1

Optimization of Flexural capacity of Reinforced fibrous Concrete

Beams Using Genetic Algorithm SUJI.D,NATESAN S.C, MURUGESAN.R

Biography: Suji .D is a Research Scholar at Sathyabama Institute Of Science and

Technology, Deemed University, Chennai, India. She is a Life member of Indian society for

Technical Education and Member of Institution of Engineers, India. She is having 18 years of

teaching experience.

Dr.Natesan S.C is the principal , VLB. Janakiammal College of Engineering and

Technology, Coimbatore, India. He has been actively involved in teaching, research and

consultancy work since last 34 years. His field of interest is Fiber reinforced concrete and

Ferro cement. He was awarded as the best outstanding corporate fellow member for the year

2001 in Civil Engineering division by the Institution Of Engineers, Coimbatore Chapter.

Dr.Murugesan.R is a Senior lecturer in the Department of Civil engineering at Government

Polytechnic College , Coimbatore, Chennai, India. He is actively involved in teaching for the

past 25 years. His field of interest is Maintenance and Rehabilitation of structures.

ABSTRACT

In this paper formulation and solution technique using Genetic algorithms (GAs) for

Optimizing the flexural capacity of steel fiber reinforced concrete beams, with random

orientated steel fibers, is presented along with identification of design variables, objective

function and constraints. The most important factors which influence the ultimate load

carrying capacity of FRC are the volume percentage of the fibers, their aspect ratios and bond

characteristics. Hence an attempt has been made to analyze the effective contribution of

fibers to bending and shear strength of reinforced fiber concrete beams. Equations are derived

to predict the ultimate strength in flexure of steel fiber reinforced concrete beams with

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uniformly dispersed and randomly oriented fiber reinforcement. Predicted strengths using the

derived expressions have been compared with the experimental data .Computer coding for

GA has been developed based on the formulations. Using the results obtained the influence of

various parameters on the ultimate strength are discussed. Particular attention. is given to the

construction practice as well as the reduction of searching space.

Key words: optimization; fiber reinforced concrete; flexure; Shear; Genetic algorithms;

INTRODUCTION

In the area of Structural Engineering the method of optimization has been steadily applied to

various structural problems. Distinguishable linear and non linear optimization techniques

have been successfully developed for finding optimum set of the material, topology,

geometry or cross-sectional dimensions of different type of structures subject to particular

loading systems. Along the main stream of linear programming and non linear programming

techniques refined algorithms have been branched out in order to take into account for the

discrete nature of structure, fabricated standardized structural components, for example.

Although great success has been achieved during the past decades in structural optimization,

these techniques generally have difficulties in avoiding local minima and results are

sometimes dependent upon the choice of the initial values in the design space. With recent

advances of computer technology, combinatorial optimization techniques have emerged.

Genetic algorithm (GA) and simulated annealing (SA) are quite popular among them and

they can efficiently solve the optimization problems with higher probability.

GAs are search procedures based on the mechanics of natural genetics and natural selection.

GAs are very effective and useful to treat the optimization problems concerning with discrete

variables. They are stochastic search procedures that have their philosophical basis in

Darwinian’s postulate of the “survival of the fittest”.(Goldberg 1989). They combine the

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concept of artificial survival with genetic operators, abstracted from nature, to form a robust

search mechanism. The main advantages of GA over conventional optimization techniques

can be summarized as: (1) GAs does not require gradient computations (2) GAs does not

require that the constraints be expressed explicitly in terms of design variables (3) GAs take

advantage of carry out optimization procedures in a stochastic frame work and (4) GAs are

not limited by restrictive assumptions about search spaces, such as continuity or the existence

of derivatives.

RESEARCH SIGNIFICANCE

Several investigations have shown that the presence of steel fibers in beams reinforced with

high strength deformed bars increases the ultimate strength9.To achieve efficiency in

performance and economy optimization techniques could be used. The main objective of this

study is to accommodate the usefulness of GA in practically optimizing reinforced steel

fibrous concrete beams with due considerations given to the construction practice.

Realizations of much of the code provisions in regard to strength requirements as well as

structural constraints have been considered.

LITERATURE REVIEW

In spite of performing various optimization techniques for reinforced concrete structures by

various researchers the algorithms developed using GAs for structural optimization of fiber

reinforced concrete structural elements are much limited. Optimization techniques for the

element level of reinforced concrete structures have been presented by different researchers

2.These methods were based on sequential linear programming, continuum- type optimality

criteria, and nonlinear programming such as Powell’s algorithm. Recently the discrete

optimization of structures has been performed using Genetic Algorithms13. Very little

literature is available in the field of fiber reinforced concrete structural optimization because

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design methods for FRC are yet to be fully developed, though some guidelines are available

for its applications to airfield pavements and some hydraulic structures4.So is the case with

standard test procedures to be adopted for testing and evaluation of the performance of FRC

elements. Ezeldin and Hsu14 optimized reinforced fibrous concrete beams using direct search

technique. The algorithm conducts a systematic search in the space of four variables- beam

width, beam depth, fiber content, and aspect ratio of fibers to yield an optimum solution for a

given objective function. It is therefore the main objective of this research is to develop an

algorithm using GA that performs the optimum design of reinforced fibrous concrete beams.

The algorithm developed for the optimum design satisfies the specifications provided in the

ACI code1.

REVIEW OF ANALYTICAL STUDIES

FLEXURAL ANALYSIS OF FIBER-REINFORCED CONCRETE BEAMS The analysis is based on the following assumptions

1. Plane sections remain plane after bending.

2. The compressive force equals the tensile force

3. The internal moment equals the applied bending moment

It has been customary to neglect the tensile resistance of concrete in calculating the ultimate

flexural capacity of concrete beams. Kukreja 10. et al. (1980) proved that the fiber reinforced

concrete greatly increases the tensile capacity of concrete. So the contribution of fibers must

be taken into account in the flexural analysis of beams. The analysis presented in this paper is

based on the conventional compatibility and equilibrium conditions used for normal

reinforced concrete except that the effects of steel strain hardening and contribution of the

steel fibers in the tension zone are recognized (Appendix A).The analysis is based on the

compression stress blocks in ACI Code1.The actual and assumed stress and strain

distributions at failure are shown in Fig.1. The analysis was compared with the experimental

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results reported by Byung Oh.3 (1993), and Swamy 16 et al. (1981) . Details of comparison are

shown in Table.1. Column (8) shows the ratio of the results of experimental to the authors’

predicted ultimate moment. These estimates are considered reasonably close, in view of the

difficulty in establishing the peak load before an abrupt drop is recorded in their experiment.

One possible reason for these conservative estimates is the uncertainty in the value of τ .It can

be seen that the ultimate bond strength σfu is expressed in terms of fiber volume

concentration Vf, dynamic bond stress τ, and fiber aspect ratio lf/df . Experimental studies

undertaken by many investigators show the wide disparity of bond stress values in SFR

concrete16. The values depend on the response stage, concrete properties, fiber type and other

characteristics. Tests confirmed that different values of τ resulting from various types of

fibers could significantly modify the flexural behavior of SFR concrete15.

PROBLEM FORMULATION Formulation of the problem is based on the objective function, which can either be

maximized or minimized. Studies have shown that the steel fibers can effectively be used to

increase the flexural strength of the beam18.In the present study the equation derived for

ultimate flexural capacity of a fiber reinforced concrete beam containing steel fibers has been

taken as the objective function, for maximization [APPENDIX A, Eq.A12 ].The design

variables are Volume fraction of the fiber ( fv ),Width of the beam ( b ), Depth of the beam

( D ), and Aspect ratio of the fiber ( spA ).The design parameters are ultimate strength of

fibers( fuσ ) and Cylinder compressive strength of concrete ( 'cf ).

FORMULATION OF CONSTRAINTS Strength in uniaxial tension When long and strong steel fibers are incorporated in a concrete matrix as shown in

Fig .2. the strength of the composite is given by the law of mixtures as

( ) fffm VV1 σ+−σ=σ (1)

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Where fV is the volume of fibers per unit volume of the matrix, mσ , fσ and σ are the

stresses in the concrete matrix, fiber and composite respectively. When the fibers are well

bonded to the concrete, they are subjected to the same strain so that the above equation

becomes

( )[ ] εσ fffm VEVE +−= 1 (2)

whereε is the strain in the composite and mE and fE are the moduli of elasticity of the

matrix and fibers respectively. The term in the parenthesis is the effective modulus of

elasticity of the composite.

When the cracking strength σmu of the matrix is reached, the stress in the composite is

given by

( ) f'ffmucr VV1 σ+−σ=σ (3)

Where ,fσ is the stress in the fibers when the matrix cracks. As soon as the matrix cracks

the load carried by the matrix which is )1( fmu V−σ per unit area of the cross section is thrown

on to the fibers . If the bond between the fibers and matrix is inadequate, the fibers at this

stage would be pulled out of this matrix. However, if the bond is adequate the fibers will not

fail and can take additional load, leading to multiple cracking of the matrix and culminating

in the fracture of the fibers themselves. Thus the ultimate strength

ffuu Vσ=σ (4)

Where fuσ is the stress at fracture of the fibers .

Constraints on critical volume

In a plot showing the strength of the composite against the fiber volume, Equations (3) and

(4) appear as shown in Fig. (3). when the bond is adequate the following inequality is

satisfied.

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( ) f'ffmuffu VV1v σ+−σ⟩σ (5)

It is seen from Equation (5) and Fig .3 that when the fiber content is above a certain volume

fcV the strength is governed by the fracture strength of fibers. The volume

fcV is called the ‘critical volume’ . If the fiber content is above the critical volume a great

increase in the ultimate tensile stress is to be expected.

Constraints on Aspect Ratio

To utilize the fracture strength of fibers there should be excellent bond between fiber and the

matrix. If a fiber of diameter d and length l is to fracture at its mid –length, the bond length

developed over the length l/2 must be greater than the fracture strength, ie.,

fu

2

4d

2ld

σπ

≥πτ

τ

σ≥

2dl fu

(6)

Where τ is the interfacial bond stress. Equation (6) states that for the fibers to fail by fracture

their aspect ratio must be equal to or greater than, the critical value given by the right- hand

side.

Constraints on ultimate moment

To ensure safety, the ultimate moment must be greater than or equal to the applied moment.

MM u ≥ (7)

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CONSTRAINTS ON TENSION STEEL

To provide ductile failure , the member should be designed for stA less than 0.75 sbA where

sbA = area of steel required for balanced condition. It may be found by applying the

equilibrium and strain compatibility conditions (Appendix B).

sbmaxst A75.0A = (8)

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

+

+=

dDV

fVfc

bdA ffu

yc

c

y

ffubs

σεε

εσ)(

)()'7225.0( (9)

Where

'fc = compressive strength of concrete

cε = compressive strain in concrete

yε = yield strain of steel

b = width of the beam

d = effective depth of the beam

D = overall depth of the beam

and other variables defined earlier.

The member should also be provided with minimum steel to prevent from excessive cracking.

It can be obtained by equating the cracking moment of the section (using the modulus of

rupture of fiber concrete) to the strength computed as a reinforced fiber concrete section

Eq.A12. As recommended in Ref. 6, this value is taken as

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

f400bd A

ystmin

(10)

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CONSTRAINTS ON SHEAR STRENGTH

Several studies have shown that the steel fibers are particularly effective in providing

reinforcement against shear stresses in conventionally reinforced concrete11.It is evident from

the test results of various authors that stirrups and fibers can be used effectively in

combination.

The equation proposed by Narayanan and Darwish11 has been used in this study to predict the

ultimate shear strength of fiber reinforced concrete beams.

( ) 2/56.4/80'24.0 mmNFadfeV tfs ++= ρ (11)

Where

e = 1.0 when a/d > 2.8 and 2.8 (d/a) When a/d ≤ 2.8

f t, =splitting cylinder strength of fiber concrete

ρ = percentage of area of tensile steel to area of concrete

F = fiber factor (= ((l/d ) fV ) f ) where f = 0.5 for round fibers, 0.75 for crimped

fibers, and 1.0 for fibers with deformed ends

The method proposed in ACI Code 1 is used for calculating the contribution of stirrups sV to

the shear capacity, to which is added the resisting force of concrete from the added fibes

fsV obtained from Equation (11)

Sd f A V ysvs =

(12)

The constraint to check the safety of the concrete against shear is given as

unu VV ≤ (13)

where unV is ultimate shear strength of fiber concrete and uV is ultimate shear force applied.

The minimum and maximum stirrup area is taken as proposed in ACI Code1

ysvmin f

Sb 50 A = (14)

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and

( )minvmaxsv A5A = (15)

The stirrup spacing varies from d /2 down to d /4 .

Constraints on design variables

The design constraints are formulated as

minb ≤ b ≤ maxb minfV ≤ fV ≤ maxfV

minD ≤ D ≤ maxD minspA ≤ spA ≤ maxspA

minstA ≤ stA ≤ maxstA minvA ≤ v

A ≤ maxvA

minS ≤ S ≤ maxS

FITNESS FUNCTION

Genetic algorithms mimic the survival of the fittest principle. So, naturally they are suitable

to solve maximization problems. Minimization problems are usually transformed to

maximization problems by some suitable transformation. A fitness function is derived from

the objective function and used in successive genetic operations .For maximization problems,

fitness function can be considered to be the same as the objective function.

TRANSFORMATION OF CONSTRAINED OPTIMIZATION TO UNCONSTRAINED

OPTIMIZATION

GA is ideally suited for unconstrained optimization problems. As the present problem is a

constrained optimization one, it is necessary to transform it into an unconstrained problem to

solve it using GAs to handle constraints. A formulation based on the application of penalty,

whenever there is a violation of specified constraints, is used in the present study for

transformation .Unlike the minimization problems here if the design variable set violates the

constraint then a lower value of say 1.0 will be assigned and if not, a higher value say 10.0 is

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assigned as violation parameter. Because for maximization problems, the fitness function and

objective function are considered to be the same .The violation coefficient “ Ф” is computed

as follows

∑=

=Φm

1l Фi (16)

Фi = a * gi (x) if gi (x) < 0; Фi = 0 if gi (x) ≥ 0

Where

m = number of constraints

a = penalty parameter and

g(x) = constraint function.

The modified objective function is given as

Z1 = Z + Ф (17)

Where

( )[ ] )Dk425.0d(A2

Dk15.0DDkDbVZ 1sty11

ffu −σ++−

σ= (18)

WORKING OF GENETIC ALGORITHM

With an initial population, the population for the next generation is to be generated which are

the off springs of the next generation. The reproduction operator selects the fit individuals

from the current population and places them in a mating pool where as the lesser ones get

fewer copies. As the number of individuals in the next generation is also same the worst fit

individuals die off eventually. The factor (Fave/F) for all the individuals is calculated, where F

is the average fitness. The factor is the expected count of individuals in the mating pool. It is

then converted into an actual count by appropriately rounding off so that individuals get

copies in the mating pool proportional to their fitness. This process of reproduction confirms

the Darwinian principle of survival of the fittest.

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To effect crossover, a set of crossover parameters are generated randomly. The first step in

crossover is finding a match for individuals. Once the pairs are decided it is necessary to find

the cross over sites. The sub string between the cross-sites is swapped from one individual in

the pair to other. To effect mutation, for each bit of a string, the random number generated is

checked for. If it is greater than the mutation rate specified then the bit conversion occurs

otherwise it remains as such.

The Genetic algorithm repeats the same process by generation of a new population and

evaluating its fitness. Proceeding with more generations, there may not be much

improvement in the population’s fitness barring a few because of mutation operation and the

best individual may not change for subsequent population. As the generation advances, the

population gets filled by more fit individuals with only slight deviation from the fitness of the

best individual So for found and the average fitness comes very close to the fitness of the best

individual, if and only if, there are no mutation operation. Criteria have to be evolved to

decide the termination of the process. In the present study the number of generations as 50 is

chosen for the purpose.

GA PARAMETERS

Fixing up GA parameters is very crucial in an optimization problem. In the present problem

the following GA parameters have been given as input values.

Number of parameters 4

Total string length 40

Population size 25

Maximum generation 50

Mutation probability 0.001

Cross over probability 0.9

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String length for variables

X [0] to X [4] 10

Overall Depth of the Beam (D) 450.0 mm

Upper and Lower bound for variables:

Volume fraction of fiber X [0] 0.03 -005

Width of the beam (mm) X [1] 300.0-230.0 mm

Depth of the beam X [2] 450.0 -300.0 mm

Aspect ratio of fiber X [3] 100.0 -50.0

DESCRIPTION OF ALGORITHM

The objective function for optimization is maximization of ultimate moment for reinforced

fiber concrete beams subjected to bending and shear. It can be formulated as

( )[ ] )Dk425.0d(A2

Dk15.0DDkDbVZ 1sty11

ffu −σ++−

σ=

and the variables explained earlier. Fig .4 presents the flow chart for the proposed algorithm.

The program is designed to read the required data- limiting values of variables, and GA

parameters. The program searches for a maximum for the objective function in the space of

four variables, namely Volume fraction of the fiber )( fV ,Width of the beam )(b , Depth of the

beam ( D ), and Aspect ratio of the fiber ( spA ). The stirrup spacing varies from d /2 down to

d /4, while the stirrup area increases from the minimum allowed by the ACI Code1( minvA =

50 b S / yf ) up to five times this value. The maximum of the objective function is recorded if

all the constraints are satisfied. If any of the constraint is violated, the penalty for violation is

given and the objective function is modified. The modified objective function incorporating

the constraint violation is given in Equation (17).

Genetic Algorithms are normally begun with a population of strings created randomly.

Therefore, each string in the population is evaluated. The population is then operated by three

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main operators’, namely, reproduction, crossover and mutation to create a better population.

The population is further evaluated and tested for termination. If the termination criteria are

not met, the population is again operated by the above three operators and evaluated further.

This procedure is continued till the termination criteria are met. One cycle of these operations

and the evaluation procedure is known as generation in GA terminology. In the proposed

algorithm the above criteria is taken as No of generations and is equal to 50. The flow chart

for the proposed algorithm is given in Fig.4.

CONCLUSIONS

Based on the formulation for Genetic Algorithm based optimal design of Reinforced

Concrete Beams along with identification of design variables, objective function and

constraints the following conclusions are arrived.

1. The developed GA reinforced with three basic operators (reproduction, crossover and

mutation) successfully led the randomly distributed initial design points in the design space to

the local optimum design point.

2. Predicted strengths using the derived expressions were compared with experimental

data. Good agreement was evident with different types of steel fibers, aspect ratio, and

material characteristics.

3. The outlined methods provide a simple and effective tool to assess the optimum

flexural strength of steel fiber reinforced concrete beams with randomly distributed steel

fibers.

4. The overall effect of fiber addition on ultimate strength and shear strength is studied.

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5. It must be emphasized that toughness improvement, cracking, and deflection control-

which are other reasons for use of fibers – were not considered. Future studies should

combine the optimum use of fibers for strength and serviceability criteria.

NOTATION

a = Shear span, mm

b = Width of beam, mm

d = Distance from extreme compression fiber to centroid of tension

Reinforcing bars, mm

df = Fiber diameter, mm

Em = Young’s modules of matrix, MPa

Ef = Young’s modules of fiber, MPa

F = Fiber factor

fc’ = Compressive strength (cylinder) of concrete, MPa

fy = Yield stress of reinforcing bars, MPa

ft’ = Splitting cylinder strength of fiber concrete, MPa

D = Beam overall depth, mm

l = Fiber length, mm

fsV = Shear strength of fiber concrete beams, N

Vm = Volume of matrix

Vs = Stirrups’ contribution to shear strength

Ast = Area of tension reinforcement, mm2

Asv = Area of stirrups, mm2

F = Fiber factor regulating shear strength

M = Applied moment, N-mm

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Mu = Ultimate bending strength of cross section, N-mm

rf = Radius of fiber, mm

s = Average spacing of fibers

Sv = Stirrups’ spacing, mm

ρ = Percentage of area of tensile steel to area of concrete

σ = strength of the composite

σf = Strngth of fiber, MPa

σm = Strength of matrix, MPa

σfu = Ultimate strength of fiber , Mpa

σmu = Cracking strength of matrix, MPa

σcr = Cracking strength of composite, MPa

σf’ = Stress in fiber when the matrix cracks, MPa

σu = Ultimate strength of composite, MPa

Vfc = Critical volume of fibers

Vu = Ultimate shear force applied, N

τ = Interfacial bond stress, MPa

Asb = Area of steel required for balanced condition, mm2

εc = Strain in concrete, mm

εy = Strain in steel, mm

REFERENCES

1. ACI Commmittee 318, “Building Code Requirements For Reinforced Concrete and

Commentary (ACI 318-89/ACI 318-89),” American Concrete Institute, Detroit, 1989, pp.353.

2. Adamu, A., and Karihaloo, B.L. (1994), “Minimum cost design of reinforced concrete beams using

continuum-type optimality criteria.” Struct.Optim., 7,91-102.

3. Byung Hwan Oh, (1993), “ Flexural Analysis of reinforced Concrete Beams Containing steel

17

Fibers”, J .Struct. Eng., V. 118, No. 10 ,Oct . 1992 . pp. 2821-2835.

4. Cohn, M.Z., and Lounis, Z. (1994), “Optimal design of structural concrete bridge systems”,

J.Struct.Eng., ASCE, 120 (9), 2653-2674.

5. Cox, H.L. (1952), “The elasticity and strength of paper and other fibrous materials”, British Journal

of Applied Physics” , 3, 72-79.

6. Craig, R., “ Flexural Behaviour and Design of Reinforced Fiber Concrete Members”,

SP-105, American Concrete Institute , Detroit, 1987,pp 517-563.

7. Dwarakanath, M.V. , and Nagaraj, T.S. , “ Flexural Behaviour of reinforced Fiber Concrete Beams”,

Proceedings of the International symposium on Fiber Reinforced Concrete, Dec. , 1987, Madras, Vol.

1, pp. 2.49-2.58.

8. Goldberg, D. “ Genetic algorithms in Search,optimization & Machine learning” , 1989.

9. Henager, C.N., and Doherty, T.J., “ Analysis of Reinforced Fibrous Concrete Beams”,

Journal of the Structural Division, ASCE, V. 102, No, ST!, Jan, 1976, pp. 177-188.

10. Kukreja, C.B., Kaushik, S.K., Kanchi, M.B., and Jain, O.P. (1980), “Tensile strength of steel fibre

reinforced concrete”, Indian concrete Journal, July. 184-188.

11. Narayanan, R., and Darwish, I. Y.S. , “ Use of Steel Fibers as Shear Reinforcement” , ACI

structural Journal, Vol. 84, No. 3, May. -Jun. , 1987, pp 216- 227.

12. Rajagopalan, K.,Parameshwaran, V.S., and Ramasway, G.S. (1974), “Strength of steel fibre

Reinforced Concrete beams”, Indian Concrete Journal, Jan. 17-25.

13. Rajeev, S., and Krishnamoorty, C.S. (1992), “Discrete optimization of structures using Genetic

Algorithms”, J .Struct. Eng.,118(5), 1233-1250.

14. Samer Ezeldin , A., and Cheng- Tzu Thomas Hsu . (1992), “Optimization of Reinforced Fibrous

Concrete Beams”, J.Struct.Eng., ACI, 89 (1) , 106-114.

15.Soroushian, P., and Bayasi, Z. (1991), “Fiber type effects on the performance of steel fiber

reinforced concrete”, Mat. J. ACI , 88(2), 129-134.

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16. Swamy, R.N., and Al-Ta’ an, S.A., “ Deformation and Ultimate Strength in flexure of Reinforced

concrete beams made with Steel Fiber Concrete”, ACI Journal , Proceedings V. 78, no. 5, Sept.- Oct.

1981 , pp. 395-405.

17. Wafa, F.F., and Ashor, S.A. (1992). “Mechanical properties of high strength fiber reinforced

concrete”, Mat. J. ACI , 88(6), 54-60.

APPENDIX A

DERIVATION FOR ULTIMATE MOMENT CAPACITY OF REINFORCED STEEL

FIBROUS CONCRETE BEAMS

The concept of composite materials may be introduced to describe the mechanical behaviour

of fiber-reinforced concrete. The strength of fiber reinforced composite material may be

described as the sum of matrix strength and fiber strength as follows.

ffmm VV σ+σ=σ (A1)

in which σ = strength of fiber-reinforced composite; mσ = strength of matrix;

fσ = strength of fibers; mV = volume of matrix (= fV−1 ); and fV = volume of fibers per unit

volume of the matrix. When the fibers are well bonded to the concrete, they are subjected to

the same strain. so that the above equation becomes,

( )[ ] ε+−=σ fffm vEV1E (A2)

where ε is the strain in the composite, and mE and fE are the modulus of elasticity of the

matrix and fibers respectively. The term in parenthesis is the effective modulus of elasticity

of the composite. When the cracking strength crσ of the matrix is reached, the stress in the

composite is given by

( ) fffmucr VV '1 σσσ +−= (A3)

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where 'fσ is the stress in the fibers when the matrix cracks. As soon as the matrix cracks, the

load carried by the matrix which is )1( fmu V−σ per unit area of the cross section is thrown

on to the fibers. If the bond between the fibers and the matrix is inadequate, the fibers at this

stage would be pulled out of the matrix; however if the bond is adequate, the fibers will not

fail and can take additional load, leading to multiple cracking of the matrix and culminating

in the fracture of the fibers themselves. It is reasonable here to assume that the strength

contribution of the concrete matrix at ultimate state may safely be neglected due to tensile

cracking .The first term of the right –hand side of Eqn. (A3) then vanishes. Thus the ultimate

strength of the composite is given by

ffuu Vσ=σ (A4)

where fu

σ is the stress at the fracture of the fibers. Since the orientation, length, and bonding

characteristics of fibers will influence the strength of fiber–reinforced concrete, these

parameters must be incorporated in Eqn.(4).

ffub10u Vσααα=σ (A5)

in which , 0α , 1α and bα are orientation factor, length - efficiency factor and bond efficiency

factor of fibers respectively. The fiber strength fuσ may be derived from bonding

characteristics of fibers as follows.

⎟⎟⎠

⎞⎜⎜⎝

⎛τ=σ

f

ffu d

l2

(A6)

in which τ = bond strength of matrix.

The ultimate strength uσ of fiber-reinforced concrete is now summarized as

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⎟⎟⎠

⎞⎜⎜⎝

⎛ααα=σ τ

f

ffbl0u d

lv2

(A7)

The orientation factor 0α is known to be about 0.41 for uniformly distributed fiber-

reinforced concrete, and the bond efficiency factor bα is about 1.0 for straight fibers (Henager,

C.N., and Doherty, T.J., 1976) .The present study exploits Cox’s (1952) results for length-

efficiency factor as follows.

⎟⎟⎠

⎞⎜⎜⎝

⎛ β

⎟⎟⎠

⎞⎜⎜⎝

⎛ β−

=α

2l

2l

tanh1

f

f

l

(A8)

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=β

fnff

m

rslAE

G2

(A9)

ff

f

dVl

S 25= (A10)

in which mG =shear modulus of concrete matrix; fE =elastic modulus of fiber; fA =cross

sectional area of fiber; S=average spacing of fiber; fr =radius of fiber ; fd =diameter of fiber;

fl = Length of fiber and fV =volume ratio of fiber. Eqn. (A7) of fiber–reinforced composite

may now be employed to derive the flexural capacity of concrete beams containing steel

fibers. The strain profile as shown in Fig.1 has been assumed for a cracked section in pure

bending. The concrete has reached its ultimate compressive strain ecu. The stress block in the

compression zone is the one commonly assumed in ultimate strength calculations. It has been

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adopted under the assumption that the behaviour of the fiber-reinforced compression zone is

similar to that of one without fiber-reinforcement. Hence k1 can be obtained from equilibrium

conditions

)'7225.0( ffu

styfful VfcbD

AbDVk

σσσ++

= (A11)

in which 'cf,=Cylinder compressive strength of concrete; b =Width of the beam; D=Overall

depth of the beam; fV =Volume fraction of the fibers; and

fuσ =2 0α 1α bα τ ( fl / fd ) = Ultimate fiber strength incorporating orientation, length and

bond efficiency factor. yσ = yield strength of tensile steel; stA = Area of tensile steel.

The flexural capacity is then derived as follows;

( ) ( )[ ] ( )DkdADkDDkDM styfuu 11

1 425.025.1

−++

−= σσ (A12)

APPENDIX B

DETERMINATION OF TENSION STEEL FOR BALANCED STRAIN CONDITION

( )

yc

cBal

dDk

εεε+

=1 (B1)

From Force equilibrium

( )( ) ( ) '11 7225.0 cBalffuvsb fDkbDkDbVfA =−+σ (B2)

From B1 and B2

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛

++=

dDV

ff

bDA ffu

yc

cfuc

ysb

σεε

εσ'7225.0 (B3)

22

TABLES AND FIGURES

List of Tables:

Table 1 – Comparision With Results Of Swamy And Al-Ta’ An16 (1981)

Table 2 – Comparision With Results Of Buing O’ 3 (1993)

Table 3 – Parametric Study Results Of GA

List of Figures:

Fig. 1 –Strain And Stress Distribution At Cross Section Of SFRC Beam

Fig. 2 –Composite In Uniaxial Tension

Fig. 3– Concept of Critical Volume

Fig. 4–Flow Chart for the Proposed Algorithm

Table 1 – Comparison With Results Of Swamy And Al-Ta’ An16 (1981)

Beam No

Fiber aspect ratio

Volume fraction of fiber , %

Compressive strength,

ksi (kN/mm2)

Interfacial bond stress ,

ksi (kN/mm2)

MuAuthor

x 106 lb-in. (kNm)

MuExpt

x 106 lb-in. (kNm)

MuExpt / MuAuthor

(1) (2) (3) (4) (5) (6) (7) (8)

DR11 100 0.5 5.50 (0.03079)

0.354 (0.00244)

0.159 (18.00)

0.206 (23.25)

1.29

DR12 100 1.0 4.53 (0.03120)

0.354 (0.00244)

0.176 (20.00)

0.210 (23.81)

1.19

DR21 100 0.5 4.22 (0.02909)

0.354 (0.00244)

0.245 (27.68)

0.298 (33.68)

1.21

DR22 100 1.0 4.55 (0.03135)

0.354 (0.00244)

0.283 (32.00)

0.310 (35.06)

1.09

DR31 100 0.5 4.48 (0.03091)

0.354 (0.00244)

0.225 (25.46)

0.253 (28.62)

1.12

DR32 100 1.0 4.62 (0.03188)

0.354 (0.00244)

0.222 (25.13)

0.273 (30.83)

1.22

23

Table 2 – Comparision With Results Of Buing O’ 3 (1993)

Table 3 – Parametric Study Results Of GA

Gen. No.

Volume fraction,

(%)

Width of beam,

in.

(mm)

Depth of

beam, in .

(mm)

Aspect Ratio,

Spacing of stirrup,

in.

(mm)

Area of Stirrup,

in.2

(mm2)

Area of tension steel, in.2

(mm2)

Ultimate moment,

x 106 lb-in.

(kNm)

Ultimate shear,

kips (kN)

10 1.299 11.379 (289)

11.988 (304)

70 2.846 (72)

0.0314 (20)

0.286 (184)

0.509 (57)

7.975 (35)

20 1.033 10.280 (261)

14.875 (377)

79 3.57 (90)

0.0350 (22)

0.3318 (214)

0.660 (74)

8.764 (39)

30 1.614 9.184 (297)

16.279 (413)

50 3.99 (99)

0.0381 (24)

0.400 (258)

0.889 (100)

9.534 (42)

40 1.749 9.184 (233)

14.927 (379)

97 3.58 (91)

0.048 (31)

0.287 (185)

0.760 (85)

12.049 (53)

48 1.212 8.956 (227)

14.07 (357)

65 3.53 (89)

0.0496 (32)

0.400 (258)

0.606 (68)

11.87 (52)

49 1.223 9.000 (228)

14.00 (355)

67 3.54 (90)

0.046 (30)

10.171 (258)

0.616 (69)

11.056 (49)

50 1.224 9.000 (228)

14.00 (355)

67 3.54 (90)

0.046 (30)

10.171 (258)

0.616 (69)

11.056 (49)

Beam No

Fiber aspect ratio

Volume fraction of fiber , %

Compressive strength ,

Ksi (kN/mm2)

Interfacial bond stress ,

ksi (kN/mm2)

Mu Author,

x 106 lb-in. (kNm)

Mu Expt,

x 106 lb-in. (kNm)

Mu Expt/ Mu Author

(1) (2) (3) (4) (5) (6) (7) (8) S1V1 57 1.0 6.241

(0.043) 0.354

(0.00244) 0.130 (14.7)

0.135 (15.229)

1.03

S1V2 57 2.0 6.937 (0.0478)

0.354 (0.00244)

0.115 (13.0)

0.159 (17.963)

1.38

S2V1 57 1.0 6.241 (0.043)

0.354 (0.00244)

0.181 (20.4)

0.200 (22.638)

1.10

S2V2 57 2.0 6.937 (0.0478)

0.354 (0.00244)

0.177 (20.0)

0.207 (23.373)

1.16

24

b

D

ecu

esf

K1D

Ast

Cross section Strain Diagram Stress Distribution

Actual Assumed

C

T

0.85fc’

0.85K1D

Fig. 1–Strain and Stress Distribution at cross section of SFRC Beam

25

σ σ

Fig.2–Composite In Uniaxial Tension

26

ffuu Vσ=σ

( ) f'ffmu VV1 σ+−σ=σ

Val

ues o

f σ

0 Vfc 1.0

Values of Vf

Fig. 3– Concept of Critical Volume

27

STAR T

Inp ut: N um ber o f d esig n var iabl es su b-str ing s leng th, ma x. a nd M in . b ou nd po pulati on siz e no. of g e nerati ons SFR C dat a and GA Para met ers

Gen eratio n = 1

Ra ndom ly G e ne ra te popula tion

C omp ute co nstrains

Assig n viol atio n p ara met er

Evalua te Indi vi dual F it ness

Store best In di vidu al

C reate M a ting Po ol

C reate po pula tion for n e xt g e nerati on by a ppl ying g ross o ver an d mut atio n opera tor

Gen eratio n = Gen eratio n + 1

Pr int b est I ndi vid ual

If g ener ation is < n

?

STOP

yes no

Fig. 4– Flow chart for the Proposed Algorithm