{"source_id": "deep-surface-sec-00001", "section_title": "Chapter 1: Introduction and Fundamental Concepts", "content": "The introduction and the fundamental concepts are as follows.", "tables": [], "figures": [], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["chapter 1: introduction and fundamental concepts"], "topic_tags": ["chapter 1: introduction and fundamental concepts"]}
{"source_id": "deep-surface-sec-00002", "section_title": "1.1 Introduction", "content": "Eurocode standards provide the limits and design guidelines for reinforced concrete columns, ensuring that axial loads, bending moments, and other effects are accounted for within safe and efficient design parameters. These standards define the permissible strength, stability criteria, and interaction limits necessary for structural integrity.\\n\\n\n\nFor designs requiring unique reinforcement configurations or custom arrangements, custom N-M (axial force-moment) interaction diagrams or surfaces can be produced. This approach is particularly useful in both standard symmetrical layouts and when the reinforcement layout deviates from standard configurations, namely, asymmetrical arrangements or other non-standard placements, allowing the design to be tailored to specific project or design requirements. This method can accommodate a wide range of reinforcement setups, from the simplest configurations to complex arrangements, including asymmetrical or multi-layered reinforcement layouts. By generating custom interaction diagrams or directly utilizing them, one can accurately capture the behaviour of columns under various load scenarios and reuse these diagrams for similar projects in the future. This method provides a high degree of flexibility and precision, ensuring the design aligns with specific safety requirements and structural demands.\\n\\n\n\nColumns are designed based on their loading behaviour, which can be classified into three main types, axial, uniaxial and biaxial. The Axial loading involves forces applied along the central axis of the column, resulting in uniform compression or tension throughout the column without inducing bending moments. The uniaxial loading applies a combination of axial load and bending about one principal axis, either $x$ or $y$. This causes bending in a single direction while the column also experiences axial compression or tension. The biaxial loading combines an axial load with bending moments about both principal axes, $x$ and $y$. This requires a more complex analysis, as the column must resist bending moments in both directions simultaneously. Each loading behaviour impacts the design approach, influencing the choice of methods and calculations necessary to ensure stability and strength.\\n\\n\n\nIn addition to those loading behaviours, columns may experience eccentric, torsional, and combined loading in certain structural configurations. Eccentric loading occurs when the axial load is applied off-centre, creating additional bending moments around one or both principal axes. This eccentricity introduces additional stresses and must be considered when loads are not perfectly aligned with the centre of the column. Torsional loading involves twisting forces that cause the column to rotate about its longitudinal axis. Although torsion is typically not a primary design concern for columns, it can arise in asymmetrical or irregular structures under lateral loads such as wind or seismic forces. Eurocode 2 readily addresses torsional effects as part of its comprehensive design guidelines, particularly within normal loading conditions. It provides interaction formulas that account for the combined effects of axial, bending, shear, and torsional stresses without the need for separate calculations unless torsion is unusually high or critical. Combined loading refers to scenarios where a column experiences a mix of axial, bending, shear, and possibly torsional forces simultaneously, which is common in real-world structures. The integrated approach of Eurocode considers these combined effects within standard design checks, provided that the loading conditions remain within normal, expected ranges. This ensures that columns are designed to withstand complex loading scenarios efficiently without the need for isolated torsional calculations, simplifying the design process while maintaining structural integrity.\\n\\n\n\nThe methods of designing columns revolve around the calculation of N-M interactions, the relationship between axial load (N) and bending moment (M) in a column. This interaction illustrates how these forces combine to impact the capacity of the column and stability. Columns rarely experience purely axial loads. They typically encounter a mix of vertical loads and bending moments due to lateral forces, asymmetry, eccentric loading, and forces transferred from attached beams. Understanding the N-M interaction is essential for assessing the ability of the column to withstand these combined forces without compromising structural integrity.\\n\\n\n\nTo ensure safe design limits and code compliance, the N-M interaction diagrams and surfaces are utilized. This method is suitable for manual calculations, spreadsheets and software applications, which provides a precise assessment of the resistance of the column on axial force and bending moments, and allowing a check to see if the reinforcement setup of the column lies within a safe zone. These help verify that columns can handle the intended loads and bending moments effectively, contributing to a stable and reliable structural design.\\n\\n\n\nMoreover, the orientation of the cross-section defines principal axes, commonly labelled as the $x$ and $y$ axes. In this book, the $y$ axis is considered the long axis of the column, regardless of whether the column is oriented horizontally or vertically in the structure. Accordingly, the $x$-axis (short side) is treated as the major principal axis due to its higher bending stiffness, while the $y$-axis (long side) is the minor principal axis. This convention helps maintain consistency when referring to bending about the principal axes.", "tables": [], "figures": [], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.1 introduction"], "topic_tags": ["1.1 introduction"]}
{"source_id": "deep-surface-sec-00003", "section_title": "1.2 Fundamental Concepts", "content": "Understanding fundamental concepts, namely, stress, strain, and material behaviour is essential to grasp how reinforced concrete components function under load. These concepts form the foundation to analyse both concrete and steel reinforcement, which help determine structural behaviour and ensure safety and reliability.", "tables": [], "figures": [], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.2 fundamental concepts"], "topic_tags": ["1.2 fundamental concepts"]}
{"source_id": "deep-surface-sec-00004", "section_title": "1.2.1 Stress", "content": "Stress is a measure of how much force is applied to a certain area. In other words, if you pull on or push on a material, the material experiences a force spread out over its cross section. The amount of force divided by the area where the force is applied is called stress, which is the force per unit area. This is shown in the Equation 1.1.\\\\\n\nExamples:\\\\\n\n\\textbf{Tension (Pulling)}\\\\\n\nImagine pulling on the ends of a rope. The rope resists this pulling force, and the cross section of the rope “feels” the force. What the rope feels in every square millimetre is called tensile stress.\\\\\n\n\\textbf{Compression (Pushing)}\\\\\n\nImagine pressing both ends of a rubber eraser, between your fingers. The eraser “feels” the force pushing inward. What the eraser feels in every square millimetre is called compressive stress.", "tables": [], "figures": [], "formulas_latex": [{"formula_number": "1.1", "latex": "\\text{Stress} (\\sigma) = \\frac{\\text{Force}}{\\text{Area}}"}], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.2.1 stress"], "topic_tags": ["1.2.1 stress"]}
{"source_id": "deep-surface-sec-00005", "section_title": "1.2.2 Strain", "content": "Strain is the ratio of deformation of a material when stress is applied. It represents how much a material stretches or compresses relative to its original length. This is shown in the Equation 1.2.\n\nExample: A rubber band gets stretched (strained) when you pull it (stressed).\n\n\\textbf{Where,}\\\\\n$\\varepsilon$: Strain (unitless)\\\\\n$\\Delta L$: Change in length ($\\mu$m, mm, m, etc.)\\\\\n$L_0$: Original length ($\\mu$m, mm, m, etc.) \\\\", "tables": [], "figures": [], "formulas_latex": [{"formula_number": "1.2", "latex": "\\varepsilon = \\frac{\\Delta L}{L_0}"}], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.2.2 strain"], "topic_tags": ["1.2.2 strain"]}
{"source_id": "deep-surface-sec-00006", "section_title": "1.2.3 Significance of Stress and Strain", "content": "Stress and strain are essential to understand how materials behave under forces. They help determine whether a material can withstand a given load without breaking or deforming excessively.\\\\\n\nOne key reason we use them is that they allow us to calculate the resistive force a material provides when deformed. This is determined by utilizing the elastic modulus (E) of the material, which represents its stiffness, and its cross-sectional area (A).\\\\\n\nInitially, stress is determined using the relationship between strain ($\\varepsilon$) and Young’s modulus (E). This is shown in the Equation 1.3.\n\n\\textbf{Where,}\\\\\n$\\sigma$: Stress \\\\\n$\\varepsilon$: Strain\\\\\n$E$: Young's modulus (i.e.: Stiffness)\\\\ \n\nSince stress is force per unit area, the total resistive axial force (N) is determined as shown in Equation 1.4.\n\n\\textbf{Where,}\\\\\n$N$: Resistive axial force\\\\\n$\\sigma$: Stress \\\\\n$\\varepsilon$: Strain\\\\\n$A$: Cross sectional area which the force is applied\\\\\n\nConsequently, it is utilized to determine the resistive moment as shown in the Equation 1.5.\n\n\\textbf{Where,}\\\\\n$M$: Resistive bending moment\\\\\n$N$: Resistive axial force\\\\\n$La$: Lever arm\\\\", "tables": "", "figures": [], "formulas_latex": [{"formula_number": "1.3", "latex": "\\sigma = \\varepsilon \\cdot E"}, {"formula_number": "1.4", "latex": "N = \\sigma \\cdot A"}, {"formula_number": "1.5", "latex": "M = N \\cdot La"}], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.2.3 significance of stress and strain"], "topic_tags": ["1.2.3 significance of stress and strain"]}
{"source_id": "deep-surface-sec-00007", "section_title": "1.2.4 Strain at Peak Stress and Ultimate Strain", "content": "Strain at peak stress ($\\varepsilon _c$) is the axial strain at which concrete reaches its maximum compressive stress ($\\sigma c_{max}$). In Eurocode design models, this maximum compressive stress is limited by the design compressive strength ($f_{cd}$), which is derived from the characteristic compressive strength ($f_{ck}$). From this point onward, the material does not provide any additional strength but continues to deform until it reaches the ultimate compressive strain ($\\varepsilon_{cu}$). \\\\\n\nUltimate compressive strain ($\\varepsilon_{cu}$) is the final strain a material experiences before failure. It represents the maximum deformation the material can undergo before breaking or collapsing. \\\\\n\nIn Eurocode standards, specific notations, namely,  '$\\varepsilon_{c3}$' and '$\\varepsilon_{cu3}$' are utilized for concrete. This notation appears in Eurocode and throughout this book. The subscript '$c$' indicates the strain at peak stress and '$cu$' indicates ultimate compressive strain, while additional numbers like ‘$3$’ differentiate between stress-strain models defined in Eurocode 2, for different levels of design accuracy.\\\\\n\nWhen using the parabolic-rectangular stress-strain model, '$\\varepsilon_{c2}$' and '$\\varepsilon_{cu2}$' is utilized. For the bilinear stress-strain model, '$\\varepsilon_{c3}$' and '$\\varepsilon_{cu3}$' is utilized instead. Throughout this book, the bilinear stress-strain model is utilized, thereby, '$\\varepsilon_{c3}$' and '$\\varepsilon_{cu3}$' are applied in all calculations. This is shown in Tables 1.1 and 1.2. \\\\\n\n\\textbf{Where,}\\\\ \n'C50/60' represents concrete at 28 days of age, with a characteristic cylinder strength of 50 N/mm\\textsuperscript{2} (50 MPa) and a characteristic cube strength of 60 N/mm\\textsuperscript{2} (60 MPa).\\\\\n\nThese values are specified in Eurocode 2 (EN 1992-1-1), Table 3.1.", "tables": [{"table_number": "1.1", "table_title": "Strain Values at Peak Stress for Concrete", "columns": ["Strain Type", "$\\varepsilon_{c}$ up to C50/60", "$\\varepsilon_{c}$ from C51/61 to C90/105"], "rows": [["$\\varepsilon_{c1}$", "0.0018 to 0.00245", "0.00246 to 0.0028"], ["$\\varepsilon_{c2}$", "0.0020", "0.0021 to 0.0026"], ["$\\varepsilon_{c3}$", "0.00175", "0.00176 to 0.0023"]]}, {"table_number": "1.2", "table_title": "Ultimate Strain Values for Concrete", "columns": ["Strain Type", "$\\varepsilon_{cu}$ up to C50/60", "$\\varepsilon_{cu}$ from C51/61 to C90/105"], "rows": [["$\\varepsilon_{cu1}$", "0.0035", "0.0034 to 0.0028"], ["$\\varepsilon_{cu2}$", "0.0035", "0.0034 to 0.0026"], ["$\\varepsilon_{cu3}$", "0.0035", "0.0034 to 0.0026"]]}], "figures": [], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.2.4 strain at peak stress and ultimate strain"], "topic_tags": ["1.2.4 strain at peak stress and ultimate strain"]}
{"source_id": "deep-surface-sec-00008", "section_title": "Concrete Strength Classifications", "content": "Concrete strength classifications, for instance, C50/60 indicate the characteristic compressive strength of concrete as defined in Eurocode 2 (EN 1992-1-1). The first number, 'C50' represents the characteristic compressive strength of a cylinder specimen in N/mm\\textsuperscript{2} (MPa). The second number '60' represents the characteristic compressive strength of a cube specimen in N/mm\\textsuperscript{2} (MPa). This is shown in Figure 1.1.\\\\\n\nExample:\\\\\nC50/60 denotes, 50 N/mm\\textsuperscript{2} of cylinder strength and 60 N/mm\\textsuperscript{2} of cube strength.", "tables": [], "figures": [{"figure_number": "1.1", "figure_title": "Comparison of cylinder and cube strength classification.", "image_path": "images/fig-001.png", "image_width": "0.4\\textwidth"}], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["concrete strength classifications"], "topic_tags": ["concrete strength classifications"]}
{"source_id": "deep-surface-sec-00009", "section_title": "1.2.6 Yielding in Materials and its Role in Structural Design", "content": "When a material is subjected to force, it first deforms elastically, meaning it returns to its original shape when the force is removed. If the force exceeds a certain limit, known as the yield point, the material enters plastic deformation, where the change in shape becomes permanent.\\\\\n\nFor instance, a paperclip bent slightly springs back to its original shape (elastic deformation), but if bent too far, it stays deformed (plastic deformation). This transition is called yielding, and it plays a crucial role in structural design, especially in reinforced concrete structures. In these, the embedded steel reinforcement bars primarily undergo longitudinal stretching or shortening, due to axial and bending loads acting on the column. While the reinforcement also follows the curvature of the concrete section and may experience slight bending, this effect is considered structurally negligible. Axial strain is primarily used to analyse the structural behaviour of the reinforcement.", "tables": [], "figures": [], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.2.6 yielding in materials and its role in structural design"], "topic_tags": ["1.2.6 yielding in materials and its role in structural design"]}
{"source_id": "deep-surface-sec-00010", "section_title": "1.2.6.1 Steel Yielding in Reinforced Concrete", "content": "To ensure safety, reinforced concrete structures are designed so that steel yields before concrete crushes. Steel is ductile, meaning it can deform significantly before failing, while concrete is brittle and fails suddenly. This behaviour allows steel to act as a warning mechanism, preventing catastrophic failure.\\\\\n\nBy designing structures so that steel yields before concrete crushes, engineers create ductile failure mechanisms that provide visible warning signs, namely, cracking and deformation, before structural failure. This is why the yield strength ($f_y$) of steel is used in both tension and compression, simplifying calculations while maintaining safety and reliability.\\\\\n\nSince yield strength of steel ($f_y$) marks the transition between elastic and plastic behaviour, structural calculations focus on the yield strain ($\\varepsilon_{y}$), rather than ultimate strain of steel ($\\varepsilon_{su}$). Although, steel does not break at the yield point, this is the critical value used in design to ensure predictable structural performance.", "tables": [], "figures": [], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.2.6.1 steel yielding in reinforced concrete"], "topic_tags": ["1.2.6.1 steel yielding in reinforced concrete"]}
{"source_id": "deep-surface-sec-00011", "section_title": "1.2.6.2 Rationale for Using Yield Strength of Steel in Both Compression and Tension", "content": "In reinforced concrete design, steel reinforcement is assigned the same yield strength ($f_y$) for both tension and compression, unlike concrete, which has distinct tensile and compressive strengths. This approach reflects the fundamental behaviour of steel under load. \\\\\n\nConcrete has a defined compressive strength, but its tensile strength is often ignored due to its insignificant contribution to tensile resistance. Steel, however, does not have a well-defined compressive strength independent of its tensile yield strength. Instead, its ability to resist compression is directly derived from its tensile yield strength.\\\\\n\nThis is not a simplification but a necessity, as steel does not fail in compression the way concrete does. Concrete crushes at its ultimate strain, whereas steel undergoes significant plastic deformation in compression without a sudden rupture. Since there is no clear compressive failure mechanism for steel, it is impractical to define a separate compressive strength. Instead, design standards universally apply the tensile yield strength ($f_y$) for both tensile and compressive reinforcement.", "tables": [], "figures": [], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["1.2.6.2 rationale for using yield strength of steel in both compression and tension"], "topic_tags": ["1.2.6.2 rationale for using yield strength of steel in both compression and tension"]}
{"source_id": "deep-surface-sec-00012", "section_title": "Chapter 2: Area of Concrete Compression Zone", "content": "The area of the concrete compression zone in a biaxial concrete column plays a crucial role in determining the strength and stability of the column. This zone resists compressive forces and directly influences load-bearing capacity, stiffness, and failure mechanisms. Its shape and size vary with axial load and biaxial bending moments, impacting the neutral axis position and moment capacity.", "tables": [], "figures": [], "formulas_latex": [], "book_title": "Deep Surface: Unlocking N–M Biaxial Interaction in Rectangular Concrete Columns to Eurocode 2", "author_name": "Dr. Harshana S. P. Wattage", "publication_year": 2025, "book_url": "https://www.wattagepublishers.com/deep-surface", "reference_code": "EN 1992-1-1:2004", "difficulty_level": "Advanced", "language": "English", "keywords": ["chapter 2: area of concrete compression zone"], "topic_tags": ["chapter 2: area of concrete compression zone"]}