Computer Aided Optimum Design in Engineering IX
235
Optimum design of structures with
limited ductility
F. Giambanco, L. Palizzolo & A. Caffarelli
Department of Structural Engineering and Geothecnics,
Palermo University, Italy
Abstract
A minimum volume design problem formulation for elastic perfectly plastic
structures, subjected to loads quasi-statically variable within a given domain, is
proposed. In particular, the actions are defined as the combination of fixed loads
and perfect cyclic loads. The optimal design problem is formulated according to
a plastic shakedown criterion so that the incremental collapse and the
instantaneous collapse are certainly prevented when the structure is subjected to
the prescribed amplified loads. In addition, further appropriate constraints on the
plastic deformations characterizing the structure response are imposed. In
particular, suitable limits on the plastic deformations occurring at the limit state
of the plastic shakedown are imposed and some chosen measures of the plastic
deformations related to the transient phase structural response are suitably
bounded. For computational purposes an appropriate solution procedure is
utilized. The present optimum design formulation shows very promising features
in order to be numerically applied.
Keywords: bounds, Bree diagram, structural optimization, ductility, plastic
strains, plastic shakedown, elastic plastic structures.
1
Introduction
As it is well known, the external mechanical and/or kinematical actions which a
structure must suffer during its lifetime are usually variable both in time and in
space, and furthermore they are sometimes of very high intensity, so that
structures in these cases find their selves above the elastic limits exhibiting an
elastic plastic behaviour.
Obviously, such occurring permanent deformations must be limited in
engineering practice, in order to avoid the exceeding of the ductility capacity of
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
236 Computer Aided Optimum Design in Engineering IX
the material as well as the loosing of the functionality for the whole structure or
even for a part of it.
As a consequence, the design of a structure which is expected to behave
above the elastic range when subjected to the prescribed amplified loads, must
take into account the respect of some special requirement for the relevant
occurring plastic deformations.
Recently, many researchers proposed interesting refinements and upgrading
in the framework of the optimization problems in structural mechanics,
providing both theoretical studies on the optimal design problem as well as
several original contributions related to the necessary computational procedures.
The various formulations of the design problem for elastic plastic structures
subjected to fixed and cyclic loads, usually proposed as the search for the
minimum volume structure, substantially differ for the special limiting criterion
imposed on the structure behaviour. So, the elastic optimal design (see, e.g., [1]),
the elastic shakedown optimal design (see, e.g., [2-4]), and the standard limit
design (see, e.g., [5,6]) have been developed. In addition, in order to constraint
the optimal structure to simultaneously satisfy more resistance criteria with
appropriate safety factors, suitable multicriteria optimal design formulations
(see, e.g., [7]) have been proposed.
Unfortunately, these last formulations, even if they ensure the satisfaction of
the imposed limit conditions under the prescribed amplified loads, do not
provide any useful information about the behaviour of the structure within the
range between the elastic shakedown limit and the instantaneous collapse.
Recently (see, e.g., [8,9]), some formulations of the optimal design of elastic
plastic structures subjected to a combination of fixed and cyclic loads have been
proposed, taking into account the structure behaviour within the above-described
range. In particular, the optimal structure is obtained in such a way that, above
the elastic shakedown limit, the incremental collapse is prevented as far as the
load multipliers do not exceed some given limits
Whatever the special formulation is utilized for obtaining the optimal design,
it can be very useful to know if the optimal structure, at the prescribed limit state,
fulfils appropriate limits on its ductility behaviour. Actually, in the above
referred formulations possible limits on the ductility behaviour of the structure
have been disregarded.
To the author’s knowledge, some contributions on this topic have been
proposed in the case of elastic shakedown design (see, e.g., [10]), in the case of
standard limit design (see, e.g., [11]) and making reference to a special
multicriteria optimal design (see, e.g., [12]).
The present paper is devoted to propose a formulation of the optimal design of
elastic perfectly plastic structures subjected to a combination of fixed and cyclic
loads, according to a plastic shakedown criterion. The acting cyclic loads will be
conceived as perfect cyclic load, namely, for each basic load condition an
opposite one exists in the load space. Behaving the optimal structure in condition
of plastic shakedown, when subjected to the prescribed amplified loads, it
exhibits a periodic elastic plastic steady-state response eventually occurring after
a transient phase (usually elastic plastic as well). So that, in addition, further
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
Computer Aided Optimum Design in Engineering IX
237
constraints on the plastic strains will be considered, related to the steady-state
response at the limit state of plastic shakedown as well as on some suitably
chosen measures of the plastic deformations related to the transient phase.
The relevant search problem is a strongly non-linear mathematical
programming one and, as a consequence, suitable solution procedures must be
utilized. The effected numerical applications, regarding steel structures and here
not reported for the sake of brevity, confirm the theoretical expectations in terms
of behavioural features of the obtained optimal designs.
2
Structural model and elastic plastic behaviour
Let us consider a structure discretized into n finite elements exhibiting an elastic
perfectly plastic behaviour and such that, for each element, plastic deformations
can occur just at the plastic nodes (usually not coincident with the element
nodes), which are conceived as sources from which plastic strains spread within
the element volume, according to fixed shape functions. The elastic plastic
behaviour of the typical structural element can be described in terms of
generalized variables (stresses and strains).
Let us suppose that the loads acting on the structure, F = F (t ) , are variable
in time quasi-statically and that they are defined within the time interval
0 ≤ t ≤ t f . The time t is not the physical time, but just some monotonically
(
)
increasing parameter aimed at specifying the loading sequence.
In the hypothesis of small displacements and homogeneous kinematical initial
conditions, assuming that the elastic domain of the typical element is a convex
and temperature-independent hyperpoliedric function, the elastic plastic
behaviour of the structure at time t is described by the following equations:
K u−B p = F
~
P = Bu − Dp + P ∗
~
~
ϕ = NP − R ≤ 0 , λ ≥ 0 , ϕ~ λ = 0 , ϕ λ = 0
p = N λ
t
p = ∫ N λ(t ) d t
0
(1a)
(1b)
(1c)
(1d)
(1e)
where u is the structure node displacement vector, p the generalized plastic
~
strain vector evaluated at the strain points, K = C De C the external stiffness
matrix with C the compatibility matrix and De the block diagonal element
~
internal stiffness matrix, B = C De G p the pseudo-force matrix, being G p a
matrix which applied to plastic strains provides element nodal displacements,
F = F + F * the equivalent nodal load vector with F representing the vector of
the loads directly acting upon the structure nodes and F * representing the nodal
load vector equivalent to the actions applied upon the elements, P the
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
238 Computer Aided Optimum Design in Engineering IX
generalized stress response vector evaluated at the strain points, P ∗ the
generalized stress response vector evaluated at the strain points but due just to
~
the loads directly acting upon the structural elements, D = G p De G p the block
diagonal stiffness matrix related to the strain points, ϕ the yield function vector
which also plays the role of plastic potential, N the block diagonal matrix of
unit external normals to the yield surface, R the plastic resistance vector and,
finally, λ the vector of plastic activation intensities.
Operating with the appropriate replacement, the solving set is obtained in the
following form, that must be satisfied for all t 0 ≤ t ≤ t f :
(
(
)
) (
)
t
~ ~
~ ~
− ϕ = R − N BK −1F + P ∗ − N BK −1 B − D N ∫ λ (t )dt
(2a)
~
− ϕ ≥ 0 , λ ≥ 0 , ϕ~ λ = 0 , ϕ λ = 0
(2b)
0
where just the independent unknown vectors ϕ and λ appear.
Problem (2) refers to a structure discretized into finite elements and with a
discrete yield surface, but it is yet continuous with respect to the time t. At least
in principle eqs. (2), plus the appropriate initial conditions at t = 0 , can be
integrated with respect to time t in order to obtain the unknown vectors ϕ and
λ , and, therefore through eqs. (1), the elastic plastic structural response.
Anyway, in practice, in order to obtain a numerical solution to problem (2), it is
necessary to discretize the problem also with respect to the time. Therefore,
subdividing the time axis into a chosen number of time subintervals and adopting
a suitable plastic activation intensity modelling, within each defined subinterval
k, equations (2) can be satisfied in the following integrated, holonomic form:
Z k ≡ −ϕ k = SY k + b k
~
Zk ≥ 0, Yk ≥ 0, Y kZk = 0
where:
(
)
(
(3a)
(3b)
)
~
~ ~
~~
S = − N B K −1 B − D N = − N G p De C K −1C De − De G p N
(4a)
k −1
~~
~ k
b k = R + S ∑Y j − N B K −1F k − N P ∗
(4b)
and
j =1
being S is a time independent symmetric structural matrix which transforms the
plastic activation intensities Y k into the opposite of the plastic potentials ϕ k ,
and b k a known vector defined as the sum of the pertinent loading at step k and
the increments of the plastic activation intensity vectors accumulated at step k-1,
as well as the constant plastic resistance R.
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
Computer Aided Optimum Design in Engineering IX
239
In virtue of the effected time discretization, problem (2) transforms into the
following sequence of linear complementarity problems:
~
Z k ≥ 0 , Y k ≥ 0 , Y k Z k = 0, (k = 1,2 ,...., f )
(5)
In the present case, matrix S is semi-defined positive, and as a consequence,
neither the existence of a bounded solution Y k , nor its uniqueness is guaranteed.
If eqs. (5) admit an unbounded solution Y k (at least somewhere in the structure),
instantaneous collapse occurs; on the contrary, if they admit a vanishing solution
Y k , the full structure is elastic; finally, if they admit a finite no vanishing
solution Y k , the structure exhibits an elastic plastic behaviour. In this last case,
any two solutions to the same problem can differ at least by a stressless (i.e.
compatible, corresponding to a mechanism) set of plastic deformations.
Very often, engineering practice structures are subjected to the
contemporaneous action of fixed and cyclic loads. Therefore, let us suppose that
the acting load F = F (t ) , 0 ≤ t ≤ t f , is represented by the combination of a
(
)
reference mechanical fixed load F0 (t ) = F0 and a reference mechanical cyclic
load Fc (t ) of period ∆ t (Fig. 1).
F0
F0
t
Fc
Fc
t
-Fc
Figure 1:
∆t
Fixed and cyclic load history.
In addition, let us assume that the cyclic load identifies with a convex
polygonal shaped loading path with vertices corresponding to an even number b
of mutually independent load vectors, denoted with Fci , ∀ i ∈ I (b ) ≡ {1,2 ,...,b} .
Furthermore, let us assume the hypothesis that the cyclic load is a perfect one,
namely for each basic load condition an opposite one exists in the load space.
Finally, let us introduce the two scalars ξ 0 ≥ 0 and ξ c ≥ 0 , which represent the
fixed and the cyclic load multiplier, respectively, so that ξ 0 F0 and ξ c Fc are the
amplified fixed and cyclic loads.
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
240 Computer Aided Optimum Design in Engineering IX
As it is well known, in the described load condition the response of the
relevant structure follows two subsequent phases: first a transient (short-term)
response and eventually a steady-state (long-term) response. The latter exhibits
the same periodicity features as the cyclic loads and it is independent of the
initial conditions and of the initially chosen special load path. Actually, for each
cycle of the loading history, the steady-state response just depends on the
sequence of the b amplified basic load conditions Fi = ξ 0 F0 + ξ c Fci ,
∀i ∈ I (b ) , obtained as combination of the amplified fixed and perfect cyclic
loads.
As a consequence, the elastic plastic steady-state response of the structure in
the cycle can be obtained by an analysis affected just for the b basic conditions,
i.e.:
Z i = SYi + bi
~
Z i ≥ 0 , Yi ≥ 0 , Yi Z i = 0
∀i ∈ I (b )
(6a)
∀i ∈ I (b )
(6b)
∀i ∈ I (b )
(6c)
where
~~
~
bi = R − N B K −1Fi − N Pi∗
and Yi is the vector of plastic activation intensities related to the i-th basic load
condition.
For the purposes of the present paper it is very useful to consider the steadystate elastic plastic response of the structure subjected just to the amplified
perfect cyclic loads ξ c Fci , ∀i ∈ I (b ) , and separately the elastic response of the
same structure to the amplified fixed loads ξ 0 F0 ; moreover, in order to describe
the shakedown behaviour of the structure it can be useful to determine, on the
Bree diagram, the borderline between the (elastic and plastic) shakedown
domains (zones S+F of the Bree-diagram) and the incremental/instantaneous
collapse regions (zones R+I of the Bree-diagram) of the relevant structure,
solving the following problem
Kuci − Fci = 0
∀i ∈ I (b )
~
∗
Pci = Buci + Pci
∀i ∈ I (b )
a ~
− ϕ ci = R − ξ c N Pci + SYci
∀i ∈ I (b )
~
∀i ∈ I (b )
− ϕ ci ≥ 0 , Yci ≥ 0 , Yci ϕ ci = 0
Ku0 − F0 = 0
~
P0 = Bu0 + P0∗
( ) = (max
ξ
ξ
)
ξ ca
,
ξ 0A
0
Y0
0
subject to
~
− ϕ iS = −ϕ ci − ξ 0 NP0 + SY0 ≥ 0 , Y0 ≥ 0
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
(7a)
(7b)
(7c)
(7d)
(8a)
(8b)
(8c)
∀i ∈ I (b )
(8d)
Computer Aided Optimum Design in Engineering IX
241
for a suitably chosen number of assigned cyclic multiplier values ξ ca , such that
0 ≤ ξ ca < ξ cU , being ξ cU the limit cyclic load multiplier value above which the
structure subjected just to the amplified perfect cyclic load instantaneously
collapses (Fig. 2).
ξ
ξ
ξ
C
C
ξU
U
C
C
I
I
ξa
C
F
R
ξS
ξS
C
C
ξa
C
a)
Figure 2:
R
F
E
S
S
E
ξA ξU
0
0
ξ
0
b)
ξA
0
ξU
0
ξ
0
Bree-like diagrams: a) determination of the elastic shakedown
limit; b) determination of the plastic shakedown limit.
In eqs. (7)-(8), besides the already defined symbols, Pci and uci are the
purely elastic response just to the reference cyclic loads in terms of generalized
stresses evaluated at the strain points and in terms of structure node
displacements, being Pci∗ the generalized stress response vectors evaluated at the
strain points due to the cyclic loads directly acting upon the structural elements,
while ϕ ci and Yci are the analogous of ϕ i and Yi but related to the purely
cyclic load. Furthermore, P0 , u0 and P0∗ are the analogous of Pci , uci and
Pci∗ , but related to the reference fixed load, ϕ iS is the vector of plastic potentials
for the structure at the limit state of (elastic/plastic) shakedown (depending on
value of ξ ca ) and Y0 is a time independent vector of plastic multipliers related
with the selfstress field at the (elastic/plastic) shakedown limit.
If 0 ≤ ξ ca ≤ ξ cS is assumed, being ξ cS the elastic shakedown limit load
multiplier related to the purely cyclic load (Fig. 2a), then eqs. (7) admit the
vanishing solution Yci = 0 , ∀i ∈ I (b ) , and in the steady-state phase the whole
structural behaviour is eventually elastic. In this case the couple of values
ξ 0A ξ ca ,ξ ca , deduced solving problem (8), represents a point of the borderline
between the elastic shakedown domain and the incremental/instantaneous
collapse regions. Otherwise, if ξ cS < ξ ca < ξ cU is assumed (Fig. 2b), then eqs. (7)
[ ( ) ]
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
242 Computer Aided Optimum Design in Engineering IX
admit a non-vanishing solution, Yci , at least for some i ∈ I (b ) , and the structure
eventually exhibits a steady-state elastic plastic behaviour, so that the couple of
values ξ 0A ξ ca ,ξ ca represents a point of the borderline between the plastic
shakedown domain and the incremental/instantaneous collapse regions. In this
last case, the increment of plastic strain in the cycle is nought.
The results obtained by means of the previously described analyses are very
consistent but, unfortunately, they are not complete enough in order to give a
definitive judgement on the structural safety with respect to some ductility and/or
functionality limits. Actually, as already stated, especially during the design
stage of a structure it is necessary to have the possibility of limiting some elastic
plastic response quantities. In particular, if a plastic shakedown behaviour is
expected, then it is appropriate to prescribe suitable limits on the plastic
deformations occurring during the steady-state phase, as well as on the plastic
strains which the structure exhibits during the transient phase.
Making reference to the previously exposed formulations, the constraints on
the plastic deformation occurring during the steady-state phase are easy enough
to be imposed, while an assessment over the amount of the plastic strain related
to the transient phase can be provided by means of suitable bounding techniques
[13].
For the purposes of the present paper a suitable measure of the plastic
deformation, related to the elastic shakedown, will be bounded as indicated
hereafter:
[ ( ) ]
~
1 ~
Yˆ 0 SYˆ 0
2ω
ω>0
ms ≤
(9a)
(9b)
where m s is the quantity to be bounded, ω > 0 the perturbation multiplier and
Ŷ0 a plastic multiplier vector related to the perturbed yield domain.
3
Minimum volume design formulation
Let us consider the structure described in the previous section and let us assume
that the ν -th element geometry is fully described by the s components of the
~ ~ ~
~
~
vector d ν (ν = 1,2 ,..., n ) , so that d = d1 , d 2 , ..., d ν , ..., d n represents the n × s
supervector collecting all the design variables.
We want now to determine the minimum volume design of the structure able
to plastically shakedown for fixed and cyclic loads amplified by ξ 0 and ξ c ,
respectively. In addition, we impose that some suitably chosen measure of the
plastic deformation occurring in the elements at the limit state of plastic
shakedown be not greater than some assigned value m i and that a suitably
computed bound on a chosen measure of the plastic deformations related to the
transient phase response not exceed a given limit m s .
[
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
]
243
Computer Aided Optimum Design in Engineering IX
min V
 d ,u ,u ,Z ,Y ,Yˆ ,ω 

0 ci i i 0 

d −d ≥ 0
Td − t ≥ 0
subjected to:
~
P0 = Bu0 + P0∗ ,
Ku0 − F0 = 0
~
∗
Pci = Buci + Pci ,
Kuci − Fci = 0
∀i ∈ I (b )
~
Z i = R − ξ c N Pci + SYi ∀i ∈ I (b )
~
Z i ≥ 0 , Yi ≥ 0 , Yi Z i = 0 ∀i ∈ I (b )
~
~
ˆ −ξ N
ˆ
ˆ
− ϕˆ i ≡ R − ωR
c Pci − ξ 0 N P0 + SYi + SY0 ≥ 0, Y0 ≥ 0 ∀i ∈ I (b )
M i Yi + m i ≤ 0 ∀i ∈ I (b )
~
1 ~
Yˆ 0 SYˆ 0 − m s ≤ 0
2ω
ω>0
(10a)
(10b)
(10c)
(10d)
(10e)
(10f)
(10g)
(10h)
(10i)
(10j)
(10k)
where, besides the already known symbols, V is the structure volume to be
minimized, eqs. (10b,c) represent some appropriate technological constraints,
ϕˆi
is the perturbed yield domain and R̂ is the perturbation vector. It is worth
noticing that suitably choosing the perturbation vector R̂ it is possible to obtain
bounds on different quantities related to the actual process, while the value of ω
influence the stringentness of the bounds to be computed.
The optimization problem (10) is a strongly non-linear mathematical
programming one and its solution can be pursued by utilizing suitable iterative
techniques.
4
Conclusions
The plastic shakedown minimum volume design of elastic perfectly plastic
structures subjected to a combination of fixed and cyclic loads has been studied.
The search problem has been formulated in order to prevent
incremental/instantaneous collapse with appropriate safety factors and suitable
constraint the plastic deformations.
In particular, some suitable limits on the plastic deformations related to the
steady-state response occurring at the limit state of the plastic shakedown have
been imposed as well as analogous limits related to the transient phase of the
elastic plastic structural response. For this last case a chosen measure of the
plastic deformation related to the relevant transient phase is suitably bounded,
according with a special bounding theorem formulation based on the
perturbation method.
The proposed minimum problem is a strongly non-linear mathematical
programming one and, as a consequence, many efforts must be required in the
computational stage.
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
244 Computer Aided Optimum Design in Engineering IX
In particular, some appropriate iterative procedure can be utilized, here not
discussed for the sake of brevity, very promising in terms of numerical
applicability.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
Cinquini, C., Guerlement, G. & Lamblin, D., Finite element iterative
methods for optimal elastic design of circular plates. Computers &
Structures, 12(1), pp. 85-92, 1980.
König, J.A., On optimum shakedown design. Optimization in structure
design. Edited Sawczuk A. & Mroz Z., Springer-Verlag: Berlin, pp. 405414, 1975.
Giambanco, F., Palizzolo, L. & Polizzotto, C., Optimal shakedown design
of circular plates. Journal of Engineering Mechanics, ASCE, 120(12), pp.
2535-2555, 1994.
Giambanco, F., Palizzolo, L. & Cirone, L., Computational methods for
optimal shakedown design of FE structures. Structural Optimization,
15(3/4), pp. 284-295, 1998.
Rozvany, G.I.N., Optimal design of flexural systems, Pergamon Press,
Oxford, England, 1976.
Save, M. & Prager, W., Structural Optimization, Plenum Press, New
York, U.S, 1985.
Palizzolo, L., Optimization of continuous elastic perfectly plastic beams.
Computers & Structures, 82(4-5), pp. 397-411, 2004.
Caffarelli, A., Giambanco, F. & Palizzolo, L., Optimal plastic shakedown
design of FE structures. VII Int. Conf. on Computational Plasticity,
Barcelona, Spain, April 7-10, 2003.
Giambanco, F., Palizzolo, L. & Caffarelli, A., Computational procedures
for plastic shakedown design of structures. Journal of Structural and
Multidisciplinary Optimization, 27, pp. 1-13, 2004.
Polizzotto, C., Shakedown analysis and design in presence of limited
ductility behaviour. Engineering Structures, (6), pp. 80-87, 1984.
Kaneko, I. & Maier, G., Optimum design of plastic structures under
displacement constraints. Computational Methods in Applied Mechanics
Engineering, (27), pp. 369-391, 1981.
Benfratello, S., Cirone, L. & Giambanco, F., An optimal design of FE
structures with limited ductility. The International Conference of
Computational Methods, Singapore, 2004.
Giambanco, F., Palizzolo, L. & Caffarelli, A., Bounds on Plastic
Deformations for Structures in Plastic Shakedown, in Computational
Mechanics, WCCM VI, Beijing, China, 2004.
WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)