Method for modeling beam or column and method for evaluating their reuse

Abstract

To provide a method for modeling a beam or column including an end region and a general region having different bending stiffness from each other, as a wire having one bending spring at each end, without being restricted by the requirement that the strength or stiffness of one of a relocated portion and an end surface of the beam must be designed to be significantly high.SOLUTION: Bending springs 15 are set at the end surfaces of the beam and at a relocated portion 7, which is the boundary between an end region 3 and a general region 4, and the two bending springs 15 are combined into one using a geometric equation based on the rotation angle of the two bending springs 15 and the ratio of the length of the end region 3.SELECTED DRAWING: Figure 2

Description

[Technical field]

[0001] The present invention relates to a method for modeling a beam or column that includes end regions provided at both ends in the extension direction and a general region extending between the two end regions, wherein the end regions and the general region have different bending rigidities, and to an evaluation method for reuse that utilizes this modeling method, which is a method for evaluating a beam or column that is a precast concrete member that is pressure-bonded to another member by an unbonded tendon. [Background technology]

[0002] When performing dynamic response analysis of a structure, the structure is modeled. In beams whose stiffness changes midway along their extension direction, such as a non-uniform cross-section reinforced concrete beam or a composite structural beam of steel frame and reinforced concrete, the behavior changes significantly at the position where the stiffness changes. It is possible to model such beams as wire rods having bending springs at both the end face of the beam and the position where the stiffness changes, but some software for performing dynamic response analysis of structures cannot set two bending springs on one side of the beam. As a means of modeling such beams as wire rods having one bending spring at each end, there are two cases: one in which the strength and stiffness of the part where the behavior changes are designed to be quite high, and the beam is modeled as a wire rod having a bending spring at the end (end face) of the beam, and one in which the strength and stiffness of the beam end are designed to be quite high, and the beam is modeled as a wire rod having a bending spring at the part where the behavior changes (relocated part) (for example, Patent Document 1). The latter method is effective, for example, by setting the bending yield strength of the beam end to 1.5 times or more than the bending yield strength of the relocated part, and is a method known as hinge relocation. Columns can also be modeled in the same way as beams. [Prior art documents] [Patent documents]

[0003] [Patent Document 1] JP 2015-30985 A Summary of the Invention [Problem to be solved by the invention]

[0004] The two modeling methods mentioned above are limited in that either the relocated part or the end face of the beam must be designed to have a significantly higher strength or stiffness. A method is needed to model beams and columns as wire rods with one bending spring at each end without such a limitation.

[0005] Furthermore, in order to ensure sustainable production and consumption patterns, there is a need to significantly reduce waste generation through prevention, reduction, recycling, and reuse. Among building materials such as beams and columns, the reuse of steel frame members has progressed relatively well, but the reuse of reinforced concrete members has not progressed as quantitative evaluation of restoring force characteristics, etc. has not been established. As cement, the material in concrete, emits carbon dioxide during its production, the reuse of reinforced concrete members also contributes to reducing carbon dioxide emissions.

[0006] In view of the above background, the present invention aims to provide a method for modeling a beam or column including an end region and a general region with different bending stiffness as a wire having one bending spring at each end, without being restricted by the requirement to design the strength or stiffness of either the relocated portion or the end face of the beam to be significantly higher. Another aim of one embodiment of the present invention is to provide a method for quantitatively evaluating the behavior of a beam or column that includes such an end region and a general region and is a precast concrete member of a reinforced concrete structure that is reused, against an earthquake force. [Means for solving the problem]

[0007] In order to solve the above-mentioned problems, one aspect of the present invention is a method for modeling a beam (1) or column including end regions (3) provided at both ends in the extension direction and a general part region (4) extending between the two end regions, wherein the two end regions have equal lengths in the extension direction and equal bending stiffnesses different from those of the general part regions, the method comprising the steps of regarding the beam or column as a wire having bending springs at its end face in the extension direction and at a relocated part (7) which is the boundary between the end region and the general part region, and converting the bending spring of the end face and the bending spring of the relocated part into a single composite bending spring on each side of the beam or column in the extension direction using the following equations (13) and (15), respectively.

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[0008] According to this embodiment, by combining two bending springs into one, for a beam or column that includes an end region and a general region with different bending stiffness, it is possible to model the beam or column as a wire having one composite bending spring at each end, without being restricted by the requirement to design the strength or stiffness of either the relocated portion or the end face of the beam to be significantly higher. Equations (13) and (15) are established from a geometric relationship, and the rotation angle ( r R', e R') is the rotation angle of the two bending springs before they are combined ( r R, eThis means that equations (13) and (15) hold not only when elastic, but also after the stiffness of each of the two bending springs has decreased. In other words, equations (13) and (15) hold at any time. In other words, the rotation angle ( r R, e R) can include the effects of each stiffness reduction.

[0009] In the above aspect, the above-mentioned restoring force characteristic model or hysteretic restoring force characteristic model may be used for the beam or column.

[0010] According to the above embodiment, it is possible to provide a restoring force characteristic model that corresponds well to the experimental results. Furthermore, since the formulas (13) and (15) are established from a geometric relationship, they are also established when unloading. Therefore, the above modeling method can be applied not only to the restoring force characteristic model, but also to the hysteretic restoring force characteristic model.

[0011] In the above-mentioned aspect, the method is used for dynamic response analysis of the beam (1) or column during reuse, the beam or column is a precast concrete member, the end region (3) includes an expanded portion (8) that is expanded with respect to the general portion region (4), and the beam or column is configured to be pressure-connected to another member (2) by tension of a tendon (9) having one end fixed to the expanded portion, and the converting step is configured to convert the formula (13) and the formula (15) into the formula (16). r R may include reflecting residual deformation of the relocated portion caused by an earthquake.

[0012] In this manner, a beam or column subjected to a seismic force can be modeled.

[0013] In the above aspect, the converting step is e The method may further include making R reflect the decrease in Young's modulus of concrete at the end faces of the beams or columns caused by an earthquake.

[0014] In this manner, a beam or column subjected to a seismic force can be modeled.

[0015] In the above-mentioned aspect, the method is used for dynamic response analysis of the beam (1) or column during reuse, the beam or column is a precast concrete member, the end region (3) includes an expanded portion (8) that is expanded with respect to the general portion region (4), and the beam or column is configured to be pressure-connected to another member (2) by tension of a tendon (9) having one end fixed to the expanded portion, and the converting step is configured to convert the formula (13) and the formula (15) into the formula (16). e This may include having R reflect the reduction in Young's modulus of concrete at the end faces of the beams or columns caused by an earthquake.

[0016] In this manner, a beam or column subjected to a seismic force can be modeled.

[0017] One aspect of the present invention is a method for evaluating a beam or column to be reused, comprising the steps of: modeling the beam (1) or column by the modeling method of the above aspect; and evaluating the behavior of the beam or column to be reused against seismic forces by performing an analysis of the restoring force characteristics or hysteretic restoring force characteristics of the modeled beam or column.

[0018] According to this aspect, the behavior of the reused beam or column against earthquake forces can be quantitatively evaluated.

[0019] In order to solve the above problem, one aspect of the present invention is a method for modeling a reinforced concrete beam (1) or column, which includes end regions (3) provided at both ends in the extension direction and a general region (4) extending between the two end regions, and the two end regions have the same length in the extension direction and the same bending rigidity different from that of the general region, comprising the steps of: setting an equivalent rigid region (16) having an infinite second moment of area on the outer end side of the end region on each side of the extension direction of the beam or column, calculating a ratio of the length of the equivalent rigid region in the extension direction to the length of the beam or column in the extension direction by the following formula (8), and calculating the length of the equivalent rigid region in the extension direction based on the ratio, and regarding the beam or column as a wire having a bending spring at the inner end of the equivalent rigid region in the extension direction. Here, the "outer end side" of the end region refers to the side of the end point located opposite to the end point (inner end) facing the center of the beam or column.

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[0020] According to this embodiment, by setting a bending spring at the tip inside the equivalent rigidity zone, a beam or column including an end region and a general region with different bending rigidity can be modeled as a wire having one bending spring at each end without being restricted to designing the strength or rigidity of either the relocated portion or the end face of the beam to be significantly higher. By considering the equivalent rigidity zone in the end region, a restoring force characteristic model of hinge relocation can be set. In other words, by considering the length of the equivalent rigidity zone, the relationship between the bending moment and the rotation angle can be expressed by only the bending spring of the relocated portion. Effect of the Invention

[0021] According to the above aspects, for a beam or column that includes an end region and a general region with different bending stiffness, a method can be provided for modeling the beam or column as a wire having one bending spring at each end, without being restricted by the requirement that the strength or stiffness of either the relocated portion or the end face of the beam must be designed to be significantly higher. [Brief description of the drawings]

[0022] [Figure 1] FIG. 1 is a perspective view showing a joint portion of a beam to a column according to an embodiment; [Diagram 2] Non-uniform cross-section beam model [Diagram 3] Definition of the restoring force model of uniform cross-section beam (a: end section, b: general section) [Figure 4] Definition of rotation angle of non-uniform cross-section beam [Diagram 5] Definition of beam rotation angle considering equivalent rigid area [Figure 6] Example of restoring force model for a non-uniform cross-section beam (a: beam end, b: relocated part, c: equivalent rigid zone end, d: load-displacement relationship) [Figure 7] Ratio relationship between equivalent rigid zone length coefficient and bending stiffness [Figure 8] Restoring force characteristics and rigidity during reuse (Upper left: Restoring force characteristics of the beam end, Lower left: Stress-strain relationship of the beam end, Upper right: Restoring force characteristics of the relocated part, Lower right: Residual deformation of the relocated part and rigidity toward the unloading start point) [Figure 9] Test specimen diagram [Figure 10] Distortion and cracking of concrete at corners during compression [Figure 11] Loading device and measurement location [Figure 12] Calculation of the separation point in the experiment [Figure 13] Destruction state after R=±1 / 50rad loading (left half: top surface of beam, right half: bottom surface of beam) [Figure 14] Load-deformation angle relationship [Figure 15] Relationship between the ratio of rotation angles of the end region and the general region and the deformation angle [Figure 16] Strain distribution in PC steel strands [Figure 17] Strain distribution of beam main bars [Figure 18] Comparison of load-displacement relationship and restoring force model [Figure 19] Stress-strain relationship [Figure 20] Comparison of restoring force model after bending yield strength and experimental results [Figure 21] Comparison of load-deflection angle relationship and initial restoring force characteristics [Figure 22] Comparison of the load-displacement angle relationship when subjected to the second earthquake force and the restoring force characteristics when reused, which reflect the reduction in stiffness due to bending cracks caused by the first earthquake force DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0023] Hereinafter, an embodiment of the present invention will be described with reference to the drawings.

[0024] FIG. 1 is a perspective view showing a joint of a beam 1 to a column 2 according to an embodiment. The beam 1 is made of reinforced concrete and includes end regions 3 provided at both ends of the beam 1 in the extension direction and a general region 4 extending between the two end regions 3. FIG. 1 shows one end region 3 of the beam 1 and a part of the general region 4, but the beam 1 is substantially mirror-symmetrical with respect to a plane passing through the center of its extension direction and perpendicular to the extension direction, that is, substantially bilaterally symmetrical in front, back, plan, and bottom views. The beam 1 includes beam reinforcement bars 5 extending parallel to the extension direction of the beam 1 in the end region 3 and the general region 4, shear reinforcement bars (see FIG. 9) arranged so as to surround the beam reinforcement bars 5 perpendicular to the extension direction of the beam reinforcement bars 5, and pretensioning type tendons (PC steel strands in FIG. 9) extending parallel to the beam reinforcement bars 5 over substantially the entire length of the beam 1. A mechanical anchoring hardware 6 is attached to the end of the beam main reinforcement 5, which is fixed to the end of the beam 1. The beam 1 is a precast concrete member.

[0025] The end region 3 has a predetermined length in the extension direction of the beam 1, is joined to the column 2 at the end face of the beam 1 which is the outer end in the length direction, and is connected to the general region 4 at a relocated portion 7 which is the inner end in the length direction. The end region 3 includes a widened portion 8 which is formed so as to be wider in the beam width direction than the general region 4. Since the end region 3 includes the widened portion 8, the bending rigidity of the end region 3 is higher than the bending rigidity of the general region 4.

[0026] The beam 1 and the column 2 are joined together by a tension member 9. The tension member 9 is, for example, a PC steel bar, PC steel wire, PC steel strand, or a fiber-reinforced plastic member such as aramid fiber, carbon fiber, or glass fiber. The tension member 9 extends in the extension direction of the beam 1 and penetrates the column 2 and the widened portion 8 in the end region 3 of the beam 1 that faces each other across the column 2. The tension member 9 is tensioned and fixed by fixing devices 10 to the surface facing inward in the extension direction of the beam 1 in the widened portion 8 at both ends. A joint material 11 such as mortar is injected between the beam 1 and the column 2. The tension member 9 is used in an unbonded state, that is, in a state in which a filler such as grout is not filled in a sheath 12 that is embedded in the column 2 and the widened portion 8 and through which the tension member 9 is inserted. Therefore, the beam 1 can be removed from the column 2 by removing the fixing device 10 to release the tension of the tension member 9, and can be reused.

[0027] The column 2 includes a column body 13 extending in the vertical direction, and a joint member 14 that joins the column body 13 and the beam 1. The column body 13 and the joint member 14 are precast concrete members.

[0028] <Restoring force model for non-uniform cross-section beams> <Calculation of equivalent rigid zone length and elastic rigidity> Beam 1 is a dumbbell-shaped variable cross-section beam when viewed from above, and since its bending restoring force characteristics are affected by each cross-section that constitutes beam 1, a method of expressing it with multiple bending springs 15 can be considered for modeling. Therefore, the variable cross-section beam is modeled as beam 1 having bending springs 15 at the end of each cross-section as shown in Figure 2(b), and the elastic rigidity is calculated assuming that an antisymmetric bending moment acts on the variable cross-section beam (Equations (1) to (6) also include the case where beam 1 is asymmetric). The Young's modulus of concrete is E, and the equivalent second moments of area of ​​the general area 4 and the left and right end areas 3 are I0, I1, and I2. Using the coefficient n that indicates the position of the inflection point, the lengths of the left and right end areas 3 are λ1l and λ2nl (λ1, λ2: end area length coefficients), and the material end moment is M a , nM aThen, the bending moment distribution, curvature distribution, and deflection curve will be as shown in Figure 2(d) to (f), respectively. From Mohr's theorem, the deflection angle θ (the angle between the tangent at any position on the deflection curve and the material axis before deformation) and deflection δ can be expressed by equations (1) to (4).

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[0029] In this embodiment, assuming that the beam end (end face of the beam 1) is not separated significantly and a plastic hinge occurs in the relocated portion 7, it is considered that there is a portion that can be regarded as a rigid zone on the end side in the end region 3, as shown in Fig. 2(c). This is called an equivalent rigid zone 16, and the length of the equivalent rigid zone 16 is λ e1 l, λ e2 Let nl. λ e1 and λ e2 is called the equivalent rigid zone length coefficient. Assuming that the second moment of area in the equivalent rigid zone 16 is infinite, C1' to C8' are respectively obtained when I1 →∞ and I2 →∞ in C1 to C8. If the deflection angles at both ends of the beam 1 are equal when the bending spring 15 is considered at the end of each zone (Fig. 2(b)) and when the equivalent rigid zone 16 is considered (Fig. 2(c)), then from equations (1) and (3), C1 = C1' and C5 = C5', and these can be rearranged as equations (5) and (6).

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[0030] Equivalent rigid zone length coefficient λ e1 and λ e2 can be obtained by simultaneously solving equations (5) and (6), but convergence calculations are required to find the solution. Therefore, we will simplify the equations below. In order to reuse components, it is rational to use symmetrical components and to have fewer types of beam components that make up a building. Therefore, in equations (5) and (6), I1=I2=I, λ1=λ2=λ, λ e1 = λ e2 = λ e Then, due to the symmetry of the member, n = 1 (the center of the beam is the inflection point),

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[0031] The elastic stiffness K' in the bending moment M-rotation angle (the rotation angle of the bending spring 15 when modeled as in Figure 2(b) or (c)) R relationship is calculated from the deflection angle at x = 0 (or X = 0) as shown in formula (9). This is the elastic stiffness of the composite bending spring of a bilaterally symmetrical variable cross section beam when the equivalent rigidity area 16 is not considered.

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[0032] When modeling the restoring force characteristics of a variable cross-section beam, if the bending spring 15 of the beam end and the relocated part 7 can be combined into one, it will be easier to handle in static nonlinear load incremental analysis and the structural performance of the variable cross-section beam will be easier to understand. Here, we take up a cantilever beam 17 with a variable cross-section having an end region 3 and a general region 4, and combine the bending spring 15 from a geometric approach to the behavior of each of the two elements of the end region 3 and the general region 4 to evaluate the deformation in the restoring force model. At this time, the boundary conditions for the displacement at the tip of the cantilever beam 17 where the load acts are satisfied. Here, modeling is performed for a member in which bending deformation is more prominent than shear deformation.

[0033] The restoring force characteristics of a uniform cross-section beam having an end section (a section corresponding to end region 3) or a general section (a section corresponding to general region 4) over the entire length of the member are given as shown in Figure 3. e R is used, and in the case of a variable cross section, e A prime (') is added to the upper right of the symbol, such as R'. The subscripts e and r at the beginning of the symbol represent the initials of the cross-sectional positions, such as the end of the beam (base end of the cantilever beam 17) and the relocated part 7, and represent the bending moment M and rotation angle R at each position. e α y and r α y is the yield stiffness reduction rate of the beam end and relocated part 7. The member lengths of the uniform cross-section beam of the end section and general section are l and (1-λ)l.

[0034] Since the deformation characteristics of the end region 3 and the general region 4 of the variable cross section beam are different, we assume a simplified deformation state in which the rotation angle changes significantly at the relocated part 7 as shown in Figure 4, and set the respective angles when a load P acts on the tip of the cantilever beam 17 as shown in the figure. The rotation angle of the end region 3 is e R is the rotation angle between the end and the tip of the cantilever 17. e R' is the angle between the axis of the end region 3 and the axis of the general region 4. rLet R'. Figure 4 shows a variable cross section beam without considering the equivalent rigid area 16, whereas Figure 5 shows the rotation angle when the equivalent rigid area 16 is taken into account and the bending spring 15 is at the tip of the equivalent rigid area 16.

[0035] The displacement of the tip of the cantilever beam 17 is calculated from equation (9), and the displacement due to the rotation of the end region 3 is subtracted from it to obtain r R' can be calculated as shown in formula (10). This can be considered as the difference between the rotation angles when a load P acts on the tip of a uniform cross-section beam with bending stiffnesses EI0 and EI with a member length of (1-λ)l. r R is expressed as the angle of the part shown in FIG.

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[0036] Bending moment at the end of the beam e M and the relocate part r For M Since the relationship of TIFF2025014768000022.tif9168 holds, the restoring force model of the variable cross section beam in this embodiment can be constructed by applying existing models shown in "Guidelines for Performance Evaluation Design and Construction of Prestressed Concrete Buildings (Draft) and Commentary" (Architectural Institute of Japan, 2015) (hereinafter referred to as "PC Performance Evaluation Guidelines") and "Guidelines for Earthquake Performance Evaluation Design of Reinforced Concrete Buildings Based on Equivalent Linearization Method (Draft) and Commentary" (Architectural Institute of Japan, 2019) (hereinafter referred to as "RC Equivalent Linearization Method Guidelines") to the restoring force models of the equal cross section beams of the end section and general section (Figure 3). For example, if the gap that occurs at the beam end is the first stiffness reduction point of the variable cross section beam, the rotation angle of the beam end at that time can be calculated from equation (15):

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[0037] <Restoring force model of hinge relocation beam considering equivalent rigid zone 16> In the previous section, we showed that the rotation angle of each bending point of the restoring force model when the equivalent rigid zone 16 is not considered is expressed by Equation (15). In this section, as shown in Figure 5, we develop a restoring force model of hinge relocation that considers the equivalent rigid zone 16 in the end region 3. First, the bending moment of the beam end eM and rotation angle e Consider the relationship of R'. Figure 6(a) shows the restoring force model of a uniform cross-section beam in which the cross-section of the end region is uniform over the entire length of the member using thin lines, and that of a variable cross-section beam using thick lines. Here, a four-line restoring force model in which stiffness decreases in the order of the first bend point (r1) of the relocated part 7, the first bend point (e1) of the beam end, and the second bend point (r2) of the relocated part 7 is shown. From equation (15), the rotation angle of each bend point is expressed by the following three equations.

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[0038] Next, the bending moment of the relocated part r M and rotation angle r The rotation angle of each break point of the relationship of R' is given by (13).

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[0039] For the hinge relocation beam, the equivalent rigid area 16 in the end region 3 is considered. Using equation (8) for various λ, λ e The relationship between EI0 / EI is shown in Figure 7. When EI0 / EI=1, λ e = 0, and the equivalent rigid zone 16 is not formed. On the other hand, when EI0 / EI = 0, λ e = λ, and the entire end region 3 is regarded as an equivalent rigid region 16. When the cross-sectional difference between the end region 3 and the general region 4 is not large as in this embodiment, 0<λ e / λ<1, and an equivalent rigid zone 16 is formed on the end side of the end region 3. Equation (8) is valid when elastic, and it is thought that EI and EI0 will also decrease if the rigidity of the end region 3 or the general region 4 decreases. However, since it would be cumbersome to change the length of the equivalent rigid zone 16 in the model each time depending on the magnitude of the applied bending moment, the equivalent rigid zone length coefficient is calculated using equation (8), λ e was kept constant.

[0040] Taking into account the rigid zone, if I→∞, e K=∞, e R=0, λ=λ e Therefore, equations (13) and (16) become

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[0041] From the above, when the member's strength is reached by the yield of the beam end, the rotation angle of the end increases and it is considered that it cannot be considered as a rigid zone. Therefore, in the case of hinge relocation beams, the shear span is set to (1-λ e A restoring force characteristic model or a hysteretic restoring force characteristic model of a non-uniform cross-section beam is constructed using equations (27), (28) and (30) taking into account the equivalent rigid area 16 as .

[0042] <Reuse of Beam 1> As shown in Figures 1 and 2, the bending spring 15 at the end of the beam is an unbonded PC (prestressed concrete) cross section that uses unbonded tendons 9, and is expressed by the relationship between the bending moment and rotation angle of the unbonded PC. The bending spring 15 at the relocated part 7 is a bending spring 15 of a pretensioned member (a pretensioned tendon that extends over almost the entire length of the beam 1 (PC steel strand in Figure 9)), and is a bonded PC cross section that is expressed by the relationship between the bending moment and rotation angle of the bonded PC.

[0043] A nonlinear elastic model is applied to the bending spring 15 at the end of the beam. This shows a relationship between bending moment and rotation angle such that the beam returns to the origin when the load is removed, even if the stiffness is reduced due to separation (not bending cracks, but because beam 1 tilts relative to column 2 during an earthquake, it separates at the boundary between column 2 and beam 1 (joint 11)) specific to unbonded PC sections and bending yield strength (not the true yield strength of tendon 9 is reached, but the point at which strain becomes large to a certain extent) (Fig. 8(a)).

[0044] The bending spring 15 of the relocated section 7 is applied with the relationship between the bending moment and the rotation angle, which is characteristic of prestressed concrete members. In other words, the residual deformation when unloaded is small. The difference with an unbonded PC section is that the deformation does not return to the origin (it does not return to 0), but rather the restoring force characteristics are such that some deformation remains (Figure 8(d)).

[0045] Even with an unbonded PC section, if the cross section of the beam end opens and closes many times, such as during an earthquake, the Young's modulus of the edge of the cross section (compression edge) will decrease. If bending cracks occur in the general area 4, the bending spring 15 of the relocated section 7 will also have residual deformation when unloaded. Therefore, the decrease in Young's modulus of the end (end face in the extension direction of the beam 1) and the residual deformation of the relocated section 7 are taken into consideration in the restoring force characteristics when reused.

[0046] Specifically, the right-hand sides of equations (13) and (15) e The decrease in Young's modulus is expected for R (the rotation angle of the beam end), r A value that takes into account the residual deformation of the relocated portion 7 is substituted for R (the rotation angle of the relocated portion 7). Furthermore, when expressions (13) and (15) are expressed in terms of rigidity, they become expressions (16) and (17).

[0047] In equations (13) and (15), e The decrease in Young's modulus of R (the rotation angle of the beam end) is due to the e The change in K can be expressed as the change in the stress and strain at the end of the beam. Figure 8(c) shows the relationship between stress and strain at the end of the beam. The reloading stiffness after the last unloading, i.e., the slope of the line showing the relationship between stress and strain, is the change in the reloading stiffness after the last unloading. eIt only affects (decreases) the Young's modulus E contained in K ( e K=3EI / l). In Fig. 8(a), the parallel lines show the initial restoring force characteristics (skeleton curve), and the thick lines show the restoring force characteristics after reuse.

[0048] In equations (13) and (15), r The residual deformation of R (the rotation angle of the relocated portion 7) is expressed by the following equations (16) and (17): r It can be expressed as a change in K. In the relocation part 7, the deformation of the residual deformation is slid by that amount when reused. This is equivalent to returning the residual deformation in the figure to the origin, and the residual deformation is absorbed when the joint mortar is recast. Furthermore, considering the hysteresis restoring force characteristic model of the relocation part 7, after experiencing bending cracks, deformation remains at the time of unloading, but if the earthquake continues, or if the next earthquake occurs from that point, it will move from the residual deformation point to the unloading start point (Figure 8 (d)). This is a decrease in stiffness compared to the initial stiffness, and the restoring force characteristic at the time of reuse is set with the decreased stiffness as the apparent initial stiffness at the time of reuse. Therefore, the first break point of the restoring force characteristic at the time of reuse is the unloading start point, and the second break point is the position where the yield point of the restoring force characteristic model of the original virgin material is slid by the residual deformation (Figure 8 (b)).

[0049] A model of the hysteretic restoring force characteristics during reuse can also be constructed by following the above rules. EXAMPLES

[0050] Example 1: Bending-shear experiment of unbonded PCaPC (precast prestressed concrete) beam Bending shear experiments were conducted on unbonded cantilever PCaPC beam members with pressure joints at the beam end widening section. In order to continuously evaluate a range of specimens, from those aimed at remaining undamaged after an earthquake so that the member can be reused as is, to those for hinge relocation, which require relatively large-scale repairs, bending margin was used as the main parameter. The bending margin was changed according to the length of the widening section and the amount of steel at the beam end and relocated section. The bending margin was calculated by the bending yield strength of the relocated section. e M ryBending yield strength at the end of the beam for e M y The bending yield strength of the beam end was set to 0.9 times the ultimate bending strength, and the ultimate bending strength was calculated based on the reference 10). Specifically, the tension side PC tendon tension was set to the Takemoto method (for vertical loads) 11), the compression side PC tendon tension was set to the tensile force at the start of loading, and the ACI stress block 12) was used to calculate the compressive stress distribution in the concrete. The stress block coefficient was k1 = 0.85-0.05(σ B -28) / 7(0.65≦k1≦0.85), k2=0.5k1, k3=0.85. Here, σ B is the concrete compressive strength (N / mm 2 The bending yield strength of the relocated parts was calculated based on the "PC Performance Evaluation Guidelines" for PC beams and the "RC Equivalent Linearization Method Guidelines" for RC beams.

[0051] 1. Test specimen plan The diagram and list of the test specimens are shown in Figure 9 and Table 2, respectively. The test specimens were non-uniform cross-section beams with widened sections at the ends, and the widened sections were crimped to the stubs with PC steel bars. The PC steel bars were arranged in four or two rows on both sides of the widened section. The scale of the test specimen was approximately 1 / 2, and the shear span l from the beam end to the loading point was 1400 mm (l / D = 3.1). The inner perimeter reinforcement in the widened section was the same steel material as the shear reinforcement in the general beam section. In the widened section, the outer perimeter reinforcement was arranged in the assembly bars at the four corners, and only the second outer perimeter reinforcement from the relocation section to the beam end side was arranged to surround the second and third rows of PC steel bars in order to suppress the crack width in the inside corner. The amount of reinforcement in the widened section in the beam depth direction was set to be equal to or greater than that of the general section. In order to bear the peeling stress in the beam width direction that occurs at the inside corner at the end of the widening section due to the prestress of the PC steel bars, the amount of peripheral reinforcement in the beam width direction (end) was set to 0.2 to 0.3 times the effective prestress force of the PC steel bars based on the literature 13), with only M-1.4s being set to less than 0.2 times. [Table 2]

[0052] The distance from the relocation part to the inside of the anchor plate of the mechanical anchoring hardware of the main reinforcement (anchorage length la) is based on the Architectural Institute of Japan's "Standards and Commentary for Calculating Reinforced Concrete Structures" 14) (Hereinafter referred to as the "RC Standards") The ratio of anchorage within members according to Article 16 (study on ensuring safety) or the required anchorage length lab according to Article 17 was planned to be significantly less than 1 for the S series with an expanded section length of 260 mm, around 1 for the M series with an expanded section length of 350 mm, and greater than 1 for the L series with an expanded section length of 500 mm. The PC steel strands were arranged in a straight line in two rows of five at the center of the cross section. The PC steel bars and PC steel strands were arranged symmetrically above, below, left and right so that they were not eccentric to the beam axis. The PC steel strands were secured within the expanded section to a required anchorage length of at least 45φ (φ: wire diameter) as stipulated in the "PC Performance Evaluation Guidelines," and were anchored within the expanded section by adhesion between the PC steel strands and the concrete.

[0053] The target compressive strength of concrete is 60N / mm 2 The maximum aggregate particle size is 20 mm. The material test results for concrete and mortar are shown in Table 3, and the material test results for steel are shown in Table 4. The Young's modulus of concrete was set to 1 / 3 secant stiffness. The concrete was poured from the top of the beam. [Table 3] [Table 4]

[0054] 2. Prestressing using the pretensioning method and pressure bonding method A self-reacting steel jig was used to tension the PC steel strands. One end of the 10 PC steel strands was fixed to the jig, which was then passed through the test piece and fixed to the jig on the other side. Two PC steel rods passed through the jig were tensioned with two hydraulic jacks, so that all 10 PC steel strands were simultaneously tensioned. The fixed end was at the end of the beam (widened section) and the tensioned end was at the tip of the beam. The strain on all the PC steel strands was observed while tensioning in order to apply equal tension to all 10 strands. After about 24 hours had passed since the initial tensioning, the strands were tensioned again to the specified tension, and the concrete was poured. Five to six days after pouring, the concrete strength reached 50 N / mm 2 After this level of stress was reached, prestress was introduced. The target amount of prestress was set at about 0.7 times the standard yield strength of the PC steel strand.

[0055] Non-shrink mortar is poured into the 20mm thick joint between the precast beam and the stub, and the compressive strength of the beam, joint mortar and stub is 50N / mm 2 After the above was achieved, the widened section was pressed onto the stub by the PC steel bar. The PC steel bars at the diagonal positions of the cross section were tensioned two at a time to half the required tension, and after half of the tension had been applied to all the PC steel bars, the tension was applied in the same order up to the required tension. The target amount of tension applied was the standard yield strength P of the PC steel bar. y The diameter of the holes for the PC steel bars in the widened section is 38 mm.

[0056] During compression, as shown in Figure 10, a large tensile strain occurred in the corner of the relocated part, and a crack occurred in the diagonal direction. Table 5 shows the crack width at the time of crack occurrence in the corner and at the start of loading. The inventors have shown through two-dimensional finite element analysis that cracks are unavoidable in the corner during compression. 13) The results were similar in this experiment. However, no cracks could be clearly seen by visual inspection in specimens L-0.7p and L-0.3p, which is thought to be due to the influence of prestress from the PC steel strands. [Table 5]

[0057] 3. Loading and measurement methods The loading device is shown in Figure 11. A concrete block was fixed on top of the reaction floor, and the test specimen was rotated 90 degrees on top of it, with the stub fixed to the reaction floor. The tip of the beam was repeatedly loaded left and right with a 1000kN jack, alternating positive and negative loads. The horizontal displacement δ at the loading point was divided by the shear span l to calculate the deformation angle R, and R=±1 / 3200rad was loaded for one cycle, and ±1 / 1600, 1 / 800, 1 / 400, 1 / 200, 1 / 100, 1 / 75, 1 / 50, 1 / 33, 1 / 25, and 1 / 20rad were loaded for two cycles each. Positive loading was defined as when the top surface of the beam was in tension.

[0058] Displacement measurements included the relative displacement in the axial direction of the top and bottom surfaces of the beam, the relative displacement in the diagonal direction on the side of the widening section, and the displacement in the loading direction (shear force direction) of the beam end and relocated section (Fig. 11). Steel strain was measured on the PC steel bars, main reinforcement, shear reinforcement and PC steel strands, while concrete strain was measured on the underside of the beam and at the inside corners of the widening section (Fig. 10). The tension of the PC steel bars was also measured with a load cell attached to the stub side. The positions of the strain gauges for the main reinforcement and PC steel strands are as shown in Fig. 9.

[0059] 4. Failure state and load-deformation angle relationship The calculation method of the separation point in the experiment is shown in Fig. 12, the destruction state at the end of the loading of R = ±1 / 50rad is shown in Fig. 13, the load P-deformation angle R relationship is shown in Fig. 14, and the experimental and calculated values ​​are listed in Table 6. Since it is extremely difficult to visually confirm the start of separation at the beam end, it was estimated using a displacement gauge for measuring separation installed on the top and bottom surfaces of the beam at the beam end. As shown in Fig. 12, the separation point in the experiment was determined when the average strain at the tensile edge of the 70 mm section of the beam end, obtained by considering the distance from the top and bottom surfaces of the beam to the displacement gauge, reached the tensile strain at bending tensile strength (for pressure joints) ("PC performance evaluation guidelines"), and is shown in the experimental value column in Table 6. However, the load at the separation point estimated by the above method was about 1 / 3 to 1 / 2 of the calculated value, and there is room for consideration of the estimation method, so the separation point was not listed in Fig. 14. The concrete crush point was determined when the value of the strain gauge attached to the bottom surface of the beam reached the strain at compressive strength. Since no strain gauges were attached to the top surface of the beam, only the crushing point of the concrete on the positive load side is shown in Fig. 14. The deformation angle when the strength is reduced to 80% of the maximum load is called the limit deformation angle R. u In all specimens, the diagonal crack at the relocated corner first propagated and reached near the axis of the beam. In all specimens, almost no slippage was observed at the end of the beam until the end of loading. [Table 6]

[0060] (1) Test specimen with bending margin exceeding 1.0 After the bending cracks occurred in the relocated section, the cracks penetrated in a semicircular shape to connect the left and right corners, and then spread across several layers toward the end of the beam. Along the way, the main reinforcement in the relocated section reached its yield strain. The yield strain of the steel material is the value obtained by dividing the yield strength by the Young's modulus, and the point at which this value was reached was the yield point. Some specimens did not show the yield point of the main reinforcement (◇ point) in the load-deformation angle relationship, but the strain gauges were only attached in one row in the vertical direction of the cross section (Figure 9), so it is possible that other main reinforcement had yielded. The larger the bending allowance, the less damage there was in the widened section, and the tendency for the bending crack width in the relocated section to become larger was observed. Specimen M-1.4 maintained its strength up to R = ±1 / 50rad, just like M-1.4w and M-1.4s, and in the cycle toward R = +1 / 33rad, destruction was observed in which the tensile side main reinforcement pushed up the top surface of the widened section, and the strength decreased. There was almost no change in tension of the PC steel bars until the end of the loading, and the separation of the beam ends was not noticeable with the naked eye.

[0061] (2) Comparison of test specimens M-1.2 and M-1.0 Specimen M-1.0 suffered less damage in the widened area than the specimens with a bending margin of over 1.0 mentioned above, and the separation of the beam ends was visually confirmed. Specimen M-1.2 had a larger bending margin than M-1.0, but damage progressed in the widened area early on, and a drop in strength occurred when loading R = +1 / 75rad. If the ratio of the rotation angle of the relocated part to the rotation angle of the beam end is expressed as ρ, the relationship between ρ and the deformation angle R is as shown in Figure 15. There was a tendency for ρ to become smaller as the bending margin became smaller. This is thought to be because the main reinforcement of the relocated part of M-1.0 reached the yield strain, and as the strain increased, the deformation angle increased due to the separation of the beam ends, which suppressed the bending deformation of the widened part and caused less damage.

[0062] (3) Test specimen with long widening section (L series) In the case of specimen L-0.3p, no cracks occurred except on the side of the widening end and in the corner, and the residual cracks after unloading were barely visible. From Fig. 15, L-0.3p was always about ρ = 1.0 except when the deformation angle was small, so the rotation angle of the beam end eIt can be seen that R was dominant, and the entire beam rotated almost as a rigid body. The residual deformation was almost 0 up to R = ±1 / 33 rad cycles, but after that, slip properties became noticeable. This is because the tension decreased due to the yielding of the PC steel bars, and the compression force completely disappeared when the load was removed, causing the beam to rotate until the PC steel bars were once again bearing the tensile force, and almost no slip displacement was observed on the compression surface.

[0063] In specimen L-0.7p, cracks occurred in the relocated section and in some of the widened sections, but the number and width of the cracks were considerably smaller than in the other specimens, and the residual cracks in the widened sections, excluding the cracks in the inside corners, were a maximum of 0.06 mm even after loading at R=±1 / 50rad. Since the openings at the beam ends were dominant, damage to the widened sections was suppressed compared to the other series. Although concrete collapsed at the beam ends when loaded at R=+1 / 75rad, the strength did not decrease significantly. The gradual decrease in strength from R=+1 / 33rad onwards is thought to be due to the progression of collapse at the beam ends after the PC steel bars yielded.

[0064] Damage to the widened section of specimen L-1.0 was less than that of specimen M-1.0, which had a similar bending margin, especially near the mechanical anchoring hardware at the end of the beam's main reinforcement. No damage was observed in which the tensile main reinforcement pushed up the top surface of the widened section, and the load-deformation angle relationship showed that it was a hinge relocation beam with excellent deformation performance. Whereas bending rigidity was significantly reduced due to tensile yielding of the main reinforcement in the relocated section, bending rigidity at the beam end did not decrease significantly even if it was separated until the PC steel bar yielded or the concrete collapsed, which is thought to be why deformation in the relocated section progressed.

[0065] (4) Test specimen with short widening section (S series) In specimen S-1.3, the crack width in the widened area increased early on, and after bending cracks appeared in the relocated area, separation of the beam end was visually confirmed. Since diagonal cracks extending from the inside corner to the beam end were dominant, it is believed that the effective anchorage length (the distance from the diagonal crack to the anchorage plate) of the main bars other than those at the four corners was significantly shorter than that of the main bars at the outermost edges.

[0066] 5 Strain in PC steel strands and main bars The strain distribution of the PC steel strand is shown in Figure 16. This is the distribution of the average value of the strain gauges on both sides of the PC steel strand in row A shown in Figure 9, and also shows the distribution of the difference from the strain just before the introduction of prestress. The strain at the position of 100 mm and 500 mm from the beam end of the upper strand of L-0.7p was missing from the middle. The strain of the PC steel strand in rows B to E in the relocated part (only immediately after introduction, at the start of loading, and at maximum load) is also shown. From the difference in strain in row A immediately after introduction (black triangle in the figure) in the lower part of Figure 16, it is estimated that the anchorage length from the beam end is about 600 mm, but from the strain in rows B to E, it can be seen that it was anchored at about 500 mm. This is because the PC steel strand in row A had strain gauges attached to both sides at nine places, while rows B to E had them attached to one side only in the relocated part, and it is thought that the lead wires and protective tape affected the adhesion properties. The difference in the anchorage length between the upper and lower ends is thought to be due to the fact that the bond stress was smaller at the upper end due to the settlement of the concrete, and that the strain was more difficult to reduce at the lower end due to friction between the bottom plate of the formwork and the concrete of the test specimen when prestressing was introduced.

[0067] After concrete was poured, the strain in the PC strands decreased by about 200-300μ, which is thought to be due to the shrinkage of the concrete and the relaxation of the PC strands. There was also almost no change in the strain in the PC strands in the widened section before and after compression. The prestressing force of the PC strands at the start of loading in the relocated section cross section was 245kN at the top and 251kN at the bottom for specimen L-0.7p, and 252kN at the top and 278kN at the bottom for L-0.3p. This was calculated from the average of the four strands in rows B to E.

[0068] Specimen L-0.3p sustained little damage during loading, and it is believed that the strain fluctuations at both the upper and lower ends were small. In specimen L-0.7p, the strain in the PC strands decreased due to adhesion deterioration associated with cracks near the relocated section, and the anchorage length expanded by about 1D toward the loading point. During loading, there was almost no change in the strain in the PC strands at the beam end side in the widened section, resulting in a downward convex strain distribution in the relocated section. The widened section was more voluminous than the general section, and the distance from the PC strands to the concrete surface was long, so it is believed that the restraint by the concrete was large.

[0069] Figure 17 shows the strain distribution of the first main reinforcement at the top of specimens M-1.4 and L-0.3p. For L-0.3p, the change from just before the end was crimped is shown, and no correction was made to take into account the effect of prestress from the PC steel strands. If the prestress is uniformly introduced to each cross section, a compressive strain of about 120μ in the general section and about 60μ in the widened section is added to the strain distribution in Figure 17. For specimens with a bending allowance of over 1.0, the main reinforcement reached the yield strain near the relocated section, and the yield region then expanded. On the other hand, for L-0.3p, the main reinforcement never reached the yield strain. For specimens with a bending allowance close to 1.0, the PC steel bar reached the strain (▽ point) calculated by the Takemoto formula almost at the same time as the main reinforcement in the relocated section reached the yield strain, showing the behavior expected in the original design. In addition, there were test specimens in which the strain in the main reinforcement at the end of the beam within the widened section immediately after compression became compressive strain.

[0070] 6. Verification of consistency with the resilience model The first bending point of the beam end and relocation part as a uniform cross-section beam ( e M cr and r M cr ) and the second break point ( e M y and r M y ) The values ​​of the beam end are calculated using reference 10), and the values ​​of the relocated parts are calculated with reference to the "Guidelines for RC Equivalent Linearization Method" and "Guidelines for PC Performance Evaluation." The consistency of the restoring force model shown in the "Form for implementing the invention" and the experimental results is verified.e M cr When calculating the bending tensile strength σ tb The equivalent rigid zone length coefficient λ was calculated using the formula (8). e is as shown in Figure 7 and Table 6 depending on the length of the widening section.

[0071] Figure 18 shows a comparison of the load P-displacement δ relationship and the restoring force model in this experiment. The thick line indicates the restoring force model that does not consider the equivalent rigid zone, and the dotted line indicates the model that does. The gap of the hole for the PC steel bar in the widening section was considered for the second moment of area I and cross-sectional area of ​​the beam end. The initial stiffness was slightly smaller than the calculated elastic stiffness value. This is thought to be because a crack extending diagonally from the inside corner of the relocated part had occurred at the time of compression (before loading). The initial stiffness of specimen M-1.8 was larger than the calculated value compared to the other specimens because a crack had occurred in the relocated part on the top surface of the beam (the tensile side during positive loading) before loading. For specimens with a bending allowance of less than 1.0, it corresponds well with the model that does not consider the equivalent rigid zone. On the other hand, for specimens with a bending allowance of 1.0 or more, it corresponds well with the model that considers the equivalent rigid zone, but it also matches to some extent with the model that does not consider the equivalent rigid zone. Based on the damage conditions during the experiment, damage to the widening section has a large effect in the large deformation region after the main reinforcement yields, so it is considered that the effect on the shape of the restoring force model is small up to the deformation region of about 0.9 times (e2 point) the ultimate bending strength according to the beam reinforcement yield (r2 point) or the Takemoto formula11). However, when the anchorage length of the beam reinforcement is significantly shorter than the required anchorage length, as in S-1.3, it also has an effect during small deformation, and the bending yield strength is not reached, and the second gradient after the reduction in stiffness due to bending cracks (r1 point) also progresses at a significantly lower level than the model.

[0072] 7. Verification of the consistency of bending points after bending yield strength In the test specimens with a bending margin of less than 1.0, as shown in Fig. 18(i) and (j), a certain degree of stiffness remains even near the calculated strength of point e2 and beyond it in the experiment, so it is considered that there is room for the strength to increase further from point e2. e M yThe Takemoto formula11) used to calculate pc and its validity pb Ratio of L pc / d pb This was proposed based on experimental data in the range of 15 to 40, but L pc / d pb For this experiment, where the strain coefficient was small at 5.5 to 7.4, it is believed that the Takemoto formula underestimated the tension increment of the PC steel bars. In fact, the outermost PC steel bars of specimens L-0.7p and L-0.3p reached their yield strain, which greatly exceeded the tension increment calculated by the Takemoto formula. Therefore, the strain of the PC steel bars was calculated using the strain compatibility coefficient F 15),16) The bending strength at the time of yield of the PC steel bar is expressed using the following formula: e M py In order to calculate the e3 point (defined as point e3), a cross-sectional analysis was performed using a fiber model.

[0073] The F value for unbonded PC members is 0.25 in reference 10) and around 0.1 to 0.2 in reference 17), but these are values ​​for cases where the strain increment of the PC steel does not become large, such as when the PC steel is laid across multiple spans and the distance between the anchorage ends is long. This experiment used a cantilever type and the distance between the anchorage ends of the PC steel bars was short, so the strain of the PC steel bars is prone to fluctuations and it is considered appropriate to take an F value larger than the above value. Here, the hinge rotation angle R at the ultimate bending strength of PC beam members in the "PC Performance Evaluation Guidelines" is used. h Refer to the F value when calculating the shear span ratio (l / d pb )2 to 5 and F=0.6 was applied when round steel bars were used for the PC tendons.

[0074] The material constitutive law for cross-sectional analysis is shown in Fig. 19. The concrete stress σ c -Strain ε c The relationship is linear elasticity, and since the concrete used in this experiment is high strength, the compressive strength and beyond are ignored. In addition, it is assumed that the concrete does not bear tensile stress. Stress σ of the PC steel bar p -Strain ε pThe relationship between tension and compression was bilinear as shown in Fig. 14(b). The yield points (black inverted triangles) of the PC steel bars in Fig. 14 and Fig. 20 are determined by the strain of the outermost PC steel bar being the yield strain ε py This represents the point at which

[0075] For specimen L-0.7p, after the bending yield strength of the beam end, the outermost PC steel bar reached the yield strain, and then the strength gradually decreased as the beam end collapsed. Here, the point where the strength decreased is defined as the safety limit point (point e4), and the bending strength at the safety limit e M sl teeth e M py 0.8 times 10) From equation (15), the rotation angles of points e3 and e4 of L-0.7p are respectively

number

number

number

number

number

number

[0076] From the above calculations, points e3 and e4 were calculated, and a comparison of the restoring force model and the experimental results is shown in Figure 20. Point e3 was slightly underestimated at the time of yielding of the outermost PC steel bar (point marked with a black inverted triangle) at L-0.7p, but was roughly consistent at L-0.3p. In the specimens used in this experiment, the concrete compressive edge stress did not reach close to the compressive strength, but in cases where it is appropriate to take plasticity into account, it is necessary to appropriately consider the stress-strain relationship. Point e3 is R h or R III Regardless of which method is used, the consistency with the experimental results does not change significantly, and the e4 point e M py Since the calculation was made at 0.8 times, the evaluation was made on the conservative side.

[0077] For specimen M-1.0, point r2 appears immediately after point e2, so the strength after that reaches a plateau and only the deformation increases. When calculating in the same way as the <Restoring force model of hinge relocation beam considering equivalent rigid zone> in [Form for carrying out the invention] considering the equivalent rigid zone, points e1 and e2 are both located on the line segment between points rz1 and rz2 (Figure 18(d)). However, since the beam end is deformed up to point e2, it is considered that an equivalent rigid zone is not formed, and therefore, compared to the restoring force model when the equivalent rigid zone is not considered, the evaluation was able to be performed with good accuracy up to the third rigidity after point e1.

[0078] Example 2: Comparison of restoring force characteristic model during reuse with experimental results Figure 21 shows a comparison of the load-displacement relationship of a beam when an earthquake force is applied 1 to 4 times with the restoring force characteristic model for a virgin material beam. As shown in Figure 21(a), the restoring force characteristic model shown by the thick line almost matches the load-displacement relationship for the first earthquake force. However, as shown in Figures 21(b) to (d), the load-displacement relationship deviates more from the restoring force characteristic model as the number of times the beam is subjected to earthquake force increases to the second, third, and fourth times.

[0079] Figure 22 shows the results of Eqs. (13) and (15) for a beam subjected to a single earthquake load. r A comparison of the load-displacement relationship for the second earthquake force with the restoring force characteristic model obtained by reflecting the residual deformation of the relocated section is shown in Figure 21(b). The deviation of the load-displacement relationship for the second earthquake force for the restoring force characteristic model for reuse, shown by the thick line, is smaller than the deviation of the load-displacement relationship for the second earthquake force for the restoring force characteristic model for virgin material (Figure 21(b)).

[0080] Although the description of the specific embodiment is finished above, the present invention is not limited to the above embodiment and can be widely modified. The modeling method of the present invention can be applied not only to a non-uniform cross-section reinforced concrete beam, but also to a beam having different bending stiffness between the end region and the general region, for example, a beam having a general region made of steel members and an end region having a composite structure of the steel members and the reinforced concrete surrounding the steel members, or a reinforced concrete beam having different amounts of main reinforcement between the end region and the general region. The present invention can also be applied to columns, not just beams.

[0081] References 10) Architectural Institute of Japan: Future Space Creation by Unbonded PC, Panel Discussion of Annual Meeting, 2021 (in Japanese) Architectural Institute of Japan Structural Committee Prestressed Concrete Structure Steering Committee: Old and new - Future space creation woven by unbonded PC, Panel Discussion Materials for the Structural Division (PC) of the 2021 Annual Meeting of the Architectural Institute of Japan (Tokai), 2021 11) Y. Takemoto: Ultimate Tendon Stresses in Unbonded Partially Prestressed Concrete Members, Report of Obayashi Corporation Technical Research Institute, No.28, pp.49-54, 1984 (in Japanese) 12) American Concrete Institute: Building Code Requirements for Structural Concrete (ACI318-19) and Commentary (ACI318-19R), 2019 13) M. Tani et al.: Reinforced Steel Design behind Anchorage of Prestressed Concrete Beam Post-tensioned at Widened Beam End, Proceedings of the Symposium on Developments in Prestressed Concrete (30th), pp.171-176, 2021 (in Japanese) 14) Architectural Institute of Japan: AIJ Standard for Structural Calculation of Reinforced Concrete Structures, 2018 (in Japanese) 15) ALL Baker et al.: A plastic theory of design for ordinary reinforced and prestressed concrete including moment re-distribution in continuous members, Magazine of Concrete Research, Vol.1, pp.57-66, June, 1949 (DOI: https: / / doi.org / 10.1680 / macr.1949.1.2.57) 16) H. Muguruma et al.: Study on Flexural Failure Strength of Unbonded PC Beams, Architectural Institute of Japan Kinki Chapter Research Reports, pp.69-72, 1978(in Japanese) 17) H. Muguruma et al.: Ultimate Flexural Capacity of Unbonded PC Members, Journal of Prestressed Concrete, Vol.26, No.1, pp.10-16, 1984 (in Japanese) 18) K. Sugimoto et al.: Development of a large-span Precast concrete structural system with ease of construction using prestressed connections, Part8, Summaries of Technical Papers of Annual Meeting, Architectural Institute of Japan, C-2, Structure IV, pp.665-666, 2008 (in Japanese) [Explanation of symbols]

[0082] 1: Beam 3: End area 4:General area 7: Relocate section 8: Widening section 9: Tensile material 15: Bending spring 16: Equivalent rigid area

Claims

1. A method for modeling a beam or column including end regions provided at both ends in an extension direction and a general region extending between the two end regions, the two end regions having equal lengths in the extension direction and equal bending stiffnesses different from those of the general region, A step of regarding the beam or column as a wire having bending springs at an end face in the extension direction and at a relocated portion which is a boundary between the end region and the general portion region; converting the bending spring of the end face and the bending spring of the relocated portion into one composite bending spring on each side of the beam or column in the extension direction by the following formulas (13) and (15). The modeling method comprises: ##EQU00013## ##EQU00015## Where: ｒ R': rotation angle in the relocated part after synthesis ｅ R': rotation angle at end face after synthesis ｒ R: Rotation angle at the relocated part before synthesis ｅ R: Rotation angle at the end face before synthesis λ: Ratio of the extension length of the end region to 1 / 2 of the extension length of the beam or column

2. The modeling method according to claim 1 , which is used for a restoring force characteristic model or a hysteretic restoring force characteristic model of the beam or column.

3. used for dynamic response analysis of said beam or column during reuse, The beam or column is a precast concrete member; The end region includes a widened portion that is wider than the general region, The beam or column is configured to be crimped and joined to another member by tension of a tension member having one end fixed to the widening portion, The converting step is carried out by: ｒ The method of claim 1 , further comprising: making R reflect residual deformation of the relocated portion caused by an earthquake.

4. The converting step is carried out by: ｅ 4. The method of claim 3, further comprising: making R reflect an earthquake-induced decrease in Young's modulus of concrete at the end faces of the beams or columns.

5. used for dynamic response analysis of said beam or column during reuse, The beam or column is a precast concrete member; The end region includes a widened portion that is wider than the general region, The beam or column is configured to be crimped and joined to another member by tension of a tension member having one end fixed to the widening portion, The converting step is carried out by: ｅ 2. The method of claim 1, further comprising: making R reflect an earthquake-induced decrease in Young's modulus of concrete at the end faces of the beams or columns.

6. Modelling the beam or column by the modelling method according to any one of claims 3 to 5; A step of evaluating the behavior of the beam or column to be reused against an earthquake force by performing an analysis of a restoring force characteristic or a hysteretic restoring force characteristic on the modeled beam or column; A method for evaluating a beam or column to be reused, comprising:

7. A method for modeling a reinforced concrete beam or column including end regions provided at both ends in an extension direction and a general region extending between the two end regions, the two end regions having equal lengths in the extension direction and equal bending stiffnesses different from those of the general region, a step of setting an equivalent rigid zone having an infinite second moment of area on the outer end side of the end region on each side of the extension direction of the beam or column, and calculating a ratio of the length of the equivalent rigid zone in the extension direction to the length of the beam or column in the extension direction by the following formula (8); calculating a length of the equivalent rigid zone in the extension direction based on the ratio, and regarding the beam or column as a wire having a bending spring at an end of the equivalent rigid zone on the inside in the extension direction; The modeling method comprises: [0080] Where: λ ｅ : Ratio of the extension length of the equivalent rigid area to 1 / 2 of the extension length of the beam or column λ: Ratio of the extension length of the end region to 1 / 2 of the extension length of the beam or column E: Young's modulus of concrete I ０ : Equivalent moment of area of ​​general area I: Equivalent moment of area of ​​the end region

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