Optimization of Flexural capacity of Reinforced fibrous Concrete Beams Using Genetic Algorithm

Transcription

1 Optimization o Flexural capacity o Reinorced ibrous Concrete Beams Using Genetic Algorithm SUJI.D,NATESAN S.C, MURUGESAN.R Biography: Suji.D is a Research Scholar at Sathyabama Institute O Science and Technology, Deemed University, Chennai, India. She is a Lie member o Indian society or Technical Education and Member o Institution o Engineers, India. She is having 18 years o teaching experience. Dr.Natesan S.C is the principal, LB. Janakiammal College o Engineering and Technology, Coimbatore, India. He has been actively involved in teaching, research and consultancy work since last 34 years. His ield o interest is Fiber reinorced concrete and Ferro cement. He was awarded as the best outstanding corporate ellow member or the year 2001 in Civil Engineering division by the Institution O Engineers, Coimbatore Chapter. Dr.Murugesan.R is a Senior lecturer in the Department o Civil engineering at Government Polytechnic College, Coimbatore, Chennai, India. He is actively involved in teaching or the past 25 years. His ield o interest is Maintenance and Rehabilitation o structures. ABSTRACT In this paper ormulation and solution technique using Genetic algorithms (GAs) or Optimizing the lexural capacity o steel iber reinorced concrete beams, with random orientated steel ibers, is presented along with identiication o design variables, objective unction and constraints. The most important actors which inluence the ultimate load carrying capacity o FRC are the volume percentage o the ibers, their aspect ratios and bond characteristics. Hence an attempt has been made to analyze the eective contribution o ibers to bending and shear strength o reinorced iber concrete beams. Equations are derived to predict the ultimate strength in lexure o steel iber reinorced concrete beams with 1

2 uniormly dispersed and randomly oriented iber reinorcement. Predicted strengths using the derived expressions have been compared with the experimental data.computer coding or GA has been developed based on the ormulations. Using the results obtained the inluence o various parameters on the ultimate strength are discussed. Particular attention. is given to the construction practice as well as the reduction o searching space. Key words: optimization; iber reinorced concrete; lexure; Shear; Genetic algorithms; INTRODUCTION In the area o Structural Engineering the method o optimization has been steadily applied to various structural problems. Distinguishable linear and non linear optimization techniques have been successully developed or inding optimum set o the material, topology, geometry or cross-sectional dimensions o dierent type o structures subject to particular loading systems. Along the main stream o linear programming and non linear programming techniques reined algorithms have been branched out in order to take into account or the discrete nature o structure, abricated standardized structural components, or example. Although great success has been achieved during the past decades in structural optimization, these techniques generally have diiculties in avoiding local minima and results are sometimes dependent upon the choice o the initial values in the design space. With recent advances o computer technology, combinatorial optimization techniques have emerged. Genetic algorithm (GA) and simulated annealing (SA) are quite popular among them and they can eiciently solve the optimization problems with higher probability. GAs are search procedures based on the mechanics o natural genetics and natural selection. GAs are very eective and useul to treat the optimization problems concerning with discrete variables. They are stochastic search procedures that have their philosophical basis in Darwinian s postulate o the survival o the ittest.(goldberg 1989). They combine the 2

3 concept o artiicial survival with genetic operators, abstracted rom nature, to orm a robust search mechanism. The main advantages o 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 o design variables (3) GAs take advantage o carry out optimization procedures in a stochastic rame work and (4) GAs are not limited by restrictive assumptions about search spaces, such as continuity or the existence o derivatives. RESEARCH SIGNIFICANCE Several investigations have shown that the presence o steel ibers in beams reinorced with high strength deormed bars increases the ultimate strength 9.To achieve eiciency in perormance and economy optimization techniques could be used. The main objective o this study is to accommodate the useulness o GA in practically optimizing reinorced steel ibrous concrete beams with due considerations given to the construction practice. Realizations o much o the code provisions in regard to strength requirements as well as structural constraints have been considered. LITERATURE REIEW In spite o perorming various optimization techniques or reinorced concrete structures by various researchers the algorithms developed using GAs or structural optimization o iber reinorced concrete structural elements are much limited. Optimization techniques or the element level o reinorced concrete structures have been presented by dierent 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 o structures has been perormed using Genetic Algorithms 13. ery little literature is available in the ield o iber reinorced concrete structural optimization because 3

4 design methods or FRC are yet to be ully developed, though some guidelines are available or its applications to airield pavements and some hydraulic structures 4.So is the case with standard test procedures to be adopted or testing and evaluation o the perormance o FRC elements. Ezeldin and Hsu 14 optimized reinorced ibrous concrete beams using direct search technique. The algorithm conducts a systematic search in the space o our variables- beam width, beam depth, iber content, and aspect ratio o ibers to yield an optimum solution or a given objective unction. It is thereore the main objective o this research is to develop an algorithm using GA that perorms the optimum design o reinorced ibrous concrete beams. The algorithm developed or the optimum design satisies the speciications provided in the ACI code 1. REIEW OF ANALYTICAL STUDIES FLEXURAL ANALYSIS OF FIBER-REINFORCED CONCRETE BEAMS The analysis is based on the ollowing assumptions 1. Plane sections remain plane ater bending. 2. The compressive orce equals the tensile orce 3. The internal moment equals the applied bending moment It has been customary to neglect the tensile resistance o concrete in calculating the ultimate lexural capacity o concrete beams. Kukreja 10. et al. (1980) proved that the iber reinorced concrete greatly increases the tensile capacity o concrete. So the contribution o ibers must be taken into account in the lexural analysis o beams. The analysis presented in this paper is based on the conventional compatibility and equilibrium conditions used or normal reinorced concrete except that the eects o steel strain hardening and contribution o the steel ibers in the tension zone are recognized (Appendix A).The analysis is based on the compression stress blocks in ACI Code 1.The actual and assumed stress and strain distributions at ailure are shown in Fig.1. The analysis was compared with the experimental 4

5 results reported by Byung Oh. 3 (1993), and Swamy 16 et al. (1981). Details o comparison are shown in Table.1. Column (8) shows the ratio o the results o experimental to the authors predicted ultimate moment. These estimates are considered reasonably close, in view o the diiculty in establishing the peak load beore an abrupt drop is recorded in their experiment. One possible reason or these conservative estimates is the uncertainty in the value o τ.it can be seen that the ultimate bond strength u is expressed in terms o iber volume concentration, dynamic bond stress τ, and iber aspect ratio l /d. Experimental studies undertaken by many investigators show the wide disparity o bond stress values in SFR concrete 16. The values depend on the response stage, concrete properties, iber type and other characteristics. Tests conirmed that dierent values o τ resulting rom various types o ibers could signiicantly modiy the lexural behavior o SFR concrete 15. PROBLEM FORMULATION Formulation o the problem is based on the objective unction, which can either be maximized or minimized. Studies have shown that the steel ibers can eectively be used to increase the lexural strength o the beam 18.In the present study the equation derived or ultimate lexural capacity o a iber reinorced concrete beam containing steel ibers has been taken as the objective unction, or maximization [APPENDIX A, Eq.A12 ].The design variables are olume raction o the iber ( v ),Width o the beam ( b ), Depth o the beam ( D ), and Aspect ratio o the iber ( A sp ).The design parameters are ultimate strength o ibers( ) and Cylinder compressive strength o concrete ( ' ). u FORMULATION OF CONSTRAINTS Strength in uniaxial tension When long and strong steel ibers are incorporated in a concrete matrix as shown in Fig.2. the strength o the composite is given by the law o mixtures as c m ( 1 ) + (1) 5

6 Where is the volume o ibers per unit volume o the matrix, m, and are the stresses in the concrete matrix, iber and composite respectively. When the ibers are well bonded to the concrete, they are subjected to the same strain so that the above equation becomes [ E ( ) + E ] ε 1 (2) m whereε is the strain in the composite and Em and E are the moduli o elasticity o the matrix and ibers respectively. The term in the parenthesis is the eective modulus o elasticity o the composite. When the cracking strength mu o the matrix is reached, the stress in the composite is given by cr mu ( 1 ) + ' (3) Where, is the stress in the ibers when the matrix cracks. As soon as the matrix cracks the load carried by the matrix which is 1 ) per unit area o the cross section is thrown mu ( on to the ibers. I the bond between the ibers and matrix is inadequate, the ibers at this stage would be pulled out o this matrix. However, i the bond is adequate the ibers will not ail and can take additional load, leading to multiple cracking o the matrix and culminating in the racture o the ibers themselves. Thus the ultimate strength u u (4) Where is the stress at racture o the ibers. u Constraints on critical volume In a plot showing the strength o the composite against the iber volume, Equations (3) and (4) appear as shown in Fig. (3). when the bond is adequate the ollowing inequality is satisied. 6

7 u v mu ( 1 ) + ' (5) It is seen rom Equation (5) and Fig.3 that when the iber content is above a certain volume c the strength is governed by the racture strength o ibers. The volume c is called the critical volume. I the iber 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 racture strength o ibers there should be excellent bond between iber and the matrix. I a iber o diameter d and length l is to racture at its mid length, the bond length developed over the length l/2 must be greater than the racture strength, ie., τ π d 2 l π d 4 2 u l d u 2τ (6) Where τ is the interacial bond stress. Equation (6) states that or the ibers to ail by racture 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 saety, the ultimate moment must be greater than or equal to the applied moment. M u M (7) 7

8 CONSTRAINTS ON TENSION STEEL To provide ductile ailure, the member should be designed or A st less than 0.75 A sb where A sb area o steel required or balanced condition. It may be ound by applying the equilibrium and strain compatibility conditions (Appendix B). A 0.75 A st max sb (8) A s b bd ( c' + y u ) ( ε ) c c ( ε + ε ) y u d D (9) Where ' c compressive strength o concrete ε c compressive strain in concrete ε y yield strain o steel b d D width o the beam eective depth o the beam overall depth o the beam and other variables deined earlier. The member should also be provided with minimum steel to prevent rom excessive cracking. It can be obtained by equating the cracking moment o the section (using the modulus o rupture o iber concrete) to the strength computed as a reinorced iber concrete section Eq.A12. As recommended in Re. 6, this value is taken as A stmin 400 bd y (10) 8

9 CONSTRAINTS ON SHEAR STRENGTH Several studies have shown that the steel ibers are particularly eective in providing reinorcement against shear stresses in conventionally reinorced concrete 11.It is evident rom the test results o various authors that stirrups and ibers can be used eectively in combination. The equation proposed by Narayanan and Darwish 11 has been used in this study to predict the ultimate shear strength o iber reinorced concrete beams. Where 2 ( 0.24 ' + 80 d / a) 4.56 F N / mm s e t ρ + (11) e 1.0 when a/d > 2.8 and 2.8 (d/a) When a/d 2.8 t, splitting cylinder strength o iber concrete ρ percentage o area o tensile steel to area o concrete F iber actor ( ((l/d ) ) ) where 0.5 or round ibers, 0.75 or crimped ibers, and 1.0 or ibers with deormed ends The method proposed in ACI Code 1 is used or calculating the contribution o stirrups s to the shear capacity, to which is added the resisting orce o concrete rom the added ibes s obtained rom Equation (11) s A sv y d S (12) The constraint to check the saety o the concrete against shear is given as u un (13) where un is ultimate shear strength o iber concrete and u is ultimate shear orce applied. The minimum and maximum stirrup area is taken as proposed in ACI Code 1 A svmin 50 b S y (14) 9

10 and A 5 sv max ( A ) v min (15) The stirrup spacing varies rom d /2 down to d /4. Constraints on design variables The design constraints are ormulated as b min b max D min D max b min D spmin A Asp max A spmax A st min A st A st max v min A A v A vmax S min S S max FITNESS FUNCTION Genetic algorithms mimic the survival o the ittest principle. So, naturally they are suitable to solve maximization problems. Minimization problems are usually transormed to maximization problems by some suitable transormation. A itness unction is derived rom the objective unction and used in successive genetic operations.for maximization problems, itness unction can be considered to be the same as the objective unction. TRANSFORMATION OF CONSTRAINED OPTIMIZATION TO UNCONSTRAINED OPTIMIZATION GA is ideally suited or unconstrained optimization problems. As the present problem is a constrained optimization one, it is necessary to transorm it into an unconstrained problem to solve it using GAs to handle constraints. A ormulation based on the application o penalty, whenever there is a violation o speciied constraints, is used in the present study or transormation.unlike the minimization problems here i the design variable set violates the constraint then a lower value o say 1.0 will be assigned and i not, a higher value say 10.0 is 10

11 assigned as violation parameter. Because or maximization problems, the itness unction and objective unction are considered to be the same.the violation coeicient Ф is computed as ollows Φ m l 1 Ф i (16) Ф i a * g i (x) i g i (x) < 0; Ф i 0 i g i (x) 0 Where m number o constraints a penalty parameter and g(x) constraint unction. The modiied objective unction is given as Z 1 Z + Ф (17) Where ( D k D)[ D k D] b 1 1 Z u + y Ast ( d k1d ) 2 (18) WORKING OF GENETIC ALGORITHM With an initial population, the population or the next generation is to be generated which are the o springs o the next generation. The reproduction operator selects the it individuals rom the current population and places them in a mating pool where as the lesser ones get ewer copies. As the number o individuals in the next generation is also same the worst it individuals die o eventually. The actor (F ave/ F) or all the individuals is calculated, where F is the average itness. The actor is the expected count o individuals in the mating pool. It is then converted into an actual count by appropriately rounding o so that individuals get copies in the mating pool proportional to their itness. This process o reproduction conirms the Darwinian principle o survival o the ittest. 11

12 To eect crossover, a set o crossover parameters are generated randomly. The irst step in crossover is inding a match or individuals. Once the pairs are decided it is necessary to ind the cross over sites. The sub string between the cross-sites is swapped rom one individual in the pair to other. To eect mutation, or each bit o a string, the random number generated is checked or. I it is greater than the mutation rate speciied then the bit conversion occurs otherwise it remains as such. The Genetic algorithm repeats the same process by generation o a new population and evaluating its itness. Proceeding with more generations, there may not be much improvement in the population s itness barring a ew because o mutation operation and the best individual may not change or subsequent population. As the generation advances, the population gets illed by more it individuals with only slight deviation rom the itness o the best individual So or ound and the average itness comes very close to the itness o the best individual, i and only i, there are no mutation operation. Criteria have to be evolved to decide the termination o the process. In the present study the number o generations as 50 is chosen or the purpose. GA PARAMETERS Fixing up GA parameters is very crucial in an optimization problem. In the present problem the ollowing GA parameters have been given as input values. Number o parameters 4 Total string length 40 Population size 25 Maximum generation 50 Mutation probability Cross over probability

13 String length or variables X [0] to X [4] 10 Overall Depth o the Beam (D) mm Upper and Lower bound or variables: olume raction o iber X [0] Width o the beam (mm) X [1] mm Depth o the beam X [2] mm Aspect ratio o iber X [3] DESCRIPTION OF ALGORITHM The objective unction or optimization is maximization o ultimate moment or reinorced iber concrete beams subjected to bending and shear. It can be ormulated as ( D k D)[ D k D] b 1 1 Z u + y Ast ( d k1d ) 2 and the variables explained earlier. Fig.4 presents the low chart or the proposed algorithm. The program is designed to read the required data- limiting values o variables, and GA parameters. The program searches or a maximum or the objective unction in the space o our variables, namely olume raction o the iber ( ),Width o the beam (b), Depth o the beam ( D ), and Aspect ratio o the iber ( A sp ). The stirrup spacing varies rom d /2 down to d /4, while the stirrup area increases rom the minimum allowed by the ACI Code 1 ( A vmin 50 bs/ y ) up to ive times this value. The maximum o the objective unction is recorded i all the constraints are satisied. I any o the constraint is violated, the penalty or violation is given and the objective unction is modiied. The modiied objective unction incorporating the constraint violation is given in Equation (17). Genetic Algorithms are normally begun with a population o strings created randomly. Thereore, each string in the population is evaluated. The population is then operated by three 13

14 main operators, namely, reproduction, crossover and mutation to create a better population. The population is urther evaluated and tested or termination. I the termination criteria are not met, the population is again operated by the above three operators and evaluated urther. This procedure is continued till the termination criteria are met. One cycle o these operations and the evaluation procedure is known as generation in GA terminology. In the proposed algorithm the above criteria is taken as No o generations and is equal to 50. The low chart or the proposed algorithm is given in Fig.4. CONCLUSIONS Based on the ormulation or Genetic Algorithm based optimal design o Reinorced Concrete Beams along with identiication o design variables, objective unction and constraints the ollowing conclusions are arrived. 1. The developed GA reinorced with three basic operators (reproduction, crossover and mutation) successully 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 dierent types o steel ibers, aspect ratio, and material characteristics. 3. The outlined methods provide a simple and eective tool to assess the optimum lexural strength o steel iber reinorced concrete beams with randomly distributed steel ibers. 4. The overall eect o iber addition on ultimate strength and shear strength is studied. 14

15 5. It must be emphasized that toughness improvement, cracking, and delection controlwhich are other reasons or use o ibers were not considered. Future studies should combine the optimum use o ibers or strength and serviceability criteria. a Shear span, mm b Width o beam, mm NOTATION d Distance rom extreme compression iber to centroid o tension Reinorcing bars, mm d Fiber diameter, mm E m Young s modules o matrix, MPa E Young s modules o iber, MPa F Fiber actor c Compressive strength (cylinder) o concrete, MPa y Yield stress o reinorcing bars, MPa t Splitting cylinder strength o iber concrete, MPa D Beam overall depth, mm l Fiber length, mm s Shear strength o iber concrete beams, N m olume o matrix s Stirrups contribution to shear strength A st Area o tension reinorcement, mm 2 A sv Area o stirrups, mm 2 F Fiber actor regulating shear strength M Applied moment, N-mm 15

16 M u Ultimate bending strength o cross section, N-mm r Radius o iber, mm s Average spacing o ibers S v Stirrups spacing, mm ρ Percentage o area o tensile steel to area o concrete strength o the composite Strngth o iber, MPa m Strength o matrix, MPa u Ultimate strength o iber, Mpa mu Cracking strength o matrix, MPa cr Cracking strength o composite, MPa Stress in iber when the matrix cracks, MPa u Ultimate strength o composite, MPa c Critical volume o ibers u Ultimate shear orce applied, N τ Interacial bond stress, MPa A sb Area o steel required or balanced condition, mm 2 ε c Strain in concrete, mm ε y Strain in steel, mm REFERENCES 1. ACI Commmittee 318, Building Code Requirements For Reinorced Concrete and Commentary (ACI /ACI ), American Concrete Institute, Detroit, 1989, pp Adamu, A., and Karihaloo, B.L. (1994), Minimum cost design o reinorced concrete beams using continuum-type optimality criteria. Struct.Optim., 7, Byung Hwan Oh, (1993), Flexural Analysis o reinorced Concrete Beams Containing steel 16

17 Fibers, J.Struct. Eng.,. 118, No. 10,Oct pp Cohn, M.Z., and Lounis, Z. (1994), Optimal design o structural concrete bridge systems, J.Struct.Eng., ASCE, 120 (9), Cox, H.L. (1952), The elasticity and strength o paper and other ibrous materials, British Journal o Applied Physics, 3, Craig, R., Flexural Behaviour and Design o Reinorced Fiber Concrete Members, SP-105, American Concrete Institute, Detroit, 1987,pp Dwarakanath, M.., and Nagaraj, T.S., Flexural Behaviour o reinorced Fiber Concrete Beams, Proceedings o the International symposium on Fiber Reinorced Concrete, Dec., 1987, Madras, ol. 1, pp Goldberg, D. Genetic algorithms in Search,optimization & Machine learning, Henager, C.N., and Doherty, T.J., Analysis o Reinorced Fibrous Concrete Beams, Journal o the Structural Division, ASCE,. 102, No, ST!, Jan, 1976, pp Kukreja, C.B., Kaushik, S.K., Kanchi, M.B., and Jain, O.P. (1980), Tensile strength o steel ibre reinorced concrete, Indian concrete Journal, July Narayanan, R., and Darwish, I. Y.S., Use o Steel Fibers as Shear Reinorcement, ACI structural Journal, ol. 84, No. 3, May. -Jun., 1987, pp Rajagopalan, K.,Parameshwaran,.S., and Ramasway, G.S. (1974), Strength o steel ibre Reinorced Concrete beams, Indian Concrete Journal, Jan Rajeev, S., and Krishnamoorty, C.S. (1992), Discrete optimization o structures using Genetic Algorithms, J.Struct. Eng.,118(5), Samer Ezeldin, A., and Cheng- Tzu Thomas Hsu. (1992), Optimization o Reinorced Fibrous Concrete Beams, J.Struct.Eng., ACI, 89 (1), Soroushian, P., and Bayasi, Z. (1991), Fiber type eects on the perormance o steel iber reinorced concrete, Mat. J. ACI, 88(2),

18 16. Swamy, R.N., and Al-Ta an, S.A., Deormation and Ultimate Strength in lexure o Reinorced concrete beams made with Steel Fiber Concrete, ACI Journal, Proceedings. 78, no. 5, Sept.- Oct. 1981, pp Waa, F.F., and Ashor, S.A. (1992). Mechanical properties o high strength iber reinorced concrete, Mat. J. ACI, 88(6), APPENDIX A DERIATION FOR ULTIMATE MOMENT CAPACITY OF REINFORCED STEEL FIBROUS CONCRETE BEAMS The concept o composite materials may be introduced to describe the mechanical behaviour o iber-reinorced concrete. The strength o iber reinorced composite material may be described as the sum o matrix strength and iber strength as ollows. m m + (A1) in which strength o iber-reinorced composite; m strength o matrix; strength o ibers; m volume o matrix ( 1 ); and volume o ibers per unit volume o the matrix. When the ibers are well bonded to the concrete, they are subjected to the same strain. so that the above equation becomes, [ E ( 1 ) + E v ] ε m (A2) where ε is the strain in the composite, and E m and E are the modulus o elasticity o the matrix and ibers respectively. The term in parenthesis is the eective modulus o elasticity o the composite. When the cracking strength cr o the matrix is reached, the stress in the composite is given by cr mu ( 1 ) ' + (A3) 18

19 where ' is the stress in the ibers when the matrix cracks. As soon as the matrix cracks, the load carried by the matrix which is 1 ) per unit area o the cross section is thrown mu ( on to the ibers. I the bond between the ibers and the matrix is inadequate, the ibers at this stage would be pulled out o the matrix; however i the bond is adequate, the ibers will not ail and can take additional load, leading to multiple cracking o the matrix and culminating in the racture o the ibers themselves. It is reasonable here to assume that the strength contribution o the concrete matrix at ultimate state may saely be neglected due to tensile cracking.the irst term o the right hand side o Eqn. (A3) then vanishes. Thus the ultimate strength o the composite is given by u u (A4) where u is the stress at the racture o the ibers. Since the orientation, length, and bonding characteristics o ibers will inluence the strength o iber reinorced concrete, these parameters must be incorporated in Eqn.(4). u α 0 α 1 α b u (A5) in which, α 0, α 1 and αb are orientation actor, length - eiciency actor and bond eiciency actor o ibers respectively. The iber strength u may be derived rom bonding characteristics o ibers as ollows. u l 2τ d (A6) in which τ bond strength o matrix. The ultimate strength u o iber-reinorced concrete is now summarized as 19

20 u 2α 0 α α v l b τ l d (A7) The orientation actor α 0 is known to be about 0.41 or uniormly distributed iber- reinorced concrete, and the bond eiciency actorα is about 1.0 or straight ibers (Henager, C.N., and Doherty, T.J., 1976).The present study exploits Cox s (1952) results or length- eiciency actor as ollows. b α l β l 1 tanh 2 β l 2 (A8) β E 2π Gm A ln s r (A9) l S 25 (A10) d in which G m shear modulus o concrete matrix; E elastic modulus o iber; sectional area o iber; Saverage spacing o iber; r radius o iber ; A cross d diameter o iber; l Length o iber and volume ratio o iber. Eqn. (A7) o iber reinorced composite may now be employed to derive the lexural capacity o concrete beams containing steel ibers. The strain proile as shown in Fig.1 has been assumed or a cracked section in pure bending. The concrete has reached its ultimate compressive strain e cu. The stress block in the compression zone is the one commonly assumed in ultimate strength calculations. It has been 20

21 adopted under the assumption that the behaviour o the iber-reinorced compression zone is similar to that o one without iber-reinorcement. Hence k 1 can be obtained rom equilibrium conditions k l u bd + y Ast (A11) bd c' + ) ( u in which, c ' Cylinder compressive strength o concrete; b Width o the beam; DOverall depth o the beam; olume raction o the ibers; and u 2 α0 α 1 αb τ ( l / d ) Ultimate iber strength incorporating orientation, length and bond eiciency actor. y yield strength o tensile steel; The lexural capacity is then derived as ollows; M ( ) [( D + 1.5k )] 1D D k D + A ( d 0. k D ) u u y st 1 A st Area o tensile steel (A12) 2 APPENDIX B DETERMINATION OF TENSION STEEL FOR BALANCED STRAIN CONDITION ( k D) 1 d Bal ε c ε + ε c y (B1) From Force equilibrium sb v u (( D k1d) ) b( k1d) ' A + b (B2) From B1 and B2 A sb 0 Bal c bd ( ) ε D c u c' u (B3) y εc + ε y d 21

22 TABLES AND FIGURES List o Tables: Table 1 Comparision With Results O Swamy And Al-Ta An 16 (1981) Table 2 Comparision With Results O Buing O 3 (1993) Table 3 Parametric Study Results O GA List o Figures: Fig. 1 Strain And Stress Distribution At Cross Section O SFRC Beam Fig. 2 Composite In Uniaxial Tension Fig. 3 Concept o Critical olume Fig. 4 Flow Chart or the Proposed Algorithm Table 1 Comparison With Results O Swamy And Al-Ta An 16 (1981) Beam No Fiber aspect ratio olume raction o iber, % Compressive strength, ksi (kn/mm 2 ) Interacial bond stress, ksi (kn/mm 2 ) Mu Author x 10 6 lb-in. (knm) Mu Expt x 10 6 lb-in. (knm) Mu Expt / Mu Author (1) (2) (3) (4) (5) (6) (7) (8) DR ( ) ( ) (18.00) (23.25) DR ( ) ( ) (20.00) (23.81) DR ( ) ( ) (27.68) (33.68) DR ( ) ( ) (32.00) (35.06) DR ( ) ( ) (25.46) (28.62) DR ( ) ( ) (25.13) (30.83) 22

23 Table 2 Comparision With Results O Buing O 3 (1993) Beam No Fiber aspect olume raction o Compressive strength, Interacial bond stress, Mu Author, Mu Expt, Mu Expt/ Mu Author ratio iber, % Ksi (kn/mm 2 ) ksi (kn/mm 2 ) x 10 6 lb-in. (knm) x 10 6 lb-in. (knm) (1) (2) (3) (4) (5) (6) (7) (8) S (0.043) ( ) (14.7) (15.229) S (0.0478) ( ) (13.0) (17.963) S (0.043) ( ) (20.4) (22.638) S (0.0478) ( ) (20.0) (23.373) Table 3 Parametric Study Results O GA Gen. No. olume raction, Width o beam, in. (%) (mm) (289) (261) (297) (233) (227) (228) (228) Depth o beam, in. (mm) (304) (377) (413) (379) (357) (355) (355) Aspect Ratio, Spacing o stirrup, in. (mm) (72) (90) (99) (91) (89) (90) (90) Area o Stirrup, in. 2 (mm 2 ) (20) (22) (24) (31) (32) (30) (30) Area o tension steel, in. 2 (mm 2 ) (184) (214) (258) (185) (258) (258) (258) Ultimate moment, x 10 6 lb-in. (knm) (57) (74) (100) (85) (68) (69) (69) Ultimate shear, kips (kn) (35) (39) (42) (53) (52) (49) (49) 23

24 b e cu 0.85c C K 1 D 0.85K 1D D e s T Ast Actual Assumed Cross section Strain Diagram Stress Distribution Fig. 1 Strain and Stress Distribution at cross section o SFRC Beam 24

25 Fig.2 Composite In Uniaxial Tension 25

26 alues o u u mu ( 1 ) + ' 0 c 1.0 alues o Fig. 3 Concept o Critical olume 26

27 START Inp ut: N umber o design variables sub-strings leng th, ma x. a nd M in. b ou nd po pulati on siz e no. o g e nerati ons SFR C dat a and GA Parameters Gen eratio n 1 Random ly Generate population C omp ute co nstrains Assig n viol atio n p ara meter Evalua te Indi vi dual Fitness Store best In di vidu al C reate M a ting Po ol C reate po pula tion or n e xt g e nerati on by a ppl ying g ross o ver an d mut atio n opera tor Generation Gen eratio n + 1 STOP yes I g ener ation is <n? no Print b est I ndi vid ual Fig. 4 Flow chart or the Proposed Algorithm 27