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Multi-model reliability-based design optimization of structures considering

the intact configuration and several partial collapses

Article in Structural and Multidisciplinary Optimization · March 2018

DOI: 10.1007/s00158-017-1789-y

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Multi-model reliability-based design
optimization of structures considering the
intact configuration and several partial
collapses
C. Cid, A. Baldomir, S. Hernández & L. E. Romera

Structural Mechanics Group, School of Civil Engineering, University of A Coruña

(+34) 981 167 000
clara.cid.bengoa@udc.es, abaldomir@udc.es

Abstract: Some structures require keeping a specific safety level even if part of their elements have
collapsed. The aim of this paper is to obtain the optimum design of these structures when uncertainty
in some parameters that affects to the structural response is also considered. A Reliability-Based
Design Optimization (RBDO) problem is formulated in order to minimize the mass of the structure
fulfilling probabilistic constraints in both intact and damaged configurations. The proposed
methodology combines the formulation of multi-model optimization with RBDO techniques
programmed in a Matlab code. Two application examples are presented consisting of a two-
dimensional truss structure with stress constraints as well as a curved stiffened panel of an aircraft
fuselage subjected to buckling constraints.

Keywords: fail-safe design, damaged configurations, uncertainty, reliability index,

optimization, multi-model, RBDO.

1. Introduction

In civil and aerospace engineering, designs are subjected to very demanding

requirements, since structural damages may lead to loss of goods and even human
lives. Therefore, the possibility of a partial collapse must be contemplated into the
design process of these structures, being able to guarantee their survival during
repairing tasks or evacuation procedures, among others. This idea is supported with
a considerable number of codes and design regulations which provide guidance for
compliance, establishing the limit states that a structure must satisfy. Some
examples in civil engineering are the specifications of the Post-Tensioning Institute
of U.S. with regards to cable-stayed bridges, establishing that “cable-stayed bridges
shall be capable of withstanding the loss of any one cable without the occurrence

1
of structural instability” (2007) or the case of high-speed train bridges formed by
steel-lattice, where the structure must be safe after a bar breaking due to an impact.
In Aerospace industry, the Federal Aviation Administration published the Advisory
Circular 25.905-1 (2000) specifying that “the airplane must be capable of
successfully completing a flight during which likely damage occurs …”. This
design regulation aims to minimize the hazards that could occur to an airplane if a
propeller blade fails and its impact causes a loss of structure in the fuselage.
On the other hand, uncertainties are inherent to real-world systems, being the
applied loads or the material properties some of the main sources of variability.
Nevertheless, this uncertainty can be characterized through random variable
distributions allowing to include these real data into the structural analysis. As a
result, more reliable structural responses can be obtained, leading to better designs.
Traditionally, uncertainty was taken into account by using partial safety factors
based on the engineer experience, not being possible to quantify the structural
reliability. On the contrary, Reliability Analysis (RA) provides information about
the probability of failure of a structure with regards to a limit state when uncertainty
of some structural parameters is known. Several RA methods have been developed
including moment-based methods and sampling methods (Choi et al., 2007; Zhao
and Ono, 2001) or approximation techniques such as Polynomial Chaos Expansion
(Choi et al., 2004). In RA literature there is a wide range of applications to very
different structures as marine structures (Young et al. 2010), bridges (Baldomir et
al. 2013) or aircrafts (Hu et al. 2013).
Statistical information on certain structural parameters can be also included into
the optimization process giving rise to the probabilistic optimization methods, also
known as Reliability-Based Design Optimization (RBDO). They combine
Reliability Analysis and Design Optimization, seeking for the best compromise
between cost and safety. The objective is to obtain the optimum design of a structure
guaranteeing a fixed safety level with reference to several limit states. A precise
overview of these techniques can be found in Aoues et al. (2010). RBDO has been
successfully applied to several multidisciplinary areas, as automotive (Moustapha,
M. et al. 2016 or Cid Montoya et al. 2015), where the RBDO with surrogate models
was conducted in a crashworthiness analysis. In civil engineering, Kusano et al.
(2014, 2015) minimize the weight of the main cable and the bridge deck of long-
span suspension bridges considering probabilistic flutter constraints and Saad et al.

2
(2016) where the RBDO is performed on the life cycle cost of reinforced concrete
bridge elements under coupled fatigue-corrosion deterioration processes. In
aerospace some examples are the probabilistic optimization of the transmission loss
of a composite ribbed panel (D’Ippolito R. et al. 2012) or the application of RBDO
to composite stiffened panels in post-buckling regime using discrete design
variables (Lopez C. et al. 2016). This last research considers the progressive failure
of the composite stiffened panel as a probabilistic constraint, but does not consider
the loss of material associated to a partial collapse of the structure.
As exposed previously, a set of additional configurations need to be considered
into the design process of certain structures apart from the intact model, since they
represent different critical situations where structural responses must be controlled.
In such case, several finite element models need to be simultaneously considered
into the optimization process leading to a significant increment of structural
responses to be computed. This strategy, also known as multi-model optimization
technique, has been successfully applied to different fields. For example, Baldomir
et al. (2015) carried out the multi-model optimization of cable weight in multi-span
cable-stayed bridges, considering two stages of the bridge construction: the critical
cantilever stage and the full bridge model. Moreover, Baldomir et al. (2012) also
conducted the multi-model deterministic size optimization of shell structures
considering several incomplete configurations, checking the structural safety using
partial safety factors. Zhou and Fleury (2016) proposed a multiple model
optimization technique to carry out the topology optimization of general 3D
structures, introducing the concept of “fail-safe design”, firstly used by Jansen et
al. (2014).
In the present research RBDO techniques have been combined with multi-model
optimization. As a result, the optimum design meets the regulations that
contemplate accidental situations where part of the structure is lost. The novelty of
this research is the strategy proposed to solve a structural engineering problem
which aims to minimize the weight of a structure considering the existence of
several possible partial collapses and at the same time setting the maximum value
of the probability of failure when uncertainty in some parameters is known. For that
purpose a set of existent techniques need to be assembled. In order to perform the
RBDO procedure with multi-model optimization, the Sequential Optimization
Reliability Assessment method (SORA) (Du X. et al., 2004) has been implemented

3
in a MATLAB Code (MATLAB 2013). All the structural responses are obtained
through the commercial software Abaqus (2014) or OptiStruct (Altair OptiStruct
2013). The Sequential Quadratic Programming algorithm (SQP) implemented in
the MATLAB function fmincon was used as optimization tool method in one of the
examples. In the other one, the optimization algorithm SQP implemented in
Optistruct was used. For the second stage of the SORA method, the RA loop was
implemented in Matlab using the Advanced Mean Value method (AMV) (Wu 1990,
1994).
It must be highlighted that in this research no new algorithms have been
developed, but a useful optimization strategy is proposed to solve a novel
engineering problem where existent RBDO methods are combined with structural
models that contemplate the possibility of partial collapses. This approach allows
to minimize the weight of the structure under these failure situations considering
the present uncertainty in some parameters that affects to the structural responses
of both intact and damaged configurations. The methodology proposed has been
tested in a two-dimensional truss structure and in a curved stiffened panel of an
aircraft fuselage.

2. General formulation of the RBDO problem

considering multi-model configurations

Consider a finite element (FE) model and D incomplete configurations of a

structure, as shown in Figure 1. An identification number d is assigned to each
model where d=0 corresponds to the intact configuration. A vector d is defined with
the set of design variables associated with the structural properties to be designed.
Since only the intact structure is going to be optimized the design variables are
associated with this configuration. Nevertheless, constraints are imposed on the
entire set of configurations, being necessary to obtain the structural responses in all
of them. This fact forces us to link the properties associated with each design
variable with the same properties of the incomplete configurations. Therefore, when
the optimization algorithm changes a property of the intact structure, the same
change is performed in the damaged models.

4
 d=0 d=1 d=2 … d=D

Figure 1. Intact and damaged configurations of a generic FE model.

On the other hand several uncertainty parameters are taken into account into the
optimization process through the definition of a vector x with the set of random
variables. In addition to the deterministic constraints gjd≤0, probabilistic constraints
are established over the limit states Gid considered into the RBDO problem. Finally
the probability of failure of each limit state must be set, being possible to require
different values Pf i depending on the limit state and the structural configuration.
Thus, the general multi-model RBDO problem can be defined as:

(
min F d0 , x0 ) (1a)
subject to


( )
P Gi d dd , xd ≤ 0 < Pfd
 i
i=
1,..., md ; d = 0,..., D (1b)

( )
g j d dd ≤ 0 j=
md + 1,..., M d ; d = 0,..., D (1c)

being F the objective function, d the superscript associated with a specific

configuration, that varies from 0 (intact model) to D (number of damaged
configurations), md the number of probabilistic constraints for the configuration d
and Md the total number of deterministic and probabilistic constraints for the
configuration d. Additionally, P[ - ] denotes the probability operator and Pf is the
probability of failure of the probabilistic constraint.
The probability of failure Pf associated to a limit state is directly linked to the
reliability index β defined by Cornell (1969). This index is defined as the shortest
distance between the mean value of the random variables and the limit state surface
G(x)=0. The point on the limit state surface is denoted as the Most Probable failure
Point (MPP). That is, the most probable value of the random variables that can
produce the overcoming of a limit state. When the random variables follow a normal
distribution, the limit state function is linear and the random variables are not
correlated, G(x) can be considered as normally distributed. After doing the
transformation to the standard normalized u-space, Pf can be obtained as expressed
in Eq. (2), being Φ the standard normal distribution of the limit state function G.

5
 =Pf P G ( x ) ≤ 0  =
Φ ( −β ) (2)

Among the methods to solve the RBDO problem of Eq. (1) it has been
implemented the Sequential Optimization Reliability Assessment method (SORA)
(Du X. et al., 2004) due to its decoupled nature. That is, the RBDO problem is
transformed into a sequence of Deterministic Optimization (DO) and Reliability
Analysis (RA) until convergence of both iterative processes. This decoupled
method avoid the use of traditional nested optimization loops, like Reliability Index
Approach method (RIA) (Enevoldsen I., et al., 1994) or Performance Measure
Approach method (PMA) (Tu J. et al., 1999), and significantly reducing the
computational effort. The flowchart of this algorithm is presented in Figure 2 along
with the corresponding formulation of the DO problem as well as the RA problem
solved through the Advanced Mean-Value method (AMV) (Wu 1990, 1994). The
above mentioned RBDO methods need to perform a reliability analysis phase in
order to evaluate the probabilistic constraints. MPP for each iteration of the
optimization algorithm is obtained by linearizing the limit state function in the point
of study making possible the use of these methods even when the limit state function
in nonlinear, as in most engineering problems.

Figure 2. General flowchart of SORA method to perform the multi-model RBDO.

6
 It must be highlighted that the optimum solution of the RBDO problem defined
in Eq. (1) can result from active constraints in different structural configurations,
being necessary to include simultaneously the constraints associated with the intact
and damaged configurations into the optimization process.
In order to clarify the purpose of the multi-model RBDO technique, Figure 3
shows a generic 2D graphical example representing the optimum solutions for the
following cases: Deterministic Optimization (DO) of the intact model -point A-,
DO considering partial collapses -point B-, and RBDO with partial collapses -point
C-, being the design variables d1, d2, and the random variables x1, x2. In the
particular case that the design variables are the random variables at the same time,
the reliability analysis phase can be represented in the same space that the
optimization phase.
Two configurations have been considered in Figure 3: the intact structure and
one partial collapse, referenced as 0 and 1, respectively. In this case, two limit states
1 and 2 of the previous configurations have been represented. For the deterministic
optimum solution considering the intact model exclusively (A), two limit states
associated to the intact model (g10 and g20) are active. When a partial collapse is
introduced, the optimum solution for Deterministic Optimization (B) has active
constraints in both the intact and damaged model (g10 and g21). As a more restrictive
constraint is satisfied (g21), the final design B is heavier than A. By performing the
probabilistic optimization with partial collapses, the optimum solution corresponds
to C, being the active constraints G10 and G21, represented through dashed lines. As
can be observed, a target safety index βT is guaranteed with respect to the
deterministic constraints g10 and g21, resulting in a weight increment compared to
B solution. The points C’ and C’’ correspond to the Most Probable failure Points
(MPPs) associated with the active constraints in the optimum design C. The point
C’ is the MPP associated to the limit state 1 of the intact model 0 and C’’ to the
MPP associated to the limit state 2 of the damaged configuration 1.

7
Figure 3. Illustrative example of DO design of intact model (A), with damages (B) and multi-model
RBDO design considering active constraints in both the intact and the damaged configuration (C).

3. Implementation of the RBDO algorithm

considering multi-model configurations

The RBDO problem exposed in Eq. (1) has been solved by programming a
MATLAB code (MATLAB 2013) implementing the SORA method, which
performs two separated steps sequentially until convergence, as can be seen in
Figure 2. The convergence criterion of the problem is defined as the absolute
difference in the objective function within two consecutive RBDO iterations, as
expressed in Eq. (3).
F k +1 − F k ≤ ε (3)

where k and k+1 are two consecutive iterations of the RBDO algorithm and ε is the
maximum convergence criterion value, which is set to ε = 1·10-3 in the following
application examples. It has been verified that this convergence criterion provides
good enough results when comparing to those obtained with a value of ε=1·10-5.
The steps required to perform the RBDO method are discussed below:

8
 I. Multi-model Deterministic Optimization loop (DO)
In this iterative process only the design variables (d) are modified, involving
both the intact and damaged configurations. Throughout the whole procedure the
values of random variables (x) remain constant, corresponding to their mean values
(x0=μx) for the initial iteration (k=0). In the following iterations of the RBDO
algorithm these values will be the MPP of each limit state i, obtained in the
corresponding RA, apart from the initial values of the design variables, that will
correspond to those obtained in the previous multi-model DO.
In the first example of this research the DO problem is solved using the
commercial software OptiStruct (Altair OptiStruct 2013). In the other example the
DO problem is solved through the gradient-based SQP algorithm implemented in
the MATLAB function fmincon. When the convergence criterion is achieved the
algorithm stops. The MATLAB code makes external calls in parallel to a FEM
analysis software in order to obtain the structural responses for evaluating the
design constraints in both intact and damaged configurations.

II. Multi-model Reliability Analysis loop (RA)

It was implemented using the Advanced Mean-Value method (AMV), which is
an inverse MPP search algorithm where the safety index βT is imposed while the
limit state G must be positive so that the random variables are in the safe region.
The gradients of the limit state functions are calculated using the finite difference
method. In this iterative process only the random variables (x) are modified, being
the design variables (d) those obtained from the previous DO step in the same
iteration k of the RBDO algorithm.
This method aims to obtain the new MPPs associated to each limit state i, which
will be used in the next DO iteration of the SORA cycle. It is important to remark
that it is necessary to assess the reliability analysis for each limit state i and
configuration d, performing a total of i x d RA loops, leading to a high
computational cost. As a result, they can be implemented more efficiently in
parallel to speed up the process, reducing considerably the computational effort.
The AMV (formulated in Figure 2) requires a previous transformation of the
random variables to the uncorrelated normalized u-space (Hasofer and Lind 1974).
The initial value of random variables corresponds to its mean value. After doing the
transformation to the u-space, the limit state function and their gradients are
evaluated through the structural responses externally obtained from the FEM

9
analysis software. These tasks are easily parallelizable for the whole set of limit
state functions in order to reduce the computing time of the procedure. The
normalized gradient vector is calculated as:

n =
r ( )
∇G u r
(4)
∇G ( u ) r

being r the current iteration of the AMV algorithm. Afterwards, the new design
point is calculated through the expression:

u r +1 = − β T ⋅ n r (5)
This iterative process is repeated until convergence in the value of u. After
convergence of the AMV algorithm the final value ur+1 associated to the i limit state
is stored as the MPP value for the next iteration k+1 of the SORA cycle. The
convergence of the AMV is reached when the absolute difference of the limit state
function between two consecutive iterations of the AMV algorithm is reached.
G r +1 − G r ≤ ξ (6)

where ξ is the maximum convergence criterion value, which is set to ξ = 1·10-3 in

the following examples. As in the DO phase, it has been verified that this
convergence criterion is enough to provide accurate results.
A more detailed flowchart of the algorithm implementation is shown in Figure
4. The k index makes reference to the global loop iteration of the SORA method,
while the r index refers to the internal loop iteration of both DO and RA. When the
RBDO algorithm stops, the final values of the design variables and the MPPs
associated to the active constraints (G(x)=0) are obtained, verifying that the
remaining conditions are passive constraints (G(x)>0). In order to give a more clear
explanation of this sentence Figure 5 was added, using the same notation as in
Figure 3.

10
Figure 4. Flowchart of SORA to perform the multi-model RBDO.

11
Figure 5. Graphic representation of the MPPs obtained after convergence of the RBDO process.

In this Figure, the RBDO design corresponds to the point C. The active limit
states are g1 and g2, since the reliability index β1 and β2 are equal to the target
reliability index βT demanded in the probabilistic optimization. The shortest
distance between the point C and the active limit states g1 and g2 results in the Most
Probable failure Points (MPPs) C’ and C’’, respectively. If we evaluate the active
constraints in their corresponding MPPs (g1 in C’ and g2 in C’’) the limit state values
correspond to g1=0 and g2=0.
On the other hand, g3 is a passive constraint, since β3> βT. Indeed, the point C’’’
obtained from the Reliability Analysis does not represent the MPP of g3, because if
we evaluate g3 in C’’’ the value of the limit state corresponds to g3>0 (safe region
in the reliability analysis). As can be seen the MPP of g3 correspond to CIV. This
point is not obtained through the SORA method since the RA is performed using
the AMV where the reliability index is imposed contrary to the RIA method, where
the beta index is obtained at each iteration.
As mentioned most of the time consuming tasks were performed in parallel
using the High Performance Computing Cluster (HPCC) of the Structural
Mechanics Group at the University of A Coruña, which has 928 cores, a physical
memory of 1.8 TB and a theoretical peak performance of 7.6 TFLOP’s,
considerably reducing the computational cost of the problem.
12
4. Application examples

4.1. Two-dimensional truss structure

The first example corresponds to the well-known 25-bar truss structure shown
in Figure 6. The original optimization problem was studied by Pedersen (1972)
obtaining the minimum volume of the structure subject to stress constraints. Three
load cases are considered as a result of applying a vertical load in nodes of the half
length of the bottom chord, due to the symmetry of the structure. The design
variables (d) are the cross sectional area of each bar.

Figure 6. 25-bar truss structure. Geometry and load cases. Dimensions in metres.

Accepting that this structural model corresponds to a highway steel bridge, the
hypothesis of a vehicle impact on one of the bars can be contemplated in the design
phase. It is assumed that only the inner bars (vertical and diagonal bars) can be
affected by the impact of vehicles causing their total loss of resistance. As a result,
and making use of the model symmetry, there are 7 possible damaged
configurations corresponding to the removal of bars 13, 14, 15, 18, 19, 20 and 21.
Load values of the three load cases defined in Figure 6 are chosen as random
variables (x). It has been required that the intact model supports the ultimate load
(P0), whereas the damaged configurations must support only service loads (Pd), that
is, a reduced value that guarantee the execution of repairing operations. These two
values are considered as random variables with a mean value of μP0=300 kN for the
intact model and μPd=200 kN for the damaged configurations. Both are defined as
normally distributed with a coefficient of variation of 10%, resulting in a stardard
deviation of σP0=30 kN and σPd=20 kN for the intact and damaged models,
respectively. The target values of safety index is set to βT=3.7190, corresponding to
a probability of failure of Pf=0.0001.

13
 The multi-model RBDO can be formulated as follows:
25
=
min V ∑
i =1
d i ⋅li (7a)
subject to


( )
P σ MAX − σ i0,h P0 ≤ 0 ≤ Pf ;
 i
for ( )
σ i0,h P0 > 0 i=
1,...,25 (7b)

P σ i0,h

( )
P0 − σ MIN ≤ 0 ≤ Pf ;
 i
for σ i0,h (P ) < 0
0
i=
1,...,25 (7c)


( )
P σ MAX − σ id ,h P d ≤ 0 ≤ Pf ;
 i
for σ id ,h (P ) > 0
d
i⊂Md d=
1,...,7 (7d)


( )
P σ id ,h P d − σ MIN ≤ 0 ≤ Pf ;
 i
for σ id ,h (P ) < 0
d
i⊂Md d=
1,...,7 (7e)
lb ≤ xi ≤ ub =
lb 0 =
ub 0.01 i⊂Md =d 0,...,7 (7f )
with h = 1,2,3 σ MAX = 130000kPa σ MIN = −104000kPa
where V is the structure volume, di the ith bar area, li the ith bar length and h the
load case. Additionally, P0 and Pd are the load values for the intact and damaged
configurations, respectively, σi0 is the ith bar axial stress in the intact model, σid the
ith bar axial stress for the damaged configuration d and Md the bar subset that has
not collapsed in the model d.
The objective is to get the minimum structural volume of the intact structure
considering as design variables (d) the cross-sectional area of bars, resulting in a
total of 13 (corresponding to the left half of the structure). The final design must
satisfy probabilistic stress constraints in both intact and damaged configurations
with a target reliability (Eq. 7b-7e), as well as deterministic side constraints for the
cross-sectional area values (Eq. 7f).
The problem has been solved through the SORA method programmed in a
MATLAB code, using the commercial software OptiStruct to solve the DO loop
and to obtain the necessary structural responses to evaluate the limit state functions
and their gradients, obtaining the MPPs in the RA loop, as explained in section 3.
The optimum design for the multi-model RBDO is shown in Table 1,
corresponding to a final volume of 0.37845 m3. The active constraints are shown in
Figure 7 and 8 and Table 2. The blue bars are associated to active stress constraints
for the load case 1 (P1), the red bars to the load case 2 (P2) and the green bars to the
load case 3 (P3). Additionally, it is also presented the results for the following
optimization cases: DO of intact model, multi-model DO, and RBDO of intact
model exclusively. In brackets (Table 1) appears the variation percentage referred
to the deterministic optimum solution of the intact model.
14
Table 1. Numerical results of the optimization for the 25-bar truss structure.

Cross-sectional DO intact RBDO intact model Multi-model RBDO

Multi-model DO
area (m2) model βT= 3.7190 βT= 3.7190

d1 0.004348 0.004349 (+0.02 %) 0.005957 (+37.01 %) 0.005962 (+37.12 %)

d2 0.001664 0.001611 (-3.19 %) 0.002285 (+37.32 %) 0.002205 (+32.51 %)
d3 0.00364 0.003711 (+1.95 %) 0.004976 (+36.7 %) 0.005073 (+39.37 %)
d4 0.000588 0.000988 (+68.03 %) 0.000808 (+37.33 %) 0.001355 (+130.44 %)
d5 0.00332 0.003416 (+2.89 %) 0.004558 (+37.29 %) 0.004693 (+41.36 %)
d6 0.000389 0.00053 (+36.2 %) 0.000524 (+34.68 %) 0.000731 (+87.84 %)
d13 0.001166 0.001551 (+33.02 %) 0.001602 (+37.39 %) 0.002128 (+82.5 %)
d14 0.001225 0.001329 (+8.49 %) 0.001678 (+36.98 %) 0.001815 (+48.16 %)
d15 0.001319 0.001159 (-12.13 %) 0.001809 (+37.15 %) 0.001568 (+18.88 %)
d18 0.0005582 0.00145 (+159.76 %) 0.000768 (+37.62 %) 0.001987 (+255.97 %)
d19 0.001705 0.002115 (+24.05 %) 0.002331 (+36.72 %) 0.002895 (+69.79 %)
d20 0.001084 0.001358 (+25.28 %) 0.001492 (+37.64 %) 0.001861 (+71.68 %)
d21 0.0006821 0.00118 (+73 %) 0.000937 (+37.37 %) 0.001624 (+138.09 %)
3)
Volume (m 0.238612 0.27622 (+15.76 %) 0.32703 (+37.06 %) 0.37845 (+58.6 %)

Figure 7. Active constraints at the optimum for the RBDO considering only the intact structure.

Figure 8. Active constraints at the optimum for the multi-model RBDO.

15
Table 2. Active constraints for the intact and the multi-model RBDO of the 25-bar truss structure.
RBDO intact model βT= 3.7190 Multi-model RBDO βT= 3.7190
Load Case Bars with active constraints Model Bars with active constraints
1 5, 7, 15, 21, 22 d=0 5, 7, 15
d=0 3
2 3, 5, 10, 14, 20 d =2 20
d=6 6, 14, 21
d=0 1, 2
3 1, 2, 4, 8, 13, 18 d=1 19
d=5 4, 13, 18

The MPPs of the random variables associated to each active stress constraint
correspond to a load value of 411.57 kN for the intact model and 274.38 kN for the
damaged configurations.
From the results it can be observed that the optimum values of the cross-
sectional areas are influenced by active stress constraints in both intact and damaged
configurations at the same time. Consequently, a 15.76 % increase in volume is
necessary when damaged configurations are included into the deterministic
optimization problem. On the other hand, the effect of considering uncertainty in
load values leads to a rise of the volume increase up to 37.06 % when only intact
model is considered. This value reaches 58.60 % if the whole set of configurations
are included into the RBDO problem. It is important to remark that this value is the
minimum penalty volume needed to satisfy the stress constraints in the whole set
of structural configurations with a target reliability index of βT = 3.7190 when
uncertainty in load values is assumed.
In this example, with 579 limit states, a total number of 61956 limit-state
function evaluations have been required in order to obtain the optimum solution of
the multi-model RBDO problem.
The evolution of the objective function for the RBDO problems is shown in
Figure 9. Only three cycles of the SORA method have been required.
0.6
Objective function

0.5
(kg/m2)

Multi-model RBDO
0.4
RBDO of intact model
0.3

0.2
0 1 2 3
Cycle of SORA method

Figure 9. Evolution of the objective function in the RBDO problem with SORA method.

16
 Parameter variation in the RBDO problem and comparison of results
The previous example was solved for a target reliability index of βT=3.7190 that
corresponds to a probability of failure Pf =0.0001 and setting a coefficient of
variation of 10%. In order to know the influence of these parameters in the optimum
solution the following study was performed. Table 3 shows the numerical results
for different values of probability of failure, from Pf =0.01 to Pf =0.00001. On the
other hand, Table 4 shows the influence of considering an increase in the standard
deviation of random variables. RBDO results are summarized in Table 3 for a
coefficient of variation of 10%, 20% and 30%. In brackets appears the change in
volume with respect to the multi-model DO design.
As can be observed in Table 3, the higher the reliability index, the higher the
values of cross-sectional areas, in order to guarantee the security imposed.
Moreover, in Table 4, the higher standard deviation of the random variable P, the
biggest is the final design. This is due to the presence of more uncertainty in the
load values, leading to a volume increase of the final structure necessary to
guarantee the same safety level.

Table 3. Optimum design for the intact model and the 7 damaged configurations for different values
of the target probability of failure.
DETERMINISTIC
PROBABILISTIC OPTIMIZATION
OPTIMIZATION

Cross-
- Pf =0.01 Pf =0.001 Pf =0.0001 Pf =0.00001
sectional
area
- βT =2.3263 βT =3.0902 βT =3.719 βT =4.2649
(m2)
d1 (m2) 0.004349 0.005348 (22.97 %) 0.005682 (30.65 %) 0.005962 (37.09 %) 0.006186 (42.24 %)
d2 (m2) 0.001611 0.001981 (22.97 %) 0.002108 (30.85 %) 0.002205 (36.87 %) 0.002293 (42.33 %)
d3 (m2) 0.003711 0.004563 (22.96 %) 0.004846 (30.58 %) 0.005073 (36.7 %) 0.005283 (42.36 %)
d4 (m2) 0.000988 0.001219 (23.38 %) 0.001292 (30.77 %) 0.001355 (37.15 %) 0.001408 (42.51 %)
d5 (m2) 0.003416 0.004216 (23.42 %) 0.004481 (31.18 %) 0.004693 (37.38 %) 0.004885 (43 %)
d6 (m2) 0.0005298 0.000654 (23.5 %) 0.000696 (31.31 %) 0.000731 (37.98 %) 0.000758 (43.05 %)
d13 (m2) 0.001551 0.001916 (23.53 %) 0.002031 (30.95 %) 0.002128 (37.2 %) 0.002212 (42.62 %)
d14 (m2) 0.001329 0.001628 (22.5 %) 0.001729 (30.1 %) 0.001815 (36.57 %) 0.001886 (41.91 %)
d15 (m2) 0.001159 0.001414 (22 %) 0.001497 (29.16 %) 0.001568 (35.29 %) 0.001629 (40.55 %)
d18 (m2) 0.00145 0.001785 (23.1 %) 0.001894 (30.62 %) 0.001987 (37.03 %) 0.002068 (42.62 %)
d19 (m2) 0.002115 0.0026 (22.93 %) 0.002764 (30.69 %) 0.002895 (36.88 %) 0.003012 (42.41 %)
d20 (m2) 0.001358 0.001672 (23.12 %) 0.001773 (30.56 %) 0.001861 (37.04 %) 0.001934 (42.42 %)
d21 (m2) 0.00118 0.00145 (22.88 %) 0.001546 (31.02 %) 0.001624 (37.63 %) 0.001691 (43.31 %)
Volume
0.27622 0.33984 (23.03 %) 0.36102 (30.7 %) 0.37845 (37.01 %) 0.39351 (42.46 %)
(m3)

17
Table 4. Optimum design for the intact model and the 7 damaged configurations for different values
of standard deviation.
DETERMINISTIC
PROBABILISTIC OPTIMIZATION
OPTIMIZATION

Cross- - βT=3.7190
sectional
area (m2) - σ= 0.1 ⋅ P
P
σ= 0.2 ⋅ P
P
σ= 0.3 ⋅ P
P

d1 0.004349 0.005962 (37.09 %) 0.007562 (73.88 %) 0.009177 (111.01 %)

d2 0.001611 0.002205 (36.87 %) 0.002823 (75.23 %) 0.003406 (111.42 %)
d3 0.003711 0.005073 (36.7 %) 0.006465 (74.21 %) 0.007827 (110.91 %)
d4 0.000988 0.001355 (37.15 %) 0.001728 (74.9 %) 0.002092 (111.74 %)
d5 0.003416 0.004693 (37.38 %) 0.005967 (74.68 %) 0.007258 (112.47 %)
d6 0.0005298 0.000731 (37.98 %) 0.000925 (74.59 %) 0.001125 (112.34 %)
d13 0.001551 0.002128 (37.2 %) 0.002693 (73.63 %) 0.003285 (111.8 %)
d14 0.001329 0.001815 (36.57 %) 0.002314 (74.12 %) 0.0028 (110.68 %)
d15 0.001159 0.001568 (35.29 %) 0.001997 (72.3 %) 0.002417 (108.54 %)
d18 0.00145 0.001987 (37.03 %) 0.002527 (74.28 %) 0.003073 (111.93 %)
d19 0.002115 0.002895 (36.88 %) 0.003682 (74.09 %) 0.004462 (110.97 %)
d20 0.001358 0.001861 (37.04 %) 0.002369 (74.45 %) 0.002866 (111.05 %)
d21 0.00118 0.001624 (37.63 %) 0.002067 (75.17 %) 0.002513 (112.97 %)
Volume (m3) 0.27622 0.37845 (37.01 %) 0.48133 (74.26 %) 0.58388 (111.38 %)

4.2. Curved stiffened panel of an aircraft fuselage

As mentioned in the introduction there are aerospace regulations which

contemplate the possibility of accidental damages in aircrafts that produce loss of
structural parts in flight. When this situation occurs, the aircraft structure must
ensure the “get-home load” capability, that is, a reduced load case for safe landing
at the next airport. In that sense, the Federal Aviation Administration published the
Advisory Circular 25.571-1D (2011) specifying that if an accidental damage occurs
“the remaining structure can withstand reasonable loads without failure or
excessive structural deformation until de damaged is detected”. It is also mentioned
that “the load conditions will not be exceeded on the flight during which the
specified incident occurs”, specifying “a 70% limit flight-maneuver loads, a 40%
of the limit gust velocity …” .
The curved stiffened panel presented in this section is composed by four frames
(curved stiffeners), four stringers (longitudinal stiffeners) and the skin. The panel
is 1,450 mm length and the curved edge length is 580 mm, with a radius of curvature
of value r=1,500 mm as presented in Figure 10. In Figure 11 appears the geometry
and the notation used for the definition of stringers and frames profiles, as well as
the skin thickness.
18
 Shear and compression loads presented in Figure 10 are used to perform the
buckling analysis of the structure through a FE model defined in Abaqus (2014). In
this FE model the skin is modelled with shell elements and the stringers and frames
with bar elements. The material used was aluminum type AL2024-T3 for the skin
and aluminum type AL7075-T6 for the stiffeners.

0.5kN/m

1kN/m

0.5kN/m

0.5kN/m

Figure 10. Geometry and loads. Units in kN and m.

Figure 11. Transversal cross section of frames (left), stringers (right) and skin thickness (bottom).

The idea is to define a set of possible damages and find the minimum weight of
the structure that satisfies a set of constraints. Five damaged models (d=1,…,5)
were arbitrarily generated as a result of the removal of part of the panel structure.
The design variables (d) considered are the set of cross-sectional dimensions of
frames, stringers and skin shown in Figure 11. In Table 5 appears the notation used,
a brief description, and the initial and limit values of the design variables defined
in the optimization problem as side constraints.

19
Table 5. Lower bound, initial value and upper bound of the design variables. Dimensions in mm.
Variable Description Min. Initial Max.
Skt Skin thickness 0.4 1.5 3
F1 Bottom flange length of the Frames 5 25 50
F2 Top flange length of the Frames 5 15 50
FA Web height of the Frames 20 60 100
FB Vertical border height of the Frames 1 8 20
Fe Thickness of the Frames elements 0.5 2 5
S1 Bottom flange length of the Stringers 5 22 50
S2 Top flange length of the Stringers 5 15 50
SA Web height of the Stringers 10 25 50
SB Vertical border height of the Stringers 1 6 20
Se Thickness of the Stringers elements 0.5 2 5

The proposed incomplete configurations can be seen in Figure 12, with the
corresponding identification number d and the first buckling factor for each model
considering the initial values of the eleven design variables presented in Table 5. It
can be observed the dramatic decrease of the buckling factor due to the loss of
material in the panel.

Figure 12. Finite element model of the panel structure (d = 0) and partial collapses (d = 1,…,5).

The random variables (x) considered are the values of the compression (Pc0, Pcd)
and shear (Ps0, Psd) loads defined in the buckling analysis, apart from the elasticity
modulus of the skin (Eskin) and the stiffeners (Estiffeners) material. The superscript 0
is associated with the intact model and the superscript d with the damaged
configurations. The six random variables are considered normally distributed with
a standard deviation corresponding to a 10% of the mean value μx presented in
Table 6.

20
Table 6. Mean and standard deviation of the normal random variables.
0 d 0 d
Pc (kN/m) Pc (kN/m) Ps (kN/m) Ps (kN/m) Eskin (MPa) Estiffeners (MPa)
µx 1 0.6 0.5 0.3 72000 71000
σx 0. 1 0.06 0.05 0.03 7200 7100

In this table, load values for damaged configurations are reduced to include the
“get-home loads” concept defined at the beginning of this section. The target safety
index is set βT=3.7190, corresponding to a probability of failure of Pf=0.0001.
The objective is to get the minimum structural mass of the panel satisfying
probabilistic buckling constraints in both intact and damaged configurations with a
specific target reliability, as well as deterministic slenderness constraints for the
cross section of the stiffeners. The objective function is the mass of panel per unit
area of skin.
Thus, the multi-model RBDO problem can be formulated as:

F=
( MF + MS + MSk )  kg 
(8a)
1.45 ⋅ 0.58  m2 
 
subject to
 λ 0 ( d, PC 0 , PS 0 , Eskin , Estiffeners ) 
P 1 − ≤ 0 ≤ Pf0 (8b)
 λ 0 
 min



P 1−
(
λ d d, PC d , PS d , Eskin , Estiffeners
≤ 0

≤P ) d=
1,..., D (8c)
 λ mind  fd
 

P 
i (
 w 0 d, P 0 , P 0 , E , E
C S skin stiffeners ) 
− 1 ≤ 0 ≤ Pf i i =
1,..., nS (8d)
wmax 0
 

P 
i (
 w d d, P d , P d , E , E
C S skin stiffeners ) 
− 1 ≤ 0=


wmax  ≤ Pf i i 1,...,=
nS
d
d 1,...D (8e)
 

FA F2 FB 
11.423 ≤ ≤ 15.714; 11.423 ≤ ≤ 15.714; 3.214 ≤ ≤ 4.286 
Fe Fe Fe 
 (8f )
SA S2 SB
11.423 ≤ ≤ 15.714; 11.423 ≤ ≤ 15.714; 3.214 ≤ ≤ 4.286 
Se Se Se 

being MF, MS and MSk the mass in kg of the frames, stringers and skin,
respectively, and F the objective function expressed in kg/m2 (Eq. 8a). The
probabilistic constraints are established on the buckling factor λd of each structural
configuration of the panel (Eq. 8b and 8c) as well as the buckling mode amplitude
wid of the stiffeners (Eq. 8d and 8e). This last constraint prevents global buckling
of the panel and guarantees that the buckling occurs between two stiffeners. In Eq.
(8d) and (8e), i corresponds to the node number of the stiffener where the constraint
21
is defined, being ns the total number of nodes in stringers and frames. The
slenderness constraints of the cross section in stiffeners (Eq. 8f) are included
according to the recommendations of Eurocode 9 as deterministic constraints
(Eurocode 9, 2007). In addition, side constraints imposed on the design variables
appear in Table 5.

The probabilistic optimization of Eq. (8) has initially been solved for the intact
model exclusively, d=0. Then, more damaged models were added progressively to
the existent ones, increasing the size of the probabilistic optimization problem. As
a result, five different problems have been defined, named as follows: I) RBDO of
the intact model (d=0); II) RBDO of the intact model considering the damaged
configurations d=1 and 2 (D=2); III) Intact model and d=1, 2 and 3 (D=3); IV)
Intact model and d=1, 2, 3 and 4 (D=4); V) Intact model and d=1, 2, 3, 4 and 5
(D=5). The aim is to analyze the evolution of the design variables and the objective
function value, as well as the active design constraints related to the buckling factor
and the buckling mode amplitude for an increasing number of damaged models.
The minimum values of buckling factors imposed in Eq. (8b) and Eq. (8c) are
λ0min=125 and λdmin =75 respectively, and the maximum displacement allowed in
the stiffeners (Eq. 8d and Eq. 8e) must be wmax=0.05, that is, less than 5% of the
maximum displacement of the buckling mode. The target safety index is set
βT=3.719, corresponding to a probability of failure of Pf = 0.0001.
A summary of the results obtained is presented below. Table 7 shows the values
of the optimum design variables for the five RBDO problems defined above. The
buckling factors for each optimal configuration are shown in the Table 8 and the
values of the maximum displacement in the stiffeners (expressed in units of the
maximum displacement produced in the corresponding buckling mode) are shown
in Table 9. The values of λd and wdmax related to the active constraints in each model
appear in bold. It must be noticed that when considering partial collapses in the
optimization process the active constraints at the optimum correspond to different
structural configurations. In Figure 13 appear two views of the buckling mode for
each configuration, in the optimal design for the last case D=5. In one of them the
skin has been removed in order to visualize clearer the stiffeners displacements.

22
Table 7. Results of the RBDO considering only the intact model (d=0) and the multi-model
configurations D=2, D=3, D=4 and D=5.
Design
D=2 D=3 D=4 D=5
variables d=0
(d = 0,1,2) (d = 0,1,2,3) (d = 0,1,2,3,4) (d = 0,1,2,3,4,5)
(mm)
Skt 1.9871 2.0806 (4.71 %) 2.4385 (22.72 %) 2.4999 (25.81 %) 2.5023 (25.93 %)
F1 5 5 (0 %) 5 (0 %) 5 (0 %) 7.972 (59.44 %)
F2 14.5386 14.5386 (0 %) 20.2624 (39.37 %) 14.5386 (0 %) 25.3471 (74.34 %)
FA 20 20 (0 %) 20.2624 (1.31 %) 20 (0 %) 34.8687 (74.34 %)
FB 4.0906 4.0906 (0 %) 4.1443 (1.31 %) 4.0906 (0 %) 7.1317 (74.34 %)
Fe 1.2728 1.2728 (0 %) 1.2895 (1.31 %) 1.2728 (0 %) 2.219 (74.34 %)
S1 5 5 (0 %) 5 (0 %) 5 (0 %) 7.0317 (40.63 %)
S2 19.8293 20.3639 (2.7 %) 22.8302 (15.13 %) 23.4465 (18.24 %) 23.5314 (18.67 %)
SA 27.2781 28.0135 (2.7 %) 31.4063 (15.13 %) 32.2541 (18.24 %) 32.3709 (18.67 %)
SB 5.5792 7.2098 (29.23 %) 6.4236 (15.13 %) 7.1994 (29.04 %) 6.6209 (18.67 %)
Se 1.7359 1.7827 (2.7 %) 1.9986 (15.13 %) 2.0526 (18.24 %) 2.06 (18.67 %)

Objective
7.6865 8.0877 (5.22 %) 9.5459 (24.19 %) 9.7951 (27.43 %) 10.711 (39.35 %)
function (kg/m2)

Table 8. Buckling factors after RBDO considering only the intact model (d=0) and the multi-model
configurations D=2, D=3, D=4 and D=5.
D=2 D=3 D=4 D=5
Buckling factor d=0
(d = 0,1,2) (d = 0,1,2,3) (d = 0,1,2,3,4) (d = 0,1,2,3,4,5)
λ0 196.83 224.51 348.16 376.23 377.84
λ1 - 117.86 183.84 199 201.82
λ2 - 121.06 192.95 205.39 218.08
λ3 - - 116.65 124 132.10
λ4 - - - 116.71 117.10
λ5 - - - - 168.10

Table 9. Maximum displacements in stiffeners after RBDO considering only the intact model (d=0)
and the multi-model configurations D=2, D=3, D=4 and D=5. Units of maximum displacement of
the buckling mode.
Maximum D=2 D=3 D=4 D=5
d=0
displacement (d = 0,1,2) (d = 0,1,2,3) (d = 0,1,2,3,4) (d = 0,1,2,3,4,5)
0
wmáx 0.0493 0.04933 0.04925 0.04927 0.04055
w1máx - -0.03179 -0.03597 -0.03744 -0.02971
2
w máx - 0.03094 0.03302 0.03163 0.03223
3
wmáx - - 0.01023 0.01106 0.005999
4
wmáx - - - 0.03237 0.0258
5
wmáx - - - - 0.05

23
Figure 13. Buckling mode of each configuration (left column) and the same result removing the skin
from the view (right column), for D=5.
24
 For the larger size problem, D=5, with 114 limit states, a total number of 79926
limit-state function evaluations have been required in order to obtain the optimum
solution for the multi-model RBDO problem. The computing time of the
optimization procedure resulted in 15 hours using the High Performance
Computing Cluster (HPCC) mentioned in section 3, performing the structural
analyses in parallel. Otherwise, the total time would have increased up to 19 days.
For this case D=5, where a highest number of damaged configurations are
involved, the Performance Measure Approach (PMA) was also used in order to
validate the results obtained through SORA method. The PMA is a two level RBDO
approach where the design optimization is performed in the outer loop while the
reliability analysis is carried out in the inner loop, leading to nested optimization
problems. As the reliability analysis must be performed at each iteration of the
optimization process, it results in a higher computational effort compared to
decoupled methods. The final mass of the panel corresponds to 10.6772 kg/m2 (-
0.3%). A total number of 497760 limit-state function evaluations were needed using
PMA with a computing time of 55 hours performing the analysis in parallel,
substantially higher than using the SORA method.
Figure 14 summarizes the evolution of the objective function after each cycle
of the SORA method in the five optimization problems described before. As can be
seen, the case D=5 corresponds to the solution with higher mass of the panel, where
six models are simultaneously involved (the intact configuration and five partial
collapses). In this figure the numbers 1, 2, 3 and 4 represents the final design
achieved after each cycle of the SORA method for the case D=5. As can be observed
in Figure 15 the number of iterations needed for each deterministic optimization
(DO) loop was 30, 14, 5 and 2, respectively. In this figure it was also represented
the evolution of the objective function using PMA method.
As explained in the flowchart of Figure 4, after each DO phase as many
reliability analyses are required as limit states defined in the problem. Figures 16
and 17 are referred to the Reliability Analysis (RA) loop performed after each
deterministic optimization of case D=5. They represent the evolution of the two
active limit states at the final RBDO solution (buckling factor in the configuration
d=4 and buckling mode amplitude of stiffeners in the configuration d=5). The MPP
values associated to these active constraints are listed in Table 10.

25
 14
13
d=0

Objective function (kg/m2)

12
2 3 4
11 D=2
10 D=3
1
9 D=4
8
D=5
7
6
0 1 2 3 4 5 6
Cycle of SORA method

Figure 14. Evolution of the objective function for the different sets of damaged configurations.

20

18 1 2 3 4
Objective function (kg/m2)

16

14

12

10

6
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52
Iteration
SORA it 1 SORA it 2 SORA it 3 SORA it 4 PMA

Figure 15. Evolution of the objective function for D=5 in SORA and PMA method.

1.00 1.10
0.80 0.90
0.60 0.70
G(x)

0.40
G(x)

0.50
0.20
0.30
0.00
0.10
-0.20 0 1 2 3 4 5
-0.10 0 1 2 3
-0.40
RA iteration RA iteration

Figure 16. Normalized limit state function Figure 17. Normalized limit state function
associated to the buckling factor in the associated to the buckling mode amplitude of
configuration d=4, for D=5. stiffeners in the configuration d=5, for D=5.

Table 10. MPPs associated to active constraints in damaged configurations, when D=5.
d d
Limit state Pc (kN/m) Ps (kN/m) Eskin (MPa) Estiffeners (MPa)
G(x) buckling factor d=4 0.6759 0.3351 48635 66926
G(x) buckling mode amplitude d=5 0.6182 0.3084 90839 89535

26
 On the other hand, Figure 18 represents the mass increase (%) referring to the
optimum mass in the RBDO of the intact model, for each value of D.
45.00 39.35
40.00
Increase in mass (%) 35.00
30.00 27.43
24.19
25.00
20.00
15.00
10.00
5.22
5.00
0.00
0 1 2 3 4 5 6
Number of damaged configurations (D)

Figure 18. Mass increase for each value of D with respect to the optimum mass for the RBDO of the
intact model.
It can be observed that the penalty mass of the panel is not linear with the
number of damages considered. The most important factor in the mass increase is
the magnitude of the damage and not the number of configurations considered. It
could be observed a noticeable change of tendency in the case of considering three
and five damaged configurations. In the first one, the damaged model d=3 has a
buckling factor for the initial values of design variables (λ3=32.33) much lower than
the intact model and the damaged configurations 1 and 2 (λ1=53.78, λ2=55.49), and
consequently the values of the design variables grow to a greater extent to satisfy
the buckling design constraint. In the case of considering five damaged
configurations, this growth is produced because the damage concerns both to a
stringer and a frame, something that does not happen in the remaining
configurations, hence the rigid connection between them no longer exists and do
not collaborate in stiffening the skin panel.
From Table 7 it can be drawn that although the optimum mass of the panel
increases progressively by adding damaged configurations, the same does not
happen with the value of the design variables. For instance, comparing the optimum
design for D=2 and D=3, all the values of the design variables increase except the
vertical border height of the stringers (SB). Moreover, the values of the design
variables F2, FA, FB and Fe when D=4 decrease considerably in comparison with
the corresponding values when D=3.

27
5. Conclusions

A new approach has been presented to include in the structural optimization

process the effect of having uncertainty in several structural parameters combined
with the presence of possible damaged configurations. The standard optimization
process of a single structural model does not guarantee the accomplishment of the
design constraints in a predefined set of incomplete configurations as a result of
losing part of its structural elements. This is due to the fact that the set of active
constraints at the optimum design can be associated to different structural
configurations as stated in the presented examples.
The decoupled method SORA has been successfully implemented in a multi-
model configuration being possible to parallelize several tasks improving the
computational cost.
Two application examples were presented in this work: a two-dimensional truss
structure and a curved stiffened panel of an aircraft fuselage, with different sets of
damaged configurations. In the first one only loads were considered as random
variables, whereas in the curved panel both loads and material properties were
assumed random. The results show a mass increase at the optimum as a
consequence of considering uncertainty data and several damaged configurations
in the RBDO procedure.
In the first example of a two-dimensional truss structure, both deterministic and
probabilistic optimizations were carried out considering first the intact model
exclusively and then performing the multi-model optimization including several
partial collapses. The deterministic optimum design considering the intact model
exclusively corresponds to a volume of 0.238612 m3. Referred to this value, the
volume increases up to 0.27622 m3 (15.76%) when 7 damaged configurations are
included into the deterministic optimization problem. When uncertainty in loads is
taken into account the multi-model RBDO design reaches a volume of 0.37845 m3
(58.6%). This final design verifies the accomplishment of a specific probability of
failure of Pf=1·10-4 against stress constraints for the intact configuration and 7
partial collapses simultaneously. Moreover parametric studies were carried out in
order to know the influence in the final design of two parameters: the target
reliability index βT and the standard deviation of the random variable P. Analyzing
the results, it can be concluded that the higher the reliability index, the higher the
values of cross-sectional areas, since the safety level demanded is higher. For
28
example, imposing a Pf=1·10-3 the final design increases up to 0.36102 m3 (30.7%)
and with Pf=1·10-5 the final design reaches 0.39351 (42.46%) with respect to the
deterministic optimization of the intact model. On the other hand, an increase in the
standard deviation of the random variables leads to a heavier design, reaching a
volume of 0.48133 m3 (74.26%) for a standard deviation of the 20% of the load
value and 0.58388 m3 (111.38%) for a standard deviation of the 30% of the load
value. From these results it can be concluded that the presence of more uncertainty
leads to a volume increase in order to guarantee the same safety level.
In the example of a curved stiffened panel of an aircraft fuselage the influence
in the number of damaged configurations into the multi-model RBDO problem was
analyzed. It has been demonstrated how the penalty mass of the panel is not linear
with the number of damages considered, being the determining factor the
magnitude and location of the damage. This phenomenon can be appreciated in the
change of tendency when including the partial collapse 3 and 5, leading to a
significant increment in the optimum volume. Moreover, including additional
damaged configurations into the optimization process does not entail an increase in
the value of every design variable, even decreasing in some of them. This is due to
the growth of some design variables in order to satisfy the new limit states makes
possible the reduction of other ones. For instance, the values of the design variables
F2, FA, FB and Fe for D=4 decrease considerably in comparison with the
corresponding values for D=3.
To summarize, this research has demonstrated the advantages of including
possible partial collapses that could occur during the life cycle of a structure. The
proposed optimization strategy makes possible to meet the regulations that
contemplate accidental damages from a probabilistic approach, guaranteeing safe
structures with the minimum material cost.

Acknowledgments

The research leading to these results is part of the research project DPI2016-
76934-R financed by the Spanish Ministry of Economy and Competitiveness.
Further funding has been received from the Galician Government (including
FEDER funding) with reference GRC2013-056.

29
6. References
Abaqus (2014) Abaqus 6.14.2. Documentation.
Altair OptiStruct (2013). User Manual Version 12, Altair Engineering Inc.
Aoues, Y. and Chanteauneuf, A. (2010), Benchmark study of numerical methods for reliability-
based design optimization, Structural and Multidisciplinary Optimization, 41(2):277-294.
Baldomir A., Hernández S., Romera L. E. and Díaz J. (2012), Size optimization of shell structures
considering several incomplete configurations, 8th AIAA Multidisciplinary Design
Optimization Specialist Conference, Sheraton Waikiki Honolulu, Hawaii.
Baldomir A., Kusano I., Hernandez S., Jurado J. A. (2013), A reliability study for the Messina
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