Limit and Shakedown
Analysis under Uncertainty
Manfred Staat1
ABSTRACT
Structural reliability analysis is based on the concept of a limit state function separating failure
from safe states of a structure. Upper and lower bound theorems of limit and shakedown
analysis are used for a direct definition of the limit state function for failure by plastic collapse
or by inadaptation. Shakedown describes an asymptotic and therefore time invariant structural
behavior under time variant loading. The limit state function and its gradient are obtained from
a mathematical optimization problem. The method is implemented into a general purpose FEM
code. Combined with FORM/SORM robust and precise analyses can be performed for structures
with high reliability. This approach is particularly effective because the sensitivities which are
needed by FORM/SORM are derived from the solution of the deterministic problem.
Key words: Shakedown, Direct plasticity, FEM, Mathematical programming, Structural reliability,
FORM/SORM
1. INTRODUCTION
Designing and assessing of structures the engineer has to
make decisions under uncertainty of the actual load carrying
capacity of the structure. Uncertainty may originate from
limited information about idealizations in the structural model
and from random fluctuations of significant physical properties.
The mechanical deterministic model and stochastic model
depend on the definition of limit states. For instance, if the limit
state of the structure is defined with respect to plastic collapse,
then Young’s modulus, hardening modulus and residual stress
need not be modeled as random variables, because they all
do not influence the limit load. Conversely, elastic buckling is
governed by Young’s modulus, residual stress, and geometry
imperfections. Structural reliability analysis deals with all
these uncertainties in a rational way. Safety assessment of
structures requires deterministic mechanical models and
analysis procedures that are capable of modeling limit states
accurately. Additionally a full stochastic model of the random
variables and numerical methods are also necessary for a
reliability assessment.
σther.
σ0
σ
σ
ε
ε
ratchetting
σ
low cycle fatigue
2
σ
collapse
ε
ε
1
elastic shakedown
pure elastic
behaviour
0
σmech.
σ0
1
Figure 1: The interaction diagram of the pressurized thin walled tube
under thermal loading is called Bree-Diagram [1], [7]
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The probabilistic procedures depend on the underlying
deterministic approach. Damage accumulation in low cycle
fatigue (LCF) or plastic strain accumulation in ratcheting
are evolution problems. In principle the possible structural
responses, which are presented as icons in the Bree-Diagram,
see Figure 1, may be reproduced in a detailed incremental
plastic analysis. However, this assumes that the details of the
load history (including any residual stress) and of the constitutive
equations are known. Often it is economically not justified or
even impossible to obtain all these necessary information.
Therefore, it is an advantage of limit and shakedown analysis
that it is a direct plasticity method which only considers the
limiting state. The upper bound method calculates the least
Khoa học & Ứng dụng
45
failure load and the lower bound method gives the largest
safe load. Both solutions converge to the same exact load
bearing capacity of the discretized structure (which may
not be precisely the limit for the continuum problem). In
more detail limit and shakedown analysis are optimization
problems with an objective function to be minimized
or maximized under some constraints. We present the
solution of the deterministic shakedown problem in a
form which has been developed in [2], [3], [4]. This kind
of primal dual shakedown analysis has been extended to
bounded linearly hardening plastic material in [5], [6] and
to shell problems in [7], [8]. Alternative numerical methods
are Sequential Quadratic Programming (SQP) with basis
reduction for the lower bound problem [9], [10], interior
point methods [36], Second Order Cone Programming
(SOCP) [11], [12] or the Linear Matching Method (LMM) [13].
Shakedown analysis is an exact method of classical
plasticity. However, geometrical nonlinearities [14], bounds
on inelastic deformations, and temperature dependent
Young’s modulus pose some open problems [15].
The reliability of most structures under time variant loads
such as cyclic loading will decrease or in other words
the failure probability will increase. One could model
the loading as a stochastic process and calculate the
probability of this process to outcross the safe load domain.
Shakedown analysis renders the reliability problem timeinvariant. This not only simplifies reliability analysis greatly
but also cuts costs of the analysis because only the convex
hull of the load domain and only little key information
on material behavior are needed. Pioneering work on
probabilistic limit and shakedown analysis has been
collected in [16]. Follow-up work was firstly restricted to
stochastic limit analysis of frames and linear programming
[17], [18], [19], [20]. We have used shakedown analysis to
calculate the so-called limit state function and reliability
methods to compute failure probabilities by calling the
limit state function in an inner loop. First Order Methods
(FORM) and Second Order Methods (SORM) are wellestablished analytical methods which approximate the
limit state function at the so-called design point (point
most likely failure) and calculate exact failure probabilities
for this approximation. FORM is also used to define partial
safety factors in the Eurocodes for a reliability management
with standard procedures using safety factors [37]. For
several reasons they appear best suited for shakedown
analysis: FORM is exact for linear limit state functions and
for shakedown analysis the limit state function is linear or
only weakly nonlinear, FORM/SORM solve optimization
problems like shakedown analysis does and are very
effective for high reliability and few uncertain variables. The
costly sensitivities of a nonlinear FEM problem need not
be provided because all needed sensitivities are already
available during the solution of the shakedown problem.
Therefore there is little argument to use modern stochastic
simulation methods instead of FORM/SORM here except
maybe of the potential occurrence of multiple design
points. Our probabilistic approach was first established
for the lower bound problem in [1], [9], [21] and then for
the upper bound problem for shells [7], [22], [23]. Other
46
Khoa học & Ứng dụng
contributions are [24], [25].
Stochastic programming is an alternative approach to
limit and shakedown analysis under uncertainty. Under
uncertainty the shakedown problem can be stated with
random objective function or with random constraints. In the
second approach, called chance constraint programming, a
probability is set with which the constraint has to be satisfied.
This has been considered for limit analysis of frames in
[26] and for any structure in [1]. For large FEM problems
the solution of the problem seems to be only feasible
if all stochastic variables are normally distributed and a
deterministic equivalent problem can be stated. The limit
load corresponding to a defined probability is obtained.
Engineering structures are no typical mass products
and thus the stochastic approach to uncertainty may be
questionable. One interesting alternative are fuzzy models
which are similar to human reasoning but may be even
more questionable for engineering structures. However,
fuzzy programming has been proposed for limit analysis in
[27], [28].
2. SHAKEDOWN FORMULATION
2.1. Upper Bound Problem
Inelastic structures under variable repeated or cyclic
loading may work in four different regimes, which are
presented in the Bree-diagram (figure 1) together with the
evolution of the structural response: elastic, shakedown
(adaptation), inadaptation (non-shakedown), and limit
state (plastic collapse). Since for the elastic regime there
are no plastic effects at all, whereas for the adaptation
regime the plastic effects are restricted to the initial loading
cycles and then they are followed by asymptotically elastic
behavior, both regimes are considered as safe working
ones and they constitute a foundation for the structural
design. We do not consider elastic failure such as buckling
or high cycle fatigue. The inadaptation phenomena such
as low cycle fatigue and or ratchetting should be avoided
since they lead to a rapid structural failure. At the limit
load the structure loses instantaneously its load bearing
capacity. Limit and shakedown analyses deal directly with
the calculation of load capacity or the maximum load
intensities that the structure is able to support.
Consider a convex polyhedral load domain and a special
loading path consisting of all load vertices (
) of . At each load vertex the kinematical condition may
not be satisfied, however the accumulated generalized
strains
over a load cycle in the time interval (
) must
be kinematically compatible. Let the fictitious elastic stress
be denoteWd by . In upper bound shakedown analysis
we calculate the exterior approximation of the shakedown
load domain α , i.e. we determine the minimum load
factor
for failure by non-shakedown. It converges to the
same load factor as the interior approximation of the lower
bound analysis which seeks the maximum load factor
for survival by shakedown:
.
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According to Koiter’s theorem, shakedown cannot occur
if a kinematically admissible velocity field can be found
so that the exterior work is larger than the internal plastic
dissipation [38]. In a weaker form the shakedown conditions
are considered over a cycle and the shakedown limit is the
smaller one of the low cycle fatigue limit, and the ratcheting
limit may be found by the following minimization:
(3)
Then numerical form of eq. (1) is
(4)
where
is the weight at Gaussian point ,
(the
superscript p is omitted for simplicity) is the strain vector
corresponding to load vertex , at point
(5)
(1)
where the total plastic energy dissipation over a cycle in a
perfectly plastic structure is
is the fictitious elastic stress vector corresponding to load
vertex k, at point i, u is the nodal displacement vector and
is deformation matrix. D and
are square matrices,
(2)
The upper bound of the shakedown multiplier
is the
solution of constraint nonlinear programming problem
(1). For the numerical solution we regularize the nondifferentiable dissipation function with a very small value ,
where
, such that
.
Constraint (1b) is the definition of plastic strain
accumulation. The plastic strain rate may not necessarily
be compatible, but
must be compatible. This is
expressed by constraints (1d) and (1e). Constraint (1c) is the
incompressibility condition and (1g) is the normalized work
of the external actions.
2.2. Problem Discretization
The whole structure is discretized into
finite elements
with
Gaussian points, where
is number
of Gaussian points in each element. If the load domain
is convex, it is sufficient to check if shakedown will happen
at all vertices of . So the load domain can be discretized
into a finite number of load combinations ,
, and
is the total number of vertices of . By these
discretizations, the shakedown analysis is reduced to
checking shakedown conditions at all Gaussian points and
all load vertices , instead of checking for whole structure
and all load histories in the domain . For the later
probabilistic analysis we write the yield stress as product
of a reference value
and a stochastic stress variable ,
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(6)
For the sake of simplicity, we define some new plastic
strain , fictitious elastic stress , deformation matrix ,
regularization parameter respectively as
(7)
Then eq. (4) becomes
(8)
Dealing with the nonlinear constrained optimization
problem above, let us write the penalty function for the
compatibility condition (8b) and the incompressibility
condition (8c)
Khoa học & Ứng dụng
47
equilibrium condition
and is a time
invariant residual stress. The structural shakedown takes
place due to development of permanent residual stresses
which, imposed on the actual stresses, shift them towards
purely elastic behaviour. Residual stresses are a result of
kinematically inadmissible plastic strains introduced to
the structure by overloads. They clear out effects of all
preceding smaller loads. They also avoid any plastic effects
in the future provided that the loads are smaller than the
initial overload. Therefore, in limit and shakedown analyses
the knowledge of the exact load history is not necessary.
Only the load (limits) count and the envelopes should
be taken into consideration which makes the reliability
problem time-invariant.
(9)
where is a penalty parameter such that
sake of simplicity, let be constant.
. For the
Following (8) the modified kinematic formulation (8)
becomes
(10)
The corresponding Lagrange function of (9) is
By duality
(11)
(14)
both optimization problems yield the exact solution
because the maximum safe load and the minimum
overload are the same.
The so-called Karush-Kuhn-Tucker (KKT) conditions and
the strict complementary slackness must hold at the
optimal point in the form
3. STRUCTURAL RELIABILITY ANALYSIS
3.1. Limit State Function
(12)
By applying Newton’s method to solve the system (12) we
obtain the Newton directions
and
, which assure
that suitable steps along them will lead to a decrease of
the objective function
in (8). If the relative improvement
between two steps is smaller than a given constant, the
algorithm stops and the shakedown limit factor is obtained.
Details of the iterative algorithm can be found in [7].
If
= 1 the problem reduces to limit analysis and
denotes the limit load factor. Otherwise an upper bound of
the shakedown factor
is obtained.
The behavior of a structure is influenced by various typically
uncertain parameters (loading type, loading magnitude,
dimensions, or material data, ...). All parameters are
described by random variables
which are collected
in the n-dimensional random vector of basic variables
. We will restrict us to those basic variables
for which the joint density
exists and the
joint distribution function
is given by
(15)
The deterministic safety margin R - S is based on the
comparison of a structural resistance (threshold) R and
loading S . With R, S functions of the structure fails for any
realization with non­positive limit state function
, i.e.
2.3. Lower Bound Problem
There is the dual lower bound problem which can be
derived from Melan’s lower bound theorem [39], [40]. The
discretized problem is
(13)
Here the fictitious elastic stress
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Khoa học & Ứng dụng
is calculated from the
(16)
Different definitions of limit state functions for various
failure modes are suggested in Table 1. The limit state
function
defines the limit state hypersurface
which separates the failure region
from the
safe region. Figure 2 shows the densities of two random
variables R, S which are generally unknown or difficult to
establish. The failure probability
is the
probability that
is non-positive. This means Số 19 - 2012
(FORM/SORM)
Figure 2: Basic R - S problem in
,
(17)
presentation on one axis,
In general it is not possible to calculate analytically from
the failure integral, because of the complex structure of the
failure region V . Additionally, it is not necessary that the
limit state function is given explicitly but only in algorithmic
form. A FE-analysis of structures with one initial data set
gives only one value of the limit state function.
For the probabilistic shakedown problem the loads
,
, and the yield limit
are considered as
random variables. Let us restrict ourselves to the case of
homogeneous material, where the yield limit is the same
at every point of the structure. Then we always can write
where is a constant reference value and Y is a
random variable.
For the general cases, there are analytical methods like
FORM/SORM and several simulation methods to compute
the failure probability . Direct Monte Carlo Simulation
(MCS) becomes increasingly expensive with the increase
of the structural reliability. Acceptable failure probabilities
might be in the range of
to
. They are even
much lower in nuclear reactor technology. For a validation
that the failure probability is less than an accepted limit
, the sample size required for direct MCS must be at
least
leading to a minimum sample size in the range
of
to
. Such a large number exceeds particularly
for complex FE‑models, available resources by far. The
numerical effort can be reduced considerably by Response
Surface Methods (RSM) and by Importance Sampling,
Directional Sampling [41], Latin Hypercube Sampling or
a combination of these variance reduction methods. An
obvious strategy is to concentrate the simulation around
the design point so that about 50% of the simulations fall
into the failure region. However, the most effective analysis
is based on First- and Second-Order Reliability Methods
(FORM/SORM) if gradient information is available.
First and Second Order Reliability Methods (FORM/SORM)
are analytical probability integration methods. Therefore,
the defined problem has to fulfill the necessary analytical
requirements (e.g. FORM/SORM apply to problems, where
the set of basic variables is continuous). The numerical
effort depends on the number of stochastic variables but
not on (contrary to MCS).
The failure probability is computed in three steps.
Transformation of the vector of basic random variables
into a vector of uncorrelated basic random variables ,
Linear or quadratic approximations
region in the U-space,
of the failure
3.2. First- and Second-Order Reliability Methods
Table 1: Different limit state functions in structural reliability
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Khoa học & Ứng dụng
49
Computation of the failure probability due to the
approximations
.
FORM/SORM is an exact solution of the approximated
stochastic problem.
Transformation
The basic variables
are transformed into standard
normally distributed variables
(
:
normal distribution with zero means,
, unit variance,
, and independent). Such a transformation is always
possible for continuous random variables. If the variables
are mutually independent, with distribution functions
, each variable can be transformed separately by the
Gaussian normal distribution
into
(definition of
in eq. (20) below). For dependent
random variables other isoprobabilistic transformations
(generalized Nataf [29] or Rosenblatt [30]) can be used.
The function
is the corresponding limit
state function in U-space. The dimension of the U-space
depends on the dependencies of the random variables
and is not necessarily equal to the dimension of the
X-space. However, the transformation to U-space is exact
and not an approximation [31].
Approximation
In FORM,
expansion
with
called design point
is approximated by its Taylor
at the so(so that
)
(18)
The failure region V is linearly approximated by
The vector
Figure 3: Safe and failure regions with their linear and quadratic
approximations in U-space.
For a linear limit state function FORM gives the exact
failure probability
. If the limit state function
is not linear in U-space a quadratic approximation of
the failure region V gives closer predictions of . These
second order methods (SORM) may be either based on
a nonlinear optimization algorithm or on correction of a
FORM analysis. FORM/SORM give the exact solution of an
approximate problem.
A quadratic approximation
generated in SORM [7] and
of the failure region V is
(21)
where H ist the scaled Hessian matrix of the limit state
function and
(19)
is proportional to the sensitivities
The
failure event
is equivalent to the event
such that an approximation of the failure probability
is
,
(22)
(20)
because the random variable
is normally distributed.
The failure probability depends only on , such that
it is called safety (or reliability) index. If it is possible to
derive analytically from the input data, the probability
is calculated directly from the cumulative distribution
function of the standard normal distribution (Gaussian
distribution function) .
50
Khoa học & Ứng dụng
with are n - 1 principle curvatures at the design point. The
calculation of
normally needs the second derivatives of
the limit state function.
3.3. Calculation of the Design Point
In order to apply FORM/SORM, the design point must be
identified. This leads to a nonlinear, inequality constrained
optimization problem as follows
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(23)
where a coefficient 1/2 is added for technical reasons.
Many algorithms have been suggested to deal with this
problem. In and [1], [9], [19], [21] good results have been
obtained with Rackwitz’s simple gradient search algorithm,
which is based on a linearization of the limit state function
at each step. However, this algorithm is only guaranteed to
converge towards a locally most likely failure point in each
sequence of points on the failure surface if the safe region
is quasi-convex or concave.
A more general algorithm is the Sequential Quadratic
Programming (SQP), [32], [33]. The SQP method, also
known as successive or recursive quadratic programming,
employs Newton’s method (or quasi-Newton methods) to
solve the KKT conditions for the original problem directly.
As a result, the accompanying subproblem turns out to
be the minimization of a quadratic approximation to the
Lagrangian function optimized over a linear approximation
of the constraints.
Consider the inequality constrained nonlinear optimization
problem
(24)
where x is an n-dimensional parameter vector containing
the design variables and j the set of active constraints.
A basic SQP algorithm optimizes the Lagrangian
function
over a linear
approximation of the constraints
, where
is a matrix of Lagrange multpliers.
In each iterate
an appropriate search
direction
is defined as the solution to: minimize
linearized contraints.
(25)
is obtained and must be solved in each iteration. Let be
the optimal solution,
the corresponding multiplier of
this subproblem, then the new iterate is obtained by
where
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(26)
is a suitable step length parameter.
(27)
The KKT conditions for this problem are
(28)
Applying Newton’s method for solving these equations
leads to the system
(29)
with the solutions
(30)
as the Lagrange multiplier and search direction for the next
iteration. If
then
together with the Lagrange
multiplier
yields the optimal solution for the problem
(23), i.e. the design point is actually found. The calculation
of necessary derivatives is considered as the sensitivity
analysis and will be discussed in the next section.
There is only the potential occurrence of multiple design
points which need extra consideration in FORM/SORM [7],
[18], [19].
4. SENSITIVITY OF THE LIMIT STATE FUNCTION
under the
Then a quadratic programming subproblem of the form
Applying SQP for solving the optimization problem (22),
the quadratic programming sub-problem (25) at iteration k
simplifies with
and
:
The reliability analysis described above can be carried out
now with the help of a probabilistic limit and shakedown
analysis. From the results of the finite element analysis,
the necessary derivatives of the limit state function based
shakedown analysis can be determined analytically. This
represents a considerable reduction of computing time
comparing with the other methods, e.g. the difference
approximation, and makes such an efficient and cost-saving
calculation of the reliability of the structure. Contrary to the
numerical calculation, the analytical calculation is faster
and more exact. The necessary data for the calculation
of the derivatives are available after the execution of the
deterministic shakedown analysis since they are based on
the upper bound load factor . The derivatives must be
calculated at each iteration in the physical x space and then
transformed into the standard Gaussian u space by
Khoa học & Ứng dụng
51
(31)
4.1. Definition of the Limit State Function
where
is the stress vector at Gauss’ point i and load
vertex k due to the
load case. The derivatives of the
limit load factor
versus yield stress variable Y can be
determined in the same way and have the form
As mentioned above, the limit state function contains
the parameters of structural resistance and loading. If we
defined the limit load factor
as follows
(32)
where , are limit load and actual load of the structure.
For the sake of simplicity, the limit state function can be
normalized with the actual load and then becomes
(33)
4.3. Second Derivatives of the Limit State Function
In the SORM algorithm, as discussed above, the Hessian
matrix which summarizes second partial derivatives of the
limit state function is needed. They can be obtained from a
direct analytical derivation of the first derivatives. By taking
derivatives of (38) versus
, Y and referring to (37) one has
The load factor
can be calculated by the nonlinear
program (8). It can be seen that the limit state function is
the function of yield stress variable and load variables.
The actual load is defined in n components by using the
concept of a constant reference load
as follows
(35)
From (6) and (35), the corresponding normalized stress
fields are obtained
,
(39)
(40)
(34)
where
is the realization of the
basic load variable
(
). The corresponding actual stress field can also
be described in the same way
(36)
For
we refer to Tran et al. It is obvious
from (40) that for the case of homogeneous material the
shakedown load factor is actually a linear function of
the yield stress, i.e. the optimization variables , u are
independent of Y. Moreover, the Lagrange multiplier is
a linear function of Y.
A flow chart of the algorithm is presented in Figure 4 which
also shows the derivatives which are needed because of
the transformation in the u space. The sensitivity analysis
and algorithm for the lower bound method are presented
in [1], [9].
4.2. First Derivatives of the Limit State Function
4.4. Special Case of Probabilistic Shakedown Analysis
In order to get the sensitivities of the limit state function, one
Consider a special case of probabilistic shakedown
analysis, where is a stochastic one-parameter load, i.e.
with a reference load
we have
and
where is the stress vector due to referent load . In this
case, the nonlinear optimization (10) becomes
requires that
i.e. the partial derivatives of the limit load factor
must
be available. Let
be the solutions of the
optimization problem (8). At this optimal point, the first
derivative of the limit load factor versus
load variable
can be calculated as follows [7]
52
Khoa học & Ứng dụng
(37)
(38)
(41)
where NV = 2 for shakedown analysis and NV = 1 for limit
analysis. If we now define a new strain vector
Số 19 - 2012
(46)
These above derivatives can also be easily obtained by using
the same procedure as applied for the material variable
Y. By this way, it is easy to prove that
,
, and then we find again the above expressions
finally.
5. NUMERICAL APPLICATIONS
The probabilistic limit and shakedown algorithm described
above has been programmed and implemented in the
finite element package Code_Aster in [7]. The 4-noded
quadrangular isoparametric flat shell element, which is
based on Kirchhoff’s hypothesis, was applied. Higher
order shell elements were not available in former versions
of Code_Aster, [34]. In all numerical examples, the structures
are made of elastic-perfectly plastic material. For each
test case, some existing analytical and numerical solutions
found in literature are briefly represented and compared.
The finite element discretisations were realized by the
personal pre- and post-processor GiD, [35].
5.1. Square Plate with a Central Hole
A square plate with central circular hole as shown in Figure
5 is considered. This is a standard test for deterministic
shakedown analysis. Under uncertainty it was also studied
analytically and numerically in [1], [9], [21] using lower
bound limit and shakedown analysis, FORM with Rackwitz’s
algorithm and volume finite elements. The two following
cases are examined.
Figure 4: Flowchart of the probabilistic limit and shakedown analysis, [7]
and new displacement vector
rewritten as follows
, then (41) can be
(42)
with
(43)
Since the constraint is now no longer dependent on the
load variable X, it follows that
(44)
(45)
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Khoa học & Ứng dụng
53
=
β
Figure 5: FE-mesh and geometrical dimensions of square plate, [7]
Limit load analysis
In case of a square plate of length L with a hole of diameter
2R (see Figure 5) and R/L = 0.2 subjected to the uniaxial
tension the exact limit load for plane stress is given by
with the yield stress .
Thus the limit load
depends linearly of the realization
of the yield stress basic variable X. The load S is a
homogeneous uniaxial tension on one side of the plate.
The magnitude of the tension is the second basic variable
Y. The limit load of every realization of Y is
(47)
(1 − R L ) µr − µs
=
2
(1 − R L ) σ r2 + σ s2
0.8µr − µ s
0.64σ r2 + σ s2 (51)
In Table 2 the probability
of a plastic collapse from the
numerical limit analyses are compared with the analytic
values resulting from the exact solution
for the case that both variables are normally distributed
with standard deviations
and
. The
different accuracies of the deterministic limit analyses
seem to cause the small differences between the FORM
solutions. The numerical error involves both deterministic
shakedown analysis and reliability analysis and is very small
and acceptable.
for
because
the density of the normal distribution is symmetric. For
both normal and log-normal distributions SORM solutions
are almost the same of FORM due to weak linearity of
the limit state function. The weak nonlinearity may occur
in the case of multi-parameter loads or by extreme value
distribution of the random variables. But to our experience
the improvement by SORM is rarely needed.
Shakedown load analysis
For this case, the tension
varies within the range
and only the amplitudes but not the uncertain
complete load history influences the solution. Consider
the case where the maximal magnitude
is a random
variable and the minimal magnitude is held zero. From
the
deterministic analysis we got the
shakedown load . The numerical probability of failure
for normally distributed variables are presented in Table
2, compared with the analytical solutions, which are
calculated from
The limit state function is defined by
(48)
The normally distributed random variables X and Y with
means
and variances
respectively, yield with
and
the transformed limit
state function
(49)
With realizations
U it may be written
of the new random variable
(50)
such that the safety index
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Khoa học & Ứng dụng
(with R/L =
0.2) is
(52)
It is worth to note that the shakedown probabilities of failure
are considerably larger than those of limit analysis. Thus,
the loading conditions should be considered carefully
when assessing the load carrying capacity of the structure.
5.2. Pipe-Junction under Internal Pressure
A pipe-junction is considered under internal pressure
which can be constant or vary within the range
.
The FE mesh and the geometrical data are introduced in
Figure 6. Both yield limit
and pressure are considered
as random variables. Numerical deterministic analyses lead
to a collapse pressure
and a shakedown
limit
. If both material resistance and
load (stress) random variables are supposed to be normally
distributed with means
,
and standard deviations
, respectively, then the reliability index may be given [8]
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[7]
[7]
[1], [9]
Table 2: Numerical results and comparison for normal distributions (
[7]
[7]
)
(53)
Numerical failure probabilities for limit analysis
(probabilities of plastic collapse) for normal distributions
are presented in Table 3 compared with the corresponding
analytical solutions. Both random variables have standard
deviations
. It is shown that the numerical
results of FORM and SORM are very close to the exact
ones. Besides normal and log-normal distributions, the
Weibull distribution is frequently used for modeling loads
and resistance. Table 3 also shows the numerical results
of FORM/SORM for the case that both random variables
have a Weibull distribution. The Weibull distributions lead
to much larger failure probabilities. No analytical results
are known. The limit state function is weakly nonlinear
after the transformation into the u space. In this case
SORM gives slightly improved results. Table 4 shows the
failure probabilities for shakedown analysis which are
considerably larger than those of limit analysis. Comparing
the two distributions similar observations as for limit
analysis are made.
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D=39
s=3.44
d=15 t=3.44
Figure 6: FE-mesh and geometrical dimensions of pipe-junction, [7]
6. CONCLUSIONS
A probabilistic model of finite element limit and shakedown
analysis has been presented, in which the loading and
strength of the material are considered as random
variables. The procedure involves a deterministic limit and
shakedown analysis for each probabilistic iteration, which
is based on the kinematical approach. Different kinds
of distribution of basic variables can be considered for
calculation of the failure probability of the structure with
FORM/SORM. A nonlinear optimization was implemented,
which is based on the Sequential Quadratic Programming
for finding the design point. Nonlinear sensitivity analyses
are also performed for computing the Jacobian and the
Hessian of the limit state function.
Khoa học & Ứng dụng
55
Table 3: Numerical results for limit analysis,
Table 4: Numerical results for shakedown analysis, The advantage of the method is that sensitivity analyses
are obtained directly from a mathematical optimization
with no extra computational cost. Two examples were
tested against literature with analytical method and with a
numerical method using volume elements. The proposed
method appears to be capable of identifying good
estimates of the failure probability, even in the case of very
small probabilities. The limit state function is linear with
56
Khoa học & Ứng dụng
[7]
[7]
the limit and shakedown analysis. Then the use of FORM is
sufficient and SORM is only necessary if the linearity is lost
after the transformation into the U-space.
1Faculty of Medical Engineering and Technomathematics,
Aachen University of Applied Sciences
Heinrich-Mußmann-Str. 1, 52428 Jülich, Germany
e-mail: [email protected], www.fh-aachen.de
Số 19 - 2012
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