JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

42, 3, pp. 585-608, Warsaw 2004

RELIABILITY BASED OPTIMIZATION OF STEEL FRAMES

UNDER SEISMIC LOADING CONDITIONS USING

EVOLUTIONARY COMPUTATION

Manolis Papadrakakis

Institute of Structural Analysis and Seismic Research, National Technical University of Athens,

Athens, Greece; e-mail: mpapadra@central.ntua.gr

Yiannis Tsompanakis

Division of Mechanics, Department of Applied Sciences, Technical University of Crete, Chania

Crete, Greece; e-mail: jt@mechanics.tuc.gr

Nikos D. Lagaros

Michalis Fragiadakis

Institute of Structural Analysis and Seismic Research, National Technical University of Athens,

Athens, Greece; e-mail: nlagaros@central.ntua.gr; mfrag@central.ntua.gr

Earthquake-resistant design of structures using probabilistic analysis and
performance-based design criteria is an emerging field of structural engine-
ering. These new analysis and designmethodologies are aimed at improving
the existing practice and design codes for better prediction of the structural
performance. In this paper, a robust and efficient methodology is presented
for performing reliability-based structural optimum design of steel frames
under seismic loading. The optimization part is realisedwith evolution stra-
tegies,while the reliability analysis is carriedoutwith theMonteCarlo simu-
lationmethod incorporating the latin hypercube sampling technique for the
reduction of the sample size. The probability of failure of the frame struc-
tures, in terms of interstorey drift limits, is determined via the multi-modal
response spectrum analysis.

Key words: structural optimization, reliability analysis, Monte Carlo simu-
lation, evolutionary computation, multi-modal response spectrum analysis

1. Introduction

The inherent probabilistic nature of geometry, material properties and lo-
ading conditions involved in structural analysis is an important factor that



586 M.Papadrakakis

influences structural safety. Reliability analysis leads to safety measures that
a design engineer has to take into account due to the aforementioned un-
certainties. The modern conceptual approach towards seismic structural de-
sign follows the so-calledPerformance-BasedEarthquakeEngineering (PBEE)
([12], [2], [27], Fajfar and Krawinkler, 1997). The most important ingredient
of PBEE is the structural reliability: straightforward consideration of all un-
certainties and variabilities that arise in structural design, construction and
serviceability in order to calculate the level of confidence about the structure
ability to meet the desired performance goals.

Within this probabilistic framework, the seismic hazard is typically expres-
sed in terms of occurrence of earthquakes having a certain (or bigger) intensity
over a specific time period, which is normally 50 years. The structural per-
formance in PBEE is measured as the probability that damages caused by
a certain seismic hazard level are kept under a specified level. For example,
one PBEE goal would be to calculate the probability to have collapse preven-
tion if the earthquake has 2% probability of exceedance in 50 years period.
Obviously, according to PBEEmethodology, one can obtain all levels of confi-
dence for various combinations of the structural capacity and seismic demand
levels. In recent years a number of publications have appeared dealing with
the performance and reliability-based optimum structural design (Alimoradi,
2003; Beck et al., 1997; Collins et al., 1996; Ganzerli et al., 2000; Hasan and
Grierson, 2002; Wen, 2000).

Due to the uncertain nature of earthquake loading, the structural design is
often based on the design response spectrum of the region of interest and on
some simplified assumptions of the structural behaviour under earthquakes. In
the case of direct consideration of the earthquake loading, the optimization of
structural systems requires multiple solution of dynamic equations of motion
which can be orders ofmagnitudemore computationally intensive than a case
of static loading. In the present study the reliability-based sizing optimization
ofmulti-story steel frames under seismic loading is investigated. The objective
function is the weight of a structure while constraints are both deterministic
(stress and displacement limitations imposed by the design codes) as well as
probabilistic (limitation on the overall probability failure of the structure).
Randomness of excitation due to ground motion and material properties are
taken into consideration in the reliability analysis using Monte Carlo simula-
tion (MCS). The probability of failure of frame structures, in terms of inter-
storey drift limits, is determined viamulti-modal response spectrum analysis.
The optimization part is solved using the Evolution Strategies (ES) method,
which in most cases is more robust and present better global behaviour than



Reliability based optimization of steel frames... 587

mathematical programming methods (Papadrakakis et al., 1999; Lagaros et
al., 2002).

2. Structural reliability analysis

In the design of structural systems, limiting uncertainties and increasing
safety is an important issue to be considered. The structural reliability, which
is defined as the probability that a system meets some specified demands for
a specified time period under specified environmental conditions, is used as
a probabilistic measure to evaluate the reliability of structural systems. The
performance function of a structural system must be determined to descri-
be the system behaviour and to identify the relationship between the basic
parameters in the system. It should be noted that in the earthquake loading
environment the uncertainties related to seismic demand and structure capa-
city are strongly coupled.

The probability of failure pf can be determined using the time invariant
reliability analysis procedure with the following relationship

pf = p[R < S] =

∞∫

−∞

FR(t)fS(t) dt =1−
∞∫

−∞

Fs(t)fR(t) dt (2.1)

where R denotes the structure bearing capacity and S the external loads.
The randomness of R and S can be described by known probability density
functions fR(t) and fS(t), with FR(t)= p[R < t], FS(t)= p[S < t] being the
cumulative probability density functions of R and S, respectively.

Most often, the limit state function is defined as G(R,S)= S−R and the
probability of structural failure is given by

pf = p[G(R,S)­ 0]=
∫

G­0

fR(R)fS(S) dR dS (2.2)

It is practically impossible to evaluate pf analytically for complex and/or
large-scale structures, especially in the case of dynamicReliability-BasedOpti-
mization (RBO) problems that are considered in the present study. In such
cases, the integral of Eq. (2.2) can be calculated only approximately using
either simulationmethods, such as theMonte Carlo Simulation (MCS), or ap-
proximation methods like the first order reliability method (FORM) and the



588 M.Papadrakakis

second order reliability method (SORM), or response surfacemethods (RSM)
(Gasser and Schueller, 1997; Huh and Haldar, 2000; Gupta and Manohar,
2004). Despite its high computational cost, MCS is considered as an efficient
method and is commonly used for the evaluation of the probability of failure
in computational mechanics, either for comparison with other methods or as
a standalone reliability analysis tool.

2.1. Monte Carlo simulation

In reliability analysis, the MCS method is often employed when the ana-
lytical solution is not attainable and the failure domain can not be expressed
or approximated by an analytical form. This ismainly the case in problems of
complex nature with a large number of basic variables where all other reliabi-
lity analysismethods are not applicable. Expressing the limit state function as
G(x) < 0, where x = [x1,x2, ...,xM]

> is the vector of the random variables,
Eq. (2.2) can be written as

pf =

∫

G(x)­0

fx(x) dx (2.3)

where fx(x) denotes the joint probability of failure for all random variables.
Since MCS is based on the theory of large numbers (N∞) an unbiased esti-
mator of the probability of failure is given by

pf =
1

N∞

N∞∑

j=1

I(xj) (2.4)

in which I(xj) is an indicator for considering successful or unsuccessful simu-
lations, defined as

I(xj)=




1 if G(xj)­ 0

0 if G(xj) < 0
(2.5)

thus in every violation a successful simulation is encountered and the failure
counter is increased by 1.

It is important,while using simulationmethods in the structural reliability,
to efficiently and accurately evaluate the probability of failure for a given
performance function. In order to estimate pf, an adequate number of Nsim
independent random samples is produced using a specific, usually uniform,
probability density function of the vector x. The value of the failure function



Reliability based optimization of steel frames... 589

is computed for each random sample xj and the Monte Carlo estimation of
pf is given in terms of the sample mean by

pf ∼=
NH
Nsim

(2.6)

where NH is the number of successful simulations.

2.2. Latin hypercube sampling

Although the mathematical formulation of MCS is relatively simple and
has the ability of handling practically every possible case, regardless of its
complexity, the computational effort involved in conventional MCS is excessi-
ve. For this reason a lot of sampling techniques, also called variance reduction
techniques, have been developed in order to improve the computational effi-
ciency of themethod byminimizing the sample size and reducing the statisti-
cal error that is inherent inMCS. Among them are the importance sampling,
adaptive sampling technique, stratified sampling, latin hypercube sampling,
antithetic variate technique, conditional expectation technique (Kamal and
Ayyub, 2000). Latin Hypercube Sampling (LHS) is generally recognized as
one of the most efficient size reduction techniques (McKay et al., 1979; Stein,
1987). The basis of LHS is full stratification of the sampleddistributionwith a
random selection inside each stratum. In consequence, sample values are ran-
domly shuffled among different variables. Apart from the standard LHS there
are also improvedLHSschemes,which combineLHSwithdescriptive sampling
methods (Ziha, 1995) or adaptive importance sampling (Olsson et al., 2003),
in order to further increase the efficiency of this sampling procedure.

In the LHSmethod, the range of probable values for each randomvariable
is divided into M non-overlapping segments of equal probability of occurrence.
Thus, the whole parameter space, consisting of N parameters, is partitioned
into MN cells. For example, for the case of 3 parameters and 5 segments, the
parameter space isdivided into 53 cells.Then, the randomsamplegeneration is
performed,bychoosing M cells fromthe MN spacewith respect to thedensity
of each interval, and the cell number of each random sample is calculated. The
cell number indicates the segment number the sample belongs to with respect
to each of the parameters. For example, cell number (3,2,1) indicates that the
sample lies in the segments 3, 2 and 1 with respect to the first, second and
third parameter, respectively. In LHS, the sampling is realized independently,
whereas, the matching of random samples is performed either randomly or
in a restricted manner. All necessary random samples are produced and they
are accepted only if they do not agree with any previous combination of the



590 M.Papadrakakis

segment numbers.Theadvantage of theLHSapproach is that randomsamples
are generated from all ranges of possible values, thus giving a more thorough
insight into the tails of the probability distributions.

3. Structural design under seismic loading

The equations of equilibrium for a finite element system inmotion can be
written in the usual form

M(si)üt+C(si)u̇t+K(si)ut =Rt (3.1)

where M(si), C(si), and K(si) are the mass, damping and stiffness matrices
for the ith design vector si; Rt is the external load vector, while ut, u̇t
and üt are the displacement, velocity, and acceleration vectors of the finite
element assemblage, respectively. The design approach based on the multi-
modal response spectrum analysis, which is based on the mode superposition
approach, will be considered in the following section.

3.1. Multi-Modal Response Spectrum analysis

The Multi-Modal Response Spectrum (MMRS) analysis is based on sim-
plification of the mode superposition approach in order to avoid time history
analyseswhich are requiredby both, the direct integration andmode superpo-
sition approaches. In the case of the multi-modal response spectrum analysis
Eq. (3.1) is modified according to the modal superposition approach in the
following form

mj(si)üt+ c
j(si)u̇t+k

j
(si)ut = rt (3.2)

where
m
j
i =(φ

j
i)
>
Miφ

j
i c

j
i =(φ

j
i)
>
Ciφ

j
i

k
j

i =(φ
j
i)
>
Kiφ

j
i rt =(φ

j
i)
>Rt

(3.3)

are thegeneralizedvalues of the correspondingmatrices and the loadingvector,
while Φi is the eigenmode shapematrix. For simplicity, the matrices m

j(si),

cj(si), k
j
(si) are denoted by m

j
i , c
j
i, k
j

i, respectively. According to the modal
superposition approach, the system of N simultaneous differential equations
(j = 1,2, ...,N), which are coupled with the off-diagonal terms in the mass,
damping and stiffness matrices, is transformed to a set of N independent
normal-coordinate equations. Thedynamic response can therefore be obtained



Reliability based optimization of steel frames... 591

by solving separately for the response of each normal (modal) coordinate and
by superposing the response in the original coordinates.
In theMMRSanalysis, a number of different formulas have been proposed

to obtain reasonable estimates of themaximumresponse based on the spectral
values without performing time history analyses for a considerable number of
transformed dynamic equations. The simplest and the most popular one is
the Square Root of Sum of Squares (SRSS) of modal responses. According
to this estimate the maximum total displacement for a degree of freedom is
approximated by

ui,max =
√
(u1i)

2+(u2i)
2+ . . .+(uNi )

2 (3.4)

where the subscript i denotes the design vector; u
j
i corresponds to the ma-

ximum displacement of the jth transformed dynamic equation over the com-
plete time period. The use of Eq. (3.4) permits this type of dynamic analysis
by knowing only the maximummodal dsiplacement u

j
i .

The MMRS analysis method is summarized in the following steps, where
the subscript i refers to the si design vector

• Obtain first m < N eigenfrequencies and the corresponding eigen-
mode shape matrices, which are classified in the following order
(ω1i ,ω

2
i , . . . ,ω

m
i ) and Φi = [φ

1
i ,φ
2
i , . . . ,φ

m
i ], respectively; ω

j
i , φ

j
i are the

jth eigenfrequency and eigenmode, respectively. N is the total number
of modes, while m – the number of important modes considered. The
number of important modes is specified by the condition that the sum
of modal masses considered must be equal or greater than 90% of the
total participating mass of the system.

• Calculate the modal masses, according to the following equation

m
j
i =(φ

j
i)
>
Miφ

j
i (3.5)

calculate the coefficients L
j
i, according to the following equation

L
j
i =(φ

j
i)
>
Mir (3.6)

where r is the influence vector, which represents the displacements of
the vibratingmasses resulting from static application of the unit ground
displacement along the direction of the seismic excitation.

• Calculate the modal participation factors Γ ji , according to

Γ
j
i =

L
j
i

m
j
i

(3.7)



592 M.Papadrakakis

• Calculate the effective modal mass for each design vector and for each
eigenmode, by the following equation

m
j
eff,i
=
L
j
i

2

m
j
i

(3.8)

• Calculate the spectral accelerations Rd(T ij) for each period of the m
modes considered. For this step, the knowledge of the design response
spectrum is necessary.

• Finally, obtain the modal displacements according to the relations

(SDi)j =
Rd(T

i
j)

ωij
2
=
Rd(T

i
j)T
i
j
2

4π2

(3.9)

u
j
i = Γ

j
iφ
j
i(SD

i)j

The total maximumdisplacement is obtained by superimposing thema-
ximummodal displacements using the SRSS rule of Eq. (3.4).

3.2. Load combinations

In the Eurocode earthquake, the loading is taken as a random action,
therefore it must be considered for the structural design with the following
loading combination [7]

Sd =
∑

Gkj ”+”Ed ”+”
∑

ψ2iQki (3.10)

where ”+” implies ”to be combined with”,
∑
implies ”the combined effect

of”, Gkj denotes the characteristic value of the permanent action j, Ed is the
design value of the seismic action, and Qki refers to the characteristic value
of the variable action i, while ψ2i is the combination coefficient for the quasi
permanentvalueof thevariable action i, here takenas0.30.Designcode checks
are implemented in the optimization algorithm as constraints. Each structural
member should be checked for actions that correspond to themost severe load
combination obtained fromEq. (3.10) and the persistent load combination

Sd =1.35
∑

Gkj ”+”1.50
∑

Qki (3.11)

It should be pointed out that the seismic action is obtained from the elastic
response spectrum reduced by the behaviour factor q. This is done because
the structure is expected to absorb the energy through inelastic deformation.
The maximum values of the q-factor are suggested by design codes and vary
according to the material and type of the structural system. For the framed
steel structures considered in this study q =4.0.



Reliability based optimization of steel frames... 593

3.3. Probabilistic definition of Seismic Response Spectra

The most common approach for the definition of the seismic input is the
use of the design code response spectrum.This is a general approach which is
easy to implement. However, if higher precision is required the use of spectra
derived from natural earthquake records is more appropriate. Since a signifi-
cant dispersion on the structural response due to the use of different natural
records has been observed, these spectra must be scaled to the same desired
earthquake intensity. The most commonly applied scaling procedure is based
on the peak ground acceleration (PGA).

Fig. 1. Natural record response spectra and their median

In this study, a set of twenty natural accelerograms, shown in Table 1,
is used. It can be seen that each record corresponds to different earthquake
magnitudes and soil properties. The records of this set correspond to a wide
rangeofPGAandpeakacceleration overpeakdisplacement ratio (a/v) values.
The latter parameter is considered to describe the damage potential of the
earthquakemore reliably thanPGA.The records are scaled, to the samePGA
according to Eurocode 8 in order to ensure compatibility between the records.
The response spectrum for each scaled record are shown in Figure 1. It has
been observed that the response spectra follow the lognormal distribution
(Chintanapakdee andChopra, 2003). Therefore, themedian spectrum x̂, also



5
9
4

M
.
P
a
p
a
d
r
a
k
a
k
is

Table 1. List of natural accelerograms

Earthe name
(Date)

Site \ Soil
conditions

Orien-
MS

PGA PGV a/v

tation [g] [cm/s] [s]

1 VictoriaMexico (06.09.80) Cerro Prieto \Alluvium 45 6.40 0.62 31.57 19.30
2 Kobe (16.01.95) Kobe \Rock 0 6.95 0.82 81.30 9.91
3 Imperial Valley (19.05.40) El Centro Array \CWB: D, USGS: C 180 7.20 0.31 29.80 10.32
4 Duzce (12.11.99) Bolu \CWB: D, USGS: C 90 7.30 0.82 62.10 12.99
5 San Fernando (09.02.71) Pacoima dam \Rock 164 6.61 1.22 112.49 10.69
6 Gazli (17.05.76) Karakyr \CWB: A 90 7.30 0.72 71.56 9.83
7 Friuli (06.05.76) Bercis \CWB: B 0 6.50 0.03 1.33 21.17
8 Aigion (17.05.90) OTE building \ Stiff soil 90 4.64 0.20 9.76 20.00
9 Central California (25.04.54) Hollister City Hall \CWB: D, USGS: C 271 – 0.05 3.90 12.77
10 Alkyonides (24.02.81) Korinthos OTE building \ Soft soil 90 6.69 0.31 22.70 13.34
11 Northridge (17.01.94) Jensen filter Plant \CWB: D, USGS: C 292 6.70 0.59 99.10 5.86
12 Athens (07.09.99) Sepolia (Metro Station) \Unknown 0 5.60 0.24 17.89 13.32
13 CapeMendocino (25.04.92) Petrolia \CWB: D, USGS: C 90 7.10 0.66 89.72 7.24
14 Erzihan, Turkey (13.03.92) Erzikan East-East Comp \CWB: D, USGS: S 270 6.90 0.49 64.28 7.56
15 Kalamata (13.09.86) Kalamata, Prefecture \ Stiff soil 0 5.75 0.21 32.90 6.41
16 Iran (16.09.78) Tabas \CWB: S 0 7.40 0.85 121.40 6.89
17 Loma Prieta1 (18.10.89) Hollister Diff Array \CWB: D 255 7.10 0.28 35.60 7.69
18 Loma Prieta2 (18.10.89) Coyote Lke dam \CWB: D 285 7.10 0.48 39.70 11.95
19 Mammoth Lakes (27.05.80) McGee Creek \CWB: D 0 5.00 0.33 8.55 37.29
20 Irpinia, Italy (23.11.80) Sturno \Unknown 270 6.50 0.36 52.70 6.66
MS: surface momentmagnitude



Reliability based optimization of steel frames... 595

shown in Fig.1, and the standard deviation are calculated from the above set
of spectra using the following expressions

x̂ =exp
{1
n

n∑

i=1

ln[Rd,i(T)]
}

(3.12)

δ =

√√√√ 1
n−1

n∑

i=1

[lnRd,i(T)− ln x̂]2

where Rd,i(T) is the response spectrum value for a period equal to T of the
ith record (i =1, ...,n, where n =20 in this study). For a given period value,
the acceleration Rd is obtained as a randomvariable following the log-normal
distribution whose mean value is equal to x̂ and standard deviation is equal
to δ.

4. Evolutionary computation in structural optimization

The two most widely used optimization algorithms belonging to a class
of the evolutionary computation that imitates the nature by using biologi-
cal methodologies are the Genetic Algorithms (GA) and Evolution Strategies
(ES). The ES method was proposed for parameter optimization problems in
the seventies (Schwefel, 1981). In this work, ES are used as the optimization
tool for addressing RBO problems under earthquake loading. Both GA and
ES imitate biological evolution in the nature and have three characteristics
that make them differ frommathematical optimization algorithms

(i) in the place of usual deterministic operators they use randomised ope-
rators

(ii) instead of a single design point they work simultaneously with a popu-
lation of design points

(iii) they can handle continuous, discrete andmixed optimization problems.

The second characteristic allows for a natural implementation ofES in parallel
computing environments (Papadrakakis et al., 1999).

In studies byPapadrakakis et al. (1999), Lagaros et al. (2002) it was found
that probabilistic searchmethods are computationallymore efficient thanma-
thematical programmingmethods, even thoughmore optimization steps were
required in order to reach the optimum. In the former case, the optimization



596 M.Papadrakakis

steps were computationally less expensive than in the latter case since the-
re was no need for gradient information. This property of the probabilistic
search methods is of greater importance in the case of RBO problems since
the calculation of derivatives of reliability constraints is very time-consuming.
Furthermore, the probabilistic methodologies can be considered, due to their
random search, as global optimization methods because they are capable of
finding the global optimum, whereas the mathematical programming algori-
thmsmay be trapped in local optima.
The ES optimization procedure initiates with a set of parent vectors, and

if any of these parent vectors gives an infeasible design then it ismodifieduntil
it becomes feasible. Subsequently, the offspring design vectors are generated
and checked if they are in the feasible region. According to (µ+λ) selection
scheme in every generation, the values of the objective function of the parent
and the offspring vectors are compared and the worst vectors are rejected,
while the remaining ones are considered to be the parent vectors of the new
generation. This procedure is repeated until the chosen termination criterion
is satisfied. The ES algorithm for structural optimization applications under
seismic loading can be stated as follows

1. Selection step – selection of si (i =1,2, ...,µ) parent design vectors

2. Analysis step– solve M(si)üt+C(si)u̇t+K(si)ut =Rt (i =1,2, ...,µ)

3. Constraints check – all parent become feasible

4. Offspring generation – generate sj (j = 1,2, ...,λ) offspring design
vectors

5. Analysis step – solve M(sj)üt+C(sj)u̇t+K(sj)ut =Rt
(j =1,2, ...,λ)

6. Constraints check – if satisfied continue, else go to Step 4

7. Selection step – selection of the next generation parent design vectors

8. Convergence check – If satisfied stop, else go to Step 4

5. Reliability-based structural optimization under earthquake

loading

In deterministic sizing optimization problems themain goal is tominimize
the weight of the structure under certain deterministic behavioral constraints
usually on stresses and displacements. In the RBO design, additional proba-
bilistic constraints are imposed in order to take into account various random



Reliability based optimization of steel frames... 597

parameters. So far, many articles have been devoted to the RBO design rese-
arch field and efficient methods have been presented (Alimoradi, 2003; Beck
et al., 1997; Collins et al., 1996; Ganzerli et al., 2000; Gasser and Schueller,
1997; Hasan andGrierson, 2002; Tsompanakis andPapadrakakis, 2004; Papa-
drakakis and Lagaros, 2002; Wen, 2000).

5.1. Performance based earthquake engineering

Over the last ten years various design codes and guidelines ([12], [2], [27],
Fajfar andKrawinkler, 1997) have introduced performance-based engineering
concepts to the evaluation or improvement of the existing structures and the
analysis anddesign of newones. Themain objective of this effort is to increase
the safety of old and new buildings against natural hazards and in the case
of earthquakes tomake them having a predictable and reliable seismic perfor-
mance. In other words, the structures should be able to resist earthquakes in
a quantifiable manner and hold possible damages desired within levels.

Fig. 2. Performance objectives [27]

A typical limit-state based design according to the modern codes concept
can be viewed as a two-level approach: serviceability (damage control) and ul-



598 M.Papadrakakis

timate strength (life safety) limit-states. On the other hand, the Performance-
Based Earthquake Engineering (PBEE) is a multi-level design approach whe-
re various levels of the structural performance are encountered ([12], [2], [27],
Fajfar and Krawinkler, 1997): operational, immediate occupancy, life-safety,
collapse-prevention as it is shown inFig.2. In otherwords, taking into account
all the important uncertainties related to seismic demandand structural capa-
city, PBEEdesign criteria try to define certain levels of structural performance
for various levels of seismic hazard. In Table 2, the description of the perfor-
mance levels is shown, and the relation of these levels with the inter-storey
drift limits is depicted in Table 3, whereas the various seismic risk levels are
presented in Table 4. The structural performance can be measured either in
terms of stresses, or displacements. Since the latter approach provides a better
indicator of damages it is usually preferred, especially in terms of drift limits.

Table 2. Structural performance levels

Performance level
NEHRP Vision Description
Guidelines 2000

Opera-
tional

Fully func-
tional

No significantdamagehasoccurred to structural
and non-structural components. The building is
suitable for normal intended occupancy anduse.

Immediate
occupancy

Opera-
tional

No significant damage has occurred to the
structure, which retains nearly all of its
pre-earthquake strength and stiffness. Non-
structural components are secure and most wo-
uld function, if utilities available. The building
may be used for intended purpose, albeit in an
impairedmode.

Life safety Life safe Significant damage to structural elements, with
substantial reduction in stiffness, however, mar-
gin remains against collapse. Non-structural ele-
ments are secured but may not function. Occu-
pancymay be prevented until repairs can be in-
stituted.

Collapse
prevention

Near
collapse

Substantial structural andnon-structural dama-
ge. Structural strength and stiffness substantial-
ly degraded. Little margin against collapse. So-
me falling debris hazards may have occurred.



Reliability based optimization of steel frames... 599

Table 3.Performance levels, corresponding damage state and drift limits

Performance level Damage state Drift limits

Fully operational No
damage

< 0.2%
Immediate occupancy

Operational
Damage control Repairable < 0.5%
Moderate

Life safe – Damage state Irreparable < 1.5%

Near collapse
Limited safety Severe < 2.5%
Hazard reduced

Collapse Collapse > 2.5%

Table 4.Earthquake hazard levels

Earthquake Return period Probability of
frequency [years] exceedance

Frequent 72 50% in 50 years
Occasional 225 20% in 50 years
Rare 475 10% in 50 years
Very rare 970 5% in 50 years
Extremely rare 2475 2% in 50 years

In the aforementioned guidelines, the use of various types of analysis me-
thods is suggested: linear static, non-linear static, linear dynamic, non-linear
dynamic, etc. A commonly used approach is the non-linear static, or push-over
analysismethod.According to thismethod the groundmotion is projected to a
diagramwhere the base shear seismic force vs lateral drift is plotted. Inmany
cases (Freeman, 1998; Fajfar, 1999), the Capacity Spectrum method is used,
where the push-over curve is plotted in a diagramhaving Sa–Sd (i.e. spectral
acceleration and spectral displacement) axes. In this plot, the response spec-
tra of the seismic excitation, for different hazard levels, are drawn in order to
determine if the basic relation: structural capacity > seismic demand, is true
or not. The response spectrum(in terms of Sa–Sd) of the seismic excitation is
produced by transforming the peak ground acceleration (PGA)which is given
by Probabilistic Seismic Hazard Analysis (PSHA).

PSHA quantifies the probability of exceeding a certain level of seismic
excitation at a specific site. Apart from its probabilistic considerations, it
consists of twomain parts: specification of the sourcemodel of the earthquake



600 M.Papadrakakis

as well as the modeling of the groundmotion. Usually, the description of the
sourcemodels is related tomagnitude, location and rate of occurrence (annual
or periodical). The groundmotionmodel is given by the so-called attenuation
relationships, where the PGA (or the ln(PGA) since PGA is considered to
follow logarithmic distribution and ln(PGA) normal orGaussian distribution)
is givenwith regard to themagnitude anddistance. It is also given in terms of
local conditions which sometimes can amplify the groundmotion. Once these
two basic aspects are specified, probabilistic calculations can be applied in
order to calculate the probability of exceedance of a certain ln(PGA) value
in annual or periodical terms and produce seismic hazard curves for various
scenarios of earthquake excitations (Field, 2003; [28]).

5.2. Formulation of RBO problems

In the present study the reliability-based sizing optimization of multi-
storey framed structures under earthquake loading is investigated. In deter-
ministic sizing optimization problems the aim is to minimize the weight of a
structure under certain deterministic behavioral constraints usually imposed
on stresses and displacements. In reliability-based optimal design additional
probabilistic constraints are imposed in order to take into account various ran-
dom parameters and to ensure that the probability of failure of the structure
is within acceptable limits. The probabilistic constraints enforce the condition
that the probability of failure of the system is smaller than a certain value.
In this work, the overall probability of failure of the structure, as a result
of multi-modal response spectrum analysis, is taken as the global reliability
constraint. The failure is detected when the maximum interstorey drift exce-
eds the threshold value, here considered as 4% of the storey height. Due to
engineering practice demands, the members are divided into groups having
the same design variables. This linking of the elements results in a trade- off
between the use ofmorematerial and the need of symmetry and uniformity of
structures due to practical considerations. Furthermore, it has to be to taken
into account that, due to manufacturing limitations, the design variables are
not continuous but discrete since the cross-sections belong to a certain set.

A discrete RBO problem can be formulated in the following form:

min F(s)

subject to gj(s)¬ 0 j =1, ...,m
si ∈ Rd i =1, ...,n
pf(dr > dal)¬ pa

(5.1)



Reliability based optimization of steel frames... 601

where F(s) is the objective function, s is the vector of design variables which
can take values only from the given discrete set Rd, gj(s) are deterministic
constraints and pf is the probability of failure of the structure, i.e. the proba-
bility that the interstorey drift dr exceeds the allowable value dal for various
structural performance levels. Most frequently, deterministic constraints of
a structure are member stresses and nodal displacements or the inter-storey
drifts.For rigid frameswith W-shapecross sections, as in this study, thedesign
constraints were taken from the design requirements specified by Eurocodes 3
[6] and 8 (Olsson et al., 2003).

Two safety limit states have been considered in this work: ultimate and
serviceability limit states. In the ultimate limit state the following equations
should be verified. For beams the capacity design against shear requires that
the following condition is satisfied

VG,Sd+VM,Sd
Vpl,Rd

¬ 1
2

(5.2)

where VG,Sd is the shear force due to non seismic actions and VM,Sd is the
shear force due to the application of resisting moments with opposite signs
at the extremities of the beam.Moreover, the applied moment should be less
that Mpl,Rd while the axial load should be less than the 15% of Npl,Rd.

For columns subjected to bending with the presence of an axial load the
following formula should be satisfied

Nsd
χminNpl,Rd

+
κyMsd
Mpl,Rd

¬ 1 (5.3)

where χmin factor is taken equal to 0.7 and κy equal to 1.Moreover, the shear
capacity should be two times greater than the applied shear force. The plastic
capacities for each member section are determined from the expressions

Mpl,Rd =
Wplfy
γM0

Npl,Rd =
Afy
γM1

Vpl,Rd =
1.04htwfy√
3 γM0

(5.4)

where γM0 and γM1 are considered equal to 1.10. The second order effects
are not considered since the following condition is assumed to be fulfilled in
all storeys

Ptotdr
Vtoth

¬ 0.10 (5.5)

where Ptot is the total gravity load at the storey considered, dr is the inter-
storey drift, Vtot is the total seismic shear and h is the storey height.



602 M.Papadrakakis

For the serviceability limit state the interstorey drift should be limited to

dr
ν
¬ 0.006h (5.6)

where ν is a reduction factor for the serviceability limit state (taken equal to
2.5 for the test example considered in this study). Different drift limits are
adopted for the probabilistic and deterministic constraints, since the failure
is supposed to take place for considerably higher deformations. Furthermore,
the strength ratio of the column to beam is calculated and also a check of
whether the sections chosen are of class 1, as EC3 suggests, is carried out. The
later check is necessary in order ensure that the members have the capacity
to develop their full plastic moment and rotational ductility, while the former
is necessary in order to have a design consistent with the strong column-weak
beam design philosophy.
The proposed reliability-based sizing optimization methodology proceeds

with the following steps

1. At the outset of the optimization procedure the geometry, boundaries
and loads of the structure under investigation are defined.

2. The constraints are defined in order to formulate the optimization pro-
blem as in Eq. (5.1).

3. The optimization phase is carried out with evolution strategies where
feasible designs should be generated at each generation. The feasibility
of the designs is checked for each design vector with respect to both
deterministic and probabilistic constraints of the problem.

4. The satisfaction of the deterministic constraints is monitored through a
MMRS analysis of the structure.

5. The satisfaction of the probabilistic constraints is realized with the re-
liability analysis of the structure using the MCS technique in order to
evaluate its probability of failure.

If the convergence criteria for the optimization algorithm are satisfied then
the optimum solution has been found and the process is terminated, else the
whole process is repeated from step 3 with a new generation of the design
vectors.

6. Numerical results

One test example has been considered in the present study in order to il-
lustrate the efficiency of the proposedmethodology for reliability-based sizing



Reliability based optimization of steel frames... 603

optimization problems under earthquake loading. This test example is a four-
bay, three-storeymoment resisting the plane frame shown inFig.3. The frame
has been previously studied in Gupta and Krawinkler (2000) where a deta-
iled description of the structure is given. The frame consists of rigid moment
connections and fixed supports. Each bay has a span of 9.15m (30ft), while
each storey is 3.96m (13ft) high. The permanent action considered is equal to
5kN/m2 while the variable action is equal to 2kN/m2, both distributed along
the beams. The frame is considered to be a part of a 3D structure where each
frame is 4.5m (15ft) apart. Themedian spectrum used for the determination
of the base shear corresponds to a peak ground acceleration of 0.32g. The
structuralmembers are divided into five groups, as shown inFig.3, correspon-
ding to five design variables of a discrete structural optimization problem.The
cross-sections are W-shape beamand column sections available frommanuals
of the American Institute of Steel Construction. The objective function is the
weight of the structure that is to be minimized.

Fig. 3. Test example – geometry andmember grouping

The deterministic constraints are those discussed in Section 5. The proba-
bilistic constraint is imposed on the probability of structural collapse which is
set equal to pf =0.001.Theprobability of failure causedbyuncertainties rela-
ted to seismic loads andmaterial properties of the structure is estimated using
bothMCS andLHS techniques. The earthquake groundmotion parameter, as
described in Eq. (5.2), yield stress and elastic modulus are considered to be
randomvariables. The type of probability density functions,mean values, and
variances of the random parameters are shown in Table 5. The seismic action
follows a log-normal probability density function,while the rest of the random



604 M.Papadrakakis

variables follow a normal probability density function. Formore details on the
probabilistic formulations of uncertainties the reader is referred to [19].

Table 5.Characteristics of the random variables

Random Probability Mean Standard
variable density function value deviation

E N 2.1 ·106MPa 0.10E
σy N 235MPa 0.10σy
Seismic load logN x̂, (3.12)1 δ, (3.12)2

For this test case the (µ+λ)−ES approach is usedwith µ = λ =5,while a
sample size of 5000 and1000 simulations is taken forMCSandLHStechniques,
respectively. Table 6 depicts the performance of the optimization procedure
for this test case. As it can be seen, the probability of failure corresponding to
the optimum computed by the deterministic optimization procedure is much
larger than the specified value of 10−3. In this example, the increase in the
optimum weight achieved, when the probabilistic constraints are considered,
is approximately 26% compared to the deterministic one, as it canbe observed
in Table 6.

Table 6.Performance of the methods

Optimization ES
pf

Optimum
procedure generations volume [m3]

DBO 157 9.32 ·10−2 15.95
RBO-MCS (5000 siml.) 65 0.08 ·10−2 21.44
RBO-LHS (1000 siml.) 72 0.10 ·10−2 21.30

7. Concluding remarks

In most cases, the optimum design of structures is based on determini-
stic parameters and is focused on the satisfaction of associated deterministic
constraints. Since there aremany random factors that affect the design,manu-
facturing and performance of a structure during its lifetime, the deterministic
optimum is not indeed a safe optimum. In order to find the real optimum the
designer has to take into account all necessary random parameters, and via
reliability analysis of the structure to determine its optimumdesign taking in-
to account the desired level of probability of the structural failure. Only after



Reliability based optimization of steel frames... 605

forming and solving thisRBOproblem, evenwith an additional cost inweight
and computing time, a global and realistic optimum structural design can be
found.

The aim of the proposed RBO procedure is twofold. To increase the safe-
ty margins of structures optimized under various uncertainties, while at the
same time minimizing their weight, and to reduce substantially the required
computational effort. The solution to realistic RBO problems in structural
mechanics is an extremely computationally intensive task. As it can be obse-
rved fromnumerical results, the computational cost of the solution to realistic
RBO problems can be order(s) of magnitude larger than the corresponding
cost of the deterministic optimization. Due to the size and complexity of RBO
problems, a stochastic optimization method, such as ES, appears to be the
most suitable choice.

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Niezawodnościowo zorientowana optymalizacja ram stalowych poddanych

obciążeniom sejsmicznym za pomocą ewolucyjnej techniki obliczeniowej

Streszczenie

Konstruowanie budowli o zwiększonej odporności na trzęsienia ziemi poprzezwy-
korzystanie rachunku prawdopodobieństwa i kryteriów eksploatacyjnych jest nowa



608 M.Papadrakakis

rozwijająca się dziedzina inżynierii konstrukcji. Nowametodologia i procedury postę-
powania celują w udoskonalanie istniejących programów i pakietów obliczeniowych
przewidujących zachowanie się danej budowli w zadanych warunkach. W pracy za-
prezentowano sztywną i wydajną metodologię optymalizacji zorientowaną na nieza-
wodność ram stalowych poddanych obciążeniom sejsmicznym. Optymalizację oparto
na obliczeniach ewolucyjnych, natomiast analizę niezawodności zrealizowanometodą
Monte-Carlo, w której do redukcji wymiarowości zagadnienia wykorzystano techni-
kę LHS (Latin Hypercube Sampling). Prawdopodobieństwo uszkodzenia konstrukcji,
w sensie przekroczenia granicznychprzemieszczeńmiędzykondygnacyjnych, obliczono
za pomocą wielomodalnej analizy widma odpowiedzi konstrukcji.

Manuscript received January 7, 2004; accepted for print March 16, 2004