A reliability analysis method for reinforced concrete columns

Abstract

The invention relates to a reliability analysis method for reinforced concrete columns, and belongs to the technical field of reliability analysis of engineering structures. The method comprises the following steps: importing design parameters into a design formula of the reinforced concrete column and performing dimensionless transformation, obtaining a plurality of relative compression zone heights according to the probability characteristics of each action effect coefficient, then correspondingly selecting a function function of the reinforced concrete column, and solving the function function to obtain a reliability probability, a failure probability, and a reliability index; the invention performs dimensionless form transformation on the design formula of the reinforced concrete column, fully considers the failure mode of the reinforced concrete column, calculates the change of its function function with conditions, improves the accuracy of the reliability analysis result, and accurately reveals the reliability control level of the reinforced concrete column design method, especially for the situation where the relative compression zone height of the cross section in the design is close to the limit failure; the analysis method of the invention has high accuracy, a wide range of applications, strong practicality, and is worthy of promotion.

Description

Technical Field

[0001] The invention belongs to the technical field of engineering structure reliability analysis, and particularly relates to a reliability analysis method for reinforced concrete columns. Background Art

[0002] Reinforced concrete columns are currently the primary load-bearing components in building structures. Seismic damage surveys and statistics on building structures show that failure is mostly concentrated at the column ends, with the strong-column-weak-beam yield mechanism rarely occurring. While this phenomenon is certainly related to the greater destructive power of earthquakes, the high incidence of RC frame column failure is also related to the low design reliability of their bearing capacity. To improve the safety of building structures, reliability analysis of reinforced concrete compressive components is necessary. Current reliability analyses mostly focus on components in civil buildings or hypothetical components without engineering background. Furthermore, these analyses often rely on enumeration of selected engineering examples, limiting the reliability level of reinforced concrete columns to typical design scenarios. Furthermore, the actual failure mode of components does not necessarily correspond to the designed mode. Previous studies using the JC method have only selected a single functional function, oversimplifying the reliability analysis process for reinforced concrete columns and resulting in inaccurate reliability analysis results. Summary of the Invention

[0003] In view of this, the present invention provides a reliability analysis method for reinforced concrete columns to address the deficiencies in the prior art.

[0004] The technical solution of the present invention is: a reliability analysis method for reinforced concrete columns, comprising the following steps:

[0005] S1. Select a set of design parameters for reinforced concrete columns based on the reinforcement ratio ρ;

[0006] S2. Import the selected design parameters into the reinforced concrete column design formula, and then perform dimensionless form transformation to obtain formula (1):

[0007]

[0008] Among them, v N is the axial pressure ratio, v M is the relative bending moment, γ0 is the structural importance coefficient, γ0 is 0.9-1.1, γ G is the dead load partial factor, γ G is 1.3, h0 is the effective height of the section, G k is the standard value of constant load, f c is the design value of concrete axial compressive strength, b and h are the cross-sectional dimensions of reinforced concrete columns, is the live load partial factor, is 1.5, γ Li , γ Lj is the design service life variable load adjustment factor, γ Li , γ Lj is 0.9~1.1, a0, a i 、a j , b0, b i 、b j is the effect coefficient, Q k (i≠j) is the live load, i, j are serial numbers, ψ cj is the load combination coefficient, ψ cj is 0.7;

[0009] S3. According to the probability characteristics of each action effect coefficient in formula (1), obtain N random values ​​v of the axial force of the reinforced concrete column section N , and then N random values ​​v N Import them into formula (2) to obtain N relative compression zone heights ξ:

[0010]

[0011] Wherein, α1 is a coefficient. When the concrete strength does not exceed C50, α1 is 1.0. When the concrete strength is C80, α1 is 0.94. The values ​​in between are obtained by linear interpolation.

[0012] S4. Import the obtained N relative pressure zone heights ξ into formula (3) respectively to obtain N Z values;

[0013]

[0014]

[0015] Among them, f y 、f′ y are the tensile and compressive strength values ​​of steel bars, respectively, and a′ s is the distance from the resultant point of the longitudinal ordinary steel bars in the compression zone to the compression edge of the section, η s is the bending moment amplification factor, e is the distance from the axial pressure point to the resultant force point of the longitudinal tensile reinforcement, ξ b is the height of the relative limit compression zone, e′ is the distance from the axial pressure action point to the resultant force point of the longitudinal reinforcement in the compression zone, e i is the initial eccentricity; e i =e0+e a , e0 is the eccentricity of the axial pressure to the center of gravity of the section, taken as e a The additional eccentricity is the maximum value of 20mm and 1 / 30 of the cross-sectional dimension in the eccentric direction.

[0016] S5. Count the number of Z values ​​greater than or equal to 0 and record it as N r , and then N, N r The reliability probability p of reinforced concrete columns is obtained by Monte Carlo simulation by introducing equations (4) and (5) respectively. r , failure probability p f ;

[0017] p r =N r / N (4)

[0018] p f =1-p r (5)

[0019] S6. Obtain the failure probability p of reinforced concrete columns f , import formula (6) to obtain the reliability index β of reinforced concrete column:

[0020] β=Φ(p f ) (6)

[0021] Where Φ(·) is the standard normal distribution function.

[0022] Preferably, the design parameters of the reinforced concrete column include the concrete strength f c , steel bar strength f y 、f′ y , cross-sectional dimensions b, h, constant load G k , live load Q k , bending moment increase coefficient η s .

[0023] Preferably, selecting a set of design parameters of reinforced concrete columns according to the reinforcement ratio ρ comprises the following steps:

[0024] Based on the design parameters of reinforced concrete columns, the total axial compression ratio v is obtained using formula (1): N ;

[0025] Based on the total axial pressure ratio v N , use formula (2) to obtain the relative pressure zone height ξ;

[0026] Based on the obtained relative compression zone height, the reinforcement ratio ρ is obtained using formula (7):

[0027]

[0028]

[0029]

[0030] Compared with the prior art, the reliability analysis method of reinforced concrete columns provided by the present invention transforms the design formula of reinforced concrete columns into a dimensionless form, establishes a functional function of the dimensionless model, fully considers the failure mode of reinforced concrete columns, calculates the changes of their functional function with conditions, improves the accuracy of reliability analysis results, and accurately reveals the reliability control level of the reinforced concrete column design method, especially for the situation where the height of the relative compression zone of the cross section in the design is close to the limit of failure; the analysis method of the present invention has high accuracy of results, a wide range of applications, strong practicality, and is worthy of promotion. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 is a flow chart of the analytical method of the present invention;

[0032] Figure 2 It is a flow chart of selecting design parameters of the present invention. DETAILED DESCRIPTION

[0033] The present invention provides a reliability analysis method for reinforced concrete columns. Figures 1 to 2 The present invention is described with reference to a structural schematic diagram of FIG.

[0034] Example 1

[0035] like Figure 1 As shown, a reliability analysis method for reinforced concrete columns includes the following steps:

[0036] S1. Select a set of design parameters for reinforced concrete columns based on the reinforcement ratio ρ;

[0037] S2. Import the selected design parameters into the reinforced concrete column design formula and change the force design value N of the reinforced concrete column to d , design value of bending moment M d After being expressed by the partial factor design method, the design values ​​of the axial force and bending moment of the reinforced concrete column member section expressed by the partial factor design method fully consider special issues such as the correlation between the bending moment and axial force and basic variables, and the random change of eccentricity;

[0038]

[0039]

[0040] Then, the dimensionless form transformation is performed to obtain formula (1). By performing the dimensionless form transformation on the design formula of reinforced concrete columns, the performance function of the dimensionless model is established, the failure mode of reinforced concrete columns is fully considered, and the variation of its performance function with conditions is calculated, thereby improving the accuracy of the reliability analysis results and accurately revealing the reliability control level of the reinforced concrete column design method, especially for the case where the height of the cross section relative to the compression zone in the design is close to the limit failure;

[0041]

[0042] Among them, v N is the axial pressure ratio, v M is the relative bending moment, γ0 is the structural importance coefficient, γ0 is 0.9-1.1, γ G is the dead load partial factor, γ G is 1.3, h0 is the effective height of the section, G k is the standard value of constant load, f c is the design value of concrete axial compressive strength, b and h are the cross-sectional dimensions of reinforced concrete columns, is the live load partial factor, is 1.5, γ Li , γ Lj is the design service life variable load adjustment factor, γ Li , γ Lj is 0.9~1.1, a0, a i 、a j , b0, b i 、b j is the effect coefficient, Q k (i≠j) is the live load, i, j are serial numbers, ψ cj is the load combination coefficient, ψ cj is 0.7;

[0043] S3. According to the probability characteristics of each action effect coefficient in formula (1), obtain N random values ​​v of the axial force of the reinforced concrete column section N , and then N random values ​​v N Import them into formula (2) to obtain N relative compression zone heights ξ:

[0044]

[0045] Wherein, α1 is a coefficient. When the concrete strength does not exceed C50, α1 is 1.0. When the concrete strength is C80, α1 is 0.94. The values ​​in between are obtained by linear interpolation.

[0046] S4. Import the obtained N relative pressure zone heights ξ into formula (3) respectively to obtain N Z values;

[0047]

[0048]

[0049] Among them, f y 、f′ y are the tensile and compressive strength values ​​of steel bars, respectively, and a′ s is the distance from the resultant point of the longitudinal ordinary steel bars in the compression zone to the compression edge of the section, η s is the bending moment amplification factor, e is the distance from the axial pressure point to the resultant force point of the longitudinal tensile reinforcement, ξ b is the height of the relative limit compression zone, e′ is the distance from the axial pressure action point to the resultant force point of the longitudinal reinforcement in the compression zone, e i is the initial eccentricity; e i =e0+e a , e0 is the eccentricity of the axial pressure to the center of gravity of the section, taken as e a The additional eccentricity is the maximum value of 20mm and 1 / 30 of the cross-sectional dimension in the eccentric direction.

[0050] S5. Count the number of Z values ​​greater than or equal to 0 and record it as N r , and then N, N r The Monte Carlo simulation is introduced into equations (4) and (5) respectively, and the failure probability is calculated using the Monte Carlo method to further improve the accuracy of the reliability analysis results and obtain the reliability probability p of the reinforced concrete column. r , failure probability p f ;

[0051] p r =N r / N (4)

[0052] p f =1-p r (5)

[0053] S6. Obtain the failure probability p of reinforced concrete columns f , import formula (6) to obtain the reliability index β of reinforced concrete column:

[0054] β=Φ(p f ) (6)

[0055] Where Φ(·) is the standard normal distribution function.

[0056] Preferably, the design parameters of the reinforced concrete column include the concrete strength f c , steel bar strength fy 、f′ y , cross-sectional dimensions b, h, constant load G k , live load Q k , bending moment increase coefficient η s .

[0057] like Figure 2 As shown, preferably, the step of selecting a set of design parameters for reinforced concrete columns according to the reinforcement ratio ρ comprises the following steps:

[0058] Based on the design parameters of reinforced concrete columns, the total axial compression ratio v is obtained using formula (1): N ;

[0059] Based on the total axial pressure ratio v N , use formula (2) to obtain the relative pressure zone height ξ;

[0060] Based on the obtained relative compression zone height, the reinforcement ratio ρ is obtained using formula (7):

[0061]

[0062]

[0063]

[0064] This invention uses a reinforced concrete open-air trestle column with a simple load pattern but capable of reflecting the main load characteristics as an example for illustration. When designing reinforced concrete open-air trestle columns for ultimate load capacity, the main consideration is the simultaneous action of permanent loads and crane loads (vertical loads and horizontal loads). Therefore, the following points should be noted when combining loads:

[0065] 1) Consider only the combination of dead load and crane load (vertical load and transverse horizontal load);

[0066] 2) P acting on the same column bracket caused by the vertical load of the crane max and P min , only one of them can be selected when combining;

[0067] 3) Crane lateral horizontal load H max When acting on the column, its direction of action is sometimes to the left and sometimes to the right, but in the analysis only one of the directions is taken into account in the combination;

[0068] 4) Within the same span, when combining the internal forces generated by vertical loads, it is not necessary to combine the internal forces generated by transverse horizontal loads. When combining the internal forces generated by transverse horizontal loads, it is necessary to combine the internal forces generated by vertical loads. For example, transverse horizontal loads only generate bending moments but not axial forces, so they participate in the combination of bending moments but not axial forces.

[0069] 5) Since it is unlikely that multiple cranes are fully loaded at the same time and the trolleys are in the most unfavorable position at the same time, the Code for Loads on Building Structures (GB 50009-2012) stipulates that when multiple cranes are combined, the standard values ​​of the crane loads (vertical loads and lateral horizontal loads) should be reduced.

[0070] Example 2

[0071] The present invention will be further described with reference to the reinforced concrete columns under industrial plants.

[0072] The reinforced concrete column is designed as follows: the concrete strength grade is C30, the steel bar type is HRB400, the cross-sectional dimensions are b×h=400mm×600mm, the column height h0 is 7.2m, and the load combination is a combination of crane load and floor dead load. The concrete strength and steel bar strength both obey the normal distribution, and the mean and coefficient of variation are The standard value of floor live load is G K =30kN / m, ρ1 is the ratio of the crane's vertical load to the permanent load, ρ2 is the ratio of the crane's horizontal load to the permanent load, η is the ratio of the crane's width to the crane's beam span, and ε is the ratio of the crane's wheelbase to the crane's width. The specific parameter changes are shown in Table 1.

[0073] (1) From the above parameters, the concrete strength f c , steel bar strength f y 、f′ y , cross-sectional dimensions b, h, constant load G k , live load Q k , bending moment increase coefficient η s Take the value and use formula (1) to obtain the axial pressure ratio v N Then, substitute into formula (2) to obtain the relative compression zone height ξ, and then substitute into formula (7) to obtain the reinforcement ratio ρ;

[0074] (2) The obtained reinforcement ratio ρ and ρ min For comparison, when ρ≥ρ min When ρ<ρ, the design parameters of the reinforced concrete column meet the design requirements, and the reinforced concrete column can be analyzed for reliability. min , the design parameters of the reinforced concrete column do not meet the design requirements, and the design parameters need to be readjusted until the reinforced concrete column can be analyzed for reliability. min 5%;

[0075] (3) Due to the concrete strength f c , steel bar strength f y 、f′ y, cross-sectional dimensions b, h, constant load G k , live load Q k , bending moment increase coefficient η s Generally, it obeys normal distribution, while crane load generally obeys extreme value type I distribution. Therefore, according to the probability model obeyed by each action effect coefficient in formula (1), N random numbers of each action effect coefficient are obtained through MATLAB program;

[0076] (4) By obtaining N random numbers of each effect coefficient, N axial pressure ratios ν are obtained using formula (1) N and relative bending moment ν M The random value of , and then use formula (2) to obtain N relative compression zone heights ξ;

[0077] (5) The obtained N relative pressure zone heights ξ are respectively compared with ξ b Compare and select the corresponding function Z, then substitute the corresponding random number into the function Z and determine the positivity of the function Z, and count the number of Z≥0 and record it as Nr;

[0078] (6) Based on the obtained Nr and N values, the reliability probability p of reinforced concrete column components is obtained by Monte Carlo simulation. r , failure probability p f and the reliability indicator β.

[0079] From (5) in the above embodiment 2, it can be seen that for the design of reinforced concrete columns, the relative height of the cross section relative to the compression zone ξ is not greater than but close to ξ b When the load is designed for large eccentricity but actually suffers from small eccentricity, the probability of this happening is also given in this analysis method. The results are shown in Table 2.

[0080] As shown in Table 2, the ultimate limit state design reliability control level of reinforced concrete open-air trestle columns meets the requirements of my country's "Uniform Standard for Reliability Design of Building Structures" (GB 50068-2018). The maximum value of the design reliability index is 5.61 (heavy-duty system 30 / 5t), and the minimum value is 3.22 (heavy-duty system 5t). The maximum value of the coefficient of variation of the design reliability index is 0.09, and the minimum value is 0.05. The overall mean of the calculated results is 4.65. Large eccentric failure is ductile failure, and the standard stipulates that its target reliability index is 3.2; small eccentric failure is brittle failure, and its target reliability index is 3.7, both of which are less than 4.65. At the same time, in order to study the influence of a certain variable on the calculation results of the reliability index, the present invention adopts the control variable method. Taking the intermediate crane A4-A5 as an example, the statistical proportion of failure types under the change of the ratio ρ1 of the crane vertical load to the permanent load is listed. The results are shown in Table 3.

[0081] Table 1 Value ranges of relevant parameters

[0082]

[0083] Table 2 Reliability results of reinforced concrete open-air trestle columns

[0084]

[0085] Table 3 Proportion of each damage type under the influence of ρ1

[0086]

[0087]

[0088] The above disclosure is only a preferred specific embodiment of the present invention. However, the embodiments of the present invention are not limited thereto. Any changes that can be conceived by those skilled in the art should fall within the scope of protection of the present invention.

Claims

1. A reliability analysis method for reinforced concrete columns, characterized in that: The following steps are involved: S1. According to the reinforcement ratio Select a set of design parameters for reinforced concrete columns; S2. Import the selected design parameters into the reinforced concrete column design formula, and then perform dimensionless transformation to obtain formula (1): （1） in, is the axial pressure ratio, is the relative bending moment, is the structural importance coefficient, 0.9-1.1, is the dead load partial factor, is 1.3, is the effective height of the section, is the standard value of constant load, is the design value of the concrete axial compressive strength, is the cross-sectional size of the reinforced concrete column, is the live load partial factor, is 1.5, 、 is the variable load adjustment factor for the design service life, 、 0.9~1.1, 、 、 、 、 、 is the effect coefficient, is the live load, is the serial number, , is the load combination value coefficient, is 0.7; S3. According to the probability characteristics of each action effect coefficient in formula (1), the axial force of the reinforced concrete column section is obtained. Random values , and then Random values Import them into formula (2) to obtain Relative compression zone height : （2） in, is the coefficient. When the concrete strength does not exceed C50, is 1.0, when the concrete strength is C80, is 0.94, with values ​​taken by linear interpolation; S4. The obtained Relative compression zone height Import them into formula (3) respectively to obtain indivual value; （3） in, are the tensile and compressive strength values ​​of steel bars, respectively. is the distance from the resultant force point of the longitudinal ordinary steel bars in the compression zone to the compression edge of the section, is the bending moment multiplication factor, is the distance from the axial pressure point to the resultant force point of the longitudinal tensile reinforcement, , is the relative limit compression zone height, is the distance from the axial pressure action point to the resultant force point of the longitudinal reinforcement in the compression zone, , is the initial eccentricity; , is the eccentricity of the axial pressure to the center of gravity of the section, taken as , The additional eccentricity is the maximum value of 20mm and 1 / 30 of the cross-sectional dimension in the eccentric direction. ; S5. Statistics indivual The number of values ​​greater than or equal to 0 is recorded as , and then 、 Import equations (4) and (5) respectively and use Monte Carlo simulation to obtain the reliability probability of reinforced concrete columns , failure probability ; （4） （5） S6. Obtained failure probability of reinforced concrete columns , import formula (6) to obtain the reliability index β of reinforced concrete column: （6） in, is the standard normal distribution function.

2. The reliability analysis method of reinforced concrete columns according to claim 1, characterized in that: The design parameters of reinforced concrete columns include concrete strength , steel bar strength , cross-sectional dimensions , constant load , live load , bending moment increase factor .

3. The reliability analysis method of reinforced concrete columns according to claim 1, characterized in that: According to the reinforcement ratio Selecting a set of design parameters for reinforced concrete columns involves the following steps: Based on the design parameters of reinforced concrete columns, the total axial compression ratio is obtained using formula (1): ; Based on the total axial pressure ratio obtained , using formula (2) to obtain the relative pressure zone height ; Based on the relative compression zone height obtained, the reinforcement ratio is obtained using formula (7): , （7） 。

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