System and program for calculating ferroconcrete bending member section

Abstract

To address the problem of impossibility of accurately taking all reinforcements into account during modeling and accurately taking bending moment of an oblique direction for a member section into account in section calculation of a beam, a column, and a reinforcement exposed columnar leg which are ferroconcrete bending members.SOLUTION: Generally implemented has been a method for calculating a section by solving an algebraic equation in which the position of a neutral axis is unknown. Thus, provided is a method for identifying the position of a neutral axis through numerical analysis from a neutral axis basic expression. Sectional performance of all individual reinforcements is calculated for reinforcements, and a concrete section is divided into some basic figures to calculate each sectional performance for concrete sections. Overall sectional performance is calculated by totaling these. Then, using the neutral axis basic expression substituted with the sectional performance, the correct position of the neutral axis is obtained through repeated calculation. The stress state of the section is obtained from the position of the neutral axis, and sectional calculation can be carried out.SELECTED DRAWING: Figure 8

Description

[Technical field]

[0001] The present invention relates to a cross-sectional calculation system and program for beams and columns, which are bending members in reinforced concrete structures, and exposed column bases in steel-frame structures. [Background technology]

[0002] For architectural structural design, if the building is of a certain size or larger, a procedure called confirmation application is carried out. When submitting a confirmation application, the review body evaluates the correctness of the content. As a rule, the Building Standards Act and the Building Standards Act Enforcement Order must be strictly adhered to as standards for review, but as these laws and regulations indicate the principles, the details are based on documents such as the "Reinforced Concrete Structural Calculation Standards and Commentary" (hereinafter referred to as the "RC Standards") and the "Explanation of Technical Standards for Building Structures". Furthermore, generally accepted methods or methods approved by designated performance evaluation bodies are also acceptable. For standard beams and columns, calculations are based on the RC standard and are incorporated into the integrated design system for structural design. In addition, the "Guidelines for Design of Steel Structure Joints" state that for column bases, "the size of the base plate is taken as the cross section, and the tension anchor bolts can be calculated as reinforced concrete columns, with the reinforcement bars considered." Although this invention is not commonly used and is not incorporated into a consistent design system, it is a highly versatile cross-sectional calculation system based on the conditions set out in the RC standards.

[0003] A brief description of the members that are the subject of the present invention will be given below. Figure 1(1) is an example of a reinforced concrete frame. The components in reinforced concrete construction are beams and columns, with beams being horizontally arranged components and columns being vertically arranged components. Both are made up of concrete with reinforcing bars embedded in it. Figure 2 shows the arrangement of reinforcing bars in beams and columns. Reinforcing bars are arranged parallel to the axis of the components.

[0004] Here, we will summarize the forces that occur in bending members in structural design and the types of reinforcing bars that correspond to them in design. Bending members such as beams and columns are designed to handle three types of forces: axial force, bending moment, and shear force. Reinforcing bars used in bending members are classified into main bars and shear reinforcement bars, and in cross-sectional calculations, axial force and bending moment are handled by main bars, and shear reinforcement is handled by shear reinforcement bars. Therefore, in reality, shear reinforcement bars arranged in a direction perpendicular to the main bars are also embedded, but since the cross-sectional calculations targeted by this invention are related to bending moment and axial force, they are excluded from the scope of the present invention, and Figure 2 shows only main bars and not shear reinforcement bars. Figure 3 shows the relationship between the types of forces that occur in bending members and the corresponding reinforcing bars.

[0005] FIG. 1(2) is an example of a steel frame structure. The column base of interest in steel frame structures is located at the bottom of a steel column. There are three types of column bases: exposed column bases, root-wrapped column bases, and embedded column bases. The present invention targets exposed column bases, where the base plate is exposed and the upper steel column is fastened to the lower reinforced concrete with anchor bolts. FIG. 2 shows the state of the anchor bolts in an exposed column base. The anchor bolts are embedded in the foundation concrete, so the situation is similar to that of a reinforced concrete column. For this reason, the anchor bolts are regarded as reinforcing bars, and the cross-sectional calculation is performed using a method similar to that for column cross-sectional calculations. The method of the present invention can also be used to deal with this issue by making similar substitutions, and they are the target components of the present invention.

[0006] [Allowable stress calculation] Structural design of buildings is largely classified into primary design based on allowable stress calculations and secondary design based on ultimate strength calculations, and the methods of setting external forces and calculating component stresses are different. The difference lies in the purpose of the design. The primary design aims to "prevent damage to major structural members from rare earthquakes," while the secondary design aims to "prevent the building from collapsing or collapsing from rare major earthquakes." Allowable stress calculations are used to design members such as reinforcing bars and concrete so that the stress caused by external forces such as earthquakes is below their respective allowable stress levels. This invention is intended for primary design using allowable stress calculations.

[0007] Explain the difference between stress and stress intensity. Stress is the force acting inside components such as reinforcing bars and concrete, while stress intensity is obtained by dividing stress by the cross-sectional area, and is defined as "stress acting per unit area of ​​the cross section." Each component has an allowable stress level for each material, such as reinforcing bars or concrete, and design using allowable stress calculations involves checking safety by comparing the stress level generated inside the component with the allowable stress level of that material.

[0008] In order to clarify the differences between beams, columns, and exposed column bases, we will first explain the difference between beams and columns. In the RC standards, calculation formulas are derived based on the common basic assumption that beams and columns are bending members. In general buildings, the difference between the two in terms of structural mechanics is that axial force is applied to columns, but not to beams. This is also clear from the contents of the RC standards, as the headings of the articles are "Cross-sectional calculation for bending of beams" in Article 13 and "Cross-sectional calculation for axial force and bending of columns" in Article 14.

[0009] Next, we will explain the difference between columns and exposed column bases. A column and an exposed column base are both subjected to bending moments and axial forces. The difference between a column and an exposed column base is the force that occurs in the rebar on the compression side. In a column, tension and compression forces occur in the rebar, but in an exposed column base, only tension forces, not compression forces, occur in the anchor bolt. Therefore, no stress occurs in the rebar on the compression side. This is because rebar has protrusions to attach to the concrete, but anchor bolts do not have protrusions, so the two do not deform as a unit. However, as shown in Figure 4, the structure is such that the concrete is sandwiched between the upper base plate and the lower fixing plate, so the compressive force of the concrete and the tensile force of the anchor bolt balance, and tensile force occurs in the anchor bolt. As a result, the anchor bolt only experiences tensile force.

[0010] From the above, beams are a special case where the axial force in columns is set to zero, and exposed column bases can be handled by always setting the stress of the reinforcing bars on the compression side of the columns to zero. In other words, columns are general components when it comes to bending materials in reinforced concrete structures, while beams and exposed column bases are special components with additional conditions. Therefore, the explanation of this invention will focus on the calculation of the cross section of columns.

[0011] In the RC Code, Article 12 specifies basic assumptions for bending members, followed by specific methods for calculating the cross-section of beams in Article 13 and columns in Article 14. In the explanation of this invention, the cross-section calculations specified in Articles 13 and 14 are referred to as "RC Code cross-section calculations." The point to note here is whether the calculation results are rigorous solutions to the basic assumptions. The RC Code cross-section calculations can be said to be approximate cross-section calculations rather than rigorous solutions, in that they involve modeling that aggregates rebars and cannot directly analyze diagonal bending moments.

[0012] [Neutral axis] The neutral axis, which is important for understanding the present invention, will now be explained. An explanation of the neutral axis is shown in Figure 5. The upper figure is an explanatory diagram of a cross section of a bending member with the neutral axis inserted. The neutral axis is an axis perpendicular to the direction of the bending moment, and indicates that the neutral axis is the boundary between the tension side and the compression side. The hatching in the figure indicates the compression side range in the cross section. The lower figure is a stress diagram showing the stress distribution in the cross section. The stress becomes 0 at the neutral axis, and the magnitude of the stress is proportional to the distance from the neutral axis. In the design of reinforced concrete bending members, the tensile force of the concrete is ignored, so the parts of the concrete where no stress occurs are shown with dotted lines. This stress distribution follows the basic assumptions in the cross-sectional calculation of bending members described later. The position of the neutral axis is mainly determined by the combination of bending moment and axial force, and it may exist either inside or outside the cross section. In Figure 5 (1), the neutral axis is on the tension side outside the cross section, and only compression forces occur in the cross section. In Figure 5 (2), the neutral axis is inside the cross section, and both tension and compression forces occur in the cross section. In Figure 5 (3), the neutral axis is on the compression side outside the cross section, and only tension forces occur in the cross section. Both tension and compression forces occur in the rebars, with the rebars on the tension side experiencing tension and the rebars on the compression side experiencing compression. The magnitude of the stress in the rebars is proportional to the distance from the neutral axis, just like concrete.

[0013] In the cross-sectional calculation of the RC code, the position of the rebar is concentrated at one point on both the tension side and compression side of both beams and columns. This makes it possible to formulate an equation with the neutral axis as an unknown, and by solving this equation, the position of the neutral axis is found and the stress of the concrete and rebar that make up the member is calculated. This method often produces a solution that is acceptable for practical use and is used in normal structural design, but it is an approximate solution that lacks rigor in the following respects: (1) If all rebars are located at the same distance from the neutral axis, there is no problem with consolidating them. However, if the positions are different, it is impossible to model the area so that both the moment of inertia and the moment of inertia, which will be described later, obtain correct values. (2) The equations are treated differently for reinforcing bars on the tension side and reinforcing bars on the compression side. Therefore, the modeling must be changed depending on the position of the neutral axis. If the reinforcing bars are distributed, the modeling must be divided into many cases, making it difficult to deal with them precisely in practice. A typical example of modeling a beam is shown in Fig. 6, and a typical example of modeling a column is shown in Fig. 7. Reinforcing bars that are taken into account in the modeling are shown as filled circles, reinforcing bars that are not taken into account in the modeling are shown as open circles, and the center of gravity of the reinforcing bars is shown with an x.

[0014] Generally, all rebars are considered for beams because they are basically arranged in a concentrated manner on the tension side and compression side, but in columns, some rebars perpendicular to the calculation direction are generally not considered because they are arranged effectively in two directions, the X direction and the Y direction. Also, if all rebars are not equidistant from the neutral axis, the model will lack accuracy. The following shows typical examples of cases where accurate modeling is possible and cases where accurate modeling is not possible for beams and columns. (1) Second and third stage reinforcement of beams When rebar is arranged in multiple rows, it is modeled by concentrating the rebar at the center of gravity on both tension and compression sides. (2) Orthogonal reinforcement of columns In general columns, rebars arranged in the perpendicular direction are not taken into account. Conversely, they are taken into account in calculations in the perpendicular direction. (3) Column bracing The rebars are placed closer to the target direction among the rebars placed in the orthogonal direction. The rebars, including the rebars, are modeled at the center of gravity on both the tension side and compression side.

[0015] [Diagonal bending moment] The RC standards show the calculation method when a bending moment is applied only in the same direction as the axis of the cross section, and do not give a direct calculation method when a bending moment is applied in a diagonal direction. It explains this as a biaxial bending problem in which bending moments are applied in two directions simultaneously, with the bending moments calculated separately along the material axis of the cross section. Specifically, the two axes are the X-axis and Y-axis, and the strength is calculated separately, and an approximate formula using these strengths is given. The calculation formula for evaluating two axes simultaneously is shown in Equation 1. When bending is applied simultaneously on two axes, this is the same as when a bending moment is applied in a diagonal direction, in other words, when the combined bending moment is applied in a direction that does not coincide with either of the two axes.

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[0016] [Non-Patent Document 1] Reinforced Concrete Structural Calculation Standards and Commentary 2018 (Architectural Institute of Japan) [Non-Patent Document 2] Steel structure joint design guidelines (Architectural Institute of Japan) Summary of the Invention [Problem to be solved by the invention]

[0017] The present invention is a method for calculating the cross-section of a bending member with a rectangular cross-section, and the calculation is performed in a manner that strictly adheres to basic assumptions regarding the following two points that are not strictly addressed in conventional methods. -Section calculation considering all rebars - Calculation of cross section of member where bending moment occurs diagonally

[0018] [Consider all rebars] The cross-sectional calculation of the RC standard used in the structural design of general bending members is calculated by solving an algebraic equation based on basic assumptions with the position of the neutral axis as an unknown. In the process of deriving this algebraic equation, the reinforcing bars on both the tension side and compression side are concentrated at one point. This is a model that lacks rigor in calculating the cross-sectional performance. Also, since reinforcing bars are more effective on the outside, the reinforcing bars on the inside are sometimes modeled as not existing. In contrast, in this invention, all reinforcing bars are considered in the same way.

[0019] [Diagonal bending moment] In normal structural design, the X and Y directions are treated separately. However, in the following cases, a diagonal bending moment occurs. (1) In the design of a building, external forces such as earthquake forces may be applied to the building at a 45-degree angle. In such cases, a diagonal bending moment is generated in the columns. In many cases, in a consistent design, the direction of the earthquake force can be set at a 45-degree angle. (2) If some of the columns are arranged diagonally, a diagonal bending moment will be generated in one of the columns. (3) If three-dimensional analysis reveals that the entire building is twisted, a diagonal bending moment will be generated in the columns. (4) Normally, beams only experience vertical stress, but beams subjected to earth pressure in basement floors also experience horizontal stress. In that case, a diagonal bending moment is generated as a result of the force. Even if a diagonal bending moment occurs, it is possible to confirm safety based on the section calculation in the RC code. On the other hand, the section calculation of this invention directly calculates bending moments in all directions using the same method. [Means for solving the problem]

[0020] In architectural design, the cross-sectional calculation of reinforced concrete bending members is performed based on the RC Code. In the RC Code, the basic assumptions for bending members are stated in Article 12 as "Basic assumptions for the cross-sectional calculation of bending members." The basic assumptions are as follows: (1) Tensile stress in concrete is ignored (2) Each cross section of the bending member remains flat even after the member is bent, and the compressive force of the concrete is proportional to the distance from the neutral axis. (3) The ratio of Young's modulus of steel to concrete, n, shall be the same regardless of the type of concrete or the long-term or short-term load, and shall be a value corresponding to the design strength Fc of the concrete. (4) For reinforcing bars that are not perpendicular to the calculation cross section, the effective cross-sectional area shall be determined by multiplying the cross-sectional area by cosθ.

[0021] The following explains the long-term and short-term loads that appear in the basic assumptions of bending members. The representative types of loads are shown below. Dead load: Building weight Live load: The load of furniture, people, etc. inside the building Earthquake load: The force acting horizontally on a building during an earthquake; also known as seismic force. Wind load: The force exerted on each surface of a building by the wind. It is also called wind pressure. Design is carried out based on combinations of these loads, which are classified as long-term and short-term. Long term: Fixed load + live load Short term: Fixed load + live load + earthquake load, Fixed load + live load + wind load Fixed loads + live loads are vertical downward forces, but earthquake loads and wind loads mainly act horizontally on the building, generating upward axial tensile forces on the columns and column bases. In addition, there are long-term and short-term allowable stresses for concrete and reinforcing bars. Safety is confirmed by the long-term allowable stress for stresses caused by long-term loads, and the short-term allowable stress for stresses caused by short-term loads.

[0022] The RC Code lays out the method for calculating the cross-section of beams and columns based on the above-mentioned Article 12. However, the RC Code also describes the possibility of methods other than the cross-section calculation in the RC Code, so we will quote it here. The following sentence is written in Article 14 of the RC Code on the cross-section calculation for axial force and bending of columns: "As long as you comply with the basic assumptions in Article 12, the same results will be obtained regardless of the calculation procedure used, and the results can be easily obtained through numerical analysis using a computer, which is actually more convenient in practice, and further..." Although "the same results can be obtained" is slightly different from the explanation of this invention, this can be interpreted as "an approximation to the correct answer can be obtained in the same way" if we assume that the results will differ slightly depending on the modeling and calculation method. Furthermore, the numerical analysis method that complies with the basic assumptions of Article 12 of the RC Standards can be said to be consistent with this invention. In addition, the fact that at this stage, methods that take into account all rebars or methods that directly solve for diagonal bending moments are not commonly used was used as one of the grounds for evaluating the novelty and inventive step of this invention.

[0023] The following formulas 2 to 6 are written in the RC standard and are derived from basic assumptions. Equation 2 shows the formula for the moment of inertia. The first half is a term related to concrete, and the second half is a term related to reinforcing bars. The moment of inertia is the cross-sectional performance related to axial force.

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[0024] The important process in the cross-sectional calculation of bending members of reinforced concrete structures is to determine the correct position of the neutral axis. Once the position of the neutral axis is determined, the stress of the reinforcing bar and concrete can be determined from the basic assumptions and the equilibrium equation. Figure 8 shows the general cross-sectional calculation flow, but the process of "calculating the position of the neutral axis" in the figure differs between the calculation method in the cross-sectional calculation of the RC code and the calculation method of the present invention. In the RC code, the position of the neutral axis is calculated by setting up an equation with the position of the neutral axis as an unknown and solving it. In contrast, the cross-sectional calculation of the present invention first calculates the cross-sectional performance that appears in the basic equation of the neutral axis more precisely as a numerical value. Then, the correct position of the neutral axis is determined by repeated calculations using the basic equation of the neutral axis without setting up an equation. The examination flow for determining the position of the neutral axis of the present invention is shown in Figure 9.

[0025] In the present invention, the position of the neutral axis is found by strictly treating the "basic equation of the neutral axis" of Equation 6, which is derived from the basic assumptions, and using a solution method that calculates an accurate solution. The method for precisely calculating the first moment of area Sn and the second moment of area In that appear in the basic equation for the neutral axis is shown below. (1) For concrete, the cross section is divided into multiple basic shapes and the cross-sectional properties are calculated. (2) Regarding reinforcing bars, calculate the cross-sectional properties of each bar. (3) The calculation results of the concrete and reinforcing bars are added together to calculate the overall cross-sectional performance. This method is based on the fact that the section performance can be calculated individually for all elements that make up the section, and then the overall value can be calculated by adding them up. In Formula 2 and Formula 3, which are the formulas for the section performance, all the mathematical symbols indicate addition. Concrete is treated as a continuous quantity, while reinforcing bars are treated as a discrete quantity, which conforms to the general method of structural calculation for architecture. Once the method for calculating the cross-sectional performance has been decided, the position of the neutral axis that satisfies the "fundamental formula for the neutral axis" is investigated. The formula for the cross-sectional performance can be calculated no matter where the neutral axis is. However, the "fundamental formula for the neutral axis" is only satisfied when it is in the correct position, and is not satisfied otherwise. In this invention, an evaluation value is defined from the "fundamental formula for the neutral axis", and the position of the neutral axis where this evaluation value is 0 is searched for, thereby calculating the position of the neutral axis that can be approximated as the correct position. The segment line method is used as the search method. Effect of the Invention

[0026] The effects of the present invention will be summarized. The objectives of the development of the present invention are as follows: (1) When calculating the cross-sectional area of ​​reinforced concrete beams, columns, and exposed column bases, all reinforcing bars and anchor bolts can be taken into account while strictly adhering to the basic assumptions of the RC code. (2) When calculating the cross-sections of reinforced concrete beams, columns, and exposed column bases, diagonal bending moments can be calculated directly. Although not listed as an objective of the development of the present invention, the following effects are also achieved. (3) A common algorithm can be used to calculate beams and columns, which are specified as bending members in the RC standards, as well as exposed column bases, whose cross-sections can be calculated in a similar manner to columns. This has the effect of making it possible to calculate the cross-section as follows for beams, which previously had an axial force of 0. (4) The cross-section of a beam with axial force can be calculated. (5) The cross-section of a beam subjected to lateral forces such as earth pressure and axial forces can be calculated. Exposed column bases have the same effects as columns, plus the following: (6) The cross-section of exposed column bases can be calculated by using reinforcing bars instead of anchor bolts. [Brief description of the drawings]

[0027] [Figure 1] 1 is a diagram showing the positions of members that are the subject of the present invention, in which the beams and columns are reinforced concrete members, and the exposed columns are steel frame members. [Diagram 2]This is a diagram showing that the member targeted by this invention is composed of concrete and reinforcing bars embedded in it. Reinforcing bars include main bars and shear reinforcement bars, but since only the main bars are related to the cross-sectional calculation targeted by this invention, only the main bars are drawn in the diagram. [Diagram 3] FIG. 1 is a diagram showing the relationship between the types of forces generated in bending members and the types of reinforcing bars that correspond in design. [Figure 4] FIG. 13 is a diagram explaining the principle by which tensile force is generated in an anchor bolt of an exposed column base. [Diagram 5] FIG. 1 is a diagram illustrating a neutral axis and stress intensity. [Figure 6] FIG. 13 is a diagram illustrating modeling of reinforcing bars of a beam. [Figure 7] FIG. 13 is a diagram illustrating modeling of reinforcing bars in a column. [Figure 8] This is the flow for calculating the cross section of a general bending member. [Figure 9] 4 is a flow chart for searching the position of the neutral axis of the present invention. [Figure 10] FIG. 1 is a diagram showing the configuration of a computer for implementing the present invention. [Figure 11] FIG. 13 is a diagram for explaining a bending moment in an oblique direction. [Figure 12] FIG. 13 is a diagram showing the division of a rectangular cross section when a diagonal bending moment is applied to the rectangular cross section. [Figure 13] This is a diagram showing the division of cross sections when a bending moment occurs in a square cross section and a rectangular cross section, changing the angle of the bending moment direction. [Figure 14] FIG. 13 is a diagram showing the types of divided cross sections when a diagonal bending moment occurs and the cross section is divided along a neutral axis. [Figure 15] The formulas for the section performance of rectangles, parallelograms, trapezoids, and triangles are shown below. The formulas are for when the neutral axis passes through the centroid of the section. [Figure 16] FIG. 13 is a diagram for explaining a correction formula for cross-sectional performance when the neutral axis does not pass through the centroid of the cross section. [Figure 17] FIG. 13 is a diagram illustrating the distance y used to calculate the cross-sectional properties of a reinforcing bar. [Figure 18] FIG. 13 is a diagram illustrating the secant method used to search for the position of the neutral axis. [Figure 19] FIG. 13 is a diagram for explaining the convergence state of the secant method when a diagonal bending moment occurs in a rectangular cross section. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0028] Figure 10 is a diagram showing the configuration of a computer that runs this system. The computer is composed of a CPU, which is an arithmetic processing unit, memory, which is a main storage device, and auxiliary storage devices such as HDD and SSD, and is connected to the main unit as input devices such as a keyboard and mouse, and as output devices, a display and a printer. It is also connected to a network that can access an external server. All calculations in the present invention are written in source code in a programming language. The source code is converted by a compiler into an executable file that can be run by a computer. The source code and executable file are stored in the computer's auxiliary storage device or on a server connected via the Internet. The executable file is called up into memory and run by the CPU. The memory is used as a work area etc. when the CPU is running the program. When the user starts up the system, the executable file is called up into memory. By selecting the calculation menu for beams, columns, etc., inputting data in the appropriate format, and pressing the start calculation button, the system will calculate the cross section. Data is entered using the keyboard and mouse, and can also be saved in an auxiliary storage device and recalled for further use. The cross section calculation results are shown on the display, and can also be output to a printer if necessary.

[0029] The equations for the first moment of area and the second moment of area are arranged for the purpose of explaining the present invention. The formula for the moment of inertia of the calculated section is shown in Equation 7. The moment of inertia of the calculated section is the sum of the moment of inertia of the concrete and the moment of inertia of the reinforcing steel.

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[0030] All of Equations 9 to 12 include the distance y from the neutral axis. The direction of the neutral axis is perpendicular to the direction of the bending moment, but its position is not self-evident. If the bending moment is in the X-axis direction, the neutral axis is perpendicular to the X-axis, and if it is in the Y-axis direction, the neutral axis is perpendicular to the Y-axis. If the bending moment is in an oblique direction that does not match either the X-axis or the Y-axis, the neutral axis is perpendicular to the direction of the bending moment. When both the X-axis component and the Y-axis component are obtained as a result of the three-dimensional analysis, the position of the neutral axis is determined on each axis and a cross-sectional calculation is performed. In this invention, in order to show a method for directly calculating the cross-sectional area of ​​the oblique bending moment that is a combination of the X-axis component and the Y-axis component, a case where the neutral axis is oblique is assumed.

[0031] The cases where the bending moment and neutral axis are parallel or perpendicular to the axis of the cross section are shown in Figure 11 (1) and (2). In this case, the rectangular cross section performance is used. Next, the case where the bending moment and neutral axis are oblique to the axis of the cross section is shown in Figure 11 (3), and its perspective view is shown in Figure 11 (4).

[0032] FIG. 12 shows a modified view of FIG. 11(3). Figure 12(1) is the same as Figure 11(3). Figure 12(2) shows Figure 12(1) rotated so that the neutral axis is vertical, without changing the relationship between the cross section and the neutral axis. Figure 12(1) and Figure 12(2) are the same because the relationship between the direction of the bending moment and the direction of the neutral axis with respect to the cross section is the same. Figure 12 (2) is a general representation of when a diagonal bending moment occurs on a rectangular cross-section, and a special case can be said to be when the neutral axis is parallel to one of the sides of the rectangle.

[0033] [Sectional performance of concrete] The cross-sectional properties of concrete are derived for the case where the neutral axis is oblique, as in Figure 12 (2). When the shape is not simple, such as an H-shaped steel, the cross-sectional properties can be determined by dividing it into several basic shapes. A basic shape is one for which there are formulas for area, second moment of area, etc., and in addition to rectangles, triangles, parallelograms, trapezoids, circles, etc. are used. Figure 12(3) is a diagram in which Figure 12(2) is divided into three figures by a dividing line parallel to the neutral axis. There are countless ways to divide, but in order to reduce the number of divisions, the figure is divided by a line passing through the vertices. Figure 12(3) divides the figure into two triangles and one parallelogram. This division is an example of division when the neutral axis is oblique, but it is the basic division method of the present invention.

[0034] [Basic division] Figure 13 shows the case where the angle of the neutral axis is changed for Figure 12(3). The top row is a square cross section, and the bottom row is a rectangular cross section. The shape of the divided figure changes depending on the angle, but it can be seen that it is basically divided into three figures by two dividing lines, just like Figure 12(3). There are two types of three figures: triangles on both sides and a parallelogram in the middle. When the angle is 0 degrees, it is divided into one rectangle without division, and when the two dividing lines coincide, it is divided into two triangles, but this is a special case and can be treated in the same way, so a rectangular cross section can be divided into three or less basic figures. Since the cross-sectional properties can be calculated for each divided basic shape and then added together, it is necessary to be able to calculate the cross-sectional properties of rectangles, triangles, and parallelograms.

[0035] [Divide by neutral axis] If the neutral axis is located within the cross section, the shape must also be divided at the neutral axis. This is because concrete only experiences compression, so the cross-sectional performance is calculated only for the compression part and not the tension part. Figure 14 shows the pattern of further dividing the basic divided cross section according to the position of the neutral axis. The neutral axis can be located either inside or outside the cross section, so the patterns are organized including outside the cross section. If the neutral axis is located within the parallelogram part as in Figure 14 (3), the shape is divided into two parallelograms, but if the neutral axis is located within the triangular part as in Figure 14 (2), the shape is divided into two shapes: a triangle and a trapezoid. Therefore, a trapezoid is added as a type of basic shape. As a result, it is necessary to calculate the cross-sectional performance of a rectangle, a triangle, a parallelogram, and a trapezoid. For these shapes, there are commonly known formulas, which are shown in Figure 15. In the figure, A is the cross-sectional area, I is the second moment of area, and c is the symbol representing the position of the centroid.

[0036] [Sectional performance formula] In the formula in Figure 15, the rectangle and parallelogram have exactly the same formula. It can be confirmed that the triangle has the same formula when the upper base of the trapezoid is set to a = 0. It can also be confirmed that the rectangle and parallelogram have the same formula when the upper and lower bases of the trapezoid are set to a = b. Therefore, it is possible to represent all basic figures with the formula for a trapezoid.

[0037] The formula in Figure 15 is for when the neutral axis passes through the center of gravity. If the axial force is 0, the neutral axis passes through the center of gravity, but in the following cases, the neutral axis does not pass through the center of gravity. (1) When axial force is applied (2) When the cross-sectional shape is a composite of multiple basic shapes (3) In the case of materials that only undergo compression, such as concrete (4) In the case of a cross section made of multiple composite materials, such as reinforced concrete Therefore, the conversion formula in Figure 16 is used. This formula can be calculated regardless of the type of base figure, as long as the cross-sectional area, second moment of area, and the position of the centroid relative to the neutral axis are known.

[0038] Here is a supplementary explanation of the formulas in Figures 15 and 16. The formula for the second moment of area in Figure 16 contains the second moment of area Io when the neutral axis passes through the centroid, but the formula for the first moment of area does not contain anything equivalent. This is because the first moment of area is 0 when the neutral axis passes through the centroid, which is consistent with the fact that substituting y=0 into the formula in Figure 16 gives S=0 and I=Io. For that reason, the formula for the first moment of area is not included in Figure 15.

[0039] [Sectional performance of rebar] The formula in the RC Code itself can be used to handle rebars. The distance from the neutral axis of each rebar can be calculated from the coordinates, regardless of the arrangement pattern of the rebars. Equation 10 and Equation 12 are the product of the cross-sectional area a, the distance from the neutral axis y, and the Young's modulus ratio n. According to the basic assumptions of the RC Code, the Young's modulus ratio n does not change within a member, but the cross-sectional area and neutral axis can differ for each rebar, so subscripts are added to indicate the cross-sectional area of ​​the rebar as ai and the distance from the neutral axis as yi. The following are formulas for Equation 10 and Equation 12, substituting the cross-sectional area of ​​each rebar with ai and the distance from the neutral axis with yi. Equation 13 is an expression for the first moment of area of ​​a reinforcing bar, with the subscript i added to distinguish each individual reinforcing bar.

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[0040] An explanatory diagram of ai and yi for one rebar is shown in Figure 17(1). An explanatory diagram of yi for all rebars placed within a cross section is shown in Figure 17(2) and Figure 17(3). Figure 17(2) is the case where the neutral axis is in the same direction as the axis of the cross section, while Figure 17(3) is the case where the neutral axis is diagonal to the axis of the cross section, in other words, where a diagonal bending moment is occurring. It can be calculated from the coordinate of the bending moment direction of the rebar and the position of the neutral axis. With the above steps, we can now perform all the calculations required for Equation 6, which is the basic equation for the neutral axis.

[0041] [Neutral axis evaluation] The values ​​of Sn and In can be calculated even if the neutral axis is in any position, but the condition for the position of the neutral axis to be correct is that it satisfies the basic equation for the neutral axis. The basic equation for the neutral axis is transformed into an equation with the value "=0". The transformed equation is shown in Equation 15.

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[0042] The secant method is a method for searching for values ​​close to the true solution by repeated calculations. The next approximate solution is set based on the results of the previous two calculations, and by repeating this process, it is possible to search for values ​​close to the true solution. FIG. 18(1) is an explanatory diagram for determining xn that approaches F=0 from a line connecting the (n-2)th calculation point (xn-2, Fn-2) and the (n-1)th calculation point (xn-1, Fn-1), assuming that these two points are known. Figure 18 (2) is a plot of the calculation point (xn, Fn) consisting of xn obtained in (1) and the evaluation value Fn calculated from xn. Since xn is not a correct value during the calculation, the evaluation value Fn will be a value other than 0. FIG. 18(3) is an explanatory diagram for finding xn+1 that approaches F=0 from a straight line connecting the (n-1)th calculation point (xn-1, Fn-1) and the previously obtained nth calculation point (xn, Fn). For the first two points, it is necessary to determine distinct x1 and x2 and set two points (x1, F1) and (x2, F2), but from the third point onwards, the true solution can be approached by repeating the same calculations. The calculations end when a solution with a set level of accuracy or higher is confirmed. If the accuracy required for architectural structural design is at most about three digits, it has been confirmed that convergence occurs in about five tries, so this amount of calculation is not a problem in practice.

[0043] In Figure 18, a graph of the evaluation value F is included for ease of explanation, but the change in evaluation value F in actual calculations is unknown. Figure 19 shows the convergence status when a diagonal bending moment is applied to a rectangular column. Each graph plots calculation points consisting of the neutral axis position and evaluation value on an orthogonal coordinate system, with the neutral axis position on the abscissa and the evaluation value on the ordinate. They are arranged from top to bottom in the order of iterative calculations, and the first two points are set as the initial values. The horizontal line is the line where the evaluation value F is 0, and it can be seen that the position of evaluation value F approaches 0 as you move downwards. It is determined that convergence has ended on the fourth iteration, and the position of the neutral axis at that point is taken as the neutral axis where convergence has been confirmed. Once the position of the neutral axis is determined, the maximum concrete stress, σc, can be calculated using the axial force balance equation in Equation 4, or the bending moment balance equation in Equation 5. Once σc is determined, the stress level at the neutral axis is 0, so the stress level for the entire cross section can be determined by linear interpolation. Then, the stress level for all reinforcing bars can be calculated using the Young's modulus ratio n in Basic Assumption (3).

[0044] Basically, the procedure for finding the position of the neutral axis is the same for columns and beams. However, if the axial force is zero, formulas 4 and 5, which are formulas for axial force N, cannot be applied directly to find the stress in the reinforcing bars and concrete after determining the neutral axis. If the axial force is zero, formula 4, which is derived from the balance of axial forces, has no meaning, so formula 5, which is derived from the balance of bending moments, is converted to a formula using bending moment M. What is used here is the relational formula M=N·e between bending moment M, axial force N, and eccentricity distance e. Equation 5 is transformed into Equation 17.

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[0045] On the tension side, there is no problem because the sectional performance of concrete is not taken into account, but on the compression side, if the formula in Figure 15 is used for the concrete sectional performance, the area of ​​the reinforcing steel will be a double addition of the reinforcing steel and the concrete. Therefore, n-1 is included in the term for compression reinforcing steel in the RC Code section calculation formula. This -1 is used to subtract the sectional performance of the concrete part that overlaps with the reinforcing steel. Therefore, in this invention, n is replaced with n for the tension side in Formula 13 and Formula 14, and n is replaced with n-1 for the compression side. It is also possible to define the Young's modulus ratio on the tension side as nt and the Young's modulus ratio on the compression side as nc, and set nt = n and nc = n-1.

[0046] In the case of exposed column bases, the basic procedure should be the same as for bending members in reinforced concrete construction, and the stress of the anchor bolt on the compression side should be set to 0. This can be achieved by using conditional branching processing within the program, but it can also be achieved by defining the Young's modulus ratio n separately for the compression side and tension side, as described above. If nt=n, nc=0, the anchor bolt on the compression side will not be taken into account. Also, if nt=n, nc=-1, the anchor bolt on the compression side will not be taken into account, and the cross-sectional performance of the concrete in that area will be subtracted. The process of reducing the Young's modulus ratio by 1 allows for a more precise calculation of the cross-sectional performance, but the effect on the results is thought to be small. However, since this is taken into account in the calculation formula for cross-sectional calculation in the RC code, the present invention also allows for the same consideration.

Claims

1. This is a system that can calculate the cross-sections of beams and columns, which are bending members in reinforced concrete structures, and exposed column bases in steel-frame structures.It complies with the basic assumptions for calculating the cross-sections of bending members in reinforced concrete written in the RC standards, takes into account all reinforcing bars, and is characterized by its ability to directly calculate when a diagonal bending moment occurs to the cross-section of a component.

2. 2. The cross-sectional calculation system according to claim 1, wherein when calculating the cross-sectional performance of concrete in the cross-sectional calculation, the concrete cross-section is divided into basic figures by straight lines parallel to a neutral axis perpendicular to the bending moment, the cross-sectional performance is calculated for each divided basic figure, and the overall cross-sectional performance is calculated by adding up the cross-sectional performances.

3. 2. The cross-section calculation system according to claim 1, wherein when calculating the position of the neutral axis in the cross-section calculation, an evaluation value based on a basic equation of the neutral axis is calculated, and the position of the neutral axis where the evaluation value is 0 is searched for by a secant method.

4. A program for executing the above-mentioned section calculation system, which causes a computer to input data necessary for section calculation, perform section calculation using the method described in claims 2 and 3, and output the results.

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