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INNOVATIVE SEISMIC DESIGN OPTIMIZATION WITH RELIABILITY CONSTRAINTS

ANARGYRI TH. GARAVELAS, NIKOS D. LAGAROS, MANOLIS PAPADRAKAKIS*

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

9, Iroon Polytechniou Str., Zografou Campus, 157 80 Athens, Greece e-mail: {cv01053,nlagaros, mpapadra)@central.ntua.gr

ABSTRACT: Performance-Based Design (PBD) methodologies is the contemporary trend in designing better

and more economic earthquake-resistant structures where the main objective is to achieve more predictable and

reliable levels of safety and operability against natural hazards. On the other hand Reliability-Based Optimization

(RBO) methods directly account for the variability of the design parameters into the formulation of the

optimization problem. The objective of this work is to incorporate PBD methodologies under seismic loading into

the framework of RBO in conjunction with innovative tools for treating computational intensive problems of

real-world structural systems. Two types of random variables are considered: Those which influence the level of

seismic demand and those that affect the structural capacity. Reliability analysis is required for the assessment of

the probabilistic constraints within the RBO formulation. The Monte Carlo Simulation (MCS) method is

considered as the most reliable method for estimating the probabilities of exceedance or other statistical quantities

albeit with excessive, in many cases, computational cost. First or Second Order Reliability Methods (FORM,

SORM) constitute alternative approaches which require an explicit limit-state function. This type of limit-state

function is not available for complex problems. In this study, in order to find the most efficient methodology for

performing reliability analysis in conjunction with performance-based optimum design under seismic loading, a

Neural Network approximation of the limit-state function is proposed and is combined with either MCS or with

FORM approaches for handling the uncertainties. These two methodologies are applied in RBO problems with

sizing and topology design variables resulting in two orders of magnitude reduction of the computational effort.

KEY WORDS: Performance-based design, reliability-based optimization, response surface, Monte Carlo, neural

networks, sizing-topology design variables, earthquake resistant structures

*Corresponding Author

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1. INTRODUCTION

In deterministic optimization problems randomness and uncertainty are considered through

several safety factors introduced into the constraints of the optimization problem. On the other

hand, non-deterministic performance measures are increasingly being taken into consideration

in many contemporary engineering applications that involve various reliability requirements.

In structural optimization, non-deterministic performance measures can be taken into account

using two distinct formulations: Robust Design Optimization (RDO) [1,2] and

Reliability-Based design Optimization (RBO) [3-5].

The modern conceptual approach to structural design under seismic loading is based on

the principal that a structure should meet performance-based objectives for a number of

different hazard levels, ranging from earthquakes of small intensity with small return period, to

more destructive events with large return period. This approach constitutes the

Performance-Based Design (PBD) concept which has been introduced in order to increase the

safety against natural hazards. According to PBD the structures should be able to resist

earthquakes in a quantifiable manner and to preset levels of desired possible damage. The

current state of practice in performance-based engineering can be found in US guidelines such

as FEMA-350 [6] and FEMA-356 [7].

The structural performance during an earthquake depends highly on a number of

structural parameters which are inherently uncertain. Such parameters are, among others, the

material properties, the workmanship, the hysteretic behaviour of structural members and

joints. The intensity and the earthquake ground motion characteristics are also random.

Furthermore, uncertainty is also involved in the analysis procedure that would be adopted and

in the numerical simulation of the structure. In order to account for as many as possible of the

above uncertainties, a reliability-based in conjunction with a performance-oriented approach

should be considered. For earthquake engineering applications, the reliability problem can be

defined as a problem of two separate types of variables, representing the demand and the

capacity. This concept is adopted in the reliability-based design procedure suggested in the

SAC/FEMA project [8]. The SAC/FEMA methodology allows the consideration of

uncertainties on both capacity and demand. However, for the calculation of the limit-state

probabilities, prior knowledge of the distribution and the variance of the capacity and the

demand is necessary.

A limited number of studies have been published in the past, where the PBD concept is

implemented in a structural optimization problem considering uncertainties. These studies are

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restricted to relatively small-scale structures due to the increased computational cost. Αleatory

and epistemic uncertainties are introduced in a structural optimization environment by Beck et

al. [9]. In the work by Wen [10], the issue of the proper consideration of the uncertainty in the

demand and capacity and the balance of reliability against cost is investigated based on the

minimization of the expected lifecycle cost. The concept of performance-based design within

the context of robust design optimization for the design of structures is examined by Lagaros

and Fragiadakis [11]. In the work by Foley et al. [12] a state-of-the-art model code

performance-based design methodology is proposed. This design methodology is applied to

multiple-objective optimization problems for single storey and multi-storey structural

frameworks with fully and partially restrained connections.

Structural reliability analysis can be performed either with simulation methods, such as

the Monte Carlo Simulation (MCS) method, or with other approximation methods. First and

second order reliability methods (FORM, SORM) require prior knowledge of the mean and the

variance of each random variable. Furthermore, these methods require a differentiable failure

function. On the other hand, although the major advantage of MCS is that accurate solutions

can be obtained for almost every problem, yet it requires excessive computational cost in many

cases. Variance reduction techniques, such as Importance Sampling, Directional Simulation,

Antithetic Variates or Adaptive Sampling, have been proposed in order to reduce the

computational effort of MCS. The disadvantage of these methods is that they require prior

knowledge of the behaviour of the structure in order to determine the most effective sampling

region, which for many practical problems is not clearly identifiable. Recent results [13] reveal

that variance reduction techniques still require significant number of the system response

evaluations to estimate failure probabilities of the order less than 10−3. Other recently proposed

simulation methods, such as Line Sampling [13] and Subset Simulation [14], were proved to be

very efficient in reducing the required sample size and the computational cost; however, their

performance and ergodicity are sensitive to the values of certain parameters which are not

known a priori.

The computational cost is the main barrier that prohibited the application of uncertain

optimization methodologies considering earthquake loading. For the application of a

computationally efficient MCS method to complex structural models it would be necessary to

have an approximate knowledge of the limit-state function g(x). Such an approximation

contributes in reducing the excessive number of repeated finite element analyses required for

the MCS. On the other hand, since complex structural reliability problems are characterized by

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the implicit nature of the limit-state functions, the implementation of FORM or SORM requires

an explicit approximation of either the entire limit-state function g(x) or of its limit-state

surface g(x)=0 in the space of the random variables x. The Response Surface (RS) method is

customarily used for approximating the limit-state function. It was found, though, that the

performance of the RS method is dependent on the experimental points required to define the

limit-state function approximation due to the rigid and non-adaptive structure of the function

implemented by the RS method [15,16]. Several attempts have been presented in the past where

the limit-state function is estimated with a Neural Network (NN) approximation. These

attempts, however, were limited to static loads [17,18]. In the present work innovative and

efficient procedures for performing RBO are implemented, within the performance-based

design framework of FEMA-356 under seismic loading, for the design of real-world structural

systems. Randomness and uncertainty are taken into consideration with two methodologies for

reducing the computational cost. The methodologies are based on the Monte Carlo and the First

Order Reliability methods and exploit the Neural Network predictions of the limit state

function. Furthermore, sizing and topology design variables are incorporated into deterministic

and probabilistic formulations of the optimization problem.

2. PERFORMANCE-BASED DESIGN

The majority of the seismic design codes belong to the category of the prescriptive design

codes, which take into consideration site selection and development of conceptual, preliminary

and final design stages. According to a prescriptive design code the strength of the structure is

evaluated at one limit state, between life-safety and near collapse, using a response spectrum

based loading corresponding to one design earthquake [19]. In addition, the serviceability limit

state is usually checked in order to ensure that the structure will not deflect or vibrate

excessively. On the other hand, PBD is a different approach for the seismic design of structural

systems which includes, apart from the site selection and the consideration of the design stages,

the performance of the building after construction in order to ensure reliable and predictable

seismic performance over its life.

Prescriptive building codes do not provide acceptable levels of a building life-cycle

performance, since they only include provisions aiming at ensuring adequate strength of

structural members and, indirectly or implicitly, of the overall structural strength. The basic

philosophy of a PBD procedure is to allow engineers to determine explicitly the seismic

demands at preset performance levels by introducing design checks of a higher level.

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Performance-based seismic design has the following distinctive features with respect to a

prescriptive design requirements: (i) Allows the owner in cooperation with the structural

engineer to define the appropriate level of seismic hazard and the corresponding performance

level where the seismic demand will be evaluated. (ii) The structure is designed to meet the

requirements corresponding to a number of levels of seismic intensity.

2.1. Performance Levels and Objectives

The main part in a performance-based seismic design procedure is the definition of the

performance objectives. A performance objective is defined as a given level of performance for

a specific hazard level. In this study, three performance objectives corresponding to the

“Enhanced Rehabilitation Objectives” of FEMA-356 [7] for buildings have been considered.

The first part in the definition of a performance objective is the selection of the level of

structural performance. The following performance levels have been considered:

i) Operational level: the overall damage is characterized as very light. No permanent drift

is encountered, while the structure essentially retains original strength and stiffness.

ii) Life Safety level: the overall damage is characterized as moderate. Permanent drift is

encountered but partial or total structural collapse is avoided. Gravity-load bearing

elements continue to function while there is no out-of plane failure of the infill walls.

The overall risk of life-threatening injury as a result of structural damage is expected to

be low. It should be possible to repair the structure; however, for economic reasons this

may not be practical.

iii) Collapse Prevention level: the overall damage is characterized as severe. Substantial

damage has occurred to the structure, including significant degradation in the stiffness

and strength of the lateral-force resisting system. Large permanent lateral deformation

of the structure and degradation in the vertical load bearing capacity is encountered.

However, all significant components of the gravity load resisting system continue to

carry their gravity load demands. The structure may not be technically repairable and is

not safe for reoccupancy, since aftershock activity could induce collapse.

The second part in the definition of a performance objective is to determine the

corresponding seismic hazard level. Earthquake hazard, according to FEMA-350 [6], includes

parameters such as direct ground fault rupture, ground shaking, liquefaction, lateral spreading

and land sliding. Ground shaking is the only earthquake hazard that structural design

provisions of building codes directly address. Ground shaking hazard is defined by means of a

hazard curve, which indicates the probability that a measure of seismic intensity (e.g peak

ground acceleration or 1st mode spectral acceleration) will be exceeded over a certain period of

time. The three levels of seismic hazard considered can be defined as follows:

i) Occasional Earthquake Hazard level: with probability of exceedance 50% in 50 years

and interval of recurrence 72 years.

ii) Rare Earthquake Hazard level: with probability of exceedance 10% in 50 years and

interval of recurrence 475 years.

iii) Maximum Considered Event Earthquake Hazard level: with probability of exceedance

2% in 50 years and interval of recurrence 2475 years.

According to the Enhanced Objectives of FEMA-350 the following three performance

objectives are considered: (i) Operational level corresponding to Occasional Earthquake

Hazard level, (ii) Life Safety level in connection to Rare Earthquake Hazard level, (iii)

Collapse Prevention level correlated to the Maximum Considered Event Earthquake Hazard

level.

2.2. Structural analysis phase – Evaluation of structural capacity

According to FEMA-356 four alternative analytical procedures, based on linear and nonlinear

static and dynamic structural response, are recommended for the structural analysis of

buildings under earthquake loading. In this study the Nonlinear Static analysis Procedure

(NSP) of FEMA, also known as pushover analysis, is used to assess the structural capacity. In

order to assess the capacity of the structure, for every performance level considered, a lateral

load distribution that follows the fundamental mode is adopted. For the PBD procedure, prior to

any structural analysis, the column to beam strength ratio is calculated and is examined whether

the sections chosen are of class 1 , as Eurocode 3 (EC3) [20] suggests. Class 1 cross-sections

are those which can form a plastic hinge with the rotation capacity required from plastic

analysis without reduction of the resistance. The column to beam strength ratio is calculated as:

1 20M pl ,column

M pl ,beam.

∑

∑≥ (1)

where ΣMpl,column and ΣMpl,beam is the sum of the design values of plastic moment resistances of

the structural members at each joint.

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The PBD procedure consists of the following steps: (i) All EC3 checks must be satisfied

for the gravity loads; (ii) if the checks of Step (i) are satisfied then NSP is performed in order to

explicitly calculate the demand for the defined intensity levels. The structural capacity is

associated to the maximum interstorey drift values θ, and the acceptance criteria of Step (ii) are

confirmed if satisfied or not in order to accept or revise the design. NSP is terminated as soon as

a target displacement that corresponds to 1.5 times the 2% in 50 years (2/50) probability of

exceedance earthquake is achieved [7]. A flowchart for the PBD procedure employed in this

study is shown schematically in Figure 1.

For every tentative design the capacity is assessed at three performance levels using the

displacement coefficient method [6]. The structure is “pushed” with a triangular distribution of

the lateral loads until the target displacement is reached. The target displacement can be

obtained from the FEMA-356 formula:

2

0 1 2 3 24e

tTC C C C Sa gδπ

= (2)

where C0, C1, C2, C3, are modification factors. C0 relates the spectral displacement to the likely

building roof displacement. C1 relates the expected maximum inelastic displacements to the

displacements calculated for linear elastic response. C2 represents the effect of the hysteresis

shape on the maximum displacement response and C3 accounts for P-Δ effects. Te is the

effective fundamental period of the building in the direction under consideration. Sa is the

response spectrum acceleration corresponding to the Te period.

The applicability of NSP has limitations; it provides very good estimates of global as well

as local inelastic deformation demands in the case of structures that vibrate primarily in their

fundamental mode. NSP is capable to expose the design weaknesses that might remain hidden

when an elastic analysis is considered. Such weaknesses include story mechanisms, excessive

deformation demands, strength irregularities, and overloads on potentially brittle elements,

such as columns and connections. However, as higher mode contributions become more

significant, NSP overestimates the maximum displacements computed during the analysis

procedure. High-rise buildings or buildings with irregular distribution of strength and/or mass

are not suitable for the application of the NSP. On the contrary, low to mid-rise or regular

buildings vibrate primarily in the fundamental mode and their response is reasonably predicted

by NSP. This type of building is considered in the present study. Furthermore, to take into

account the effect of simultaneous ground shaking in two orthogonal directions, the

recommendation of FEMA-350 is employed, where multidirectional excitation effects are

accounted for by combining 100% of the response due to loading in the longitudinal direction

with 30% of the response due to loading in the transverse direction, and vice versa. The most

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severe maximum interstorey drift value obtained when both of these load combinations are

applied is used to obtain the seismic demand.

3. RELIABILITY-BASED STRUCTURAL DESIGN OPTIMIZATION

Reliability theory is introduced in structural engineering and structural optimization in order to

consider, in a more rational way, all existing uncertainties which are not known with the

desirable degree of precision and can influence structural response. Reliability is recognized as

a safety constraint in structural engineering, therefore any optimum design under reliability

constraints should balance both cost and safety. In the case of deterministic optimum design,

stress and displacement constraints are considered in accordance with the design code safety

factors, while in reliability based optimum design additional probabilistic constraints related to

the behaviour of the structure are considered.

The main goal of Reliability-Based design Optimization (RBO) is to perform a safe and

economic design with respect to extreme loading events. In RBO problems, probabilistic

constraints are incorporated into the optimization procedure leading to unbiased estimates of

the structural performance and, subsequently, to the determination of designs that are located

within a range of target failure probabilities. In the case of the incorporation of PBD approach

into the RBO formulation, the objective function is assumed to be the weight of the structure,

while the deterministic constraints correspond to serviceability and ultimate limit state checks

and the probabilistic constraints to the probabilities of violation of the ultimate limit state

checks. In particular, the probabilities of exceeding the limit states, which correspond to each

hazard level, are taken as the probabilistic constraints. The probabilistic constraints enforce the

condition that the probability violation is smaller than a certain threshold value.

The RBO problem can be formulated in the following form:

(3)

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a

min ( , )subject to g ( , ) 0 (serviceability checks) g ( , ) 0 (ultimate limit state checks) P(g ( , ) 0) P (probabilistic checks)

where

IN x

EC x

PBD x

PBD

C∈

≤≤

> ≤

s s μs μs μ

s x

F

n

2

R N( , )∼ x x

∈sx μ σ

The vectors s, x and μr represent the design, the random variable and the mean value vectors,

respectively. F is the feasible region, where both the serviceability gEC3 and the ultimate limit

state constraint functions gPBD are satisfied, Pa is the allowable probability of violating the

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ultimate limit state constraints, while CIN corresponds to the objective function representing the

initial construction cost.

4. NEURAL NETWORKS

Artificial Neural Networks (NN) are biologically inspired, since they are composed by

elements that perform in a manner analogous to the elementary functions of a biological

neuron. These elements are known as artificial neurons. NN are organized in a way that is

related to the anatomy of the brain and they exhibit a surprising number of the brain’s

characteristics such as: learning from experience, generalizing from previous examples and

abstracting essential characteristics from sets of inputs containing irrelevant data. The use of

hidden layers and nonlinear activation functions enhance the ability of the NN to “learn” the

complicated relationship between a set of input and a set of output data.

A feed-forward NN attempts to create a desired mapping between the inputs and the

targets of a training set. The training set is composed by m input-target pairs D=[xm, tm]. A

Neural Network architecture A consists of a specific number of layers, a number of units in

each layer and a type of activation function. If a set of weight parameters w is assigned to the

connections of the network, a mapping y(xm; w, A) is defined between the input vector xm and

the output vector y. The quality of this mapping is measured using the following error function:

( 2

D1( | , ) = ( ; , )2

m

m−∑D w )my x w tE A A (4)

A learning algorithm tries to determine the weight parameters w in order to minimize the error

function ED. Iterative minimization algorithms are therefore used to obtain the optimum weight

parameters w. For the solution of the minimization problem the operator O is applied, resulting

to the following iterative formula:

(5) ( +1) ( ) ( ) ( ) = ( ) = +Δt t tw w wO tw

t

Most of the numerical minimization methods are based on the above expression, while to

initiate the algorithm a starting vector of weight parameters w(0) is necessary. The changing part

of the algorithm Δw(t) can be further decomposed in:

(6) ( ) ( )Δ att=w d

where d(t) specifies the direction of search and at is the corresponding step size.

Learning algorithms are classified to local or global algorithms. Global algorithms make

use of the knowledge about the state of the entire network, such as the direction of the overall 9

weight update vector. In the widely used back-propagation global learning algorithm the

gradient descent algorithm is used. In contrast, local adaptation strategies are based on specific

information of the weight values such as the temporal behaviour of the partial derivative of the

weights. The local approach is better related to the NN concept of distributed processing, where

the computations are performed independently. Moreover, it appears that for many applications

local strategies achieve faster and more reliable predictions than global techniques. The

Resilient backpropagation learning algorithm, abbreviated as Rprop [21], is adopted in this

study. Rprop is a local algorithm, based on an adaptive version of the Manhattan-learning rule

that has been proved very efficient in the past [22].

5. SOLVING THE OPTIMIZATION PROBLEM

Evolutionary Algorithms (EA) are population based, probabilistic, direct search optimization

algorithms gleaned from principles of Darwinian evolution. Starting with an initial population

of μ candidate designs, an offspring population of λ designs is created from the parents using

variation operators. Depending on the manner in which the variation and selection operators are

designed and the spaces in which they act, different classes of EA have been proposed. In the

EA algorithm employed in this study, each member of the population is equipped with a set of

parameters:

10

c

ncRn

(7) γd

c σ

d c d

nnd

n n nc

[( , , ( , , )] (Ι ,Ι )

Ι =D R

Ι =R R [ π,π] a

+

+

= ) ∈

×

× × −

a s γ s σ α

where sd and sc are the vectors of discrete and continuous design variables defined in the

discrete and continuous design sets , respectively. Vectors γ, σ and α are the

distribution parameter vectors taking values in

n and dDn n, and [ , ] ,aR Rγ σ+ + −π π respectively. Vector γ

corresponds to the variances of the Poisson distribution. Vector corresponds to the

standard deviations (1 ≤ n

σnR +∈σ

σ ≤ nc) of the normal distribution. Vector is related to the

inclination angles (n

n[ π,π] a∈ −α

α = (nc-nσ/2)(nσ-1)) defining linearly correlated mutations of the continuous

design variables sc, where n = nd + nc is the total number of design variables.

Let P(t) = {a1,…,aμ} denotes a population of individuals at the t-th generation. The

genetic operators used in the EA method are denoted by the following mappings:

(8)

μ λd c d c

λ λd c d c

k k μμ d c d c

rec : (Ι ,Ι ) (Ι ,Ι ) (recombination)

mut : (Ι ,Ι ) (Ι ,Ι ) (mutation)

sel : (Ι ,Ι ) (Ι ,Ι ) (selection, k {λ, μ+λ})

→

→

→ ∈

A single iteration of the EA, which is a step from the population to the next parent

population is modelled by the mapping

( )P tp

( )P t+1p

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μ (9) μEA d c t d c t+1opt : (Ι ,Ι ) (Ι ,Ι )→

In Figure 2 a pseudo-code of the EA algorithm is depicted. At the beginning of the

procedure in generation t = 0 the initial parent population , composed by μ design vectors,

is generated randomly (step 3 of the pseudo-code). Steps 5 to 12 correspond to the main part of

the EA algorithm, where in every generation λ offspring vectors are generated by means of

recombination and mutation. D

( )P tp

l is a sub-population with two members selected from the parent

population of the current generation (Step 6) which is used by the recombination operator.

Recombination and mutation operators, described in steps 7 to 10, act on both design variable

vectors s

( )P tp

l and distribution parameter vectors (distribution parameter vectors are denoted as yl in

the pseudo-code). In step 11 the objective and constraint functions are calculated in order to

assess the design vectors in terms of the objective function value and feasibility.

6. SOLVING THE RELIABILITY ANALYSIS PROBLEM

The advancements in structural reliability theory during the last twenty years and the

attainment of more accurate quantification of the uncertainties associated with structural loads

and resistances have stimulated the interest in the probabilistic treatment of structural systems

[23]. The reliability of a structure or its probability of failure is an important factor in the design

procedure since it investigates the probability of the structure to successfully accomplish its

design requirements. Reliability analysis leads to safety measures that a design engineer has to

take into account due to the aforementioned uncertainties. Although from a theoretical point of

view the field has reached a stage where the developed methodologies are becoming

widespread, from a computational point of view serious obstacles have been encountered in

practical implementations. First and Second Order Reliability Methods (FORM and SORM),

that have been developed to perform structural reliability, although they lead to elegant

formulations, they require prior knowledge of the means and variances of the component

random variables and the definition of a differentiable limit-state function. On the other hand

the Monte Carlo Simulation (MCS) method is not restricted by the form and the knowledge of

the limit-state function but is characterized by the high computational cost.

6.1. Monte Carlo simulation

The MCS method is applied in stochastic mechanics when an analytical expression of the

limit-state function is not attainable. This is mainly the case in problems of complex nature with

a large number of random variables, where all other stochastic analysis methods are not

applicable. In structural stochastic analysis problems, the probability of violation of the

behavioural constraints can be written as:

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d (10) ( ) 0

( )viol xg

p f≥

= ∫x

x x

where fx(x) denotes the joint probability of violation for the random variables, the limit-state

function g(x)<0 defines the safe region and x is the vector of the m random variables.

Considering that MCS is based on the theory of large numbers (N∞) an unbiased estimator of

the probability of violation is given by:

1

1 ( )∞

=∞

= ∑N

viol jj

p IN

x (11)

where xj is the j-th vector of the random structural parameters, and I(xj) is an indicator for

successful and unsuccessful simulations defined as:

1 if ( ) 0

( )0 if ( ) 0

gI

g≥⎧

= ⎨ <⎩

jj

j

xx

x (12)

In order to estimate pviol an adequate number of Nsim independent random samples is produced

using a specific, uniform probability density function of the vector xj. The value of the violation

function is computed for each random sample xj and the Monte Carlo estimation of pviol is given

in terms of sample mean by:

H

sim

NNviolp ≅ (13)

where NH is the number of successful simulations and Nsim the total number of simulations.

The basic MCS is simple to use and has the capability of handling practically every

possible case regardless of its complexity. However, for typical structural reliability problems

the computational effort involved becomes excessive due to the enormous sample size

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required. To reduce the computational effort more elaborate simulation methods, called

variance reduction techniques, have been developed, their efficiency though is limited for

larger probability values. Moreover, despite the improvements achieved on the efficiency of

computational methods for treating structural analysis problems of large scale structures, they

still require disproportional computational effort for reliability analysis of realistic problems.

This is the reason why very few successful numerical investigations are known in estimating

the probability of failure and are mainly restricted to simple elastic frames and trusses [24].

In a previous work by the authors [17] two methodologies have been proposed where

Neural Networks have been incorporated into the RBO problem in order to reduce the

computational cost of MCS. Both methodologies take advantage of the global approximation

capabilities of the neural networks. The first one exploits the global approximation capabilities

over the design variables space while the second operates on the space of the random variables.

In the first methodology the NN is trained once outside the RBO phase, while in the second one

a new NN is trained for every new candidate design inside the RBO procedure. As it can be

seen from Figure 3a, three steps are required for the first implementation. The first step

corresponds to the construction of the training set, where N design vectors are generated using

the Latin Hypercube Sampling (LHS) technique [25] and their feasibility with respect to both

deterministic and probabilistic constraints is evaluated. The training phase of the NN is

performed in step 2. This phase consists of the selection of a suitable NN architecture and the

training/testing procedures. The trained NN is used to predict the feasibility of new design

vectors both in terms of deterministic and probabilistic constraint checks. The RBO procedure

is performed using Evolutionary Algorithms in the last step of the methodology where the

trained NN is used to assess the feasibility and the quality of every new candidate design

selected by the Evolutionary Algorithm.

In the second methodology, shown schematically in the flowchart of Figure 3b, for every

new candidate design inside the optimization procedure, a new NN is trained in order to replace

the analyses required during the MCS. Figure 3b describes the procedure for assessing the λ

offsprings of an optimization cycle (generation) of the Evolutionary Algorithm. As soon as the

NN is trained over a sample of M random vectors generated using LHS, the trained NN is

applied to predict the structural performance for every new vector of random variables

encountered during the MCS. The second methodology has been proved more efficient [17] in

terms of robustness and computational effort and for this reason it is implemented in the current

work. Special care has been taken in this implementation to alleviate this extrapolation

drawback of NN by generating the training sample using the Latin Hypercube Sampling

method in the range of μi ± 6 σi for, where μi and σi are the mean value and standard deviation of

the i-th random variable.

6.2. First-Order Reliability method

In the general case of a nonlinear limit state function the main objective of the First Order

Reliability Method (FORM) is to calculate the reliability index β. The Hasofer-Lind reliability

index β [26] is calculated by a process of minimization, and the probability of violation is

approximated by:

(14) violp = Φ(-β)

where Φ is the standard normal cumulative distribution function. This equation is exact when

the failure criterion is linear and all random variables have normal distributions. Given a vector

of basic variables x, a failure surface ∂ω on which the failure criterion g(x)=0 is satisfied and a

safe region denoted by g(x)>0, the vector of the reduced variables z is defined as follows:

(15) -1x ( - )≡ ⋅z S x μx

where Sx, is a diagonal matrix of the standard deviations and μx is the vector of mean values.

Then the Hasofer-Lind reliability index β is defined as:

T

ωβ min

∈∂≡

zz z (16)

The point on the failure surface g(x)=0, where its transformation to the z space satisfies

equation (16), is called design or most probable point and will be denoted as zD. The design

point zD is located on the limit-state surface, g(x)=0 and has minimum distance from the origin

in the standard normal space. For applying either first or second-order methods to complex

structural models it is necessary to have an explicit expression or an approximation of either the

entire limit-state function g(x) or of its limit-state surface g(x)=0 in the space of the random

variables x. This is because these methods require not only knowledge of the function but also

of its gradient in the vicinity of its limit-state surface. In the case of unknown expression, the

limit-state function is usually approximated by the Response Surface method.

6.2.1. The Response Surface (RS) method

The RS method was originally proposed by Box [27] as a statistical tool, to find the operating

conditions of a chemical process at which some response was optimized. In order to reduce the

computational effort while maintaining an acceptable accuracy, two important issues should be

14

considered when applying the RS method to the failure probability: (i) The definition of

experimental points for defining the approximation of the limit-state function, (ii) The

analytical expression of the Response Surface function [15]. Usually, a quadratic RS function is

assumed:

m m

2i i i i

i=1 i=1g( ) a + b x + c x= ∑ ∑x (17)

defined in an m-dimensional random variable space where the constants a, bi and ci are

determined by evaluating g(x) at certain specified experimental points, while xi, i=1, …,m are

the random variables. Higher order functions are not generally used for conceptual as well as

for computational reasons.

Bucher and Bourgund [28] proposed an interpolation scheme for the solution of structural

reliability problems, where the quadratic RS function of Eq. (17) is defined with 2m+1

experimental points. It was suggested the experimental points xi, to be taken as xi = μi ± f σi,

where μi and σi are the mean value and standard deviation of the i-th random variable and f is an

arbitrary factor taken equal to 3. Figures 4a and 4b depict the sampling method for defining the

2m+1 experimental points where xi,low = μi - 6 σi and xi,up = μi + 6 σi. In order to avoid the

undesirable case of Figure 4a, where unrealistic values of the random variables are generated

(i.e. a negative value of the modulus of elasticity), a correction of the experimental points is

performed by moving all trial points to an acceptable region as shown in Figure 4b. It was later

shown [15,29] that using a constant value of f = 3 may not always yield good solutions,

particularly when g(x) is highly nonlinear. A better fit of the RS function was obtained by

updating the centre point:

i,D ii,M i

D

(x μ ) (x = μ

( - (− )

+) )

gg g

μμ x

(18)

which helps to locate the new centre point closer to the actual g(x)=0, and by decreasing the

value of the parameter f in subsequent cycles of updating. Then the interpolation scheme of Eq.

(17) is repeated using xM as a new center point. A number of modifications for better estimating

the RS through the generation of the experimental points have been proposed [30-38] in order

to alleviate the shortcomings of the method.

6.2.2. Neural network approximation of the limit-state function

In the previously mentioned implementations of the RS method, it was found that the

performance of the method is significantly influence by the polynomial representation and the

15

16

way the experimental points are generated. In an effort to improve the robustness of the

approximation procedure the polynomial representation of RS is replaced by an approximation

produced by neural networks. In this study local approximation capabilities of the neural

networks in the vicinity of the centre point xM are implemented with the intention to reduce the

number of the training patterns required by the global NN approximator described in Section

6.1.

Hurtado [16] confirmed with statistical learning concepts the sensitivity of the RS

approximation via polynomial expressions compared to the neural network approximations.

The global capabilities of the neural network approximator were examined [39-42] in the

framework of FORM for simple analytic limit-state functions or academic structural reliability

problems. The local approximation capabilities of the neural networks in conjunction with

reliability analysis methods are examined in a simple portal frame by Deng et al. [43].

In this work a novel neural network based methodology, denoted as FORM-NN, is

proposed for the solution of reliability analysis problems in conjunction with structural

optimization considering performance objectives. In the proposed methodology neural

networks are implemented as a local approximator in the vicinity of the centre point xM in a

similar way that the RS based FORM is used. More specifically a neural network

approximation of the limit-state function replaces the polynomial expression of RS. The steps

of the proposed FORM-NN methodology are given in Figure 5. In the first step of the proposed

implementation a number of experimental points are generated through the LHS technique.

Figures 6a and 6b demonstrate schematically the LHS technique with a correction procedure.

This sampling method can generate a variable number of samples well distributed over the

entire range of interest. On the other hand, the number of samples generated with the standard

sampling procedure [28] (Figure 4) is fixed to 2m+1 experimental points. According to LHS

the distribution function of each random variable is divided into a number of stratums of equal

marginal probability. These stratums are randomly selected for each random variable and

randomly shuffled among different variables to define the experimental points. The main

advantage of the LHS technique is that it produces well distributed samples of almost any

sample size of experimental points. This characteristic feature of LHS is demonstrated in the

framework of FORM-NN in the numerical studies where it is shown that a reduced number of

experimental points are required for adequately defining the NN approximation. It has to be

emphasized that FORM-NN does not suffer from the extrapolation drawback since any point

outside the range of interest is corrected as it demonstrated in Figures 6a and 6b.

In the next step the experimental points are used to define the NN approximation of the

limit state function according to the following equation:

k m

H,ij O,ij ki=1 i=1

g( ) = w × w × x⎛ ⎞⎛⎜ ⎜

⎝ ⎠⎝ ⎠∑ ∑x f f ⎞

⎟⎟ (19)

where k is the number of the hidden layer neurons, wH and wO are the weight parameters of the

hidden and output layers, respectively, and f(x) is the sigmoid transfer function:

1(y) =1+ exp(-γy)

f (20)

The coefficient γ defines the slope of the sigmoid function. By the time the NN is trained and

the weight parameters are defined, FORM is performed in order to define the reliability index β

and the corresponding design point xD. Hereafter, the new centre point xM is defined according

to Eq. (18). If there is no convergence with respect to the reliability index β and the

corresponding design point xD, new experimental points are generated with LHS for the new

centre point xM, in order to define the new approximation of limit-state function (Eq. (19)). The

advantages of the proposed NN approximation scheme are demonstrated in the numerical test

section, where it is shown, that contrary to the quadratic polynomial approximation of RS, this

methodology is not affected by the number of the interpolation points and the area from which

the points are generated as defined by the coefficient f.

7. NUMERICAL TESTS

The seven-storey 3D steel building, shown in Figure 7, has been considered in order to perform

both deterministic and reliability-based design optimization considering two types of design

variables: size of member cross-sections and the topology of the vertical structural elements.

All columns are made of X-shapes obtained by welding two HEB profices along the

longitudinal axis. This solution avoids the engagement of the column web in moment resistant

connections as well as the development of high stress connections along the beam flanges

welds which occur in the case of box-columns, due to the transmission of the beam web stresses

through the out of plane bending of the column wall plates [44]. The frame members are

modelled with the force-based fiber beam-column element [45], where four integration points

are considered for each structural element. The initial FE model of the 3D frame consists of 9×6

columns, 1,029 beam-column finite elements with 2,268 degrees of freedom and a slab

thickness of 10 cm. The mean value of the modulus of elasticity is 210 GPa and of the yield

17

stress is fy=235 MPa (S235). The slab distributed loading is G=5.4kPa corresponding to the

permanent action and Q=5 kPa for the live load [46].

The columns and the beams of the structure have double I-shaped cross-sections, and are

grouped into five sets, each corresponding to an independent design variable. The columns on

the corners correspond to the 1st group, the intermediate columns of the perimeter of the plan in

the x-direction belong to the 2nd group, while the intermediate columns in the y-direction of the

plan are grouped in the 3rd set. All remaining columns belong to the 4th group, while the beams

constitute the 5th group. In the case of the combined sizing and topology optimization, two

additional design variables are considered corresponding to the number of the columns along

the two structural dimensions. All optimization runs were performed using the (μ+λ) EA

optimization scheme with μ=10 parents and λ=10 offsprings.

7.1. Random variables

Two types of random variables are considered: randomness of the ground motion excitation

which influences the level of seismic demand, and randomness of the material properties which

affects the structural capacity. The structural stiffness is directly connected to the modulus of

elasticity E, while the strength is influenced by the yield stress fy. For each group of structural

elements three random variables are considered; the modulus of elasticity E, the yield stress fy

and the hardening parameter b of the stress-strain curve. Overall fifteen random variables

related to the structural capacity are considered.

The most common approach to determine seismic actions is the use of a response

spectrum defined by the design code. This approach is general and easy to implement.

However, a more realistic representation of seismic loading can be obtained if a number of

response spectra derived from natural records are used instead. Furthermore, since significant

dispersion on the structural response may arise when different natural records are used, it is

advisable to apply properly scaled seismic records. In this study, the scaling procedure adopted

is based on the peak ground acceleration (PGA) of the seismic records. Three sets of natural

records with both longitudinal and transversal components are used. The records have been

selected from the database of Somerville and Collins [47] and correspond to three hazard

levels, with 50%, 10% and 2% probability of exceedance in 50 years. The records are scaled to

PGA values obtained from hazard curves produced for a specific region. In this work the hazard

curves derived by Papazachos et al. [48] (Table 1) are implemented. Moreover, based on the

assumption that seismic data follow the lognormal distribution [49], the median spectrum x

18

and the standard deviation δ of the spectral acceleration can be calculated using the following

expressions:

n

d,ii 1ln( ( ))

x expR T

n=

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

∑ (21)

( )

1 22nd,ii 1

ˆln( ( )) ln(x)1

R Tδ

n=

⎡ ⎤−⎢ ⎥=

−⎢ ⎥⎣ ⎦

∑ (22)

where Rd,i(T) is the spectral acceleration value of the i-th record (i=1,…,n) for a period equal to

T, n is the number of records which depend on the hazard level. In this work, n is taken equal to

22 for the 50% in 50 years hazard level, n = 19 for the 10% in 50 years and n = 6 for the 2% in

50 years [48]. Since three hazard levels are considered in this study, three random variables are

assigned to the problem related to the seismic demand. Figure 8 depicts the two median spectra

for the three hazard levels taken into consideration. The properties, probability density

function, mean value and variance, of each random parameter are given in Table 2.

7.2. Deterministic optimization

In the first part of the numerical studies two deterministic formulations of the optimization

problem are examined, depending on the type of the design variables. The Deterministic Sizing

Optimization (DSO) and the Deterministic combined Sizing-Topology Optimization (DSTO).

The feasibility of every design tested during the optimization procedure is assessed in the

serviceability and ultimate limit states, while three hazard levels are employed for performing

the ultimate limit state checks. According to these checks the maximum inter-storey drift of the

structure for each hazard level should be less than a permissible drift value in agreement with

HAZUS [50]. Both DSO and DSTO problems can be expressed in the following form:

19

s

EC8

50/50

10/50

2/50

min ( )subject to

g ( ) 0θ ( ) (2/3)1.2%θ ( ) (2/3)3.0%θ ( ) (2/3)8.0%

where

INC∈

≤

≤≤

≤

s

ssss

F

dis R , i = 1,..., n ∈

(23)

where CIN(s) is the objective function to be minimized (the volume of the structure), s is the

design variables vector that can take values from a discrete design set Rd, gEC8(s) are the

serviceability constraints, θt/50(s) correspond to the maximum inter-storey drift for the t/50

hazard level, while F is the feasible set. According to HAZUS [50] for a low-rise steel building

the drift limits for the moderate, extensive and complete damage states are defined to 1.2%,

3.0% and 8.0%, respectively. The drift limits for the mid-rise building in question are taken

equal to 2/3 times of the corresponding low-rise ones [50].

The design variables for the DSO formulation are the dimensions of the cross sections of

all structural elements, whereas for the DSTO formulation two additional design variables are

taken into consideration: the number of columns in the horizontal x and y directions of the

structure. The deterministic behavioural constraints, under which the objective function is

minimized, as well as the design variables fulfil the demands of feasibility which are expressed

in terms of displacements. The optimum designs obtained through the two deterministic

formulations are shown in Table 3 where the cross sections and the number of columns are

depicted. One significant observation is the fact that the calculated volume for the DSTO

optimum design is almost half of that obtained with the DSO formulation with fixed topology

of the columns.

7.3. Verification

The proposed NN based methodologies are assessed with a parametric study performed on the

optimum design obtained through the DSTO formulation. Table 4 demonstrates the violation

probabilities with the corresponding reliability indices of the three performance objectives

estimated after 106 Monte Carlo simulations. Table 5 shows the performance of the polynomial

approximation of RS with respect to the value of the parameter f required for the definition of

the 2m+1 experimental points. Moreover, two different methodologies, for defining the

20

21

experimental points, are examined. The standard sampling, where the experimental points are

generated according to the scheme described in Figure 4, and the sampling based on Latin

Hypercube, where the range xi,M ± f σi for each of the m random variables is divided into 2m

non-overlapping segments of equal marginal probability. The results of Table 5 confirm the

observations of other researchers that, the probabilities of violation of the three performance

objectives vary with reference to the value of the coefficient f, irrespective of the procedure

used for generating the sample of experimental points. Worth mentioning is also the fact that

the probability of violation of the performance objective, corresponding to the 50/50 hazard

level, calculated with the FORM (Table 5), is orders of magnitude less compared to the

corresponding one calculated with the MCS (Table 4).

The performance of the two neural network based approximation procedures described in

Figures 3b and 5 is depicted in Tables 6 to 8. The MCS-NN procedure under different number

of training patterns and simulations is shown in Table 6. The neural network employed is the

18×40×3 fully connected network, where the 18 input nodes correspond to the 15 random

variables related to the capacity of the structure plus 3 related to the seismic demand of the

three hazard levels. The three nodes of the output layer correspond to the structural response in

terms of interstorey drift for the three performance objectives of the corresponding hazard

level. Since each simulation run of the NN approximation is computationally inexpensive, two

estimations of the violation probabilities are computed after 106 and 1010 simulations. It can be

observed that the probabilities (and the corresponding reliability indices) estimated after 106

simulations are very close of those given in Table 4 with the conventional MCS which confirms

the global approximation capabilities of NN. The violation probability of 50/50 hazard level

estimated after106 MC simulations is one order of magnitude different compared to the

estimated value after 1010 simulations. As can be seen, 500 training patterns are adequate for an

acceptable approximation of the violation probability by the NN scheme. This size of the

training set is used in the subsequent numerical studies.

Tables 7 and 8 depict the performance of the FORM-NN. Table 7 shows the performance

of FORM-NN with respect to the value of the coefficient f used to define the training set. The

neural network employed is the 16×30×1 fully connected network where the 16 input nodes

correspond to 15 random variables related to the capacity of the structure, plus one node related

to the seismic demand in the corresponding hazard level. The single node of the output layer

corresponds to the performance objective of the structural response in terms of interstorey

drifts. It can be seen that, contrary to the sensitivity observed in the computation of the

22

statistical quantities with respect to the values of f for the conventional FORM, the FORM-NN

demonstrates a relative invariance in the computations of the violation probabilities for f>0.5.

The sensitivity in the computation of the statistical quantities of FORM-NN is examined in

Table 8 with respect to the size of the training set for f=1.0. It is verified that, acceptable

approximations are observed with a sample size greater or equal to m. In the next part of the

numerical studies, m experimental points and f=1.0 are used for the FORM-NN methodology.

For every hazard level, m=16 training samples times the iterations of the FORM-NN procedure

as indicated in Figure 5 are required for training the local approximator. In this case 2 to 3

iterations are adequate for the convergence of the FORM-NN. Thus 16×3(iterations)×3(hazard

levels)=144 training patterns are required compared to the 500 training patterns required for the

MCS-NN methodology.

7.4. Reliability-Based design Optimization

Reliability considerations are recognized as safety constraints in structural engineering and an

optimum design considering these type of constraints should balance cost and safety. In the

case of deterministic optimum design, stress and displacement constraints are taking into

consideration in accordance with the design code safety factors without introducing reliability

as explicit design constraints. In the case of the reliability based design optimization additional

probabilistic constraints related to the strength of the structure are considered. The main goal of

RBO methods is to design for safety and economy with respect to extreme events. The

probabilistic constraints, which are incorporated into the optimization procedure, lead to

unbiased estimates of the structural performance and subsequently, to the determination of

design points that are located within a range of target failure probabilities. The RBO problem

for the examined test case can be formulated, in the framework of PBD, in the following form:

(24)

s

EC8

50/50

10/50

2/50

min ( , )subject to g ( , ) 0 θ ( , ) (2/3)1.2% θ ( , ) (2/3)3.0% θ ( , ) (2/3)8.0%and

IN x

x

x

x

x

C∈

≤

≤

≤

≤

F s μ

s μs μs μs μ

50/50

10/50

2/50

di

P(θ ( , ) > (2/3)1.2%) 0.1% P(θ ( , ) > (2/3)3.0%) 0.05% P(θ ( , ) > (2/3)8.0%) 0.001%where s R , i = 1,..., n

≤

≤

≤

∈

s xs xs x

2x x N( , )∼x μ σ

where CIN(s,μx) is the objective function to be minimised, s is the vector of the design variables,

x is the vector of random variables, P(θt/50(s,x)>θallowable) is the probability of violation of the

constraints which should be less than a specified allowable probability. The probabilistic

constraint enforces the condition that the probability of violating the performance objectives is

less than a certain value. Similarly to the deterministic formulation, two different types of

probabilistic formulation of the optimization problem are examined depending on the type of

the design variables; the Reliability based Sizing Optimization (RSO) and the Reliability based

combined Sizing-Topology Optimization (RSTO).

The optimum designs obtained according to the two probabilistic formulations are shown

in Table 9 where the cross sections and the number of columns are provided. Similarly to the

observation for the two deterministic formulations, the total optimized volume for the RSTO

optimum design is almost half of that obtained with the RSO formulation with fixed topology

of the columns and similar reliability indices. The comparison of the deterministic and

probabilistic optimum designs, with respect to the volume, violation probabilities and

computational cost, is shown in Tables 10 and 11. As can be seen, by examining the violation

probabilities corresponding to the two DBO optima, none of the two DBO formulations (DSO

and DSTO) leads to an optimum that fulfils the reliability constraints. Worth noticing is also the

fact that the RBO designs require 25 to 35 percent more material than the corresponding DBO

and needs three orders of magnitude more computational effort.

In the implementation of the RSO and RSTO formulations both methodologies proposed

for the NN approximation are considered. For the conventional MCS method with 50,000

23

24

samples generated with the LHS method, 1.5 to 2.5 years of computation will be required on a

single processor computing environment. On the other hand, the computational effort is

reduced by two orders of magnitude when NN approximation is considered. This improvement

will be further enhanced when the required number of samples by the conventional MCS is

larger as the allowable violation probabilities become smaller. Moreover, the FORM-NN

methodology, where neural networks are implemented as an approximator of the limit-state

surface in the random variables space, outperforms the MCS-NN where the global

approximation capabilities are required, since it needs almost half the corresponding

computational time.

8. CONCLUSIONS

In most cases the optimum design of structures is based on deterministic formulations that are

focused on the satisfaction of the associated deterministic constraints. In order to find a realistic

optimum design it is required to take into account all necessary random parameters that might

affect the performance of the structural system and incorporate reliability constraints into the

formulation of the optimization problem. Despite the improvements achieved on the methods

dealing with reliability analysis problems the solution of realistic Reliability-Based

Optimization problems in structural mechanics remains an extremely computationally

intensive task. The computational cost is further increased when non-linear structural response

under seismic loading conditions is considered for assessing the behaviour of the tentative

designs encountered during the optimization procedure. In this work the main goal was to

solve, in a computational efficient manner, a performance based design optimization problem

under seismic loading considering probabilistic constraints.

Two distinctive methodologies, which are based on a Neural Network approximator, are

proposed. In the first one the Neural Networks are implemented as a global approximator and

the NN predictions are incorporated into the Monte Carlo simulation method, while in the

second one, the NN are used as a local approximator of the limit-state surface and are

implemented into the First Order Reliability Method framework. Both methods outperform the

conventional implementation of the Monte Carlo simulation method, combined with the Latin

Hypercube sampling, by two or more orders of magnitude in computation time depending on

the required number of MCS samples. Moreover it was found that the implementation of the

Neural Networks as a local approximator seems to be computationally more efficient than its

global counterpart requiring half the corresponding computational time. This is due to the fact

25

that much more training patterns are required for training a Neural Network to perform as a

global approximator over the random variable space, than those required when the Neural

Network is used as a local approximator.

Apart from these two implementations of the Neural Network approximator the quadratic

Response Surface approximator was also examined and was found to be influenced by the

value of the coefficient f used for defining the experimental points of the Response Surface.

This variation of the Response Surface was stabilised with the Neural Network implementation

where the FORM-NN demonstrated a relative invariance in the computation of the violation

probabilities with respect to the coefficient f.

Furthermore, two formulations were examined with reference to the type of the design

variables. It was found that a combination of sizing and topology optimization is considered

has a profound effect on the material volume required compared to the optimum designs

achieved with sizing design variables only.

ACKNOWLEDGEMENTS

The second author acknowledges the financial support of the John Argyris Foundation.

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[44] Mazzolani FM, Piluso V, A simple approach for evaluating performance levels of moment-resisting steel frames. In: Fajfar P, Krawinkler H (Eds), Seismic design methodologies for the next generation of codes, 241-252, Balkema, Rotterdam (1997).

[45] Papaioannou I, Fragiadakis M. Papadrakakis M. Inelastic Analysis of framed structures using the fiber approach, Proceedings of the 5th International Congress on Computational Mechanics (GRACM 05), Limassol, Cyprus, 29 June-1 July, (2005).

[46] Eurocode 1: Actions on structures - Part 1-1: General actions – Densities, self-weight, imposed loads for buildings, CEN, prEN 1991-1-1:2001, July 2001.

[47] Somerville P, Collins N. Ground motion time histories for the Humboldt bay bridge, Pasadena, CA, URS Corporation, 2002, www.peertestbeds.net/humboldt.htm.

[48] Papazachos BC, Papaioannou ChA, Theodulidis NP. Regionalization of seismic hazard in Greece based on seismic sources, Natural Hazards, 8(1): 1-18, (1993).

[49] Shome N, Cornell CA. Probabilistic seismic demand analysis of nonlinear structures, Rep. No. RMS-35, Dept. of Civil Engineering, Stanford Univ., Stanford, California, 320, 1999.

[50] FEMA-National Institute of Building Sciences. HAZUS-MH MR1, Multi-hazard Loss Estimation Methodology Earthquake Model, Washington, DC, 2003.

TABLES

Table 1: Seismic hazard levels [48]

Event Recurrence Interval Probability of Exceedance PGA (g)

Frequent 21 years 90% in 50 years 0.06 Occasional 72 years 50% in 50 years 0.11

Rare 475 years 10% in 50 years 0.31 Very Rare 2475 years 2% in 50 years 0.78

Table 2: Properties of the random variables

Random variable Probability density function Mean value Standard

deviation Es (MPa) (5 random variables) N 2.10 E+05 0.10E fy (MPa) (5 random variables) N 235 0.10fy

Hardening (5 random variables) N 1.0% 0.10% Seismic Load (3 random variables) Log-N x (Εq. 21) δ (Εq. 22)

Table 3: Comparison of optimum designs for the Deterministic Sizing Optimization (DSO) and the

combined Sizing-Topology Optimization (DSTO)

Design DSO DSTO

Columns 1 HEB360 HEB500 Columns 2 HEB360 HEB400 Columns 3 HEB400 HEB550 Columns 4 HEB400 HEB400 Beams 5 IPE200 IPE240 No columns x 9 6 No columns y 6 4

Volume (m3) 38.32 22.86

28

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Table 4: Violation probabilities and reliability indices estimated with MCS after 106 simulations

Hazard level Reliability Index (β) Pviol(%)

50/50 4.76 1.00E-04 10/50 0.47 3.20E+01 2/50 0.10 4.60E+01

Table 5: Violation probabilities and reliability indices β (in parentheses) estimated with FORM-RS. The

quadratic form of RS is implemented with two sampling procedures for generating the experimental

points and different values of f

Hazard level f=0.5 f=1.0 f=2.0 f=3.0

Standard sampling

50/50 1.47E-08 (6.301) 2.08E-08 (6.248) 8.46E-19 (9.280) 1.01E-08 (6.360) 10/50 2.93E+01 (0.546) 2.85E+01 (0.567) 3.04E+01 (0.512) 3.05E+01 (0.511) 2/50 4.51E+01 (0.122) 4.49E+01 (0.128) 4.45E+01 (0.138) 4.45E+01 (0.139)

Latin Hypercube sampling

50/50 4.59E-12 (7.452) 2.79E-08 (6.202) 1.25E-12 (7.622) 6.84E-05 (4.830) 10/50 2.99E+01 (0.528) 2.87E+01 (0.563) 3.13E+01 (0.488) 3.16E+01 (0.479) 2/50 4.42E+01 (0.145) 4.45E+01 (0.137) 4.60E+01 (0.100) 4.52E+01 (0.121)

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Table 6: Violation probabilities and reliability index β (in parentheses) estimated with MCS-NN for

different number of training patterns and MC simulations

Number of training patterns Hazard level 100 200 500 1000

Estimated for 106 simulations

50/50 4.61E-05 (4.908) 8.43E-05 (4.788) 9.41E-05 (4.766) 8.75E-05 (4.780) 10/50 1.67E+01 (0.965) 2.60E+01 (0.643) 3.07E+01 (0.506) 3.15E+01 (0.480) 2/50 3.63E+01 (0.351) 4.43E+01 (0.144) 4.58E+01 (0.105) 4.54E+01 (0.116)

Estimated for 1010 simulations

50/50 3.44E-06 (5.394) 1.73E-05 (5.096) 9.90E-06 (5.201) 1.09E-05 (5.183) 10/50 1.69E+01 (0.959) 2.88E+01 (0.560) 3.13E+01 (0.478) 3.19E+01 (0.472) 2/50 3.63E+01 (0.347) 4.48E+01 (0.134) 4.59E+01 (0.103) 4.56E+01 (0.107)

Table 7: Violation probabilities and reliability indices β (in parentheses) estimated with FORM-NN, for

2m experimental points and different values of f

Hazard level f=0.5 f=1.0 f=2.0 f=3.0

50/50 1.25E-09 (6.674) 1.21E-05 (5.164) 1.20E-05 (5.165) 1.09E-05 (5.184) 10/50 3.18E+01 (0.474) 3.14E+01 (0.483) 3.11E+01 (0.493) 3.13E+01 (0.488) 2/50 4.56E+01 (0.109) 4.57E+01 (0.108) 4.56E+01 (0.110) 4.55E+01 (0.112)

Table 8: Violation probabilities and reliability indices β (in parentheses) estimated with FORM-NN, for

f=1.0 and different number of experimental points

Number of the experimental points Hazard level m 2m 3m 4m 50/50 4.48E-06 (5.347) 1.21E-05 (5.164) 9.88E-06 (5.202) 1.02E-05 (5.195) 10/50 3.13E+01 (0.487) 3.14E+01 (0.483) 3.13E+01 (0.488) 3.19E+01 (0.472) 2/50 4.50E+01 (0.126) 4.57E+01 (0.108) 4.58E+01 (0.106) 4.58E+01 (0.106)

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Table 9: Comparison of optimum designs of the Reliability Sizing Optimization (RSO) and the

Reliability combined Sizing-Topology Optimization (RSTO)

Design RSO RSTO

Columns 1 HEB800 HEB450 Columns 2 HEB550 HEB500 Columns 3 HEB400 HEB600 Columns 4 HEB500 HEB700 Beams 5 IPE220 IPE330 No columns x 9 6 No columns y 6 4

β50/50 11.144 11.546 β10/50 3.959 3.413 β2/50 4.415 4.331

Volume (m3) 47.71 31.25

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Table 10: Performance of the optimum designs

DSO DSTO RSO RSTO

Volume (m3) 38.32 22.86 47.71 31.25

Pviol(50/50)/ Pallowable=1.0E-01(%) 6.45E-05 (%) 1.09E-05 (%) <10-7(%) <10-7(%)

Pviol(10/50)/ Pallowable=5.0E-02(%) 2.51E+01 (%) 3.19E+01 (%) 3.76E-03 (%) 3.21E-02 (%)

Pviol(2/50)/ Pallowable=1.0E-03(%) 4.28E+01 (%) 4.56E+01 (%) 5.05E-04 (%) 7.42E-04 (%)

Table 11: Computational effort of optimization formulations

DSO DSTO RSO MCS

RSO MCS-NN

RSO FORM-NN

RSTO MCS

RSTO MCS-NN

RSTO FORM-NN

Time (hours) 8.00E+00 3.38E+00 2.25E+04* 2.25E+03 6.48E+02 1.44E+04* 1.44E+03 4.14E+02 Time (years) 9.13E-04 3.85E-04 2.57E+00* 2.57E-01 3.70E-02 1.64E+00* 1.64E-01 2.36E-02 * Estimated for 50,000 simulations

FIGURES

Figure 1: Flowchart of the Performance-Based Design (PBD) procedure

33

1. Begin 2. : 0t =3. ( ) ( ) ( ) ( )( )( ){ }( )0 0 0 0initialize : , , , 1,...,p m m mP y s F s m μ= =

4. Repeat 5. For To λ Do Begin : 1l =6. ( )( ): marriage t

l pD P=

7. ( ): s_recombinationl ls D=

8. ( ): y_recombinationl ly D=

9. ( ): s_mutationl ls s=

10. ( ): y_mutationl ly y=11. ( ):l lF F s=

12. End

13. ( ) ( ) ( ) ( )( )( ){ }: , , , 1,...,t t t to l l lP y s F s l λ= =

14. Case selection_type Of

15. ( ) ( ) ( )( )1, : : selection ,t tp oP Pμ λ μ+ =

16. ( ) ( ) ( ) ( )( )1: : selection , ,t t tp oP Pμ λ μ++ = pP

17. End 18. : 1t t= +19. Until termination_criterion 20. End

Figure 2: Pseudo-code of the Evolutionary Algorithm (EA) optimization procedure

34

(a)

(b)

Figure 3: MCS-NN - (a) Methodology 1, (b) Methodology 2

(a) (b) Figure 4: Standard sampling method [28] (a) before and (b) after correction

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Figure 5: The FORM-NN methodology

(a) (b) Figure 6: LHS technique (a) before and (b) after correction

36

(a)

(b)

Figure 7: 7-storey frame, (a) plan view and (b) side view

37

0

5

10

15

20

25

0 1 2 3 4 5 6

Period T (sec)

Acc

eler

atio

n (m

/sec

2)Mean-X (50/50)

Mean-Y (50/50)

Mean-X (10/50)

Mean-Y (10/50)

Mean-X (2/50)

Mean-Y (2/50)

Figure 8: Median response spectra for three hazard levels

38