JP7415474B2 - Calculation method, program and system - Google Patents

Description

The present invention relates to a calculation method, program, and system.

As a conventional technique, a method is known in which a concrete structure is modeled using finite elements, and the performance of the concrete structure is evaluated by performing finite element analysis on this model (Patent Document 1).

Japanese Patent Application Publication No. 2004-171190

K. Naganuma, et. al.: Simulation of Nonlinear Dynamic Response of Reinforced Concrete Scaled Model Using Three-dimensional Finite Element Method, 13th World Conference on Earthquake Engineering, Vancouver, B.C., Canada, Paper No.586, 2004.8 H. Nakamura and T. Higai: Compressive Fracture Energy and Fracture Zone Length of Concrete, Seminar on Post-Peak Behavior of RC Structures Subjected to Seismic Load, JCI-C51E, Vol.2, pp.259-272, 1999.10 Architectural Institute of Japan: Guaranteed toughness seismic design guidelines and commentary for reinforced concrete buildings, 1999 Architectural Institute of Japan: Maintenance management guidelines and commentary for buildings in nuclear facilities, 2008

When attempting to evaluate the performance of concrete structures by performing finite element analysis (hereinafter also referred to as FEM analysis) using conventional technology, it was difficult to quantitatively calculate the cracks that occurred in the concrete structures. .

In view of the above problems, the present invention provides a method that can quantitatively calculate cracks in concrete structures and can be used for performance evaluation of concrete structures, or a program or system using the method.

In order to solve the above problems, one aspect of the present invention is to perform finite element analysis on a finite element model of a concrete structure, and to calculate the product of the initial area and the crack-equivalent strain in the direction orthogonal to the crack in the region on the finite element model. Provided is a method for calculating a crack area of at least a portion of the concrete structure based on the above.

As one aspect of the present invention, output data of a finite element analysis for a finite element model of a concrete structure is acquired, and based on the product of the initial area in the region on the finite element model and the crack equivalent strain in the direction orthogonal to the crack, A program is provided that causes a computer to execute a process of calculating a crack area of at least a portion of the concrete structure.

Further, one aspect of the present invention includes a storage device that stores a program for executing a finite element analysis of a concrete structure modeled using finite elements, and a control device capable of executing the program, The apparatus provides a system for calculating a crack area of at least a portion of the concrete structure based on a product of an initial area in a region on the finite element model and a crack-equivalent strain in a direction perpendicular to the crack.

According to the present invention, it is possible to provide a method that quantitatively calculates the amount of cracks in a concrete structure and can be used for performance evaluation of the concrete structure, or a program or system using the method.

(a) An outline of a method for calculating a crack area in a micro region, (b) an outline of a method for calculating a crack area in the entire region, and (c) a mechanical strain-stress relationship graph of concrete. It is an outline of the calculation method of the crack area concerning a modification. (a) An example of a concrete structure, (b) A diagram showing a finite element analysis model of the reinforced concrete structure, (c) A diagram showing the state of loading and deformation of the analysis model, (d) Analysis model according to the embodiment Diagram showing a method for calculating the crack area of an element, (e) Calculation formula for the crack area of the analytical model element, (f) Calculating method of the crack area of the analytical model, and (g) Another calculation of the crack area of the analytical model element. FIG. 2 is a diagram illustrating the method. FIG. 3 is a diagram showing a flow of performance evaluation according to the embodiment. It is a drawing of a test specimen used in an experiment. It is a figure showing specifications of a test object, and shows (a) dimensions and reinforcement arrangement, (b) mixture of concrete, (c) material test results of concrete, and (d) tensile test results of reinforcing bars. FIG. 2 is a diagram showing a mesh drawn on a test specimen and reinforcing bar strain measurement positions. It is a schematic diagram showing (a) a method of calculating a deformation angle, and (b) a method of measuring a crack area in a test specimen. 3 is a graph showing the horizontal load versus deformation angle among the experimental results. FIG. 2 is a diagram showing an analytical model of a test specimen so that the state of element division can be understood. This is a graph comparing the relationship between horizontal load and deformation angle of a shear wall in experiments and analyses. This is a graph comparing the crack area with respect to the deformation angle of the shear wall in experiments and analyses. Figure (a) shows the cracks in the walls and columns obtained through the experiment. Figure (b) shows the cracks in the walls and columns obtained from the analysis. (a) A state in which the element is deformed due to thermal strain, and (b) a diagram showing a state in which the element is deformed under biaxial stress. FIG. 1 is a block diagram of a computer system according to the present embodiment.

A method for evaluating the performance of a concrete structure according to an embodiment will be described below with reference to FIGS. 1 to 15. In addition, various concrete structures such as reinforced concrete, steel plate concrete, and concrete-filled steel pipes are assumed as specific examples of concrete structures.

In addition, performance evaluation includes evaluation of leakage rate (airtightness/watertightness) of concrete structures, evaluation of negative pressure maintenance function, and evaluation of radiation shielding performance. Furthermore, evaluating the damage that has occurred to the concrete structure is also included in the performance evaluation in the present invention.

<Embodiment>
The performance evaluation method according to the embodiment is performed by modeling a concrete structure using finite elements and calculating a crack area. The crack area is calculated by deriving a crack area calculation formula. First, the procedure for deriving the crack area calculation formula will be explained below. Note that the crack area refers to the total area of cracks that occur within a set area in a concrete structure. In addition, in an analytical model, it refers to the total area of cracks that occur within a region set by the analyst, such as a predetermined element or the entire model.

Furthermore, the targets of crack area calculation are not limited to cracks that appear on the surface or side surfaces of structures or analysis model elements, but also include cracks in cross sections of concrete structures and element cross sections.

In deriving the crack area calculation formula, first assume a region S with an arbitrary shape as shown in Figure 1(a) on concrete or its model, and consider calculating the crack area within this region S. . Further, in the region S, the crack length direction (the direction in which the crack extends) is defined as the x-axis direction, and the direction orthogonal to the crack is defined as the y-axis direction.

When a rectangular minute area extending in the x-axis and y-axis is extracted in the area S, the crack width w _cr in this minute area is expressed as in equation (1) in FIG. 1(a). Furthermore, since the length of this micro region in the x-axis direction is equal to the length of the crack, the crack area a _cr can be calculated as shown in equation (2) in FIG. 1(a). Here, the strain equivalent to cracking in the y-axis direction is defined as shown in FIG. 1(c). That is, it is the mechanical strain generated in the concrete in the direction perpendicular to the crack, minus the strain at the time of crack occurrence (crack strain). Note that the definition of the total strain in Equation (6) is an example. It is clear that, in addition to thermal strain and creep strain, shrinkage strain and the like can be added to the right side of equation (6) depending on the load and strain that actually occur in the concrete structure.

The crack area in the region S is obtained by adding the crack areas in the micro regions over the entire region S. That is, when the number of minute elements in the region S is j, the crack area is expressed as in equation (3) in FIG. 1(b).

Equation (3) is converted into Equation (4) for double integration in the x-axis direction and the y-axis direction by reducing the size of the minute region to the minimum, and Equation (5a) is obtained as a result. As shown in Equation (5a), the crack area in the region S is expressed as the product of the average value of crack-equivalent strain in the region S and the initial area of the region S. The initial area is the area in a state where no strain is generated in the region S.

In addition, if the cracking equivalent strain is a constant value or if the area S is sufficiently small, a representative point within the area S may be determined, and the cracking equivalent strain at this representative point may be used as the representative value to obtain the crack area ( Formula (5b)). Furthermore, the representative value of strain equivalent to cracking may be determined based on a plurality of representative points. As an example of the representative point, if the region S is an element of finite element analysis, an integration point may be selected. The method of setting the representative value is not limited to using the average value in the area S, but also the method of using the value of the representative point as is, or applying Simpson's formula or Gaussian integral formula to multiple representative points. Various methods can be considered, such as determining the

<Modified example>
It is also possible to derive a crack area calculation formula by considering the following. This will be explained as a modified example.

In a modified example, as shown in FIG. 2, the crack length in the microelement in the area S is considered to be equal to the actual length of the microelement in the x-axis direction. Since the crack area is calculated by crack width x crack length, it can be considered that it is affected not only by the strain in the direction perpendicular to the crack, but also by the strain in the direction of the crack length. In this case, the crack area in the microelement can be expressed as in equation (8) in FIG. 2 using the total strain in the x-axis direction. Equation (8) can be obtained by using (1+ε _tot,x )Δx instead of Δx as the width of the minute element in Equation (2) of FIG.

As a result, the crack area in region S is expressed as equations (9a) and (9b) in FIG. The derivation process is similar to that performed in the above embodiment (FIG. 1). Equations (9a) and (9b) correspond to equations (5a) and (5b) in FIG. 1. Note that the average value or representative value in the region S may be used for the total strain in the x-axis direction in equations (9a) and (9b), especially when the total strain is not uniform in the region S.

Further, by transforming the right sides of these equations (9a) and (9b) and passing through equation (10), equations (11a) and (11b) are obtained. As shown in equations (11a) and (11b), these crack areas are determined by the increase or decrease in crack area based on the change in crack length from the initial state with respect to the crack area obtained by equations (5a) and (5b). It can be seen that the amounts are added.

<Performance evaluation flow>
Based on the formula obtained above, a method for evaluating the performance of concrete structures will be described below. Here, a case is assumed in which a rectangular reinforced concrete shear wall 100A is modeled as an analysis model 100 as shown in FIG. 3, and performance evaluation is performed after finite element analysis. The reinforced concrete shear wall 100A is a flat member extending in the horizontal and vertical directions. The lower end of the reinforced concrete shear wall 100A is simply supported by rollers and pins, as shown in FIG. 3(a).

The analytical model 100 is composed of a plurality of nodes 1 to 9 and a plurality of rectangular elements 11 to 22 defined by each node, as shown in FIG. 3(b). Similar to the reinforced concrete shear wall 100A, a horizontal force is applied to the node 7 of the analytical model 100 (FIG. 3(c)). The lower end of the analytical model 100 is supported by rollers and pins, similar to the reinforced concrete shear wall 100A. It is assumed that a distributed crack model (non-patent document 1, also referred to as smeared crack model) is used as the analysis model. Appropriate constitutive laws for concrete and reinforcing bars are selected in consideration of actual physical properties.

Performance evaluation is performed based on the flow shown in FIG. First, in step S1, preset conditions such as load and displacement are input to the nodes 11 to 22, and the response of the analytical model 100 to these conditions is determined. The response is obtained as a numerical solution based on a general finite element analysis method. That is, this is a method of determining the overall stiffness matrix of the analytical model 100 from the stiffness matrices of each of the elements 11 to 22, and using this to obtain an approximate solution.

In step S2, the strain and stress of each element in the analytical model 100 are output. Note that output may be performed for each analysis step. In subsequent steps, the crack area is calculated by acquiring the strain. There are various methods of calculating strain, such as a method of extracting it from the analysis process and a method of calculating it from nodal displacement, and the method to be selected is determined as appropriate depending on the purpose of the analysis.

In step S3, the crack area is calculated. For calculating the crack area, equations (12a) and (12b) in FIG. 3(e), which were derived based on equations (5a) and (5b) in FIG. 1, are used. Formulas (12a) and (12b) are obtained by replacing region S with element mn in formulas (5a) and (5b) in FIG. As shown in equations (12a) and (12b), in step S3, the crack area for each element is first calculated as shown in FIG. 3(d). Next, as shown in equation (13) in FIG. 3(f), by calculating the sum of the crack areas in each element, the crack area for the entire element analysis model can be obtained. After that, the crack rate is calculated from the ratio of the sum of the crack areas of each element (that is, the crack area of the entire element analysis model) and the sum of the initial areas of the elements (Fig. 3 (f), Equation (14) ).

Note that, as shown in FIG. 3(g), a method may be used in which the element is further divided into a plurality of regions and the crack area in each region is determined. For example, in FIG. 3(g), element mn is divided into four regions, and the crack area is calculated for each of these regions. The crack area of element mn in this case can be expressed as the sum of the crack areas in these four regions. Alternatively, a region formed by a plurality of elements may be determined, and the crack area may be calculated using the average value of crack-equivalent strain within this region.

Finally, in step S4, the performance of the reinforced concrete shear wall 100A is evaluated. Performance evaluation of the reinforced concrete shear wall 100A, such as leakage rate evaluation, is performed using the crack area or crack rate obtained in step S3.

Note that in performing performance evaluation, it is not necessarily necessary to obtain the crack area for the entire analytical model. The crack area may be calculated for some regions or elements of the analytical model, and the performance of the reinforced concrete shear wall 100A may be evaluated.

<Comparison of experiment and analysis>
In order to verify the validity of the above performance evaluation method, an experiment was conducted using a test specimen, and the experimental results were compared with the calculation results according to the present invention. The details are explained below.

[Test specimen]
FIG. 5 shows the shape of the specimen, and FIG. 6 shows the specifications of the specimen. The test specimen was a reinforced concrete shear wall. The test specimen had a seismic wall (hereinafter also referred to as wall plate) in the center, and side columns were placed on the left and right sides of it. Additionally, stubs extending left and right are fixed to the top and bottom of the seismic walls. To make it easier to observe cracks, the surfaces of the walls and pillars were painted with white paint, and a 100 mm square mesh was drawn (Figure 7).

The wall reinforcement ratio of the test specimen was 0.5% (Fig. 6(a)). The wall portion of the test specimen and the core of the columns on both sides were aligned, and the wall transverse reinforcement was fixed in a straight line (fixed length 30 d) within the column (Figure 5). The wall vertical reinforcements were fixed by welding to the steel plate at the bottom of the foundation stub, and fixed in a straight line (fixed length 50 d) to the load-bearing stub.

As shown in FIG. 6(b), ordinary cement was used as the cement, and the maximum dimension of the coarse aggregate was 9 mm. Figure 6(c) shows the material test results for concrete, and Figure 6(d) shows the tensile test results for reinforcing bars. Three specimens were used in each material test, and each figure shows their average value.

[Applying method, measurement method]
The method of applying force will be described. In the experiment, as shown in Figure 5, a column axial force of 146 kN was first applied to the tops of both columns. The column axial force was calculated so that the axial force ratio to the concrete compressive strength of the test specimen was 0.1.

Next, while the column axial force was held constant, a gradually increasing horizontal force was applied in alternating positive and negative directions to the height center position of the force-applying stub. During application, the deformation angle was calculated from the horizontal relative displacement of the horizontal force loading position with respect to the wall leg (Figure 8(a)), and was 0.1%, 0.15%, 0.2%, and 0.25%. After repeated positive and negative loading (twice for 0.2%, once for others), it was pushed in the positive direction.

[Measurement of crack width and crack area]
The crack width was measured when the first crack occurred, at each deformation peak during repeated loading, when unloading from each peak, and when the deformation angle reached +0.3%.

The crack area was calculated for each crack by multiplying the crack width by the crack length. In this way, the crack area was calculated for all the cracks in the shear wall, and the sum of these was taken as the crack area in the shear wall.

In detail, the crack area (Acr) was determined by the method and formula shown in FIG. 8(b). Here, w _i is the crack width (millimeters, mm) measured at point i. w _S and w _E are the crack widths (mm) at the starting point and the ending point, respectively, and were taken as the crack widths at the closest measurement point (in the example of FIG. 8(b), w _S = w ₁ , w _E = w ₄ ). L _i→i+1 is the crack length (mm) between points i and i+1, and is not the straight line distance between two points, but the length of the trajectory of a multipoint polygonal line tracing the crack. This was calculated for all cracks and the total was taken as the crack area. Note that if the measured crack width was smaller than the minimum graduation of 0.03 mm of the crack scale used in this experiment, w _i = 0 mm was set.

[Experiment results summary]
Figure 9 shows the relationship between horizontal load and deformation angle. The initial crack initiation point was the point where diagonal cracks in the wall were first visually confirmed. In addition, the yield point of the wall reinforcement was determined from the values of strain gauges (Fig. 7) attached at 150 mm intervals to the vertical and horizontal wall reinforcement installed in the center of the earthquake-resistant wall (wall plate), and the point where the yield strain was reached first appeared. It is a point.

The first crack occurred around the deformation angle of ±0.02%. Later, as the deformation increased, the cracks in the wall increased and the stiffness decreased. In the test specimen, the wall longitudinal reinforcements yielded when the deformation angle reached -0.18%. In addition, in the final push-off loading, the wall transverse reinforcement yielded when the deformation angle exceeded +0.5%, and the maximum load was reached near the deformation angle +0.6%.

Figure 13(a) shows the cracks in the wall and column at a deformation angle of +0.2%. The first crack was a crack extending diagonally downward from the vicinity of the third or fourth mesh from the top of the left and right ends of the shear wall. In addition, in this test, the force was applied from both the left and right sides of the force-applying stub, so there were almost no cracks in the upper parts of the left and right sides of the wall on which no force was applied.

〔Analysis model〕
Next, the test specimen was modeled using a finite analysis model M, and the analysis results were compared with the experimental results. The details are explained below.

FIG. 10 shows the element division of the analytical model M. The analysis model M was a three-dimensional model in which 1/2 of the wall thickness was modeled in consideration of symmetry conditions. The concrete was modeled using hexahedral elements, the main column reinforcements were modeled using wire rod elements, and the wall and column reinforcements were modeled using embedded reinforcing bars. A spring element was inserted between the concrete and the main reinforcement of the column, and the adhesive slip characteristics were taken into consideration. The reinforcement arrangement was modeled in the same way as the test specimen, and the wall reinforcement ratio was set to 0.5%.

In the analysis model M, displacement in all directions was restrained at all nodes on the bottom surface, and nodal displacement in the wall thickness direction was restrained at the cut plane on the axis of symmetry. As in the experiment, after applying an axial force to the top of the column, alternating positive and negative loading was performed using forced displacement at the center height of the applied stub.

The material model was selected with reference to Non-Patent Document 1. Concrete was assumed to be an orthotropic body based on equivalent uniaxial strain, and cracks occurring in multiple directions were considered using a non-orthogonal distributed crack model. The stress-strain relationship on the compression side was expressed by a modified Ahmad model, taking into account the reduction in compressive strength after cracking. The concrete failure criterion was expressed by Ottosen's four-parameter model, and after reaching the maximum compressive strength, a softening gradient based on the compressive failure energy of Nakamura et al. (Non-Patent Document 2) was defined. The stress-strain relationship on the tensile side was linear up to the cracking strength, and the properties after the strength were expressed by the model of Naganuma et al. (Non-Patent Document 1). The shear transfer characteristics on the cracked surface were expressed by the model of Naganuma et al. (Non-Patent Document 1).

The stress-strain relationship of reinforcing bars was expressed using a complete elastic-plastic model. The model of Ciampi et al. was applied to the hysteresis characteristics under repeated stress. The adhesion strength of the column main reinforcement was the adhesion reliability strength calculated in accordance with the toughness guideline (Non-Patent Document 3), and the slip amount at the time of strength was assumed to be 1.0 mm.

[Comparison between experiment and analysis: load-deformation relationship]
Figure 11 shows the load-deformation relationship in the experiment and FEM analysis. Although the load in the analysis was slightly higher than in the experiment in the negative side, the FEM analysis was able to accurately reproduce the experimental results in terms of the skeletal curve of the load-deformation relationship, the timing of the first crack in the shear wall, etc.

[Comparison between experiment and analysis: wall crack situation]
The crack area over the entire area of the shear wall was compared with the area of the shear wall calculated from the analysis model M. Figure 12 shows a comparison between experiment and analysis of the crack area versus the deformation angle of the shear wall. In calculating the crack area in the analysis model M, first, the crack area of each element was determined based on equation (10) in FIG. 3(d). Thereafter, by calculating the sum of the crack areas of each element, the crack area of the entire analytical model M was obtained (FIG. 3(e), equation (11)). The crack area calculated from the FEM analysis was comparable to the experimental results both at the peak time and at the time of unloading. In addition, it increased approximately linearly with the increase in wall deformation, which also coincided with the increasing trend of crack area in the experimental results.

FIG. 13(b) shows the results of FEM analysis regarding the cracking state of walls and columns at a deformation angle of 0.2% (first time). It should be noted that the crack distribution shown in FIG. 13 is the result of FEM analysis according to the prior art, and does not reflect the results of calculation of the crack area, etc. FIG. 13 shows that the FEM analysis can qualitatively represent the experimental results (FIG. 13(a)), such as the direction of cracks. However, on the other hand, it can be seen that the conventional FEM analysis cannot accurately represent quantitative results such as crack width and spacing, unlike the results shown in FIG.

As mentioned above, it can be seen that the crack area calculated by the method according to the present embodiment matches well with the experimental results.

<Effect>
In the above calculation formula, the crack area within the region can be calculated by using the initial area and crack equivalent strain. Therefore, even when load conditions such as temperature load and creep and strain are complex, it is possible to calculate accurate values.

For example, when thermal strain occurs as shown in FIG. 14(a), the element undergoes deformation due to the thermal strain (indicated by dotted lines in the figure) in addition to deformation due to cracks. Furthermore, as shown in FIG. 14(b), in cases where loads are applied in two axial directions, there are cases where the element area does not change even if the element is deformed. One possible method for calculating the crack area is, for example, to calculate the crack area of a concrete structure based on the area increment of the element. It may be difficult to directly obtain the crack area by simply measuring it. Even in such cases, by using the above method, it is possible to accurately calculate the crack area of the concrete element or concrete structure. In other words, this method has a wider range of applicable conditions and ranges than existing methods.

Note that the calculation method shown in the above embodiment is just an example. The calculation of the crack area according to the present invention is not necessarily limited to the method of calculating the crack area for each element. Furthermore, when evaluating the performance of a concrete structure, the crack area or performance may be evaluated by multiplying the crack area calculation formula by an arbitrary coefficient.

Further, although a hexahedral linear element is used as a finite element in the above embodiment, this is merely an example. A concrete structure is modeled with arbitrary finite elements including triangular primary elements, triangular quadratic elements, quadrilateral primary elements, quadrilateral secondary elements, tetrahedral primary elements, tetrahedral secondary elements, and hexahedral secondary elements, and each The area may be calculated using an appropriate method depending on the factors. Furthermore, the number of finite elements to be used is not limited to one type, and the area may be calculated for a model made up of multiple types of finite elements.

In the above, the crack area of the element is explained as the area of the surface or side surface of the element as shown in each figure, but the present invention is not limited to this. Especially when considering three-dimensional elements, it is possible to assume a cross section perpendicular to the crack surface and derive the crack area in this cross section.

Note that the performance evaluation method or analysis method of the present invention can be automatically, simply, and quickly executed by constructing it as a program that can function on a computer and making it function on the computer. For example, as shown in the block diagram of FIG. 15, it is possible to configure a computer system C that includes a storage device 30 that stores the program and a control device 40 so as to be able to communicate with each other, and to execute the above method. . Furthermore, by visualizing the crack area in a format such as that shown in FIG. 13 through post-processing, performance evaluation becomes easier.

Although the embodiments for carrying out the present invention have been described above, the above embodiments are intended to facilitate understanding of the present invention, and are not intended to be interpreted as limiting the present invention. Further, the present invention may be modified and improved without departing from the spirit thereof, and the present invention also includes equivalents thereof.

100, M Analysis model 100A Concrete structure

Claims (6)

Performing a finite element analysis on a finite element model of the concrete structure based on the physical properties of the concrete structure ,
A method of calculating a crack area of at least a portion of the concrete structure based on the product of an initial area in a region on the finite element model and a crack-equivalent strain in a direction perpendicular to the crack in the region . The method according to claim 1, wherein the crack area is calculated by adding to the product an increase/decrease in crack area based on a change in crack length from an initial state. The method according to claim 2, wherein the crack area of at least a portion of the concrete structure is calculated based on Equation 1 .
The method according to any one of claims 1 to 3, wherein the crack area is used to evaluate the performance of the concrete structure. Obtaining output data of a finite element analysis based on the physical properties of the concrete structure for a finite element model of the concrete structure,
A program that causes a computer to execute a process of calculating a crack area of at least a portion of the concrete structure based on the product of an initial area in a region on the finite element model and a crack-equivalent strain in a direction perpendicular to the crack in the region. . a storage device that stores a program for performing finite element analysis on a finite element model of a concrete structure based on the physical properties of the concrete structure ;
A control device capable of executing the program,
The control device includes:
A system that calculates a crack area of at least a portion of the concrete structure based on a product of an initial area in a region on the finite element model and a crack-equivalent strain in a direction perpendicular to the crack in the region .

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