption M Kj.)Now, the scheme can be rearranged in the following, ex-
plicit form:j+1 = j rTj u/pTuuj+1 = uj + K
1rj + (j+1 j)uUsing arguments similar to the loadcontrolled scheme, we
can derive the following (sucient) convergence condition:
0 < Kt[u] 0
for each nonnull collapse mechanism uc. Then, the direction
of singularity of the operator Kt[u] does not lie in the or-
thogonal subspace and the global convergence of the scheme
is assured.
39
-
3.13 Adaptive solution strategy
An ecient incremental process should have an adaptive be-
haviour; that is, it should be able, on the basis of autonomous
choices, to vary its internal parameters in order to reduce the
computational cost and increase the accuracy of the analysis.
In particular, the following features are required:
the process should tune the step length: that is, it shouldincrease such length where the equilibrium curve is weekly
nonlinear, and reduce it in the strongly nonlinear parts.
This allows either a better description of the curve and
the computation of a smaller number of equilibrium points.
The process should tune the iteration matrix K too, inorder to follow the evolution of Kj: for instance, we
should use the full elastic stiness matrix only in the
nearly elastic incremental steps, and a reduced stiness
in the plastic steps.
The adaptivity feature must be computationally cheapand, anyway, must preserve the reliability of the overall
analysis.
40
-
3.14 Some implementation details
It is convenient to introduce two adaptive parameters: and
.
The rst one controls the initial extrapolationu1 = u0 + u01 = 0 + 0
where u0 and 0 are the total increments attained
in the previous step.
The second one controls the evaluation of the iterationmatrix K, which is assumed to be proportional to KE
K =1
KE
(The use of this scalar does not bring additional compu-
tational costs: in fact, being K1 = K1E , it is possibleto use the matrix KE which is assembled and decom-
posed once for all)
The following formulas can be used:
in the jth iteration loop,
j+1 = jrTj uj
(rj rj+1)Tuj with the limits 0 < < 2
in the kth step of the incremental process,k+1 = kt
nn2nn
where n is the number of iteration loops performed in
the previous step, n is the number of desired loops (usu-
ally 410) and t is the relative tolerance required for the
equilibrium.
41
-
Chapter 4
Finite element discretization
The presence of the discontinuities related to the elastoplas-
tic behaviour requires an approach dierent from the one
generally used for elastic problems, for which the solution is
characterized by an high degree of regularity.
Discretization methods (nite element modeling) which
are ecient for elastic problems can be not so convenient
for plastic ones.
In this part a rapid sketch on this topics is given, with the
aim to describe some nite elements potentially suitable for
elastoplastic problems.
42
-
4.1 Some initial remarks
Within plastic zones, because of the discontinuities inthe strain eld (introduced by the plastic behaviour),
the nite element discretization error depends linearly
on the size h of the mesh, even using high interpolation
elements.
Therefore the use of complex elements, characterized bya high number of variables per element, is not advanta-
geous.
The accuracy must be attained through the mesh rene-ment.
All this consideration suggests the use of simple elements,with few variables per element, and ne meshes.
An accurate evaluation of the stress eld is more impor-tant than in elasticity, because the material behaviour is
strictly related to the level of stress.
From this point of view, compatible elements, which eval-uate displacements more accurately than stresses (the
latter are only obtained by derivation from the former),
could represent a bad choice.
Mixed or Hybrid elements, for which the accuracy fordisplacements and stresses can be balanced, seem to be
more convenient.
43
-
4.2 Further comments
Since plastic deformations are typically localized on slip(yield) planes, above all when the load approaches the
collapse level, it is convenient the use of elements suitable
to allow discontinuity both in the tension and in the
strain eld.
With respect to kinematics, elements should assure thecontinuity, on contact surfaces, of normal displacements
but not of tangential ones (just the opposite of what
usually happens).
With respect to statics, elements should assure the conti-nuity of normal and tangential stresses but not of transver-
sal ones (just the opposite of what usually happens).
A compromise between dierent continuity requirementsis not easy and does not lead to a simple algebraic for-
mulation of the element. Moreover, the use of discon-
tinuous elds could be inappropriate for the part of the
body which remains elastic.
Apparently, the only alternative is the use of very nemeshes, such that discontinuities can develop in a band
of small thickness (comparable to the mesh size).
44
-
4.3 Simplex mixed elements
These elements are triangular (more generally, a tetra-hedron with n+ 1 vertices, n being the space dimension
of the problem).
Both displacement and stress elds are linearly interpo-lated on the basis of the nodal values.
both displacements and stresses are continuous on theelement interfaces. Neverthless, if the ratio n.of elements
/ n.of vertices is 2, the model requires 2.5 variables per
element, and the algebra of the problem is very simple.
This allows the use of ne meshes.
It may be convenient to combine 4 triangular elementsinto a single quadrangular element (the quadrangle is di-
vided along the two diagonals). In this case we have 5
variables (2 displacements and 3 stresses) for each ele-
ment.
A convenient variant is the assumption of a constantstress within the inuence area of each node.
45
-
4.4 Fluxlaw elements
These are, usually, quadrangular elements (but triangu-lar versions are also possible), endowed with topological
regularity (regular geometry is even better).
Stress and displacement variables are associated to thetotal ux which passes through the interfaces.
The balance equations (equilibrium and kinematic com-patibility) are written in absolute form (rigid body equi-
librium and mass conservation law). The discretization
error is related to the internal stress interpolation used
to dene the discrete form of the strain energy.
We have 6 variables per element (2 displacements and 4stresses). The element is suitable for ne meshes.
The algebra is very simple for regular geometry (rectan-gular elements)
46
-
4.5 HC elements
HC stands for High Continuity (of the displacementeld).
These elements are quadrilateral, and require a regulartopology (regular geometry is even better).
The displacement is interpolated by means of biquadraticfunctions, using control nodes located at the center of the
element. The fullllment of the displacement and dis-
placement gradient continuity allows to reduce the total
number of variables to only one node per element (hence
there are two displacement variables).
The (almost) compatible element type require 5 stress(Gauss) points per element. A mixed version can be de-
ned with stress nodes located at the vertices of elements
(the nodes of the mesh).
The overall interpolation allows C1 continuity, using aminimum number of displacement variables. So this ele-
ment is suitable for very ne meshes.
The element algebra is very easy for regular meshes (rec-tangular elements). However it becomes rather complex
for meshes described in curvilinear coordinates.
The element is particularly convenient for the analysis ofregular domains, using regular meshes.
It is suitable for large displacement analysis.
47
-
4.6 Flat elements
This element is a mixed, triangular one, and is used forplates and shells.
For the displacements a linear intepolation is used, whilestresses are constant over the element. The exural strain
is taken into account introducing the relative rotations
on the element edges.
The element is quite rough, neverthless it has a very easyalgebra. It requires very ne discretizations.
It is suitable for large displacement analysis. Dierent mixed variants are possible, using dierent in-terpolations for the stress (linear over the element, con-
stant on the nodal area, localized on the edges).
It is suitable for analysis based on explicit algorithms(e.g., pseudodynamical simulation) or when the solu-
tion is obtained using rened meshes (e.g., multigrid so-
lutions).
48
-
4.7 Superconvergence, Multigrid solutions and
much more...
The use of appropriate elements, ne meshes, regulartopology and regular (at least, nearly regular) geometry
leads to a phenomenon called superconvergence.
That is, the errors produced in the single elements com-
pensate each other on the overall mesh, so that the total
error is smaller than the expected one (the magnitude
order can be relevant, e.g., from h2 to h6).
Neverthless, the use of really ne meshes implies a highnumber of variables and this, even if a very simple al-
gebra is involved for the single element, can make the
overall analysis extremely expensive, if traditional solu-
tion strategies (based on explicit assemblage and Gauss
decomposition of the stiness matrix) are used. This last
operation alone, for a problem with n variables, would
involve O(n2) computational arithmetic operations.
In situations like these, adaptive multigrid strategies arevery useful. They are based on the simultaneous use of
a sequence of discretization meshes increasingly rened,
and allow to catch the solution which corresponds to the
nest mesh, working essentially on the coarse ones. We
obtain therefore a very powerful tool, which can solve
the problem performing less than O(n) operations, that
is to say with an eciency of some order greater of the
one obtained through traditional procedures.
An eciency even greater is obtained using adaptive (se-lective) renement (the solution process automatically
controls the renement).49