1

SEISMIC DESIGN OF RC STRUCTURES: A CRITICAL ASSESSMENT IN

THE FRAMEWORK OF MULTI-OBJECTIVE OPTIMIZATION

NIKOS D. LAGAROS

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

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

SUMMARY

The assessment of seismic design codes has been the subject of intensive research work in an effort to

reveal weak points that originated from the limitations in predicting with acceptable precision the response

of the structures under moderate or severe earthquakes. The objective of this work is to evaluate the

European seismic design code, in particular the Eurocode 8 (EC8), when used for the design of 3D

reinforced concrete buildings, versus a Performance Based Design (PBD) procedure, in the framework of a

multi-objective optimization concept. The initial construction cost and the maximum interstorey drift for

the 10/50 hazard level are the two objectives considered for the formulation of the multi-objective

optimization problem. The solution of such optimization problems is represented by the Pareto front curve

which is the geometric locus of all Pareto optimum solutions. Limit-state fragility curves for selected

designs, obtained from the pareto front curves of the EC8 and PBD formulations, are developed for

assessing the two seismic design procedures. It was found that a linear analysis in conjuction with the

behavioural factor q of EC8 cannot capture the nonlinear behaviour of an RC structure. Consequently the

corrected EC8 Pareto front curve, using a nonlinear static procedure, differs significantly with regard to the

corresponding Pareto front obtained according to EC8. Furthermore, similar designs, with respect to the

initial construction cost, obtained through the EC8 and PBD formulations were found to exhibit different

maximum interstorey drifts and limit state fragility curves.

KEY WORDS: prescriptive design procedures, performance-based design, limit state fragility,

multi-objective optimization, reinforced concrete buildings.

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1. INTRODUCTION

In the past, the requirements and provisions of the seismic design codes for buildings

have been based on experience and experimental observations and they were periodically

revised after disastrous earthquakes. Most of the current seismic design codes define a

single design earthquake that is used for assessing the structural performance against

earthquake hazard. As a consequence, these codes have many inherent assumptions built

in the design procedure regarding the behaviour of the structure against earthquake

loading. Severe damages caused by recent earthquakes made the engineering community

to question the reliability of the current seismic design codes [1,2]. Given that the

primary goal of contemporary seismic design is the protection of human life in

connection to economic design, it is evident that additional performance targets and

earthquake intensities should be considered in order to assess the structural performance

in many hazard levels. In the last decade the concept of performance-based structural

design (PBD), subjected to seismic loading conditions, was introduced [3-5]. In PBD

more accurate analysis procedures are implemented based on nonlinear structural

response. Most of the current seismic design codes belong to the category of the

prescriptive design procedures (or limit state design procedures), where if a number of

checks, expressed in terms of forces, are satisfied, then the structure is considered safe

and no collapse will occur. A typical limit state based design can be viewed as one (i.e.

ultimate strength) or two (i.e. serviceability and ultimate strength) limit state approach.

All contemporary seismic design procedures are based on the concept that a structure will

avoid collapse if it is designed to absorb and dissipate the kinetic energy that is induced in

it during the seismic excitation. Most of these seismic codes assume the capability of the

structure to absorb energy, through inelastic deformation by allowing a reduction of

external forces by the behaviour factor q.

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ATC-40 [3] and FEMA-273 [4] were the first guidelines for performance-based

seismic rehabilitation of existing buildings, while in the report Vision 2000 [5] these

ideas were extended to the design process of new buildings. The main objective of this

kind of design procedures is to achieve more predictable and reliable levels of safety and

operability against natural hazards. According to PBD procedures, the structures should

be able to resist earthquakes in a quantifiable manner and to present target performance

levels of possible damages. PBD procedures are multi-level design approaches where

various levels of structural performance are considered. FEMA-356 [6] suggests, for

example, the following performance levels: operational level, immediate occupancy, life

safety and collapse prevention. For assessing the structural performance the guidelines

reccomend the use of various types of analysis methods: linear static, nonlinear static,

linear dynamic and nonlinear dynamic. The most commonly used approach is the

nonlinear static procedure (NSP). NSP allows for the direct evaluation of the

performance of the structure at each limit state as opposed to the prescriptive design

procedures of Eurocode 8 (EC8) [7], where the structure is designed for the ultimate

strength limit state implementing a linear analysis procedure.

The main objective of this study is to perform a critical assessment of the EC8

prescriptive design code versus a performance-based seismic design procedure in the

framework of a two-objective optimization concept. The proposed PBD procedure

employs three performance objectives leading to Pareto type optimization. The two

design procedures are compared with respect to the limit-state fragilities corresponding

to selected designs obtained from the two respective Pareto front curves.

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2. PROGRESS ON STRUCTURAL DESIGN OPTIMIZATION OF RC

STRUCTURES

A number of studies have been published in the past dealing with structural optimization

of RC structures. One of the earliest studies on this subject is the work by Frangopol [8]

where the general formulation of the deterministic optimization problem is reviewed and

some of the main features of two general purpose deterministic optimization programs

are presented in developing the reliability-based optimization approach for the design of

both steel and RC framed structures. In a work by Kanagasundaram and Karihaloo [9] it

is demonstrated how the design process can be simulated mathematically to achieve

designs of RC structures which conform to the requirements of the Australian Standard

AS3600-1988 minimizing the construction cost. In the work by Moharrami and Grierson

[10], a computer-based method for the optimal design of RC buildings is presented,

where the width, depth and longitudinal reinforcement of member sections are

considered as design variables.

The optimization of 3D RC frames is discussed by Balling and Yao [11], where the

validity of the assumption that optimum concrete-section dimensions are insensitive to

the number, diameter, and longitudinal topology of reinforcing bars is studied. In a work

by Koskisto and Ellingwood [12] a decision model is presented for minimizing the

life-cycle cost of prefabricated concrete elements and structures. The decision model

utilizes principles of engineering economic analysis under uncertainty in considering

costs and benefits of construction, maintenance, repair, and consequences of failure. A

review of papers on cost optimization of concrete structures can be found in [13], where it

was concluded that there is a need to perform research on cost optimization of realistic

three-dimensional structures, especially large structures with hundreds of members

where optimization can result in substantial savings.

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Performance-based optimum design of reinforced concrete buildings is a relatively

new field of research. The performance criteria are imposed as constraints that affect the

initial construction cost that has to be minimized. Based on this approach Ganzerli et al.

[14] proposed an optimization methodology for seismic design considering performance

based constraints. Lagaros et al. [15] proposed an automated procedure for the

minimization of the eccentricity between the mass center and the rigidity center.

Furthermore, in the work by Li and Cheng [16], the optimal decision model of the target

value of performance-based structural system reliability of RC frames is established

according to the cost-effectiveness criterion.

Chan and Wang [17] presented a numerical approach based on a nonlinear cracking

analysis methods in the framework of an optimization algorithm for the stiffness-based

optimum design of tall RC buildings. A general approach to the multiobjective

reliability-based optimum designs of prestressed concrete beams is presented in the work

by Barakat et. al [18], where all behavior and side constraints specified by the American

Concrete Institute code for prestressed concrete are incorporated. In the work by Chan

and Zou [19] an optimization technique is presented for the elastic and inelastic drift

performance-based design of reinforced concrete buildings.

3. SEISMIC DESIGN PROCEDURES

The majority of the seismic design codes follow the standard procedure of site selection

and establishement of conceptual, preliminary and final design stages. According to a

prescriptive design code the strength of the structure is evaluated against earthquake

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

corresponding to one design earthquake [7]. In addition, serviceability limit state is

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

during its lifetime. On the other hand, PBD follows a different design concept which

takes into consideration, apart from the seismic characteristics of the site selection and

the establishement of the design stages, the construction process and the maintenance of

the building, in order to achieve a more reliable and predictable seismic performance over

its life.

3.1. Prescriptive seismic design procedures

According to the European design codes a number of checks must be considered in order

to ensure that the structure will meet the code requirements. Each candidate design is

assessed with respect to these constraints. All Eurocode 2 (EC2) [20] checks must be

satisfied for the gravity loads using the following load combination

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kiiQ

kiiQ

1.35 " "1.50d kjjS G= +∑ ∑ (1)

where “+” implies “to be combined with”, the summation symbol “Σ” implies “the

combined effect of”, Gkj denotes the characteristic value “k” of the permanent action j and

Qki refers to the characteristic value “k” of the variable action i. If all required constraints

are satisfied for the gravity loads, a multi-modal response spectrum analysis is performed

and earthquake loading is considered using the following load combination

2" " " "d kj d ijS G E ψ= + +∑ ∑ (2)

where Ed is the design value of the seismic action for the two components (longitudinal

and transverse), respectively and ψ2i is the combination coefficient for the

quasi-permanent action i, here taken equal to 0.30.

3.2. Performance-based design procedure

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

the prescriptive design codes: (i) Allows the owner, and structural engineer to choose

both the appropriate level of seismic hazard and the corresponding performance level of

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the structure. (ii) The structure is designed to meet a series of combinations of hazard

levels in conjunction with corresponding performance levels. In PBD performance

targets are incorporated in the conceptual design process, consequently performance

assessment becomes more of a verification process of an efficient design rather than a

design improvement process that may require radical changes of the initial design

concept [21].

The applied PBD concept [22] is a displacement-based design procedure where the

design criteria and the capacity demand levels are expressed in terms of displacements

rather than forces [23,24]. The main part in a performance-based seismic design

procedure is the definition of the performance objectives that will be used. The

implemented in this work PBD process can be described with the following two steps:

1) Proportioning of the longitudinal and transverse steel reinforcement of all

members on the basis of the serviceability limit state.

2) Use of the NSP in order to estimate the structural capacities of the structure in the

different intensity levels employed. Revise the steel reinforcement and the

dimension of the members cross-section so that the capacities exceed the seismic

demands [24].

The completion of Step 1 is prerequisite for Step 2 as the structural capacity depends both

on steel reinforcement and the dimensions of members cross-section.

The constraints considered for Step 2 of the PBD procedure are related to the

maximum interstorey drift limits θ, which is the largest value of the height-wise peak

interstorey drift ratios for each hazard level. This is a commonly used measure of both

structural and non-structural damage because of its close relationship to plastic rotation

demands on individual beam-column connection assemblies.

In this study three performance objectives are considered corresponding to hazard

levels 50, 10 and 2 percent probabilities of exceedance in 50 years. The drift limits θ, for

the three performance objectives considered, are equal to 0.4, 1.8 and 3.0 percent,

respectively, according to Ghobarah [25]. The θ values obtained at the three hazard levels

are also used for the calculation of the limit state cost, as will be described in a following

section. The flowchart of the PBD procedure applied in this study is shown schematically

in Figure 1.

4. MULTI-OBJECTIVE OPTIMIZATION

In practical applications of sizing optimization problems, the material weight itself rarely

gives a representative measure of the performance of the structure. In fact, several

conflicting and usually incommensurable criteria usually exist in real-life design

problems that have to be dealt with simultaneously. This situation forces the designer to

look for a good compromise among the conflicting requirements. Problems of this kind

constitute multi-objective optimization problems. In general, a multi-objective

optimization problem can be stated as follows:

8

s

Tmax,10 /50

8

min [ ( ), ( )]where ( ) ( ) ( ) ( )

subject to ( ) 0 =1,...,

or ( ) 0 =1,...,

IN

IN b sl cl

ECj

PBDj

CC C C C

g j k

g j k

θ∈

= + +

≤

≤

s s ss s s

s

s

F

(4)

where s represents the design vector, F is the feasible region where all the constraint

functions gEC8 or gPBD are satisfied for the EC8 and the PBD formulations, respectively.

The objective functions considered are the initial construction cost CIN and the maximum

interstorey drift for the 10% in 50 years hazard level θmax,10/50. CIN(s) refers of the total

initial construction cost for the structure, while Cb(s), Csl(s) and Ccl(s) refer of the total

initial construction cost for the beams, the slabs and the columns respectively. The term

initial cost of a new structure refers to the cost during its construction stage. The initial

cost is related to the material and the labour cost for the construction of the building

which includes concrete, steel reinforcement, labour cost and the non-structural

components cost.

Several methods have been proposed for treating structural multi-objective

optimization problems [26,27]. In this work, the Nondominated Sorting Evolution

Strategies II (NSES-II) algorithm, proposed by Lagaros and Papadrakakis [28], is used in

order to handle the two-objective optimization problem at hand. This algorithm is

denoted as NSES-II(μ+λ) or NSES-II(μ,λ), depending on the selection operator, as will be

discussed in a following paragraph.

When the ES algorithm is used for the solution of a mixed (having both continuous

and discrete design variables) single-objective structural optimization problem, each

candidate design is termed as ‘an individual’ and is equipped with a set of parameters:

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c

γd

c σ

d c d

nnd

n n nc

[( , , ( , , )] (Ι ,Ι )

Ι =D R

Ι =R R [ π,π]+

+

= ) ∈

×

× × − a

p s γ s σ α

(5)

where sd and sc are the vectors of discrete and continuous design variables, respectively.

Vectors γ, σ and α are distribution parameter vectors. Vector γ corresponds to the

variances of the Poisson distribution used for the mutation operator of the discrete

variables, vector corresponds to the standard deviations (1 ≤ nσ ≤ nc) of the

normal distribution used for the mutation operator of the continuous variables, while the

vector corresponds to the inclination angles (nα = (nc-nσ/2)(nσ-1)) defining

linearly correlated mutations of the continuous design variables sc. A more detailed

description of the ES algorithm can be found in the study of Lagaros et. al [29].

σnR +∈σ

n[ π,π] a∈ −α

Let ( )1 μp { , , }= …B g p p denote a parent population of individuals pi that belong to

the g-th generation. If μ and λ denote the number of parents and offspring of a population,

respectively, the three genetic operators used in this mixed ES method can be described

by the following mappings:

d c d c

d c d c

d c d c

rec : (Ι ,Ι ) (Ι ,Ι ) recombination operatormut : (Ι ,Ι ) (Ι ,Ι ) mutation operatorsel : (Ι ,Ι ) (Ι ,Ι ) selection operator, { or }

μ λ

λ λ

k k μμ k λ μ+ λ

→→→ ∈

(6)

In the selection operator when the (μ+λ) selection scheme is adopted, the parents of the

next generation are chosen from a population that consists of the μ parents and the λ

offspring of the current generation (k=μ+λ), while when the (μ, λ) selection scheme is

adopted the parents of the next generation are the best μ offspring of the current

generation (k=λ). Based on Eq. (6) a single iteration of the mixed discrete ES, which is a

step from the parent population ( )gpB to the next generation parent population ( )1g

p+B is

defined by the mapping:

μES d c d copt : (Ι ,Ι ) (Ι ,Ι )→ μ (7)

The main part of the NSES-II algorithm is the Fast-Nondomination-Sort procedure

according to which a population is sorted in non-dominated fronts. The complexity of the

sorting procedure is O(mn2), where m is the number of objectives and n is the population

size. The pseudo-code of the procedure is given in Figure 2. For each design p, two

entities are calculated (lines 2 to 10): i) np represents the number of designs that dominate

design p and ii) Sp corresponds to the set of designs in which design p dominates. Every

design p, that belongs to the first non-dominated front, is dominated by no other design

and therefore np = 0. The criterion whether or not a population belongs to the first 10

nondominated front is given in lines 11-14 of the pseudocode, while in lines 17-28 the

remaining fronts are created.

The pseudo code of the NSES-II(μ+λ) algorithm is shown in Figure 3. The first

twelve lines correspond to the ES(μ+λ) part of the algorithm. The next twelve lines

correspond to the NSES-II part of the algorithm. After the offspring population is created

(line 13), the combined parent and offspring population is sorted by the

Fast-Nondomination-Sort (line 15). After the sorting is complete, the first nondominated

front F1 will contain the nondominated designs. If the amount of designs contained in F1

is less than μ, then all the designs are copied to the parent population ( )1gp+B of the next

generation. If the number of nondominated designs in ( )1gp+B is less than μ, then ( )1g

p+B is

populated from the designs of the second nondominated front, the third nondomination

front and so on. The process continues until the number of designs in ( )1gp+B is equal to or

greater than μ (lines 17 to 21). If the number of designs in ( )1gp+B is equal to μ, the

algorithm continues with the next generation. Usually, ( )1gp+B will contain more than μ

solutions, so we must chose which of the designs of the last nondominated front that

entered ( )1gp+B , will be kept. This procedure takes place in lines 22 to 23, where the last

front is sorted according to the crowded comparison operator and then an appropriate

number of the least crowded designs are chosen so that the size of population ( )1gp+B is

exactly μ.

5. NUMERICAL STUDY

The two storey 3D RC building, shown in Figure 4, has been considered in order to

perform the critical assessment of the seismic design procedures. In the framework of this

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study, the RC building has been optimaly designed to meet the Eurocode (EC2 and EC8)

or the PBD requirements. The Pareto fronts obtained, following these two formulations,

have been used for this assessment. In the case of the EC8 the lateral forces were derived

from the design response spectrum (5%-damped elastic spectrum divided by the

behaviour factor q=3.0) at the fundamental period of the building. Concrete of class

C16/20 (nominal cylindrical strength of 16MPa) and class S500 steel (nominal yield

stress of 500MPa) are assumed. The base shear is obtained from the response spectrum

for soil type B (stiff soil θ = 1.0, with characteristic periods Τ1 = 0.15sec and Τ2 =

0.60sec) and a PGA of 0.31 g. Moreover, the importance factor γI was taken equal to 1.0,

while damping correction factor is equal to 1.0, since a damping ratio of 5% has been

considered. The elastic response spectra corresponding to the three hazard levels, with

50, 10 and 2 percent probability of exceedance in 50 years are derived from the PGA

values obtained from hazard curves given by Papazachos et al. [30] (Table 1).

The slab thickness is equal to 15 cm and is considered to contribute to the moment

of inertia of the beams with an effective flange width. In addition to the self weight of the

beams and the slab, a distributed dead load of 2 kN/m2, due to floor finishing and

partitions and imposed live load with nominal value of 1.5 kN/m2, is considered, in the

combination with gravity loads (“persistent design situation”). Nominal dead and live

loads are multiplied by load factors of 1.35 and 1.5, respectively (Eq. 1). Following EC8,

in the seismic design combination, dead loads are considered with their nominal value,

while live loads with 30% of their nominal value (Eq. 2).

5.1. Solving the multi-objective optimization problem

In the first part of this study the two-criteria design optimization problem is considered,

and the efficiency of the NSES-II(μ+λ) algorithm is examined. The initial construction

cost CIN and the maximum interstorey drift for the 10/50 hazard level θmax,10/50 constitute

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the two criteria. For all test cases examined, a NSES-II(100+100) optimization scheme is

employed. The Pareto front curves obtained after 10, 50, 100, 250 and 500 generations

based on EC8 and PBD formulations are shown in Figures 5 and 6, respectively. As can

be seen, 100 generations are sufficient to obtain a good quality Pareto front curve. For

this reason the two Pareto front curves obtained after 100 generations are used for the

comparative part of this study.

5.2. Assessment of design based on EC8 and PBD

In the second part of this work the difference between EC8 and PBD formulations is

demonstrated in terms of the Pareto front curves and limit state fragilities of selected

designs. The EC8 formulation implements a linear analysis procedure where the

behavioural factor q is used to take into account the inelastic behaviour of the structure.

Most of the contemporary seismic design codes rely to the ability of the structure to

absorb energy through inelastic deformation using the reduction or behaviour factor q.

The capacity of a structure to resist seismic actions in the nonlinear range generally

permits the design seismic loads to be smaller than the loads corresponding to a linear

elastic response. Thus, the seismic loads are reduced by the behaviour factor q. The

numerical validation of the behavior factor has become a subject of research [31,32] in

order to check the validity of design assumptions and to make structural performance

more predictable from engineering point of view. According to EC8 the nonlinear

deformation of the structure is equal to q times the corresponding deformation of the

linear analysis. In order to make equivalent the Pareto front curves, obtained through the

EC8 and PBD formulations, the EC8 Pareto front curve is corrected with respect to the

maximum interstorey drifts obtained by the NSP used for the calculation of the structural

response in the case of PBD. All three Pareto front curves, i.e. the EC8, PBD and

EC8_Corrected Pareto front curves, are depicted in Figure 7. Alhtough the EC8 and EC8

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Pareto front curves almost coincide, the EC8_Corrected Pareto front curve is much

different than the original one since the nonlinear behaviour is not accounted reliably

through the behavioural factor q.

In the second part of this assessment five designs are selected distributed along the

EC8_Corrected and the PBD Pareto front curves, in order to compare them with respect

to the limit state fragility curves. Two of the designs, i.e. designs A and E, correspond to

the two extreme ends of the two Pareto front curves. Designs PBD_E and EC8_E

correspond to the optima, where the initial construction cost (CIN) is the dominant

criterion. Furthermore, designs PBD_A and EC8_A correspond to the optimum designs

obtained if the maximum interstorey drift (θmax,10/50) is the dominant criterion. The other

three designs are distributed along the Pareto front curves and have been selected with

respect to their initial construction cost. The cross sections of the designs A to E obtained

for the EC8 and the PBD formulations, along with the initial construction cost and the

maximum interstorey drift for the 10/50 hazard level, are given in Tables 2 and 3,

respectively. As can be seen in Table 2 there are two values for the maximum interstorey

drift for the designs of the EC8 formulation. Those obtained through the linear design

procedure and the corrected ones obtained using a NSP. Comparing the corresponding

designs of EC8 and PBD formulations, it can be seen that, although they correspond to

similar values of the initial construction cost, the corresponding maximum interstorey

drift values of the EC8 designs are two times larger than those of the PBD formulation.

Figures 8a to 8e depict the limit state fragility curves for low-rise RC buildings for

the High Code Design Level of the earthquake loss estimation methodology (HAZUS)

[33] for the Slight, Moderate, Extensive and Complete structural damage states. The

damage states, defined with respect to the drift limits according to HAZUS for this type

of structure, are equal to 0.5%, 1.0%, 3.0% and 8.0% for Slight, Moderate, Extensive and

Complete structural damage states, respectively. In these Figures, the PGA value for the

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design earthquake for the EC8 formulation is denoted with a bold vertical line and the

corresponding probabilities of exceedance of the four damage states are given in Tables 2

and 3, respectively. Designs A and B, which were selected from the part of the Pareto

front curves where the maximum interstorey drift (θmax,10/50) is the dominant criterion,

obtained through the two formulations depict similar performance. On the other hand

design C, D and E depict different performance with respect to the formulation employed.

It can be seen that although the designs with respect to the CIN obtained with the EC8 and

PBD formulations are similar, they differ significantly with respect to their response

since, as a general comment, the probability of exceedance for all damage states of the

EC8 designs is larger than the corresponding of the PBD designs. Worth mentioning is

the observation that, although, the probability of exceedance of the PBD_E for the Slight

damage state is almost the same with the corresponding of EC8_E design (82% versus

98%), the probability of exceedance for the Complete damage state of the PBD_E design

is two orders of magnitude less than the corresponding probability of the EC8_E design

(6.5% versus 0.06%).

6. CONCLUSIONS

In this work a multi-objective optimization problem of reinforced concrete structures is

considered in order to assess the designs obtained using either the European seismic

design code or a performance-based design procedure where three performance

objectives are considered. In the case of the European seismic design code, a linear static

analysis procedure has been used, while in the performance-based design procedure the

nonlinear static analysis procedure has been performed in order to determine the

structural capacity levels for different earthquake intensities. The two design procedures

have been applied for the optimum design of a 3D RC building. It has also been

demonstrated how the concept of performance-based design can be integrated into a

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structural optimum design procedure in order to obtain optimum designs that fulfill the

provisions of a contemporary framework for seismic design of RC structures, like the

three performance objectives employed in this study.

Two are the most significant findings of this study:

(i) The corrected Pareto front curve, with respect to the maximum interstorey drift, for

the 10 percent in 50 years hazard level implementing a nonlinear static procedure

obtained through the Eurocode 8 formulation, is much different than the

corresponding Pareto front obtained with a linear analysis procedure. The

numerical tests revealed specific weakness in using the behaviour factor q adopted

by the Eurocode 8. This was revealed by the inability of the hebavioural factor q,

recommended by Eurocode 8, to reliably represent the nonlineat behaviour of the

structure.

(ii) Furthermore, for the test example considered it has been shown that similar

designs, in terms of initial construction cost, designed in accordance to the

European seismic design code, are more vulnerable to future earthquakes compared

to the corresponding designs obtained with the performance-based design

procedure. This vulnerability increases for designs selected from the part of the

Pareto front curves where the initial construction cost is the dominant criterion.

Even though these conclusions cannot be generalized, they provide an indication of

the quality of the designs obtained according to a prescriptive design code and to a

performance-based design procedure.

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20. European Committee for Standardization. Draft for Development of Eurocode 2: Design of concrete structures. ENV 1992-1-1, 1992.

21. Krawinkler, H., Zareian, F., Medina, R.A., Ibarra, L.F., Decision support for conceptual performance-based design, Earthquake Engineering Structural Dynamics 2006; 35(1), 115-133.

22. Lagaros N.D., Fotis A.D., Krikos S.A., Assessment of seismic design procedures based on the total cost, Earthquake Engineering Structural Dynamics 2006; 35(11): 1381-1401.

23. Sullivan, T.J., Calvi, G.M., Priestley, M.J.N., Kowalsky, M.J., The limitations and performances of different displacement based design methods, Journal of Earthquake Engineering, 2003; 7(1): 201-241

24. Panagiotakos, T.B., Fardis, M.N., A displacement-based seismic design procedure for RC buildings and comparison with EC8, Earthquake Engineering & Structural Dynamics 2001; 30, 1439-1462.

25. Ghobarah, A., On drift limits associated with different damage levels, International Workshop on Performance-Based Seismic Design, June 28-July 1, 2004.

26. Coello Coello C.A., An updated survey of GA-based multi-objective optimization techniques. ACM Computing Surveys 2000; 32(2): 109-143.

27. Marler R.T., Arora J.S., Survey of multi-objective optimization methods for engineering, Structural and Multidisciplinary Optimization 2004; 26(6): 369-395.

28. Lagaros, N.D., Papadrakakis, M., Robust seismic design optimization of steel structures, Structural and Multidisciplinary Optimization 2006; (in print).

29. Lagaros ND, Fragiadakis M, Papadrakakis M Optimum design of shell structures with stiffening beams. AIAA Journal 2004; 42(1): 175-184.

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30. Papazachos, B.C., Papaioannou, Ch.A., Theodulidis, N.P., Regionalization of seismic hazard in Greece based on seismic sources, Natural Hazards 1993; 8(1): 1-18.

31. Fajfar, P., Towards nonlinear methods for the future seismic codes, In Seismic Design Practice into the Next Century, Booth (ed.), Balkema, 1998.

32. Mazzolani, F.M., Piluso, V., The theory and design of seismic resistant steel frames, E & FN Spon, 1996.

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

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Tables

Table 1: Seismic hazard levels [30]

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: Selected designs from the Pareto front of the EC8 formulation.

Design EC8_A EC8_B EC8_C EC8_D EC8_E

h1 0.70 0.60 0.55 0.50 0.25 b1 1.20 1.05 0.65 0.30 0.25 h2 1.20 1.20 1.00 0.65 0.60 b2 0.55 0.35 0.25 0.25 0.25 h3 0.30 0.30 0.25 0.12 0.30 b3 1.20 1.20 1.20 1.00 0.65

CIN (€) 25875.75 23016.52 18434.22 15508.38 14271.22 θmax,10/50-Linear (%) 0.055 0.064 0.10 0.25 0.54 θmax,10/50-NSP (%) 0.11 0.15 0.28 0.81 2.21

Limit state Probability of exceeding the limit state (%)

Slight 0.60 1.58 17.87 77.60 98.29 Moderate 0.01 0.05 1.98 44.36 87.24 Extensive 0.00 0.00 0.00 12.03 37.94 Complete 0.00 0.00 0.00 00.38 6.48

Table 3: Selected designs from the Pareto front of the PBD formulation.

Design PBD_A PBD_B PBD_C PBD_D PBD_E

h1 0.55 0.50 0.25 0.25 0.25 b1 1.20 1.20 1.20 0.40 0.30 h2 0.70 0.65 0.90 0.75 0.55 b2 0.70 0.50 0.25 0.25 0.25 h3 0.75 0.50 0.45 0.25 0.25 b3 1.20 1.10 0.80 1.20 0.80

CIN (€) 26362.64 23107.72 18365.51 15474.84 14396.11 θmax,10/50 (%) 0.065 0.084 0.13 0.30 0.75

Limit state Probability of exceeding the limit state (%)

Slight 0.05 0.24 2.03 21.29 82.24 Moderate 0.00 0.00 0.07 2.83 39.00 Extensive 0.00 0.00 0.00 0.01 5.20 Complete 0.00 0.00 0.00 0.00 0.06

Figures

Figure 1: Flowchart of the proposed PBD design procedure

20

1 For each ( )g∈p R2 p = ∅S3 0pn =4 For each ( )g∈q R5 If Then ≺p q6 { }p p= ∪S S q7 Else if ≺q p8 1p pn n= +9 End 10 End 11 If Then 0pn =12 1rank =p13 { }1 1= ∪F F p14 End 15 1i =16 While Do i ≠ ∅F17 = ∅Q18 For each i∈p F19 For each p∈q S20 1q qn n= −21 If Then 0qn =22 1rank i= +q23 { }= ∪Q Q q24 End 25 End 26 End 27 1i i= +28 i =F Q29 End 30 End

Figure 2: Fast-nondominated-sort – pseudocode

21

1 Begin 2 : 0g =

3 ( ) ( ) ( ) ( )( )( ){ }( )0 0 0 0initialize : , , , 1,...,p m m mf m μ= =B y s y

4 Repeat 5 For To λ Do Begin : 1l =6 ( )random_indexk μ=

7 ( )( ): SelectParent ,gl p k=p B

8 ( ): GetStrategyl l=s p9 ( ): GetDesignl l=y p

10 ( ): s_mutationl l=s s11 ( ): y_mutationl l=y y12 End

13 ( ) ( ) ( ) ( )( )( ){ }: , , , 1,...,g g g go l l lf l λ= =B y s y

14 ( ) ( ) ( )g gp o= ∪R B B g

15 ( ) ( )( )fast nondominated sortg g= − −S R

16 ( )1 , 1gp i+ = ∅ =B

17 Until ( )1gp i μ+ + ≤B S Do

18 ( )icrowding distance assignment− − S19 ( ) ( )1 1g g

p p+ += ∪B B iS

20 1i i= +21 End

22 ( )icrowded comparison operator− − S

23 ( ) ( ) ( )( )1 1 1 :g g gp p i pμ+ + +1⎡ ⎤= ∪ −⎢ ⎥⎣ ⎦B B S B

24. : 1g g= +25. Until termination_condition 26. End

Figure 3: NSES-II(μ+λ) – pseudocode

22

B4 30x60

B1 30x60

C.A4 h1xb1

4.00 m

5.0

0 m

B6 3

0x6

0C.C2 h3xb3

C.A1 h1xb1

5.0

0 m

B7 3

0x6

0

C.B2 h2xb2

C.A3 h1xb1

4.00 m

C.C1 h3xb3

B2 30x60

C.B1 h2xb2

B8 3

0x6

0

C.A2 h1xb1

B5 30x60

B3 30x60

B9 3

0x6

0

(a)

(b)

Figure 4: Geometry of the two storey 3D building (a) layout, (b) front and side views

23

12500

15000

17500

20000

22500

25000

27500

30000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

θmax,10/50

CIN

(Eur

o)

GEN 10GEN 50GEN 100GEN 250GEN 500

Figure 5: Pareto front curve obtained for EC8 formulation

12500

15000

17500

20000

22500

25000

27500

30000

0 0.2 0.4 0.6 0.8 1 1.2

θmax,10/50

CIN

(Eur

o)

GEN 10GEN 50GEN 100GEN 250GEN 500

Figure 6: Pareto front curve obtained for PBD formulation

24

12500

15000

17500

20000

22500

25000

27500

30000

0 0.5 1 1.5 2 2.5

θmax,10/50

CIN

(Eur

o)

3

EC8EC8-CorrectedPBD

EC8_A

EC8_C

EC8_D

EC8_E

PBD_E

PBD_D

PBD_C

EC8_BEC8_B

PBD_A

Figure 7: Comparison of the Pareto front curves obtained for EC8 and PBD formulation

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

PGA (m/sec2)

Prob

abili

ty o

f exc

eedi

ng th

e lim

it st

ate

8

PBD_A(Slight)EC8_A(Slight)PBD_A(Moderate)EC8_A(Moderate)PBD_A(Extensive)EC8_A(Extensive)PBD_A(Complete)EC8_A(Complete)

Design Earthquake PGA=3.1

(a)

25

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

PGA (m/sec2)

Prob

abili

ty o

f exc

eedi

ng th

e lim

it st

ate

8

PBD_B(Slight)EC8_B(Slight)PBD_B(Moderate)EC8_B(Moderate)PBD_B(Extensive)EC8_B(Extensive)PBD_B(Complete)EC8_B(Complete)

Design Earthquake PGA=3.1

(b)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

PGA (m/sec2)

Prob

abili

ty o

f exc

eedi

ng th

e lim

it st

ate

8

PBD_C(Slight)EC8_C(Slight)PBD_C(Moderate)EC8_C(Moderate)PBD_C(Extensive)EC8_C(Extensive)PBD_C(Complete)EC8_C(Complete)

Design Earthquake PGA=3.1

(c)

26

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

PGA (m/sec2)

Prob

abili

ty o

f exc

eedi

ng th

e lim

it st

ate

8

PBD_D(Slight)EC8_D(Slight)PBD_D(Moderate)EC8_D(Moderate)PBD_D(Extensive)EC8_D(Extensive)PBD_D(Complete)EC8_D(Complete)

Design Earthquake PGA=3.1

(d)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

PGA (m/sec2)

Prob

abili

ty o

f exc

eedi

ng th

e lim

it st

ate

8

PBD_E(Slight)EC8_E(Slight)PBD_E(Moderate)EC8_E(Moderate)PBD_E(Extensive)EC8_E(Extensive)PBD_E(Complete)EC8_E(Complete)

Design Earthquake PGA=3.1

(e)

Figure 8: Fragility curves.

27