w Proceedings of CENG’12 the 7th International Symposium on Civil and Environmental Engineering Symposium Head: 7th ARCHENG-2012 International Architecture and Engineering Symposiums Symposium Theme: CREATING THE FUTURE Symposium Venue: European University of Lefke Faculty of Architecture and Engineering Gemikonağı – TRNC Mersin-10, Turkey Symposium Period: 29-30 November, 2012 ii Editors of CENG’12 Proceedings: Khaled H. Marar İbrahim Yitmen Associate Editors: Ferhat Türkman Hakan Yalçıner ISBN: 978-975-98897-8-4 Copyright © 2012 European University of Lefke Faculty of Architecture and Engineering Gemikonağı − Lefke TRNC http://www.eul.edu.tr No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form by any means, electronic, mechanical, photocopying, recording or otherwise, with the prior written permission of the editor of CENG’12. iii STEERING COMMITTEE Ahmet Bülend Göksel, Rector Akın Cellatoğlu, Vice Rector Mehmet Yalçın, General Secretary General Chair and Coordinator of 7th ARCHENG Karuppanan Balasubramanian Chair of CENG’12: Ferhat Türkman ORGANISING COMMITTEE of CENG’12 Chair: Ferhat Türkman Khaled H. Marar İbrahim Yitmen Ayşen Türkman Hakan Yalçıner Symposium Sponsor Cyprus Science Foundation iv SCIENTIFIC COMMITTEE of CENG’12 A. Alkan, Dokuz Eylül University, Turkey A. Dikbaş, Istanbul Technical University, Turkey A. Müezzinoglu, Dokuz Eylül University, Turkey A. Özer, Dokuz Eylül University, Turkey A.Ş. Kayalar, Dokuz Eylül University, Turkey A. Şenol, Istanbul Technical University, Turkey A. Türkman, European University of Lefke, TRNC B. Kaya, Dokuz Eylül University, Turkey C. Atalar, Near East University, TRNC D. Özdaglar, Dokuz Eylül University, Turkey E. Benzeden, Dokuz Eylül University, Turkey E. Küçükgül, Dokuz Eylül University, Turkey F. Piroğlu, Istanbul Technical University, Turkey F. Türkman, European University of Lefke, TRNC G.Elkıran, Near East University, TRNC G. Özden, Dokuz Eylül University, Turkey G. Rao, National Institute of Technology, India H.Yalçıner, European University of Lefke, TRNC İ. Yitmen, European University of Lefke, TRNC M.A. Yurdusev, Celal Bayar University, Turkey M.A. Taşdemir, Istanbul Technical University, Turkey M.Düzgün, Dokuz Eylül University, Turkey M.Ergil, Eastern Mediterranean University, TRNC M.Tokyay, Middle East Technical University, Turkey M.V. Akpınar, Karadeniz Technical University, Turkey K. Marar, European University of Lefke, TRNC K.Ramyar, Ege University, Turkey K.Tuncay, Middle East Technical University, TRNC M. Semih Yücemen, Middle East Technical University, Turkey N. Harmancioglu, Dokuz Eylül University, Turkey O. Baykan, Pamukkale Üniversitesi, Turkey Ö. Eren, Eastern Mediterranean University, TRNC O.Fıstıkoğlu, Dokuz Eylül University, Turkey S. Özkul, Dokuz Eylül University, Turkey Ş. Güney, Dokuz Eylül University, Turkey S. Küçükarslan, Istanbul Technical University, Turkey S. Arsoy, Kocaeli University, Turkey T. Çelik, Middle East Technical University, TRNC U. Türker, Eastern Mediterranean University, TRNC Contact Email ID: symposium@eul.edu.tr Future Correspondence: K. Balasubramanian, General Chair 7th ARCHENG-2012 International Architecture and Engineering Symposiums EUROPEAN UNIVERSITY OF LEFKE Gemikonagı – Lefke, TRNC, Mersin-10. Turkiye mailto:symposium@eul.edu.tr� v 7th ARCHENG-2012 International Architecture and Engineering Symposiums Abstract of the Addressing by Board of Trustees’ Chairman Dear Guests, Colleagues, and Friends, May I take this opportunity, as Chairman of the Board of Trustees, to welcome you to our university, the European University of Lefke, and to our Symposium, the Architecture and Engineering Symposium. Organized and hosted by the European University of Lefke, the 2012 Architecture and Engineering Symposium is the 7th of its kind. This international symposium has been running for 12 years now, and was first organized in the year 2000 to mark 10 years since the establishment of the European University of Lefke. The phrase “Creating the Future” has become our Symposium’s motto, and in fulfilling this motto, the Symposium has endeavored to encourage the rapid dissemination of research contributions that push back the frontiers of technology in the areas of both Architecture and Engineering. I am a Civil Engineer by profession, and so I am well aware of the importance of providing our students with the necessary skills and knowledge for a successful career in both Engineering and Architecture. It is of paramount importance to our local community, as well as to our society. Since the establishment of the university, over 20 years ago, we have seen a large number of developments in the university, including the addition of Faculties, Schools and Departments, as well as improvements to the infrastructure and faculties of the university. Recently the University’s scientific and engineering laboratories have had a major overhaul, and I am pleased to say that EUL is now equipped with some of the most sophisticated and useful scientific equipment our students could wish for. The Faculty of Architecture and Engineering at EUL has made significant strides in moving this institution forward. Their vision is to compete to be the best faculty of its kind in the country. To this end, it has increased its scientific contributions to the body of knowledge in Engineering and Architecture. This has been achieved by increasing the number of publications of scientific articles in prestigious journals; and through paper presentations by Faculty members in some of the most important international conferences in the world. Our staff are also involved in various computer and electronic engineering projects in the Turkish Republic of Northern Cyprus. Civil engineering professors are involved in TUBITAK projects, and our architecture staff has been involved in a number of architecture workshops. Many of the undergraduate degree programs offered by our University have been accredited by other Nation’s Governments and prominent educational organizations and bodies. This to me is testament to the fact that EUL is a world class university with world class degree programs. Our students can most definitely be assured that an education from the European University of Lefke is one that will stand them in good stead for whatever they decide to do in the future. I trust that you will enjoy your time with us at this symposium, and I am sure that you will be able to take away with you lots of inspiration and ideas for future research. Mr. Mehmet ZAFER Chairman of Board of Trustees European University of Lefke vi 7th ARCHENG-2012 International Architecture and Engineering Symposiums Abstract of Opening Speech by Rector Dear friends and colleagues from North Cyprus, Turkey, and the rest of the world. On behalf of the Steering, Organizing and Scientific Committees, and the European University of Lefke which is hosting this event, I warmly welcome you to Lefke, and to this Symposium - our 7th Architecture and Engineering Symposiums - "Creating the Future". Our symposiums cover the three major disciplines of our Faculty of Architecture and Engineering, namely Architecture and Civil Engineering, Electrical Engineering, and Computer Engineering. Since we last met in 2010, there have been many developments within the Faculty of Architecture and Engineering. As well as the faculty growing, with new talented staff, we have passed some major milestones. We have received accreditation from the Pakistan Engineering Council (PEC) for our undergraduate degrees in Computer Engineering and Electrical Engineering. We have also started a PhD program in Computer Engineering, and have our first intake of PhD students this semester. Our university has become an authorized CUDA teaching centre. Our staff is currently engaged in research using the CUDA GPU from NVIDIA in our supercomputer facility. Our objective is to teach students parallelization of computer intensive algorithms using the new generation computing hardware. Our first symposium was held in the year 2000. Twelve years later, our current symposium deals with the timely and challenging theme, "Creating the Future". Without research and innovation, the economies of the world would stagnate, so as engineers and scientists in the research community, we have our part to play in creating the future world of our nations. As educators, we also create the workforce of tomorrow. Including research in our daily activities at the European University of Lefke allows us to teach at the highest academic level and our students take advantage of this. I would like to offer my sincere congratulations to the staff of our Faculty of Architecture and Engineering for their earnest efforts and contributions to academic and research growth at our university. I wish for our Faculty a long future with exciting research and much collaboration. I would like to convey a special thank you to all the presenters, distinguished scholars, and younger researchers that are taking part in our symposium. Finally, I would like to express my cordial thanks to the Scientific Committee and the Organizing Committee of each Symposium, for their tireless work in the preparation of the symposium, and the smooth running of our symposium throughout its duration. Thank you everyone for listening, I hope that you will enjoy our symposium, and take with you some new ideas for your own organizations and your own research. Prof. Dr. Ahmet Bülend Göksel Rector European University of Lefke. vii 7th ARCHENG-2012 International Architecture and Engineering Symposiums Abstract of Welcome Speech by General Chair In our efforts to organize our traditional biennial international symposium series on Architecture and engineering now we have moved on to conduct the 7th symposium series in the campus of European University of Lefke. As in the past, this 7th ARCHENG international symposium hosts three symposiums, namely, EEECS’12- International symposium on Electrical and computer Systems, CENG’12- International Symposium on Civil and Environmental Engineering and ARCH’12- International Symposium on Architecture and Interior Architecture. While the first international symposium was conducted when celebrating the 10th anniversary of European University of Lefke, in Nov 2000, the previous 6th ARCHENG international symposium was conducted during Nov 2010 on the celebration of 20th anniversary of EUL. This symposium series is favoring the research contributions of academics and researchers in the chosen fields of Architecture and Engineering. The objective of each symposium is to bring together the researchers in the emerging fields of engineering or architecture providing them with a forum where they can present their current work leading to innovations for future inventions and thereby contributing to the theme of the symposium, “Creating the Future”. We have gained experience in the past symposiums and we trust that we have contributed to science. Tremendous growth of technology in electronics, computer and communication engineering has been witnessed globally and we have a pleasure to have a tiny share from our symposiums. The growth in building construction technology is also remarkably increasing where improved building materials and newer recycled materials are being launched. Today’s research we do will come to market tomorrow and will reach the common man. Therefore, the education must be updated to keep in pace with the research outcomes setting new trends. In the campus, IEEE chapter and ACM club have been taking active parts in organizing talks, attending special lectures and undertaking projects as to stimulate research minds in the young engineers of EUL. The student community is gaining academic and research maturity by these extra curricular activities. This symposium series is hoping to concede another step in their growth of knowledge and research experience for the challenges they face tomorrow. On behalf of the organizing committees of the symposium series I express my gratitude to authors who submitted their innovative works based on analogical thinking, case studies and problems that require solutions in the chosen fields of the symposium. We extend our hearty welcome to all participants and contributing authors who have chosen to come over here to take part in the presentations and discussions. We are thankful to the reviewers for their support in exercising their responsibilities to choose the quality papers for presentation and for inclusion in the proceedings of the symposium. We thank all our co-sponsors who have associated with us in organizing this symposium with their contributions. We hope you have an interesting and thought provoking experience with us, and that we will all come together again for future symposiums. Prof. Dr. K.Balasubramanian General Chair, 7th ARCHENG-2012 & Dean, FAE viii CONTENTS No. Author Title Page Theme 1: Structural and Materials Engineering 1 Soner Şeker Erkan Doğan Performance Testing of Hunting Search Algorithm in Finding The Optimum Solution of Engineering Design Problems 1 2 Gökhan Şakar Carbon Fiber Reinforced Polymer Shear Strengthening Of RC Beams Subjected to Cyclic Load 7 3 Hilal M. Atalay Şevket Ozden Erkan Akpınar Hakan Erdoğan Structural Precast Concrete in Turkey: Development and Earthquake Performance in Last Decades 13 4 Semih Küçükarslan Mehmet A. Karaca The Homotopy Perturbation Method for Complex Linear Schrödinger Wave Equations 20 5 Khaled H. Marar Özgür Eren Hakan Yalçıner Compressive Strength And Modulus of Elasticity of Fiber Reinforced Concrete 23 6 Engin Gücüyen R.Tuğrul Erdem A.Uğur Öztürk Computer Based Phase Analysis of Cement Mortars with Chemical Admixtures by Image Processing 30 7 Özgür Eren Khaled H. Marar Abdulmecit Altay Mehmetali Okman Properties of Mortar Containing Waste Glass as Fine Aggregates 35 8 JC. Naito Mustafa Kaya R. Horwhat Bond Performance of Self Compacting Concrete and Prestressing Strand Using Direct Tension Pull-Out Test Technique 41 Theme 2: Geotechnical Engineering 9 Akbar Pashazadeh Morteza Chekaniazar Amin Hessami Fatemeh Samimi-asl Application of Geo-Synthetics in Controlling Landfill Leachate 45 10 Olcay Polat Erdinç Keskin Cüneyt Yılmaz Mehmet Özgür Sami Arsoy Effect of Elevated Temperature on Behavior of Cohesive Soil 51 11 Erol Şadoğlu Hakan A. Kamiloğlu Optimization of Gravity Retaining Walls in Cohesionless Soils 57 ix 12 Mehmet Özgür Sami Arsoy TDR Application in Civil Engineering 63 13 Soheil Khadr Sajjad Mirsalihi Hüriye Bilsel Characterization of Tire Powder Improved Swelling Soil 69 Theme 3: Construction Management and Education in Civil Engineering 14 Alireza Rezaei Project Management in The Iranian Construction Industry 73 15 Ary S. Korsheed İbrahim Yitmen Understanding Project Complexity in Construction 80 16 Changiz Ahbab Navid Sanei Sistani Alireza Rezaei Adopting BIM as a New Technology into Civil Engineering Education 86 17 Onye Buchi Mogbo İbrahim Yitmen Khaled H. Marar Balkız Yapıcıoğlu Innovative Strategies for Transport Policies in Infrastructure Development 90 18 Nesrin Baykan N. Orhan Baykan Ferhat Türkman Conceptual/Theoretical Approach to Education in Civil Engineering 95 19 Ülker G. Bacanli N. Orhan Baykan Nesrin Baykan Engineering Ethics and its Meaning Among Disciple- Students 100 Theme 4: Hydraulics and Environmental Engineering 20 Levent Yilmaz The Gravel-Bed River Reach Properties Estimation in Bank Slope Modelling 104 21 N. Orhan Baykan Nesrin Baykan Ferhat Türkman Onur Abay Population Estimation of Ancient Cities by Using Methods of Water Engineering 111 22 Nesrin Baykan N. Orhan Baykan Murat Erdem Effect of Climate Change to the Collapse of Ancient Civilizations 117 23 Rasoul Daneshfaraz Farzaneh Sayyadzadeh Ali Keshavarzi Maryam Menazadeh Flood Zoning of V-Sections In GIS Using Hec-Ras Hydraulic Model (Case Study: Ghale Chay River) 125 24 Mutlu Yaşar Sedef Genç Ferhat Türkman Environmental Water and Turkey’s Applications 129 x 25 Nik Daud, N. N. Azmi, J. A. G. Remediation of Heavy Metals in Contaminated Soil by Using Electrokinetic Technique 135 26 Şevket Tulun İsmet Arslan Melayib Bilgin Renewable Energy Sources and Solar Energy Solar Energy Potential in Turkey 140 27 U. Tezcan Un S. Eren Ocal E. Oduncu Copper Removal From Synthetic Waste Water with Different Design Reactor by Electrocoagulation 143 28 Bülent Keskinler Ayhan Çelik Nadir Dizge Fatma Ertan Gizem Güneç The Production of Citric Acid from Waste Molasses by Yeast 146 1 PERFORMANCE TESTING OF HUNTING SEARCH ALGORITHM IN FINDING THE OPTIMUM SOLUTION OF ENGINEERING DESIGN PROBLEMS Abstract —This study presents a hunting search based optimum design algorithm for the solution of benchmark problems. Hunting search algorithm is a numerical optimization method inspired by group hunting of animals such as wolves, lions, and dolphins. Each of these hunters performs hunting in a different way. However, they are common in that all of them look for a prey in a group. Prey is encircled and the ring of siege is tightened gradually until it is caught. Hunting search is employed for the automation of optimum design process, during which size variables are selected in such a way that the objective function value of the design problem is the minimum and the design constraints are satisfied. Two numerical design examples namely welded beam design problem and spring design problem are solved by the presented algorithm to demonstrate its efficiency. These results are then compared with the ones obtained with particle swarm and harmony search algorithms. Results reveal that hunting search shows great performance compare to the other stochastic search techniques taken into account in the present study. Keywords: : Metaheuristic search techniques, harmony search algorithm, particle swarm method, hunting search algorithm, optimization problems. 1. INTRODUCTION In recent years, as an alternative to mathematical programming based methods, several meta-heuristic or evolutionary algorithms have been developed, which combine rules and randomness by mimicking natural phenomena, including biological evolutionary processes (evolutionary algorithm and genetic algorithms) [1, 2] animal behavior and intelligence of swarms (ant colony and particle swarm optimizers) [3, 4], the physical annealing process (simulated annealing) [5] and the musical process of searching for a perfect state of harmony (harmony search) [6]. The aim of researchers introducing these methods is to overcome above mentioned shortcomings of traditional mathematical programming techniques in solving optimization problems. What makes these techniques quite robust and simple compared to other classical methods is the fact that they do need neither the gradient information nor the convexity of the objective function and constraints functions. The optimum structural design algorithms based on these techniques are quite effective in finding the solution of discrete programming problems. The common features of these algorithms are that they all employ random number and incorporate a set of parameters that require to be adjusted initially. They show different performance depending on the problem under consideration and the predefined values of these parameters. One of the recent additions to these novel optimization algorithms is the hunting search algorithm [7], which is inspired by group hunting of animals such as lions, wolves, and dolphins. Hunters involved in the hunting group encircle and catch their prey abiding by the certain strategies. For instance, wolves can hunt animals bigger or faster than themselves by relying on this kind of hunt. One prey is selected and the hunting group gradually moves toward it. The hunters avoid standing in the wind such that the prey senses their smell. This concept is used in the constrained problem to avoid prohibited regions. In optimization process, each of the hunters indicates one solution for a particular problem. Similar to animals cooperate to find and catch the prey, the optimum design process seeks to find the optimum solution. 2. MATHEMATICAL FORMULATION OF AN OPTIMIZATION PROBLEM One of the most difficult parts encountered in practical engineering design optimizations is the constraint handling. Real-world limitations frequently introduce multiple, non-linear and non- Soner Şeker Faculty of Engineering Celal Bayar University Manisa , Turkey soner.seker@cbu.edu.tr Erkan Doğan Faculty of Engineering Celal Bayar University Manisa , Turkey erkan.dogan@cbu.edu.tr mailto:soner.seker@cbu.edu.tr� mailto:erkan.dogan@cbu.edu.tr� 2 trivial constraints on a design. A general engineering optimization problem can be defined as follows; Minimize f(x), x={x1, x2,…,xNd} which is subjected to gi(x)≤0, i=1,2,…,p and hj(x)=0, j=1,2,…,m where Lxk≤x≤Uxk, k=1,2,…,Nd. Here, f(x) is the objective function, x denotes the decision solution vector, Nd is the number of decision variables, Lxk and Uxk, are the lower and the upper bound of each decision variable, respectively. p is the number of inequality constraints and m is the number of equality constraints. 3. HUNTING SEARCH OPTIMIZATION (HSO) Hunting search algorithm is instigated by Oftadeh, et. al. [7]. This algorithm is inspired by group hunting of animals such as lions, wolves and dolphins. The common part in the way of hunting of these animals is that they all hunt in a group. They encircle the prey and gradually tighten the ring of siege until they catch the prey. Each member of the group corrects its position based on its own position and the position of other members during this action. If a prey escapes from the ring, hunters reorganize the group to siege the prey again. The hunting search algorithm is based on the way as wolves hunt. The steps of the algorithm are given in the following: 1. Initialize the parameters: The parameters of hunting search algorithm are required to be initialized. These are hunting group size (number of solution vectors in hunting group, HGS), maximum movement toward the leader (MML) and hunting group consideration rate (HGCR) which varies between 0 and 1. The parameters MML and HGCR are used to improvise the hunter position (solution vector). 2. Initialize the hunting group: Based on the number of hunters (HGS), the hunting group matrix is filled with feasible randomly generated solution vectors. The values of objective function are computed for each solution vector and the leader is defined depending on these values. 3. New hunters’ positions (new solution vectors) { }1 2, ,......., nx x x x′ ′ ′ ′= are generated by moving toward the leader (the hunter that has the best position in the group) as follows ( )i i i ix x rand MML x x′ = + × × − The MML is the maximum movement toward the leader, rand is a uniform random number [0,1] and ix′ is the position value of the leader for the ith variable. For each hunter, if the movement toward the leader is successful, the hunter stays in its new position. However, if the movement is not successful (its previous position is better than its new position) it comes back to the previous position. This provides two advantages. First, the hunter is not compared with the worst hunter in the group to allow the weak members to search for other solutions. They may find better solutions. Secondly, for prevention from rapid convergence of the group the hunter compares its current position with its previous position; therefore, good positions will not be eliminated. The value of MML varies depending on the problem under consideration. The range within 0.05 to 0.4 gives good results. 4. Position correction- cooperation between members: The cooperation among the hunters is required to be modelled in order to conduct the hunt more efficiently. After moving toward the leader, hunters (based on other hunter positions and some random factors) choose another position to find better solutions. Hunters correct their position either following “real value correction” or “digital value correction”. In real value correction, the new hunter’s position { }1 2, ,......., nx x x x′ ′ ′ ′= is generated from HG, based on hunting group considerations or position corrections. For instance, the value of the first design variable for the jth hunter 1 jx ′ for the new vector can be selected as a real number from the specified 1 2( , ,......, )HGS i i iHG x x x or corrected using HGCR parameter (chosen between 0 and 1). The variable is updated as follows: =1,….,n j=1,.HGS The parameter HGCR is the probability of choosing one value from the hunting group stored in the HG. It is reported that selecting values between 0.1 and 0.4 produces better results. Ra is an arbitrary distance radius for the continuous design variable. It can be fixed or reduced during optimization process. Several functions can be selected for reducing Ra. The following is used in [7]. max min minR ( ) R ( ) expi i max min Ra n it Ra a it a x x itm     ×    = −          where it is the iteration number. max ix and min ix are the maximum and minimum possible values for ix . Ramax and Ramin are the maximum and minimum of relative search radius of the hunter, 3 respectively, and itm is the maximum number of iterations in the optimization process. In digital value correction, instead of using real values of each variable, the hunters communicate with each other by the digits of each solution variable. For example, the solution variable with the value of 23.4356 has six meaningful digits. For this solution variable, the hunter chooses a value for the first digit (i.e. 2) based on hunting group considerations or position correction. After the quality of the new hunter position is determined by evaluating the objective function, the hunter moves to this new position; otherwise it keeps its previous position. 5. Reorganizing the hunting group: In order to prevent being trapped in a local optimum they must reorganize themselves to get another opportunity to find the optimum point. The algorithm does this in two independent conditions. If the difference between the values of the objective function for the leader and the worst hunter in the group becomes smaller than a preset constant ε1 and the termination criterion is not satisfied, then the algorithm reorganizes the hunting group for each hunter. Alternatively, after a certain number of searches the hunters reorganize themselves. The reorganization is carried out as follows: the leader keeps its position and the other hunters randomly choose their position in the design space max min( ) exp( )i i i ix x rand x x ENα β′ = ± × − × × − × where ix is the position value of the leader for the ith variable. rand is a uniform random number between [0,1]. max ix and min ix are the maximum and minimum possible values of variable xi, respectively. EN counts the number of times that the group has been trapped until this step. As the algorithm goes on, the solution gradually converges to the optimum point. Parameters α and β are positive real values. 6. Termination: Steps 3-5 are repeated until maximum number of iterations is satisfied. The pseude code of the algorithm is given in Fig.1 4. DESIGN EXAMPLES 4.1. Welded Beam Design Problem A rectangular cantilever beam made of low carbon steel is selected as first design example. The geometric view and the dimensions of the beam are illustrated in Fig. 2. The beam is designed to carry a certain P load acting at the free tip with minimum overall cost of fabrication. Design variables of the optimization problem can be listed as in the following. 1xh = : the thickness of the weld 2xl = : the length of the welded joint 3xt = : the width of the beam 4xb = : the thickness of the beam Fig. 1 Pseudo code for hunting search algorithm. Fig. 2 Welded beam design. The optimization problem can be stated as follows: Minimize the cost function; )0.14(04811.010471.1)( 2432 2 1 xxxxxxf ++= Subject to: 0)()( max1 ≤−= ττ xxg : shear stress 0)()( max2 ≤−= σσ xxg : bending stress in the beam 0)( 413 ≤−= xxxg : side constraint 05)0.14(04811.010471.0)( 243 2 14 ≤−++= xxxxxg : side constraint 0125.0)( 15 ≤−= xxg : side constraint 0)()( max6 ≤−= δδ xxg : end deflection of the beam 0)()(7 ≤−= xPPxg c : buckling load on the bar Where 4 2''2'''2' )( 2 2)()( τττττ ++= R x x 21 ' 2 xx P =τ       +==τ 2 xLPM, J RM 2'' 2 31 2 2 2 xx 4 xR       + +=                       + += 2 31 2 2 21 2 xx 12 xxx22J 2 344 3 3 3 6)(,4)( xx LPx xxE LPx == σδ ) G4 E L2 x 1( L 36 )xx( E013.4 )x(P 3 2 6 4 2 3 c −= 6 6 max max max 6000 , 14 ., 30 10 , 12 10 13,600 , 30,000 , 0.25 . P lb L in E psi G psi psi psi in τ σ δ = = = × = × = = = The side constraints for the design variables are given as follows: 1 2 3 4 0.1 2.0, 0.1 10, 0.1 10, 0.1 2.0 x x x x ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ The optimum designs obtained by the metaheuristic techniques are tabulated in Table 1. The optimum solution belongs the hunting search algorithm which is 1.724941. This is followed by particle swarm algorithm as 1.724966 and the harmony search algorithm found the third best value as 1.77192. The design histories of each algorithm are shown in Fig. 3. It is clear from Fig. 3 that hunting search has the best convergence rate and the best objective function value obtained is 1.724941. Table 1. Optimum designs of welded beam problem obtained by three metaheuristic techniques. Fig. 3 Design histories of three algorithms for welded beam problem. 4.2 Spring Design Problem Spring design problem aims to minimize the weight of tension/compression spring and is subjected to four constraints. These are minimum deflection, shear stress, surge frequency constraints and limits on outside diameter and design variables. The design problem shown in Fig. 4 has three design variables which are; 1xd = : the wire diameter 2xD = : the mean coil diameter 3xN = : the number of active coils Fig. 4. Spring design. The mathematical model of the problem can be expressed as follows; Minimize; 2 123 )2()( xxxxf += Subjected to; 1,7 1,8 1,9 2 0 50000 100000 150000 200000 250000 B es t f ea si bl e so lu tio n Number of function evaluation Particle Swarm Hunting Search Harmony Search Meta- heuristic Search Techniques Hunting Search Harmony Search Particle Swarm f 1.724941 1.77192 1.724956 x1 0.205731 0.18461 0.205730 x2 3.47112 3.95998 3.471079 x3 9.036624 9.10543 9.036796 x4 0.205730 0.20626 0.205729 5 0 71785 1)( 4 1 3 3 2 1 ≤−= x xx xg 01 5108 1 )(12566 4)( 2 1 4 1 3 12 21 2 2 2 ≤−+ − − = xxxx xxxxg 045.1401)( 3 2 2 1 3 ≤−= xx xxg 01 5.1 )( 21 4 ≤− + = xxxg And side constraints; 152,3.125.0,205.0 321 ≤≤≤≤≤≤ xxx The optimum solutions obtained by there metaheuristic algorithms considered in these review are tabulated in Table 2. It is clear from the table that except harmony search algorithm the remaining metaheuristic algorithms has obtained the optimum solutions that are very close to each other. The best solution for the spring design problem is determined by the hunting search algorithm which is equal to 0.012665. Table 2. Optimum designs obtained by three metaheuristic techniques. Fig. 5 Design histories of three algorithms for spring design problem. 5. CONCLUSION The highly promising outcome of this research suggests that hunting search algorithm is an effective altenative for solving engineering optimization problems. In view of the results obtained, it can be concluded that hunting search algorithm can be extended to other real-world optimization problems in manufacturing and design area. Spring design problem and the welded beam design problem taken from the literature are solved by using three metaheuristic techniques namely harmony search, particle swarm and hunting search and their performance is evaluated and compared. It is noticed that hunting search algorithm shows great performance in finding the solution of design optimization problems. REFERENCES [1] Keane, A. 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Berlin, Germany: Springer. p. 591– 600. [12] Kochenberger, G.A. (2003). Glover F. Handbook of meta- heuristics. Kluwer Academic Publishers. [13] Bonabeau, E., Dorigo M. and Theraulaz, G. (1999). Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press, U.K. [14] Saka, M.P. and Erdal, F. (2009). Harmony Search Based Algorithm for The Optimum Design of Grillage Systems 0,0126 0,0128 0,013 0,0132 0,0134 0 60000 120000 180000 240000 300000 B es t f ea si bl e so lu tio n Number of function evaluation Particle Swarm Hunting Search Harmony Search Meta- heuristic Search Techniques Hunting Search Harmony Search Particle Swarm f 0.012665 0.012835 0.012666 x1 0.051695 0.054665 0.051935 x2 0.356869 0.432354 0.362666 x3 11.27982 7.934532 10.94829 6 To LRFD-AISC. Structural and Multidisciplinary Optimization, 38, 1, 25-41. [15] Saka, M.P. (2007). Optimum Geometry Design of Geodesic Domes Using Harmony search Algorithm. Advances in Structural Engineering, 10, 6, 595-606. [16] Kaveh, A. and Talatahari, S. (2009). Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures. Computers and Structures, 5-6, 87, 267-83. [17] Lee, K.S. and Geem, Z.W. (2004). A New Structural Optimization Method Based on theHarmony Search Algorithm. Computers and Structures, 87, 5-6, 284-302. [18] Omran, M.G.H. and Mahdavi, M. (2008). Global-best harmony search. Applied Mathematics and Computation, 198(2):643–656. [19] Geem, Z.W. (2009). Particle-Swarm Harmony Search for Water Network Design. Engineering Optimization, 41(4):297-311. [20] Hasançebi, O., Erdal, F., and Saka, M.P. (2010). An Adaptive Harmony Search Method for Structural Optimization. Journal of Structural Engineering, ASCE, 136,4, 419-431. 7 CARBON FIBER REINFORCED POLYMER SHEAR STRENGTHENING OF RC BEAMS SUBJECTED TO CYCLIC LOAD Gökhan Şakar Civil Engineering Department Dokuz Eylul University İzmir, Turkey gsakar@hotmail.com Abstract—An experimental investigation was conducted to study the effect of composite carbon fabric shear reinforcement on the ultimate strength and behaviour of reinforced concrete beam. Shear deficient specimens were strengthened by using side- bonded and wrapped CFRP straps. Eight beams were fabricated and tested under the cyclic loads. The main objective of the study is to obtain ductile flexural behaviour from the shear deficient RC beams. To verify the reliability of shear design equations and guidelines, experimental results were compared with all common guidelines and published design equations. Keywords: Shear strengthening, CFRP strip, Cyclic Load, Reinforced concrete beam, Anchorage. 1. INTRODUCTION Many of the earlier studies about shear strengthening were concerned with the proof of effectiveness of CFRP as shear strengthening material. These studies indicated that using CFRP for shear strengthening is an effective and convenient method for improving member’s strength and/or stiffness. In these studies, the behaviour of shear deficient beams strengthened with CFRP was also investigated [1]. The shear capacity of the strengthened RC beam depends on the compressive strength of concrete, the yield strength of shear and longitudinal reinforcement, the shear span to depth ratio and the area of composite fabrics [2, 3]. Bonding is critical for the effectiveness of CFRP strengthening. The effectiveness of strengthening techniques employing FRP relies on the adhesion between the FRP strap/sheet and the concrete surface of the element to be strengthened. One important aspect, peculiar to this technique, concerns the anchorage failure that occurs in a strap/sheet bonded to a concrete surface. Many studies, both theoretical and experimental, have been carried out on FRP-concrete adhesion [4]. For preventing debonding failure, anchorage details must be developed and applied at the end of the CFRP strips. But, limited number of studies is currently encountered about developed anchorage details [5, 6]. However, behaviour of the CFRP was not examined under cyclic load as monotonic loading was applied in all these studies. Experimental results were compared with the analytical approaches that were suggested by ACI-440 Committee Report [7], Concrete Society TR55 [8] and Fib Bulletin 14 [9]. 2. EXPERIMENTAL PROGRAMME Eight rectangular RC cantilever beams with different CFRP orientations were tested subjected to cyclic load. All specimens had the same geometry and materials as shown in Fig. 1. Longitudinal reinforcement consists of four 20 mm diameter steel rebars in the bottom and top of the beam section. Beams had no internal shear reinforcing rebars in the shear span. Stirrups were, however, utilized outside the shear span to block local cracks that might occur under the applied load. A single concrete mix was used for all specimens to achieve a similarity in strength. The concrete was a mixture of water, cement, sand and aggregate with the ratio of 0.68:1:2:3 by mass. The measured average cylinder compressive strengths of concrete cylinder were approximately 25 MPa. The yield stress of the longitudinal reinforcing steel was 414 MPa. Beam-1 was the reference of shear deficient beams, tested without strengthening, and used as a baseline comparison to evaluate the enhancement in strength provided by the strengthening material. The remaining shear deficient RC beams were strengthened with CFRP strips. Beam-2 was strengthened with 50 mm wide CFRP strips spaced at 80 mm. Strips were applied to the both sides of the beam symmetrically. Beam- mailto:gsakar@hotmail.com� 8 3 was strengthened with 50 mm wide CFRP strips spaced at 120 mm. The beam-4 strengthening scheme was identical to beam-3. But the top and bottom ends of each of the strips were anchoraged to the beam with two anchorages. Beam-5 was strengthened with 50 mm wide strips spaced at 160 mm and anchorages were applied to both the top and bottom ends of the beam web. Strengthening scheme of beam-6 was identical to beam-5 but without any anchorage applied for strengthening material. Beam-7 was strengthened with 100 mm wide CFRP strips spaced at 160 mm. Beam-8 was strengthened with 50 mm wide CFRP strips spaced at 120 mm. CFRP was wrapped around the entire cross-section. CFRP strips were applied to both sides for all except beam-8. A A B B Section A-A Section B-B Stirrup: Ø10/110 12 00 1700400 225 757575 50 0 400 Ø 20 Ø 20 V Longitudinal rebars: 10Ø16 Ø 16 Ø 10 Ø 6 a d 5.0 Specimen Beam 1 to 8 a=1600 11 01 10 350 350 325 325 7 7 7 7 Dimensions in mm. 4Ø 20 35 0 d= 32 0 4Ø 20 20 Fig. 1 Reinforcement details of specimen beams. Detail A CFRP Strip Ancorage Details of Specimens Detail A Ø 12 CFRP Strip Ø 10 Steel Bolt 50x50x5 Steel plate Dimensions in mm Beam Section Fig. 2 Anchorage details for specimen beam. 9 Different mechanical anchorage details were developed in the study. Anchorages were included as 50x50x5 mm square steel plates and 10 mm diameter threaded rods with the anchorage details presented in Fig. 2. The location at which threaded rods were to be placed was marked for the anchorage application. The bottom and top marks at the beams’ web were drilled with 12 mm diameter drills up to 50 mm depth. All the drilled holes were cleaned with pressured air. Threaded rods of 10 mm diameter were placed at the bottom and top drilled holes by taking the bonding procedure of CFRP into account. While placing threaded rods, extra care was taken not to damage the continuity of fibre directions. After curing at room temperature, developed anchorage details were performed on the beam specimens. Before applying to composite, some pre- treatment steps for the beams were necessary. First, the beams were sand-blasted to remove outer weak surface of the concrete and then the surfaces were vacuum cleaned to remove loose particles and dust. Epoxy primer was coated all throughout the beam according to the manufacturer’s suggestions. CFRP strips were then placed on the coated epoxy primer and constant pressure was applied on the sheet surface by a roller. Another layer of epoxy was then put on top of the fabric and the extreme resin was cleaned. Specimens were cured for a minimum of 15 days under laboratory conditions. The properties of CFRP and the epoxy resin are presented in Table 1. Table 1. Properties of CFRP and Resin. 3. TEST SETUP All cantilever specimens were tested under cyclic load. To perform cyclic load to the specimen, a loading column was designed with hinges by the beam’s free end. The loading column contained two hinges, a load cell and a hydraulic jack. The capacity of the hydraulic jack was 500 kN while the load cell’s capacity was 400 kN. Load was applied in cycles of loading and unloading. The same loading cycles were applied to all specimens. Loading was increased up to the yield of the flexural reinforcements or until the failure of the specimen. For each load increment, displacements, loads and strains were recorded simultaneously by means of an automatic data acquisition system. Four linear variable differential transformers (LVDTs) were used to monitor displacements. The LVTDs are located at the end of the beam for maximum displacement, under the rigid support to calculate the undesired displacement and finally on the rigid support to calculate the rotation. Strain gauges help to determine the shear cracks before propagation with the help of increase in strains. Strain gauges were attached to CFRP strap along the fiber direction and were attached to straps that were situated at distances between 80 to 1600 mm apart from the rigid support. Eight strain gauges were used for each specimen. 4. TEST RESULTS All strengthening types significantly increased the shear strength of the beams. The first crack always appeared as a flexural crack at the maximum bending moment region for all beams. Shear cracks followed the flexure cracks at the unstrengthened part of the specimens i.e. between CFRP strips. Later, shear cracks developed towards the strips due to the increasing load. Because of the different strengthening scheme, first flexure crack propagation was observed between 34 kN and 40 kN load level for all specimens. 5. EVALUATION OF TEST RESULTS The test results pointed out that CFRP is an efficient material to increase the shear capacity of RC beams. Table 2 summarizes of the results of specimens during test. Comparing to the reference beam, a minimum increase of 20% and a maximum of 75% in shear capacity was acquired from the strengthened specimens. Beam-2 showed 75% more strength than beam-1. Beam-3 showed 23% more strength than the reference beam. By decreasing the amount of the strengthened area (increasing the spacing of CFRP strips from 80 to 120), a decrease of 42% at contribution to the shear resistance was obtained according to beam-2. The beam-4 was strengthened in a manner similar to that of beam-3. The ends of the strips in beam-4 were anchoraged. Apart from beam-3, beam-4 had reached 100 kN loading cycle but failed in 83 kN load. The specimen showed 9% more strength than beam-3. Anchorages did not provide any substantial increase in shear strength but they obtained remarkable increment in the displacement capacity of beam-4. Properties of CFRP Remarks of SikaWrap 230C Fiber Orientation 00 (unidirectional) Construction 99% Wrap, 1% weft Thickness (mm) 0.12 Tensile Strength (MPa) 4 100 Elastic Modulus (MPa) 231 000 Ultimate Tensile Strain (%) 1.7 % Properties of Resin Remarks of Resin Tensile Strength (MPa) 30 Elastic Modulus (MPa) 3800 10 Table 2. Experimental Results. *Forward loading step was described with positive mark and backward loading step was described with negative mark in Table Table 3. Comparison of Experimental and Analytical Results. Specimen Wf (mm) Sf (mm) Anchorage Arrangements Failure Load (kN) Maximum load at the forward cycle Maximum load at the backward cycle Failure Mode at Ultimate Beam- 1 Control ------ ------ ------ ------ 61.7 62.4 -63.1 Shear Beam- 2 Strengthening 50 80 ------ Side bonding 108.0 108.0 -99.2 Shear Beam- 3 Strengthening 50 120 ------ Side bonding -76.1 92.5 -89.0 Shear Beam- 4 Strengthening 50 120 Yes Side bonding -83.0 99.6 -89.7 Shear Beam- 5 Strengthening 50 160 Yes Side bonding -74.0 89.5 -79.8 Shear Beam- 6 Strengthening 50 160 ------ Side bonding 84.0 84.0 -80.1 Shear Beam- 7 Strengthening 100 160 ------ Side bonding -85 89.5 -85 Shear Beam- 8 Strengthening 50 120 ------ Wrapped 102.0 102.0 -100.6 Flexure Specimen Experimental Strengths Calculated Strengths for ACI-440 Calculated Strengths for Concrete Society, TR55 Calculated Strengths for Fib Bulletin 14 Experimental/calculated Vexp. (kN) Vcal. (kN) Vcal. (kN) Vcal. (kN) Vexp./Vcal. for ACI-440 Vexp./Vcal. for TR55 Vexp./Vcal. for Fib Beam-2 108 66.15 114.15 77.29 1.63 0.95 1.40 Beam-3 76 60.32 93.89 73.81 1.26 0.81 1.03 Beam-4 83 61.04 95.10 74.53 1.36 0.87 1.11 Beam-5 74 53.21 76.83 67.19 1.39 0.96 1.10 Beam-6 84 52.16 75.21 66.08 1.61 1.12 1.27 Beam-7 85 68.60 80.87 79.53 1.24 1.05 1.07 Beam-8 102 62.84 128.41 84.59 1.62 0.79 1.21 11 Beam-5 had more strip space than aforementioned strengthened specimens. Hence, the least strength improvement was observed at that beam among the strengthened specimens. It was showed 20% more strength than the control beam. Beam-5 was showed 14% less strength than beam-6 that was strengthened with same CFRP arrangement without anchorages. This result might be arising from abrupt debonding failure of the specimen. Beam-6 showed 36% more strength than beam-1. It was minimum displacement capacity beam of the strengthened specimens. Beam-7 was nearly the same shear capacity as beam-6. The specimen had double strip width as to beam-6 but that application did not improve the shear strength. As a result of the use of wrapping, a significant increase in the shear capacity was achieved in beam-8. Furthermore, the failure mode at ultimate changed from CFRP debonding to flexural failure mode. 5.1 Comparison of Test Results and Design Equations Results from both analytical and experimental study are shown in Table 3. The analytical shear force capacities of specimens were calculated according to guidelines. For all these methods, shear strength of a strengthened RC section could be expressed as the sum of the three components. Vn = Vc + Vs + Vf (1) Where, Vc is the contribution of concrete, Vs is the contribution of internal steel shear reinforcement and finally Vf is the contribution of CFRP in equation (1). Concrete and/or steel stirrups can not utilize their ultimate strengths when CFRP is used as a strengthening material, as demonstrated in equation (1). Common design codes, however, still use the relative contribution of these materials to compute the total shear strength. It must be considered that these parameters interact with each other. For this reason, analytical and experimental results must be evaluated according to the aforementioned considerations. Differences were determined between analytical shear load capacities that were suggested by ACI-440 committee report, The Concrete Society TR55, Fib Bulletin 14 and experimental results. The reasons for obtaining different results in analytical calculation and experiments from the expected were that the guidelines did not consider a/d ratio and the effects of anchorage. The analytical shear force capacities of beams were calculated according to ACI-440 committee report, Concrete Society TR55 and Fib Bulletin 14 suggestions. Maximum divergences between analytical and experimental shear load capacities were calculated from ACI-440 results. Moreover, the Concrete Society TR55 calculated closer results for the specimens. The closest results of the experimental programme were obtained for beam-7. The shear capacity of beam-7 was 24% less than that of the experimental result for ACI-440 and 5% and 7% less than The Concrete Society TR55 and Fib Bulletin 14, respectively. The experimental and analytical results were not too close. Thus, the impact of the ratio a/d, the anchorage effect and the interaction of internal shear reinforcement to shear capacity due to CFRP were not included in the guideline. The proposed design results would be more realistic if the unconsidered parameters were measured while improving the preceding equations. 6. CONCLUSIONS Strengthening of reinforced concrete beams in shear using CFRP straps appears to be a highly efficient technique. Developed anchorage detail worked efficiently under the cyclic load. Anchorages prevented the peeling of the CFRP from concrete. However, bottom anchorages were efficient as they prevent the premature peeling of the CFRP. All CFRP applications improved the strength and behaviour of the specimens in a different way. The strength was increased between 1.20 and 1.75 times compared to reference beam. Besides the displacement capacity of the strengthened beams was improved meaningfully. In addition, Beam-8 that was strengthened with the wrapped CFRP strips behaved like a ductile flexural beam. Significant differences were determined between the analytical shear force capacities that were suggested by ACI-440 committee report, The Concrete Society TR55, Fib Bulletin 14 and experimental results. The reasons for obtaining different results in analytical calculation and experiments from the expected were that the guidelines did not consider a/d ratio and the effects of anchorage. ACKNOWLEDGEMENT The author wishes to thank Dokuz Eylul University for supporting this project. The study was also supported by Afaprefabrik Precast Concrete Ind. & Trade Co., Inc., Izmir, Turkey. REFERENCES [1] Sakar, G., Tanarslan, H. M., Alku, O.Z., An Experimental Study on Shear Strengthening of RC T-section Beams with CFRP Plates Subjected to Cyclic Load, Magazine of Concrete Research, 61/1, (2009), 43-55. [2] Triantafillou, T.C. “Shear Strengthening of Reinforced Concrete Beams Using Epoxy-Bonded FRP Composites”, ACI Structural Journal, 95/2, (1998), 107-115. 12 [3] Swamy, R. N., Mukhopadhyaya, P., Lynsdale C. J., “Strengthening for Shear of RC Beams by External Plate Bonding”, Structural Engineering, 77/12, (1999), 19-30. [4] Fib, “Retrofitting of Concrete Structures by Externally Bonded FRPs, with Emphasis on Seismic Applications”, Fib Bulletin 35, (2006), Technical Report. [5] Khalifa, A., Alkhrdaji, T., Nanni, A., Lansburg, A. “Anchorage of Surface Mounted FRP Reinforcement”, Concrete International, ACI, 21/10, (1999), 49-54. [6] El-Mihilmy, M., Tedesco, J., “Prediction of anchorage failure of RC beams strengthened with fiber-reinforced polymer plates”, ACI Structural Journal, 98/ 3, (2001), 301-314. [7] American Concrete Institute, “Guide for the Design and Strengthening of Externally Bonded FRP Systems for Strengthening Concrete Structures”, ACI, Detroit, (2001), ACI Committee 440 Report. [8] Concrete Society, “Design Guidance for Strengthening Concrete Structures Using Fibre Composite Materials”, The Concrete Society, Camberley, (2000), Technical Report 55, UK. [9] Fib, “FRP as Externally Bonded Reinforcement of RC Structures: Basis of Design and Safety concept”, Fib Task Group 9.3, Fib Bulletin 14, International Federation for Structural Concrete, Switzerland, (2001). 13 STRUCTURAL PRECAST CONCRETE IN TURKEY: DEVELOPMENT AND EARTHQUAKE PERFORMANCE IN LAST DECADES Hilal Meydanli Atalay Department of Civil Engineering, Kocaeli University, Kocaeli, Turkey, hilal.meydanli@kocaeli.edu.tr Şevket Ozden Civil Eng. M.Sc., Ph.D., Istanbul, Turkey sevketozden@yahoo.com Erkan Akpınar Department of Civil Engineering, Kocaeli University, Kocaeli, Turkey, erkan.akpinar@kocaeli.edu.tr Hakan Erdoğan Department of Civil Engineering, Kocaeli University, Kocaeli, Turkey, hakan.erdogan@kocaeli.edu.tr Abstract—In recent decades, the precast concrete structures become increasingly popular in Turkey. Besides its cost efficient and time-saving nature, sufficient earthquake resistance should be provided at the design stage for proper and safe utilization of those structures. For this purpose, prime importance should be given to joint locations such as beam- column connections where considerable amount of energy dissipation is required to ensure earthquake resistance. In this paper, the development of the structural precast concrete in Turkey over the last decades is discussed with special emphasis on the research and developments in the area of beam-to- column connections. In addition, the performance of the existing precast concrete structures in recent earthquakes is also reported in order to highlight the significance of know-how transfer to avoid the repetitive errors both at design and construction stages. Keywords: Precast concrete structure, Beam-column connection, Seismic performance. 1. INTRODUCTION Structural precast concrete technology provides various advantages compared to the conventional cast in- place construction method. These well-known advantages may be listed as the minimized construction duration, higher compatibility with the architectural details, lower labor and material cost due to mass production process in a factory or a manufacturing site, reduced risks caused by the local climatic conditions by spending less time in construction site, easier and reliable quality-control process at prefabrication stage compared to in-situ construction [1]. In this technology, the structural elements of load-bearing system such as columns, beams and slabs were first produced and stocked up in a factory plant. Afterwards, the prefabricated structural elements were transported to construction site and assembled. The most critical requirement of this construction method is the specially automated system of industrial plant, since the design and production stages have to be well- organized together in an efficient manner. Precast concrete structures were classified into two main groups according to their load bearing systems named as frame and panel systems respectively. Frame systems generally consist of beams, columns and floor elements whereas panel systems composed of wall and floor panels. The systems constructed using both frame and panel systems were called as composite systems [2]. The structural performance of precast concrete systems generally depends on the connection types between the precast concrete members [3]. Pin connected precast concrete systems were mostly preferred for industrial one-story structures with large girder span lengths. The well-known drawback of such systems is the non-moment resisting behavior that leads to unrepairable damages under lateral loading such as earthquake. On the other hand, moment resisting frame connection types may be divided into two groups namely dry and wet connections [4]. In the early years of application, precast concrete structures were generally used in low seismicity regions. By time, these type of structures spread through the seismically active regions depending on the growing interest. Accordingly, the 14 rigidity, strength, ductility and energy dissipation capacities of the structures become more critical in the design of such structures. For this purpose, the studies in the literature generally focused on obtaining a behavior similar to monolithic frame systems without sacrificing strength, ductility and energy dissipation capacities at the connection regions [5]. In this study a detailed review about the development and improvement of precast concrete systems in Turkey was presented and the performance of these systems under recent earthquakes is discussed. 2. PRECAST CONCRETE STRUCTURES IN TURKEY Manufacture of State Fez building is known as the first structure that was built by precast technology in Turkey. The structure was built in 1868 and having 8.000 m2 closed area. The load- bearing system of this building was constructed with shed roof sitting on thin walled cast iron cylindrical columns. Bulgarian Orthodox Church between Balat and Fener districts in Istanbul was reported as the first full prefabricated structure of the Ottoman period. Those buildings were built by steel precast structural systems due to poor soil conditions. The structural members were connected with bolts, nuts, rivets and welding. In 1955, first prefabricated concrete structural elements, (centrifugal concrete poles used for street lightning purposes) were manufactured in order to balance the overproduction of the cement plants in Turkey. This may be accepted as the pioneering investment in concrete prefabrication industry in Turkey. Later on, first precast concrete structural constructions were built as one-story school buildings in the first half of 1960’s. These buildings were constructed by using semi-precast systems. First, the surrounding walls were constructed. Afterwards, 7.20m wide span was covered by truss roof system that consists of iron rods and concrete top chord. The prefabricated roof panels were placed on the roof truss. A total of 234 school buildings were constructed in same manner in three months time and opened to public service (Fig. 1). Meanwhile, the residential houses of Ereğli Iron Steel Factory were constructed by a well known precast concrete panel system called Larsen-Nielsen System that was developed in Denmark in 1948. The structural components of this system were wall and slab panels. The primary objective of Larsen- Nielsen System is to minimize in-situ construction process. The wall and floor panels were first assembled by bolt connections and then those connection regions were filled with mortar to ensure continuity which is a good example of wet type of connections. Total of 446 apartment were constructed by using mentioned method in one year time. It is possible to register the year 1961 as the birth of precast concrete were using in public buildings and residence (Fig. 1). Fig. 1 School buildings and the residency of Ereğli Iron Steel Factory [6]. The utilization of precast concrete systems in construction of industrial buildings has been started in the second half of 1960s. The industrialization thrust in those years triggered the heavy and urgent demand for rapidly constructed industrial structures for storing and sheltering the machinery safely. The rapid construction of those buildings has prime importance due to loan cost of the machinery. In 1968, The Ministry of Industry decided to establish small industrial sites at different districts to promote industrial development. In the late 1970s, Tekel tobacco warehouse was built by fully prefabricated system from foundation to roof. A total of 32 dormitories were built with prefabrication between the years of 1984 and 1989. These buildings were constructed with either panel 15 system consisting of hollow core slabs and wall panels or frame system consisting of prefabricated column, beam and double tee flooring unit. The popularity of precast concrete systems in construction of residential buildings increased significantly in 1980s. Bingöl earthquake houses and Kocaeli immigrant houses were the significant examples of residential buildings constructed by precast concrete systems in 1980s (Fig. 2). In 1990s, technological improvements in conventional methods such as widespread production and use of ready mixed concrete with increasing number of ready mix concrete plants and achievement of tunnel formwork concrete construction suddenly paused the growing interest in prefabrication especially for residential buildings [7]. Turkish Precast Concrete Association was founded in 1984 with the collaboration and supports of manufacturer companies. Beyond its representative role in order to promote the use of precast concrete, Fig. 2 Bingöl earthquake houses [6]. Turkish Precast Concrete Association also put emphasis on the research and development activities, technical publications and regulations, etc. TS 9967 (1992) regulation including analysis principles and manufacturing and assembly rules of reinforced concrete prefabricated structural elements and structural systems was published by Turkish Standards Institute with the contributions of Turkish Precast Concrete Association [8]. The restrictive and router rules in design of precast structures were explained in Turkish Earthquake Code that activated in 1998 (TEC, 1998). The guidelines about precast concrete structures were then modified in Turkish Earthquake Code 2007 (TEC, 2007) according to performed research and development by different institutes. 3. RESEARCHES ON PRECAST CONCRETE SYSTEMS IN TURKEY Precast concrete systems are most preferred construction method for industrial buildings, 85 percentages, in Turkey. It is vise versa in the case of residential structures, only 4 percentages [9]. Typical industrial structures are single-story and consist of simple portal frames of which connections fixed at the bottom side and hinged at the top side. Such precast framing systems which have hinged beam-to-column connections could be used in multi-story structures in case of that seismic load would be carried by cast in place reinforced concrete walls. Moreover, precast concrete structures with moment resisting connections are applicable for earthquake prone regions. It should be proven that such connections could provide equivalent strength and ductility as monolithic connections under reversed cyclic loading [10]. Basically, there are two methods to build moment resisting connections for precast concrete systems namely dry connection and wet connection. Outcomes of the experimental studies in Turkey on moment resisting connections for precast concrete beam-to-column joints are summarized in this section. 3.1 Dry Connections In application of typical dry connections, steel rods and plates are embedded at both ends of precast beams during casting in order to connect those beams to columns by bolts or welding in construction site. Most of the time welding was preferred compared to bolted connections. Another type of dry connections was implementation of post tensioned steel for assembling purposes [4]. 3.1.1 Welded connection The first comprehensive research program that focused on the moment-resisting connection of precast structures in Turkey was performed in the Structural Mechanics Laboratory of METU. Precast beam-column connections, designed in multi-storey buildings located in seismic area, were modified to resist for earthquake loading [3]. Later on, the dry beam-column connections located away from column face were tested by Ersoy and Tankut. The connection between the column bracket and the precast beam consisted of steel plates, those were welded to plates embedded in the members on both top and bottom faces. According to test results, a modification was proposed to obtain a behavior similar to monolithic connections. In this modification, double steel plates were used on both top and bottom faces for moment transfer and additional double plates were used on both sides to provide shear transfer. The test results indicated remarkable increases in terms of strength, rigidity and energy dissipation capacity of the specimens [11]. Korkmaz and Tankut investigated the behavior of moment resisting precast concrete beam-to-beam connections under reversed cyclic loading. The connections consist of a middle 16 precast beam that is placed on a cantilever beam connected to the column. Within the connection region, the continuity of the bottom reinforcement was provided by welding them to the steel plate at the junction of middle and cantilever beams. On the other hand, continuity of top reinforcement was satisfied by lap-splicing and in-situ concrete casting. However, the performance of those specimens was questionable. Therefore, the insufficient anchorage due to lap-splicing on top was eliminated by welding the reinforcing bars. The test results indicated that the quality of workmanship in welding the bars has prime importance to avoid premature failure in the vicinity of welded region [12]. 3.1.2 Bolted connection Ertas et al. [13] developed a special bolted connection detail to minimize the duration of assembling in site. The connection detail consisted of rectangular steel boxes allowing dimensional advantages during construction. This detail was proposed especially for short span and low level of shear forces formed by vertical loads. In addition, steel plates were placed at the top and bottom of the beam section to delay crushing of the concrete adjacent to the column surface. Steel bars were welded around the steel boxes and rods passing through the box section to eliminate any possible sliding of the steel boxes with respect to concrete beam. Bolts were pre-tensioned during assembly of connection and specimens were tested under reversed cyclic loading. The test results revealed that the performance of bolted connection is better than monolithic connection in terms of strength, ductility and energy dissipation capacity. 3.1.3 Post-tensioned connection Post-tensioned connection detail was first developed and tested by Pınarbaşı in Turkey. The effect of post tensioning steel ratio on behavior of precast connections was investigated. The test results indicated significant increase in strength, ductility, rigidity and energy dissipation capacity by application of post-tensioning[14]. Another post- tensioned connection detail was tested by Kaya and Arslan. The effect of diameter of post-tension strands was investigated in their experimental study [15]. Ozden and Ertas also proposed a special hybrid connection to improve the moment resistance of precast beam-column connections. The main objective of the study was to investigate the effect of mild steel reinforcement ratio on behavior of post-tensioned precast concrete connections. The performance of hybrid connections improved significantly by increase of mild steel reinforcement ratio such that the capacity of companion monolithic subassembly was almost reached in terms of strength, stiffness and energy dissipation [16]. 3.2 Cast in Place Connections The most common moment resisting connection detail using in precast concrete structures is cast in place connection in other words wet connection. The well-known type of wet connection is composite connection of which reinforcement continuity provided by welding or bolts. In general, tension due to positive moment is transferred through the welding or bolts while tension due to the negative moment is transferred through reinforcing steel bars longitudinally placed in cast in place concrete. It is revealed that adequate performance parameters such as strength, ductility and energy dissipation capacity can be provided with the composite connections. One of the well- known study was conducted by Ersoy and Tankut. The flexural reinforcement at the bottom of the connection was supplied by welding of steel plates embedded into joining portions of the beam and the corbel on the column. Besides the positive findings obtained from the tests, it was revealed that the observed damage accumulates in the connection region even though the connection is designed to be stronger than the joining members [17]. The composite connection is the most preferred cast in place connection type used in Turkey as basically described above. The composite connections was also examined in the study done by Ertas et al.. It was concluded that the viability of the connection in seismic areas were provided in terms of strength and energy dissipation capacity while ductility of connection was less than that of the monolithic one. It was thought that the low ductility level of the composite connection was caused by the adverse and overturning effect of the welding process on material characteristics [13]. In addition to composite connection, there are several types of cast in place connections. It is known that the connection region could be arranged in different locations in beam-to-column joint even outside of the joint zone. Two different cast in place connection techniques were summarized and tested in the research which was carried on by Ertas et al.. There is not any cast in place topic concrete along the upper side of beams in those wet connection types namely cast in place in column and cast in place in beam. Only concrete casting process is limited in either joint zone in column for "cast in place in column connections" or joining end of the beam for "cast in place in beam connections". The precast concrete beams protruding U shaped reinforcing bars which serves as both positive and negative flexural reinforcement at the connection region considering the anchorage issue are used in the either techniques. There is a gap of which height is precisely equal to beam depth at mid- height of precast concrete column so that precast 17 concrete beam can be easily seated and aligned in the assembling process in the cast in place in column connections. On the other hand in the cast in place in beam connections, there is no gap inside the column and U shaped reinforcing bars protruding from the column like joining beams are placed at the mid-height of the precast columns. Single leg ties are placed into joining zone and steel fiber reinforced concrete is used for filling the empty space in connection region. It is recommended that both precast concrete connections are suitable for high seismic areas in terms of strength properties and energy dissipation [13]. 4. EARTHQUAKE DAMAGE IN PRECAST CONCRETE STRUCTURES Severe and catastrophic earthquakes occurred frequently in Turkey as well as in the world in last decades. 92% of Turkish territory is located in a seismically active region which means that 95% of the total population and 98% of industrial facilities are placed in disaster prone and hazardous regions. Therefore excessive damage was observed in earthquakes occurred in Turkey and such events result in lots of fatalities and breakdown in industrial production [18]. The precast concrete structures exist in Turkey can be classified in three different categories which are panel systems, pin connected frame systems, moment resisting frame systems. The concrete precast structures exposed to Ceyhan Earthquake in 1998 for the first time in Turkey. 1999 Marmara Earthquakes and 2011 Van Earthquake caused to different level of damage on the precast structures as well. Kocaeli immigrant houses and dormitory building have been built as concrete precast panel system which consists of slab panels and wall panels. There was not any significant damage in these buildings after Marmara Earthquakes in 1999. It was concluded that the buildings were fully operational and occupied after the earthquakes (Fig. 3). Fig. 3 Kocaeli immigrant houses. Majority of the single-story precast industrial buildings are constructed with pin connections at beam-to-column joints. Severely damage was observed in these kinds of buildings in Marmara Earthquakes in 1999 [19, 20]. According to the field studies after the earthquakes, it was concluded that the main reason of the encountered excessive damage was inadequate and improper pin connection production in assembly phases of the prefabricated structural members. Encountered pin connection failures were caused by lack of transverse reinforcement around the pin holes, low strength and poor quality mortar usage for filling the holes and no washer, nut or welding detail consideration at the extruded ends of pins. It should be mentioned that overmuch lateral tip displacements of the columns in such structures due to improper cross-sectional design and lack of lateral rigidity also resulted in overloading and poor behavior at connections. Similar damage pattern was observed in only three pin-connected precast industrial structures which were under construction during the Van Earthquake in 2012. However, the level of damage was minimum in most of the precast industrial buildings. It is considered that the existence of external walls reduce the inter-storey drift, while the metallic roof cover results in a sort of diaphragm action (Fig. 4). There were several examples of multi-story precast structures which have moment resisting connections subjected to recent earthquakes in Turkey. Dormitory building in Bolu could be given as an example of this kind of precast structures (Fig. 5). Fig. 4 Undamaged Industrial building in the Van earthquake. Fig. 5 Dormitory building in Bolu. 18 Moment resisting connections provided by welding steel plates which already placed at the contact points of the prefabricated beam and column at the bottom of the connection and additional longitudinal reinforcement passing through the connection region placed into cast in place topic concrete at the top of the connection in general. There was not any collapsed precast building which has moment resisting connections in Marmara Earthquakes in 1999. However, slight to moderate damages i.e. cracking in walls, plastification at end of structural members were observed such structures during the site surveys (Fig. 5). Another way to build a moment resisting connection for precast systems is introducing the post-tensioning technique to the beam-to-column joints with special constructional details. During the site survey in Van Earthquake in 2012, such a multi-story concrete precast building with post- tensioned moment resisting connections examined closely. There were not any damage in precast structural members, the post tensioned connections and post tensioning ducts. In addition, flexural or shear cracks did not occur in column to foundation connections. On the other hand, shear cracks took place on some of the inner and outer walls due to incompatible displacement characteristics of the structural system and the walls (Fig. 6). Fig. 6 Residential building in Van. 5. CONCLUSION Prefabrication which is simply assembling of the precast concrete members in site and quick way to construct a concrete structure is a well known construction technology all over the world. The application area of the concrete precast systems got wider in Turkey last decades. Research issues and development are still in progress for the earthquake performance of precast systems. It is obvious that the performance of the connections dedicates the performance of the prefabricated structure since the structural members are produced individually in the factory under quality assurance. Several research projects were done in Turkey on performance and deficiencies of precast connection resulted in detailed and reliable connection design criteria. Precast concrete systems are generally used for construction of industrial type of structures especially in Turkey. Most of these structures are built as single story with 6-9 meter height and composed non-moment resisting connections which constructed with pin connections which transfer only shear and axial loads among the structural elements. On the other hand, few examples of concrete precast multi-story structures which have moment resisting connections are in service for residential purposes. It is observed that these buildings were achieved quit good performance in the past earthquakes in Turkey. Even though, earthquake resistant and easily applicable moment resisting connection details should be investigate to build fast and reliable residential multi-story buildings in earthquake prone areas in Turkey. In fact these kinds of studies on concrete precast systems are in progress almost all over the world, the comprehensive multi-partner research project of which actors should be government, universities and construction industry must be conducted for development of more reliable moment resisting connection details for earthquake resistant precast structures. It should be a lead to advance of both national and international knowledge about the precast systems but also result in widespread usage of the faster and safer construction method in earthquake prone regions. REFERENCES [1] Gül Polat and Atilla Damcı, "Factors affecting the use of precast concrete systems in the Turkish construction sector", 4th Construction Management Congress, October 2005, Istanbul, Turkey. [2] Y. Ayaydın, Prefabricated Structural Wall Structures, Yılmaz Offset Printing, 1987, Istanbul, Turkey. [3] Ersoy, U., Tankut, T., Ozcebe, G., "Seismic Behavior of Precast Concrete Connections", TUBITAK Report No. INTAG-504, 1997, Ankara, Turkey. [4] Hakan Ataköy, "Prefabricated structural system connection techniques", Workshop of Prefabricated Structures and Earthquake, Turkish Precast Concrete Association, 1998, Ankara, Turkey. [5] Stanton, J. F., Havkins, N. M., Hicks, T. R., "PRESSS Project 1.3: Connections classifications and evaluations" PCI Journal, Vol. 36, No.5, 1991, pp. 62-71. [6] Sinasi Acar, History of Concrete Prefabrication in Turkey, Turkish Precast Concrete Association, 2006, Ankara, Turkey. [7] Turkish Precast Concrete Association, Ankara, Turkey. [8] M. Tuncag, "TS 9967: The scope of the preparation of standard related to prefabricated buildings", Turkey Engineering News, Vol. 371, 1995, pp. 58-59. [9] H. Ataköy, "17 August Marmara Earthquake and the precast concrete structures built by TPCA Members" Concrete Prefabrication, Vol.52-53, 2000, pp.5-14. 19 [10] Turkish Earthquake Code (TEC). Regulations on structures constructed in disaster regions. Ministry of Public Works and Settlement, Ankara; 2007. [11] Ersoy, U. and Tankut, T., "Precast concrete members with welded plate connections under reversed cyclic loading", PCI Journal, Vol.38, No.4, 2003, pp. 94-100. [12] Korkmaz, H. H. and Tankut, T., "Performance of a precast concrete beam-to-beam connection subject to reversed cyclic loading", Engineering Structures, Vol.27, No.9, 2005, pp. 1392–1407. [13] Ertaş, O., Özden, Ş., Özturan, T., "Ductile connections in precast concrete moment resisting frames", PCI Journal, Vol.51, No.3, 2006, pp.2-12. [14] S. Pınarbaşı, "Development and seismic performance of a precast concrete beam-column connection by post- tensioning", M.S. Thesis, METU, 2000. [15] Kaya, M. and Arslan, A.S. "The effect of the diameter of prestressed strands providing the post-tensioned beam-to- column connections, Materials & Design, Vol.30, No.7, 2009, pp. 2604-2617. [16] Ozden, S. and Ertas, O. "Behavior of unbonded, post- tensioned, precast concrete connections with different percentages of mild steel reinforcement", PCI Journal, Vol.52, No.2, 2007, pp.32-44. [17] Ersoy, U. and Tankut, T., "Seismic behavior of a precast beam-column concrete connection used in Turkey", Concrete Prefabrication, Vol.22, 1992, pp. 5-15. [18] Adalier K. and Aydingun O., "Structural engineering aspects of the June, 1998 Adana–Ceyhan (Turkey) earthquake", Engineering Structures, Vol. 23, 2001, pp. 343–55. [19] Ozden, S. and Meydanlı, H., "Seismic response of pre- cast industrial buildings during 1999 Kocaeli Earthquake", Skopje Earthquake 40 Years of European Earthquake Engineering, August 2003, Skopje, Macedonia. [20] Posada, M. and Wood, S.L., "Seismic performance of precast industrial buildings in Turkey",http://www.nd.edu/~linbeck/wood1.pdf. http://www.nd.edu/~linbeck/wood1.pdf� 20 THE HOMOTOPY PERTURBATION METHOD FOR COMPLEX LINEAR SCHRÖDINGER WAVE EQUATIONS Semih Küçükarslan Department of Civil Engineering İstanbul Technical University İstanbul,Turkey kucukarslan@itu.edu.tr Mehmet Ali Karaca Mathematics Department İstanbul Technical University İstanbul,Turkey karacam@itu.edu.tr Abstract—In this article, the analysis of the complex partial wave equations is presented by using the Homotopy Perturbation Method (HPM) and least square method. For this investigation, firstly complex linear Schrödinger equation and then complex nonlinear Hirota equation is studied. Keywords: Dynmacis of structures, Homotopy perturbation method (HPM). 1. INTRODUCTION The analytical and/or numerical solutions of the linear and nonlinear partial differential equations (PDE) are required in the many branches of the science and engineering fields. Main purpose is to obtain first an analytical solution. If it is not possible, many researcher tries to find an accurate numerical solution. In the last two decades, researches progresses with this direction since analytical solutions to NPDEs are limited or sometimes unavailable. Complex varied linear and nonlinear PDEs are also interest of these fields. In the past, the traveling wave solutions of linear and nonlinear PDEs such as Schrödinger [1] and Hirota equation [2,3] is obtained by Fan [4] for Hirota equation. The mathematical model equation for the Schrödinger and Hirota equation is given in the following form, respectively. 0t xxiU U+ = , (1a) 2 22 6 0t xx xxx xiU U U U i U i U Uα α+ + + + = (1b) The Homotopy Perturbation Method (HPM) is proposed by He [5] and further improved by He in [6-10] for better results. It provides a fast convergence for the studied problems. It shows a good accuracy and a fast convergence to the solutions of the linear and nonlinear partial differential equations. In this article, the HPM is applied for an accurate numerical solution of Schrödinger and Hirota equations. The efficiency and convergence of the numerical results are compared with exact ones. 2. HOMOTOPY PERTURBATION METHOD One considers the following nonlinear partial differential equation to represent the algorithm of the HPM, ( ) 0Au f r− = , r ∈ Ω (2) with the boundary conditions of ( , ) 0uB u n ∂ = ∂ , r ∈ Γ (3) where A and B are general differential operator and boundary operator, respectively. Γ is the boundary of the domain Ω , and ( )f r is a given and known analytical function. After dividing the general operator into linear part (L) and nonlinear part (N), one can write the equation (2) as ( ) 0Lu Nu f r+ − = (4) By constructing the homotopy procedure to equation (4), one can rewrite a homotopy in the form 0( , ) (1 )[ ( ) ( )] [ ( ) ( )] 0H V p p L V L u p A V f r= − − + − = [0,1]p ∈ , r ∈Ω (5) where [0,1]p ∈ is an embedding parameter, 0u is an initial approximation of the equation (2) which satisfies the equation (3). In the HPM, one use the embedding parameter as a small parameter. Thus, the solution of equation (5) can be given a power series of p in the form, 2 1 2 ...oV V pV p V= + + + (6) 21 By giving 1p = , one can get an approximate solution of the equation (2) as, 1 21 lim ...op u v V V V → = = + + + (7) The combination of a small parameter with a homotopy is called homotopy perturbation method. 3. EXAMINATION OF COMPLEX LINEAR SCHRÖDINGER EQUATION WITH HPM To test the accuracy of the proposed method, a comparison will be done by using two-dimensional finite elements having 4 nodes rectangular elements. In this part, the numerical solution of the Schrödinger equation [1] is examined by using the HPM. By selecting U u iv= + , one can separate equation (1a) into real and imaginary parts. Thus, one gets (1+1) dimensional coupled system in the following 0t xxv u− = 0t xxu v+ = (8) The exact solutions to equation (8) are given [1] by 2 1 2 2( )u c Cos c x c t= − 2 1 2 2( )v c Sin c x c t= − (9) (0, ) 0xu t = , (1, ) 0tu t = , ( ,0) 0tu x = , ( ,1) 0tu x = , (0, ) 0v t = , (1, ) 0tv t = , ( ,0) 0tv x = , ( ,1) 0tv x = for 0 1x≤ ≤ , 0 1t≤ ≤ . where 1 2,c c are any arbitrary constants. For the simplicity, 1 2 1c c= = are used in the analyses. Let us assume an initial trial function as 0 ( , ) ( )ou u x t Cos ax= = and 0 ( , ) ( )ov v x t Sin bx= = for the equation (9) that satisfies the given boundary conditions and a, b are the unknown constants to be determined optimally. Substituting equation (6) into equation (9) and using equation (5), one gets a system of equation with n+1 terms that needs to be solved simultaneously. Since computations are dependent on the value of ou and ov , a minor modification gives flexibility to choose the initial ou [10,12,13]. For this purpose, the following homotopy is constructed 0 0( ) ( ) 0t t xx tw u p r u− + + = (10) 0 0( ) ( ) 0t t xx tr v p w v− + − + = where 2 0 1 2 ...w w pw p w= + + + and 2 0 1 2 ...r r pr p r= + + + The variables u and v can be obtained similar to equation (7) as 1 21 lim ...op u w w w w → = = + + + 1 21 lim ...op v r r r r → = = + + + (11) After expanding with powers of p the equation (10) for n=3, i.e. the third power of p, one gets 0 :p 0 0t tw u= , 0 0t tr v= 1 :p 1 0 0 0t xx tw r u+ + = 1 0 0 0t xx tr w v− + = 2 :p 2 1 0t xxw r+ = 2 1 0t xxr w− = 3 :p 3 2 0t xxw r+ = (12) 3 2 0t xxr w− = The analytical solution of the equation (12) can be obtained as, 0 ( )w Cos ax= 0 ( )r Sin bx= 2 1 ( )w b tSin bx= 2 1 ( )r a tCos ax= − 4 2 2 1 ( ) 2 w a t Cos ax− = 4 2 2 1 ( ) 2 r b t Sin bx− = 6 3 3 1 ( ) 6 w b t Sin bx− = 6 3 3 1 ( ) 6 r a t Cos ax− = (13) The optimal identification of the unknown parameters a and b in the trial function is determined using the least squares method over the given domain as in [10]. 1 1 1 10 0 0, RR dxdt a ∂ = ∂∫ ∫ 1 1 2 20 0 0. RR dxdt b ∂ = ∂∫ ∫ (14) where 1 t xxR v u= − , 2 t xxR u v= + 22 From the equation (21), one calculates 1a = and 1b = . After substituting these values in the equation (11), one can write the optimal homotopy solution. Fig. 1 Analytical solution of ( , )u x t for the intervals 0 1x≤ ≤ and 0 1t≤ ≤ . Fig. 2 HPM solution of 0 1 2 3( , )u x t w w w w≅ + + + for 0 1x≤ ≤ and 0 1t≤ ≤ . Fig. 3 Analytical solution of ( , )v x t for the intervals 0 1x≤ ≤ and 0 1t≤ ≤ . Fig. 4 HPM solution of 0 1 2 3( , )v x t r r r r≅ + + + for 0 1x≤ ≤ and 0 1t≤ ≤ . In Fig. 1, the analytical distribution of the ( , )u x t for the intervals 0 1x≤ ≤ and 0 1t≤ ≤ is given. The approximated solution of the ( , )u x t is given for the first four terms in the Fig. 2 for the same interval. In the Fig. 3, the analytical distribution of the ( , )v x t for the intervals 0 1x≤ ≤ and 0 1t≤ ≤ is given. The approximated solution of the ( , )v x t is shown for the first four terms in the Fig. 4 for the same interval. 4. CONCLUSION The numerical analysis of complex PDEs were presented by using the Homotopy perturbation method (HPM) and the calculated numerical results were compared with the available exact solutions. The numerical HPM results were obtained by selecting a trial function and determining the unknown constants with the least squares method. This algorithm allowed an accurate and efficient use of the HPM with minimum numbers of iteration. REFERENCES [1] A.D. Poyanin, V.F. Zaitsev, Handbook of Nonlinear PDES, Chapman Hall, 2004. [2] ML Wang, Physics Letters A 215, (1996) 279. [3] M Boiti, J Leon, F Pempinelli , Iverse Probl. 13, (1987) 371. [4] E Fan, J. Physics A: Math. And General, 35 (2002) 6853. [5] J.H. He, Comput. Methods Appl. Mech. Engrg. 178 (3/4) (1997) 230. [6] J.H. He, Comput. Method Appl. Mech. Engrg. 167 (1–2) (1998) 57. [7] J.H. He, Int. J. Nonlinear Mech. 35 (1) (2000) 37. [8] J.H. He, Appl. Math. Comput. 135 (2003) 73. [9] J.H. He, Appl. Math. Comput. 156 (2004) 527. [10] J.H. He, Topological Methods in Nonlinear Analysis, 31 (2008) 205. [11] S. Abbasbandy, Phys. Lett A 361, (2007) 478. [12] M.Refai, D.D. Ganji, H.R M. Daniali, H. Pashaei, Physics Letter A, 364 (2007) 1. [13] J.H. He, Int. J. of Modern Physics B, 20 (2006) 1141. hpmu hpmv exactv 23 COMPRESSIVE STRENGTH AND MODULUS OF ELASTICITY OF FIBER REINFORCED CONCRETE Khaled H. Marar Department of Civil Engineering European University of Lefke, Gemikonağı-Lefke, TRNC, Mersin 10 Turkey kmarar@eul.edu.tr Özgür Eren Department of Civil Engineering Eastern Mediterranean University, Gazimağusa, TRNC, Mersin 10 Turkey ozgur.eren@emu.edu.tr Hakan Yalçıner Department of Civil Engineering European University of Lefke, Gemikonağı-Lefke, TRNC, Mersin 10 Turkey hyalciner@eul.edu.tr Abstract—Compressive strength and modulus of elasticity of concrete are two important properties involved in the design of reinforced concrete members and sections. The main objective of this investigation was to investigate the effect of fiber aspect ratio (l/d) and fiber volume on the compressive strength and modulus of elasticity of fiber reinforced concrete (FRC), and to study the relationship between compressive strength and modulus of elasticity of FRC. In this investigation, 19 fiber reinforced concrete mixes were produced by adding three different hooked-end steel fibers of aspect ratio of 60, 75, and 83 and also six different volumes of fibers namely 0.5, 1.0, 1.25, 1.5, 1.75, and 2.0 % by volume of concrete were added for each aspect ratio. As a result, good relationships between compressive strength of FRC and the fiber reinforcement index (FRI = Vf .l/d) between modulus of elasticity and FRI, and between compressive strength and modulus of elasticity of FRC were obtained. Keywords: steel fibers, compressive strength, modulus of elasticity, fiber reinforced concrete, fiber reinforcement index. 1. INTRODUCTION The strength of a material is defined as the ability to resist stress without failure, and failure is identified with the appearance of cracks. Strength of concrete is related to the stress required to cause fracture and is signification with the degree of failure at which the applied stress reaches its maximum value [1]. The compressive strength is the most important overall measure of the quality of concrete, because strength is directly related to the structure of the hardened cement paste. Strength is not a direct measure of concrete dimensional stability and durability, but it has a strong relationship to the w/c ratio of the concrete. The w/c ratio affects durability, dimensional stability and other properties of the concrete by controlling porosity. Therefore, compressive strength is commonly used in specifying, controlling, and evaluating the quality of concrete [2]. The strength of concrete is affected by several factors including: degree of hydration, rate of loading, method of testing, testing conditions, specimen geometry, and the properties and proportions of the constituent materials such as the quality of fine and coarse aggregate, cement paste, and the paste-aggregate bond characteristics. As an example for testing conditions and specimen size effect: the strength of saturated specimens can be 15 to 20 % lower than that of dry specimens, and cube specimens can be 20 to 25 % higher strengths than that of cylindrical specimens. In general, larger specimens show lower strengths than that of smaller specimens [3]. Modulus of elasticity is the ratio of stress to the strain in the elastic region. Modulus of elasticity is important in structural member design. Modulus of elasticity depends on several factors, and these factors from different investigations are: (a) modulus of elasticity decreases as w/c ratio increases [4, 5], (b) modulus of elasticity (Ec) of concrete increases with its compressive strength, also increases with concrete age [6], (c) the elastic modulus increases with an increase in rate of loading [7], (d) concrete specimens tested in wet condition can exhibit 15 % higher modulus of mailto:kmarar@eul.edu.tr� mailto:hyalciner@eul.edu.tr� 24 elasticity than tested in dry condition [3], (e) modulus of elasticity of high strength hardened cement paste has a very much higher modulus of elasticity than normal hardened cement paste, as a result, concrete has a more linear stress-strain curve and may exhibit an increased brittleness [8], (f) values of compressive strength and modulus of elasticity are influenced by the size of the cylindrical specimens. The relationship for calculating modulus of elasticity of 150 x 300 mm cylinder from 100 x 200 mm cylinders is in the from [9]: Ec150 = 1.05 Ec100, (g) modulus of elasticity of HSC is influenced by the elastic properties of coarse aggregates [10], (h) modulus of elasticity of concrete is affected by the type of aggregates used in concrete [11], (i) elastic modulus is dependent on the aggregate characteristics including the mineralogy, aggregate content, surface texture, particle size, and age of the cement paste [12]. There are many investigations on the relationship between modulus of elasticity and compressive strength of plain concrete [13-24], on the other hand, there are few investigations on the relationship between compressive strength and modulus of elasticity of fiber reinforced concrete [25-29]. In these investigations, all the models proposed did not account for the fiber reinforcement index which involves the volume and the aspect ratio of fibers. On contrary, in this investigation the proposed model of modulus of elasticity for the fiber reinforced concrete involves the compressive strength, volume of fibers and aspect ratio of fibers. 2. EXPERIMENTAL STUDY 2.1 Materials The w/c ratio used for the stee