Castilho Lima StructConcrete v8 p111-118 2007

8/12/2019 Castilho Lima StructConcrete v8 p111-118 2007

http://slidepdf.com/reader/full/castilho-lima-structconcrete-v8-p111-118-2007 1/8

Cost optimisation of lattice-reinforced joist slabs using

genetic algorithmsV. C. de Castilho and M. C. V. de Lima

Genetic algorithms (GA), a search method inspired by Darwin’s theory of evolution, offer an optimisation tool has

been used very successfully to solve a variety of engineering problems. The search process it implements starts with

a set of one or more chromosomes (initial population) and, by applying selection and reproduction operators,

iteratively ‘evolves’ the population into hopefully better ones, until a stopping criteria is reached. This article

investigates lattice-reinforced joist slab cost optimisation problems using a GA with continuous variables.

The problem considered concerns one-way slabs, continuous over two spans, which only the in-situ concrete

characteristics and joist spacing are varied. The design variables are: concrete layer thickness, concrete layer

strength, reinforcement, distance between joists and degree of redistribution of the continuous slabs’ negative

moments. The search for a solution includes an investigation into the use of discrete variables for data

representation. To obtain results that allow for a comparative empirical analysis, these problems are also evaluated

by a conventional optimisation method. The results indicate that the GA method is a viable optimisation tool for

solving lattice-reinforced joist slab cost minimisation problems.

Vanessa Cristina deCastilhoUniversidade Federal deUberla ˆ ndia, Brazil

Maria Cristina Vidigalde Lima

Universidade Federal deUberla ˆ ndia, Brazil

Notation

h degree of redistribution

M ap negative bending moment after the

redistribution

M el negative bending moment for

elastic-linear materialf ( x ) cost function, in R$/m2

x 1 cast-in-place concrete, in m

x 2 compressive strength of the con-

crete, in MPa

x 3 distance between joists axes, in m

x 4 degree of redistribution, in %

Introduction

Researchers in the field of structural engineer-

ing have always attempted to devise optimised

design solutions. Indeed, few topics have

received more attention in structural analyses

than that of optimisation. In the area of

precast concrete structures, the optimisation

of elements is of major interest due to the

way these elements are produced. In Brazil,

one of the most common applications of

precast concrete elements is in buildings

slabs. These slabs usually consist of lattice-

reinforced joists, prestressed concrete joists,TT panels and hollow core panels.

In precast concrete elements, part of the

problem is the transitory stages of production,

transportation and assembly, which may

impose more unfavourable loads on these

elements than on cast-in-place structures.

These stages consist of:1

(a) production – execution of precast

concrete elements;

(b) transportation – moving from the

production to the building site;

(c ) assembly – placement of the elements in

their permanent position and execution

of the connections.

A robust solution to the problem of cost

minimisation of precast concrete structures

requires that all these stages be taken into

account.

This article analyses the cost optimization of

lattice-reinforced joist slabs based on genetic

algorithms (GAs).2,3 The problem variables are

the height and strength of the concrete, the

reinforcement, the distance between joists

and the coefficient of redistribution of the

negative moments of continuous slabs. Based

on these factors, the influence of the continu-

ous reinforcement is evaluated in the struc-

ture’s final cost.4,5 The variables are defined

initially as continuous and later as discrete,

since it is much easier to associate a discrete

set of values to the variables values because

they are more easily applied. To investigate

the potential of the GA, the same problemswere addressed using a conventional method,

the Excel Solver, which uses the nonlinear

optimisation code Generalized Reduced

Gradient (GRG2).

Genetic algorithm

The genetic algorithm (GA) is an optimisation

and search method based on the concepts of

genetics, i.e. the evolutionary mechanisms of

populations of live beings. GAs were inspired

on the principle of natural selection and

survival of the fittest established in 1859 by

Charles Darwin in his book The Origin of the

Species. According to Darwin’s theory, in any

given population, the individuals possessing

‘good’ genetic characteristics have better

chances for survival and reproduction than

the less ‘fit’ individuals, who tend to disappear

over time.

The GA simulates biological evolution by

means of a multidirectional search within the

space of potential solutions for the problem.

This algorithm maintains a constant number

of potential solutions (population), modifying

the population in each successive generation

SC5150 Techset Composition Ltd, Salisbury, U.K. 2/1/2007

1464–4177 # 2007 Thomas Telford and fib



so that ‘good’ solutions can ‘reproduce’ and

pass on to the next generation, while ‘bad’solutions are discarded. GA generally uses

probabilistic transition rules to select some

solutions for reproduction and others to be

discarded. The basic principles of GA basic

were established in Holland6 and are men-

tioned in many bibliographical references.7–11

Each individual in a population (called a

‘chromosome’) usually corresponds to a point

in the search space and represents a possible

solution to the problem – this solution is also

called a hypothesis. The GA can explore the

space of possible solutions to seek the ‘best’one by applying its reproduction mechanism

on the individuals of the current population.

Instead of starting from a single point (or

potential solution) in the search space, a GA

is initialised with a population of potential sol-

utions. These potentials solutions are normally

generated randomly and represent dispersed

points in the search space.

A typical GA uses three operators –

selection, crossover and mutation – to guide

the population (through several generations)

toward a convergence at the global optimal

point. After the selection, crossover and

mutation operators have been applied, a new

population is formed. The process is repeated

until a given number of generations have

been created or another stop criterion is

reached.

These characteristics, organised into a pro-

cedure, can be rewritten as the pseudo-code

shown in Figure 1.

This optimisation tool is widely employed in

the search for solutions to innumerable pro-

blems in the field of engineering.12–17

Generalised reduced gradient

(GRG2)

The GRG2 is a nonlinear optimisation program

used by the Microsoft Excel Solver to solve

minimisation and maximisation problems.18

Basically, GRG2 uses an implementation of the

generalized reduced gradient algorithm (GRG).

To generalise the GRG algorithm, the func-

tions gradient is used, thus ensuring that,

regardless of the expression, a linear relation

among the variables is always obtained.

A typical optimisation problem with restric-

tions addressed via nonlinear programmingcan be equated as shown in Figure 2. In the

presence of inequations, the equations are

transformed using fictitious variables.

The solution to the optimisation problem is

found starting from this typical algorithm and

applying the characteristics of the GRG itself. In

addition to all these characteristics for solving

optimisation problems, unlike the GA, an initial

solution must also be supplied, whether it is

feasible or not.

Ribbed slabs with lattice-reinforced joists

Lattice-reinforced joist slabs can be treated as

monolithic structures because they are joined to

cast-in-place concrete. The behaviour of these

slabs is similar to that of conventionally designed

slabs, and these precast concrete elements playa

rational role, representing low-cost and fast con-

struction (EF-9619 and NBR 611820).

Lattice-reinforced joists are structures com-

posed of steel bars joined together by electrofu-

sion welding at various points to form a space

lattice. In the transitory phases, the diagonals

of these lattices confer rigidity on the set, as

well as excellent transport conditions and hand-

ling, and connections between the cast-in-place

concrete and the precast concrete base. More-

over, the lattices can be used as transversal

reinforcements to offset shear stresses.21–23

The concrete base is moulded in metal

moulds to ensure the quality of the concrete,which is applied in 2– 3 cm thickness, using

concrete with small rich aggregates in a aggre-

gate cement paste to avoid the vibration

operation (Figure 3).

Lattice-reinforced joists are usually pro-

duced in lengths varying from 80– 300 mm,

with possible one-centimetre variations. Each

type of joist is identified by a set of symbols,

as illustrated in the example below,24 where

TR-08634: TR characterises the lattice-

reinforced joist; 08 indicates the 8 cm high

lattice reinforcement; 6 indicates the f 6 mmgauge of the lattice’s upper bar; 3 indicates

the f 3 . 4 mm gauge of the diagonals; and

4 indicates the f 4 . 2 mm gauge of the lower

bars.

Lattice-reinforced joists can be produced by

order to include additional reinforcements

introduced in the concrete base without

complicating the production process.

El Debs1 points out that the use of this type

of lattice-reinforced joist favours the appli-

cation of bi-directional reinforcement slabs.

The Brazilian Civil Construction Committee

is currently studying the elaboration of the

Code for Prefabricated Slabs. Therefore, the

calculations normally used by design engineers

to dimension and check lattice-reinforced joist

slabs are those published by manufacturers of

precast elements, according to Diniz,25 Lima26

and Pereira.27

S (t ) – population of chromosomes in generation

t .

initialize S (t )

evaluate S (t )

while (not termination-condition)

begin

select S (t ) from S (t – 1)

crossover S (t )

mutation S (t )evaluatedS (t )

end

4 Figure 1 Typical genetic algorithm

minimize f (x )

Such that:

g i (X ) = 0, i = 1, neq

L j < X j < U j, j = 1, n

where L j and U j are the lower and upper

limits from x j.

4 Figure 2 Typical optimization algorithm

diagonals

concrete base

lower bars

additionalreinforcement

upper bar

h

4 Figure 3 Lattice-reinforced joists4

Structural Concrete 2007 8 No 1

2 de Castilho and de Lima



The continuous reinforcement of the

lattice-reinforced joist was determined using

as the variable the degree of the bending

moment’s redistribution, taking care not toexceed the slab’s bearing capacity.28

The degree of redistributionh , expressed as

a percentage (Figure 4), is given by

h ¼ 1 M ap

M el

100(%)

where M ap is the negative bending moment

after the redistribution and M el is the negative

bending moment for elastic-linear material.

Solution of the cost optimizationproblem of lattice-reinforced

joists slabs

Costs and input for lattice-reinforced loists

The definition of the cost minimisation function

for the precast concrete elements studied here

was based on the costs of fabrication, external

transport and application. Listed below are the

partial values of the inputs in each stage. Note

that these values correspond to an initial analy-

sis of the problems evaluated in this work.

All prices are given in Brazilian Real (R$). The

conversion rate is 1.00USD ¼ 2.14BRL and

1.00EUR¼ 2.79BRL (at 16 May 2006). The

costs considered here are the same as those

adopted by Castilho:29

(a) Fabrication costs. The costs involved in the

production of precast concrete elements

correspond to the cost of raw material

(concrete, reinforcements, infill blocks),

factory activities (the cost of this phase

corresponds to the activities that take

place after moulding but before delivery

of the product), and administrative costs

(the overall costs involved in administrative

tasks and the salaries of the people

involved in the job). A breakdown of

these costs is given in Table 1.

(b) External transport cost . This is the cost of

transporting the joist from the factory to

the construction site, comprising labour,

trucks, fuel, insurance and maintenance

costs. For the purposes of this study, we

considered a construction site located

100 km from the factory.

External transport costs (R$=m3) ¼ 52

(c ) Assembly costs. The costs involved in the

assembly refer to the assembly of the

joist, the cast-in-place concrete, the comp-

lementary reinforcement and administra-

tive costs, as indicated in Table 2.

Definition of the problem

The solution for the problem of lattice-

reinforced joist slab cost optimization was

investigated here using genetic algorithms

(GAs). In addition to the traditional variables

of cast-in-place concrete strength, concrete

layer height and distance between joists, the

degree of redistribution variable was also ana-

lysed. The analysis comprised three lattice

reinforcement cases: the TR-08634 discussed

by Magalha ˜ es,4 and the TR-12645 and TR-

16645 available used in Brazilian market, all

with the same loading conditions.

The lattice-reinforced joist was optimised

under the same loading conditions adopted

by Magalha ˜ es:4

(a) Distributed uniformly along the element,

the loads were based on the concrete’s

dead weight (g c ¼ 25 kN/m3), expanded

polystyrene (EPS) infill blocks

(g e ¼ 0.12 kN/m3), a permanent load of

0.5 kN/m2 and a live load of 2.0 kN/m2.

(b) The slabs were dimensioned considering a

concrete base compressive strength of

20 MPa and a CA 60 reinforcement (yield

strength equal to 552 MPa).

(c ) The resistance factor used for concrete is

1.4 and 1.15 for steel. The load factor

used in the analysis is 1.4.

Figures 5 and 6 depict, respectively, the

static scheme and the slab section. The slab’s

uniform

distributedload

M apM el

4 Figure 4 Difference between M el and

M ap

Table 1 Fabrication costs

Material Labour Equipment

Raw material

Concrete: R$/m3 123.75 4.40 4.00

Reinforcement: R$/kg 1.80 0.25 –

Infill blocks: R$/m3 2.00 2.00 1.20

Factory activity: R$/m3 – 4.40 1.67

Administration: R$/m3 10%of theoverall costs involved in raw material, factory activity

Table 2 Assembly costs

Material Labour Equipment

Erection: R$/m3 – – 10.4

Cast-in-place concrete: R$/m3 (24.75 . f ck,capaþ 74.25) 110.75 –

Reinforcement: R$/kg 8.18 – –

Administration: R$/m3 20% of the overall costs involved in the assembly,

cast-in-place concrete and complementary reinforcement

4 m4 m

4 Figure 5 Static outline (span length 8 m)


Cost optimisation of lattice-reinforced joist slabs 3



final section was completed with EPS infill

blocks and a layer of concrete.

The cost function was represented by

four variables: x 1 representing the height of

the cast-in-place concrete; x 2 representing

the concrete’s compressive strength; x 3

representing the distance between joists;

and x 4 representing the degree of

redistribution.

Three joists with heights (H e) of 8 cm,

12 cm and 16 cm (TR-08634, TR-12645 and

TR-16645) were analysed to find a solution

for the optimization problems of lattice-

reinforced joists loaded as mentioned earlier.

The height of the infill blocks is a function of

the slab’s total span:

(a) for a total span of 8 m (4 m–4 m):

H EPS¼ 8 cm; and

(b) for a total span of 12 m (6 m–6 m):

H EPS¼ 12 cm.

The process of designing the slab with

lattice-reinforced joists was the same as that

described by Magalha ˜ es.4 The design criteria

were based on the joist’s service limit state

and ultimate limit state.

Total cost function

Thevarious costs were added up to findthe func-

tion that represents the joist’s total production

cost, considering the stages of execution, trans-

portation and application. Further details regard-

ing the final expression of the function (equation

(1)) are described by Castilho.29

f ( x )¼1 115

x 3þ

(0 00122þ0 127 x 1 x 3

0 0127 x 1 þ0 0158 x 3

0 00395 x 3(100 x 4)0 01)2

(0 219þ 25 x 3 x 1

2 50 x 1 þ2 50 x 3)

( x 1 þ0 065)100

264

375

þ

(12 50 x 1 x 3 þ 0 000624

1 248 x 1)

x 3

þ120 (2 475 x 2 þ74:25)

12 50 x 1 x 3 þ0 000624

1 248 x 1 x 3

0

BB@

1

CCA

þ

34 3 x 1 þ14 57 x 2 x 3

(88 8 x 1 þ5 457)

(22 4 x 1 þ1 456)

((100 x 4)0 01=

4633 x 1 þ284 713)

x 1 x 3 þ0 000050 1 x 1(1)

where x 1 is cast-in-place concrete, in m; x 2 is com-

pressivestrengthof theconcrete,in Mpa; x 3 isdis-

tance between joists axes, in m; and x 4 is degree

of redistribution, in %.

Note that, after several tests, this function

was found to incorporate all the problem vari-

ables, offering the best representation of the

lattice-reinforced joist cost minimisation

Table 3 Inequalities for continuous and discrete variables

Continuous

variables

Discrete

variables

0.040 x 1 0.100: m x 1 ¼ f0.040, 0.050, 0.060, 0.070, 0.080, 0.090, 0.100g

15 x 2 30: MPa x 2 ¼ f15, 16, 17, 18, 19, . . . , 30g

0.300 x 3 0.600: m x 3 ¼ f0.300, 0.310, 0.320, 0.330, 0.340, . . . , 0.600g

0 x 4 40: % x 4 ¼ f0, 1, 2, 3, 4, 5, 6, 7, . . . , 40g

Table 4 Values of variables, reinforced areas and cost function for two spans2

TR-08634

TR08634

Span – 8 m Span – 12 m

GA GA Excel

Solver

GA GA Excel

Solver(cont_var) (disc_var) (cont_var) (disc_var)

x 1: m 0.040 0.040 0.040 0.094 0.100 0.085

x 2: Mpa 19.3 20.0 19.0 23.0 21.0 21.8 x 3: m 0.598 0.600 0.600 0.373 0.370 0.378

x 4: % 40 40 40 22 38 40

Cost function: R$/m2 19.96 20.58 20.34 39.55 38.84 35.56

Negative reinforcement:

1024 m2

1.90 1.97 1.96 2.76 2.07 2.12

Additional reinforcement/ joists: 1024 m2

1.19 1.19 1.19 1.20 1.23 1.56

EPS EPS EPS

10 10

H EPSx 1

EPS3

2 5

1·5

13

3

H e

additionalreinforcement

(a) slab cross-section (b) joist section (H e = joist length)

4 Figure 6 Slab cross-section and joist section (in cm)





problem. These results are part of an initial

study of the negative moments in continuous

slabs with lattice-reinforced joists.

Therefore, the slab cost optimisation

problem boils down to the minimisation

problem f ( x ) ¼ ( x 1, x 2, x 3, x 4), which is subject

to service limit state and ultimate limit state

restrictions. In addition to these restrictions,

the continuous and discrete variables should

satisfy the inequalities presented in Table 3.

Description of the experimentsand analysis of the results

In this section we investigate the search for the

solution to the cost optimisation problem via

GA, based on the following characteristics:

elitism (1 individual), a population of 200

individuals, representation by real numbers,

uniform crossover, competition selection

strategy, and stop criterion delimited in 3000

generations. Since the GA is highly sensitive

to the initial population, the data describing

the results of each experiment represent the

average values obtained using five randomly

selected initial populations (average of five

runs).

Tables 4, 5 and 6 present the values

obtained for each variable and for the cost

function, using GA through continuous

(cont_var) and discrete (disc_var) variables.

The analysis involved TR-08634, TR-12645

and TR-16645 lattice-reinforced joists

with heights (H e) of 8 cm, 12 cm, and 16 cm,respectively, for two slab spans: 8 m

(4 m–4 m) and 12 m (6 m–6 m). For purposes

of reference, these tables also list the values of

the variables for the conventional optimization

method (Excel Solver), and the values evaluated

of the final areas of the negative (at middle

support) and additional reinforcement (one

joist) for each case. The reinforcement areas

(negative and additional) are not variables of

the optimisation method. Transverse reinforce-

ment is not required.

Tables 4, 5 and 6 indicate that, for largerspans of the same joist, there is:

(a) an increase in the height of the concrete

layer (in the case of TR-08634), which

was expected, since this increase leads to

an increase of the compression flange,

thereby enhancing the slab’s strength;

(b) an increase in the concrete layer strength.

Thus, increasing the span requires increas-

ing the slab’s strength, which is also

achieved by increasing the strength of

the cast-in-place concrete;

(c ) a reduction in the distance between joists,

in the case of the TR-08634 joist, which

was also expected, although there was

no alteration in the distance between the

other joists. The increased height of the

compression flange resulted in a shorter

distance between joists and, hence, a

reduction in the area of concrete, which

was reflected in a reduction of the cost

of the cast-in-place concrete;

(d ) an increase in the negative and additional

reinforcements, as expected.

A further analysis of the results shown in

the above tables indicates that, when the

height of the lattice is increased, the results

for the same span are inverted. As can be seen:

(a) in the case of the TR-08634 and TR-12645

joists with a 12 m span, the height of the

concrete layer decreased, and the distance

between joists increased. Conversely, as

the height of the compression flange

increased, the distance between joists

decreased to obtain the lowest cost for

concrete;

Table 5 Values of variables, reinforcedareas and cost function for twospans – TR-12645

TR-12645


GA GA Excel

Solver

GA GA Excel


x 1: m 0.040 0.040 0.040 0.040 0.040 0.040

x 2: Mpa 18.2 19.0 18.2 19.5 20.0 20.0 x 3: m 0.599 0.600 0.600 0.597 0.580 0.600

x 4: % 40 40 40 40 38 31

Cost function: R$/m2 21.82 22.06 21.48 23.60 24.60 22.85


1024

m2

2.47 2.59 2.48 2.64 2.72 3.13

Additional reinforcement/ joists: 1024 m2

0.72 0.86 0.86 1.86 1.84 1.81

Table 6 Values of variables, reinforced areas and cost function for two spans2TR-16645

TR-16645


GA GA Excel

Solver

GA GA Excel


x 1: m 0.040 0.040 0.040 0.040 0.040 0.040

x 2: MPa 16.5 17.0 17.0 18.5 19.0 18.9

x 3: m 0.599 0.600 0.600 0.599 0.600 0.600

x 4: % 40 40 40 40 40 40

Cost function: R$/m2 22.66 22.08 22.08 25.19 25.47 25.32


1024 m2

2.86 2.95 2.95 3.21 3.30 3.28

Additional reinforcement/ joist: 1024 m2

0.68 0.68 0.68 1.47 1.47 1.47





(b) a reduction in the strength of the concrete

layer since, under the same load situation,

a lower slab strength was found to be

necessary to offset the loads.

Figures 7, 8 and 9 illustrate the behaviour

of the cost function for each generation,

using the GA (cont_var), for the three cases

listed in the tables. Note that the curves

depict the same tendency: up to the generation

of 500 individuals, the values of the cost func-

tion remained practically unchanged.

The results indicate that most of the

values obtained both with the GA, in the

case of continuous variables, and with the

conventional optimization method were

practically the same. It is worth noting

that the values obtained by the GA with

discrete variables were compatible with

those of the Excel Solver, leading us to con-

clude that the GA was effective in finding

the solution for the lattice-reinforced joist

cost optimisation problems.

Most conventional optimisation methods do

not work with discrete variables, which is a sig-nificant disadvantage. Another disadvantage

involved the initial points of the design vari-

ables. In the conventional method, the initial

values of the variables must be provided for

the initial processing. In our analysis of these

problems, the initial values corresponded to

those obtained via the GA (continuous

variables). Different initial values were

adopted to evaluate the sensitivity of the

Excel Solver optimisation method. In general,

the results (GA and Excel Solver) converged

to an optimum. In some cases, the algorithmdid not converge or showed no significant

improvement at all using GA with discrete

variables.

Conclusions

This work investigated the use of GAs to find

solutions for the cost optimisation of lattice-

reinforced joists. The problems involved slabs

with lattice-reinforced joists for TR-08634, TR-

12645 and TR-16645 lattices with span

lengths of 8 m and 12 m. In addition, an

analysis was made of the design variables,

some continuous and others discrete. For

purposes of comparison, the same problems

were evaluated using a conventional method

(Excel Solver).

The results of the cost function and the vari-

ables, in the case of TR-08634, indicate that

there was an increase in the height of the con-

crete layer, leading to an increase of the com-

pression flange and an increment in the slab’s

strength.

The longer span led to increased strength in

the concrete layer. This, in turn, led to the need

for increasing strength in the slab, which was

also obtained by increasing the strength of

the cast-in-place concrete. Moreover, the dis-

tance between these same joists decreased.

Increasing the height of the compression

flange led to reduction in the distance

between joists and, hence, a smaller area of

concrete, resulting in a reduction of the cost

of cast-in-place concrete.

The other lattices showed no change in

the distance between joists. All the lattices

analysed showed an increase in the negative

and additional reinforcements.

TR-08634 – cost x generation

15

20

25

30

35

40

45

generation

c o s t : R $ / m 2

span = 8 m

span = 12 m

0 500 1000 1500 2000 2500 3000 3500

4 Figure 7 Average cost function values for TR-08634


20

21

22

23

24

25

26

generation

c o s t : R $ / m 2

span = 8m

span = 12m

0 500 1000 1500 2000 2500 3000 3500






Based on these analyses, we found that

most of the values of the variables for the GA

(discrete variables) and for the Excel Solvertended toward the same results. Therefore,

we concluded that the GA optimisation tool

efficiently solved the cost optimization

problems of the lattice-reinforced joists ana-

lysed here.

References

1. El Debs, M. K. Concreto pre -moldado: funda-

mentos e aplicaco es. (In English: Precast con-

crete: fundamentals and applications). Projeto

REENGE, EESC-USP, Sa ˜ o Carlos, 2000.

2. Castilho, V. C., Nicoletti, M. and El Debs, M. K.

An investigation of the use of three selection-

based genetic algorithm families when minimiz-

ing the production cost of hollow core slabs.

Computer Methods in Applied Mechanics

and Engineering, 2005, 194, No . 45 – 47,

4651–4667.

3. Castilho, V. C. et al. Aplicacao de Algoritmos

Gene ticos na otimiz aca o estrutural dos elemen-

tos de concreto pre -moldado. ( In English:

Genetic Algorithms application on precast con-

crete elements optimization). In XXIX Jornadas

Sudamaricanas de Ingenieria Estructural , PontaDel Leste, Anais (CD-ROM), Ponta Del Leste,

2000.

4. Magalha ˜ es, F. L. Estudo dos momentos fletores

negativos nos apoios de lajes formadas por

elementos pre -moldados tipo nervuras com

armaca o trel icada. (In English: Study of negative

bending moment of one-way slabs continuous

over two spans with lattice-reinforced joists).

Masters thesis, Universidade de Sa ˜ o Paulo, Sa ˜ o

Carlos, 2001.

5. Merlin, A. J. Momentos fletores negativos nos

apoios de lajes formadas por vigotas de concreto

protendido. ( In English: Study of negative

bending moment of one-way slabs continuous

over two spans with prestressed concrete

joists). Masters thesis, Escola de Engenharia de

Sa ˜ o Carlos, Universidade de Sa ˜ o Paulo, Sa ˜ o

Carlos, 2002.

6. Holland, J. H. Adaptation in Natural and Artificial

Systems. University of Michigan Press, Ann

Arbor, MI, 1975.7. Goldberg, D. E. Genetic Algorithms in Search,

Optimization and Machine Learning. Addison-

Wesley, Reading, MA, 1989.

8. Michalewicz, Z. Genetic Algorithmsþ Data

Structures ¼ Evolution Programs. Springer,

Berlin, 1996.

9. Gen, M. and Cheng, R. Genetic Algorithms and

Engineering Design. John Wiley & Sons,

New York, 1997.

10. Beasley, D. et al. An overview of genetic algor-

ithms: Part 2. Research Topics, University Com-

puting, 1993, 15, No. 4, 170–181.

11. Beasley, D. et al. An overview of genetic

algorithms: Part 1. Fundamentals, University

Computing, 1993, 15, No. 2, 58–69.

12. Coello, C. C. et al. Optimal design of reinforced

concrete beams using genetic algorithm. Expert

Systems with Applications, 1997, 12, No. 1,

101–108.13. Koskisto, O. J. and Ellingwood, B. R. Reliability-

based optimization of plant precast concrete

structures. Journal of Structural Engineering,

ASCE, 1997, 123, No.3, 298–304.

14. Lemonge, A. C. C Aplicaca o de algoritmos

gene ticos em otimizaca o estrutural . (In English:

Genetic algorithms applications on structural

optimization). PhD Thesis, COPPE, Universidade

Federal do Rio de Janeiro, 1999.

15. Rajev, S. and Krishnamoorthy, C. S. Discrete

optimization of structures using genetic algor-

ithm. Journal of Structural Engineering, ASCE,

1992, 118, No. 5, 1233–1250.

16. Koumousis, V. K. and Arsenis, S. J. Genetic algor-

ithm in optimal detailed design of reinforced

concrete members. Computer-Aided Civil and

Infrastructure Engineering, 1998, 13, 43– 52.

17. Prakash, A. et al. Optimum design of reinforced

concrete sections. Computers & Structures.

1998, 30, No. 4, 1009– 1011.

18. Lasdon, L. S. et al. Design and testing of a gener-

alized reduced gradient code for nonlinear pro-

gramming. ACM Transactions on Mathematical

Software, 1978, 4, No. 1, 34.

19. Espanha, Ministe rio de Fomento. EF-96. Instruc-

cion para el proyecto y la ejecucion de forjados

unidireccionales de hormigon armado o preten- sado. Ministe rio de Fomento, Espanha, 1997.

20. Associaca ˜ o Brasi leira de Normas Te cnicas.

ABNT NBR 6118 – Projeto de estruturas de

concreto – Procedimento, Rio de Janeiro, 2003

(Brazilian structural concrete code: Design of

Structural Concrete – Procedure).

21. Campos, D. T. A. Estudo do processo de fabrica-

ca o e execuca o das lajes trelicadas, analisando

em termos de custos com a laje macica. (In

English: Costs comparison between lattice-

reinforced joists slabs and solid slabs, considering

the fabrication process and assembly ). Graduate

thesis, Universidade Estadual do Oeste do

Parana , Parana , 2002.

22. Gaspar, R. Ana lise de seguranca estrutural das

lajes pre -fabricadas na fase de construca o.

(In English: Safety analysis on precast slabs

during construction). REIBRAC - 42 Ibracon,

2000, 1–15.

23. Borges, J. Crite rios de projeto de lajes nervuradas

com vigotas pre -moldadas. (In English: Design of

precast concrete joists slabs). Masters thesis,

Escola Polite cnica da Universidade de Sa ˜ o

Paulo, Sa ˜ o Paulo, 1997.

24. Franca, A. B. and Fusco, P. B. As lajes nervuradas

na m oderna construca o de edifı cios. (In English:

Joists slabs on actual building construction).AFALA/ABRAPEX, Sa ˜ o Paulo, 2000.

25. Dinaz, H. Lajes com armaca o em trelica. (In

English: Lattice-reinforced joists slabs). Vieira

Campos, Sa ˜ o Paulo, 1988.


20

21

22

23

24

25

26

27

28

29

30

generation

c o s t : R $ / m 2

span

=

8

mspan = 12 m

0 500 1000 1500 2000 2500 3000 3500






26. Lima, J. C. O. Sistema trelicado global. (In

English: Global lattice-reinforced system).Mediterra ˆ nea (Boletim te cnico), Campinas,

1993.

27. Pereira, V. Manual de projeto de lajes pre -molda-

das trelicadas. (In English: Design requirements

of lattice-reinforced joist slabs). Associa cao dos

fabricantes de lajes de Sa ˜ o Paulo, Sa ˜ o Paulo,

2000.

28. Droppa,Jr A. Ana lise estrutural de lajes formadas

por vigotas pre -moldadas com armaca o treli-cada. (In English: Structural analysis of lattice-

reinforced joists slabs). Masters thesis, Escola de

Engenharia de Sa ˜ o Carlos, Universidade de Sa ˜ o

Paulo, Sa ˜ o Carlos, 1999.

29. Castilho V. C. Otimizaca o de componentes de

concreto pre -moldado protendidos mediante

Algoritmos. (In English: Optimization of precast

prestressed elements using genetic algorithms).

PhD Thesis, Universidade de Sao Paulo, SaoCarlos, 2003.

30. Castilho, V. C. et al. Application of Genetic Algor-

ithm for optimization slabs of prestressed con-

crete joists. In 22nd Iberian Latin-American

Congress on Computational Methods in Engin-

eering, Campinas, Anais (CD-ROM), Campinas,

2001.



Castilho Lima StructConcrete v8 p111-118 2007

Documents

Transcript of Castilho Lima StructConcrete v8 p111-118 2007