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Nonlinear analysis: main problems and solution
methodologies
Raffaele Casciaro
Universit`
a della Calabria, Italy
Abstract
The paper describes different computational approaches and solution methodologies that
can be used in nonlinear structural analysis, in particular, the so called path–following anal-
ysis, the linearized stability analysis, the asymptotic analysis, the imperfection sensitivity
analysis and the transient dynamic analysis, for each showing the main problems, peculiar
aspects, possible failures and computational convenience. Nonlinear solutions are, by their
nature, sensitive to small variations in data, so a performance–based analysis must include
an extensive investigation, which takes into account all possible loading imperfections and
geometrical defects. Great care has to be taken to assure the reliability of the results and, if
possible, any analysis should be repeated using an alternative approach.
1 Introduction
The main qualitative difference between linear and nonlinear structural analysis lies in the loss
of the linear superposition principle. In linear analysis the structural response to a combination
of several actions is easily obtained by the sum of the their separate responses. This provides
an obvious great simplification in managing complex load combinations, but also introduces a
strong qualitative implication. In fact it states that the effects of small load perturbations are also
small, so we can consider the small differences in the loading data, due to small load fluctuations,
geometry imperfections or material defects, between the real structure and its idealized description
used in the analysis, as negligible. This is not true in nonlinear analysis where even small data
perturbations can produce significant variations in the response and produce qualitatively different
solutions. This is a crucial point, because our knowledge of the structural problem is essentially
imprecise: loads applied to the structure during its life cycle can be described only in terms of
its probability of occurrence; geometry or material imperfections can only be defined in terms of
prescribed tolerance.
Analyses based on average values are generally insufficient to obtain a reliable evaluation of
the structural safety. We then have to perform a separate analysis for each different possible load
case, and each possible imperfection. This is a lot of work which is however viable due to the
increasing power of computational hardware and, even more, due to the advances in the solution
methodologies.
In the following we will briefly describe these methodologies, trying to focus on the main
aspects of each different approach. For getting conciseness, the discussion will be limited to a
general overview and specific problems related to constitutive nonlinearities, as plasticity, fracture
or damage growing, are not treated here.
1
2 The finite element method
The discretization/solution techniques which form the so called Finite Element Method (FEM)
represent the basis of the computational approach to structural analysis. Its origins can be related
to the energy approach proposed by Castigliano [1] at the end of 19th century and to the works
by Ritz [2] and Galerkin [3] on approximate solutions (also sees [4]), in the first decade of the
20th century, and to the research into numerical solution techniques produced in the third decade
[5]–[7]. In its modern form, the method was initially developed during the Second World War,
while being considered of strategic importance and covered in military or industrial secrecy. Its
importance, as a fundamental link between continuum mechanics and numerical analysis, able
to exploit the developing digital computers technology [8]–[16], was clearly recognized after the
first published papers on the subject in the 50’s [17]–[19] (the Wright–Patterson Conference [24]
where Argyris presented is famous paper ”Continua and Discontinua” [25], could be considered
the turning point). The successive diffusion of the method as a basic tool for numerical analysis
was aided by the constant improvement and wide diffusion the computers [20]–[50]. Its continuous
evolution, which involved theoretical mechanics, mathematical founding, numerical analysis and
software technology, is too wide to be even summarized here; the reader can refer to [51]–[54] for
an essential historical review.
The general idea of the method is to refer to the potential energy
Π[u] := Φ[u]−pu (1)
ubeing the configuration of the structure (including both the displacement and stress fields) and p
the assigned external action (including both prescribed forces and distortions), which is written as
a combination of the internal strain energy Φ[u]and the external bilinear work pu. By partitioning
the structure into small elements and using appropriate shape functions, the energy Π[u, p]can be
reduced to an algebraic form so that equilibrium, expressed by its stationarity with respect to u
Φ![u]δu−pδu=0 (2)
where a prime (’) indicates Fr´
ech`
et differentiation and δuany admissible virtual change in config-
uration, could be written as a nonlinear n–dimensional vectorial equation
s[u]−p=0 (3)
vectors sand p, expressing the internal response of the structure and the external action, respec-
tively, are defined by the energy equivalences
s[u]Tδu:= Φ![u]δu , pTδu:= pδu(4)
δuand δubeing any virtual change in configuration and its discrete counterpart, respectively.
Different approaches can be taken in this discretization process, depending on the particular finite
element technology which is used (compatible, mixed or hybrid elements, X-FEM elements, PU
and meshless discretization, etc.). However, using a coherent formulation, the accuracy can always
be improved by refining the discretization mesh.
In linear analysis, Φ[u]is assumed to be quadratic in uso the response s[u]reduces to a linear
function
s[u] := Ku (5)
being related to uthrough a symmetric positive stiffness matrix K, defined by the energy equiva-
lence with the second variation of Φ[u]
δuTKδu:= Φ!!δu2(6)
2
so the equilibrium problem (3) is reduced the to the solution of a symmetric linear system
Ku =p(7)
which can easily be obtained through the Cholesky decomposition algorithm which produces a
computational convenient expression for K−1where both the sparseness and the banded structure
of matrix Kare fully exploited.
In the nonlinear case, function s[u]can be much more complicated. However, for each current
configuration u, we can define the so called tangent stiffness matrix Kt[u]through the energy
equivalence
δuTKt[u]δu:= Φ[u]!!δu2(8)
This matrix, which provides complete information of the second–order energy neighbor of uand
can be used to relate small increments ˙
uof uto the corresponding increments ˙
sof s
˙
s=Kt[u]˙
u(9)
play an important role in the analysis. In fact, equilibrium problems can be solved through the
well known Newton’s iteration scheme (e.g. see [55, 56])
uj+1 := uj−Kt[uj]−1rj,rj:= s[uj]−p,j=0,1,2. . . (10)
rjbeing the equilibrium residual corresponding to the current evaluation ujand uj+1 its improved
value. If started sufficiently near the required solution, the scheme converges, quite quickly, to an
evaluation uwhich zeroes the residual (within the prescribed tolerance) and so can be taken as
solution for the equilibrium problem.
Figure 1: Newton (a) and Modified Newton (b) iterations.
Newton scheme (10) requires the updating of matrix K−1
tat each iteration loop. This oper-
ation implies the assemblage and Cholesky decomposition of matrix Kt[uj]and usually corre-
sponds the most expensive part of the loop (roughly speaking, it costs m/4times than the eval-
uation and decomposition of the residual rj,mbeing the half–band width of the matrix, which
usually is of some hundreds). So scheme (10) is usually implemented in the modified form (see
fig.1)
uj+1 := uj−˜
K−1rj(11)
3
where ˜
Kis an approximate substitute for Kt[uj]which is taken constant during the the iteration
process. This variant allows the continuous updating of Kt[uj]−1be avoided, so it is much more
convenient from a computational point of view. Note that the convergence of the modified scheme
is related to the relative differences between ˜
Kand Kt[uj]and, while usually slower than its
original version, is always assured if the following conditions hold
0<Kt[uj]<2˜
K,j=0,1,2. . . (12)
Divergence can occur (and generally occurs) when the tangent matrix Ktloses positive defini-
tiveness (this happens in non stable equilibrium solutions) or when matrix 2˜
K−Ktis no longer
positive definite, that is, when we underestimate by a factor of more than 2the actual stiffness of
the structure in evaluating the iteration matrix ˜
K.
3 Path–following analysis
The basic idea of this analysis approach is to recover the equilibrium path u[λ]consequent to
an assigned external loading p[λ]by determining a sequence of equilibrium points {u(k),λ(k)}
sufficiently near to allow the equilibrium curve, implicitly defined by the condition
s[u[λ]] −p[λ] = 0 (13)
to be obtained by interpolation. Usually, and we refer to this in the following, we consider propor-
tional loadings
p[λ] := p0+λˆ
p(14)
so the load evolution parameter λcan be interpreted as safety multiplier for the nominal load
condition ˆ
p.
Different step–by–step strategies can be followed for characterizing the sequence of the equi-
librium points, the simplest being that of directly assigning load increments ∆λ(k)to the load
multiplier by making λ(k+1) =λ(k)+∆λ(k). The corresponding solution u(k+1) := u[λ(k+1)]is
then obtained by constructing an initial estimate u1through a direct extrapolation of the previous
step
u1:= u(k)+β(k)(u(k)−u(k−1)),λ(k+1) := λ(k)+β(k)(λ(k)−λ(k−1))(15)
βbeing a step size factor, aimed to enlarge (β(k)>1) or reduce (β(k)<1) the step size ∆λ(k)=
β(k)∆λ(k−1) according to the requirements of the analysis, and then performing a number of
modified Newton iterations (11) sufficient to reduce the residual within the required tolerance.
Usually the iteration matrix ˜
Kis updated at each step of the analysis, by using the tangent matrix
at the initial point ( ˜
K:= Kt[u(k)]) or, and this could be better, at the first evaluation ( ˜
K:=
Kt[u1]). A restart option is also generally added that, in case of convergence problems, stops the
iteration process, discards the results and repeats the step with a smaller size (β≈0.1÷0.2) in
order to mantain ˜
Knear to K[uj].
While simple, this solution strategy is also quite efficient and robust and has been widely used
in the past. It performs very well in the initial phase of the loading. However, due to its nature,
it tends to fail near limit points of the equilibrium path which are, by definition, characterized by
the singularity of tangent matrix Ktand so correspond to a loss in convergence conditions (12).
Generally we are very interested in the collapse safety of the structure, that is in estimating the
limit load, so this represents a significant defect in the strategy.
For a long time this drawback was considered an intrinsic aspect of nonlinear analysis. How-
ever, at the end of the 70’s it was recognized by Riks [57] that the drawback is actually simple
4
consequence of a loading control based on fixed increments of the load. Function u[λ]is no
longer analytical near a limit point, so it is not surprising that there are problems in these zones of
the path when describing the sequence of u(k)through increments of λ. The inconvenience can
however easily be avoided if referring to an appropriate analytical parametrization {u[ξ],λ[ξ]},ξ
being an arc–length curvilinear abscissa describing the path in the {u,λ}space, by characterizing
the {u(k),λ(k)}sequence through fixed step sizes ∆ξ(k)in spite of ∆λ(k). More precisely, and
assuming the step size be defined by
∆ξ2:= ∆uTM∆u+µ∆λ2= (∆ξ(k))2(16)
Mand µbeing an appropriate metric matrix (symmetric and positive definite) and a factor (pos-
itive), the (k)th point is defined as the solution of the nonlinear system obtained by combining
equilibrium equation (3) with the arc–length condition (16). It results in n+1nonlinear equations
in the n+1unknowns u(k+1) and λ(k+1) which can be solved through the Newton scheme:
uj+1 := uj+˙
uj,λj+1 := λj+˙
λj(17a)
where the iterative corrections ˙
ujand ˙
λjare obtained as solutions of the linear system
Jj!˙
uj
˙
λj"=!rj
gj",Jj:= #Kt[uj]−ˆ
p
∆uT
jMµ∆λj$(17b)
the increments ∆ujand ∆λjbeing defined by
∆uj:= uj−u(k),∆λj:= λj−λ(k)(17c)
and rjand gjbeing the current residuals of the equilibrium and the arc–length equations:
rj:= s[uj]−λjˆ
p,g
j:= ∆uT
jM∆uj+µ∆λ2−(∆ξ(k))2(17d)
The main feature of the Riks iteration scheme (17) is that the Jacobian matrix Jjis generally
nonsingular, even if its minor Ktis singular (possible singularities are only related to the presence
of path bifurcations), and condition Kt>0is no longer needed for convergence. Actually, the
arc–length strategy has a convergence behavior much better than that of the load controlled strategy
and allows not only limit points be simply surpassed but also the descent parts of the equilibrium
path to be followed.
Some further computational improvements are possible. We are not actually interested in a
full accuracy in the step length condition. So, providing that the initial step length is provided by
the first extrapolation (15), we can generally assume gj≈0during the iteration. Also, we can
exploit the Modified Newton strategy, using a constant iteration matrix ˜
Kin spite of Kt[uj], to
avoid the continuous updating of K−1
tand so obtain a much faster solution process (see fig. 2).
Finally, the system can be conveniently solved by partitioning. We obtain:
˙
λj:= rT
jdj
ˆ
pTdj+µ∆λj
,˙
uj:= ˜
K−1(rj−˙
λjˆ
p)(18a)
vectors rjand djbeing defined by
rj:= s[uj]−λjˆ
p,dj:= ˜
K−1M∆uj(18b)
Different variants of the solution scheme are also possible to improve its computational speed. For
instance considering that
∆uj≈∆λj˜
K−1ˆ
p(19)
5
Figure 2: Riks arc–length iteration scheme.
the scheme could be simplified into
˙
λj:= uT
rMˆ
u
ˆ
uTMˆ
u+µ,˙
uj:= ur+˙
λjˆ
u(20a)
where
ˆ
u:= ˜
K−1ˆ
p,ur:= ˜
K−1rj(20b)
In this version, which does not modify the convergence behavior of the scheme, each iteration loop
only requires a Choleski solution for vector ur, apart from the quite inexpensive scalar product
involved in the evaluation of ˙
λj, so it requires the same computational burden per loop as the
load–controlled scheme.
The convenience of the arc–length strategy was immediately recognized and it rapidly became
widely popular within numerical researchers. A large number of papers have appeared [58]–
[81] proposing further variants in the assumptions for the step–size constraint (16) and in minor
algorithmic detail which regulate the update of the iteration matrix, the restart option and the
step–size (to relate β(k)to the number of loops performed in the previous step is, in this case, the
simplest choice). The reader can refer to [84]–[88] for further details.
We have to note that the ability of the arch–length iteration to overcome the occurrence of
singularities in the tangent stiffness matrix does not avoid convergence failures related to the right
part of the convergence condition (12), this occurs when, at least in one possible incremental
displacement ˙
uthe iteration matrix ˜
Kunderestimate the actual stiffness K[uj]by a factor more
than 2, that is when
˙
uTK[uj]˙
u>2˙
uT˜
K˙
u(21)
for some increments ˙
u. As shown in [89], the occurrence of this failure condition is more frequent
than one might think and is related to a nonlinear extrapolation locking phenomenon which is
produced by a perverse interaction between strong stiffness ratios in the structural elements and
even small changes in their orientation. Slender structures, where axial stiffness is greater than
the flexural one by several orders, are particularly sensitive to this phenomenon, this is further
6
emphasized when the structure has some directions of global lability, as usually occurs near a
limit point of compressed structures or in the initial loading phase of structures which derive their
rigidity from tensional stresses, as happens in the case of suspended bridges.
The relevance of this locking phenomenon is directly related to the length of the incremental
steps and to the nonlinearity that the problem assumes in its description variables. However, as
also shown in [89], the locking is related more to the nonlinear description of structural response
than to intrinsic characteristics of the response. Compatible finite element discretization, where
the element behavior is described only in terms of nodal displacements, are usually very sensitive
to this phenomenon and can be subjected to such a pathological locking that requires such small
extrapolation steps that practically the path reconstruction process is prevented. These difficulties
however disappear when using a description based on a mixed format, in terms of both nodal
displacements and internal stress variables, and this can be actually done through minor changes
in coding which produce no computational extra cost. So, a robust mixed (stress–displacement)
variant of the arc–length iteration algorithm can be obtained which is free of locking problems.
Path–following analysis based on the arc–length iteration scheme is now widely popular within
structural analysis and can be considered the main analysis tool for structural nonlinear analysis.
It allows an accurate evaluation of the behavior of structures subjected to an assigned loading
process. However it still has some intrinsic defects. First of all, it is unable to treat problems
characterized by path bifurcations. In this case, the analysis should recognize the presence of
bifurcation points when solving for each step and then follow each branch emerging from this
point to obtain a complete answer for the overall structural behavior. Some algorithmic tricks have
been proposed for recognizing the occurrence of bifurcations and forcing the path evolution along
a selected branch [90]–[95], but none can be considered sufficiently robust to be actually reliable as
a general procedure or suitable for a complete investigation of all branches. Secondly, it provides
the response for a single loading case. A safety analysis of the structure, which is our principal
aim, should consider all possible loadings including the deviations due to load imperfections and
geometrical defects. As the single analysis is computationally quite expensive (it implies several
steps and several iteration loops in each step), to perform a complete investigation that considers all
possible imperfection shapes would be prohibitive from a computational point of view. Perhaps,
the best use of the path–following approach is that of performing a few analyses according to the
most restrictive conditions, once the worst imperfections have already been determined.
4 Linearized stability analysis
The linearized stability analysis corresponds to the implementation for complex structures of the
method originally proposed by Euler in its theory of Elastica [96, 97], that is the well known Euler
beam problem. The basic motivation for this approach is that Liapunov’s stability (see [98]) of
an equilibrium solution is strictly related to the local convexity of the potential energy and so, if
this is written in the form (1), to the local convexity of the strain energy function Φ[u]. Positive
definiteness of the second variation of the strain energy (Φ!!δu2>0,∀δu) also assures its local
convexity, so it implies stability. Conversely, the occurrence of Φ!!v2<0, for some incremental
change of the configuration v, implies instability and it is associated to a buckling collapse of the
structure. So, when considering a loading path u[λ]and remembering equivalence (8), we could
relate buckling safety to the first value of λsuch that
K[u[λ]]v=0,v$=0(22)
The numerical implementation of this criterion is quite easy when the structure does not present
noticeable rotations along the path u[λ], as happens in the Euler beam or, more generally, in purely
7
compressed structures. This is a special but frequent case for investigation.
The tangent matrix Ktdepends on the current geometry of the structure, and linearly on the
elastic factors and internal stresses. So, for fixed geometry, it can be expressed by the sum of two
contributions:
Kt=K0+λK1(23)
the first K0corresponding to the usual stiffness matrix in linear elasticity and the second λK1
accounting for the effects of the internal stresses; for fixed geometry, the stresses are linear func-
tions of λ, so the latter contribution can be explicitly taken as linear in λ. The stability check (22)
can therefore be performed through the solution of the linear eigenvalue problem
(K0+λK1)v=0,v$=0(24)
We are really interested in its principal solution, characterized by the smallest positive value of λ.
From a computational point of view, this is a simple problem and several fast and robust solution
algorithms are available for that purpose (e.g. see [99]). We can cite the well known Inverse
Power Method, the so called Subspace Iteration Method and the Lanczos Method [100, 101].
Recent variants of the latter, such as the Implicit Restarting algorithm by Sorensen [102], are
particularly efficient(see also [103]). All these methods are based on an iterative scheme which
provides successive evaluations {vj.λj},j=1,2. . . , rapidly converging to the required solution,
and make use of the inverse K−1
0of the initial stiffness matrix whose computation is generally
an important part of the total computational burden. However, the stability analysis is performed
immediately after a linear elastic analysis which uses the same matrix, whose inverse K−1
0is
already available in a Choleski factorization form from the previous analysis. This makes the
solution process computationally fast.
Linearized stability analysis is computationally fast and robust. It is also particularly rich in
information. In fact it provides not only the buckling multiplier λb(as the first eigenvalue of
problem (24)) but also the buckling mode vb(as associated eigenvector). However this informa-
tion, obtained through a drastic linearization of the equation of the problem, can be unreliable
in describing the real behavior of the structure, which can be greatly influenced by the neglected
nonlinear terms and by the data deviations deriving from the presence of even small imperfections.
As show by the classical experimental tests of Donnell on compressed cylinders [104], the actual
collapse load can be noticeably smaller (50% or more) than the theoretical Eulerian load.
It is worth mentioning that this aspect has for a long time been greatly underestimated, mainly
due to a misunderstanding of the real meaning of Eulerian critical condition (22) which were
viewed as the possibility of having a small deviation vbfrom the original position u[λb]associated
to the same load λbˆ
p. This interpretation derives from the observation that, when referring to a
Taylor expansion of equilibrium equation
%Φ![u0] + Φ!![u0]∆u+1
2Φ!!![u0]∆u2+··· &δu−(p0+∆p)δu=0 (25)
starting from an equilibrium point {u0,p
0}, by taking into account the initial equilibrium and
neglecting higher–order terms in ∆u, because of an assumption of small displacements, we get a
linear equation relating the displacement increment ∆uto the load increment ∆p:
Φ!![u0]∆uδu=∆pδu(26)
In matrix form this is written
K[u0]∆u=∆p(27)
so the interpretation cames directly by a comparison with condition (22). It is however false, as
firstly point out by von Karman and Tsien through a counterexample [105] in 1941. In fact, we
8
fundamental path
bifurcated paths
bifurcation point
fundamental path
bifurcated path
secondary bifurcation
Figure 3: interactive buckling for coincident (a) or nearly coincident (b) buckling loads.
cannot neglect the second–order term 1
2Φ!!![u0]∆u2δuas much smaller than Φ!! [u0]∆uδuif the
latter tends to zero as a consequence of condition (22). So the linearized equation (27) does not
make sense as relation between ∆uand ∆p.
A correct general nonlinear theory of elastic stability has been developed by Koiter, starting
from his famous doctoral thesis [106], in the early 40’s (the reader can refer to the review by
Budiansky [107] for a clear introduction to the theory). Koiter showed that the collapse of a
nonlinear structure subjected to instability phenomena should be investigated within bifurcation
theory, and its treatment requires not only the evaluation of the buckling load and mode but also
that of the initial slope and curvature of the bifurcated branch. Using these, a simple formula can
be used to take into account the effect of imperfections. Successive research also showed that,
when a cluster of solutions exists for equation (22) with nearly coincident values of λ(this usually
comes about as a direct consequence of the optimization of the structural design, see [109]–[110]),
a very complex interactive multimodal buckling can be produced (see fig. 3) characterized by a
strong sensitivity to imperfections [111]–[115]. All this complexity is completely lost in linearized
stability analysis.
5 Asymptotic analysis
Asymptotic analysis originates as an attempt to implement Koiter’s theory of nonlinear stability
within a finite element numerical approach. It aims to recover an asymptotic evaluation of the
equilibrium path {u, λ}implicitly defined by equation (2) by referring to a related perfect problem
characterized by bifurcation buckling. Several implementations have been proposed [116]–[161],
with minor algorithmic differences. We refer here to that proposed in [129, 142, 156], see [162]
for a general review of the topic.
When mbuckling modes occur for a value of λi,i=1· ··m, belonging to a neighborhood of
the reference buckling load multiplier λb, the equilibrium path can be recovered in the form
u[λ,ξ1,...,ξm]=λˆ
u+
m
'
i=1
ξi˙
vi+1
2
m
'
i,j=1
ξiξj¨
wij +··· (28a)
where λˆ
uis the fundamental path, representing the basic linearized or perfect solution for the
problem, ˙
vi,i=1, . . . , m, are the mlinear buckling modes and ¨
wij ,i, j =1, . . . , m, are the
9
m×mquadratic corrective modes. These quantities are characterized as follows:
1. The fundamental mode ˆ
uis defined by the solution of the linear problem
K0ˆ
u=ˆ
p(28b)
K0:= Kt[0] being the tangente stiffness matrix evaluated at the initial configuration u=
0.
2. The buckling modes ˙
viare the first msolutions of the eigenvalue problem
K[λ]v=0(28c)
where K[λ] := Kt[λˆ
u]is the tangent stiffness matrix evaluated along the fundamental path
u:= λˆ
u. Note that the assumption |λi−λb|&λbimplies buckling modes be orthogonal
each other according to the condition
Φ!!!
bˆu˙vi˙vj=−δij (28d)
where Φ!!
b:= Φ!![λbˆu]and δij is the Kronecker symbol.
3. The corrective modes ¨
wij ∈Ware determined, from buckling modes ˙
vi, as a solution of
the linear problems
Kb¨
wij =¨
s[vi,vj](28e)
where the known vectors ¨
s[vi,vj], in the right term, are defined by the energy equivalence
¨
s[vi,vj]Tδw:= Φ!!!
b˙vi˙vjδw(28f)
and W:= {w:Φ!!!
b˙viˆuw =0,i=1, . . . , m}is the subspace orthogonal to the buckling
modes vi.
Expanding equation (2) until the fourth order and assuming δu=˙vk, we obtain the following set
of algebraic nonlinear equations:
−µk[λ]−(λ−λk)ξk+1
2
m
'
i,j=1
ξiξjAijk +1
6
m
'
i,j,h=1
ξiξjξhBijhk =0,k=1, . . . , m (28g)
where
Aijk := Φ!!!
b˙vi˙vj˙vk,Bijhk := Φ!!!!
b˙vi˙vj˙vh˙vk−Φ!!
b(¨wij ¨whk +¨wih ¨wjk +¨wik ¨wjh)(28h)
represent the cubic and quartic terms in the strain energy expansion and
µk[λ] := µi
k[λ]+µg
k[λ]+µl
k[λ](28i)
the imperfection factors taking into account nonlinear terms implicitly neglected by the assumed
linearization for the fundamental path
µi
k[λ] := −1
2λ2Φ!!!
bˆuˆu˙vk(28j)
and those due to the presence of additional geometric and load imperfections. Denoting with ˜uthe
initial imperfect geometry and with λ˜pthe imperfection load, the latter can be expressed as
µg
k[λ] := −λΦ!!!
bˆu˜u˙vk,µ
l
k[λ] := λ˜p˙vk(28k)
10
Structures denoted by µk[λ] = 0 ∀kexhibit bifurcation buckling and are called perfect. Note
that when the expansion coefficients of the reference structure, with µg
k=µl
k=0∀k, have been
determined, the effects of additional imperfections can be easily taken into account by computing
the corresponding imperfection factors through (28k) and solving the algebraic equation (28g)
once more.
In spite of its apparent complexity the overall solution process (28) can be implemented in
a fast computational scheme. In fact, the fundamental mode ˆ
u, implicitly defined by eq.(28b)
is simply obtained through a linear elastic solution. This obviously requires the assemblage and
Choleski decomposition of matrix K0and will correspond to the most expensive part of the anal-
ysis.
Buckling modes ˙
vi, implicitly defined by eq. (28c), are easily provided by standard eigenvalue
algorithms, if the linearization (24) is assumed. However, the error introduced by this approxima-
tion can be unacceptable, particularly when considering that the eigensolutions {˙
vi,λi}are used
again in the computation of derived quantities through eqs. (28e)–(28h). By avoiding linearization,
eq. (28c) will result in a nonlinear eigenvalue problem which can be solved by a simple variant of
the Inverse Power Method which uses, as a basic scheme, the following iterative sequence
(vj+1 =vj−˜
K−1K[λj]vj
λj+1 =1/||vj+1|| (29)
˜
Kbeing a suitable approximation for K0(see [156] for the implementation details). The scheme
can be viewed as a convenient rewriting of the Inverse Power Method: it does not need the explicit
assemblage of matrix K1, but only that of the incremental response vector K[λj]vjwhich can
be computed by standard assemblage of the element contributions; furthermore, it proves to be
insensitive to the numerical errors introduced in evaluation of matrix K−1
0by the Choleski de-
composition process; in any case, its convergence implies that an exact solution of K[λ]v=0
has been reached. The scheme can be used also in a nonlinear context where it maintains the
fast convergence and a robust behavior proper of the Inverse Power Method, while accepting even
rough approximation in the iteration matrix ˜
K(refer to [163] for the convergence properties of
the scheme and a discussion of possible alternatives [164]–[167]).
Corrective modes ¨
wij are obtained as solutions for the linear equation
Kb¨
wij =¨
s[vi,vj](30a)
where the vector ¨
s[vi,vj]in the right term, defined by the equivalence
¨
sij δu:= Kb¨
w[vi,vj] = Φ!!!
b˙vi˙vjδu(30b)
is obtained, as vector ˙
s[v]by a standard assemblage of element contributions. This solution is
conveniently obtained through an iterative scheme using K0as iteration matrix (see [156] for
computational details).
The scalar coefficients Aijk ,Bijhk ,µi
k,µg
kand µl
kappearing in the algebraic equation (28g)
and corresponding to integrals of known functions, are finally obtained by simply summing up the
corresponding element contributions.
The nonlinear algebraic system (28g) provides a synthetic representation of the original prob-
lem (2) in terms of modal amplitudes ξi. Its dimension m, corresponding to the number of buckling
modes taken into account, is much smaller than the original one, while being highly nonlinear (ac-
tually it maintains all the nonlinear parts of the original system). It can be solved using standard
path–following methods, taking advantage of the small dimensions, and allows the accurate re-
covery of the complete post–buckling behaviour of the structure, including modal interactions and
modal jumping phenomena (see fig. 4, 5).
11
Figure 4: Thin–walled beam, geometry and loads.
w w
l
Figure 5: Results for T-Beam (6 buckling modes interaction).
We can notice analogies between the asymptotic analysis and the so called modal reduction
methods (see [169]–[171]) which are based on the expansion of the displacement field in terms
of a combination of significant modes, and on the solution of the reduced system obtained by a
Galerkin approximation of the equilibrium equations. The use of eq. (28g) has however some basic
advantages: i) the significant modes are directly provided by the method; ii) the solution of the
algebraic system in λand ξi, whose coefficients are computed once and for all, is computationally
much cheaper; iii) it uses coherent asymptotic expansions in spite of a direct reduction of the
finite equilibrium equations; the latter choice, also because of the omission of the orthogonal
corrections wij , introduces a spurious locking in the solution; iv) finally, the solution for different
imperfections only needs the solution of the same algebraic system for different µk, so quite
a negligible computational burden. These features render the asymptotic analysis much more
convenient in both computational efficiency and accuracy, especially when the investigation has to
be extended to a large number of imperfection shapes and amplitudes.
Asymptotic analysis can actually provide accurate and reliable results, as shown through both
theoretical proofs [173] (se also [172]) and numerical results [?]. It has however some potential
12
problems, apart from the obvious deterioration in accuracy in the recovering large displacement
solutions due to the use of an asymptotic expansion (generally, this is not a real problem because
we are only interested in the initial post–buckling path that implies only moderate displacements).
The difficulties essentially derive from the fact that the analysis, making use of a fourth–order
expansion of the energy, needs both the structural modeling and the finite element discretization
procedures to be also, at least, fourth–order accurate.
This requirement is not easy to satisfy. Available nonlinear modelings for structural elements
such as beams and plates are generally only aimed at providing a correct recover of the first and
second variations of the energy, which is the only information needed for deriving the response
vector s[u]and tangent stiffness matrix Kt[u], but provides an inappropriate description for third–
and fourth–order variations which are also needed by the asymptotic analysis. This introduces
kinematical incoherence in the overall modeling and can lead to noticeable errors in the results
(see [162] for a general discussion of the topic).
Care also has to be taken in the finite element discretization. Note in fact that the fourth–
order coefficients Bijhk defined by equation (28h) are obtained from a small difference between
two terms, the first depending on the buckling modes ˙
viand the second on the corrective modes
¨
wij . In slender structures, buckling modes essentially correspond to out-of-plane displacements
(i.e. transversal displacements) whereas the correction wessentially develops as plane (axial)
displacements. So an independent discretization of the axial and transversal components of the
displacement can produce large errors due to the different approximation for these displacements.
If this phenomenon is not avoided (by an appropriate combined field discretization) it will intro-
duce a spurious nonlinear interpolation locking in the solution. (see [174, 176, 177]).
Finally, an appropriate description format has to be used in order to avoid the nonlinear extrap-
olation locking phenomenon, we already cited as a possible problem in path–following analysis.
Asymptotic analysis is particularly sensitive to this locking, much more than the path–following
analysis, and great care has to be taken in an appropriate choice of the extrapolation variables.
Once more, the use of a mixed format (in both displacement and stresses) can be the right choice
[178]–[180].
6 Imperfection sensitivity analysis
Structures presenting coupled buckling, are generally very sensitive to imperfections, so that even
a small imperfection in loading or geometry can mean a marked reduction in collapse load with
respect to the theoretical bifurcation load. So an effective safety analysis should include an inves-
tigation of all possible imperfection shapes and sizes to recover (albeit in a statistical sense) the
worst case.
The asymptotic approach provides a powerful tool for performing this extensive investigation.
In fact, the analysis for a different imperfection only needs to update the imperfection factors
µg
k[λ]and µl
k[λ]through the simple formulas (28k) and solve once more the nonlinear system
(28g). Even if this system, collecting all the nonlinear parts of the original problem, proves to be
strongly nonlinear and some care has to be taken in treating the occurrence of multiple singulari-
ties, its solution through a path–following process is relatively easy because of the small number
of unknowns involved.
We do not have an explicit relation between imperfections and the corresponding reduction in
the limit load, so an effective imperfection sensitivity analysis can only be performed by a Monte–
Carlo statistical technique, where both the magnitude and the form of the imperfections are treated
as random variables. The analysis is then performed by taking the additional imperfection factors
13
in the form
µl
k[λ]+µg
k[λ]=λ)˜p˙vk−Φ!!!
bˆu˜u˙vk*:= λ¯µk,(31)
and producing a random sequence of imperfection vectors ¯
µ={¯µ1,¯µ2·· · ¯µm}, modeling pos-
sible small deviations in the loads and in the geometry of the structure, and repeating a path–
following solution of (28g) for each of these. By a statistical treatment of the obtained results we
obtain the probability distribution function for the limit load multiplier and all the other useful sta-
tistical information. This solution process, we call it full analysis, can be considered as a standard
approach for imperfection sensitivity analysis (see [156]).
Figure 6: Geodetic dome - 20 buckling modes interaction.
A
Figure 7: Geodetic dome - presence of 3 attractive paths.
A large number of different imperfections (up to several thousands) has to be considered to
obtain statistically significant results, so, while the analysis for a single imperfection can be con-
sidered an easy task, the entire solution process performed proves to be computationally expensive,
especially when a large number of coupled buckling modes have to be considered. We can, how-
ever, noticeably reduce the computational effort by exploiting information given by the knowledge
of the complete set of attractive radial paths
ξi=ξ∗
iti=1···m(32)
which are local minimizers for the cubic form
˙
λb:= 1
2
m
'
i,j,h=1 Aijh ξ∗
iξ∗
iξ∗
h=staz. ,
m
'
i=1
ξ∗
iξ∗
i=1 (33)
14
60 64 68 72 76 80
λ
lim
/
λ
b
%
0
4
8
12
fr
[
λ
lim
/
λ
b
]
%
SA
FA
Fmax
=
60 Kg
γ
max
=
0.1
%
λ
b
=
8.5
60 64 68 72 76 80
λ
lim
/
λ
b
%
0
0.2
0.4
0.6
0.8
1
F
[
λ
lim
/
λ
b
]
P
r
o
b
a
b
i
l
i
t
y
d
i
s
t
r
i
b
u
t
i
o
n
f
u
n
c
t
i
o
n
Probability cumulated function
Figure 8: Geodetic dome - statistical distribution for the collapse load multiplier - from Full and
Simplified analysis.
or for the quartic form
¨
λb:= 1
3
m
'
i,j=1 Bij hkξ∗
iξ∗
jξ∗
hξ∗
k=staz. ,
m
'
i=1
ξ∗
iξ∗
i=1 (34)
Attractive paths theory [181]–[185] can actually provide a helpful tool for driving the analysis
and reducing its total cost. In fact, it suggests that each imperfect path obtained from the solution
of (28g) will be attracted by one of the minimizing radial directions ξ∗(see figs. 6, 7). Then,
an evaluation for the limit load associated to the single imperfection vector ¯
µcan be obtained
by performing a series of different monomodal analyses, one for each minimum radial path (32),
and then taking the smallest value obtained for the limit load within all directions. The single
monomodal analysis is quite quick, so a large number of different imperfections can be investi-
gated rapidly with results, in terms of limit load distribution, equivalent to that provided by full
analysis (see fig. 8). See [186]–[188] for more details on this simplified approach to imperfection
sensitivity analysis.
Furthermore, it is worth mentioning that, once the worst imperfection shapes have already
been obtained from an imperfection sensitivity analysis, a detailed investigation can be performed
through a specialized path–following analysis, taking into account these imperfections.
7 Dynamic transient analysis
In several cases, the dynamical nature of the external actions cannot be ignored and inertia forces
have to taken into account explicitly. In this case that means equation (3) transforms into
M¨
u[t]+s[u[t],˙
u[t]] = p[t](35)
where Mis the mass matrix, ¨
uthe second derivative of uwith respect to time tand the structural
response snow explicitly depends on both uand its first derivative ˙
u.
Dynamic transient analysis corresponds to the direct numerical integration of the equation of
the motion (36). Its simplest implementation is based on the discretization of ¨
uthrough central
finite difference. We obtain the explicit recursive formula
u[tk+1]=u[tk]−u[tk−1]+∆t2(p[tk]−s[uk,˙
uk]) ,t
k+1 =tk+∆t(36)
15
allowing us to directly compute u[tk+1]as a function of the previous solutions u[tk]and u[tk−1]=
tk+∆t. Several integration formulas have been proposed, starting from the second half of the
20th century [189]–[215], the best known may be the one proposed by Newmark [192] in 1959
(originally developed within a military research on the effects of nuclear weapons).
For the present discussion, we can refer to the general expression
˙
u(k+1) −˙
u(k)=∆tM−1%α(p(k+1) −s(k+1)) + (1 −α)(p(k)−s(k))&
u(k+1) −u(k)=∆t%β˙
u(k+1) + (1 −β)˙
u(k)&(37)
where the superscripts (k)and (k+1) refer to times tkand tk+1 and αand βare appropriate weight
factors which can be assumed in the range 0. . . 1. Note that, if αβ =0the scheme correspond
to an explicit formula allowing the direct computation of both u(k+1) and ˙
u(k+1). Otherwise, it
defines them only implicitly. However the solution can be obtained by Newton’s iteration. For
instance, taking ˙
uk+1 as a main variable, we can use the iteration scheme
rj:= M(˙
uj−˙
u(k))−∆t%α(p(k+1) −sj) + (1 −α)(p(k)−s(k))&
˙
uj+1 := ˙
uj−H−1
jrj
uj+1 := −u(k)+∆t%β˙
uj+ (1 −β)˙
u(k)&
(38)
where sj:= s[uj,˙
uj]and Hjis the Jacobian matrix
Hj:= M+α∆tCj+αβ∆t2Kj(39)
which can be expressed in terms of the mass matrix Mand of the viscosity and stiffness tangent
matrices
Cj:= #∂s
∂˙
u$,Kj:= #∂s
∂u$(40)
evaluated at ˙
u=˙
ujand u=uj. Note that, as in the static path-following analysis, a Modified
Newton iteration based on a constant estimate of both stiffness and viscosity matrices is generally
computationally more convenient. Note also that, mass matrix being an important part of the total
iteration matrix, the convergence is usually much faster than in the static case, especially for small
∆t.
The numerical solution provided by the time integration scheme obviously depends on the
particular choice of factors αand β, but (at least in principle), it provides an accurate recovery of
the solution if sufficiently small time steps ∆tare used. The choice
α=1/2−1/√6,β=1/2+1/√6(41)
assures the best recovery for ∆t→0. Actual implementations can however be subjected to
some numerical phenomena which can destroy this accuracy and make the recovery completely
unreliable, as pointed out by the numerical comparison discussed in [216].
The first difficulty derives from the so called von Neumann numerical instability. A sinusoidal
motion with period Tis recovered by the scheme as a sinusoidal motion with period ˜
T≈T, if
using a small ∆t&T, but for ∆t>∆tN,∆tNbeing a limit value depending on factors αand β,
it is recovered as an exponentially increasing pulsating motion which is obviously an unacceptable
solution. An unconditionally stable scheme (∆tN=∞) is, however, obtained by the choice
α+β≥1. This choice becomes necessary in the analysis of refined FEM structural models,
characterized by a large number of degrees of freedom and by the presence of high frequency
vibrations (Tof the order of 1/1000 sec or even less), where it is practically impossible to satisfy
the condition ∆t&Tfor all the vibration modes.
16
Furthermore, when using a stable scheme, high frequency vibrations (T&2∆t) are recovered
as pulsations with period ˜
T≈2∆t. A large number of spurious vibration motions with approxi-
mately the same period are then present in the overall solution and, because of their interaction, a
beats phenomenon is produced generating a spurious low frequency motion which will overlap the
well recovered motions and produces a marked deterioration in overall accuracy. Obviously, spu-
rious motions have to be filtered by the scheme in order to obtain reliable solutions. This is simply
done by making α>1/2,β>1/2. This choice introduce a selective algorithmic dissipation
in the recovered solution which exponentially reduces the amplitudes of spurious unrecoverable
motions (T<∆t) and practically does not affect well recovered motions (T>20∆t).
Finally, we have to mention that, if using very small ∆t(this usually occurs when we want
to recover a large range in frequency of the solution) and perform the analysis for large simu-
lation times Tsim ,T, another difficulty derives from the so called Lipshitz instability. The
phenomenon originates in the way the equilibrium errors are introduced in each step of the anal-
ysis by the roundoff–error or, in the case of nonlinear structures, even more by the approximate
closure of the end–step equilibrium, act as a superimposed random excitation of amplitude propor-
tional to the ratio T/∆t. During the entire simulation the effects of these continuous perturbations
cumulate and the final relative error in the solution can be evaluated as
δ/cT
sim/2∆t(42)
δbeing the relative error produced in each step and c<1a corrective factor accounting for struc-
tural dissipation in the low–frequency range. For long–term simulations, this effect can produce a
marked progressive deterioration in the solution accuracy.
In spite of these difficulties, transient analysis represents a very powerful approach, the only
one able to account for the entire complexity in the behavior of nonlinear structures, when com-
bined with an appropriate FEM modeling. It is very effective for fast dynamics or short–term
simulation. For long simulations it tends to be less efficient due to both the deterioration in accu-
racy due to the Lipshitz instability and its heavy computational burden. In fact, while each step
is faster than in the static path–following analysis, the number of steps can be very large (usually
of the order of thousands). This could be an important drawback considering that a large number
of successive simulations have to be performed, to take into account different loading cases and
structural imperfections.
Several algorithmic expedients can be used to reduce the computational time, especially in
combination with multilevel modeling and iterative solution schemes: selective use of explicit
formulas, different time–step sizings and selective relaxing of closure tolerance of the equilibrium
equations. Some of them are included as internal tricks in the commercial simulation codes. In
any case, we have to remember that a careful attention has always to be paid to avoid divergence
errors in the solution.
8 A comment about nonlinear modeling
The availability of an appropriate expression for the total potential energy is the main point in all
the solution approaches previously described, its correctness being crucial for the reliability of the
results. With regard to this, the current state of knowledge is however not completely satisfactory.
Consider that the so called ”small displacements” assumption, allowing us to refer to relatively
simple linear equations, has played an important role in the evolution of structural mechanics.
Classical results such as the Cauchy theory of elastic three-dimensional bodies, the Saint Vennt
rod theory, the plate and shell theories are strictly related to this assumption. More recently, the
development of finite element technology has mainly addressed linear structural modeling, so a
17
lot of theoretical results and practical expertise are available for linear analysis. The experience
in deriving suitable nonlinear models and their finite element description is, however, poorer,
particularly with respect to the so called structural models, such as beams, plates and shells, which
are generally described through ad–hoc heuristic approximations.
The analysis of slender structures undergoing finite displacements requires a proper descrip-
tion of the strain energy, which must be unaffected by finite rigid body motions, i.e. it must be
objective. While being difficult to satisfy, this is a purely geometric requirement. We can clar-
ify this statement by expressing the potential energy in mixed form (this is also convenient for
computational reasons [89, 178])
Φ[u] := −Ψ[σ]+σε[d], p u := qd+εσ (43)
through an appropriate choice of configuration variables u:= {σ, d}and p:= {q, ε},σand
dbeing the stress and displacement fields, qand εthe external forces and distortions, ε[d]the
compatible strain field associated to dthrough kinematics, and Ψ[σ]the so called complementary
strain energy which is usually assumed as quadratic in σ:
Ψ[σ] := 1
2Cσ2(44)
Cbeing the symmetric positive definite bilinear compliance operator. The function ε[d], express-
ing the kinematical relationship between displacements and strains, plays an important role in the
analysis: if the ε–drelationship is assumed to be linear, we get a linear model. So we can translate
from a linear to a nonlinear modeling simply by referring to a proper nonlinear expression for ε[d].
Objectivity requires that this geometrical relationship be unaffected by rigid body motions.
This requirement is important for accuracy in path–following analysis, using either Total or Up-
dated Lagrangian formulations, and becomes even more crucial to assure the reliability of asymp-
totic analysis which requires fourth–order accuracy in kinematics. Even small geometrical in-
coherencies deriving from approximations in the continuum description or in its finite element
representation markedly affect the accuracy of the solution and can render it unreliable. This is a
real problem because geometrically exact strain expressions for structural models, such as beams,
plates and shells, are generally not available or are too complex to be used in a FEM context while
the simplified ”technical” nonlinear models usually adopted do not satisfy objectivity.
The so called Corotational Description [217]–[232] represents the main approach to overcome
these drawbacks, at least in principle. Its basic idea, of decoupling the kinematical coherency from
the elastic response, is strictly related to the earlier Argyris’ natural modes technique [25, 32, 39]
and can be summarized as follows.
e2
e3
e1
i
2
i
1
i
3
Q
c
u
R
Rd
ud
r
2
r
1
r
3
se1
si
1
Figure 9: Corotational kinematics
18
Using the classic results of the polar decomposition theorem (see [233]) the deformation gra-
dient of a body can always be decomposed in a rigid motion followed by a pure strain or in a
pure strain followed by a rigid motion. A geometrically objective strain measure must be a func-
tion of the pure strain part of the deformation gradient only. In standard corotational formulation
the rigid body motion is filtered, on average, using two reference systems for each finite element
(see fig. 9). The first is a fixed system, while the second is a moving (corotational) frame that
follows the average rigid motion of element. In the corotational frame the displacements can be
made arbitrarily small with an appropriate mesh refinement, allowing the use of technical nonlin-
ear models or even linear models. In this way, the nonlinearity of the problem is transferred from
the strain–displacement relationship to the change of frame that became a nonlinear function of
the displacement parameters. An important goal in this approach, as pointed out in the earliest
proposal by Rankin [218], is the possibility of reusing both the structural modelings and finite
element discretization procedures developed in linear analysis within the corotational framing, so
exploiting the great amount of research work spent in the linear context.
The standard corotational approach is logically simple, however the derivatives of the change
of reference immediately become complicated, especially when the finite element description uses
3d finite rotations. In fact, while it can be relatively easy to obtain the structural response vector,
that only require the first derivative of the change of reference, it becomes more difficult to ob-
tain the second derivatives needed for the evaluation of the tangent stiffness matrix [228]–[231],
especially when a vector like parametrization of the finite rotations is used. The difficulty in-
creases significantly when 3rd– and 4th–order derivatives have to be evaluated, as is necessary
in the asymptotic analysis. However, apart from these difficulties which only affect the algebraic
derivation of the FEM model, the approach is effective and suitable for robust and accurate imple-
mentations [234].
It is worth mentioning that quite different implementations of the same idea are also possible,
in which, following Biot’s approach to nonlinear continuum mechanics [235], the corotational
decomposition is applied pointwise directly to the continuum description. This produces strain
measures unaffected by rigid body motion and so, when combined with an appropriate discretiza-
tion, allows us to obtain geometrically objective nonlinear modelings. Several attempts in this
direction have been made [236]–[240] and the approach appears promising (see [241]).
9 A comment about multilevel modeling
The analysis of complex structures generally requires the use of different levels of modeling based
on different description scales (Macro–, Meso– and Micro–scale), each of them being intended
to treat phenomena at different scales. Different models can be related to each other by fine–
to–coarse homogenization procedures and coarse–to–fine stress/strain recovering techniques. A
reliable analysis should account for this mutual interaction: in fact, a solution obtained using the
global (coarse) modeling alone could neglect the influence of important details which are only
described at the local (fine) scale; conversely, an analysis based on the local scale alone can be
impracticable due to the overall complexity of the structure, especially in the presence of nonlin-
earities. A multilevel solution strategy, which combines all the modeling scales by exploiting their
different abilities, is particularly convenient in this case.
Several implementations of the Multilevel approach are available in the literature (see for
instance [242, 243] and their bibliographies). Such a technique allows a reduction in the compu-
tational costs when solving both linear and nonlinear problems by means of iterative processes
which alternate different solution schemes and different degrees of mesh refinement. We obtain
an iterative solution process where coarse solutions are used as initializers for the fine ones and,
19
conversely, fine solutions are used as a correction for improving the coarse results. The overall
solution scheme can be described by referring to two successive levels which will be indicated
with ”l” (local/fine) and ”g” (global/coarse) subscript, respectively.
The goal of the analysis is that of zeroing, through successive iterativo reductions, the local
residual
rl[ul] := sl[ul]−pl(45)
where vector ulcollects the configuration parameter at local level, slthe corresponding response
and plthe external action. Using the same symbols, the equation at global level can be written
rg[ug] := sg[ug]−pg(46)
The two levels are defined separately and then related to each other through an operator Aable to
transform the global displacements ugin terms of local ones ul, i.e.:
ul=Aug(47)
We require such a transformation to hold the energetic equivalence between the two descriptions
rT
lul=rT
gug(48)
So, the adjoint operator ATallows the local residual rlto be transformed in the global one rg:
rg=ATrl(49)
The target rl[ul] = 0 is then obtained by the following iterative scheme (j=1,2, . . . ):
1. Evaluation of the local residual:
rl,j := sl[ul,j ]−pl(50a)
2. Transfer of the local residual to the global one:
rg,j := ATrl,j (50b)
3. Global correction:
in this step the global equilibrium error is reduced by computing the global correction
dg,j := −˜
K−1
grg,j (50c)
˜
Kgbeing a suitable approximation for the global tangent matrix Ktg.
4. Transfer of the global correction to the local one:
ul,j+1 =ul,j +Ad
g,j (50d)
which is then affected by a residual mainly composed of high–frequency fluctuations, not
visible at the global level, which will be reduced by the next local correction.
5. Local correction: in this step we improve the current local solution ulby the iterative
scheme (k=1,2, . . . )
rl,k =sl[ul,k]−pl(50e)
ul,k+1 =ul,k −˜
K−1
lrl.k (50f)
where ˜
K−1
lis an iteration matrix suitable for reducing high–frequency components of the
residual as efficiently as possible (a fast Gauss–Seidel scheme is generally appropriate for
that purpose).
20
Roughly speaking, the solution is improved in two different ways: i) through a global correction
essentially aimed at reducing the low–frequency components of the residual; this lies in solving
a linear system, with relatively small dimensions, so it is conveniently obtained by a Cholesky
factorization procedure; ii) a local correction used for reducing the high–frequency components
of the residual; this involves a large number of variables, however, because of the high–frequency
distribution of the error, it is performed efficiently through a fast local Gauss–Seidel–like iterative
scheme.
Note that the scheme fully exploits both global and local models, and the solution is actually
obtained at the finest one. It is quite a different approach from that, which we can call partial
multilevel, of using the local model for defining the global one, through some homogenization cri-
teria, and then solving the problem at the global level alone, local quantities (i.e. local stresses and
displacements) being recovered from the global solution according to the assumed homogeniza-
tion. The solutions provided by the two approaches, full and simplified, are obviously different.
However, in linear analysis the differences are only quantitative, so can be accepted, providing
that an evaluation of the relative error size is possible in advance trough the same homogeniza-
tion process. In the nonlinear case, the partial use of the fine modeling can cause some important
physical phenomena, originated by this interaction, be completely neglected, and can also produce
qualitative differences. As a consequence, the use of the simplified approach could be unsafe.
10 Performance–based design of complex nonlinear structures
Structures characterized by shape stiffness, such as long–span suspended bridges, where the re-
sponse is largely influenced by the variations in geometry, are typical examples of nonlinear be-
having structures needing to be analyzed through nonlinear solution techniques.
In nonlinear analysis there are some differences with respect to the linear case: i) partial
safety factors have to be applied directly to each action and not to the results produced by these
actions; ii) the effects of a combination of several actions cannot be evaluated through the linear
superposition of their separate results; iii) which combination will be the worst one with respect
to each requested performance is not known a–priori or, at least, cannot be easily determined; so
the analysis has to investigate all possible combinations or, at least, a significant number of them
corresponding to realistic load scenarios.
Generally, the dynamical nature of the action cannot be neglected, so the analysis has to in-
clude dynamics and, due to the nonlinear behavior, should be mainly performed through a step–
by–step numerical solution. Simplified analyses based on linearized dynamics or nonlinear statics
could only be used for preliminary design or for the evaluation of the load scenarios to associate
to the single required performance. The results of these analyses should in any case be validated
by independent time–step dynamical simulations.
In the presence of slender compressed elements, the possible occurrence of local instability
phenomena makes the structure potentially sensitive to geometrical defects or load imperfections
with respect to the ideal design assumptions. As a consequence, the analysis has to be extended
in the post–buckling range, to investigate all possible imperfections (known from a statistical in-
vestigation) and to account for the interactions due to the possible modal coupling (especially
global/local couplings). This implies a really large number of different runs and very time con-
suming computations.
Asymptotic analysis, allowing us to perform imperfection sensitivity analysis efficiently and
to obtain information about the worst imperfection shapes, can help to reduce the number of runs,
but the analysis needs to be repeated and validated, according to the selected worst imperfections
and using independent path–following analyses.
21
Nonlinear solutions are, by their nature, sensitive to small variations in data, so they naturally
tend to be not sufficiently robust. Great care has to be taken to avoid unreliable results and, pos-
sibly, each analysis has to be repeated using alternative finite element modelings and independent
computer codes.
Particular care has to be taken with the simplified assumptions and all the implementation
details of the analysis, particularly the stability, accuracy and reliability of incremental schemes
used, the closure tolerances and the updating schemes. In dynamical analyses, structural dis-
sipation properties have to be carefully tuned, avoiding the introduction of spurious numerical
dissipation in the frequency ranges of interest.
The analysis of large dimension structures could imply a multilevel modelling and the use of
multilevel solution strategies. A full multilevel approach, such as that described in section 9, able
to take into account the bidirectional interaction between local (fine) and global (coarse) analysis
should be used, as a general criteria, to avoid unsafe results.
22
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34
Analisi nonlineare: problemi e metodologie di soluzione
Raffaele Casciaro
Universit`
a della Calabria, Italy
11 settembre 2007
Sommario
Vengono presentati i diversi approcci computazionali e le metodologie di soluzione uti-
lizzabili nell’analisi nonlineare delle strutture, in particolare: l’analisi incrementale al passo,
l’analisi di stabilit`
a linearizzata, l’analisi asintotica, l’analisi di sensibilit `
a alle imperfezio-
ni e l’analisi dinamica al passo. Per ciascuna di queste vengono descritte le caratteristiche
principali, le problematiche peculiari, le possibili sorgenti di errore e le convenienze compu-
tazionali. Il comportamento nonlineare delle strutture `
e, per sua natura, sensibile a piccole
variazione nei dati e pertanto una analisi a base prestazionale deve includere una indagine
estesa che tenga conto sia delle possibili imperfezioni nella disposizione dei carichi che dei
possibili difetti nella geometria della struttura. Particolare attenzione deve anche essere pre-
stata per assicurare l’affidabilit`
a delle soluzioni prodotte e, possibilmente, ciascuna analisi
deve essere ripetuta utilizzando approcci alternativi.
1 Introduzione
La principle differenza qualitativa tra l’analisi delle strutture in campo lineare e quella in campo
nonlineare risiede nella perdita del principio di sovrapposizione degli effetti. Nell’analisi lineare
la risposta della struttura ad una combinazione di azioni diverse pu`
o essere ottenuta sommando le
risposte separate corrispondenti a ciascuna delle azioni. Ci`
o determina una ovvia semplificazione
nella gestione di condizioni di carico complesse, ma anche presenta una pi`
u forte implicazione
concettuale: in effetti assicura che gli effetti prodotti da piccole perturbazioni nei carichi siano
anche essi piccoli e, pertanto, le eventuali piccole differenze fra la situazione reale e lo schema
ideale assunto nel calcolo, dovute alla fluttuazione dei carichi, a imperfezioni nella disposizione
geometrica della struttura e a difetti dei materiali, possano essere ignorati senza pregiudicare l’af-
fidabilit`
a dei risultati. Ci `
o non `
e pi`
u vero in campo nonlineare dove perturbazioni anche piccole
nei dati possono produrre variazioni notevoli nella risposta e, di conseguenza, portare a soluzioni
anche qualitativamente diverse. Il fatto riveste una importanza cruciale in quanto la nostra cono-
scenza della realt`
a strutturale `
e essenzialmente imprecisa: le azioni agenti sulla struttura, nel corso
della sua vita di esercizio, possono essere infatti valutate solo in termini di medie probabilistiche e
presentano comunque una forte fluttuazione aleatoria; imperfezioni nella geometria e nei materiali
sono anche esse essenzialmente aleatorie e definite solo in termini di valori caratteristici in base
alle prescrizioni sulle tolleranze accettabili.
In un contesto nonlineare una analisi basata su valori medi `
e generalmente insufficiente a
fornire una valutazione affidabile del soddisfacimento delle prescrizioni prestazionali e, pi`
u in
generale, della sicurezza rispetto al collasso della struttura. E’ necessario pertanto svolgere analisi
separate per ciascuna delle possibili combinazioni di carico e per tutte le possibili imperfezioni.
Questo rappresenta un impegno notevole di calcolo che `
e reso possibile solo dall’incremento delle
potenze di calcolo disponibili e, ancora di pi`
u, dallo sviluppo di metodologie di calcolo sempre
pi`
u potenti ed ottimizzate.
1
Nel seguito verranno descritte le principali metodologie attualmente disponibili, o quanto me-
no le principali fra queste. Per brevit`
a, la discussione sar`
a limitata agli aspetti principali di ciascun
approccio tralasciando aspetti pi`
u specifici legati alle nonlinearit`
a nel legame costitutivo, come
plasticit`
a, frattura e danneggiamento.
2 Il metodo degli elementi finiti
La tecnica di discretizzazione e soluzione che forma il cosiddetto Metodo degli Elementi Finiti
(FEM) rappresenta la base dell’approccio computazionale all’analisi delle strutture. Le origini
del metodo possono farsi risalire all’approccio energetico proposto da Castigliano [1] alla fine del
19osecolo ed ai lavori di Ritz [2] e Galerkin [3] su soluzioni approssimate (vedi anche [4]), nei
primi anni 20o, e dallo sviluppo di tecniche di soluzione numerica iniziato a partire dalla terza
decade del secolo [5]–[7]. Nella sua forma attuale, `
e stato messo a punto nel corso della seconda
guerra mondiale e nell’immediato dopoguerra in cui ha spesso rappresentato una risorsa strategi-
ca coperta da segreto militare o industriale. La sua importanza come collegamento fondamentale
tra la meccanica dei continui e l’analisi numerica, capace di sfruttare lo sviluppo crescente della
tecnologia del calcolo digitale [8]–[16], `
e diventata subito evidente dopo le prime pubblicazioni
sull’argomento nei primi anni ’50 [17]–[19] (la conferenza di Wright–Patterson [24] in cui Ar-
gyris present`
o la sua famosa relazione Continua and Discontinua [25], pu`
o essere considerato il
punto di svolta). La successiva diffusione del metodo come strumento base per l’analisi numerica
delle strutture `
e stata di seguito favorita dal progressivo incremento in potenza e diffusione dei
computers [20]–[50]. La continua evoluzione del metodo, che ha coinvolto la meccanica teorica,
i fondamenti matematici, gli algoritmi numerici e la tecnologia del software, `
e troppo vasta per
poter essere anche solo sommariamente descritta in questa sede; il lettore pu`
o fare riferimento a
[51]–[54] per ulteriori notizie di carattere storico.
L’idea centrale alla base del metodo FEM `
e quella di riferirsi all’energia potenziale
Π[u] := Φ[u]−pu (1)
in cui uindica la configurazione della struttura, comprensiva di entrambi i compi di spostamen-
to e di tensione, e pl’azione esterna assegnata, comprensiva di forze e distorsioni, scritta come
combinazione dell’energia interna di deformazione Φ[u]e del lavoro esterno bilineare pu. Suddi-
videndo la struttura in piccoli elementi ed usando in ciascuno di questi appropriate interpolazioni
dei diversi campi interessati, l’energia Π[u, p]pu`
o essere ricondotta ad una forma algebrica tale
che l’equilibrio, espresso dalla stazionariet`
a rispetto al campo ue cio`
e dalla condizione
Φ![u]δu−pδu=0 (2)
dove il simbolo (’) indica la differenziazione secondo Fr´
ech`
et e δula generica variazione virtuale
di configurazione, possa essere scritto come equazione vettoriale n–dimensionale
s[u]−p=0 (3)
in cui i vettori sand p, che esprimono rispettivamente la risposta strutturale interna e l’azione
esterna, sono definiti dalle equivalenze energetiche
s[u]Tδu:= Φ![u]δu , pTδu:= pδu(4)
dove δueδusono la variazione virtuale di configurazione rispettivamente al continuo ed al di-
screto. Approcci diversi possono essere seguiti in questo processo di discretizzazione, a seconda
2
della particolare tecnologia usata nella definizione degli elementi (elementi compatibili, misti o
ibridi, elementi elementi X-FEM, PU, discretizzazioni meshless o altro), ma, in ogni caso, se si
usa una formulazione coerente, l’accuratezza pu`
o essere migliorata (almeno in teoria) di quanto si
vuole infittendo il reticolo di discretizzazione.
L’analisi lineare assume che Φ[u]sia quadratica in ue quindi che la risposta s[u]possa ridursi
ad una funzione lineare
s[u] := Ku (5)
legata ad uattraverso una matrice di rigidezza simmetrica e definita positiva K, definita dalla
equivalenza energetica con la variazione seconda di Φ[u]
δuTKδu:= Φ!!δu2(6)
La condizione di equilibrio (3) si riduce pertanto ad un sistema lineare simmetrico
Ku =p(7)
la cui soluzione pu`
o essere ottenuta, con relativa facilit`
a, mediante l’algoritmo di decomposizione
di Cholesky che produce una espressione computazionalmente conveniente per l’inversa K−1
della K, di cui sfrutta pienamente alcune caratteristiche tipiche quali sparsit`
a e struttura bandata.
In analisi nonlineare, la funzione s[u]pu`
o assumere una forma sensibilmente pi`
u complicate.
Tuttavia, in corrispondenza a ciascuna configurazione corrente u, si pu`
o definire la cosiddetta
matrice di rigidezza tangente Kt[u]attraverso l’equivelenza energetica
δuTKt[u]δu:= Φ[u]!!δu2(8)
Questa matrice, che fornisce una informazione completa dell’energia Φ[u]nell’intorno di secondo
ordine di ue pu`
o pertanto essere usata per collegare piccoli incrementi ˙
udi uai corrispondenti
incrementi ˙
sdi s
˙
s=Kt[u]˙
u(9)
gioca un ruolo importante nell’analisi. Infatti, il problema (3) pu`
o essere risolto mediante il ben
noto schema iterativo di Newton (vedi [55, 56])
uj+1 := uj−Kt[uj]−1rj,rj:= s[uj]−p,j=0,1,2. . . (10)
in cui rj`
e il residuo all’equilibrio corrispondente alla valutazione corrente di ujeuj+1 `
e la
valutazione ulteriormente migliorata restituita dallo schema. Se si parte sufficientemente vicini
alla soluzione richiesta, lo schema converge, abbastanza rapidamente, ad una valutazione uche
azzera il residuo (quanto meno all’interno della tolleranza prefissata) e pu`
o essere quindi assunto
come soluzione per il problema di equilibrio.
Lo schema di Newton (10) richiede l’aggiornamento della matrice K−1
ta ciascun ciclo di
iterazione. Questa operazione implica l’assemblaggio e la decomposizione alla Cholesky della
matrice Kt[uj]e corrisponde generalmente alla parte computazionalmente pi`
u costosa dell’intero
ciclo iterativo (approssimativamente, costa m/4volte dell’impegno necessario alla valutazione e
decomposizione del residuo rj, se m`
e l’ampiezza di semibanda della matrice, spesso dell’ordine
delle centinaia). Pertanto, lo schema (10) viene usualmente implementato nella forma modificata
(vedi fig.1)
uj+1 := uj−˜
K−1rj(11)
dove ˜
K`
e una approssimazione che sostituisce la Kt[uj]nello schema ed `
e mantenuta costante
durante l’intero processo iterativo. Questa variante permette di evitare l’aggiornamento continuo
della matrice Kt[uj]−1e risulta, quindi, molto conveniente da un punto di vista computazionale.
La convergenza dello schema modificato `
e legata alla differenza relativa tra le matrici ˜
KeKt[uj]
3
Figura 1: Iterazione alla Newton (a) e Newton Modificato (b).
e, anche se generalmente pi`
u lenta della versione originale, `
e sempre assicurata se sono verificate
le seguenti condizioni
0<Kt[uj]<2˜
K,j=0,1,2. . . (12)
Si ha quindi perdita di convergenza solo se la matrice tangente Ktperde la sua caratteristica di
positivit`
a (ci`
o avviene in corrispondenza a soluzioni di equilibrio non stabile) o se ˜
K`
e tale che la
matrice 2˜
K−Ktnon risulti definita positiva, cio`
e se si `
e sottostimato per un fattore 2 o superiore
la rigidezza effettiva della struttura nella valutazione della matrice di iterazione ˜
K.
3 Analisi incrementale al passo
L’idea base di questo approccio all’analisi `
e quella di ricostruire il percorso di equilibrio u[λ]
conseguente ad un assegnato processo di carico p[λ]attraverso una sequenza di punti di equilibrio
{u(k),λ(k)}sufficientemente vicini tra loro da fornire per interpolazione la curva di equilibrio
definita dalla condizione implicita
s[u[λ]] −p[λ] = 0 (13)
Usualmente, e ci riferiremo a ci`
o nel seguito, si considera un processo di carico proporzionale
p[λ] := p0+λˆ
p(14)
in modo che il parametro λche ne regola l’evoluzione possa essere interpretato come moltiplica-
tore di sicurezza associato alla condizione nominale di carico ˆ
p.
Per caratterizzare la sequenza dei punti di equilibrio possono essere utilizzate diverse strategie
al passo, la pi`
u semplice delle quali `
e quella di assegnare direttamente gli incrementi ∆λ(k)al
moltiplicatore dei carichi, ponendo λ(k+1) =λ(k)+∆λ(k). La soluzione corrispondente u(k+1) :=
u[λ(k+1)]viene quindi ottenuta costruendo una stima iniziale u1mediante diretta estrapolazione
del passo precedente
u1:= u(k)+β(k)(u(k)−u(k−1)),λ(k+1) := λ(k)+β(k)(λ(k)−λ(k−1))(15)
4
in cui β`
e un fattore, utilizzato per ampliare (β(k)>1) o ridurre (β(k)<1) l’ampiezza del passo
∆λ(k)=β(k)∆λ(k−1) a seconda delle esigenze dell’analisi. A ci`
o segue un numero di iterazioni
Newton modificate (11) sufficiente a ridurre il residuo entro la tolleranza prefissata. Usualmente
la matrice di iterazione ˜
K`
e aggiornata ad ogni passo, usando la matrice tangente all’inizio del
passo ( ˜
K:= Kt[u(k)]) o, e questa scelta si rivela spesso pi`
u conveniente, in corrispondenza alla
estrapolazione iniziale ( ˜
K:= Kt[u1]). Spesso `
e anche aggiunta una opzione di ripristino che, in
caso si riscontrino problemi di convergenza, arresta il processo iterativo, scarica i risultati ottenuti
e ripete il passo con una ampiezza pi`
u piccola (β≈0.1÷0.2) in modo da mantenere ˜
Kvicina a
K[uj].
Anche se semplice, questa strategia di soluzione risulta efficiente e robusta ed `
e stata larga-
mente utilizzata in passato. La strategia si comporta in affetti molto bene nella fase iniziale, a
bassa nonlinearit`
a, del processo di carico. Tuttavia, per sua natura, tende a fallire vicino a punti
di carico limite, in cui la matrice tangente Ktdiventa singolare e, di conseguenza, le condizioni
di convergenza (12) non sono pi`
u soddisfatte. Generalmente si `
e particolarmente interessati alla
sicurezza a collasso della struttura, a stimarne cio`
e il carico limite, e quindi ci`
o rappresenta un
inconveniente significativo nella strategia.
Per lungo tempo, questo difetto `
e stato considerato una caratteristica intrinseca, non sanabile,
dell’analisi nonlineare. Alla fine degli anni ’70, da un articolo di Riks di seguito molto citato
[57], si `
e riconosciuto tuttavia che l’inconveniente non era altro che una conseguenza banale della
scelta fatta, di controllare cio`
e l’evoluzione del percorso di equilibrio attraverso il parametro di
carico λ. In corrispondenza a punti limite, la funzione u[λ]non `
e analitica in λe non sorprende
pertanto che il fatto di definire la sequenza u(k)attraverso incrementi ∆λ(k)possa creare pro-
blemi. Questi possono essere peraltro facilmente evitati dal semplice accorgimento di utilizzare
una parametrizzazione analitica {u[ξ],λ[ξ]}, dove ξ`
e una ascissa curvilinea che descrive la curva
di equilibrio nello spazio {u,λ}, e di definire la sequenza {u(k),λ(k)}assegnando le ampiezze
∆ξ(k)dei singoli archi di curva. Pi`
u precisamente, ed assumendo l’ampiezza del passo definita
dalla
∆ξ2:= ∆uTM∆u+µ∆λ2= (∆ξ(k))2(16)
in cui Meµsono una appropriata matrice metrica (simmetrica e definita positiva) ed un fattore
(positivo), il punto di equilibrio (k+ 1)–esimo `
e definito come soluzione di un sistema ottenuto
come combinazione delle equazioni di equilibrio (3) e del vincolo di ampiezza (16). Ne risulta un
sistema di n+1equazioni nonlineari nelle n+1incognite u(k+1) eλ(k+1) che pu`
o essere risolto
con iterazione alla Newton
uj+1 := uj+˙
uj,λj+1 := λj+˙
λj(17a)
dove le correzioni iterative ˙
uje˙
λjsono ottenute come soluzione del sistema lineare
Jj!˙
uj
˙
λj"=!rj
gj",Jj:= #Kt[uj]−ˆ
p
∆uT
jMµ∆λj$(17b)
in cui gli incrementi ∆uje∆λjsono definiti dalle
∆uj:= uj−u(k),∆λj:= λj−λ(k)(17c)
ed rjegjsono i residui correnti alle equazioni di equilibrio e di ampiezza di passo:
rj:= s[uj]−λjˆ
p,g
j:= ∆uT
jM∆uj+µ∆λ2−(∆ξ(k))2(17d)
La caratteristica pi`
u rilevante dello schema iterativo “arc–length” proposto da Riks (17) `
e che la
matrice Jacobiana Jjdel sistema resta non singolare, anche in presenza di minore Ktsingolare e
5
che, di conseguenza, la condizione Kt>0non `
e pi`
u richiesta per la convergenza dello schema.
Pi`
u in generale la strategia arc–length presenta caratteristiche di convergenza migliori di quella a
controllo di carico e permette non solo di superare facilmente i punti limite ma anche di segui-
re agevolmente le zone discendenti del percorso di equilibrio. In effetti, perde efficacia solo in
corrispondenza di punti di biforcazione, in cui lo Jacobiano Jjdiventa singolare.
Figura 2: Schema iterativo alla Riks (arc–length).
Rispetto allo schema base (17) sono possibili ulteriori miglioramenti computazionali. Non
siamo in effetti interessati ad una piena accuratezza nel rispetto della condizione di ampiezza del
passo. Pertanto, una volta che l’ampiezza desiderata sia gi`
a fornita dalla estrapolazione iniziale
(15), possiamo assumere gj≈0durante le successive iterazioni. Ancora, possiamo sfruttare la
strategia Newton modificata ed usare una matrice di iterazione costante ˜
Kin sostituzione della
Kt[uj], in modo da evitare l’aggiornamento continuo della K−1
te rendere molto pi`
u veloce il
processo di soluzione (see fig. 2). Infine, il sistema pu `
o essere convenientemente risolto per
partizionamento. Si ottiene:
˙
λj:= rT
jdj
ˆ
pTdj+µ∆λj
,˙
uj:= ˜
K
−1(rj−˙
λjˆ
p)(18a)
in cui i vettori rjedjsono definiti dalle
rj:= s[uj]−λjˆ
p,dj:= ˜
K
−1M∆uj(18b)
Altre varianti di dettaglio sono possibili per accelerare ulteriormente lo schema. Ad esempio,
considerando che
∆uj≈∆λj˜
K
−1ˆ
p(19)
lo schema pu`
o essere semplificato nella forma
˙
λj:= uT
rMˆ
u
ˆ
uTMˆ
u+µ,˙
uj:= ur+˙
λjˆ
u(20a)
6
dove
ˆ
u:= ˜
K
−1ˆ
p,ur:= ˜
K
−1rj(20b)
In questa versione, che non modifica le caratteristiche di convergenza dello schema, ciascuna
iterazione richiede solo una soluzione soluzione alla Cholesky per il vettore ur, a parte il pro-
dotto scalare richiesto dalla valutazione di ˙
λj, di costo irrilevante, e pertanto la singola iterazione
presenta lo stesso impegno computazionale della analoga nello schema a controllo di carico.
La strategia arc–length, la cui convenienza divenne subito evidente, acquist`
o rapidamente una
grande popolarit`
a fra gli strutturisti numerici generando un gran numero di articoli successivi [58]–
[81] in cui sono state proposte varianti diverse, sia nella definizione (16) dell’ampiezza di passo
che nei dettagli algoritmici che regolano l’aggiornamento dalla matrice di iterazione, le opzioni
di ripristino e la regolazione dell’ampiezza del passo (la pi`
u semplice `
e quella di legare il fattore
β(k)al numero di iterazioni svolto nel passo precedente). Per maggiori informazioni, il lettore pu `
o
riferirsi a [84]–[88].
Occorre considerare che l’abilit`
a della strategia arc–length nel superare la presenza di singo-
larit`
a nella matrice di rigidezza tangente non elimina i problemi di convergenza, legati all seconda
parte delle condizioni (12), che si presentano quando, almeno per spostamento incremento ˙
ula
matrice di iterazione ˜
Ksottostima la rigidezza effettiva K[uj]per un fattore maggiore di 2, ci`
e
quando
˙
uTK[uj]˙
u>2˙
uT˜
K˙
u(21)
per qualche incremento ˙
u. Come mostrato in [89], l’occorrenza di questa condizione `
e pi`
u fre-
quente di quanto si possa pensare ed `
e legata ad una forma di irrigidimento spurio (“nonlinear
locking”) prodotto da una interazione perversa tra elevato rapporto di rigidezza e una anche mol-
to piccola variazione di orientazione degli elementi della struttura. Le strutture snelle, dove la
rigidezza assiale pu`
o essere maggiore di quella trasversale di diversi ordini di grandezza, sono
particolarmente sensibili a questo fenomeno e questo `
e ulteriormente aggravato se la struttura
presenta direzioni globali di labilit`
a, come avviene per le strutture compresse, in vicinanza del
collasso, o per strutture che derivano la loro rigidezza dallo stato tensionale, quali ad esempio i
ponti sospesi, nella fase iniziale di caricamento.
La rilevanza di questo fenomeno di locking `
e direttamente legata alla lunghezza del passo ed
alla nonlinearit`
a del problema. Tuttavia, come anche viene mostrato in [89], il locking `
e legato
pi`
u alla descrizione in termini nonlineari della risposta s[u]che a caratteristiche intrinseche di
questa. Modellazioni basate su elementi finiti di tipo compatibile, dove la risposta dell’elemento
`
e descritta solo in termini di spostamenti nodali, sono particolarmente sensibili al fenomeno e
possono originare un locking talmente patologico da richiedere una riduzione cos`
ı forte del passo
da impedire in pratica la ricostruzione del percorso di equilibrio. Le stesse difficolt`
a scompaiono
se si usa invece una descrizione di tipo misto, che utilizza sia spostamenti nodali che tensioni
interne, e questo pu`
o essere in effetti ottenuto con variazioni secondare nel codice di calcolo che
non introducono costi aggiuntivi apprezzabili pur fornendo una variante robusta e libera da locking
dello schema iterativo.
L’analisi incrementale al passo basata, sullo schema iterativo arc–length, rappresenta la stra-
tegia attualmente pi`
u diffusa tra gli strutturisti numerici e pu`
o essere considerata lo strumento
principale per l’analisi nonlineare delle strutture. Essa permette una ricostruzione accurata del
comportamento di strutture soggette ad un assegnato processo di carico. Tuttavia presenta alcuni
difetti intrinseci. Innanzi tutto, non `
e adatta a trattare problemi in cui il percorso di equilibrio
presenti biforcazioni. In questi caso infatti, l’analisi dovrebbe riconoscere la presenza di punto di
biforcazione nell’ambito del passo corrente e quindi seguire ciascuna delle ramificazioni emergen-
ti in modo da poter dare una risposta completa del comportamento complessivo della struttura, ma
ci`
o`
e difficilmente realizzabile in pratica. Sono stati proposti diversi procedimenti algoritmici per
riconoscere la presenza di biforcazioni e reindirizzare il percorso lungo una dei rami [90]–[95], ma
7
nessuno di questi pu`
o essere considerato sufficientemente robusto ed affidabile da essere utilizzato
quale procedura generale per una indagine che abbracci l’insieme complessivo delle diramazioni.
In secondo luogo, la strategia fornisce la risposta solo in corrispondenza ad una singola condizio-
ne di carico. Una analisi che tenda a valutare la sicurezza strutturale, che resta l’obbiettivo pi`
u
importante, dovrebbe considerare tutti possibili processi di carico e tenere conto anche delle pos-
sibili fluttuazioni dovute alle imperfezioni di carico ed ai difetti geometrici. Tuttavia dato che la
singola analisi implica diversi passi e molti cicli iterativi per passo e risulta quindi computazional-
mente costosa, svolgere una investigazione completa che consideri ogni condizione di carico ed
ogni possibile forma di imperfezione diventa computazionalmente proibitiva. Pertanto il miglior
uso dell’approccio incrementale al passo resta quello di effettuare un numero limitato di analisi in
corrispondenza alle condizioni di carico pi`
u restrittive, una volta che le forme di imperfezione pi`
u
pericolose per la struttura siano state gi`
a determinate.
4 Analisi di stabilit`
a linearizzata
L’analisi di stabilit`
a linearizzata corrisponde all’implementazione a strutture complesse del me-
todo originalmente proposto da Eulero nel suo sviluppo della teoria della Elastica [96, 97], cio`
e
il ben noto problema dell’asta caricata di punta. La motivazione di base per questo approccio `
e
che la stabilit`
a secondo Liapunov (vedi [98]) di una soluzione di equilibrio `
e direttamente legata
alla convessit`
a locale dell’energia potenziale totale e pertanto, se questa `
e scritta nella forma (1),
alla convessit`
a locale dell’energia di deformazione Φ[u]. Se la variazione seconda dell’energia di
deformazione `
e definita positiva (Φ!!δu2>0,∀δu) ci`
o assicura la sua convessit`
a locale e quindi
implica stabilit`
a. D’altro canto, il fatto che risulti Φ!! v2<0per qualche variazione infinitesima
di configurazione v, implica instabilit`
a ed `
e associata al collasso per sbandamento laterale o “buc-
kling” della struttura. Pertanto, ricordando la definizione (8) e facendo riferimento ad un percorso
di carico u[λ], la sicurezza a sbandamento laterale pu`
o essere messa in relazione al primo valore
del moltiplicatore λdel carico per cui si abbia
K[u[λ]]v=0,v"=0(22)
Nel caso, particolare ma frequente in pratica, che la struttura presenti rotazioni trascurabili lungo
il percorso u[λ], come ad esempio avviene nell’asta di Eulero o, pi `
u in generale, in strutture
puramente compresse, l’implementazione numerica di questo criterio `
e abbastanza semplice.
La matrice tangente Ktdipende dalla geometria corrente della struttura e, linearmente, dalle
costanti elastiche e dalle tensioni interne. Pertanto, se la geometria resta praticamente invariata, la
matrice pu`
o essere espressa come somma di due contributi:
Kt=K0+λK1(23)
Il primo K0corrisponde alla usuale matrice di rigidezza in elasticit`
a lineare, il secondo λK1
tiene conto degli effetti delle tensioni interne e pu`
o essere assunto lineare in λin quanto, a parit`
a
di geometria, le tensioni sono funzioni lineari del carico. Il controllo di stabilit`
a (22) pu`
o essere
cos`
ı ricondotto alla soluzione del problema lineare agli autovalori
(K0+λK1)v=0,v"=0(24)
Siamo in realt`
a interessati solo alla soluzione principale, caratterizzata dal pi`
u piccolo valore po-
sitivo λe questo corrisponde ad un problema relativamente semplice da un punto di vista compu-
tazionale, risolubile con algoritmi standard particolarmente veloci e robusti (e.g. see [99]). Fra
gli altri possiamo citare il metodo delle potenze inverse, il metodo di iterazione nel sottospazio
8
ed il metodo di Lanczos [100, 101] (varianti recenti di quest’ultimo come la variante proposta da
Sorensen [102], sono particolarmente efficienti [103]). I diversi metodi sono in generale basati
su uno schema iterativo che fornisce successive valutazioni {vj.λj},j =1,2. . . , rapidamente
convergenti alla soluzione cercata, e fanno uso dell’inversa K−1
0della matrice di rigidezza inizia-
le. Il calcolo di questa rappresenta in effetti la parte computazionalmente pi`
u onerosa dell’intero
processo di soluzione, tuttavia se, come avviene di norma, l’analisi di stabilit`
a`
e condotta imme-
diatamente dopo una analisi lineare che usa la stessa la stessa matrice, la K−1
0`
e gi`
a disponibile
in forma fattorizzata dalla analisi precedente e ci`
o rende particolarmente veloce il completamento
della soluzione.
L’analisi di stabilit`
a linearizzata, oltre che computazionalmente veloce e molto robusta, `
e an-
che particolarmente ricca di informazioni in quanto fornisce sia il valore critico di carico λbas-
sociato allo sbandamento laterale (come primo autovalore del problema (24)) ma anche la forma
dello sbandamento o modo critico vb(come corrispondente autovettore). Dobbiamo ricordare tut-
tavia che questa informazione `
e stata ottenuta attraverso una drastica linearizzazione del problema
delle equazioni del problema e pu`
o rilevarsi inaffidabile nel descrivere il comportamento reale del-
la struttura, che `
e spesso fortemente influenzato dai termini nonlineari trascurati e da anche piccole
deviazioni nei dati derivanti dalla presenza di imperfezioni. I risultati della sperimentazione su ci-
lindri compressi svolta da Donnell nei primi anni ’30 [104] ha mostrato come il carico di collasso
effettivo possa essere anche molto inferiore (del 50% o ancora di meno) del valore critico fornito
dalla teoria euleriana.
Per molto tempo questo aspetto `
e stato largamente sottostimato, principalmente a causa di un
fraintendimento del significato reale della condizione critica euleriana (22) che veniva vista come
la possibilit`
a di avere piccole deviazioni vbdalla configurazione originaria u[λb]in corrispondenza
dello stesso carico λbˆ
p. Si pu`
o arrivare in effetti a questa interpretazioni osservando che, se si fa
una espansione in serie di Taylor dell’equazione di equilibrio
!Φ"[u0] + Φ""[u0]∆u+1
2Φ"""[u0]∆u2+· · · "δu−(p0+∆p)δu=0 (25)
a partire da una soluzione equilibrata {u0,p
0}e si trascurano i termini pi`
u che lineari in ∆u, in
ragione dell’ipotesi assunta di piccoli spostamenti, si ottiene una equazione lineare che lega la
variazione di configurazione ∆ualla variazione del carico ∆p:
Φ""[u0]∆uδu=∆pδu(26)
In forma matriciale ci`
o si scrive
K[u0]∆u=∆p(27)
e quindi l’interpretazione sorge spontanea dal confronto fra quest’ultima e la condizione euleriana
(22), vista come relazione tra un ∆upiccolo ma finito ed incremento nullo di carico. Questa
conclusione `
e tuttavia falsa, come gi`
a messa in luce nel 1941 da von Karman and Tsien in [105].
In effetti, nell’espansione (25) il termine di secondo ordine 1
2Φ"""[u0]∆u2δunon pu `
o certo essere
trascurato come molto pi`
u piccolo di Φ""[u0]∆uδuse quest’ultimo diventa zero in rispetto della
condizione (22) e pertanto, in queste condizioni, l’equazione linearizzata (27) non corrisponde pi`
u
ad una approssimazione valida della relazione tra ∆ue∆p.
La teoria corretta, generale nonlineare, della stabilit`
a elastica `
e stata sviluppata da Koiter, a
partire dalla sua famosa tesi di dottorato [106], agli inizi degli anni ’40 (vedi la sintesi di Budiansky
in [107] per una agevole introduzione alla teoria). Koiter ha mostrato come il collasso di una
struttura nonlineare soggetta a fenomeni di instabilit`
a debba essere investigato all’interno della
teoria della biforcazione, e che il suo trattamento richieda non solo la valutazione del carico critico
λbe del modo critico vbma anche quella del pendenza e della curvatura iniziale del percorso
diramato. Utilizzando questi ingredienti si perviene ad una formula semplice capace di mettere
9
fundamental path
bifurcated paths
bifurcation point
fundamental path
bifurcated path
secondary bifurcation
Figura 3: Buckling interattivo per modi critici coincidenti (a) o quasi coincidenti.
in conto anche gli effetti delle imperfezioni. La successiva ricerca ha inoltre mostrato che in
presenza di modi critici per valori quasi coincidenti di λ, come spesso avviene in conseguenza
dell’ottimizzazione del progetto strutturale (vedi [109]–[110]), l’interazione fra questi produce
una risposta estremamente complessa (see fig. 3) e generalmente caratterizzata da forte sensibilit`
a
alle imperfezioni [111]–[115]. Tutta questa complessit`
a si perde del tutto in una analisi di stabilit`
a
linearizzata.
5 Analisi asintotica
L’analisi asintotica nasce come implementazione della teoria nonlineare di Koiter all’interno di
un approccio numerico agli elementi finiti. Il suo obiettivo `
e quello di ricostruire una valutazione
asintotica del percorso di equilibrio {u, λ}definito dalla condizione implicita (2) a partire da un
problema perfetto associato caratterizzato da biforcazione. Diverse implementazioni sono state
proposte [116]–[161]. Ci riferiremo qu`
ı a quella proposta in [129, 142, 156]. Si veda [162] per un
ulteriore approfondimento dell’argomento.
Considerando il caso generale di pi`
u modi critici interagenti, corrispondenti a valori λi,i=
1· · · mcompresi in un intorno del valore critico di riferimento λb, il percorso di equilibrio `
e
ricostruito nella forma
u[λ,ξ1,...,ξm]=λˆ
u+
m
!
i=1
ξi˙
vi+1
2
m
!
i,j=1
ξiξj¨
wij +··· (28a)
dove λˆ
u`
e il percorso fondamentale, che rappresenta la soluzione base linearizzata del problema o
soluzione perfetta, ˙
vi,i=1, . . . , m, sono gli mmodi critici e ¨
wij ,i, j =1, . . . , m, sono m×m
correttivi quadratici. Le diverse quantit`
a sono definite come segue:
1. Il modo fondamentale ˆ
u`
e definito come soluzione del problema lineare
K0ˆ
u=ˆ
p(28b)
dove K0:= Kt[0] `
e la matrice di rigidezza tangente valutata nella configurazione u=0.
10
2. I modi critici ˙
visono le prime msoluzioni del problema ad autovalori
K[λ]v=0(28c)
dove K[λ] := Kt[λˆ
u]`
e la matrice di rigidezza tangente valutata lungo il percorso fon-
damentale u:= λˆ
u. Si noti che l’assunzione |λi−λb|"λbimplica la condizione di
ortogonalit`
a
Φ!!!
bˆu˙vi˙vj=−δij (28d)
dove Φ!!
b:= Φ!![λbˆu]eδij `
e il simbolo di Kronecker.
3. I modi correttivi ¨
wij ∈Wsono determinati, a partire dai modi critici ˙
vi, come soluzione
dei problemi lineari
Kb¨
wij =¨
s[vi,vj](28e)
dove il vettore noto ¨
s[vi,vj], a secondo membro dell’equazione, `
e definito dalla equivalenza
energetica
¨
s[vi,vj]Tδw:= Φ!!!
b˙vi˙vjδw(28f)
eW:= {w:Φ!!!
b˙viˆuw =0,i=1, . . . , m}`
e il sottospazio ortogonale ai modi critici vi.
Espandendo l’equazione (2) fino al quarto ordine ed assumendo δu=˙vk, si ottiene il seguente
sistema di equazioni algebriche nonlineari:
−µk[λ]−(λ−λk)ξk+1
2
m
!
i,j=1
ξiξjAijk +1
6
m
!
i,j,h=1
ξiξjξhBijhk =0,k=1, . . . , m (28g)
dove
Aijk := Φ!!!
b˙vi˙vj˙vk,Bijhk := Φ!!!!
b˙vi˙vj˙vh˙vk−Φ!!
b(¨wij ¨whk +¨wih ¨wjk +¨wik ¨wjh)(28h)
rappresentano i termini cubici e quartici dell’espansione dell’energia di deformazione e
µk[λ] := µi
k[λ]+µg
k[λ]+µl
k[λ](28i)
il fattore di imperfezione implicita che tiene conto dei termini nonlineari trascurati dalla lineariz-
zazione assunta per il percorso principale
µi
k[λ] := −1
2λ2Φ!!!
bˆuˆu˙vk(28j)
e del contributo di eventuali imperfezioni addizionali nella geometria o nei carichi. Indicando
con ˜ul’imperfezione geometrica iniziale e con λ˜pl’imperfezione di carico, questi ultimi possono
essere espressi come
µg
k[λ] := −λΦ!!!
bˆu˜u˙vk,µ
l
k[λ] := λ˜p˙vk(28k)
Strutture caratterizzate da µk[λ] = 0 ∀kpresentano biforcazione e sono chiamate perfette. Si noti
anche come, una volta che i coefficienti dell’espansione siano stati gi`
a calcolati per una struttura
riferimento, con µg
k=µl
k=0∀k, gli effetti di imperfezioni addizionali possono essere facilmente
messi in conto calcolando i corrispondenti fattori di imperfezione con le (28k) e risolvendo di
nuovo l’equazione algebrica (28g).
Malgrado l’apparente complessit`
a, l’intero processo di soluzione (28) pu`
o essere implementa-
to in uno schema computazionale relativamente semplice e veloce. In effetti, il modo fondamentale
ˆ
u, definito implicitamente dalla (28b) si ottiene semplicemente come soluzione di un sistema li-
neare. Questa fase richiede l’assemblaggio e la decomposizione alla Cholesky della matrice K0e
corrisponde alla parte pi`
u costosa dell’analisi.
11
I modi critici ˙
vi, definiti implicitamente dalla (28c), possono essere ricavati facilmente con
algoritmi standard agli autovalori, una volta assunta la linearizzazione (24). Tuttavia, l’errore
introdotto da questa approssimazione pu`
o rivelarsi inaccettabile, specie se si considera che le
autosoluzioni {˙
vi,λi}sono riusate nel calcolo di quantit`
a derivate attraverso le equazioni (28e)–
(28h). Se si evita la linearizzazione, la (28c) diventa un problema nonlineare agli autovalori che
comunque pu`
o essere risolto con una variante del metodo delle potenze inverse che usa il seguente
schema iterativo !vj+1 =vj−˜
K−1K[λj]vj
λj+1 =1/||vj+1 || (29)
dove ˜
Krappresenta una opportuna approssimazione della K0(si veda [156] per i dettagli imple-
mentativi). Lo schema pu `
o essere visto come una riscrittura conveniente del metodo delle potenze
inverse: non richiede infatti l’assemblaggio esplicito della matrice K1, ma solo del vettore rispo-
sta incrementale K[λj]vjottenuto in modo standard a partire dai contributi degli elementi; inoltre
risulta insensibile ad errori numerici introdotti nella valutazione della K−1
0dal processo di de-
composizione alla Cholesky, in quanto la convergenza implica il raggiungimento della soluzione
esatta cercata (K[λ]v= 0). Lo schema pu`
o anche essere usato in un contesto nonlineare dove
mantiene le caratteristiche di velocit`
a e robustezza proprie del metodo delle potenze, ed accetta
approssimazioni anche grossolane nella definizione della matrice di iterazione ˜
K(il lettore pu`
o
riferirsi a [163] per una discussione sulle propriet`
a di convergenza del metodo e sulle possibili
alternative [164]–[167]).
I modi correttivi ¨
wij sono ottenuti, a partire dai modi critici vigi`
a determinati, come soluzione
dei problemi lineari
Kb¨
wij =¨
s[vi,vj](30a)
dove i vettori ¨
s[vi,vj]a secondo membro, definiti delle equivalenze
¨
sij δu:= Kb¨
w[vi,v