JP7170992B2 - Structural calculation method and structural calculation program for linear structure - Google Patents

Description

The present invention relates to a structural calculation method for linear structures.

Structural calculations of linear structures such as subway tunnels have been conventionally performed. Such structural calculations are effectively utilized, for example, when evaluating the load-bearing performance of subway tunnels that have deformation such as subsidence in part in the axial direction.

By the way, in the case of subway tunnels, there is usually no situation in which the axial force should be considered. Therefore, it is appropriate to perform structural calculations assuming that no axial external force acts on the subway tunnel.

However, when the structural calculation is performed by the above method, the results (strain distribution, etc.) obtained by the structural calculation are sometimes greatly different from the results obtained by examining the actual structure.

SUMMARY OF THE INVENTION Accordingly, an object of the present invention is to provide a highly accurate structural calculation method that can be used for structural calculation of a linear structure in which deformation has occurred.

A structural calculation method according to a first aspect is a structural calculation method in which structural calculation of a linear structure is performed in consideration of deformation occurring in a part of the linear structure in an axial direction, Based on the structural information of the structure and the deformation information of the deformation, the extension of the linear structure caused by the occurrence of neutral axis position fluctuation factors (cracks, etc.) due to the deformation and the fluctuation of the neutral axis position is determined. , an axial force calculation step of calculating an apparent axial force when it is assumed that the linear structure is restrained from both sides in the axial direction of the deformation occurrence point; structural information of the linear structure; a result obtaining step of obtaining a structural calculation result by performing structural calculation of the linear structure based on the deformation information of the deformation and the apparent axial force.

According to the structural calculation method according to this aspect, a highly accurate structural calculation result can be obtained for a linear structure that can be said to be constrained on both sides in the axial direction across a deformed portion, such as an underground linear structure. be able to. This is because in a linear structure that can be said to be restrained at both ends in the axial direction across the deformed portion, the deformed portion cannot be stretched, and this is thought to act like an axial force. is.

Further, in the structure calculation method according to this aspect, for example, for a linear structure for which it is unclear whether it can be said that both sides in the axial direction sandwiching the deformed portion are constrained, It can also be used to determine whether or not it is constrained (constraint conditions on both sides of the linear structure in the axial direction). In other words, the calculation object of the structural calculation method of this aspect may be a linear structure, and a line that can be said to be constrained on both sides in the axial direction across a deformed location such as a linear structure in the ground. It is not limited to shaped structures.

The structural information and deformation information used in the axial force calculation step and the structural information and deformation information used in the result acquisition step may not be the same information.
In addition, in the above "structural information of the linear structure, deformation information of the deformation and the calculated apparent axial force, ... structural calculation is performed", the structural information of the linear structure The finite element model as is given with boundary conditions determined based on the deformation information and the apparent axial force. may be determined in consideration of other factors.
In addition, this structural calculation method can be used not only for structural calculations of linear structures that are actually deformed, but also for structural calculations on the assumption that deformation has not occurred at present. can also be used for

A structural calculation method for a linear structure according to a second aspect is the first aspect, wherein the neutral axis position fluctuation factor is cracks in concrete.

According to the structural calculation method according to this aspect, it is possible to obtain a highly accurate structural calculation result in the case where deformation such as subsidence occurs in a part of the tunnel in the axial direction and cracks occur in the concrete at that part. can.

A structural calculation method for a linear structure according to a third aspect is, in the first aspect, wherein the linear structure is a tunnel composed of a large number of segment rings arranged in the axial direction, and the neutral axis position variation factor is the opening of the segment ring.

According to the structural calculation method according to this aspect, a highly accurate structure can be obtained in the case where deformation such as subsidence occurs in a part of the tunnel in the axial direction, which is composed of many segment rings, and the segment rings open at that part. Calculation results can be obtained.

A structural calculation method for a linear structure according to a fourth aspect is, in any one of the first to third aspects, in the axial force calculating step, based on the deformation information of the deformation, A range in which the deformation direction side end at the occurrence point is in tension is set as the axial force calculation range, and based on the structural information of the linear structure and the deformation information of the deformation, the axial force calculation range Calculate the difference in elongation before and after the neutral axis position change of the end portion on the deformation direction side as the amount of elongation to be restrained, and based on the amount of elongation to be restrained and the axial force calculation range, the deformation direction A constrained strain amount of the side end portion is calculated, and the apparent axial force is calculated based on the constrained strain amount.

In the structural calculation method according to this aspect, since the range in which the deformation direction side end (for example, the lower end in the case of subsidence) at the deformation start point is in tension is the axial force calculation range, the deformation direction side end Structural calculation can be performed by investigating the information of displacement (deformation information).

A structural calculation method for a linear structure according to a fifth aspect is characterized in that, in any one of the first to fourth aspects, in the result acquisition step, the apparent is applied, and then forced displacement corresponding to the deformation is applied.

In the structural calculation method according to this aspect, a forced displacement corresponding to an apparent axial force is applied to both ends of the structural model of the structure in the axial direction to generate an axial force in the entire structural model of the structure. Appropriate calculation results can be obtained since corresponding forced displacements are given.

A structural calculation program for a linear structure according to a sixth aspect is a program for causing a computer to execute each step of the structural calculation method according to any one of the first to fifth aspects.

According to the structural calculation program according to this aspect, it is possible to easily obtain highly accurate calculation results for a deformed linear structure using a computer.

ADVANTAGE OF THE INVENTION According to this invention, the structural calculation method with high precision which can be used for the structural calculation of the linear structure which deformation|transformation has arisen can be provided.

It is the whole flow which shows an example of the structural calculation method of this invention. It is a schematic diagram which shows the whole subway tunnel as a linear structure. FIG. 4 is an explanatory diagram of an axial force calculation range; It is a figure which shows the difference of strain|strain by the presence or absence of a crack. FIG. 4 is an image diagram of the difference in tunnel elongation depending on the presence or absence of cracks. It is a figure which shows the result acquisition step of embodiment. FIG. 10 is a diagram showing a result acquisition step of a comparative example (an example in which apparent axial force is not considered); 1 is a diagram showing an outline of a tunnel to be considered and a situation of adjacent construction; FIG. FIG. 4 is a diagram showing cracks in a tunnel near point B; FIG. 4 is a diagram showing the results of investigation of strain in the tunnel axial direction; It is a figure which shows the image of reinforcement bending strength calculation with respect to a tunnel axial direction. FIG. 5 is a diagram showing an image of strain distribution in the tunnel height direction when cracks occur due to subsidence. FIG. 5 is a diagram showing calculation results of strain at the bottom end of a tunnel depending on the presence or absence of cracks; FIG. 10 is a diagram showing a comparison of strain distribution in the case where investigation results of strain in the tunnel axial direction and apparent axial force are taken into consideration. FIG. 10 is a diagram showing an image of modeling in the tunnel axial direction by pseudo three-dimensional; FIG. 4 is a diagram showing the relationship between the stress and strain of concrete when calculating the ultimate bending strength. FIG. 4 is a diagram showing a structural calculation model in the tunnel axial direction by pseudo three-dimensional; FIG. 10 is a diagram showing a comparison of crack diagrams (near the center of subsidence) due to differences in structural calculation models; FIG. 4 is a contour diagram of principal strain; It is a contour figure of shear strain. FIG. 5 is a diagram showing a comparison of strain distributions on sidewalls in the tunnel height direction; It is a figure which shows the relationship between settlement amount and shear strain. It is a figure which shows the structure of a structure calculation apparatus. It is explanatory drawing in the case of making a shield tunnel into calculation object.

In the following, as an embodiment of the present invention, a method of carrying out structural calculation in consideration of subsidence occurring in a portion of a subway tunnel, with the "linear structure" to be structurally calculated as a subway tunnel, will be described.

First, the target subway tunnel 10 will be described with reference to FIG. As shown in FIG. 2, the subway tunnel 10 is a reinforced concrete structure with a length of approximately 1 km. Assume that a large subsidence (for example, 100 to 300 mm) occurs at point X shown in FIG.

FIG. 1 is an overall flow showing the calculation result method of this embodiment. As shown in FIG. 1, the structural calculation method of this embodiment is roughly divided into the following steps (1) and (2).
(1) Calculate apparent axial force due to deformation (axial force calculation step)
(2) Determine boundary conditions corresponding to apparent axial force and perform structural calculation (result acquisition step)

<Axial force calculation step>
(1) Explain the axial force calculation step.
In the axial force calculation step, based on the structural information and settlement information (deformation information) of the subway tunnel, the elongation of the subway tunnel due to the occurrence of cracks due to settlement and the change in the neutral axis position is estimated at the settlement occurrence point X. Calculate the axial force assuming that it is constrained from both sides in the axial direction as the "apparent axial force".

Specifically, first, a range used for calculating the apparent axial force (hereinafter referred to as axial force calculation range L) is set (step S101).
As shown in FIG. 3, the axial force calculation range L is a range in which the lower end of the tunnel is in tension. For example, if the range in which the lower end of the tunnel is in tension is a range of 40 m, the range of 40 m is defined as the axial force calculation range L.

Since subsidence is deformation in the downward direction, the area where the lower end of the tunnel is in tension is the end of the linear structure on the deformation direction side (hereinafter referred to as the “end in the deformation direction”) that is in tension. It can be rephrased as the range in which

Note that the axial force calculation range L does not have to be the entire tensile range as long as it includes the center point of the tensile range at the ends on the deformation direction side.

Next, the strain ε when there is no crack is calculated (step S102), and the strain ε' when there is a crack is calculated (step S103).

Here, in the cross section of reinforced concrete, when cracks occur, the position of the neutral axis fluctuates, and the strain distribution also changes accordingly. When cracking occurs, the tension of the concrete can be ignored or its effect becomes small, so the neutral axis position after cracking fluctuates toward the inside of the bend. Note that cracks in concrete correspond to the "neutral axis position variation factor" of the present invention.

The strain distribution before cracking can be calculated by the following formula.

ε=φ・z

here,
ε: Strain amount at the calculated position before cracking φ: Curvature z: Length from the neutral axis position (before cracking) to the calculated position

The strain distribution after cracking can be calculated by the following formula.

ε′=φ・z′

here,
ε': Strain amount at calculated position after cracking φ: Curvature z': Length from neutral axis position (after cracking) to calculated position

Since the neutral axis fluctuates due to cracks, the calculated value of the axial length of the subway tunnel changes depending on the presence or absence of cracks. FIG. 4 shows the difference in axial length depending on the presence or absence of cracks. In addition, in the drawing, for ease of understanding, the amount of expansion and contraction is drawn as 200 times.

Next, the amount of elongation before cracking is calculated (step S104), and the amount of elongation after cracking is calculated (step S105).

The amount of elongation calculated here is the amount of elongation at the bottom of the tunnel. The amount of elongation at the bottom end of the tunnel can be calculated by the following formula from the amount of strain ε _i at the bottom end of the tunnel and the length l _i of the element.

dl = Σε _i · l _i
dl'=Σε _i '·l _i
here,
dl: Elongation at the bottom end of the tunnel (before cracking)
dl': Elongation at the bottom end of the tunnel (after cracking)
ε _i : Amount of strain at the bottom of the tunnel (before cracking)
ε _i ': amount of strain at the bottom of the tunnel (after cracking)
l _i : length of the element for which the amount of strain was calculated

FIG. 5 is an image of the difference in tunnel elongation due to the presence or absence of cracks. Assume that this difference in elongation is constrained from both axial sides of the subway tunnel. The amount of elongation at the lower end of the tunnel can be rephrased as the amount of elongation at the end portion of the linear structure in the deformation direction.

Next, the restrained strain amount is calculated (step S106).
The constrained strain amount can be calculated by the following formula.

(constrained strain amount) = (dl'-dl)/L

Finally, the apparent axial force is calculated (step S107).
Apparent axial force is calculated, for example, by the following method.

First, from the strain distribution after cracking and the strain amount to be restrained, the strain amount after restraint is calculated, for example, as follows.

(strain amount after constraint) = (maximum strain after cracking) - (constrained strain amount)

That is, for example, since the extension of the subway tunnel is restricted, the strain at the bottom of the tunnel (2836μ) where the subsidence is maximum is reduced by about 677μ, which is the amount of strain to be restricted, to 2160μ. The above strain values are values in Examples described later.

Next, the position of the neutral axis is calculated from the bending curvature near the maximum subsidence point (for example, average over a range of 5 to 20 m) and the amount of strain after restraint.
For example, if the bending curvature near the center of the tunnel is 4.52×10 ⁻⁷ (1/mm) (average of the length of 10 m near the center) and the strain at the bottom end of the tunnel is 2160 μ, the neutral axis position is 1623 mm.

Finally, the axial force (35.7 MN) at which the neutral axis position is (1623 mm) and the curvature is (4.52×10 ⁻⁷ (1/mm)) is calculated by a known ultimate bending strength calculation method. This axial force is defined as "apparent axial force due to cracks".

<Result acquisition step>
Next, the (2) result acquisition step will be described.

The result acquisition step includes steps 1 and 2 below. Note that the structural model (finite element model) is created in a partial range (for example, 100 m to 300 m) around the subsidence point X, as shown in FIG.
Step 1: Forced displacement corresponding to apparent axial force is applied from both ends of the structural model to generate axial force in the entire structural model.
Step 2: Give displacement due to subsidence as forced displacement.

Forced displacement corresponding to subsidence is given to each node at the lower end of the structural model (end in the direction of deformation), for example. The bottom end of the tunnel is vertically fixed and horizontally free. Also, the forced displacement in step 2 is preferably divided (for example, divided into 30 to 70).

Calculation results (for example, strain distribution, contour map, stress distribution) are obtained by the result obtaining step.

According to the structural calculation method of the present embodiment, highly accurate structural calculation results can be obtained for subway tunnels. In other words, as shown in FIG. 7, there is a high possibility that a more accurate result can be obtained than when structural calculation is performed assuming that no axial external force is acting on the subway tunnel.

〔supplementary explanation〕
In addition, the above-described embodiment is one embodiment of the present invention, and the present invention is not limited to the above-described embodiment as described below.

(linear structure)
In the above embodiment, an example was described in which the "linear structure" to be calculated is a subway tunnel having a certain degree of continuity in its axial direction, but the present invention is not limited to this. For example, the linear structure may be a tunnel (shield tunnel) composed of a large number of segment rings arranged in the axial direction. The axial length of each segment ring is, for example, 0.5 to 3 meters (more specifically 1 to 2 meters). In the shield tunnel, segment rings are connected to each other by joint bolts.

Further, the linear structure may be a structure whose neutral axis position changes due to deformation, or a structure whose neutral axis position can be considered to change due to deformation. In the shield tunnel, it can be considered that the neutral axis position fluctuates due to the opening of the ring (see FIG. 24).

As shown in FIG. 24, the segment ring of the shield tunnel opens due to deformation such as subsidence. It can be considered that the neutral axis position of the shield tunnel changes due to the opening of the segment ring. That is, when the linear structure is a shield tunnel, the opening of the segment ring corresponds to the "neutral axis position fluctuation factor". As shown in FIG. 24, it can be considered that the vicinity of the upper end of the segment ring is slightly compressed in the axial direction (arrow A) due to the subsidence of the shield tunnel, and the lower part of the segment ring expands greatly in the axial direction by opening. (Arrow B). That is, it can be said that the opening of the segment rings extends the axial length of each segment ring, and the shield tunnel extends in the axial direction. Therefore, if the joint portion between the segment ring and the segment ring is the crack position and the joint bolt is a reinforcing bar, it can be considered in the same way as the above embodiment. Although the illustrated segment ring has a circular cross-section, the occurrence of the above mechanism is not limited to the case where the segment ring has a circular cross-section.

Further, the linear structure to be subjected to the structural calculation is, for example, an underground linear structure as in the above embodiment. In the case of an underground linear structure, since the linear structure is buried in the ground, the extension of the range in which the position of the neutral axis is changed due to deformation is restrained from both sides in the axial direction. However, the linear structure is not limited to an underground structure, and may be, for example, a bridge. In the case of a bridge as well, depending on its structure, it can be considered that both axial sides of the bridge are constrained.

Further, in the above embodiment, an example in which the deformation is subsidence (that is, deformation in the downward direction) has been described, but the "deformation" in the present invention is not limited to subsidence. For example, it may be a horizontal deformation. The horizontal deformation means that a part of the linear structure in the axial direction is displaced in the horizontal direction, and in this case, the horizontal direction corresponds to the deformation direction.

Structural information is information representing the structure of a linear structure. Structural information is, for example, information that can obtain the neutral axis position before and after the neutral axis position fluctuation factor occurs.

Further, the deformation information is information representing deformation such as subsidence. For example, deformation information can be obtained by actually investigating the deformation locations of the linear structure. If the deformation is subsidence of a subway tunnel, the deformation information is, for example, the subsidence displacement of the bottom end of the tunnel. In this case, the deformation information can be rephrased as the displacement amount of the end portion of the linear structure on the deformation direction side.

In the above embodiment, the structural information used in the result acquisition step is a finite element model, but the present invention is not limited to this, and may be a structural model using beams, for example.

Further, in the above embodiment, in step 1, an axial force is generated in the entire structural model by applying forced displacement corresponding to the apparent axial force from both ends of the structural model, but the present invention is not limited to this. For example, the displacement corresponding to the apparent axial force may be given as the initial value. Further, for example, an axial force may be generated in the entire structural model by applying a load corresponding to the apparent axial force.

Further, in the above-described embodiment, in step 2, displacement due to subsidence is given as forced displacement, but the present invention is not limited to this. For example, step 2 may be performed by a method of applying a displacement due to subsidence with a load (response displacement).

[Structural calculation program]
Structural calculation using the structural calculation method of the present invention can also be automatically performed using a program.

That is, according to the present disclosure, an information processing device (structural calculation device 100) including axial force calculation means for performing the axial force calculation step and result acquisition means for performing the result acquisition step for causing a computer to function A structural calculation program is provided. This structural calculation program can also be rephrased as a program for causing a computer to execute an axial force calculation step and a result acquisition step.

Also, by installing this structural calculation program in a computer, as shown in FIG. A structure calculation device 100 can be obtained comprising: The structural calculation device 100 can be configured by a computer including a CPU, a RAM, and a ROM that stores the aforementioned structural calculation program and various data.

The studies conducted by the present inventors will be disclosed as examples below.
However, the present invention is not limited to the following examples. For example, in this embodiment, it was confirmed that the real structure could be reproduced with high accuracy using the pseudo three-dimensional model. The problem of the present invention is solved by calculating the already explained "apparent axial force" and considering it in the structural calculation.

(wrap up)
A rational evaluation method for the load-bearing performance of a subway cut-and-cut tunnel with partial subsidence in the axial direction was studied.
As a result of analyzing the deformation of the tunnel, it was found that the axial force acts in the direction of the tunnel axis. The mechanism by which the axial force is generated in the tunnel axial direction is that the changes in the neutral axis position and strain distribution are constrained in the tunnel axial direction by the occurrence of cracks. did. By modeling the calculated apparent axial force due to cracks and the tunnel structure in pseudo 3D, we were able to reproduce the cracks and strain tendencies measured in the existing structure. In addition, we introduced the concept of the performance check type design method to check the load-bearing performance of the tunnel, and evaluated the rational load-bearing performance corresponding to the structural calculation method of this research.

1. Purpose of Research In recent years, although there are various causes of deformation, the number of structures that require proper maintenance has been increasing year by year. Regular work such as maintenance and construction work on the subway tracks is mainly done at night when the railway service is finished, and is limited to about 3 hours a day at maximum, and the cost of countermeasures such as construction tends to be high. It is in. For this reason, measures such as reinforcement work are desired to have a light structure with good workability.
Line-shaped underground structures such as subway tunnels may partially undergo deformation such as subsidence due to the effects of adjacent construction or ground subsidence. In such cases, countermeasures such as load-bearing performance and reinforcement are often taken according to the "allowable stress design method", which is a design method at the time of construction.
Here, the "allowable stress method" is a method of evaluating safety, etc., based on the "allowable stress", which considers a certain margin in the material properties, regardless of the type or condition of the structure. There have also been reports of cases in which the allowable stress level is exceeded due to unexpected deformation during design.
If the allowable stress design method is used to deal with the deformation of existing structures, it tends to become a multi-layered structure.
On the other hand, depending on the state of deformation of the structure, even if the allowable stress level is exceeded, there may be cases where the usability of the structure is not hindered. Since it is expected that the number of structures requiring maintenance will increase in the future, rational evaluation and countermeasures according to the situation of structures will be required.

Therefore, in this study, the following investigations were carried out, taking an existing tunnel as an example, for the purpose of rationally evaluating the load-bearing performance of a subway tunnel, which is partially deformed due to subsidence in the tunnel axial direction.

・Elucidation of the mechanism of deformation in the axial direction of the tunnel In order to rationally evaluate the load-bearing performance of a tunnel, it is necessary to accurately reproduce the deformation of the tunnel. Therefore, we will clarify the mechanism of deformation from the results of investigation on deformation in the tunnel axial direction and stress state, and examine how to introduce it to the design.

・Examination of rational evaluation methods for load-bearing performance in the axial direction of tunnels In order to verify load-bearing performance, structural calculation methods and models that can accurately reproduce deformation conditions occurring in tunnels, taking into consideration the characteristics of the tunnel structure was considered.
In recent years, the "performance-based design method" has been introduced as a design method for various structures. Therefore, in order to rationally evaluate the load-bearing performance, we investigated the evaluation method of the load-bearing performance based on the concept of "performance check type design method".

2. Overview of the tunnel in the study case
(1) Overview of the tunnel structure of the study case Fig. 8 shows the outline of the tunnel used as the study case. The tunnel is about 50 years old and is a reinforced concrete structure with one layer and two spans constructed by the open-cut method. The dimensions in the cross-sectional direction of the tunnel are 8.8 m in width, 6.5 m in height, 470 mm in thickness of the side walls and lower floor slab, 410 mm in thickness of the upper floor slab, and 450 mm in thickness of the middle wall. The middle wall has a partial opening for evacuation and signal equipment installation.
The length of the target tunnel is about 1 km, and most of the tunnel is located on soft cohesive soil (lower Yurakucho layer), and soft cohesive soil is also deposited thickly below the tunnel. Below the soft and viscous ground, there are silt layer (Nana No. 7 stratum), sand gravel layer (buried terrace), Edogawa layer, etc.

(2) Overview of the Tunnel Deformation Situation in the Case Study In the tunnel that was the case study, as shown in Figure 8, two points, point A and point B, were experiencing significant subsidence. The maximum amount of subsidence measured was 177 mm at Point B. Furthermore, as shown in FIG. 9, many cracks were confirmed on the inner surface of the tunnel where the subsidence was large. Most of the cracks in the side walls are in the cross-sectional direction of the tunnel, which is considered to be due to the bending of the tunnel subsidence. Most of the cracks in the inner walls are oblique cracks, which are thought to be caused by shearing.

In response to these deformations, various investigations and monitoring, such as continuous measurement of the amount of subsidence in tunnels, are being carried out, strict management and countermeasures are being implemented for track rails, and the safety of trains is being sufficiently ensured. rice field. In addition, the following results were obtained from the results of surveys and measurements.
・The cause of the deformation is assumed to be the effect of the shield construction constructed below the tunnel in the case study.
・Deformation tended to continue for more than 10 years after completion of shield construction. This is thought to be due to the secondary consolidation of the soft, viscous soil under the tunnel, and although the degree of progress of deformation will decrease, it is expected to continue in the future.
・When the load-bearing performance in the axial direction of the tunnel was examined using a structural model using beam members, the stress level of the rebar greatly exceeded the allowable stress level, and the yield stress level also exceeded.
・Introduced the concept of "performance check type design" into the evaluation of the load bearing performance in the axial direction of the tunnel. However, shear cannot be evaluated rationally due to issues such as evaluation methods for load-bearing performance. Shear causes brittle fracture, so elucidation of the deformation mechanism and re-evaluation of load-bearing performance have been issues.

3. Investigation of deformation mechanism in tunnel axial direction
(1) Investigation of strain in the axial direction of the tunnel a) Investigation method The tunnel in the case study had large subsidence at two locations and many cracks. From the shape of the subsidence in the tunnel axial direction, it was thought that large stress and strain due to bending occurred in the tunnel near the center of the subsidence.
Therefore, in order to examine modeling and reproducibility of structural calculation, the strain in the tunnel axis direction was investigated by changing the position in the tunnel height direction. Investigation of the strain was carried out by the "stress release method", in which a strain gauge was installed on the reinforcing bar, the strain was cut, and the amount of strain generated from the fluctuating strain amount was measured. The survey was conducted at point B where the subsidence is the largest in the direction of the tunnel axis. In conducting the survey, the cross section of the tunnel was assumed to be a reinforced concrete cross section, and the distortion of the reinforcing bars due to subsidence was estimated.

b) Investigation results Fig. 10 shows the relationship between the investigation results of reinforcing bar strain and the investigation position in the height direction of the tunnel.
From the distribution of strain, it is considered that the upper part of the tunnel is compressed in the direction of the tunnel axis. A first-order approximation of this result indicates that the neutral axis position is 2423 mm from the outer edge of the tunnel, and the vicinity of the bottom of the tunnel is about the design yield strain of the reinforcing steel.

(2) Analysis of strain distribution in the tunnel height direction Fig. 10 shows the strain distribution at the time of the crack bending strength and the yield bending strength of the reinforcing bars when the axial force in the tunnel axial direction is 0 with respect to the tunnel cross-sectional direction. did. Here, the crack bending strength was calculated assuming that the entire section of concrete is effective. As shown in Fig. 11, the bending yield strength of the reinforcement was calculated as a cross section of reinforced concrete with the axial reinforcement as the main reinforcement and ignoring the tensile strength of the concrete.
As a result, the calculation results of strain distribution at crack bending strength and yield bending strength tended to differ greatly from the investigation results.
The slope of the strain distribution in the investigation results corresponds to the bending curvature. Therefore, the bending curvature (2.1 × 10 ^-7 to 4.9 × 10 ^-7 1/mm) near the investigation location obtained from the subsidence shape and the strain gradient (2.9 × 10 ^-7 1 /mm) were about the same.
From this, it is considered that the difference in the tendency of the strain distribution between the survey results and the calculation results is due to the large difference in the position of the neutral axis.

Focusing on the position of the neutral axis, the survey results tend to be positioned below the calculated values. A possible reason for this is the state in which a compressive axial force is acting in the axial direction of the tunnel. However, we could not find any factors such as close construction that exerted compressive force in the direction of the tunnel axis so that the neutral axis position would be the same as the survey results.
Therefore, it is thought that the position of the neutral axis fluctuates due to the effects of the balance of forces within the tunnel structure as the tunnel subsides.
Therefore, we performed trial calculations based on hypotheses regarding the mechanism by which the neutral axis position fluctuates, and conducted verification by comparing with measurement results.

(3) Investigation of the mechanism by which the neutral axis position fluctuates As a mechanism by which the neutral axis position fluctuates, we focused on the fact that when a crack occurs in a reinforced concrete cross section, the neutral axis position fluctuates and the strain distribution in the tunnel height direction also changes. did.
The height of the considered tunnel is 6400 mm. When bending occurs, the tension of the concrete also contributes to the rigidity at a curvature that does not cause cracking, and the neutral axis position is calculated to be 3339 mm, which is about 1/2 of the tunnel height from the top of the cross section. On the other hand, the position of the neutral axis after the occurrence of cracks is 637 mm when calculated with a cross section of reinforced concrete ignoring the tension of the concrete. All of these are calculated values when the axial force is set to 0.
From this, FIG. 12 shows an image of the difference in the neutral axis position and strain distribution depending on the presence or absence of cracks in the tunnel axis direction position.

When the tunnel subsidence is small and cracks do not occur, the neutral axis position is near the center in the tunnel height direction, and there is no difference in the tunnel axis direction. At this time, the tensile stress degree and the compressive stress degree in the tunnel height direction are in a balanced state, and since no external force is acting in the tunnel axial direction, the axial force is zero.
On the other hand, when the subsidence becomes large and cracks occur near the subsidence center, the neutral axis position fluctuates near the upper end. Since the bending curvature does not change, the strain distribution in the tunnel height direction fluctuates in the tension direction as a whole due to the change in the neutral axis position, and the cracks show behavior that extends in the tunnel axis direction. However, the tunnel is a continuous structure in the axial direction, and in places where there is no subsidence and no cracks, behavior such as expansion and contraction does not occur in the axial direction of the tunnel. Therefore, expansion in the axial direction of the tunnel due to cracking at the subsidence point is restrained from both axial ends of the tunnel where subsidence does not occur, and the subsidence point of the tunnel cannot be expanded. It was assumed that this acted like an axial force in the direction of the tunnel axis, the neutral axis position fluctuated, and the neutral axis position of the measured value was positioned lower than the calculated value.

The axial force assumed in this study does not act from outside the tunnel. Therefore, in this research, we decided to call it "apparent axial force due to cracks".

(4) Trial calculation of apparent axial force due to cracks The apparent axial force due to cracks was calculated as follows from the above mechanism. Note that the trial calculation was made for point B, where the subsidence is large, and the bending curvature for the subsidence amount of 184 mm at the time of the survey was used.

・Bending range setting Apparent axial force is calculated for the range in which the bottom end of the tunnel is in tension. At point B, which was the target of the trial calculation, the range in which the lower end of the tunnel is in tension is equivalent to the range in which the curvature is positive, which is 40 m.

・Calculation of neutral axis position and strain distribution The strain distribution in the tunnel height direction is calculated by the following formula, assuming that the neutral axis position is 0 and that it is proportional to the curvature calculated from the tunnel subsidence shape.

ε=φ・z (2)

where, ε: strain amount at the calculated position, φ: curvature, z: calculated position of strain from the neutral axis position

If there are no cracks, calculate assuming that the entire section is valid. At this time, the neutral axis position is 3339 mm from the top end of the tunnel. In the range where the lower end of the tunnel is in tension, the strain at the lower end of the tunnel was calculated from the curvature of the tunnel axial position calculated from the tunnel subsidence shape, and it was -58μ to -1506μ (compression is positive).

Next, the crack location is calculated as the cross section of the reinforced concrete shown in FIG. From this, the neutral axis position is calculated to be 637 mm from the top of the tunnel. From the neutral axis position and the curvature due to tunnel subsidence, the strain at the bottom of the tunnel in the range where cracks occur at the bottom of the tunnel was calculated to be -517μ to -2836μ.
FIG. 13 summarizes these as a relationship with the axial position of the tunnel.

・Calculation of constrained strain amount From the previously calculated strain at the bottom of the tunnel and the length of the structural calculation element, the amount of elongation at the bottom of the tunnel is calculated by the following formula.

dl = Σε _i · l _i (3)

where, dl: elongation amount at the bottom end of the tunnel, ε _i : calculated value of the strain amount at the bottom end of the tunnel, l _i : length of the element for which the strain amount was calculated

When the amount of elongation at the bottom of the tunnel is calculated for 40 m, which is the range where the bottom of the tunnel is in tension, it is 30.7 mm when using the calculated strain value before cracking, and 57 mm when using the calculated strain value after cracking. .8 mm.
This difference is the amount of elongation that is constrained after cracking occurs, and the strain that constrains the entire tunnel is calculated by the following equation.

Constrained strain amount = (57.8mm-30.7mm) / 40m = 677μ (4)

・Calculation of apparent axial force Since the elongation of the tunnel is constrained, the strain at the bottom of the tunnel (2836μ) where the subsidence is maximum is reduced by about 677μ to 2160μ. Assuming that the bending curvature near the center of the tunnel is 4.52×10 ⁻⁷ (1/mm) (average of the length of 10 m near the center) and the strain at the bottom end of the tunnel is 2160 μm, the neutral axis position is 1623 mm.
From the distribution of strain, the reinforcing bars near the bottom of the tunnel are above the yield stress. Therefore, according to the calculation method of the ultimate bending strength (Japan Society of Civil Engineers: Tunnel Standard Designation and Commentary, Open Cut Construction Method, P.55, 2016, established in 2016), the neutral axis position is 1623 mm and the curvature is 4.52 × 10 ^-7 (1 /mm) was estimated to be about 35.7 MN. This is the calculated value of the apparent axial force due to cracking.

(5) Comparison between trial calculation results of apparent axial force due to cracks and survey results The strain distribution in the tunnel height direction, which was calculated taking into account the calculated apparent axial force due to cracks, was compared with the strain survey results. FIG. 14 .
As a result, the conventional calculation method that does not consider the axial force gives a result of inconsistency between the calculation and the survey results. On the other hand, considering the apparent axial force, the calculated values tend to be closer to the measured values. From this, it is considered that the mechanism assumed in this study is appropriate.
It should be noted that the apparent axial force assumed in this study decreases as the thickness of the member becomes smaller, and is considered to be a negligible value depending on the thickness of the member. Also, considering the axial force in the load bearing performance in the axial direction results in underestimating the strain on the tension side. Therefore, it should be noted that consideration of the apparent axial force due to cracks may result in a risky evaluation.

4. Evaluation of tunnel load-bearing performance
(1) Investigation of a structural calculation model for the tunnel axis direction In previous studies, the structure in the tunnel axis direction was modeled using simple beam members. However, modeling with beam members had problems such as the inability to reproduce cracks.
Therefore, we investigated a structural calculation model that can accurately reproduce the deformation of the tunnel.
Based on the tunnel structure and crack conditions to be examined, it is also assumed that structural calculations will be modeled in three dimensions. In recent years, however, three-dimensional calculations are being applied more and more, but compared to two-dimensional calculations, they tend to be more complicated to handle and more expensive to design. Furthermore, if a result that does not satisfy the load-bearing performance is obtained, it is necessary to carry out a design such as reinforcement, and it is desirable that the design can be carried out by the same method as the load-bearing performance evaluation.
Taking these into consideration, in this study, we adopted a pseudo-three-dimensional model using WCOMD, a two-dimensional reinforced concrete structure calculation code, to evaluate load-bearing performance.
Structural calculations were modeled as follows (see Figure 15).

・Modeling of Reinforced Concrete Structure For the reinforced concrete structure, the constitutive law based on the distributed crack model of WCOMD was used.
・Modeling of the shape Side walls and inner walls are modeled with plane elements when viewed from the side in the direction of the tunnel axis. The middle wall reproduces the shape of the exit. The side wall and the inner wall are modeled with independent elements (overlapping elements) on the same plane, and the rigidity is given according to the wall thickness and the amount of reinforcing bars.
The top slab and the bottom slab are modeled as the same element in the tunnel width direction, and the stiffness is modeled by the member thickness and the amount of reinforcing bars considering the tunnel width.
・Modeling of the interaction between the side wall and the middle wall The side wall and the middle wall are connected by the top plate and the bottom plate. Therefore, the nodes of the lower end of the top plate element and the upper ends of the side walls and inner wall elements are shared. Similarly, the nodes of the upper end of the bottom plate element and the lower ends of the side walls and inner wall elements are shared.
From this, it was expected that the deformation and interaction of the top slab, bottom slab, side wall, and middle wall with respect to the deformation in the axial direction of the tunnel would be considered according to their rigidity and shape.

(2) Examination of verification items and limit values a) Verification items Verification was set with a focus on "bending" and "shearing" from the deformation conditions of tunnels and the performance required for subway tunnels.
The current tunnel deformation is subsidence due to secondary consolidation of the surrounding ground. Therefore, even if the yield stress of the tunnel material is exceeded, the tunnel is supported by the ground, so that the subsidence does not progress rapidly. However, when the ultimate bending strength is approached, the compressive strain of concrete increases and the possibility of concrete spalling increases. Therefore, for bending, the compressive strain of concrete is checked.
On the other hand, shear fracture is brittle fracture and safety cannot be ensured. In addition, even in a state where shear failure does not occur, if the shear strain increases, shear cracking may occur and the concrete may flake off. Therefore, in order to check the state of shear, we decided to check the shear strain.

b) Setting of limit values The relationship between concrete stress and strain when calculating the ultimate bending capacity is modeled as shown in Figure 16. When the compressive strain of concrete exceeds 2000μ, stress does not increase and only strain increases, and the concrete collapses at 3500μ. From this, the limit value of compressive strain was set to 2000μ.
A check for shear strain is not generally done. Therefore, for the limit value of shear strain, reference was made to the literature, etc., in which the shear failure is examined by analysis and experiment by WCOM as in this study. It is conceivable that shear failure by analysis is determined as a state in which shear strain increases sharply with respect to the degree of increase in load and the like. From the results of tests and analyses, it can be seen that when the shear strain exceeds approximately 5000μ, the shear strain tends to develop rapidly. However, considering that shearing causes brittle fracture rather than a generalized index, the limit value of shearing strain is set at 5000 μ as a guideline, and comprehensively evaluates including places and ranges where shearing strain becomes large. The policy was to

(3) Structural calculation model a) Tunnel structural calculation model A model for the axial direction of the tunnel was created as shown in Fig. 17. Based on the subsidence distribution shape, etc., the modeling range was set to a total of 200 m, centering on the point B where the subsidence was the most, and about 100 m forward and backward. As mentioned above, the top plate and bottom plate are one layer in the depth direction, the middle wall and side wall are two layers in the depth direction, and the nodes are shared at the top and bottom ends of each. was reproduced at each position along the tunnel axis. Since the tunnel width direction is bilaterally symmetrical with respect to the center of the tunnel width direction, the model is modeled with a thickness corresponding to a half section.

Axial subsidence of the tunnel is given to each node at the bottom of the model as forced displacement. In addition, the apparent axial force due to the above cracks was applied by forcibly displacing from both ends of the model toward the center of the model.
The boundary conditions of the structural model are that the lower end of the tunnel structure is vertically fixed and horizontally free. The influence of soil cover load is considered small and is not considered in the calculation.

b) Setting of physical property values The material properties used for structural calculation are set as Table 1 with reference to design documents. The element reinforcing bar ratio was set from the structural bar arrangement diagram as the ratio of the reinforcing bar cross-sectional area to the cross-sectional area of each element when each member is cut along the horizontal and vertical axes.
The structure under consideration used round bars for the reinforcing bars. Round steel cannot be expected to adhere to deformed reinforcing bars. Therefore, the tensile softening coefficient, which represents the stress load on the tension-side concrete after cracking, was set at 2.0 (generally about 0.4 in the case of deformed reinforcing bars).

c) Structural calculation method Structural calculation was performed in two steps. In step 1, forced displacement corresponding to the apparent axial force due to cracking is applied from both ends of the model to generate axial force in the entire model. In step 2, tunnel subsidence was given as forced displacement. The forced displacement was calculated as 50 divisions.
The amount of subsidence of the tunnel was set to the value corresponding to the calculation result and the comparison at the time of investigation, and the node position of the model was given by interpolating with the spline function.

d) Structural calculation case In the structural calculation, we confirmed the reproducibility of the pseudo-three-dimensional model adopted to evaluate the difference in cracks between the side wall and the inner wall. In addition, we also confirmed the difference depending on whether or not the structural calculation was considered for the apparent axial force in the tunnel axial direction and the degree of consideration.

Based on this, calculations were performed by combining the following cases, and the calculation results and survey results were compared to verify the validity of the structural calculations.

・Confirmation of the pseudo 3D model Case 1: Pseudo 3D model Case 2: Model the side wall and the inner wall separately

・Verification of apparent axial force due to cracks In order to compare the effect of apparent axial force due to cracks, the case of axial force in the axial direction of the tunnel was set as follows.
Axial force case: 0kN, 40MN, 63MN
Here, the case of the axial force of 40 MN is set as a value corresponding to the apparent axial force discussed above. 0 kN does not consider the axial force in the examination of deformation in the axial direction of the tunnel, so 63 MN, which was set in the conventional method, was set as a case of a value larger than the apparent axial force examined above.

(4) Evaluation of Structural Calculation Results a) Calculation Results of Crack Conditions Figure 18 shows both a crack diagram created from the calculation results and a sketch diagram of cracks confirmed in the tunnel.
As a result, when the axial force is 0 MN, almost no oblique direction cracks are observed in the pseudo 3D model, and only a few cracks are observed in the independent model of the inner wall. different. In the case of an axial force of 40 MN, in the case of the single inner wall model, many diagonal cracks occur over a wide range, and the tendency is different from the actual structure. In the case of the pseudo three-dimensional model, the range of occurrence of oblique cracks shows a tendency similar to the survey results.
The direction in which cracks are generated on the side wall is slightly different depending on whether or not there is an axial force. Many cracks in the actual structure are found near the bottom edge of the side wall, and there are many cracks that do not reach the top edge.
From this, it is considered that the crack condition is consistent with the survey results when the apparent axial force is set to 40 MN for the pseudo three-dimensional model.

As a result, the modeling of this research is considered to be a model that reproduces the following characteristics of the tunnel under study.
・Due to the opening of the escape door, the middle wall has lower rigidity than the side wall and is prone to diagonal cracks.
・Because there are no openings in the side walls, shear cracks do not occur, and vertical bending cracks occur near the center of subsidence.
・A structure in which the structural characteristics of the openings in the middle wall and the structural characteristics of the side walls mutually affect subsidence.
・Apparent axial force is generated due to the occurrence of cracks due to subsidence.

b) Calculation result of strain Fig. 19 is the contour diagram of the maximum principal strain, and Fig. 20 is the contour diagram of the shear strain. The figure shows cases where the apparent axial force due to cracking is 0 MN and 40 MN near the center of subsidence where deformation is large. FIG. 21 shows the relationship between the strain in the tunnel axial direction and the reinforcement strain in the pseudo three-dimensional model and the position in the height direction of the tunnel.
As a result, when the axial force is 0 MN, many cracks are generated in the side wall and the middle wall near the center of the subsidence from the bottom end to near the top end. Almost no shear strain occurs in the middle wall, and diagonal cracks seen in actual structures cannot be reproduced.
When the axial force is 63 MN, almost no transverse cracks in the sidewall occur. The generated shear strain exceeds 5000 μm in the middle wall, and it is in the state of shear failure, but the situation of shear failure in the actual structure is not observed.
At an axial force of 40 MN, the side walls exhibit some degree of transverse cracking. A shear strain of about 3000 to 4000μ is generated in the middle wall, and it is considered that the situation in which shear cracks are generated in the actual structure can be reproduced. Also, the distribution of strain near point B is considered to be most consistent with the survey results when the apparent axial force due to cracking is 40 MN.
From this result, it is considered that the situation of the actual structure is well reproduced by considering the apparent axial force of 40 MN.

c) Evaluation of Structural Calculation Model As a result of comparing the results of surveys of actual structures and the results of structural calculations, it can be said that the following items should be adopted in order to carry out highly accurate structural calculations.
・Apply a pseudo three-dimensional model that reproduces the difference in shape between the inner wall and the side wall.
・Consider the apparent axial force due to cracks that occur with subsidence.

(5) Evaluation of tunnel load-bearing performance a) Verification of bending Referring to the contour diagram of the maximum principal strain (pseudo three-dimensional model in Fig. 19, case of apparent axial force of 40 MN), the tensile strain is the position where cracks are observed. However, the compressive strain does not reach 1000μ. As a result, the compressive strain has a sufficient margin with respect to the limit value (2000μ) for bending.

b) Verification of shearing When referring to the contour diagram of shearing strain (pseudo three-dimensional model in Fig. 20, case of apparent axial force of 40MN), a large strain exceeding 3000μ is generated at the place where the diagonal crack of the inner wall can be seen. You can see many places. FIG. 22 shows the relationship between the amount of subsidence at point B and the shear strain, focusing on the elements near the shunting opening of the inner wall where the shear strain increases. As a result, the maximum value of the shear strain at the current subsidence amount is around 3000μ, but it will increase to around 3500μ as the subsidence continues in the future.
is the shear strain. It does not reach 5000μ, which is a standard for shear fracture. However, since the strain exceeding 3000 μm is generated, shear cracking may progress and spalling may occur due to the cracking.

c) Evaluation of load-bearing performance As a result of checking the load-bearing performance, it was found that there was a margin for bending. With respect to shearing, the shear strain is large, but it does not reach the strain that is the standard for shear fracture. However, it is conceivable that shear cracks may develop and spalling may occur.
Therefore, the safety of the structure is satisfied, but the usability is not satisfied due to the possibility of peeling.

5. Conclusion In this study, the following results were obtained as an evaluation of the behavior in the axial direction of the tunnel and the load-bearing performance, using the deformation that occurred in an open-cut tunnel of a subway as an example.

・It was clarified that an apparent axial force is generated in the axial direction of the tunnel due to the occurrence of cracks due to deformation such as subsidence in the axial direction of the tunnel. The mechanism of apparent axial stress caused by cracking was the change of neutral axis position and strain distribution before and after cracking. Based on the mechanism clarified in this study, a method for calculating the apparent axial force due to cracking was presented. In order to accurately reproduce the actual structure, it is necessary to consider the calculated apparent axial force due to cracks in the structural calculation.

・For deformation in the axial direction of the tunnel, structural calculations using a pseudo 3D model can accurately reproduce structural characteristics such as the opening of the shelter opening, enabling appropriate evaluation of load-bearing performance.

・Based on performance verification type design, it is possible to appropriately evaluate load-bearing performance by setting verification items and limit values based on the use, material properties, and deformation conditions of the existing structure.

In the future, it is believed that the results of this research will enable rational evaluation of load-bearing performance by appropriately evaluating load-bearing performance. In addition, it is expected that the behavior of existing structures due to adjacent construction can be reproduced with high accuracy.
Based on the results of this research, in the tunnel considered as an example, a reinforcement structure was designed using a pseudo three-dimensional model to prevent spalling and reinforce shear strength, and countermeasure construction has already been carried out. In addition, we will continue to monitor the deformation situation and continue management such as measurement.

10 subway tunnel (linear structure)
100 Structural calculation device 20 Axial force calculation means 30 Result acquisition means

Claims (6)

A structural calculation method for performing structural calculation of a linear structure in consideration of deformation occurring in a part of the linear structure in the axial direction,
Based on the structural information of the linear structure and the deformation information of the deformation, the extension of the linear structure due to the neutral axis position fluctuation caused by the neutral axis position fluctuation due to the deformation is an axial force calculation step of calculating the axial force as an apparent axial force when it is assumed that the linear structure is constrained from both sides in the axial direction of the deformation occurrence point;
a result acquisition step of obtaining a structural calculation result by performing structural calculation of the linear structure based on the structural information of the linear structure, the deformation information of the deformation, and the apparent axial force;
Structural calculation methods, including The neutral axis position variation factor is cracks in concrete,
The structure calculation method according to claim 1. The linear structure is a tunnel consisting of a large number of segmented rings arranged in the axial direction,
The neutral axis position variation factor is the opening of the segment ring,
The structure calculation method according to claim 1. In the axial force calculation step,
Based on the deformation information of the deformation, a range in which the deformation direction side end at the deformation occurrence point is in tension is set as an axial force calculation range,
Based on the structural information of the linear structure and the deformation information of the deformation, the difference in elongation before and after the neutral axis position change of the end on the deformation direction side in the axial force calculation range is regarded as the amount of elongation to be restrained. calculated,
Based on the constrained elongation amount and the axial force calculation range, calculate the constrained strain amount of the deformation direction side end,
calculating the apparent axial force based on the constrained strain amount;
The structural calculation method according to any one of claims 1 to 3. In the result obtaining step,
Giving forced displacement corresponding to the apparent axial force to both axial ends of the structural model of the linear structure,
After that, give a forced displacement corresponding to the deformation,
The structural calculation method for a linear structure according to any one of claims 1 to 4. A structural calculation program for causing a computer to execute each step of the structural calculation method according to any one of claims 1 to 5.

Images