A method for calculating the safety factor of bearing capacity of tunnel lining section with cracks

Abstract

This invention discloses a method for calculating the safety factor of the bearing capacity of a tunnel lining section with cracks. The method includes calculations for two scenarios: full-section compression and partial compression / tension of the lining section. Specifically, for both scenarios, the ultimate bearing capacity and axial force of the lining section are calculated, and the respective safety factors are obtained and compared with the standard safety factor thresholds. The calculation results are then verified through field monitoring, thus demonstrating the method's scientific validity.

Description

Technical Field

[0001] This invention belongs to the field of tunnel engineering technology, specifically relating to a method for calculating the safety factor of the bearing capacity of a tunnel lining section with cracks. Background Technology

[0002] Currently, the secondary lining of tunnels both domestically and internationally mainly uses reinforced concrete cast-in-place lining. During long-term service, tunnel linings inevitably crack and enter a state of operation with joints. However, even in this state, the reinforced concrete lining still possesses a certain load-bearing capacity. While operating reinforced concrete linings with joints is now the norm in tunnel service, a unified understanding has not yet been reached regarding how to evaluate the safety of this jointed operation phase.

[0003] Currently, methods for calculating the safety factor of reinforced concrete tunnel linings in their intact, uncracking stage and plain concrete tunnel linings in their cracked stage are relatively mature. However, methods for calculating the safety factor of reinforced concrete tunnel linings in the cracked stage are still lacking. Due to the large degree of cracking in the lining section, the structural bearing capacity and safety factor calculation methods recommended in the specifications are no longer applicable to the calculation of cracked sections, making it difficult to accurately quantify the degree of structural damage and safety. Therefore, developing a method for calculating the safety factor of cracked reinforced concrete tunnel linings is of great value for evaluating the safety and durability of cracked linings in service tunnels and can provide an important basis for tunnel operation and maintenance decisions. Summary of the Invention

[0004] This invention proposes a method for calculating the safety factor of the bearing capacity of the cross-section of tunnel lining with cracks, which fills the gap in the existing methods for calculating the safety factor of reinforced concrete tunnel lining in the cracking stage and provides important value for evaluating the safety and durability of cracked linings in service tunnels.

[0005] The present invention adopts the following technical solution:

[0006] A method for calculating the safety factor of the bearing capacity of a tunnel lining section with cracks includes the following steps:

[0007] The calculation section is treated as a rectangular doubly reinforced section, and a calculation model is established.

[0008] Calculation of axial force N in a rectangular doubly reinforced section:

[0009] Ultimate bearing capacity N of a rectangular doubly reinforced section max calculate:

[0010] Calculate N and N for two cases: full cross-section compression and one-sided compression and one-sided tension. max :

[0011] The safety factor K1 for the bearing capacity of the reinforced concrete rectangular section was calculated.

[0012] In the above method for calculating the safety factor of the bearing capacity of the tunnel lining section with cracks, in step [2], the axial force N of the rectangular doubly reinforced section is:

[0013] N = C + T S ′-T S

[0014] Where C represents the pressure exerted on the concrete; T′ s The pressure at the compression point of the reinforcing steel; T s This refers to the tensile force at the point where the steel reinforcement is under tension.

[0015] In the above method for calculating the safety factor of the bearing capacity of the tunnel lining section with cracks, the ultimate bearing capacity N of the rectangular doubly reinforced section in step [3] is... max for:

[0016] N max =f cd bx+(f s ′ d -f sd A s

[0017] In the formula f cd —Design value of axial compressive strength of concrete, f s ′ d —Design value of compressive strength of steel reinforcement, f sd —Design value of tensile strength of steel reinforcement, b—width of cross section (m), x—height of concrete compression zone (m).

[0018] In the above method for calculating the safety factor of the bearing capacity of the tunnel lining section with cracks, the condition is as follows: when the lining section with cracks is under full-section compression.

[0019]

[0020] In the formula: R a —Ultimate compressive strength of concrete; R′ s —Standard value of compressive strength of steel reinforcement; e—Distance from the point of application of axial force to the point of application of the resultant force of the lower compressed steel reinforcement; A′ s —Cross-sectional area of ​​the reinforcement in the compression zone, in m² 2 ;a′ s —Reinforcing bar A′ s The distance from the centroid of the section to the nearest edge of the section is m; h—section height, m; h0—effective section height, m, h0 = ha;

[0021]

[0022] In the formula: σ c1 —The smaller compressive stress on the concrete; σ c2 —The larger compressive stress on the concrete; σs σ′ s —The compressive stress on the reinforcing bars on both sides; A s —Cross-sectional area of ​​the reinforcement in the compression zone, in m² 2 .

[0023] In the above method for calculating the safety factor of the bearing capacity of the tunnel lining section with cracks, the case is when the lining section with cracks is under compression on one side and tension on the other side.

[0024]

[0025] In the formula: R l —Ultimate tensile strength of concrete; R s —Standard value of tensile strength of reinforcing steel; l—Height of concrete in the tension zone; x n —Height of concrete in the compression zone; A s —Cross-sectional area of ​​the reinforcement in the compression zone, in m² 2 ;

[0026]

[0027] In the formula: σ c1 —Tensile stress on the concrete; σ c2 —The compressive stress on the concrete; σ s σ′ s —Tensile stress and compressive stress on the reinforcing steel; A s —Cross-sectional area of ​​the reinforcement in the compression zone, in m² 2 a—crack depth, m.

[0028] In the above method for calculating the safety factor of the bearing capacity of the tunnel lining section with cracks, the formula for calculating the safety factor K1 of the reinforced concrete rectangular section is as follows:

[0029]

[0030] Where K is the safety factor; N is the axial force; N max This refers to the ultimate bearing capacity of the cross-section of the lining with cracks.

[0031] In the above method for calculating the safety factor of the bearing capacity of the tunnel lining section with cracks, in step [2], the method for calculating the pressure C on the concrete is as follows:

[0032]

[0033] Pressure T at the compression reinforcement s The calculation method for ′ is as follows:

[0034]

[0035] In the formula, A s′ represents the cross-sectional area of ​​the compressed steel reinforcement, a s ′ represents the distance from the compression reinforcement to the edge of the section, α Es α is the conversion factor for the cross-section of reinforced concrete members. Es =E s / E c ;

[0036] Tensile force T at the tension reinforcement s The calculation method is as follows:

[0037]

[0038] In the formula, A s Let a be the cross-sectional area of ​​the tensile reinforcement. s This is the distance from the tension reinforcement to the edge of the section.

[0039] In the above method for calculating the safety factor of bearing capacity of tunnel lining section with cracks, in step [2], the following assumptions are made when calculating the internal force of the cracked lining section: (1) the section strain remains in the plane; (2) the tensile strength of concrete is not considered; (3) before reaching the ultimate strength, the stress-strain relationship of concrete and steel is proportional.

[0040] In the above method for calculating the safety factor of the bearing capacity of the tunnel lining section with cracks, in step [3], the ultimate bearing capacity N of the section is calculated. max When the stress in the compressed concrete zone is equivalent to a rectangular stress diagram, the following conditions must be met: (1) the area of ​​the equivalent rectangular figure is equal to the area of ​​the theoretical stress figure, that is, the total pressure of the concrete remains unchanged; (2) the centroid of the area of ​​the equivalent rectangular figure coincides with the centroid of the area of ​​the theoretical stress figure, that is, the position of action remains unchanged.

[0041] The beneficial technical effects of the present invention are as follows:

[0042] I. This invention calculates the safety factor for reinforced concrete tunnel linings in the cracking stage. Due to the large extent of cracking in the lining section, the recommended methods for calculating structural bearing capacity and safety factors are no longer applicable to cracked sections, making it difficult to accurately quantify the degree of structural damage and safety. Therefore, constructing a method for calculating the safety factor of cracked reinforced concrete tunnel linings is of great value for evaluating the safety and durability of cracked linings in service tunnels and can provide important basis for tunnel operation and maintenance decisions.

[0043] Second, this invention divides the stress conditions of the lining section into two cases: full-section compression and outer compression and inner tension. The safety factor of reinforced concrete tunnel lining is calculated separately for each case, which is more scientific and reasonable. Attached Figure Description

[0044] Figure 1 This is a schematic diagram of the cracked lining of the present invention;

[0045] Figure 2 This is a schematic diagram of the seamed cracked lining of the present invention under full-section compression;

[0046] Figure 3 This is a schematic diagram of the seam-lined cracked lining of the present invention being subjected to compression on one side and tension on the other.

[0047] Figure 4 (a) is a schematic diagram of the stress-strain relationship of reinforced concrete structure materials under concrete compression;

[0048] Figure 4 (b) is a schematic diagram of the stress-strain relationship of reinforced concrete structure material under tension of steel bars;

[0049] Figure 5 (a) is a schematic diagram of the cross-sectional dimensions of the reinforced concrete structure of the present invention;

[0050] Figure 5 (b) is a schematic diagram of strain distribution in the reinforced concrete structure of the present invention;

[0051] Figure 5 (c) is a schematic diagram of the stress distribution in the reinforced concrete structure of the present invention;

[0052] Figure 6 (a) is a schematic diagram of the stress distribution in the compression zone of the concrete according to the present invention;

[0053] Figure 6 (b) is a schematic diagram of the stress distribution of the equivalent rectangular concrete according to the present invention;

[0054] Figure 7 This is a field observation diagram showing the situation under full-section compression according to the present invention;

[0055] Figure 8 This is a field observation diagram showing the situation under the condition of one side being compressed and the other side being tensile according to the present invention; Detailed Implementation

[0056] Figure 1 A structural schematic diagram of tunnel lining cracks is given. It can be seen from the diagram that the reinforcing bars on the inner and outer sides of the concrete are symmetrically distributed. When the tunnel is subjected to pressure around it, the concrete with lining cracks will exhibit two different stress conditions: being compressed on both the inner and outer sides and being compressed on one side and tensile on the other.

[0057] The Code for Design of Concrete Structures (GB50010-2010) adopts the following standards: Figure 4 (a) and 4(b) express the stress-strain relationship between concrete and steel reinforcement. The stress-strain relationship curve of concrete consists of a parabola with an ascending segment and a horizontal segment, and its expression is:

[0058]

[0059] In the formula: σ c —The compressive strain of concrete is ε c The compressive stress of concrete at that time;

[0060] f cd —Design value of axial compressive strength of concrete;

[0061] ε c0 —The compressive stress in the concrete reaches f cd The compressive strain of concrete, ε c0 =0.002+0.5(f cu,k -50)×10 -5 When the calculated value is less than 0.002, it is taken as 0.002, f cu,k This refers to the standard value of the compressive strength of a concrete cube.

[0062] ε cu —Ultimate compressive strain of concrete;

[0063] n—exponent, n = 2 - (f) cu,k -50) / 60, when n is greater than 2.0, take n = 2.0.

[0064] The stress-strain relationship of reinforcing steel exhibits a distinct yield step, and a simplified ideal elastic-plastic stress-strain relationship can be used, as expressed below:

[0065]

[0066] In the formula: σ s —The strain of the steel reinforcement is ε c The stress on the reinforcing steel at that time;

[0067] f sd —Design value of tensile strength of steel reinforcement;

[0068] ε y —The stress in the reinforcing steel reaches f sd The strain of the steel reinforcement

[0069] E s — Elastic modulus of steel reinforcement

[0070] To calculate the internal forces in the cross-section of the cracked lining structure, the following assumptions are made:

[0071] (1) The cross-sectional strain remains planar;

[0072] (2) The tensile strength of concrete is not considered;

[0073] (3) Before reaching the ultimate strength, the stress-strain relationship of concrete and steel reinforcement is directly proportional.

[0074] Based on the assumptions, we can conclude that the compressive strain of concrete at the horizontal position of the compressed steel reinforcement is directly proportional to the stress, i.e.:

[0075] σ c ′=ε c ′E c (3)

[0076] At the same time, the tensile strain of the concrete at the horizontal position of the tensioned reinforcement is proportional to the stress, that is:

[0077] σ c =ε c E c (4)

[0078] Calculate the cross-sectional dimensions, strain, and stress distribution as follows: Figure 5 As shown in the figure. In the figure, h is the cross-sectional height, b is the cross-sectional width, and A... s Let A be the cross-sectional area of ​​the tensile reinforcement. s ′ represents the cross-sectional area of ​​the compressed steel reinforcement, a s a is the distance from the tensile reinforcement to the edge of the section. s ′ is the distance from the compression reinforcement to the edge of the section, h0 is the effective height of the section, and x n This refers to the height of the concrete compression zone.

[0079] According to the principle of similar triangles, by Figure 5 (b) The strain distribution of the cross section can be used to obtain the strain of the compressed steel reinforcement:

[0080]

[0081] Similarly, the strain of the tensile reinforcement can be obtained:

[0082]

[0083] Tensile force T at the tension reinforcement s for:

[0084] T s =σ s A s (7)

[0085] Substituting equations (4) and (6) into the above equation, we get:

[0086]

[0087] In the formula, α Es α is the conversion factor for the cross-section of reinforced concrete members. Es =E s / E c .

[0088] Similarly, the pressure T at the compression reinforcement can be obtained. s 'for:

[0089]

[0090] The pressure C on the concrete is:

[0091]

[0092] For a rectangular doubly reinforced section, according to the force equilibrium condition, taking the sum of the internal and external forces along the longitudinal direction of the member as 0, the axial force N can be obtained as:

[0093] N = C + T S ′-T S (11)

[0094] For secondary lining structures, the reinforcing bars are generally arranged symmetrically, hence:

[0095]

[0096] Substituting equations (8), (9), (10), and (12) into equation (11), we obtain:

[0097]

[0098] Under ultimate stress conditions, the concrete stress in the compression zone reaches the design value of axial compressive strength, and the stress in the tensile reinforcement also reaches the design value of tensile strength. According to the relevant provisions of the "Code for Design of Concrete Structures" (GB50010-2010) and the "Code for Design of Highway Tunnels" (JTG3370.1-2018) regarding the strength verification of rectangular reinforced concrete members, concrete cannot be considered an elastic material at this point. The actual stress distribution of the cross-section is as follows: Figure 6 As shown in (a), the concrete stress in the compression zone is equivalent to a rectangular stress diagram, satisfying the following conditions: (1) the area of ​​the equivalent rectangular diagram is equal to the area of ​​the theoretical stress diagram, i.e., the total concrete pressure remains unchanged; (2) the centroid of the equivalent rectangular diagram coincides with the centroid of the theoretical stress diagram, i.e., the position of application remains unchanged. The rectangular stress distribution of the concrete in the compression zone after the equivalent stress diagram is shown in the figure. Figure 6 As shown in (b), the equivalent rectangular stress diagram can be determined by two dimensionless eigenvalues ​​α and β. The pressure C on the equivalent concrete is:

[0099] C=αf cd bx(14)

[0100] The equivalent rectangular stress diagram is calculated using the following formula to determine the height x of the compression zone:

[0101] x=βx n (15)

[0102] According to the "Code for Design of Concrete Structures" (GB50010-2010), for concrete with a strength grade lower than or equal to C50, α = 1.0 and β = 0.8.

[0103] Substituting equations (12), (14), and (15) into equation (11), we can obtain the ultimate bearing capacity N of the section. max for:

[0104] N max =f cd bx+(f s ′ d -f sd A s (16)

[0105] In the formula: f s ′ d —Design value of compressive strength of steel bars.

[0106] Through The safety factor K1 of the cross-sectional bearing capacity can be obtained. When K1 is greater than the safety threshold given by the code, the lining structure is considered safe; otherwise, the bearing capacity is considered to have reached the ultimate state and the structure tends to fail.

[0107] When the lining cracks, the height of the compression zone decreases, inevitably reducing the bearing capacity of the lining section. Therefore, it is necessary to calculate the ultimate bearing capacity of the cracked lining. When calculating the stress on the secondary tunnel lining section, based on the stress characteristics of the secondary lining, a finite-length lining structure is typically approximated as a rectangular doubly reinforced eccentrically compressed member for calculation. For a rectangular eccentrically compressed member, regardless of whether the failure is caused by large or small eccentric compression, the concrete in the compression zone reaches its ultimate compressive strain, and the same-side reinforcing steel A... s Generally, it can reach the design value of compressive strength f. s ′ d The reinforcement on the other side may be under tension or compression. Therefore, the stress condition of the cross-section can be divided into full-section compression and one-sided compression and one-sided tension. Refer to the code for the strength calculation method of eccentrically compressed rectangular cross-section members. When e≤0.2h, the cross-section is under full compression; otherwise, a tension zone exists in the cross-section. The stress distribution of the two types of rectangular cross-sections is as follows: Figure 2 and Figure 3 As shown.

[0108] (1) Full cross-section under compression

[0109] After the lining concrete cracks, the cracks tend to close under pressure, and the cracks can still withstand axial pressure under compression. Therefore, the entire cross-section can still be calculated as a complete cross-section under compression. The ultimate bearing capacity of the cross-section can be calculated using the formula given in the tunnel design code:

[0110]

[0111] Where: N—axial force at the lining section, MN;

[0112] R a —Ultimate flexural compressive strength of concrete;

[0113] R′ s —The compressive strength of the reinforcing steel;

[0114] e—the distance from the centroid of the compressed steel bar to the point of application of the axial force, in meters;

[0115] A′ s —Cross-sectional area of ​​the tension zone reinforcement, in m² 2 ;

[0116] a′ s —Reinforcing bar A′ s The distances from the centroid of each element to the nearest edge of the cross section are m.

[0117] h—section height, m;

[0118] h0—effective height of the cross section in m, h0 = ha;

[0119] b—cross-sectional width, in meters;

[0120] The stress distribution of the rectangular doubly reinforced section of the lining under full-section compression is as follows: Figure 2 As shown, the height x of the concrete compression zone n =h, the compressive stress σ on the concrete c =σ c1 +σ c2 Because the strains of the reinforcing steel and the concrete are equal, the compressive stress on the reinforcing steel is... Substituting the above equation into equation (13), we get:

[0121]

[0122] In the formula: σ c1 —The relatively small compressive stress on the concrete;

[0123] σ c2 —The concrete is subjected to significant compressive stress;

[0124] σ s σ′ s —The compressive stress on the steel bars on both sides;

[0125] A s —Cross-sectional area of ​​the reinforcement in the compression zone, in m² 2 .

[0126] (2) One side is under compression and the other side is under tension

[0127] It is unreasonable to ignore the tensile strength of concrete when it is subjected to compression on one side and tension on the other. After the lining cracks, the concrete at the crack is no longer functional, and the concrete height l in the tension zone is reduced from hx. n Become hx n -a, where a is the crack depth. At this point... Substituting the above equation into equation (16), we can obtain the ultimate bearing capacity of the section:

[0128]

[0129] In the formula: R l —Ultimate tensile strength of concrete;

[0130] R s —Standard value of tensile strength of steel bars;

[0131] The stress distribution of a rectangular doubly reinforced section of a lining under conditions of compression on one side and tension on the other is as follows: Figure 3 As shown, the compressive stress on the concrete Compressive stress on steel bars Substituting the above equation into equation (13), we get:

[0132]

[0133] In the formula: σ c1 —Tensile stress on concrete;

[0134] σ c2 —The compressive stress on the concrete;

[0135] σ s σ′ s —Tensile stress and compressive stress on the reinforcing bars;

[0136] A s —Cross-sectional area of ​​the reinforcement in the compression zone, in m² 2 ;

[0137] a—crack depth, m;

[0138] x n — Concrete height in the compression zone.

[0139] Using the Lianchengshan Tunnel on the Baoji-Hanzhong Expressway in Shaanxi Province as an example, this paper verifies and explains the correctness of the calculation method for the bearing capacity safety factor of the tunnel lining section with cracks proposed in this invention. The surrounding rock of the Hanzhong section of the Lianchengshan Tunnel is mainly chlorite schist, which has strong water-softening and compression deformation characteristics. During construction, it encountered severe compression deformation disasters, and multiple cracks appeared in the lining during operation. To ensure the safety of tunnel construction and operation, a comprehensive health monitoring system was constructed. Taking section ZK197+129 as an example, long-term measured data from this monitoring section shows that both the inner and outer sides of the secondary lining reinforcement and concrete are under pressure. In the 6th year of service (December 2023), the secondary lining cracks appeared as diagonal cracks extending from the sidewall to the foot of the wall. Figure 7 The safety factor K1 of the bearing capacity of the sidewall section is calculated based on the full cross-section under compression.

[0140] Under full-section compression, after the lining concrete cracks, the cracks tend to close under pressure, and the cracks can still withstand axial pressure under compression. Therefore, under full-section compression, it can still be calculated as a complete section. The ultimate bearing capacity of the section is calculated according to formula (17). The steel bars on both sides are symmetrically arranged, e is 0.16m, the section width b is 1m, the height of the concrete compression zone x = h0, the thickness of the concrete cover d is 6cm, the section height h is 80cm, the ultimate compressive strength of C35 concrete is 26.3MPa, the ultimate flexural compressive strength is 32.9MPa, the ultimate compressive strength of HPB400 steel bar is 540MPa, and the diameter of the steel bar is 28mm. Substituting the above parameters into formula (17) yields N. max =31250.03KN.

[0141] The axial force of the rectangular doubly reinforced section of the sidewall lining under full-section compression is calculated according to formula (18). Where σ c1 =5.303MPa, σ c2 =5.076MPa, σ s =56.419MPa, σ′ s =56.936MPa. Substituting the above parameters into equation (18), we get N =4221.4KN.

[0142] The calculated safety factor K1 for the cross-sectional bearing capacity of the sidewall is 7.40, which is higher than the safety factor threshold of 2.0 given in the standard, indicating that the tunnel structure is safe.

[0143] Long-term observations from the tunnel health monitoring system show that, after six years of service, the width of the diagonal cracks extending from the sidewall to the base of the wall in the monitored section remained unchanged, and the cracks did not propagate. Simultaneously, stress monitoring results of the secondary lining indicate that concrete stress and steel reinforcement stress remained essentially stable, without any increasing trend. Combining the secondary lining crack width monitoring results and stress monitoring results, it is clear that the secondary lining structure is stable and safe, consistent with the results obtained from the calculation method of this invention. The safety factor of the secondary lining is still within the allowable safety range, and the existing lining cracks do not affect structural safety; the tunnel lining can still serve normally.

[0144] Taking section ZK196+758 as an example, long-term measured data from this monitoring section shows that the inner side of the secondary lining reinforcement and concrete is under compression, while the outer side is under tension. In the 6th year of service (August 2023), the secondary lining cracks appeared as diagonal cracks extending from the sidewall to the foot of the wall. Figure 8 The safety factor K1 of the bearing capacity of the section at the foot of the wall is calculated based on the condition that one side is under compression and the other side is under tension.

[0145] Under the condition of lateral compression and lateral tension, the ultimate bearing capacity of the section is calculated according to formula (19). The reinforcement on both sides is symmetrically arranged, the section width b is taken as 1m, and the height of the concrete compression zone x... n =56cm, the height of the tensile zone of concrete l = 56cm, the thickness of the concrete cover d is 6cm, the section height h is 80cm, the ultimate compressive strength of C35 concrete is 26.3MPa, the ultimate tensile strength of C35 concrete is 2.47MPa, the ultimate compressive strength of HPB400 steel is 540MPa, the ultimate tensile strength of HPB400 steel is 540MPa, and the diameter of the steel is 28mm. Substituting the above parameters into equation (19) yields N. max =7191.1KN.

[0146] The axial force of a rectangular doubly reinforced section of a wall corner lining under compression on one side and tension on the other side is calculated according to formula (20). Where σ c1 =1.623MPa, σ c2 =6.6083MPa, σ s =1.4454MPa, σ′ s =3.784MPa. Substituting the above parameters into equation (20), we get N =1738.154KN.

[0147] The calculated safety factor K1 for the cross-sectional bearing capacity of the corner is 4.14, which is higher than the safety factor threshold of 2.0 given in the standard, indicating that the tunnel structure is safe.

[0148] Long-term observations from the tunnel health monitoring system show that, after six years of service, the width of the diagonal cracks extending from the sidewall to the base of the wall in the monitored section remained unchanged, and the cracks did not propagate. Simultaneously, stress monitoring results of the secondary lining indicate that the concrete stress and steel reinforcement stress remained relatively stable, showing no increasing trend. Combining the secondary lining crack width monitoring results and stress monitoring results, it is clear that the secondary lining structure is stable and safe, consistent with the results obtained from the calculation method of this invention. The safety factor of the secondary lining is still within the allowable safety range, and the existing lining cracks do not affect structural safety; the tunnel lining can still serve normally. Therefore, this invention demonstrates that the method for calculating the safety factor of the bearing capacity of a cracked lining section proposed in this invention is correct and reliable.

Claims

1. A method for calculating the safety factor of the bearing capacity of a tunnel lining section with cracks, characterized in that, Includes the following steps: [1] Treat the calculation section as a rectangular doubly reinforced section and establish a calculation model; 【2】Calculation of axial force N in a rectangular doubly reinforced section: N=C+T′ S -T S Where C represents the pressure exerted on the concrete; T′ s The pressure at the compression point of the reinforcing steel; T s This refers to the tensile force at the point where the steel reinforcement is under tension. [3] Ultimate bearing capacity N of a rectangular doubly reinforced section max calculate: N max =f cd bx+(f′ sd -f sd )A s In the formula f cd —Design value of axial compressive strength of concrete; f′ sd —Design value of steel bar compressive strength; f sd —Design value of tensile strength of steel reinforcement; b—Cross-section width, m; x—Height of concrete compression zone, m; A s —The cross-sectional area of ​​the tensile reinforcement; [4] Calculate N and N according to the two cases of full-section compression and one-sided compression and one-sided tension respectively. max : When the lining section with cracks is under full-section compression; In the formula: R a —Ultimate compressive strength of concrete; R′ s —Standard value of compressive strength of steel reinforcement; e—Distance from the point of application of axial force to the point of application of the resultant force of the lower compressed steel reinforcement; A′ s —Cross-sectional area of ​​the reinforcement in the compression zone, in m² 2 ;a′ s —Reinforcing bar A′ s The distance from the centroid to the nearest edge of the cross section, m; h—cross section height, m; h0—effective cross section height, m, h0 = ha; a—crack depth, m; In the formula: σ c1 —Minimum compressive stress on concrete; σ c2 —The maximum compressive stress on the concrete; σ s σ′ s —The compressive stress on the reinforcing bars on both sides; A′ s —Cross-sectional area of ​​the reinforcement in the compression zone, in m² 2 ; When the lining section with seams is under compression on one side and tension on the other; In the formula: R l —Ultimate tensile strength of concrete; R s —Standard value of tensile strength of reinforcing steel; l—Height of concrete in the tension zone; x n —Height of concrete in the compression zone; R a —Ultimate flexural compressive strength of concrete; R′ s —The compressive strength of the reinforcing steel; In the formula: σ c1 —Tensile stress on the concrete; σ c2 —The compressive stress on the concrete; σ s —The tensile stress on the reinforcing steel; σ′ s —The compressive stress on the reinforcing steel; [5] The safety factor K1 of the bearing capacity of the reinforced concrete rectangular section is calculated.

2. The method for calculating the safety factor of the bearing capacity of a tunnel lining section with cracks according to claim 1, characterized in that, The formula for calculating the safety factor K1 of the bearing capacity of a reinforced concrete rectangular section is as follows: Where K is the safety factor; N is the axial force; N max This refers to the ultimate bearing capacity of the cross-section of the lining with cracks.

3. The method for calculating the safety factor of the bearing capacity of a tunnel lining section with cracks according to claim 1, characterized in that, In step [2], the method for calculating the pressure C on the concrete is as follows: σ c —Concrete compressive stress; b—Liner cross-sectional width, m; x n —Height of concrete in the compression zone; Pressure T at the compression reinforcement s The calculation method for ′ is as follows: In the formula, a s ' is the distance from the compression reinforcement to the edge of the section, α Es α is the conversion factor for the cross-section of reinforced concrete members. Es =E s / E c E s —Elastic modulus of steel reinforcement; E c —Elastic modulus of concrete; Tensile force T at the tension reinforcement s The calculation method is as follows: In the formula A s —The cross-sectional area of ​​the tensile reinforcement.

4. The method for calculating the safety factor of the bearing capacity of a tunnel lining section with cracks according to claim 1, characterized in that, In step [2], the following assumptions are made when calculating the internal forces of the cracked lining structure section: (1) the section strain remains in the plane; (2) the tensile strength of concrete is not considered; (3) before reaching the ultimate strength, the stress-strain relationship of concrete and steel is proportional.

5. The method for calculating the safety factor of the bearing capacity of a tunnel lining section with cracks according to claim 1, characterized in that, In step [3], when calculating the ultimate bearing capacity Nmax of the section, the concrete stress in the compression zone is equivalent to a rectangular stress diagram, and the following conditions are met: (1) the area of ​​the equivalent rectangular figure is equal to the area of ​​the theoretical stress figure, that is, the total pressure of the concrete remains unchanged; (2) the centroid of the area of ​​the equivalent rectangular figure coincides with the centroid of the area of ​​the theoretical stress figure, that is, the position of action remains unchanged.

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