1. The document describes a genetic algorithm to optimize the seismic retrofitting of existing reinforced concrete frames. 2. A sample application of retrofitting a 3D three-storey RC frame is presented to demonstrate the algorithm. 3. The optimal retrofitting solution for the sample frame consisted of concentric steel bracings in the two plane frames along the x- and y-directions, with no need for local FRP interventions on columns.

1. Roberto FALCONE
Ciro FAELLA
Carmine LIMA
Enzo MARTINELLI
DICiv – Department of Civil Engineering, University of Salerno, IT
A Genetic Algorithm aimed at optimizing
seismic retrofitting of existing RC frames

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Summary
1. Introduction
2. Problem statement and formulation
• Representation of individuals
• Seismic Analysis
• Evolution criteria
3. Sample application and computational efficiency
4. Conclusions
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clima@unisa.it

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Introduction
“AS-BUILT ”
STRUCTURE
“MEMBER-LEVEL”
TECHNIQUES
“STRUCTURE-LEVEL”
TECHNIQUES
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
Carmine LIMA
clima@unisa.it
𝑔𝑔LS,i = 𝐶𝐶LS,𝑖𝑖 − 𝐷𝐷LS,𝑖𝑖 < 0
V
∆
V
∆ ∆
V
𝐷𝐷LS,𝑖𝑖: UNALTERED
𝐶𝐶LS,𝑖𝑖: INCREASED
𝐷𝐷LS,𝑖𝑖: REDUCED
𝐶𝐶LS,𝑖𝑖: UNALTEREDLimit State function

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A generic intervention can be conceived as a combination of the “extreme” solutions with
the aim to obtain a synergistic action in increasing seismic capacity of under-designed
members and reducing demand on the whole structure.
“MEMBER-LEVEL” “STRUCTURE-LEVEL”
GENERIC
INTERVENTION
Introduction
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
4
𝐷𝐷LS,𝑖𝑖: REDUCED𝐶𝐶LS,𝑖𝑖: INCREASED
𝑔𝑔LS,i = 𝐶𝐶LS,𝑖𝑖 − 𝐷𝐷LS,𝑖𝑖 ≥ 0
Carmine LIMA
clima@unisa.it
Each one of the potentially
infinite combinations of
member- and structure-level
interventions leads to different:
• direct costs;
• life-cycle costs;
• reliability levels;
• other quantitative/
qualitative parameters
Choosing the “fittest" possible combination
of these two techniques is clearly a
problem of structural optimisation.

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Within the present work the objective function is chosen to be
proportional to the total direct cost:
Ф is a penalty function intended at modifying the nominal cost of
intervention for those individuals that are not fit to comply with the
retrofitting objectives:
If x is the vector of design variables defining the generic intervention, the
optimal retrofitting solution may be determined by solving the following
constrained optimisation problem:
( ) ( ) ( ) ( )( )( ),maxloc glob LS i
i
f C C f g  = + ⋅Φ   x x x x
( )
( ),
arg min
0 ,LS i LS
f
g x i = 1...n
 =  
≥ ∀
x
x x
Problem statement and formulation
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
5
Carmine LIMA
clima@unisa.it

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Generating initial
population
Evaluating
fitness
Is optimization
criteria met?
SELECTION
CROSSOVER
MUTATION
Best
Individual
Translate
New
population
IN
OUT
YES
NO
Generating a new population
FLOW CHART OF
GENETIC ALGORITHM
Problem statement and formulation
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
6
Carmine LIMA
clima@unisa.it
The first step of the procedure is the random generation of an initial population of Nind
individuals. Each individual x of such a population is represented through a simple
chromosome-like array of bits.
0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0
(Ncol x 2bits)
“MEMBER-LEVEL” “STRUCTURE-LEVEL”
0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0
(Ncol x 2bits) (Nbeam x 3bits)

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In the second part, the procedure employs three bits for each bracing and, hence,
they are codified by only 23 or 8 possible phenotype solutions (from 0=absence
to 7=stiffest section) which identifying the section of steel bracings at the first
level.
In the first part, each couple of bits contains the number of FRP layers
employed for confining the corresponding column (ranging between zero
(as-built configuration) and 3 layers of FRP) and, hence, a total of 2xNcol bits
are allocated in the first part.
Confined Concrete
ε50h
ε0=0.002
εc
ε50c
0.2 f'
c
Unconfined Concrete
ε20c
θ
0.5 f'
c
ε50u
f'
c
fc
0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0
(Ncol x 2bits)
“MEMBER-LEVEL” “STRUCTURE-LEVEL”Model by Kent and Park
0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0
(Ncol x 2bits) (Nbeam x 3bits)
0 2 3 1 2 3 1 3 0 2 1 0
DECODING
DECODING
N° OF FRP LAYERS ID SECTION
6 5 5 4
Representation of individuals
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
7
Carmine LIMA
clima@unisa.it
W
A
A
A
3
1
1
3
W
W
A
2
2
, 1
1
=
=
⋅
= ⋅
⋅
∑
∑
n
j j
j k
k des n
i i
i
h W
A A
h W
A
z
The relationship between the section
of steel members at upper floors and
the section of steel bracings at the first
level stem out of a consistent design
criterion:

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V*/m*
∆*
BaseshearV
Top displacement ∆
For a given “individual” x of the population the value of function gLS,i is
simply evaluated through a Static Non Linear Analysis.
V
∆
𝐷𝐷𝑆𝑆𝑆𝑆𝑆𝑆=∆∗ � 𝛤𝛤∗
Elastic ADRS
Inelastic ADRS
𝐶𝐶𝑆𝑆𝑆𝑆𝑆𝑆
𝑔𝑔LS,i = 𝐶𝐶LS,𝑖𝑖 − 𝐷𝐷LS,𝑖𝑖 ≥ 0 ?
Seismic Analysis of individuals
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
8
Carmine LIMA
clima@unisa.it

9. The selection operator is used to select “parents” among a mating pool
solutions according to their fitness.
The third operation is mutation that allows for the possibility that non-
existing features from both parent strings may be created and passed to
their children.
The second operator, crossover, combines segments of selected strings.
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Once the total cost of all Nind individuals and their objective function gLS,i are evaluated, the
genetic algorithm evolves through three operators until the counter of population reaches a
maximum fixed number.
Is stopping
criteria met?
SELECTION
CROSSOVER
MUTATION
NO
Generating a new population
Individual Chromosomes Probability Cumulative P.
1 [010101010101100011001110] 0.24 0.24
2 [110100011101101001011000] 0.08 0.32
3 [010001110101101011000110] 0.39 0.71
4 [000101010101100011001010] 0.10 0.81
5 [010111010000101010001111] 0.19 1.00
1
24%
2
8%
3
39%
4
10%
5
19%
According the so-called “roulette-wheel” rule
the string characterized by higher fitness value
and, hence, a wider range in the cumulative
probability values has a higher probability of
being selected.
Crossover operator combines segments of
selected strings into new “offspring” solutions
by exchanging their genetic information
between successive crossover points (“multi-
point” crossover):
Old chromosome 1 0 1 0 1 1 0 0
Random numbers 0.001 0.073 0.325 0.024 0.802 0.001 0.023 0.004
New chromosome 0 0 1 0 1 0 0 0
Mutation sweeps down the string of bits and
changes the bit from 0 to 1 or otherwise if a
fixed probability test is passed. It helps to avoid
getting trapped at local optima.
SELECTIONCROSSOVERMUTATION
Counter=150?
Evolution criteria
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
9
Carmine LIMA
clima@unisa.it
( ) ( ) ( ) ( )( )( ),maxloc glob LS i
i
f C C f g  = + ⋅Φ   x x x x( ), 0 ,LS i LSg x i = 1...n≥ ∀

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A simple 3D three-storey RC frame is taken as preliminary example in
order to show the potential and the working detail of the presented
optimization algorithm.
Sample application and computational efficiency
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
10
Both the LS of Life Safety (SLV) and
Damage Limitation (SLD) are considered
X
Y
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clima@unisa.it

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Computational efficiency
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
11
Carmine LIMA
clima@unisa.it
Current research is devoted at enhancing the computational efficiency of the programming
code.
NLS analysis of 50 individuals x 150 generations x 8 pushover analysis (for each individual
according to EC8) = 60’000 Analysis
Almost 94% of the total computational time of the procedure is taken by seismic analysis in
OpenSEES, whereas the 6% only refers to pre- and post-processing operations handled by
the genetic algorithm
nonlinearity of RC members
steel bracings
floor diaphragms
Distributed plasticity (nonlinearbeamcolumn)
Concentrate plasticity (BeamWithHinges)
Accidental eccentricity
central node connectionnonlinearbeamcolumn
Buckling effect
without
Equivalent trusses
Shell elements

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Computational efficiency
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
12
Carmine LIMA
clima@unisa.it
nonlinearity of RC members
Distributed plasticity (nonlinearbeamcolumn)
Concentrate plasticity (BeamWithHinges)
BeamWithHinges
(time for performing
analyses)
40% lower
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
0 50 100 150 200 250 300 350
Vb [N]
Dtop [mm]

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Computational efficiency
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
13
Carmine LIMA
clima@unisa.it
Accidental eccentricity
central node connectionnonlinearbeamcolumn
Buckling effectsteel bracings
3D
Plain view
Initial eccentricity
3D
Plain view

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Computational efficiency
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
14
Carmine LIMA
clima@unisa.it
Accidental eccentricity
central node connectionnonlinearbeamcolumn
Buckling effectsteel bracings
0
100
200
300
400
500
600
700
800
900
0 50 100 150 200
N(kN)
force-diplacement of steel bracings
tension
compression

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Computational efficiency
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
15
Carmine LIMA
clima@unisa.it
without
Equivalent trusses
Shell elements
floor diaphragms
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
0 50 100 150 200 250 300 350
Vb [N]
Dtop [mm]
PushOver Curve X
PushOver Curve Y

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The optimal solution came up to consist of a concentric steel bracing
(realised in the two plain frame along the x- and y-direction) and no local
FRP interventions are actually required
MOST APPROPRIATE
SOLUTION
TRANSLATEBEST
INDIVIDUAL
TRANSLATEBEST
INDIVIDUAL
Results
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
16
Carmine LIMA
clima@unisa.it

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• The proposed procedure has the potential to support engineering
judgement in determining the “fittest” seismic retrofitting solution for
RC frames.
• Future developments are intended at including the aspects that are
not taken into account yet (i.e., analysis of real structures, Limit States,
indirect costs, multi-criteria objective function, among the others).
• Future developments should also aim at enhancing the computational
efficiency of the computer procedure, whose computational cost is one
of the main critical issues to be duly addressed for the proposed
method be actually feasible in real applications.
Conclusions
INTRODUCTION
PROBLEM STATEMENT
AND FORMULATION
SAMPLE APPLICATION CONCLUSIONS
17
Carmine LIMA
clima@unisa.it

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Carmine LIMA
clima@unisa.it