T.C. ISTANBUL KÜLTÜR UNIVERSITY INSTITUTE OF GRADUATE STUDIES A PROPOSED MODEL FOR ESTIMATING THE CURVATURE DUCTILITY OF REINFORCED CONCRETE SECTIONS Master of Science Thesis Mohamad MARI 1800000784 Department: Civil Engineering Program: Structural Engineering Supervisor: Prof. Dr. HÜSEYİN FARUK KARADOĞAN May 2021 T.C. ISTANBUL KÜLTÜR UNIVERSITY INSTITUTE OF GRADUATE STUDIES A PROPOSED MODEL FOR ESTIMATING THE CURVATURE DUCTILITY OF REINFORCED CONCRETE SECTIONS Master of Science Thesis Mohamad MARI 1800000784 Department: Civil Engineering Program: Structural Engineering Supervisor: Prof. Dr. HÜSEYİN FARUK KARADOĞAN Thesis Jury Members: Prof. Dr. AHMET MURAT TÜRK Dr. Öğr. ÜYESI ERDAL COŞKUN May 2021 I Preamble: I would like to acknowledge and thank the following important people who have supported me, not only during the course of this project, but throughout my master's degree. I would like to express my gratitude to my supervisor Prof. Dr. Farouk KARADĞAN for his unwavering support. Who expertly guided me and shared the excitement and enthusiasm to keep me constantly engaged with my thesis topic. All my gratefulness to İstanbul Kültür University and all the academic staff of civil engineering department for letting me fulfil my dream to get the master's degree. And finally, I would like to thank my family and all my close friends. And great thanks to my father who encourage me and believed in me. You have all helped me to focus on what has been a hugely rewarding and enriching process. II ABSTRACT Reinforced concrete is a widely used system for constructing structures all over the world. Currently, the main requirement for designing reinforced concrete structures is achieving a ductile behavior by deforming before the section fracture under the ultimate limit state. In general, to ensure a ductile behavior, an adequate moment-curvature is important. Nowadays, one of the most used methods for quantifying the ductility of the section is through curvature ductility. Previously, several analytical models were proposed to determine this parameter and its effect. However, some problems are encountered due to the major assumptions in developing these models which reduce their reliability for general applications. In this study, computer software was developed to calculate the curvature ductility of reinforced concrete columns and walls. In addition to that, an enhanced mathematical model for estimating the curvature ductility is proposed. Finally, a parametric study was conducted to evaluate the influencing parameters on the curvature ductility of different sections. The results of this study have shown a significant improvement in the proposed against the currently available one. Furthermore, the developed program was capable of defining the moment-curvature with good accuracy in comparison to both ETABS and Xtract. Moreover, the results of the parametric study have presented a considerable independency of the sectional ductility on the level of the confinement and the tensile strength of the concrete used. This study is expected to help practicing engineers in their daily works by reliably estimating the behavior of reinforced concrete sections. III ÖZET Betonarme, tüm dünyada yapıların inşasında yaygın olarak kullanılan bir sistemdir. Günümüzde, betonarme yapıların tasarımı için temel gereksinim, nihai sınır durumu altında kesit kırılmasından önce şekil değiştirerek sünek bir davranış elde etmektir. Genel olarak, sünek bir davranış sağlamak için yeterli bir moment eğriliği önemlidir. Günümüzde, kesitin sünekliğini ölçmek için en çok kullanılan yöntemlerden biri eğrilik sünekliğidir. Daha önce, bu parametreyi ve etkisini belirlemek için birkaç analitik model önerildi. Ancak, genel uygulamalar için güvenilirliğini azaltan bu modellerin geliştirilmesinde büyük varsayımlar nedeniyle bazı sorunlarla karşılaşılmaktadır. Bu çalışmada, betonarme kolon ve duvarların eğrilik sünekliğini hesaplamak için bilgisayar yazılımı geliştirilmiştir. Buna ek olarak, eğrilik sünekliğini tahmin etmek için geliştirilmiş bir matematiksel model önerilmiştir. Son olarak, farklı kesitlerin eğrilik sünekliğine etki eden parametreleri değerlendirmek için parametrik bir çalışma yapılmıştır. Bu çalışmanın sonuçları, şu anda mevcut olana karşı önerilende önemli bir gelişme göstermiştir. Ayrıca, geliştirilen program hem ETABS hem de Xtract ile karşılaştırıldığında moment eğriliğini iyi bir doğrulukla tanımlayabiliyordu. Ayrıca, parametrik çalışmanın sonuçları, kullanılan betonun sarılma seviyesi ve çekme mukavemeti üzerinde kesit sünekliğinin önemli bir bağımsızlığını ortaya koymuştur. Bu çalışmanın, betonarme bölümlerin davranışını güvenilir bir şekilde tahmin ederek mühendislere günlük işlerinde yardımcı olması beklenmektedir. IV TABLE OF CONTENTS LIST OF FIGURES ...................................................................................................................... VI LIST OF TABLES ........................................................................................................................ IX INTRODUCTION .......................................................................................................................... 1 1.1 General Introduction ........................................................................................................ 1 1.2 Aim of the Study .............................................................................................................. 1 1.3 Outline of the Thesis ........................................................................................................ 2 LITERATURE REVIEW ............................................................................................................... 3 2.1 Introduction ...................................................................................................................... 3 2.2 Curvature Ductility ........................................................................................................... 3 RESEARCH METHODOLOGY.................................................................................................... 5 3.1 Introduction ...................................................................................................................... 5 3.2 Stress-Strain Behavior Of Concrete ................................................................................. 5 3.2.1 Unconfined Concrete ................................................................................................ 5 3.2.2 Confined Concrete .................................................................................................... 6 3.2.3 Tensile Strength of Concrete .................................................................................. 12 3.3 Steel Reinforcement ....................................................................................................... 14 3.4 Moment-Curvature Curve .............................................................................................. 16 3.5 Developed Software ....................................................................................................... 20 V RESULTS AND DISCUSSIONS ................................................................................................. 31 5.1 Moment Curvature ......................................................................................................... 31 5.2 Effect of Axial Load ....................................................................................................... 32 5.3 Effect of Strength of Materials ....................................................................................... 33 5.4 Proposed Ductility Equations ......................................................................................... 37 5.5 Proposed Ductility equations vs software results ........................................................... 38 5.6 Olivia`s equation for ductility vs software results ......................................................... 40 CONCLUSIONS........................................................................................................................... 42 Appendix I: List of Figures ........................................................................................................... 46 Appendix II: Code Written for MCI Software .............................................................................. 76 VI LIST OF FIGURES Figure 1: Effect of tie volumetric ratio ........................................................................................... 7 Figure 2: Effect of tie yield strength ............................................................................................... 7 Figure 3: Effect of 𝜌𝑠. 𝑓𝑠𝑦 .............................................................................................................. 7 Figure 4: Comparison of stress-strain models (𝜎𝑐′ = 46.3 𝑀𝑃𝑎) ................................................ 10 Figure 5: Comparison of stress-strain models (𝜎𝑐′ = 84.8 𝑀𝑃𝑎) ................................................ 10 Figure 6: Comparison of stress-strain models (𝜎𝑐′ = 128 𝑀𝑃𝑎) ................................................. 10 Figure 7: Idealized stress strain behavior of steel rebar ................................................................ 15 Figure 8: Stress strain behavior of steel rebar ............................................................................... 15 Figure 9: A flowchart of the procedure for calculating moment resistance. ................................ 24 Figure 10: A screenshot of the section define window. ................................................................ 25 Figure 11: A screenshot of the material define window. .............................................................. 25 Figure 12: A flowchart of the procedure for plotting the moment curvature curve. .................... 26 Figure 13: A flowchart of the procedure for plotting the interaction curve. ................................. 27 Figure 14: A screenshot of the moment curvature curve. ............................................................. 28 Figure 15: A screenshot of the interaction curve. ......................................................................... 28 Figure 16:Moment Curvature Curves of Section 1 – Low Strength Materials ............................. 46 Figure 17:Interaction Curve of Section 1 – Low Strength Materials ............................................ 46 Figure 18:Moment Curvature Curves of Section 1 – Normal Strength Materials ........................ 47 Figure 19:Interaction Curve of Section 1 – Normal Strength Materials ....................................... 47 Figure 20:Moment Curvature Curves of Section 1 – High Strength Materials ............................ 48 Figure 21:Interaction Curve of Section 1 – High Strength Materials ........................................... 48 VII Figure 22:Moment Curvature Curves of Section 2 – Low Strength Materials ............................. 49 Figure 23:Interaction Curve of Section 2 – Low Strength Materials ............................................ 49 Figure 24:Moment Curvature Curves of Section 2 – Normal Strength Materials ........................ 50 Figure 25:Interaction Curve of Section 2 – Normal Strength Materials ....................................... 50 Figure 26:Moment Curvature Curves of Section 2 – High Strength Materials ............................ 51 Figure 27:Interaction Curve of Section 2 – High Strength Materials ........................................... 51 Figure 28:Moment Curvature Curves of Section 3 – Low Strength Materials ............................. 52 Figure 29:Interaction Curve of Section 3 – Low Strength Materials ............................................ 52 Figure 30:Moment Curvature Curves of Section 3 – Normal Strength Materials ........................ 53 Figure 31:Interaction Curve of Section 3 – Normal Strength Materials ....................................... 53 Figure 32:Moment Curvature Curves of Section 3 – High Strength Materials ............................ 54 Figure 33:Interaction Curve of Section 3 – High Strength Materials ........................................... 54 Figure 34:Moment Curvature Curves of Wall Section – Low Strength Materials ....................... 55 Figure 35:Interaction Curve of Wall Section – Low Strength Materials ...................................... 55 Figure 36:Moment Curvature Curves of Wall Section – Normal Strength Materials .................. 56 Figure 37:Interaction Curve of Wall Section – Normal Strength Materials ................................. 56 Figure 38:Moment Curvature Curves of Wall Section – High Strength Materials ...................... 57 Figure 39:Interaction Curve of Wall Section – High Strength Materials ..................................... 57 Figure 40: Moment curvature curve of section 1 .......................................................................... 58 Figure 41: Moment curvature curve of section 2 .......................................................................... 59 Figure 42: Moment curvature curve of section 3 .......................................................................... 60 Figure 43: Moment curvature curve of wall section ..................................................................... 61 Figure 44: Moment curvature curve of section 1 .......................................................................... 64 VIII Figure 45: Moment curvature curve of section 2 .......................................................................... 67 Figure 46: Moment curvature curve of section 3 .......................................................................... 70 Figure 47: Moment curvature curve of the wall section ............................................................... 73 Figure 48: The Influence of Material Properties on Curvature Ductility of Section #1 ............... 74 Figure 49: The Influence of Material Properties on Curvature Ductility of Section #2 ............... 74 Figure 50: The Influence of Material Properties on Curvature Ductility of Section #3 ............... 75 Figure 51: The Influence of Material Properties on Curvature Ductility of Wall Section ........... 75 IX LIST OF TABLES Table 1: Values of k and n ............................................................................................................ 13 Table 2: Properties of steel in standards ....................................................................................... 16 Table 3: Sections Dimensions. ...................................................................................................... 29 Table 4: Sections Reinforcement. ................................................................................................. 29 Table 5: Material Properties.......................................................................................................... 29 Table 6: The curvature ductility of section 1 calculated by MCI software. ................................. 35 Table 7: The curvature ductility of section 2 calculated by MCI software. ................................. 35 Table 8: The curvature ductility of section 3 calculated by MCI software. ................................. 36 Table 9: The curvature ductility of wall section calculated by MCI software. ............................. 36 Table 10: Curvature ductility equations for rectangular and wall sections. ................................. 37 Table 11: Comparison of curvature ductility between MCI software and the proposed equation without axial force. ....................................................................................................................... 38 Table 12: Comparison of curvature ductility between MCI software and the proposed equation with 20% axial force. .................................................................................................................... 39 Table 13: Comparison of curvature ductility between MCI software and Olivia & Mandal’s equation without axial force.......................................................................................................... 40 1 CHAPTER 1 INTRODUCTION 1.1 GENERAL INTRODUCTION Over the last decades, the philosophy behind designing of reinforced concrete (RC) structures has developed significantly. Nowadays, the main requirement for accepting the design of a given RC section is the fact that it follows a ductile behavior in order to avoid a brittle failure of the structure. This is generally done through ensuring an adequate curvature in the ultimate limit state. The definition of ductility is the ability of members to undergo deformations without a significant decrease in its flexural capacity (Park & Ruitong, 1988) [1]. Currently, one of the most common methods to quantify the ductility of RC members is through curvature ductility. Several approaches were previously introduced to the literature for calculating this parameter taking several factors into account such as the reinforcement ratio, strength of both steel reinforcement and the concrete one, and the size of the RC section. Furthermore, some parametric studies were conducted upon these computational approaches. However, there are some problems in these methods such as their capability to be applied in all reinforcement ratios including the amount of tensile reinforcement as well as compressive reinforcement. Thus, a detailed study is important to improve these models by proposing an alternative analytical model that can overcome the current lacks in these models. 1.2 AIM OF THE STUDY This study is intended to firstly develop a software that can define the moment curvature of RC columns and walls, secondly, propose an improved analytical approach for estimating the 2 curvature ductility of these sections, and finally, conduct a parametric study that can highlights the influence of several factors including the strength of steel and concrete used, the rate of axial load applied, and dimensions of the RC section on the ductility. This information is of importance for both practicing structural engineers as well as scientists in the working in the field. 1.3 OUTLINE OF THE THESIS The thesis herein consists of five main chapters of which the main scientific content is delivered. The first chapter introduce the topic by providing a general introduction and defines the aim of the study. The second chapter gives a detailed literature review of the current approaches used and highlights the findings of previous studies. Thereafter, the third chapter includes the research methodology that was followed while conducting the investigations and the main assumptions. Then the results and discussions are given in the fourth chapter of which the findings of this study will be illustrated and compared to previous ones available in the current state of the art. Finally, the major conclusions of this study are drawn in the fifth chapter and the thesis is ended with a brief information on the possible future works. 3 CHAPTER 2 LITERATURE REVIEW 2.1 INTRODUCTION In this chapter, a detailed review of the current state of the art on the analytical modeling of behavior of RC sections will be presented. Furthermore, previous approaches computing the curvature ductility will be summarized. 2.2 CURVATURE DUCTILITY The ductility of reinforced concrete sections is very important, since it is essential to avoid a brittle failure of the structure by ensuring adequate curvature at the ultimate limit state. According to Olivia & Mandal (2015) [2], ductile behavior in a structure can be achieved using plastic hinges positioned at appropriate locations throughout the structural frame. The ductility of plastic hinges can be obtained from the Moment – Curvature relationship of a reinforced concrete section by dividing the curvature at ultimate point on the curvature at yielding point. Where, yielding curvature is the point when tensile reinforcement yields, and ultimate curvature is the curvature value when the furthest compressed concrete fiber is crushed. Olivia & Mandal (2015) [2] suggested a formula for yielding curvature for a reinforced concrete section: ∅𝑦 = 𝑓𝑦 𝐸𝑠(1 − 𝑘)𝑑 𝑘 = √(𝜌 + 𝜌′)2 𝑛2 + 2 (𝜌 + 𝜌′ 𝑑′ 𝑑 ) 𝑛 − (𝜌 + 𝜌′)𝑛 4 𝜌 = 𝐴𝑠 𝑏 𝑑 , 𝜌′ = 𝐴𝑠 ′ 𝑏 𝑑 , 𝑛 = 𝐸𝑠 𝐸𝑐 where ∅𝑦 is yielding curvature, d is effective depth of tensile reinforcement, and n is modular ratio. And they suggested another formula for ultimate curvature: ∅𝑢 = 𝜀𝑐𝑢 𝛽1 𝑎 𝑎 = 𝐴𝑠 𝑓𝑦 − 𝐴𝑠 ′ 𝑓𝑦 0.85 𝑓𝑐 ′ 𝑏 where ∅𝑢 is ultimate curvature, a is the depth of equivalent rectangular stress block, b is the cross- section width, and 𝛽1 is the ratio of the height of equivalent rectangular concrete compressive stress block to neutral axis. And according to ACI 318M-08 the value of 𝛽1 can be calculated as follows: 𝛽1 = 0.85 ; 𝑓𝑐 ′ ≤ 28 𝑀𝑃𝑎 𝛽1 = 0.85 − 0.007(𝑓𝑐 ′ − 28) ≥ 0.65 ; 𝑓𝑐 ′ > 28 𝑀𝑃𝑎 The previous ultimate curvature formula assumes the compression reinforcement reaches yielding point before failure, this cannot occur except in case of a very small compression reinforcement ratio to a relatively high tensile reinforcement ratio, otherwise the formula gives greater values than actual and in the case of symmetric reinforced section without axial force and middle reinforcement, formula gives infinite ductility which does not coincide with reality. And this point will be discussed furthermore later in this research. 5 CHAPTER 3 RESEARCH METHODOLOGY 3.1 INTRODUCTION In this section detailed discussions on the material properties used in the case studies and program development stages will be provided. Furthermore, the research strategy followed in the study will be highlighted. 3.2 STRESS-STRAIN BEHAVIOR OF CONCRETE 3.2.1 Unconfined Concrete In previous studies several empirical equations were proposed to represent the stress-strain behavior of plain concrete with various compressive strengths after series of experimental tests. The proposed empirical stress-strain equations were compared with experimental results under axial compression and showed good agreements. Although, the stress-strain relationship of these equations showed a bit difference when compared with each other, and that is because of the parameters defining the relationship depend on the testing conditions. A simple formula proposed by Carreira et al. (1985) [3] for low strength and normal strength concrete: 𝑓 𝑓𝑜 = 𝑅 ( 𝜀 𝜀𝑜 ) 1 + (𝑅 − 1) ( 𝜀 𝜀𝑜 ) 𝛽 𝛽 = 𝑅 𝑅 − 1 , 𝑅 = 𝐸𝑐 𝐸𝑜 , 𝐸𝑜 = 𝑓𝑜 𝜀𝑜 6 Where R is material parameter depending on the shape of the stress-strain curve, a value of R=1.90 was proposed later as a constant value, upon this, the last equation was modified to be as follow: 𝑓 𝑓𝑜 = 1.9 ( 𝜀 𝜀𝑜 ) 1 + 0.9 ( 𝜀 𝜀𝑜 ) 2.1 3.2.2 Confined Concrete The main objectives of transverse reinforcement are (preventing buckling of longitudinal bars, resisting shear forces, and providing sufficient ductility for concrete section). In addition, confinement plays an important role in increasing the concrete compressive strength by resisting the lateral strain of the cross section. The lateral confinement of concrete resists the lateral strain by applying lateral pressure force, where the largest stress at ties location and the smallest stress at the middle distance between ties. Since lateral deformation is related to axial deformation according to Poison’s ratio, it leads to a reduction in the axial strain, which is observed as an increment in concrete compressive strength. The increment in compressive strength of concrete is determined as a proportion of the lateral confinement pressure. A model was proposed by Saatcioglu and Razvi (1992) [4] based on experimental tests then modified later by Suzuki et al. (2004) [5]. The Suzuki’s study has discussed the effect of the most important two parameters, volumetric ratio of transverse reinforcement and yield strength of transverse reinforcement as shown in Figure 1, Figure 2, and Figure 3. 7 Figure 1: Effect of tie volumetric ratio Figure 2: Effect of tie yield strength Figure 3: Effect of 𝜌𝑠 . 𝑓𝑠𝑦 Since the lateral pressure is distributed uniformly, the effective confinement index was defined as the uniform effective lateral pressure as in the following equation: 8 𝑝𝑒 = 𝑘𝑒 𝜌𝑤 𝑓𝑠,𝑐 where 𝜌𝑤 is the area ratio of transverse reinforcement; 𝑓𝑠,𝑐 is the stress in transverse reinforcement at the peak strength and 𝑘𝑒 is the effective confinement coefficient given by: 𝑘𝑒 = (1 − Σ (𝜔𝑖) 2 6 𝑏𝑐 𝑑𝑐 ) (1 − 𝑠′ 2 𝑏𝑐 ) (1 − 𝑠′ 2 𝑑𝑐 ) 1 − 𝜌𝑐𝑐 where 𝜔𝑖 is the clear spacing between adjacent longitudinal steel bars in a rectangular section; 𝑠′ is the clear spacing of ties; 𝑏𝑐 and 𝑑𝑐 are the widths of concrete core; and 𝜌𝑐𝑐 is the longitudinal reinforcement ratio in core section. The model was based on the equivalent uniform confinement pressure concept for square cross section having the same confinement pressure in two orthogonal directions. Therefore, in the case of rectangular section with different confinement pressure in both directions, the average effective pressure can be obtained after calculating the effective lateral pressure in each direction as follow: 𝑝𝑒 = 𝑝𝑒𝑥 𝑏𝑐 + 𝑝𝑒𝑦 𝑑𝑐 𝑏𝑐 + 𝑑𝑐 A regression analysis was performed on all test results to formulate the peak strength (𝜎𝑐𝑐), the strain at peak strength (𝜀𝑐𝑐), and the slope of the descending branch (𝐸𝑑𝑒𝑠) in terms of 𝑝𝑒. The results of regression analyses are presented as follows: 𝜎𝑐𝑐 𝜎𝑐𝑜 = 1.0 + 4.1 ( 𝑝𝑒 𝜎𝑐𝑜 ) 0.7 𝜀𝑐𝑐 = 𝜀𝑐𝑜 + 0.015 ( 𝑝𝑒 𝜎𝑐𝑜 ) 0.56 𝐸𝑑𝑒𝑠 = 0.026 𝜎𝑐𝑜 3 𝑝𝑒 0.4 𝜎𝑐𝑜 = 0.85 𝜎𝑐 ′ 𝜀𝑐𝑜 = 0.0028 − 0.0008 𝑘3 9 𝑘3 = 40 𝜎𝑐𝑜 ≤ 1.0 Where 𝜎𝑐𝑜and 𝜀𝑐𝑜are the peak stress and corresponding strain of unconfined concrete. The study proposed a trial-and-error method for calculating the stress in confinement ties. And for simplicity, the study proposed a direct equation: 𝑓𝑠,𝑐 = 𝐸𝑠 (0.45 𝜀𝑐𝑜 + 0.73 ( 𝑘𝑒 𝜌𝑤 𝜎𝑐𝑜 ) 0.7 ) ≤ 𝑓𝑠𝑦 The strain-strain relationship of confined concrete was plotted according to a model proposed by Shah et al. (1985) [6], where the ascending part is represented by: 𝜎𝑐 = 𝜎𝑐𝑐 (1 − (1 − 𝜀𝑐 𝜀𝑐𝑐 ) 𝛼 ) ; (0 ≤ 𝜀𝑐 ≤ 𝜀𝑐𝑐) 𝛼 = 𝐸𝑐 𝜀𝑐𝑐 𝜎𝑐𝑐 𝐸𝑐 = 3320 √𝜎𝑐𝑜 + 6900 And the descending part was defined as a straight line connecting the point of peak strength and the point at which the stress drops to 85% of peak strength: 𝜎𝑐 = 𝜎𝑐𝑐 − 𝐸𝑑𝑒𝑠 (𝜀𝑐 − 𝜀𝑐𝑐) ; (𝜀𝑐𝑐 ≤ 𝜀𝑐 < 𝜀𝑐𝑐𝑢) The study compares numerical results with experimental results for different compressive strengths of concrete, and they were in a good agreement as shown in Figure 4, Figure 5, and Figure 6. 10 Figure 4: Comparison of stress-strain models (𝜎𝑐 ′ = 46.3 𝑀𝑃𝑎) Figure 5: Comparison of stress-strain models (𝜎𝑐 ′ = 84.8 𝑀𝑃𝑎) Figure 6: Comparison of stress-strain models (𝜎𝑐 ′ = 128 𝑀𝑃𝑎) Another recent model was proposed by Legeron and Paultre (2003) [7] to describe the behavior of confined concrete based on Mander et al. (1988) [8] model. The analytical model was compared with a wide range of concrete columns with concrete strength ranging from 30 to 120 MPa confined with steel of yield strength ranging from 250 to 1,400 MPa. 11 𝑓𝑙𝑒 ′ = 𝐾𝑒(𝐴𝑠ℎ𝑦 𝑓ℎ ′) 𝑐𝑦 𝑠 𝐾𝑒 = (1 − ∑ 𝑤𝑖 2 6 𝑐𝑥 𝑐𝑦 ) (1 − 𝑠′ 2𝑐𝑥 ) (1 − 𝑠′ 2𝑐𝑦 ) 1 − 𝜌𝑐𝑐 ≥ 0 𝑓ℎ ′ = { 𝑓ℎ𝑦 0.25 𝑓𝑐 ′ 𝜌𝑠𝑒𝑦 (𝜅 − 10) ≥ 0.43 𝜀𝑐 ′ 𝐸𝑠 ≯ 𝑓ℎ𝑦 𝑓𝑜𝑟 𝜅 ≤ 10 𝑓𝑜𝑟 𝜅 > 10 } 𝜅 = 𝑓𝑐 ′ 𝜌𝑠𝑒𝑦 𝐸𝑠 𝜀𝑐 ′ 𝜌𝑠𝑒𝑦 = 𝐾𝑒 𝜌𝑠𝑦 𝑓𝑐𝑐 = 𝑓𝑐𝑜 (−1.254 + 2.254√1 + 7.94𝑓𝑙 ′ 𝑓𝑐𝑜 − 2𝑓𝑙 ′ 𝑓𝑐𝑜 ) Where 𝐾𝑒 is the geometric confinement effectiveness coefficient, which measures the effectiveness of the confinement reinforcement of confined concrete and varies from 1 for a continuous tube to 0 when ties are spaced more than half the core cross section minimum dimension; 𝐴𝑠ℎ𝑦 is the total section of confinement reinforcement for the set of ties in direction y; 𝑐𝑦 is the cross section dimension in direction y, measured center-to-center of peripheral ties; s center-to-center spacing between ties; and 𝑓ℎ ′ is the stress in the confinement reinforcement at peak stress. According to the author, the stress strain curve can be predicted by three coordinates. The ascending part between zero and the first coordinate can be derived by an equation based on Sargin et al. equation (1968) [9]: 𝜎𝑐𝑐 = 𝑓𝑐𝑐 × 𝑘𝑐 × 𝜀�̅� + (𝑘𝑐 ′ − 1) × 𝜀�̅� 2 1 + (𝑘𝑐 − 2) × 𝜀�̅� + 𝑘𝑐 ′ × 𝜀�̅� 2 ; 0 ≤ 𝜀𝑐 ≤ 𝜀𝑐𝑐𝑜 𝜀�̅� = 𝜀𝑐 𝜀𝑐𝑐𝑜 , 𝑎𝑛𝑑 𝜀𝑐𝑐𝑜 = 𝜀𝑐𝑜 × [1 + 5 ( 𝑓𝑐𝑐 𝑓𝑐𝑜 − 1)] 12 𝑘𝑐 = 𝐸𝑏𝑐𝑜 × 𝜀𝑐𝑐𝑜 𝑓𝑐𝑐 , 𝐸𝑏𝑐𝑜 = 11000√𝑓𝑐𝑐 3 , 𝑎𝑛𝑑 𝑘𝑐 ′ = 𝑘𝑐 − 1 The descending part between the first and the second coordinates can be determined by: 𝜎𝑐𝑐 = 𝑓𝑐𝑐 − 𝐸𝑠𝑐(𝜀𝑐 − 𝜀𝑐𝑐𝑜) ; 𝜀𝑐𝑐𝑜 < 𝜀𝑐 ≤ 𝜀65 𝐸𝑠𝑐 = 6 × 𝑓𝑐𝑜 2 𝑘𝑒 × 𝜌𝑠 × 𝑓𝑦ℎ 𝜀65 = 0.35 × 𝑓𝑐𝑐 𝐸𝑠𝑐 + 𝜀𝑐𝑐𝑜 where 𝐸𝑠𝑐 is the slope of the descending part and 𝜀65 is the strain at 65% of confined peak stress. After reaching the strain 𝜀65, the stress of confined concrete has a constant value of 0.65𝑓𝑐𝑐 until the ultimate strain (𝜀𝑐𝑐𝑢). 𝜀𝑐𝑐𝑢 = 0.4 𝑓𝑙 𝑓𝑐𝑜 + 𝜀𝑐𝑢 3.2.3 Tensile Strength of Concrete Concrete is a brittle material and weak under tension, therefore it is used to resist compression mainly and the tensile strength of concrete is usually neglected due to its small value compared to compressive strength. However, in this research the effect of tensile strength of concrete on bending capacity and curvature ductility of structural elements subjected to bending moment and axial force will be discussed. According to Jaber et al. (2018) [10], some laboratory split experiments were conducted, and it was noticed that split tensile strength was linearly related to compressive strength of concrete as follow: 𝑓𝑠𝑝𝑡 = 𝑘 𝑓𝑐 ′𝑛 where k and n are constants and various values have been proposed by several researchers and codes based on experimental results. The following table shows some of these values: 13 Table 1: Values of k and n Source k n ACI 318 0.56 0.50 ACI 363R 0.59 0.50 Gardner 0.47 0.59 Nihal 0.387 0.63 JCI 0.13 0.85 JSCE 0.23 0.67 CEB-FIB 0.30 0.67 Raphael 0.313 0.667 Ahmad and Shah 0.462 0.55 Oloukun et al. 0.294 0.69 The tensile concrete strength is highly affected by the shape of the applied force and the used experiment. There are three forms of tensile strength, and they are splitting tension, direct tension, and tension under bending. Karadogan et al. (2015) [11] proposed two formulae for converting direct tensile strength to splitting tensile strength and tensile strength under bending as follows: 𝑓𝑐𝑡𝑘 = 𝑓𝑐𝑡𝑘 𝑦 1.5 𝑓𝑐𝑡𝑘 = 𝑓𝑐𝑡𝑘 𝑒 2 where 𝑓𝑐𝑡𝑘 𝑦 is the splitting tensile strength, 𝑓𝑐𝑡𝑘 𝑒 is the tensile strength under bending, and 𝑓𝑐𝑡𝑘 is the direct tensile strength. After substituting one equation with the other, we get the following formula: 𝑓𝑐𝑡𝑘 𝑒 = 4 𝑓𝑐𝑡𝑘 𝑦 3 To take the effect of tensile concrete strength on bending capacity of reinforced concrete section, the value of maximum tensile strength is insufficient, thus, tensile stress-strain relationship of concrete is needed. According to Kaklauskas (1999) [12], concrete tensile stress-strain relationship 14 consists of two parts, a linear part with a slop of concrete modules of elasticity 𝐸𝑐, and it is defined by the following formula: 𝜎𝑡 = 𝐸𝑐 𝜀𝑡 The first part is limited by the strain at first crack in concrete under tension 𝜀𝑐𝑟. The second part is nonlinear and extended to the strain value of 𝛽 𝜀𝑐𝑟. The second part is defined by the following formula: 𝜎𝑡 = 0.625 𝜎𝑐𝑟 (1 − 𝜀�̅� 𝛽 + 1 + 0.6 𝛽 𝛽 𝜀�̅� ) 𝜀�̅� = 𝜀𝑡 𝜀𝑐𝑟 The value of 𝛽 depends on the reinforcement ratio 𝜌 and it is defined by: 𝛽 = 32.8 − 27.6 𝜌 + 7.12 𝜌2 3.3 STEEL REINFORCEMENT Stress-strain relationship of steel consists of two main phases, first phase is linear elastic phase with a slop of modules of elasticity 𝐸𝑠 and limited by yielding point. After yielding point steel enters plastic phase. According to Khatulistiani et al. (2020) [13], plastic phase starts with constant stress, then a hardening stage which is extended to ultimate stress, finally steel begins losing its resistance until failure, the following fig shows the stress strain curve of steel: 15 Figure 7: Idealized stress strain behavior of steel rebar The previous curve was simplified into a bi-linear relationship consists of two phases, elastic phase and plastic phase as shown in Figure8 below: Figure 8: Stress strain behavior of steel rebar Steel strength is affected by chemical composition. Various grades of steel will be used in this study, taken from the ASTM Standard. The following table shows the important characteristics of each steel grade, which are yield strength 𝑓𝑦, tensile strength 𝑓𝑢, and elongation 𝜀𝑢. Where steel modulus of elasticity 𝐸𝑠 has a constant value of 200 GPa. 16 Table 2: Properties of steel in standards Standard Yield Strength MPa Tensile Strength MPa Elongation % GR 40 ASTM A 615 280 420 10 - 12 GR 60 ASTM A 615 420 620 7 - 9 GR 75 ASTM A 615 520 690 6 - 7 GR 100 ASTM A 709 690 760 18 3.4 MOMENT-CURVATURE CURVE Moment-curvature curve of a reinforced concrete section is the relationship between bending moment resistance and its rotation around the neutral axis in terms of one per meter, either subjected to an axial force or not. Plotting the moment-curvature curve could be done by calculating the moment resistance at each curvature value, starting from a very small curvature, and increasing it gradually until failure where the moment resistance is zero. The following section discusses the derivation of curvature ductility equation in details.  Curvature at yielding point: 𝑁 + 𝑓𝑦 𝐴𝑠 = 𝑓𝑠 ′ 𝐴𝑠 ′ + 0.5 𝑓𝑐 ′ 𝐶𝑠 𝐵 ∅𝑦 = 𝜀𝑦 𝑑 − 𝐶𝑠 → 𝐶𝑠 = 𝑑 − 𝜀𝑦 ∅𝑦 17 𝜀𝑠 ′ = 𝐶𝑠 − 𝑑′ 𝑑 − 𝐶𝑠 𝜀𝑦 = (𝑑 − 𝑑′) ∅𝑦 − 𝜀𝑦 𝑓𝑠 ′ = 𝐸𝑠 𝜀𝑠 ′ = 𝐸𝑠(𝑑 − 𝑑′) ∅𝑦 − 𝐸𝑠 𝜀𝑦 = 𝐸𝑠(𝑑 − 𝑑′) ∅𝑦 − 𝑓𝑦 𝜀𝑐 ′ = 𝑑 ∅𝑦 − 𝜀𝑦 𝑓𝑐 ≅ 𝐸𝑐 𝜀𝑐 ′ = 𝐸𝑐 𝑑 ∅𝑦 − 𝐸𝑐 𝜀𝑦 𝑁 + 𝑓𝑦 𝐴𝑠 = 𝐸𝑠 𝐴𝑠 ′ (𝑑 − 𝑑′) ∅𝑦 − 𝑓𝑦 𝐴𝑠 ′ + 0.5 𝑓𝑐 𝐵 𝑑 − 0.5 𝑓𝑐 𝐵 𝜀𝑦 ∅𝑦 𝑁 ∅𝑦 + 𝑓𝑦 𝐴𝑠∅𝑦 + 𝐸𝑠 𝐴𝑠 ′ (𝑑′ − 𝑑) ∅𝑦 2 + 𝑓𝑦 𝐴𝑠 ′ ∅𝑦 − 0.5 𝑓𝑐 𝐵 𝑑 + 0.5 𝑓𝑐 𝐵 𝜀𝑦 = 0 𝑁 ∅𝑦 + 𝑓𝑦 𝐴𝑠∅𝑦 + 𝐸𝑠 𝐴𝑠 ′ (𝑑′ − 𝑑) ∅𝑦 2 + 𝑓𝑦 𝐴𝑠 ′ ∅𝑦 − 0.5 𝐵 𝐸𝑐 𝑑2 ∅𝑦 2 + 𝐵 𝐸𝑐 𝑑 𝜀𝑦 ∅𝑦 − 0.5 𝐵 𝐸𝑐 𝜀𝑦 2 = 0 [𝐸𝑠 𝐴𝑠 ′ (𝑑′ − 𝑑) − 0.5 𝐵 𝐸𝑐 𝑑2] ∅𝑦 2 + [𝑁 + 𝑓𝑦 (𝐴𝑠 + 𝐴𝑠 ′ ) + 𝐵 𝐸𝑐 𝑑 𝜀𝑦] ∅𝑦 + [−0.5 𝐵 𝐸𝑐 𝜀𝑦 2] = 0 [−0.5 𝐵 𝐸𝑐 𝑑2] ∅𝑦 2 + [𝑁 + 𝑓𝑦 𝐴𝑠 + 𝐵 𝐸𝑐 𝑑 𝜀𝑦] ∅𝑦 + [−0.5 𝐵 𝐸𝑐 𝜀𝑦 2] = 0 [−0.5] ∅𝑦 2 + [ 𝑁 + 𝑓𝑦 𝐴𝑠 + 𝐵 𝐸𝑐 𝑑 𝜀𝑦 𝐵 𝐸𝑐 𝑑2 ] ∅𝑦 + [ −0.5 𝜀𝑦 2 𝑑2 ] = 0 [−0.5] ∅𝑦 2 + [ 𝑁 + 𝑓𝑦 𝐴𝑠 𝐵 𝐸𝑐 𝑑2 + 𝜀𝑦 𝑑 ] ∅𝑦 + [0.5 ( 𝜀𝑦 𝑑 ) 2 ] = 0 [−0.5] ∅𝑦 2 + [ 𝑁 + 𝑓𝑦 𝐴𝑠 𝐵 𝐸𝑐 𝑑2 + 𝑓𝑦 𝐸𝑠 𝑑 ] ∅𝑦 + [0.5 ( 𝑓𝑦 𝐸𝑠 𝑑 ) 2 ] = 0 𝑎 = −0.5 𝑏 = 𝑁 + 𝑓𝑦 𝐴𝑠 𝐵 𝐸𝑐 𝑑2 + 𝑓𝑦 𝐸𝑠 𝑑 𝑐 = 0.5 ( 𝑓𝑦 𝐸𝑠 𝑑 ) 2 ∅𝑦 = −√𝑏2 − 4 𝑎 𝑐 − 𝑏 2𝑎 = √𝑏2 + 2 𝑐 + 𝑏 18 ∝1 2= 2 𝑐 ∝2= 𝑏 ∝1= 𝑓𝑦 𝐸𝑠 ⋅ 𝑑 ∝2= 𝑁 + 𝑓𝑦 ∙ 𝐴𝑠 𝐵 ∙ 𝑑2 ∙ 𝐸𝑐 +∝1 ∅𝑦 = √∝2 2−∝1 2+∝2  Curvature at ultimate point: 𝑁 + 𝑓𝑠 𝐴𝑠 = 𝑓𝑠 ′ 𝐴𝑠 ′ + 𝑓𝑐𝑢 + 𝑓𝑐𝑚 2 𝑦 𝐵 + 2 3 𝑓𝑐𝑚 (𝐶𝑠 − 𝑦) 𝐵 𝑦 = 𝐶𝑠 (1 − 𝜀𝑐𝑚 𝜀𝑐𝑢 ) 𝐶𝑠 = 𝜀𝑐𝑢 ∅𝑢 𝑦 = 𝜀𝑐𝑢 ∅𝑢 (1 − 𝜀𝑐𝑚 𝜀𝑐𝑢 ) = 𝜀𝑐𝑢 ∅𝑢 − 𝜀𝑐𝑢 𝜀𝑐𝑚 𝜀𝑐𝑢∅𝑢 = 𝜀𝑐𝑢 − 𝜀𝑐𝑚 ∅𝑢 𝜀𝑠 = (𝑑 − 𝐶𝑠)𝜀𝑐𝑢 𝐶𝑠 = 𝑑 𝜀𝑐𝑢 𝐶𝑠 − 𝐶𝑠 𝜀𝑐𝑢 𝐶𝑠 = 𝑑 𝜀𝑐𝑢 ( ∅𝑢 𝜀𝑐𝑢 ) − 𝜀𝑐𝑢 = 𝑑 ∅𝑢 − 𝜀𝑐𝑢 (𝐶𝑠 − 𝑦) = 𝜀𝑐𝑢 ∅𝑢 − 𝜀𝑐𝑢 − 𝜀𝑐𝑚 ∅𝑢 = 𝜀𝑐𝑚 ∅𝑢 𝜀𝑠 ′ = (𝐶𝑠 − 𝑑′)𝜀𝑐𝑢 𝐶𝑠 = 𝐶𝑠 𝜀𝑐𝑢 𝐶𝑠 − 𝑑′ 𝜀𝑐𝑢 𝐶𝑠 = 𝜀𝑐𝑢 − 𝑑′ 𝜀𝑐𝑢 ( ∅𝑢 𝜀𝑐𝑢 ) = 𝜀𝑐𝑢 − 𝑑′ ∅𝑢 19 𝑓𝑠 ′ = 𝐸𝑠 𝜀𝑠 ′ = 𝐸𝑠(𝜀𝑐𝑢 − 𝑑′ ∅𝑢) = 𝐸𝑠 𝜀𝑐𝑢 − 𝑑′ 𝐸𝑠 ∅𝑢 𝑓𝑠 = 𝑓𝑦 + (𝜀𝑠 − 𝜀𝑦)(𝑓𝑢 − 𝑓𝑦) (𝜀𝑢 − 𝜀𝑦) ≅ 𝑓𝑦 𝑁 + 𝑓𝑦 𝐴𝑠 = 𝐸𝑠 𝜀𝑐𝑢 𝐴𝑠 ′ − 𝑑′ 𝐸𝑠 𝐴𝑠 ′ + 𝑓𝑐𝑢 + 𝑓𝑐𝑚 2 𝜀𝑐𝑢 − 𝜀𝑐𝑚 ∅𝑢 𝐵 + 2 3 𝑓𝑐𝑚 𝜀𝑐𝑚 ∅𝑢 𝐵 𝑁 ∅𝑢 + 𝑓𝑦 𝐴𝑠 ∅𝑢 − 𝐸𝑠 𝜀𝑐𝑢 𝐴𝑠 ′ ∅𝑢 + 𝑑′ 𝐸𝑠 𝐴𝑠 ′ ∅𝑢 2 − (𝑓𝑐𝑢 + 𝑓𝑐𝑚)(𝜀𝑐𝑢 − 𝜀𝑐𝑚) 𝐵 2 − 2 3 𝑓𝑐𝑚 𝜀𝑐𝑚 𝐵 = 0 [𝑑′ 𝐸𝑠 𝐴𝑠 ′ ]∅𝑢 2 + [𝑁 + 𝑓𝑦 𝐴𝑠 − 𝐸𝑠 𝜀𝑐𝑢 𝐴𝑠 ′ ]∅𝑢 + [− 3(𝑓𝑐𝑢 + 𝑓𝑐𝑚)(𝜀𝑐𝑢 − 𝜀𝑐𝑚) 𝐵 6 − 4 6 𝑓𝑐𝑚 𝜀𝑐𝑚 𝐵] = 0 [𝑑′ 𝐸𝑠 𝐴𝑠 ′ ]∅𝑢 2 + [𝑁 + 𝑓𝑦 𝐴𝑠 − 𝐸𝑠 𝜀𝑐𝑢 𝐴𝑠 ′ ]∅𝑢 + [ −3(𝑓𝑐𝑢 + 𝑓𝑐𝑚)(𝜀𝑐𝑢 − 𝜀𝑐𝑚) 𝐵 − 4 𝑓𝑐𝑚 𝜀𝑐𝑚 𝐵 6 ] = 0 0.5 ∅𝑢 2 + [ 𝑁 + 𝑓𝑦 𝐴𝑠 − 𝐸𝑠 𝜀𝑐𝑢 𝐴𝑠 ′ 2 𝑑′ 𝐸𝑠 𝐴𝑠 ′ ] ∅𝑢 + [ −3(𝑓𝑐𝑢 + 𝑓𝑐𝑚)(𝜀𝑐𝑢 − 𝜀𝑐𝑚) 𝐵 − 4 𝑓𝑐𝑚 𝜀𝑐𝑚 𝐵 12 𝑑′ 𝐸𝑠 𝐴𝑠 ′ ] = 0 𝜀𝑐𝑚 = 0.5 𝜀𝑐𝑢 ~ 0.9 𝜀𝑐𝑢 𝑓𝑐𝑢 = 0.2 𝑓𝑐𝑚 ~ 0.8 𝑓𝑐𝑚 0.5 ∅𝑢 2 + [ 𝑁 + 𝑓𝑦 𝐴𝑠 − 𝐸𝑠 𝜀𝑐𝑢 𝐴𝑠 ′ 2 𝑑′ 𝐸𝑠 𝐴𝑠 ′ ] ∅𝑢 + [ −8.7 𝑓𝑐𝑚 𝜀𝑐𝑢 𝐵 12 𝑑′ 𝐸𝑠 𝐴𝑠 ′ ] = 0 0.5 ∅𝑢 2 + [ 𝑁 + 𝑓𝑦 (𝐴𝑠 + 𝐴𝑠 ′′) 2 𝑑′ 𝐸𝑠 𝐴𝑠 ′ − 𝜀𝑐𝑢 2 𝑑′ ] ∅𝑢 + [ −𝑓𝑐𝑚 𝜀𝑐𝑢 𝐵 1.38 𝑑′ 𝐸𝑠 𝐴𝑠 ′ ] = 0 𝑑′ ≅ 40 0.5 ∅𝑢 2 + [ 𝑁 + 𝑓𝑦 (𝐴𝑠 + 𝐴𝑠 ′′) 16 𝐴𝑠 ′ 106 − 𝜀𝑐𝑢 80 ] ∅𝑢 + [ −𝑓𝑐𝑚 𝜀𝑐𝑢 𝐵 22 𝐴𝑠 ′ × 106 ] = 0 𝑎 = 0.5 𝑏 = 𝑁 + 𝑓𝑦 (𝐴𝑠 + 𝐴𝑠 ′′) 16 𝐴𝑠 ′ 106 − 𝜀𝑐𝑢 80 20 𝑐 = −𝑓𝑐𝑚 𝜀𝑐𝑢 𝐵 22 𝐴𝑠 ′ × 106 ∅𝑢 = √𝑏2 − 4 𝑎 𝑐 − 𝑏 2𝑎 = √𝑏2 − 2 𝑐 − 𝑏 ∝3= − 2 𝑐 ∝4= 𝑏 ∝3= 𝜀𝑐𝑢 ∙ 𝑓𝑐𝑚 ∙ 𝐵 11 ∙ 𝐴𝑠 ′ ∙ 106 ∝4= 𝑁 + 𝑓𝑦 ∙ (𝐴𝑠 + 𝐴𝑠 " ) 16 ∙ 𝐴𝑠 ′ ∙ 106 − 𝜀𝑐𝑢 80 ∅𝑢 = √∝4 2+∝3−∝4 For Wall sections: ∝3= 5 ∙ 𝜀𝑐𝑢 ∙ 𝑓𝑐𝑚 ∙ 𝐵 𝐻𝐶 ∙ 𝐴𝑠 ′ ∙ 106 ∝4= 𝑁 + 𝑓𝑦 ∙ (𝐴𝑠 + 𝐴𝑠 " ) 2 ∙ 𝐻𝐶 ∙ 𝐴𝑠 ′ ∙ 106 − 𝜀𝑐𝑢 𝐻𝐶 Where 𝑓𝑦 is steel bars yield strength, 𝑓𝑐𝑚 is maximum strength of concrete, 𝐸𝑠 is modules of elasticity of steel, 𝐸𝑐 is modules of elasticity of concrete, 𝜀𝑐𝑢 is ultimate strain of concrete, 𝐴𝑠 is tension bars reinforcement area, 𝐴𝑠 ′ is compression bars reinforcement area, 𝐴𝑠 " is middle bars reinforcement area, 𝑑 is tension bars effective depth, 𝐵 is section width, 𝑁 is applied axial force, and 𝐻𝐶 is hidden column height of shear wall. 3.5 DEVELOPED SOFTWARE A new software for estimating and plotting the moment curvature, interaction curve, and curvature ductility of reinforced concrete sections will be developed as a part of the study. Accordingly, the MCI software was created. In general, MCI software is a program that accurately analyzes and 21 calculates rectangular reinforced concrete components as well as shear walls in order to plot the moment curvature, interaction curve, and curvature ductility of afore-mentioned sections. This is generally done using the principles of finite element method by meshing the analyzed section into several layers. The program has several features and merits that enhances its performance as compared to other available software. These features and merits can be summarized as follows:  Comparing the behavior of several sections at the same time.  Evaluating the effect of various axial loads on the performance of the sections.  Considering the effect of confinement or ignoring it for a certain section with the option of allowing a comparison of the results of both cases.  Considering the influence of concrete in tension or ignoring it with the possibility of comparing the results together.  Various material properties were already inserted in the database of the program for simplicity of defining the section properties. In addition, code-based equations were programed to simplify and facilitate the task of defining the material properties.  Analyzing the section for both positive and negative bending moments with the option of comparing both cases together.  In the case of rectangular column/beam sections, it allows the user to analyze the section in both perpendicular directions.  It is possible to automatically identify some of the critical points on the moment curvature curve such as the one when the concrete reaches its maximum resistance and when it reaches the collapse point. In addition to that, in the case of confined concrete, the program determines and specifies the points were the concrete reaches its ultimate capacity and when the concrete crashes. On the other hand, when the tensile strength of concrete is considered, the point at 22 the ultimate tensile capacity and concrete failure in tension are indicated. However, in all cases, the yielding of reinforcements and their failure points are all stored in the program.  Another important point in the developed software is its capability to present the results of the analysis after the upper layers of the concrete start crushing until total collapse of the section is achieved. This feature is not available in other software since they assume that the section has failed once crushing starts at the upper region, whereas it was found that the section can hold some capacity which can maintain its integrity for a fairly long period of time especially if the confinement is well designed until the complete collapse is reached. This case was not considered in the calculation of curvature ductility since the code presumes a total collapse once the concrete has been crushed. Moment resistance of a cross section is the sum of the moment resistance of each material’s particle around the natural axis. And because of the concrete stress strain relationship nonlinearity, the cross section will be subdivided into thin horizontal layers, assuming the stresses on each layer are uniformly distributed and equal to the stress at the middle of each layer. To find the moment resistance for a curvature value ∅, the depth of the neutral axis 𝐶𝑠 is assumed then strain on each layer is determined as follow: 𝜀𝑖 = ∅ ∙ 𝑦𝑖 where 𝑦𝑖 is the distance between neutral axis and the center of layer i. After determining strains in all layers then stresses from strass-strain relationships, the sum of the forces for each section should be satisfy the equilibrium condition. When forces throughout the cross-section are in equilibrium, the bending moment can finally be determined using trial and error method. 23 For this purpose, MCI computer program was developed to make these repeated calculations and allowing the user to increase the accuracy by decreasing the layer’s thicknesses as much as possible. The program has many options for section and material properties as shown in Figure 10, and Figure 11. In addition, it was provided with options to involve increment in concrete compressive strength due to lateral confinement and concrete tensile strength and shows their effects on bending resistance. The following figure shows flowchart of the procedure in a loop: 24 Figure 9: A flowchart of the procedure for calculating moment resistance. Where N is the applied axial force, ∅𝑗 is the curvature value, 𝐶𝑠 is the depth of neutral axis, 𝑛 is the number of layers, 𝑀𝑡 is the bending resistance of the cross section, 𝑁𝑡 is the axial resistance, 𝑦𝑖 is the distance between neutral axis and the center of layer i, 𝜀𝑖 is the strain at the center of layer i, 𝑓𝑐𝑖 is the stresses on concrete in layer i, 𝑓𝑠𝑖 is the stresses on steel in layer i, 𝐹𝑐𝑖 is the concrete resistance of layer i, 𝐹𝑠𝑖 is the steel resistance of layer i, and 𝑀𝑖 is bending resistance of layer i. 25 Figure 10: A screenshot of the section define window. Figure 11: A screenshot of the material define window. 26 After calculations are done, the previous steps will be repeated for another curvature value until the cross section is unable to carry any bending moment, the following flowchart explain the whole procedure: Figure 12: A flowchart of the procedure for plotting the moment curvature curve. Where ∅0 is curvature step, 𝜇 is the ratio between the applied axial force and the axial compressive resistance of the cross section, 𝑁𝑢 is the axial compressive resistance of the cross section, 𝑀𝑡𝑗 is the bending resistance of the cross section under a specific axial force and curvature value, and 𝑀𝑚𝑎𝑥 is the maximum bending resistance of the cross section under a specific axial force. The outputs of the program are displayed into two forms, a table and a graph show the relationship between bending moment and curvature as shown in figure 14, and the maximum point of the 27 moment-curvature curve represents the bending capacity of the cross section under the current applied axial force. The maximum bending capacity values for unlimited number of axial forces ranging between zero for pure bending moment to the ultimate compressive strength of the cross section represent the interaction curve as shown in figure 15. For plotting the interaction curve, the maximum bending resistances for many axial forces as proportions of the ultimate compressive strength of the cross section are determined, as shown in the following flowchart: Figure 13: A flowchart of the procedure for plotting the interaction curve. Where 𝑁𝑖 is the applied axial force, 𝑖 is the iteration number, and 𝑀𝑚𝑎𝑥,𝑖 is the maximum bending resistance of the cross section under a specific axial force. 28 Figure 14: A screenshot of the moment curvature curve. Figure 15: A screenshot of the interaction curve. 29 This program was designed for educational purpose to study the effects of several factors on moment curvature, interaction curve and curvature ductility. Several sections with various dimensions and material properties will be compared in this thesis. The used cross sections are listed in table 3, reinforcement distribution in table 4, and material properties in table5. Table 3: Sections Dimensions. Height Width Concrete Cover Height / Width Section 1 500 300 20 1.67 Section 2 800 500 20 1.60 Section 3 1200 600 20 2.00 Wall Section 3000 300 20 10 Table 4: Sections Reinforcement. Stirrups Top reinforcement Bottom reinforcement Middle reinforcement Diameter Spacing Diameter Number Diameter Number Diameter Number Section 1 8 100 16 3 16 3 16 2 Section 2 8 80 20 3 20 5 12 2 Section 3 10 80 25 4 25 2×4 12 2×2 Wall 8 100 20 4×2 20 4×2 12 10×2 Table 5: Material Properties. Concrete Steel 𝑓𝑐𝑚 𝜀𝑐𝑚 𝜀𝑐𝑢 𝐸𝑐 𝑓𝑦 𝑓𝑢 𝐸𝑠 𝑓𝑦𝑡 Low Strength 15 0.002 0.004 18319 280 420 200000 280 Normal Strength 35 0.002 0.0032 27983 520 690 200000 280 High Strength 80 0.0024 0.0026 42306 690 760 200000 420 30 The Moment Curvature curves of MCI software were compared with Xtract software and showed a good agreement as in Figure 16, Figure 18, Figure 20, Figure 22, Figure 24, Figure 26, Figure 28, Figure 30, Figure 32, Figure 34, Figure 36, and Figure 38. A slight shift was noticed in some sections after yielding point, and the reason of this shift is because of each software uses different formulae for concrete stress strain relationship and a different shape of mesh. Interaction curves of section 1 with low, normal, and high strength materials were compared with both Xtract and ETABS software as shown in Figure 17, Figure 19, and Figure 21 and they showed a good agreement, where Eurocode 2-2004 [14] was the used code in ETABS for column design. On the other hand, since ETABS software provides interaction curve for column sections with uniformly distributed reinforcement, thus section 2, section 3, and wall section were only compared with Xtract and showed a good agreement as in Figure 23, Figure 25, Figure 27, Figure 29, Figure 31, Figure 33, Figure 35, Figure 37, and Figure 39. 31 CHAPTER 4 RESULTS AND DISCUSSIONS 5.1 MOMENT CURVATURE As previously mentioned, MCI software was developed for estimating the curvature ductility through parametric study where different RC sections at different strength characteristics of concrete with and without the consideration of confinement, axial load, and tension in concrete was performed. Furthermore, the written code for MCI software is shown in Appendix II. The software plots the interaction curve and calculates the yielding curvature, ultimate curvature, and curvature ductility accurately. Accordingly, three various strength characteristics of concrete which are low, normal, and high strength concrete were used for conducting the parametric study as well as four different percentages of axial load of 0%, 20%, 40%, and 60% were selected. Thereafter, the results of section 1, 2,3, and wall section of MCI software were compared with Xtract and ETABS for validating the outcomes. However, only results of section 1 were compared with both ETABS and Xtract while results of the other sections were compared only with Xtract since ETABS software provides interaction curve for column sections with uniformly distributed reinforcement. As can be noticed from the results, the accuracy of MCI software is somewhat similar to Xtract and ETABS. In fact, the interaction curve increases with the increase in the strength of the material for all sections. However, the results of section 2 reflected the best accuracy of MCI in comparison to Xtract while the results of wall section relatively exhibited the lowest accuracy as illustrated in Figure 17, Figure 19, Figure 21, Figure 23, Figure 25, Figure 27, Figure 29, Figure 31, Figure 33, Figure 35, Figure 37, and Figure 39. 32 5.2 EFFECT OF AXIAL LOAD In this section, the influence of the axial load is discussed in detail. Generally, the results of this study have shown that applying an axial load on a certain reinforced concrete section reduces its ductility as shown in Figure 16,Figure 18, and Figure 20, in which the ductility of section 1 has reduced by almost 80% for 20% axial load regardless of the considerable increase in the moment as compressive axial load reduces the tensile stresses on steel and delay yielding in main bars. Regarding low strength case, the increase in moment is associated with the increase in axial load from 0% to 40% where any increase in axial load results in increase in moment. Normal strength concrete case exhibited similar results where the axial load of 40% showed the highest values in terms of the moment except the case of 20% for section 2 which showed highest values. High strength case showed consistent results for all sections where the highest moment values were at 40%. On the contrary, the decrease in curvature is associated with the increase in axial load from 0% to 60% where any increase in axial load results in decrease in the curvature for all sections at all strength characteristic cases as illustrated in Figure 16, Figure 18, Figure 20, Figure 22, Figure 24, Figure 26, Figure 28, Figure 30, Figure 32, Figure 34, Figure 36, and Figure 38. This observation can mainly be attributed to the increase in compressive stresses on concrete and the decrease in tensile stresses on steel bars. Furthermore, it can be seen in Table 6, Table 7, Table 8, and Table 9 that the ductility of the section reaches its peak in the case of axially unloaded column. Whereas it reduces between 80% to 90% when an axial load of about 20% of the ultimate capacity is applied. On the other hand, when the applied axial load is below 40% of the capacity of the section bending moment increases moderately while higher axial loads results in reducing the bending capacity of the section. 33 5.3 EFFECT OF STRENGTH OF MATERIALS In this section, the effect of selecting high strength materials on the load carrying capacity, moment capacity as well as the ductility will be discussed in detail. In general, the curvature ductility of a given reinforced concrete section is influenced by three main factors as discussed by Olivia and Mandal (2015) [2]. These factors are the variation in the loads acting on the section including the rate of axial load applied, the ratio of longitudinal and transverse reinforcements, and the elastic capacity of concrete and steel martials. Similar observation was discussed by Baji and Ronagh (2011) [15] in which they highlighted that the curvature ductility increases with respect to the section size in which larger section provides higher ductility as compared to small sections and increasing the concrete’s compressive strength results in enhanced curvature ductility. The yielding moment can be referred to the peak of moment-curvature curve of an element which is directly dependent on the strength of the materials where any increase in the strength of the concrete or reinforcements results in higher yielding moment. Regarding the load carrying capacity, it is provided by the element to produce the ultimate moment which is referred to the moment acting at the capacity (ultimate). The utilization of high strength materials leads to an increase in the load carrying capacity and hence the ultimate moment capacity. As can be seen from the results, the increase in the strength of the material is accompanied with significant increase in the moment for all sections. For example, the maximum magnitude of moment for wall section at low strength case was approximately 7400 kN M while it was 14000 kN M for normal strength and 25000 kN M for high strength case. On the other hand, ductility can be described as the ability of element to maintain plastic deformation to a high degree under the effect of tensile load before reaching fracture or failure. In addition to that, ductility can be represented as the permanent deformation using stress-strain curve. Therefore, as the strength of the materials increases, the 34 difference between the moment capacity and ultimate moment increases and hence notably reducing the ductility of the material. With respect to the curvature, the increase in the strength of the material is associated with considerable reduction in the ductility for all sections. For example, the maximum magnitude of curvature for wall section at low strength case was approximately 0.0048 1 M while it was 0.003 1 M for normal strength and 0.0022 1 M for high strength case. In general, the results of this study have shown that when the tensile strength of concrete was included into analysis, an increment in moment resistance was observed for low curvature values, then a sudden decrease in resistance and returns to the same values in the case of neglecting tension in concrete. Moreover, considering the resistance of the section is the maximum resistance when tensile effect is included may causes a sudden collapse. Therefore, it is preferable to neglect tension in design. Where curvature ductility was not significantly affected, with a slight increase in the yielding curvature value, which led to a slight decrease in the curvature ductility. The effect of lateral confinement on moment resistance and ductility was neglectable before the point at which concrete begins to collapse on compression. After this point, it was noticed a delay in total collapse and confinement causes an increase in moment resistance in some cases. Therefore, a good confinement helps delay the total section collapse, but should not be included during design. By adding axial force gradually, we get the interaction curve for the three sections and it was noticed a gradual increase in the moment capacity of the section when applying a small axial force that does not exceed 30 to 40 percent of the total axial resistance, then moment capacity begins decreasing until the section loses its moment resistance when axial force equals to the maximum axial resistance. Where curvature ductility gradually decreases with the addition of axial force until it reaches a value of one, where the collapse under moment after this point becomes brittle. 35 Table 6: The curvature ductility of section 1 calculated by MCI software. Neglecting Confinement and Tension in Concrete With Tension With Confinement ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ LS 0.0 0.0042 0.0780 18.6 0.0046 0.0769 16.7 0.0042 0.0782 18.6 0.2 0.0059 0.0210 3.6 0.0063 0.0205 3.3 0.0059 0.0245 4.2 0.4 0.0048 0.0133 2.8 0.0044 0.0128 2.9 0.0050 0.0158 3.2 0.6 0.0031 0.0098 3.2 0.0031 0.0095 3.1 0.0033 0.0115 3.5 NS 0.0 0.0074 0.0556 7.5 0.0077 0.0542 7.0 0.0074 0.0555 7.5 0.2 0.0101 0.0177 1.8 0.0103 0.0171 1.7 0.0101 0.0182 1.8 0.4 0.0096 0.0104 1.1 0.0092 0.0101 1.1 0.0100 0.0110 1.1 0.6 0.0064 0.0072 1.1 0.0063 0.0072 1.1 0.0067 0.0079 1.2 HS 0.0 0.0093 0.0564 6.1 0.0096 0.0548 5.7 0.0093 0.0563 6.1 0.2 0.0146 0.0130 0.9 0.0142 0.0126 0.9 0.0145 0.0128 0.9 0.4 0.0078 0.0074 0.9 0.0078 0.0072 0.9 0.0118 0.0072 0.6 0.6 0.0047 0.0047 1.0 0.0047 0.0047 1.0 0.0080 0.0048 0.6 Table 7: The curvature ductility of section 2 calculated by MCI software. Neglecting Confinement and Tension in Concrete With Tension With Confinement ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ LS 0.0 0.0025 0.0648 25.9 0.0029 0.0618 21.3 0.0025 0.0647 25.9 0.2 0.0036 0.0141 3.9 0.0039 0.0132 3.4 0.0036 0.0171 4.8 0.4 0.0027 0.0080 3.0 0.0026 0.0075 2.9 0.0028 0.0100 3.6 0.6 0.0019 0.0059 3.1 0.0019 0.0058 3.1 0.0019 0.0071 3.7 NS 0.0 0.0044 0.0479 10.9 0.0047 0.0462 9.8 0.0044 0.0477 10.8 0.2 0.0061 0.0111 1.8 0.0063 0.0105 1.7 0.0061 0.0119 2.0 36 0.4 0.0056 0.0063 1.1 0.0054 0.0061 1.1 0.0059 0.0069 1.2 0.6 0.0038 0.0044 1.2 0.0038 0.0044 1.2 0.0039 0.0048 1.2 HS 0.0 0.0056 0.0512 9.1 0.0058 0.0491 8.5 0.0056 0.0512 9.1 0.2 0.0079 0.0082 1.0 0.0083 0.0078 0.9 0.0085 0.0080 0.9 0.4 0.0047 0.0046 1.0 0.0047 0.0045 1.0 0.0072 0.0045 0.6 0.6 0.0029 0.0029 1.0 0.0029 0.0029 1.0 0.0050 0.0030 0.6 Table 8: The curvature ductility of section 3 calculated by MCI software. Neglecting Confinement and Tension in Concrete With Tension With Confinement ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ LS 0.0 0.0018 0.0661 36.7 0.0020 0.0651 32.6 0.0018 0.0661 36.7 0.2 0.0026 0.0105 4.0 0.0028 0.0100 3.6 0.0026 0.0131 5.0 0.4 0.0018 0.0055 3.1 0.0017 0.0052 3.1 0.0019 0.0072 3.8 0.6 0.0012 0.0040 3.3 0.0012 0.0039 3.3 0.0013 0.0050 3.8 NS 0.0 0.0031 0.0462 14.9 0.0033 0.0451 13.7 0.0031 0.0461 14.9 0.2 0.0044 0.0080 1.8 0.0045 0.0077 1.7 0.0044 0.0088 2.0 0.4 0.0038 0.0043 1.1 0.0036 0.0042 1.2 0.0040 0.0048 1.2 0.6 0.0025 0.0029 1.2 0.0025 0.0029 1.2 0.0026 0.0034 1.3 HS 0.0 0.0039 0.0473 12.1 0.0041 0.0458 11.2 0.0039 0.0472 12.1 0.2 0.0061 0.0057 0.9 0.0059 0.0055 0.9 0.0061 0.0056 0.9 0.4 0.0031 0.0031 1.0 0.0031 0.0030 1.0 0.0049 0.0030 0.6 0.6 0.0017 0.0017 1.0 0.0018 0.0018 1.0 0.0033 0.0020 0.6 Table 9: The curvature ductility of wall section calculated by MCI software. Neglecting Confinement and Tension in Concrete With Tension With Confinement 37 ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ LS 0.0 0.0007 0.0109 15.6 0.0007 0.0106 15.1 0.0007 0.0119 17.0 0.2 0.0009 0.0040 4.4 0.0010 0.0038 3.8 0.0009 0.0046 5.1 0.4 0.0007 0.0023 3.3 0.0007 0.0019 2.7 0.0007 0.0022 3.1 0.6 0.0005 0.0017 3.4 0.0005 0.0016 3.2 0.0005 0.0019 3.8 NS 0.0 0.0011 0.0098 8.9 0.0012 0.0095 7.9 0.0011 0.0101 9.2 0.2 0.0016 0.0031 1.9 0.0016 0.0029 1.8 0.0016 0.0033 2.1 0.4 0.0015 0.0018 1.2 0.0014 0.0017 1.2 0.0015 0.0019 1.3 0.6 0.0010 0.0012 1.2 0.0010 0.0012 1.2 0.0011 0.0014 1.3 HS 0.0 0.0014 0.0105 7.5 0.0015 0.0102 6.8 0.0014 0.0104 7.4 0.2 0.0020 0.0023 1.2 0.0021 0.0022 1.0 0.0020 0.0023 1.2 0.4 0.0012 0.0012 1.0 0.0013 0.0012 0.9 0.0019 0.0013 0.7 0.6 0.0007 0.0007 1.0 0.0007 0.0007 1.0 0.0013 0.0009 0.7 5.4 PROPOSED DUCTILITY EQUATIONS As mentioned earlier, curvature ductility is important for design. To calculate ductility, ultimate curvature is divided by yielding curvature. After analyzing several sections with various properties using the program that was designed for this purpose the equations shown in the following table were proposed: Table 10: Curvature ductility equations for rectangular and wall sections. For Rectangular Sections ∝1= 𝑓𝑦 𝐸𝑠 ⋅ 𝑑 ∝2= 𝑁 + 𝑓𝑦 ∙ 𝐴𝑠 𝐵 ∙ 𝑑2 ∙ 𝐸𝑐 +∝1 ∝3= 𝜀𝑐𝑢 ∙ 𝑓𝑐𝑚 ∙ 𝐵 16 ∙ 𝐴𝑠 ′ ∙ 106 ∝4= 𝑁 + 𝑓𝑦 ∙ (𝐴𝑠 + 𝐴𝑠 " ) 16 ∙ 𝐴𝑠 ′ ∙ 106 − 𝜀𝑐𝑢 80 ∅𝑦 = √∝2 2−∝1 2+∝2 ∅𝑢 = √∝4 2+∝3−∝4 𝜂∅ = ∅𝑢 ∅𝑦 ≥ 1 38 For Wall Sections ∝1= 𝑓𝑦 𝐸𝑠 ⋅ 𝑑 ∝2= 𝑁 + 𝑓𝑦 ∙ 𝐴𝑠 𝐵 ∙ 𝑑2 ∙ 𝐸𝑐 +∝1 ∝3= 5 ∙ 𝜀𝑐𝑢 ∙ 𝑓𝑐𝑚 ∙ 𝐵 𝐻𝐶 ∙ 𝐴𝑠 ′ ∙ 106 ∝4= 𝑁 + 𝑓𝑦 ∙ (𝐴𝑠 + 𝐴𝑠 " ) 2 ∙ 𝐻𝐶 ∙ 𝐴𝑠 ′ ∙ 106 − 𝜀𝑐𝑢 𝐻𝐶 ∅𝑦 = √∝2 2−∝1 2+∝2 ∅𝑢 = √∝4 2+∝3−∝4 𝜂∅ = ∅𝑢 ∅𝑦 ≥ 1 5.5 PROPOSED DUCTILITY EQUATIONS VS SOFTWARE RESULTS The proposed equation for estimating the ductility yielded similar results to MCI software with a mean error of ± 15% in the case of absence of axial force and a mean error of ± 22% in the case that axial force is available as shown in Tables 11 & 12 and Figures 48, 49, 50 & 51. In fact, the yield curvature of reinforced concrete section can be affected by the dimensions of its cross-section, the rate of the applied axial load, the characteristic strength of concrete, and slightly by the thickness of the concrete cover and the amount of longitudinal reinforcement according to Sheikh et al. (2010) [16]. Table 11: Comparison of curvature ductility between MCI software and the proposed equation without axial force. 𝜇 = 0 MCI Software Proposed Equation Error % ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ #1-LS 0.0042 0.0780 18.6 0.0041 0.0769 18.8 -2 % -1 % 1 % #1-NS 0.0074 0.0556 7.5 0.0072 0.0584 8.1 -3 % 5 % 8 % #1-HS 0.0093 0.0564 6.1 0.0091 0.0653 7.2 -2 % 16 % 18 % #2-LS 0.0025 0.0648 25.9 0.0025 0.0729 29.4 0 % 13 % 14 % #2-NS 0.0044 0.0479 10.9 0.0043 0.0547 12.6 -2 % 14 % 16 % 39 #2-HS 0.0056 0.0512 9.1 0.0055 0.0621 11.3 -2 % 21 % 24 % #3-LS 0.0018 0.0661 36.7 0.0017 0.0532 31.0 -6 % -20 % -16 % #3-NS 0.0031 0.0462 14.9 0.0030 0.0321 10.8 -3 % -31 % -28 % #3-HS 0.0039 0.0473 12.1 0.0038 0.0355 9.5 -3 % -25 % -21 % W-LS 0.0007 0.0109 15.6 0.0006 0.0112 18.5 -14 % 3 % 19 % W-NS 0.0011 0.0098 8.9 0.0011 0.0084 7.9 0 % -14 % -11 % W-HS 0.0014 0.0105 7.5 0.0014 0.0096 7.0 0 % -9 % -7 % Table 12: Comparison of curvature ductility between MCI software and the proposed equation with 20% axial force. 𝜇 = 0.2 MCI Software Proposed Equation Error % ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ #1-LS 0.0059 0.0210 3.6 0.0056 0.0279 5.0 -5 % 33 % 39 % #1-NS 0.0101 0.0177 1.8 0.0096 0.0171 1.8 -5 % -3 % 0 % #1-HS 0.0130 0.0130 1.0 0.0127 0.0149 1.2 -2 % 15 % 20 % #2-LS 0.0036 0.0141 3.9 0.0033 0.0178 5.3 -8 % 26 % 36 % #2-NS 0.0061 0.0111 1.8 0.0058 0.0117 2.0 -5 % 5 % 11 % #2-HS 0.0079 0.0082 1.0 0.0076 0.0100 1.3 -4 % 22 % 30 % #3-LS 0.0026 0.0105 4.0 0.0023 0.0112 5.0 -12 % 7 % 25 % #3-NS 0.0044 0.0080 1.8 0.0039 0.0072 1.9 -11 % -10 % 6 % #3-HS 0.0057 0.0057 1.0 0.0051 0.0063 1.2 -11 % 11 % 20 % W-LS 0.0009 0.0040 4.4 0.0009 0.0032 3.4 0 % -20 % -23 % W-NS 0.0016 0.0031 1.9 0.0016 0.0021 1.3 0 % -32 % -32 % W-HS 0.0020 0.0023 1.2 0.0021 0.0018 1.0 5 % -22 % -17 % 40 5.6 OLIVIA`S EQUATION FOR DUCTILITY VS SOFTWARE RESULTS In fact, a detailed comparison between the developed software herein and the approach suggested by Olivia & Mandal (2015) [2] has been conducted to evaluate the reliability of the proposed mathematical formulae. Generally, the results have shown a variation between both methodologies with an error of 56% and 126% for section 1 and 2 as well as the wall section respectively in the case of close values of tensile and compression reinforcements. These considerable differences can be mainly attributed to the basic assumption of these methods in which Olivia & Mandal assume that the compression reinforcements reach their yielding capacity before the crushing. This assumption cannot be generalized for all cases since it is applicable only when the ratio of compression reinforcements steel is relatively small as compared to the ratio of tensile reinforcements as seen in section 3 where the error was between 7% and 37% as shown in Table 13 and Figures 48, 49, 50 & 51. On the other hand, Olivia & Mandal’s model cannot predict the cases of axially loaded sections. Table 13: Comparison of curvature ductility between MCI software and Olivia & Mandal’s equation without axial force. 𝜇 = 0 MCI Software Olivia & Mandal Equation Error % ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ ∅𝑦 ∅𝑢 𝜂∅ #1-LS 0.0042 0.0780 18.6 0.0040 0.1155 29.1 -5 % 48 % 56 % #1-NS 0.0074 0.0556 7.5 0.0070 0.1094 15.5 -5 % 97 % 107 % #1-HS 0.0093 0.0564 6.1 0.0090 0.1243 13.8 -3 % 120 % 126 % #2-LS 0.0025 0.0648 25.9 0.0024 0.0906 37.4 -4 % 40 % 44 % #2-NS 0.0044 0.0479 10.9 0.0043 0.0858 20.0 -2 % 79 % 83 % #2-HS 0.0056 0.0512 9.1 0.0055 0.0975 17.8 -2 % 90 % 96 % #3-LS 0.0018 0.0661 36.7 0.0017 0.0385 23.1 -6 % -42 % -37 % 41 #3-NS 0.0031 0.0462 14.9 0.0029 0.0364 12.4 -6 % -21 % -17 % #3-HS 0.0039 0.0473 12.1 0.0037 0.0414 11.2 -5 % -12 % -7 % W-LS 0.0007 0.0109 15.6 0.0006 0.0205 34.5 -14 % 88 % 121 % W-NS 0.0011 0.0098 8.9 0.0011 0.0194 18.3 0 % 98 % 106 % W-HS 0.0014 0.0105 7.5 0.0014 0.0221 16.2 0 % 110 % 116 % 42 CHAPTER 5 CONCLUSIONS This thesis has focused on estimating the curvature ductility of different RC sections at different strength characteristics of concrete with and without the application of axial load, confinement, and tension in concrete using a developed software and a proposed equation. As a part of the research, a parametric study was conducted to evaluate the influencing parameters on the curvature ductility of different RC sections. In general, the developed software aimed to determine the moment curvature, interaction curve, and curvature ductility of rectangular RC sections and shear walls. The importance of this software is attributed to its enhanced features and capabilities in terms of the possibility of comparing different sections together, assessing the effect of axial load value on the section properties, considering the effects of confinement, and concrete’s tensile strength as well as simply defining the material properties of the section either in a manual way or through a code- based equation. In addition to that, it automatically highlights the crucial points of a given curve in which the material achieved its ultimate capacity or has failed. Moreover, it is based on a sophisticated approach by means of conducting the analysis in the post-crush behavior through analyzing the section beyond the state of collapse of the upper layers of the concrete. The results of this study have shown a significant improvement in the proposed approach against the currently available ones. Furthermore, the developed program was capable of analyzing and estimating the moment curvature, interaction curve, and curvature ductility of different RC sections with good accuracy in comparison to both ETABS and Xtract. In fact, the results of the 43 parametric study have presented a considerable independency of the sectional ductility on the level of confinement and the tensile strength of the used concrete. Regarding the developed mathematical formulae, the ductility equation produced a good accuracy and has a good agreement with Olivia & Mandal`s equation except in the case of high compression reinforcement ratio where some notable differences were seen since Olivia & Mandal`s equation presumes that the compression reinforcement always reaches the yielding point before crushing. This study is expected to help practicing engineers in their daily works through reliably estimating the behavior of reinforced concrete section. 44 REFERENCES 1. Park, R. & Ruitong, D. (1988). Ductility of Doubly Reinforced Concrete Beam Section. Structural Journal 85, 217-225. 2. Olivia, M., & Mandal, P. (2015). Curvature Ductility of Reinforced Concrete Beam. Journal of Civil Engineering, 6(1), 1-13. 3. Carreira, J. & Chu, K. H. (1985). Stress-Strain Relationship for Plain Concrete in Compression. ACI Journal, 82-72. 4. Saatcioglu, M. & Razvi, S. (1992). Strength and Ductility of Confined Concrete. Journal of Structural Engineering, (6). 118. 5. Suzuki, M., Akiyama, M., Hong, K. N., Cameron, I. D. & Wang, W. L. (2004). Stress- Strain Model of High-Strength Concrete Confined by Rectangular Ties. World Conference on Earthquake Engineering (13), 3330. 6. Fafitis, A. & Shah, S. P. (1985). Lateral Reinforcement for High- Strength Concrete Columns. ACI Journal, 213-232. 7. Légeron, F. & Paultre, P. (2003). Uniaxial Confinement Model for Normal- and High- Strength Concrete Columns. Journal of Structural Engineering, 129 (2). 8. Mander, J. B., Priestley, M. J. N. & Park, R. (1988). Observed Stress‐ Strain Behavior of Confined Concrete. Journal of Structural Engineering, 114 (8). 9. Sargin, M. & Handa, V. K. (1968). Structural Concrete and Some Numerical Solutions. ACM National Conference (23), 563-574. 10. Jaber, A., Gorgis, I. & Hassan, M. (2018). Relationship Between Splitting Tensile and Compressive Strengths for Self-Compacting Concrete Containing Nano- and Micro Silica. MATEC Web of Conferences, 162, 02013. 45 11. Karadoğan, F., Pala, S., Yüksel, E. & Durgun, Y. (2015). Yapı Mühendisliğine Giriş Yapısal Çözümleme Cilt: 2. Birsen Yayınları. 12. Kaklauskas, G. (1999). A New Stress-Strain Relationship for Cracked Tensile Concrete in Flexure. Statyba- Civil Engineering, 349-356. 13. Khatulistiani, U., Tavio1, T. & Raka, G. P. (2020). Stress-Strain Behavior of Steel Bars with Long Nut Connection. Journal of Physics, 1469. 14. Eurocode 2 (2004). Design of Concrete Structures. British Standards Institution. 15. Baji, H., & Ronagh, H. (2011). Investigation of Ductility of RC Beams Designed Based on AS3600. 16. Sheikh, M. N., Tsang, H. H., McCarthy, T. J., & Lam, N. T. K. (2010). Yield Curvature for Seismic Design of Circular Reinforced Concrete Columns. Magazine of Concrete Research, 62(10), 741-748. 46 APPENDIX I: LIST OF FIGURES Figure 16:Moment Curvature Curves of Section 1 – Low Strength Materials Figure 17:Interaction Curve of Section 1 – Low Strength Materials 47 Figure 18:Moment Curvature Curves of Section 1 – Normal Strength Materials Figure 19:Interaction Curve of Section 1 – Normal Strength Materials 48 Figure 20:Moment Curvature Curves of Section 1 – High Strength Materials Figure 21:Interaction Curve of Section 1 – High Strength Materials 49 Figure 22:Moment Curvature Curves of Section 2 – Low Strength Materials Figure 23:Interaction Curve of Section 2 – Low Strength Materials 50 Figure 24:Moment Curvature Curves of Section 2 – Normal Strength Materials Figure 25:Interaction Curve of Section 2 – Normal Strength Materials 51 Figure 26:Moment Curvature Curves of Section 2 – High Strength Materials Figure 27:Interaction Curve of Section 2 – High Strength Materials 52 Figure 28:Moment Curvature Curves of Section 3 – Low Strength Materials Figure 29:Interaction Curve of Section 3 – Low Strength Materials 53 Figure 30:Moment Curvature Curves of Section 3 – Normal Strength Materials Figure 31:Interaction Curve of Section 3 – Normal Strength Materials 54 Figure 32:Moment Curvature Curves of Section 3 – High Strength Materials Figure 33:Interaction Curve of Section 3 – High Strength Materials 55 Figure 34:Moment Curvature Curves of Wall Section – Low Strength Materials Figure 35:Interaction Curve of Wall Section – Low Strength Materials 56 Figure 36:Moment Curvature Curves of Wall Section – Normal Strength Materials Figure 37:Interaction Curve of Wall Section – Normal Strength Materials 57 Figure 38:Moment Curvature Curves of Wall Section – High Strength Materials Figure 39:Interaction Curve of Wall Section – High Strength Materials 58 (a) (b) (c) (d) Figure 40: Moment curvature curve of section 1 59 (a) (b) (c) (d) Figure 41: Moment curvature curve of section 2 60 (a) (b) (c) (d) Figure 42: Moment curvature curve of section 3 61 (a) (b) (c) (d) Figure 43: Moment curvature curve of wall section 62 (a) (b) (c) (d) Low Strength Materials 63 (a) (b) (c) (d) Normal Strength Materials 64 (a) (b) (c) (d) High Strength Materials Figure 44: Moment curvature curve of section 1 65 (a) (b) (c) (d) Low Strength Materials 66 (a) (b) (c) (d) Normal Strength Materials 67 (a) (b) (c) (d) High Strength Materials Figure 45: Moment curvature curve of section 2 68 (a) (b) (c) (d) Low Strength Materials 69 (a) (b) (c) (d) Normal Strength Materials 70 (a) (b) (c) (d) High Strength Materials Figure 46: Moment curvature curve of section 3 71 (a) (b) (c) (d) Low Strength Materials 72 (a) (b) (c) (d) Normal Strength Materials 73 (a) (b) (c) (d) High Strength Materials Figure 47: Moment curvature curve of the wall section 74 Figure 48: The Influence of Material Properties on Curvature Ductility of Section #1 Figure 49: The Influence of Material Properties on Curvature Ductility of Section #2 75 Figure 50: The Influence of Material Properties on Curvature Ductility of Section #3 Figure 51: The Influence of Material Properties on Curvature Ductility of Wall Section 76 APPENDIX II: CODE WRITTEN FOR MCI SOFTWARE 1. Imports System.Windows.Forms.DataVisualization.Charting 2. Public Class MainForm 3. Public section_height, section_width, concrete_cover, unconfined_layer_thickness As Double 4. Public depth_of_first_row_top_rein, depth_of_top_rein, depth_of_first_row_middle_re in, depth_of_first_row_bottom_rein, depth_of_bottom_rein As Double 5. Public top_rein_dim, middle_rein_dim, bottom_rein_dim, stirrups_dim As Double 6. Public top_rein_num_per_row, middle_rein_num_per_row, bottom_rein_num_per_row As In teger 7. Public top_rein_rows_spacing, middle_rein_rows_spacing, bottom_rein_rows_spacing, s tirrups_spacing As Double 8. Public top_rein_rows_num, middle_rein_rows_num, bottom_rein_rows_num, layers_num As Integer 9. Public layer_thickness, layer_thickness_new As Double 10. Public section_area, top_rein_row_area, middle_rein_row_area, bottom_rein_row_area, rein_area As Double 11. Public E_s, E_c, f_cm, f_ccm, f_cu, f_ccu, f_y, ep_y, ep_u, f_u, f_yt, f_ctm, f_ctu , ep_cm, ep_ccm, E_des, ep_cu, ep_ccu, ep_ccu0, ep_cu0, ep_ctm, ep_ctu, ep_ctu0, C_s, c urvature, X_m, X_mm, mio As Double 12. Public curvatures_iterations As Integer 13. Public steel_moment, steel_force, conc_moment, conc_force, applied_force, beta As D ouble 14. Public max_moment_piont, max_curvature_point, max_moment_force_piont, interaction_p oints As Integer 15. Public max_momnet_value, max_curvature_value, max_moment_force_value As Double 16. Public confined, limit_check(8), failure_at_cu, failure_at_fu, failure_at_fy, inter action As Boolean 17. Public limit_iteration(8) As Integer 18. Public concrete_reduction_factor, steel_reduction_factor, force_reduction_factor, m oment_reduction_factor, pure_force_reduction_factor As Double 19. Public limit_value(8), moment_curvature_table(1, 1), moment_force_table(1, 1) As Do uble 20. Public ConfinementFormula(2), TensionFormula(10), confinement_formula, tension_form ula, sections(12) As String 21. Dim confinment_formulae_num, tension_formulae_num As Integer 22. 23. Sub FilFormulaeSB() 24. confinment_formulae_num = 2 25. ReDim ConfinementFormula(confinment_formulae_num) 26. ConfinementFormula(0) = "Suzuki et al. (2004)" 27. ConfinementFormula(1) = "Legeron and Paultre (2003)" 28. ConfinementFormula(confinment_formulae_num) = "Neglect Confinment" 29. tension_formulae_num = 10 30. ReDim TensionFormula(tension_formulae_num) 31. TensionFormula(0) = "ACI 318" 32. TensionFormula(1) = "ACI 363R" 33. TensionFormula(2) = "Gardner" 34. TensionFormula(3) = "Nihal" 35. TensionFormula(4) = "JCI" 36. TensionFormula(5) = "JSCE" 37. TensionFormula(6) = "CEB-FIB" 38. TensionFormula(7) = "Raphael" 77 39. TensionFormula(8) = "Ahmad and Shah" 40. TensionFormula(9) = "Oloukun et al." 41. TensionFormula(tension_formulae_num) = "Neglect Tension" 42. End Sub 43. Function RoundUP(ByVal num As Double, ByVal digit As Integer) As Double 44. Dim new_num As Double 45. new_num = Math.Round(num, digit) 46. If new_num < num Then 47. new_num = new_num + 1 / (10 ^ digit) 48. End If 49. Return new_num 50. End Function 51. Sub InitialValuesSB() 52. FilFormulaeSB() 53. section_height = 600 '[mm] 54. section_width = 400 '[mm] 55. concrete_cover = 20 '[mm] 56. E_s = 200000 '[MPa] 57. f_cm = 25 '[MPa] 58. f_ccm = 50 '[MPa] 59. f_cu = 10 '[MPa] 60. f_ccu = 20 '[MPa] 61. ep_cm = 0.002 '[-] 62. ep_ccm = 0.01 '[-] 63. ep_cu = 0.0038 '[-] 64. ep_cu0 = 1.2 * ep_cu 65. ep_ccu = 0.02 '[-] 66. f_y = 240 '[MPa] 67. f_u = 300 '[MPa] 68. f_yt = 240 '[MPa] 69. ep_u = 0.2 '[-] 70. top_rein_rows_spacing = 100 '[mm] 71. bottom_rein_rows_spacing = 100 '[mm] 72. layer_thickness = 5 '[mm] 73. X_m = 0.1 '[rad/m] 74. mio = 0.9 '[-] 75. curvatures_iterations = 1000 '[-] 76. confined = True 77. top_rein_dim = 16 '[mm] 78. middle_rein_dim = 10 '[mm] 79. bottom_rein_dim = 16 '[mm] 80. stirrups_dim = 8 '[mm] 81. top_rein_num_per_row = 3 '[bars] 82. middle_rein_num_per_row = 2 '[bars] constant 83. bottom_rein_num_per_row = 4 '[bars] 84. top_rein_rows_num = 1 '[rows] 85. middle_rein_rows_num = 2 '[rows] 86. bottom_rein_rows_num = 1 '[rows] 87. concrete_reduction_factor = 1 88. steel_reduction_factor = 1 89. stirrups_spacing = 200 '[mm] 90. tension_formula = TensionFormula(10) 91. confinement_formula = ConfinementFormula(0) 92. pure_force_reduction_factor = 1 93. interaction_points = 20 94. End Sub 95. Sub TensileStrengthSB() 96. Dim f_spt, k, n, ro As Double 97. ro = rein_area / section_area * 100 98. If tension_formula = TensionFormula(0) Then 99. k = 0.56 78 100. n = 0.5 101. ElseIf tension_formula = TensionFormula(1) Then 102. k = 0.59 103. n = 0.5 104. ElseIf tension_formula = TensionFormula(2) Then 105. k = 0.47 106. n = 0.59 107. ElseIf tension_formula = TensionFormula(3) Then 108. k = 0.387 109. n = 0.63 110. ElseIf tension_formula = TensionFormula(4) Then 111. k = 0.13 112. n = 0.85 113. ElseIf tension_formula = TensionFormula(5) Then 114. k = 0.23 115. n = 0.67 116. ElseIf tension_formula = TensionFormula(6) Then 117. k = 0.3 118. n = 0.67 119. ElseIf tension_formula = TensionFormula(7) Then 120. k = 0.313 121. n = 0.667 122. ElseIf tension_formula = TensionFormula(8) Then 123. k = 0.462 124. n = 0.55 125. ElseIf tension_formula = TensionFormula(9) Then 126. k = 0.294 127. n = 0.69 128. ElseIf tension_formula = TensionFormula(10) Then 129. k = 0 130. n = 1 131. End If 132. f_spt = k * f_cm ^ n 133. f_ctm = f_spt * 4 / 3 134. f_ctu = 0 135. ep_ctm = f_ctm / E_c 136. If ro < 2 Then 137. beta = 32.8 - 27.6 * ro + 7.12 * ro ^ 2 138. Else 139. beta = 5 140. End If 141. ep_ctu = beta * ep_ctm 142. End Sub 143. Sub ResetMomentCurvatureSB() 144. max_curvature_point = curvatures_iterations 145. TrackBar1.Maximum = max_curvature_point 146. End Sub 147. 148. Sub ConfiStrenghtSB() 149. Dim b_c, d_c, k_e, wi_2, ro_cc, ro_wx, ro_wy, f_scx, f_scy, ro_ex, ro_ey , ro_e, kapa_x, kapa_y As Double 150. b_c = section_width - concrete_cover * 2 - stirrups_dim 151. d_c = section_height - concrete_cover * 2 - stirrups_dim 152. ro_cc = rein_area / b_c / d_c 153. ro_wx = (Math.PI * stirrups_dim ^ 2 / 4) * 2 / (stirrups_spacing * b_c) 154. ro_wy = (Math.PI * stirrups_dim ^ 2 / 4) * 2 / (stirrups_spacing * d_c) 155. wi_2 = (top_rein_num_per_row - 1) * ((b_c - stirrups_dim - top_rein_dim) / (top_rein_num_per_row - 1)) ^ 2 79 156. wi_2 = wi_2 + (bottom_rein_num_per_row - 1) * ((b_c - stirrups_dim - bot tom_rein_dim) / (bottom_rein_num_per_row - 1)) ^ 2 157. wi_2 = wi_2 + 2 * (top_rein_rows_num - 1) * top_rein_rows_spacing ^ 2 158. wi_2 = wi_2 + 2 * (bottom_rein_rows_num - 1) * bottom_rein_rows_spacing ^ 2 159. wi_2 = wi_2 + 2 * (middle_rein_rows_num + 1) * middle_rein_rows_spacing ^ 2 160. k_e = ((1 - wi_2 / (6 * b_c * d_c)) * (1 - stirrups_spacing / 2 / b_c) * (1 - stirrups_spacing / 2 / d_c)) / (1 - ro_cc) 161. If confinement_formula = ConfinementFormula(0) Then 162. f_scx = MinFN(E_s * (0.45 * ep_cm + 0.73 * (k_e * ro_wx / f_cm) ^ 0. 7), f_yt) 163. f_scy = MinFN(E_s * (0.45 * ep_cm + 0.73 * (k_e * ro_wy / f_cm) ^ 0. 7), f_yt) 164. ro_ex = k_e * ro_wx * f_scx 165. ro_ey = k_e * ro_wy * f_scy 166. ro_e = (ro_ex * b_c + ro_ey * d_c) / (b_c + d_c) 167. f_ccm = (1 + 4.1 * (ro_e / f_cm) ^ 0.7) * f_cm 168. ep_ccm = ep_cm + 0.015 * (ro_e / f_cm) ^ 0.56 169. f_ccu = 0.2 * f_ccm 170. ElseIf confinement_formula = ConfinementFormula(1) Then 171. kapa_x = f_cm / ((k_e * ro_wx) * E_s * ep_cm) 172. If kapa_x > 10 Then 173. f_scx = MinFN(MaxFN(0.25 * f_cm / ((k_e * ro_wx) * (kapa_x - 10) ), 0.43 * ep_cm * E_s), f_yt) 174. Else 175. f_scx = f_yt 176. End If 177. kapa_y = f_cm / ((k_e * ro_wy) * E_s * ep_cm) 178. If kapa_y > 10 Then 179. f_scy = MinFN(MaxFN(0.25 * f_cm / ((k_e * ro_wy) * (kapa_y - 10) ), 0.43 * ep_cm * E_s), f_yt) 180. Else 181. f_scy = f_yt 182. End If 183. ro_ex = k_e * ro_wx * f_scx 184. ro_ey = k_e * ro_wy * f_scy 185. ro_e = (ro_ex * b_c + ro_ey * d_c) / (b_c + d_c) 186. f_ccm = f_cm * (- 1.254 + 2.254 * (1 + 7.94 * ro_e / f_cm) ^ 0.5 - 2 * ro_e / f_cm) 187. ep_ccm = ep_cm * (1 + 5 * (f_ccm / f_cm - 1)) 188. f_ccu = 0.2 * f_ccm 189. ElseIf confinement_formula = ConfinementFormula(2) Then 190. f_ccm = f_cm 191. ep_ccm = ep_cm 192. f_ccu = 0.2 * f_ccm 193. End If 194. E_des = 0.026 * f_cm ^ 3 / ro_e ^ 0.4 195. ep_ccu = ep_ccm + 0.8 * f_ccm / E_des 196. ep_ccu0 = 1.2 * ep_ccu 197. limit_value(1) = ep_cm 198. limit_value(2) = ep_cu 199. limit_value(3) = ep_ccm 200. limit_value(4) = ep_ccu 201. limit_value(5) = ep_ctm 202. limit_value(6) = ep_ctu 203. limit_value(7) = ep_y 204. limit_value(8) = ep_u 205. End Sub 206. Sub ParametersCalcSB() 207. Dim k_3 As Double 80 208. ep_y = f_y / E_s 209. E_c = 4730 * (f_cm) ^ 0.5 210. k_3 = MinFN(40 / f_cm, 1) 211. ep_cm = 0.0028 - 0.0008 * k_3 212. ep_cu = MaxFN(0.0078 / f_cm ^ 0.25, ep_cm) 213. X_mm = X_m / 1000 '[rad/mm] 214. depth_of_first_row_top_rein = concrete_cover + stirrups_dim + top_rein_d im / 2 215. depth_of_first_row_bottom_rein = section_height - concrete_cover - stirr ups_dim - bottom_rein_dim / 2 - bottom_rein_rows_spacing * (bottom_rein_rows_num - 1) 216. middle_rein_rows_spacing = (depth_of_first_row_bottom_rein - depth_of_fi rst_row_top_rein - top_rein_rows_spacing * (top_rein_rows_num - 1)) / (middle_rein_rows _num + 1) 217. depth_of_first_row_middle_rein = depth_of_first_row_top_rein + top_rein_ rows_spacing * (top_rein_rows_num - 1) + middle_rein_rows_spacing 218. depth_of_top_rein = depth_of_first_row_top_rein + top_rein_rows_spacing * (top_rein_rows_num - 1) / 2 219. depth_of_bottom_rein = depth_of_first_row_bottom_rein + bottom_rein_rows _spacing * (bottom_rein_rows_num - 1) / 2 220. unconfined_layer_thickness = depth_of_first_row_top_rein 221. layers_num = Math.Round(section_height / layer_thickness, 0) 222. layer_thickness_new = section_height / layers_num 223. top_rein_row_area = Math.PI * top_rein_dim ^ 2 / 4 * top_rein_num_per_ro w 224. middle_rein_row_area = Math.PI * middle_rein_dim ^ 2 / 4 * middle_rein_n um_per_row 225. bottom_rein_row_area = Math.PI * bottom_rein_dim ^ 2 / 4 * bottom_rein_n um_per_row 226. rein_area = top_rein_row_area * top_rein_rows_num + middle_rein_row_area * middle_rein_rows_num + bottom_rein_row_area * bottom_rein_rows_num 227. section_area = section_height * section_width 228. applied_force = SectionAxialRsistanceFN(False) * mio 229. ConfiStrenghtSB() 230. TensileStrengthSB() 231. End Sub 232. 233. Function ConStressFN(ByVal ep_layer As Double, ByVal confined As Boolean) As Double 234. Dim stress, abs_ep As Double 235. abs_ep = Math.Abs(ep_layer) 236. If ep_layer <= 0 Then 237. If confined Then 238. If abs_ep <= ep_ccm Then 'if concrete layer is confined 239. stress = - f_ccm * (1 - (1 - abs_ep / ep_ccm) ^ (E_c * ep_ccm / f_ccm)) 240. ElseIf abs_ep <= ep_ccu Then 241. stress = -f_ccm + E_des * (abs_ep - ep_ccm) 242. Else 243. stress = -f_ccu 244. End If 245. Else 246. If abs_ep <= ep_cm Then 'if concrete layer is confined 247. stress = - f_cm * (1 - (1 - abs_ep / ep_cm) ^ (E_c * ep_cm / f_cm)) 248. ElseIf abs_ep <= ep_cu Then 249. stress = - (f_cm + (f_cu - f_cm) / (ep_cu - ep_cm) * (abs_ep - ep_cm)) 250. Else 251. stress = 0 252. End If 253. End If 81 254. Else 255. If abs_ep <= ep_ctm Then 'if concrete layer is under tension 256. stress = abs_ep * E_c 257. ElseIf abs_ep <= ep_ctu Then 258. stress = 0.625 * f_ctm * (1 - (abs_ep / ep_ctm) / beta + (1 + 0. 6 * beta) / (beta * (abs_ep / ep_ctm))) 259. Else 260. stress = 0 261. End If 262. End If 263. Return stress * concrete_reduction_factor 264. End Function 265. Function SteelStressFN(ByVal ep_depth As Double) 266. Dim stress, abs_ep As Double 267. abs_ep = Math.Abs(ep_depth) 268. If abs_ep < ep_y Then 269. stress = abs_ep * E_s 270. ElseIf abs_ep <= ep_u Then 271. stress = (f_u - f_y) / (ep_u - ep_y) * (abs_ep - ep_y) + f_y 272. Else 273. stress = 0 274. End If 275. stress = stress * Math.Sign(ep_depth) 276. Return stress * steel_reduction_factor 277. End Function 278. Sub SteelMomentForceSB(ByVal C_s As Double, ByVal X_mm As Double, check As B oolean) 279. Dim arm, centroid, ep_steel As Double 280. steel_moment = applied_force * section_height / 2 281. steel_force = applied_force 282. For i = 0 To top_rein_rows_num - 1 283. arm = - (C_s - depth_of_first_row_top_rein - top_rein_rows_spacing * i) 284. ep_steel = X_mm * arm 285. centroid = depth_of_first_row_top_rein + top_rein_rows_spacing * i 286. steel_moment = steel_moment + SteelStressFN(ep_steel) * top_rein_row _area * centroid 287. steel_force = steel_force + SteelStressFN(ep_steel) * top_rein_row_a rea 288. If check Then 289. For j = 7 To 8 290. If Math.Abs(ep_steel) > limit_value(j) And limit_check(j) = False Then 291. limit_check(j) = True 292. End If 293. Next 294. End If 295. Next 296. For i = 0 To middle_rein_rows_num - 1 297. arm = - (C_s - depth_of_first_row_middle_rein - middle_rein_rows_spacing * i) 298. centroid = depth_of_first_row_middle_rein + middle_rein_rows_spacing * i 299. ep_steel = X_mm * arm 300. steel_moment = steel_moment + SteelStressFN(ep_steel) * middle_rein_ row_area * centroid 301. steel_force = steel_force + SteelStressFN(ep_steel) * middle_rein_ro w_area 302. Next 303. For i = 0 To top_rein_rows_num - 1 82 304. arm = - (C_s - depth_of_first_row_bottom_rein - bottom_rein_rows_spacing * i) 305. centroid = depth_of_first_row_bottom_rein + bottom_rein_rows_spacing * i 306. ep_steel = X_mm * arm 307. steel_moment = steel_moment + SteelStressFN(ep_steel) * bottom_rein_ row_area * centroid 308. steel_force = steel_force + SteelStressFN(ep_steel) * bottom_rein_ro w_area 309. If check Then 310. For j = 7 To 8 311. If Math.Abs(ep_steel) > limit_value(j) And limit_check(j) = False Then 312. limit_check(j) = True 313. End If 314. Next 315. End If 316. Next 317. End Sub 318. Sub ConcMomentForceSB(ByVal C_s As Double, ByVal X_mm As Double, check As Bo olean) 319. Dim arm, centroid, ep_conc As Double 320. conc_moment = 0 321. conc_force = 0 322. For i = 0 To layers_num - 1 323. arm = -(C_s - layer_thickness_new / 2 - layer_thickness_new * i) 324. centroid = layer_thickness_new / 2 + layer_thickness_new * i 325. ep_conc = X_mm * arm 326. If layer_thickness_new / 2 + layer_thickness_new * i < unconfined_la yer_thickness Or layer_thickness_new / 2 + layer_thickness_new * i > section_height - u nconfined_layer_thickness Then 327. conc_moment = conc_moment + ConStressFN(ep_conc, False) * sectio n_width * layer_thickness_new * centroid 328. conc_force = conc_force + ConStressFN(ep_conc, False) * section_ width * layer_thickness_new 329. Else 330. conc_moment = conc_moment + ConStressFN(ep_conc, confined) * sec tion_width * layer_thickness_new * centroid 331. conc_force = conc_force + ConStressFN(ep_conc, confined) * secti on_width * layer_thickness_new 332. End If 333. If check Then 334. If ep_conc < 0 Then 335. For j = 1 To 4 336. If Math.Abs(ep_conc) > limit_value(j) And limit_check(j) = False Then 337. limit_check(j) = True 338. End If 339. Next 340. Else 341. For j = 5 To 6 342. If Math.Abs(ep_conc) > limit_value(j) And limit_check(j) = False Then 343. limit_check(j) = True 344. End If 345. Next 346. End If 347. End If 348. Next 349. End Sub 350. Function FindMomentFN(ByVal X_mm As Double) As Double 83 351. TabControl1.SelectedTab = TabPage2 352. Dim momnet, force, force_up, force_down, depth_up, depth_down As Double 353. depth_up = section_height * 0.4 354. depth_down = section_height * 0.6 355. C_s = depth_up 356. SteelMomentForceSB(C_s, X_mm, False) 357. ConcMomentForceSB(C_s, X_mm, False) 358. force_up = -steel_force - conc_force 359. C_s = depth_down 360. SteelMomentForceSB(C_s, X_mm, False) 361. ConcMomentForceSB(C_s, X_mm, False) 362. force_down = -steel_force - conc_force 363. Dim max_i As Integer = 1000 364. For i = 1 To max_i 365. C_s = (- force_up) * (depth_down - depth_up) / (force_down - force_up) + depth_up 366. If Math.Abs(C_s) > section_height * 10 ^ 6 Then 367. C_s = section_height * 10 ^ 3 368. ElseIf C_s < 0 Then 369. C_s = 0.1 370. End If 371. SteelMomentForceSB(C_s, X_mm, False) 372. ConcMomentForceSB(C_s, X_mm, False) 373. momnet = steel_moment + conc_moment 374. force = steel_force + conc_force 375. If Math.Round(force, 3) = 0 Then 376. i = max_i + 1 377. Else 378. If force < 0 Then 379. force_up = -force 380. depth_up = C_s 381. ElseIf force > 0 Then 382. force_down = -force 383. depth_down = C_s 384. End If 385. End If 386. Next 387. SteelMomentForceSB(C_s, X_mm, True) 388. ConcMomentForceSB(C_s, X_mm, True) 389. Return momnet 390. End Function 391. Function SectionAxialRsistanceFN(ByVal confined As Boolean) As Double 392. Dim resistance, core_height, core_width, core_area As Double 393. If confined Then 394. core_height = section_height - 2 * unconfined_layer_thickness 395. core_width = section_width - 2 * unconfined_layer_thickness 396. core_area = core_height * core_width 397. resistance = rein_area * f_y + (core_area - rein_area) * f_ccm + (se ction_area - core_area) * f_cm 398. Else 399. resistance = rein_area * f_y + (section_area - rein_area) * f_cm 400. End If 401. 402. Return resistance 403. End Function 404. 405. Sub MomentCurvatureSB() 406. ReDim moment_curvature_table(3, curvatures_iterations + 10) 407. Dim X_mm_i As Double 408. ResetSB() 84 409. X_mm_i = X_mm / curvatures_iterations 410. max_momnet_value = 0 411. max_moment_piont = 0 412. ProgressBar1.Value = 0 413. For i = 1 To curvatures_iterations 414. moment_curvature_table(1, i) = X_mm_i * i 415. Next 416. Dim close As Boolean = False 417. For i = 1 To curvatures_iterations 418. ProgressBar1.Value = ProgressBar1.Value + ProgressBar1.Maximum / cur vatures_iterations 419. moment_curvature_table(2, i) = FindMomentFN(X_mm_i * i) 420. moment_curvature_table(3, i) = C_s 421. 422. If moment_curvature_table(2, i) > max_momnet_value And (Math.Round(m oment_curvature_table(2, i)) > Math.Round(moment_curvature_table(2, i - 1)) * 5) = Fals e Then 423. max_momnet_value = moment_curvature_table(2, i) 424. max_moment_piont = i 425. End If 426. If Math.Round(mom