Cross section calculation system and program for bent members in reinforced concrete structures

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

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[0003]

[0001] The present invention relates to a cross-section calculation system and program for beams, columns, which are bending members made of reinforced concrete, and exposed column bases made of steel structures. <​​​​​​​​​​​​​​​​​​​Here, we will summarize the forces that occur in bending members in structural design and the types of reinforcement that correspond to them in design. In bending members such as beams and columns, the design is divided into three types of forces: axial force, bending moment, and shear force. The reinforcement used in bending members is classified into main reinforcement and shear reinforcement. In section calculation, axial force and bending moment are addressed by main reinforcement, and shear reinforcement is addressed by shear reinforcement. Therefore, in reality, shear reinforcement is also embedded, positioned perpendicular to the main reinforcement, but since the section calculation targeted by this invention concerns bending moment and axial force, it is excluded, and Figure 2 shows only the main reinforcement and does not depict shear reinforcement. Figure 3 shows the relationship between the types of forces that occur in bending members and the corresponding reinforcement.

[0005] Figure 1(2) shows an example of a steel frame structure. In steel frame construction, the column base is located at the very bottom of the steel column. There are various types of column bases, including exposed column bases, root-wrapped column bases, and embedded column bases. However, this invention focuses on exposed column bases where the base plate is exposed and the upper steel column is fastened to the lower reinforced concrete with anchor bolts. Figure 2 shows the situation of anchor bolts in an exposed column base. Since the anchor bolts are embedded in the foundation concrete, the situation is similar to that of a reinforced concrete column. Therefore, the anchor bolts are treated as reinforcing bars, and the cross-sectional calculation is performed using a method similar to that used for calculating the cross-sectional area of ​​a column. The method of this invention can also be handled by a similar substitution, and is therefore considered a target member of this invention.

[0006] [Allowable stress calculation] Structural design in architecture is broadly classified into primary design using allowable stress calculations and secondary design using ultimate strength calculations, with differences in how external forces are defined and member stresses are calculated. The difference lies in the purpose of the design. The primary design aims to ensure that structurally important members are not damaged by rare earthquakes, while the secondary design aims to ensure that the building does not collapse or fall down in the event of extremely rare major earthquakes. Allowable stress calculation is a method of designing so that the stress generated in members such as reinforcing bars and concrete due to external forces such as seismic forces is less than or equal to their respective allowable stress levels. This invention focuses on primary design using allowable stress calculation.

[0007] This explains the difference between stress and stress intensity. Stress is the force acting inside a structural member such as reinforcing steel or concrete, while stress intensity is calculated by dividing stress by the cross-sectional area and is defined as "the stress acting per unit area of ​​the cross-section." Each structural member, such as reinforcing steel or concrete, has an allowable stress intensity. Design using allowable stress calculations involves comparing the stress intensity generated inside the structural member with the allowable stress intensity of the material to confirm safety.

[0008] To clarify the differences between beams, columns, and exposed column bases, let's first describe the difference between beams and columns. In the RC standards, the calculation formulas for beams and columns are derived under the common basic assumption that they are bending members. In typical buildings, the difference between the two in terms of structural mechanics is that axial forces are generated in columns, but not in beams. This is evident from the content of the RC standards, as well as from the headings of the articles: Article 13 is titled "Sectional Calculation for Beam Bending," and Article 14 is titled "Sectional Calculation for Column Axial Force and Bending."

[0009] Next, I will explain the difference between columns and exposed column bases. Both columns and exposed column bases are subjected to bending moments and axial forces. The difference between columns and exposed column bases lies in the forces acting on the reinforcing bars on the compression side. In columns, both tensile and compressive forces are acting on the reinforcing bars, but in exposed column bases, only tensile forces are acting on the anchor bolts, and no compressive forces are acting on them. Therefore, no stress is generated on the reinforcing bars on the compression side. This is because reinforcing bars have protrusions for attaching to concrete, while anchor bolts do not, so the two do not deform as a single unit. However, as shown in Figure 4, the structure sandwiches the concrete between the upper base plate and the lower anchoring plate, so the compressive force of the concrete and the tensile force of the anchor bolts balance each other, and tensile force is generated on the anchor bolts. As a result, only tensile force is generated on the anchor bolts.

[0010] From the above, beams are a special case where the axial force in a column is set to zero, and exposed column bases can be addressed by always setting the stress on the reinforcing bars on the compression side of the column to zero. In other words, with respect to bending members in reinforced concrete structures, columns are general members, while beams and exposed column bases can be said to be special members with added conditions. Therefore, the explanation of this invention will focus on the calculation of column cross-sections.

[0011] Article 12 of the RC standard outlines the basic assumptions for bending members, followed by Article 13, which describes the specific method for calculating the cross-section of beams, and Article 14, which describes the specific method for calculating the cross-section of columns, based on these assumptions. In this description, the cross-section calculations shown in Articles 13 and 14 will be referred to as "RC standard cross-section calculations." The point of focus here is whether the calculation results are an exact solution to the basic assumptions. The RC standard cross-section calculations can be said to be approximate cross-section calculations rather than exact solutions, due to the modeling that aggregates the reinforcement and the inability to directly analyze bending moments in diagonal directions.

[0012] [Neutral axis] The neutral axis, which is important for understanding this invention, will now be explained. The neutral axis is explained in Figure 5. The upper figure is an explanatory diagram in which the neutral axis is inserted into the cross-section of a bending member. The neutral axis is an axis perpendicular to the direction of the bending moment, and it shows that the cross-section can be divided into a tension side and a compression side at the neutral axis. The hatching in the figure indicates the compression side of the cross-section. The lower figure is a stress diagram showing the stress distribution within the cross-section. The stress is 0 at the neutral axis, and the magnitude of the stress is proportional to the distance from the neutral axis. In the design of a reinforced concrete bending member, the tensile force of the concrete is ignored, so the parts where no stress occurs in the concrete are shown by dotted lines. This stress distribution follows the basic assumptions in the cross-sectional calculation of the bending member, which will be described later. The position of the neutral axis is determined primarily by a combination of bending moment and axial force, and can exist both inside and outside the cross-section. Figure 5(1) shows the case where the neutral axis is outside the cross-section on the tensile side, resulting in only compressive force on the cross-section. Figure 5(2) shows the case where the neutral axis is inside the cross-section, resulting in both tensile and compressive forces on the cross-section. Figure 5(3) shows the case where the neutral axis is outside the cross-section on the compressive side, resulting in only tensile force on the cross-section. Reinforcing bars experience both tensile and compressive forces; those on the tension side experience tensile forces, while those on the compression side experience compressive forces. The magnitude of the stress in reinforcing bars is proportional to the distance from the neutral axis, similar to concrete.

[0013] In RC standard cross-sectional calculations, the positions of the reinforcing bars on both the tension and compression sides of beams and columns are concentrated at a single point. This allows for the creation of an equation with the neutral axis as an unknown variable, and by solving this equation, the position of the neutral axis can be determined, and the stresses of the concrete and reinforcing bars constituting the member can be calculated. This method often yields practically acceptable solutions and is used in normal structural design, but it is an approximate solution that lacks rigor in the following respects. (1) If all the reinforcing bars are located at the same distance from the neutral axis, then there is no problem with consolidation. However, if their positions are different, it is impossible to create a model in which both the first moment of area and the second moment of area, which will be discussed later, will obtain the correct values. (2) The formulas are treated differently for reinforcing bars on the tension side and those on the compression side. Therefore, the model needs to be changed depending on the position of the neutral axis, and when the reinforcing bars are distributed, the number of modeling cases increases, making it difficult to handle this strictly in reality. Figure 6 shows an example of a typical beam model, and Figure 7 shows an example of a column model. Reinforcements that are considered in the model are shown as black circles, and reinforcements that are not considered in the model are shown as white circles, with the centroid of the reinforcement indicated by an "x".

[0014] While beams are generally designed with all reinforcement bars considered because they are concentrated on the tension and compression sides, columns are effectively positioned in two directions, X and Y, so it is common to not consider some reinforcement bars perpendicular to the calculation direction. Furthermore, if not all reinforcement bars are equidistant from the neutral axis, the model will lack precision. Representative examples are shown for beams and columns, illustrating cases where precise modeling is possible and cases where it is not. (1) Two-tiered and three-tiered reinforcement in beams When reinforcing bars are arranged in multiple rows, the model concentrates the bars at the center of gravity on both the tension and compression sides. (2) Orthogonal reinforcement of the column In typical column calculations, reinforcing bars placed perpendicular to the column are not considered. Conversely, they are considered in calculations involving perpendicular columns. (3) Reinforcement bars for the pillars The cluster reinforcement is created by arranging the reinforcing bars that are closest to the symmetrical direction from among the reinforcing bars arranged in orthogonal directions. The reinforcing bars, including the cluster reinforcement, are modeled at their centroid positions on both the tension and compression sides.

[0015] [Bending moment in an oblique direction] The RC standard provides a calculation method for cases where the bending moment is applied only in the same direction as the axis of the cross section, but does not provide a direct calculation method for cases where the bending moment is applied in an oblique direction. Instead, it is explained as a biaxial bending problem where the bending moment is applied simultaneously in two directions, calculated separately as two bending moments along the material axis of the cross section. Specifically, the two axes are defined as the X-axis and Y-axis, and the yield strength is calculated separately, and an approximate formula using that yield strength is presented. Equation 1 shows the calculation formula when evaluating both axes simultaneously. When bending occurs simultaneously in two axes, it is equivalent to the combined bending moment being applied in a direction that does not coincide with either of the two axes, i.e., an oblique direction.

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Non-Patent Document 1

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Summary of the Invention

Problems 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 calculates the following two points that are not strictly handled by the normal method, in a method that strictly adheres to the basic assumptions. · Cross-section calculation considering all reinforcing bars · Cross-section calculation of a member with an obliquely generated bending moment

[0018] [Consideration of all reinforcing bars] The cross-section calculation of the RC standard used in the structural design of general bending members is calculated by solving an algebraic equation with the position of the neutral axis as an unknown based on the basic assumptions. In the process of deriving the algebraic equation, the reinforcing bars are concentrated at one point on each of the tension side and the compression side. This can be said to be a modeling lacking in strictness in the calculation of cross-section performance. Also, since the reinforcing bars are more effective on the outer side, the inner reinforcing bars may be modeled as non-existent. In contrast, in the present invention, all reinforcing bars are considered in the same way.

[0019] [Bending moment in the oblique direction] In normal structural design, it is basic to handle and design the X direction and the Y direction separately. However, in the following cases, a bending moment in the oblique direction occurs. (1) In the design of a building, an external force such as an earthquake force may be applied to the building at an angle of 45 degrees. In that case, a bending moment in the oblique direction occurs in the column. In a consistent design, in many cases, the direction of the earthquake force can be set at an angle of 45 degrees. (2) If some of the columns are positioned at an angle, a bending moment will occur in one of the columns in an oblique direction. (3) In three-dimensional analysis, if the entire building undergoes a twisting deformation, a bending moment will be generated in the columns in an oblique direction. (4) While beams normally only experience longitudinal member stress, beams subjected to earth pressure in underground levels also experience transverse member stress. In such cases, a diagonal bending moment is generated as a resultant force. Even when bending moments occur in oblique directions, safety can be confirmed based on the cross-sectional calculation according to RC standards. On the other hand, the cross-sectional calculation of the present 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 curved members in reinforced concrete is performed based on the RC standard. The RC standard outlines the basic assumptions for curved members in Article 12, titled "Basic Assumptions in the Cross-Sectional Calculation of Bent Members." These basic assumptions are as follows: (1) The tensile stress of the concrete is ignored. (2) Each cross-section of the bending member remains flat even after the member has been bent, and the compressive force of the concrete is proportional to the distance from the neutral axis. (3) The Young's modulus ratio n of the reinforcement to the concrete shall be the same regardless of the type of concrete and whether the load is long-term or short-term, 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 the cross-sectional area multiplied by cosθ.

[0021] This section explains the long-term and short-term loads that appear in the basic assumptions of bending materials. Typical types of loads are shown below. Fixed load: The building's own weight Load capacity: The weight of furniture, people, etc., inside a building. Seismic load: The force that acts horizontally on a building during an earthquake; also called seismic force. Wind load: The force exerted on each side of a building by wind, also called wind pressure. The design is based on these load combinations, which are then classified into long-term and short-term categories. Long-term: Fixed load + Live load Short-term: Fixed load + live load + seismic load, fixed load + live load + wind load Dead loads plus live loads are downward forces in the vertical direction, but seismic loads and wind loads mainly apply horizontal forces to the building, and upward axial forces, or tensile forces, are generated in the columns and column bases. Furthermore, there are long-term and short-term allowable stresses for concrete and reinforcing steel, known as long-term allowable stress and short-term allowable stress. Safety is confirmed using 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 Standard outlines the method for calculating the cross-sections of beams and columns based on Article 12. However, the RC Standard also describes the possibility of methods other than those specified in the standard, which I will quote here. Article 14 of the RC Standard, concerning the calculation of cross-sections for axial forces and bending of columns, states: "As long as the basic assumptions of Article 12 are followed, the same results will be obtained regardless of the calculation procedure. Furthermore, if a computer is used, the results can be easily obtained through numerical analysis, which is actually more convenient in practice. Moreover,..." While the phrase "the same results are obtained" differs slightly from the description of this invention, it can be interpreted as "approximate solutions to the correct answer are obtained in the same way," assuming that there will be slight differences in the results depending on the modeling and calculation methods. Furthermore, the numerical analysis method that conforms to the basic assumptions of Article 12 of the RC standard is consistent with this invention. Furthermore, the fact that methods that consider all reinforcing bars or methods that directly solve for bending moments in diagonal directions are not commonly used at this stage was used as one of the grounds for evaluating the novelty and inventiveness of the present invention.

[0023] Formulas 2 through 6 shown below are written in the RC standard and are derived from the basic assumptions. Equation 2 shows the formula for the first moment of area. The first part relates to concrete, and the second part relates to reinforcing steel. The first moment of area is a measure of the sectional property with respect to axial force.

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[0024] A crucial step in calculating the cross-sectional area of ​​a reinforced concrete bending member is determining the correct position of the neutral axis. Once the neutral axis is determined, the stresses of the reinforcement and concrete can be determined from the basic assumptions and equilibrium equations. Figure 8 shows a general flow chart for cross-sectional area calculation, but the process of "calculating the position of the neutral axis" differs between the calculation method in the RC standard and the calculation method of the present invention. In the RC standard, the position of the neutral axis is determined by setting up and solving an equation with the neutral axis position as an unknown variable. In contrast, the cross-sectional area calculation method of the present invention first calculates the cross-sectional properties that appear in the basic equation for the neutral axis more precisely as numerical values. Then, without setting up an equation, the correct position of the neutral axis is determined by iterative calculations using the basic equation for the neutral axis. Figure 9 shows the consideration flow chart for determining the position of the neutral axis in the present invention.

[0025] In this invention, the position of the neutral axis is determined by rigorously handling the "basic equation of the neutral axis" in Equation 6, which is derived from the basic assumptions, and by using a solution method that calculates an accurate solution. The following shows a method for precisely calculating Sn, the first moment of area, and In, the second moment of area, which appear in the fundamental equation for the neutral axis. (1) For concrete, the cross-section is divided into multiple basic shapes and the cross-sectional properties are calculated. (2) For reinforcing bars, the cross-sectional properties of each individual reinforcing bar are calculated. (3) The calculation results for concrete and reinforcing steel are added together to calculate the overall cross-sectional performance. This method is based on the premise that the sectional properties can be calculated by individually calculating the properties for all elements constituting the cross-section and then summing them up to arrive at the overall value. In equations 2 and 3, which represent the sectional properties, all mathematical symbols signify summation. The formulas treat concrete as a continuous quantity and reinforcement as a discrete quantity, which is consistent with general structural calculation methods for buildings. Once the method for calculating the section properties is determined, the next step is to find the position of the neutral axis that satisfies the "basic equation for the neutral axis." The equation for section properties can be calculated regardless of the position of the neutral axis. However, the "basic equation for the neutral axis" is only satisfied when the neutral axis is in the correct position; otherwise, it is not satisfied. In this invention, an evaluation value is defined from the "basic equation for the neutral axis," and the position of the neutral axis that can be approximated as the correct position is calculated by searching for the position of the neutral axis where this evaluation value becomes 0. The bracketing method is used as the search method. [Effects of the Invention]

[0026] Let's summarize the effects of this invention. The following are the objectives of the development of this invention. (1) In the calculation of the cross-section of reinforced concrete beams, columns, and exposed column bases, all reinforcing bars and anchor bolts can be considered in a manner that strictly adheres to the basic assumptions of the RC standard. (2) In the cross-sectional calculation of reinforced concrete beams, columns, and exposed column bases, the bending moment in the oblique direction can be calculated directly. Although not listed as the objective of the development of this invention, the following effects are also present. (3) Beams and columns, which are designated as bending members in the RC standard, as well as exposed column bases, which can be calculated using a method similar to that for columns, can be calculated using a common algorithm. This also has the effect of enabling the following cross-sectional calculations for beams, which previously required the assumption that the axial force was zero. (4) It is possible to calculate the cross-sectional area of ​​a beam with axial force. (5) It is possible to calculate the cross-section of beams subjected to lateral forces such as earth pressure and axial forces. Exposed column bases have the same effect as columns, plus the following effects: (6) It is possible to calculate the cross-section of exposed column bases using reinforcing bars instead of anchor bolts. [Brief explanation of the drawing]

[0027] [Figure 1] This diagram shows the locations of the members covered by the present invention. The beams and columns are reinforced concrete, and the exposed columns are steel frame members. [Figure 2]This figure shows that the member targeted by this invention is composed of concrete and reinforcing bars embedded therein. Reinforcing bars consist of main bars and shear reinforcement bars, but only the main bars are relevant to the cross-sectional calculation targeted by this invention, so only the main bars are shown in the figure. [Figure 3] This diagram shows the relationship between the types of forces acting on a bending member and the types of reinforcing bars that correspond to them in the design. [Figure 4] This diagram illustrates the principle by which tensile force is generated in the anchor bolts of an exposed column base. [Figure 5] This diagram illustrates the neutral axis and stress levels. [Figure 6] This is a diagram illustrating the modeling of the reinforcing bars in a beam. [Figure 7] This is a diagram illustrating the modeling of the reinforcing bars in a column. [Figure 8] This is a general flowchart for calculating the cross-sectional area of ​​a bent member. [Figure 9] This is a flowchart for searching for the position of the neutral axis in the present invention. [Figure 10] This is a diagram showing the configuration of a computer that implements the present invention. [Figure 11] This is a diagram illustrating the bending moment in an oblique direction. [Figure 12] This diagram shows the division of a rectangular cross-section when a bending moment is applied in an oblique direction. [Figure 13] This diagram shows how the cross-sections are divided when a bending moment is applied to a square and rectangular cross-section, by changing the angle of the bending moment's direction. [Figure 14] This diagram shows the types of divided cross-sections when a cross-section is divided along the neutral axis in the event of a bending moment occurring in an oblique direction. [Figure 15] This diagram shows the formulas for the cross-sectional properties of rectangles, parallelograms, trapezoids, and triangles. The formulas are for the case where the neutral axis passes through the centroid of the cross-section. [Figure 16] This diagram illustrates the correction formula for section properties when the neutral axis does not pass through the centroid of the section. [Figure 17] This diagram illustrates the distance y used to calculate the cross-sectional properties of reinforcing bars. [Figure 18] This diagram illustrates the secant method used to find the position of the neutral axis. [Figure 19] This diagram illustrates the convergence of the secant method when a bending moment is applied to a rectangular cross-section in an oblique direction. [Modes for carrying out the invention]

[0028] Figure 10 shows the configuration of the computer running this system. The computer consists of a CPU (processing unit), memory (main memory), and auxiliary storage devices such as an HDD or SSD. Input devices such as a keyboard and mouse, and output devices such as a display and printer are connected to the main unit, and it is also connected to a network that allows access to external servers. The calculations of this invention are all written in source code using a programming language. This source code is then compiled into an executable file that a computer can run. 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 then loaded into memory, and the CPU runs the executable file. Memory is used as a work area, etc., during CPU operation. When the user starts the system, the executable file is loaded into memory. By selecting a calculation menu for beams, columns, etc., inputting data in the appropriate format, and starting the calculation, the system performs cross-section calculations. Data can be input using the keyboard and mouse, and can also be saved to an auxiliary storage device and retrieved for use. The cross-section calculation results are displayed on the screen and can be printed if necessary.

[0029] The equations for the first moment of area and the second moment of area are rearranged for the purpose of explaining the present invention. Equation 7 shows the formula for the first moment of area of ​​the calculated cross section. The first moment of area of ​​the calculated cross section is the sum of the first moment of area of ​​the concrete and the first moment of area of ​​the reinforcing steel.

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[0030] Equations 9 to 12 all include a distance y from the neutral axis. The neutral axis is perpendicular to the direction of the bending moment, but its position is not obvious. If the bending moment is in the X-axis direction, the neutral axis is perpendicular to the X-axis; if it is in the Y-axis direction, the neutral axis is perpendicular to the Y-axis. Also, if the bending moment is in an oblique direction that does not coincide with either the X or Y axis, the neutral axis is perpendicular to the direction of that bending moment. If both the X-axis and Y-axis components are obtained as a result of the stereoscopic analysis, the position of the neutral axis is determined on separate axes and the cross-section is calculated. In this invention, in order to demonstrate a method for directly calculating the cross-section of an oblique bending moment obtained by combining the X-axis and Y-axis components, we assume that the neutral axis is oblique.

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

[0032] Figure 12 shows a modified version of (3) in Figure 11. Figure 12(1) is the same figure as Figure 11(3). Figure 12(2) shows Figure 12(1) rotated so that the neutral axis is vertical, while keeping the relationship between the cross-section and the neutral axis unchanged. Figures 12(1) and 12(2) are the same figure 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 the case where a bending moment occurs in an oblique direction in a rectangular cross-section, and the case where the neutral axis is parallel to one of the sides of the rectangle can be considered a special case.

[0033] [Cross-sectional properties of concrete] As shown in Figure 12 (2), the section properties of concrete are derived when the neutral axis is oblique. When the shape is not simple, such as with H-beams, the section properties can be determined by dividing it into several basic shapes. Basic shapes are figures for which formulas for area, second moment of area, etc., and in addition to rectangles, triangles, parallelograms, trapezoids, circles, etc., are used. Figure 12(3) shows Figure 12(2) divided into three shapes by dividing lines parallel to the neutral axis. There are countless ways to divide it, but in this case, the division is done by lines passing through the vertices to minimize the number of divisions. In Figure 12(3), it is divided into two triangles and one parallelogram. This division is one example of a division when the neutral axis is diagonal, 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 compared to (3) in Figure 12. The upper row shows a square cross-section, and the lower row shows a rectangular cross-section. Although the shape of the divided figure changes depending on the angle, it can be seen that it is basically divided into three figures by two dividing lines, just like in (3) in Figure 12. The three figures are of two types: triangles on both sides and a parallelogram in the center. 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 these can be treated similarly as special cases, so a rectangular cross-section can be divided into three or fewer basic figures. Since the cross-sectional properties can be calculated for each divided basic shape and then added together, it is sufficient to calculate the cross-sectional properties of rectangles, triangles, and parallelograms.

[0035] [Division along the 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 only compression occurs in concrete, and therefore the sectional properties are calculated only for the compressive portion, without considering the tensile portion. Figure 14 shows patterns in which the basically divided cross-section is further divided according to the position of the neutral axis. Since the neutral axis can exist both inside and outside the cross-section, the patterns are organized to include the outside of the cross-section as well. If the neutral axis is located within the parallelogram portion as in (3) of Figure 14, it is divided into two parallelograms. However, if the neutral axis is located within the triangular portion as in (2) of Figure 14, it is divided into two types of shapes: a triangle and a trapezoid. Therefore, a trapezoid is added as a basic type of shape. As a result, it is sufficient to calculate the sectional properties of rectangles, triangles, parallelograms, and trapezoids. There are generally known formulas for these shapes, which are shown in Figure 15. In the figure, A represents the cross-sectional area, I represents the second moment of area, and c represents the position of the centroid.

[0036] [Formula for Sectional Properties] In the formula shown in Figure 15, the equations for rectangles and parallelograms are exactly the same. It can be confirmed that the equation for triangles is the same as that of a=0 for the upper base of a trapezoid. Furthermore, it can be confirmed that the equations for rectangles and parallelograms are the same as those for rectangles and parallelograms, assuming a=b for both the upper and lower bases of a trapezoid. Therefore, it is possible to represent all basic geometric figures with the equation of a trapezoid.

[0037] The formula in Figure 15 is for the case where the neutral axis passes through the centroid. If the axial force is 0, the neutral axis passes through the centroid, but in the following cases the neutral axis does not pass through the centroid. (1) When an axial force is present (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 materials, such as reinforced concrete. Therefore, the conversion formula shown in Figure 16 is used. This formula can be used to calculate the result regardless of the type of original figure, as long as the cross-sectional area, the second moment of area, and the position of the centroid relative to the neutral axis are known.

[0038] Let me provide some supplementary explanation regarding the formulas in Figures 15 and 16. The formula for the second moment of area in Figure 16 includes 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 include a corresponding value. This is because the first moment of area is 0 when the neutral axis passes through the centroid, which is consistent with substituting y=0 into the formula in Figure 16, resulting in S=0 and I=Io. Therefore, the formula for the first moment of area is not included in Figure 15.

[0039] [Cross-sectional properties of reinforcing bars] For reinforcing bars, the formulas in the RC standard can be used directly. The distance of each reinforcing bar from the neutral axis can be calculated from the coordinates regardless of the arrangement pattern of the reinforcing bars. Formulas 10 and 12 are in the form of multiplying 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 standard, the Young's modulus ratio n does not change within the member, but the cross-sectional area and neutral axis may differ for each reinforcing bar, so we add subscripts so that the cross-sectional area of ​​the reinforcing bar is ai and the distance from the neutral axis is yi. Formulas 10 and 12 are shown below, with the cross-sectional area of ​​each reinforcing bar replaced by ai and the distance from the neutral axis replaced by yi. Equation 13 shows the formula for the first moment of area of ​​a reinforcing bar, with the subscript i added to Equation 10 to distinguish each individual reinforcing bar.

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[0040] Figure 17(1) shows an explanatory diagram of the ai and yi of a single reinforcing bar. Figures 17(2) and 17(3) show explanatory diagrams of the yi of all reinforcing bars arranged within the cross-section. Figure 17(2) shows the case where the neutral axis is in the same direction as the axis of the cross-section, while Figure 17(3) shows the case where the neutral axis is oblique to the axis of the cross-section, i.e., an oblique bending moment is generated. This can be calculated using the coordinates in the direction of the bending moment of the reinforcing bar and the position of the neutral axis. With the above steps, we can now perform all the calculations necessary for equation 6, which is the fundamental formula for the neutral axis.

[0041] [Neutral scale evaluation] The values ​​of Sn and In can be calculated even when the neutral axis is in an arbitrary position, but the condition for the neutral axis to be in the correct position is that it satisfies the fundamental equation for the neutral axis. The fundamental equation for the neutral axis is transformed into an equation where "=0". The transformed equation is shown in equation 15.

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[0042] The secant method is a method for searching for a value close to the true solution through iterative calculations. By setting the next approximate solution based on the results of the previous two calculations and repeating this process, it is possible to search for a value close to the true solution. Figure 18(1) is an explanatory diagram that shows how to find xn approaching F=0 from the line connecting the (n-2)th calculation point (xn-2, Fn-2) and the (n-1)th calculation point (xn-1, Fn-1), which are known. Figure 18 (2) is a plot of the calculation points (xn, Fn) consisting of xn obtained in (1) and the evaluation value Fn calculated from xn. During the calculation, xn is not the correct value, so its evaluation value Fn will be a non-zero value. Figure 18 (3) is an explanatory diagram that shows how to find xn+1 that approaches F=0 from the line connecting the n-1 calculation point (xn-1, Fn-1) and the previously obtained n calculation point (xn, Fn). For the first two points, it is necessary to determine two distinct points x1 and x2 and set up two points (x1, F1) and (x2, F2). However, from the third point onward, the true solution can be approached through similar iterative calculations. The calculation is terminated when a solution with a specified level of precision or higher is confirmed. Assuming that the precision required for structural design in architecture is at most around three digits, it has been confirmed that convergence occurs in about five iterations, so the amount of computation is not problematic for practical purposes.

[0043] Figure 18 includes a graph of the evaluation value F for illustrative purposes, but the actual change in the evaluation value F is unknown in the calculation. Figure 19 shows the convergence process when a diagonal bending moment is applied to a rectangular column. Each graph plots the calculation points consisting of the neutral axis position and the evaluation value on a Cartesian coordinate system where the neutral axis position is the horizontal coordinate and the evaluation value is the vertical coordinate. The graphs are arranged from top to bottom in the order of the iterative calculations, and the first two points are the initial values ​​as they were set. The horizontal line is the line where the evaluation value F is 0, and it can be seen that the position of the evaluation value F approaches 0 as you go down. Convergence is judged to be complete after the 4th iteration, and the neutral axis position at that point is set as the neutral axis where convergence is confirmed. Once the position of the neutral axis is determined, the maximum stress σc of the concrete can be found using the axial force equilibrium equation (Equation 4) or the bending moment equilibrium equation (Equation 5). Once σc is determined, the stress at the neutral axis is 0, so the stress across the entire cross-section can be determined by linear interpolation. Then, the stress across all the reinforcement bars can be found using the Young's modulus ratio n from basic assumption (3).

[0044] Basically, the procedure for determining the position of the neutral axis is the same for columns and beams. However, when the axial force is 0, equations 4 and 5, which are equations for the axial force N, cannot be directly applied to determine the stress of the reinforcement and concrete after determining the neutral axis. When the axial force is 0, equation 4, which is derived from the equilibrium of axial forces, is meaningless, so equation 5, which is derived from the equilibrium of bending moments, is converted into an equation using the bending moment M. The relationship between the bending moment M, the axial force N, and the eccentricity distance e is M = N·e, which is used here. Transform equation 5 into equation 17.

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[0045] On the tensile side, there is no problem as the concrete's cross-sectional properties are not considered. However, on the compressive side, if the formula in Figure 15 is used for the concrete's cross-sectional properties, the area of ​​the reinforced concrete portion will be a double addition of the reinforcement and concrete areas. For this reason, the RC standard's cross-sectional calculation formula also includes n-1 in the term for the compressive reinforcement. This -1 subtracts the cross-sectional properties of the concrete portion that overlaps with the reinforcement. Therefore, in this invention, in equations 13 and 14, n is replaced with n-1 for the tensile side and n with n-1 for the compressive side in the calculation. Alternatively, the Young's modulus ratio on the tensile side can be defined as nt and the Young's modulus ratio on the compressive side as nc, and nt=n and nc=n-1.

[0046] In the case of exposed column bases, the basic treatment is the same as for bending members in reinforced concrete structures, and the goal is to ensure that the stress on the compression-side anchor bolts is zero. This can be achieved through conditional branching in the program, but it can also be handled by defining the Young's modulus ratio n separately for the compression and tension sides, as described above. Setting nt=n and nc=0 means that the compression-side anchor bolts are not considered. Furthermore, setting nt=n and nc=-1 means that the compression-side anchor bolts are not considered, and the sectional properties of the concrete in that section are also subtracted. The process of subtracting 1 from the Young's modulus ratio allows for a more precise calculation of the cross-sectional properties, although its impact on the results is thought to be small. However, since this is taken into consideration in the calculation formula for cross-sectional properties according to the RC standard, the present invention also incorporates a similar consideration.

Claims

1. A cross-sectional calculation system capable of calculating the cross-sectional sections of beams and columns, which are bending members of reinforced concrete structures, and the cross-sectional sections of exposed column bases of steel structures, wherein the cross-sections of the beams, columns, and exposed column bases are rectangular. Determine the position of a provisional neutral axis that lies on the plane including the cross-section and is perpendicular to the bending moment, and is not parallel to any side of the cross-section. The cross-section is divided into one or more types of shapes by straight lines perpendicular to the bending moment, For each of the aforementioned figures, the first moment of area and the second moment of area of ​​the concrete are calculated based on the distance from the provisional neutral axis. For each of the reinforcing bars contained within the aforementioned cross-section, the first moment of area and the second moment of area of ​​the reinforcing bar are calculated based on the cross-sectional area of ​​the reinforcing bar and the distance between the reinforcing bar and the temporary neutral axis. The sum of the first moment of area of ​​the concrete and the first moment of area of ​​the reinforcement is calculated as the first moment of area of ​​the calculated cross section, and the sum of the second moment of area of ​​the concrete and the second moment of area of ​​the reinforcement is calculated as the second moment of area of ​​the calculated cross section. Based on the first moment of area and the second moment of area of ​​the calculated cross section, it is determined whether or not the fundamental equation for the neutral axis is satisfied. If it is determined that the foundation equation for the neutral axis is satisfied, the calculation unit determines the provisional neutral axis to be the correct neutral axis and calculates the stress of the concrete and the stress of the reinforcing bars based on the correct neutral axis. A cross-section calculation system characterized by comprising the following:

2. When the provisional neutral axis passes through the cross-section, the straight line includes the provisional neutral axis, The cross-section calculation system according to claim 1, wherein if the provisional neutral axis does not pass through the cross-section, the straight line does not include the provisional neutral axis.

3. The calculation unit is: If it is determined that the aforementioned basic equation for the neutral axis is not satisfied, a new provisional neutral axis position is determined based on the secant method. A cross section calculation system according to claim 1 or 2, which uses the new provisional neutral axis to perform the following processes: division of the cross section, calculation of the first moment of area and the second moment of area of ​​the concrete, calculation of the first moment of area and the second moment of area of ​​the reinforcing steel, calculation of the first moment of area of ​​the calculated cross section, calculation of the second moment of area of ​​the calculated cross section, determination of the foundation equation of the neutral axis, determination of the correct neutral axis, and calculation of the stress of the concrete and the stress of the reinforcing steel.

4. A cross-section calculation program capable of calculating the cross-section of beams and columns which are bent members of reinforced concrete structures, and the cross-section of exposed column bases of steel structures, wherein the cross-sections of the beams, columns and exposed column bases are rectangular, On the computer, Determine the position of a provisional neutral axis that lies on the plane containing each side of the cross-section and is perpendicular to the bending moment, and is not parallel to the cross-section. The cross-section is divided into one or more types of shapes by straight lines perpendicular to the bending moment, For each of the aforementioned figures, the first moment of area and the second moment of area of ​​the concrete are calculated based on the distance from the provisional neutral axis. For each of the reinforcing bars contained within the aforementioned cross-section, the first moment of area and the second moment of area of ​​the reinforcing bar are calculated based on the cross-sectional area of ​​the reinforcing bar and the distance between the reinforcing bar and the temporary neutral axis. The sum of the first moment of area of ​​the concrete and the first moment of area of ​​the reinforcement is calculated as the first moment of area of ​​the calculated cross section, and the sum of the second moment of area of ​​the concrete and the second moment of area of ​​the reinforcement is calculated as the second moment of area of ​​the calculated cross section. Based on the first moment of area and the second moment of area of ​​the calculated cross section, it is determined whether or not the fundamental equation for the neutral axis is satisfied. If it is determined that the foundation equation for the neutral axis is satisfied, the provisional neutral axis is determined to be the correct neutral axis, and the stress of the concrete and the stress of the reinforcing bars are calculated based on the correct neutral axis. A cross-section calculation program to perform the following.

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