Bridge pier bending reliability analysis method considering bidirectional random eccentricity

Abstract

The invention discloses a bridge pier bending reliability analysis method considering bidirectional random eccentricity, which comprises the following steps of: firstly, using a Kirschner's formula as a performance function for solving a reliable index, and secondly, using material characteristics, geometric characteristics and external load of a reinforced concrete bridge pier as random variables; according to the method, the axial compression bearing capacity and the one-way bias bearing capacity required by a Kelvin formula are solved through the constitutive relation of the reinforcing steel bars and the concrete, then the reliability index solving problem is converted into the unconstrained optimization problem by using a penalty function method based on the reliability index physical significance, and finally the reliability index is solved by using a simulated annealing genetic algorithm (SA-GA). The method has the advantages that the reliable index of the bidirectional bending of the pier can be accurately calculated, the design can be checked, the bridge construction can be guided, and the application range is wide; the accuracy of the interpolation curve is guaranteed, and the axial bearing capacity and the bending moment bearing capacity during uniaxial bending of the bridge pier can be solved.

Description

Technical Field

[0001] The invention relates to the technical field of bridge structure health monitoring, and in particular to a pier compression-bending reliability analysis method considering bidirectional random eccentricity. Background Art

[0002] As the key load-bearing component of a bridge, bridge piers are often the weakest link. The safety of bridge piers can be evaluated by reliability indicators. The methods for solving reliability indicators mainly include the first-order second-moment method, the second-order second-moment method, and the Monte Carlo method. The first-order second-moment method is a simple and quick method for solving reliability indicators. This method only considers the mean and variance of the basic random variables, expands the functional function into a Taylor series and retains it to the first-order term, and uses the Taylor series containing the first-order term as the tangent plane of the limit state surface. The first-order second-moment method is widely used due to its simple calculation.

[0003] During the construction and operation of bridges, loads such as construction loads, wind loads, vehicle loads, and deadweight will produce axial force and bending moment effects on bridge piers. Due to the different distribution types of loads such as construction loads, wind loads, vehicle loads, and deadweight, the axial force and bending moment are not completely related, so the axial force eccentricity has uncertainty. Therefore, the uncertainty of bidirectional eccentricity needs to be considered when conducting reliability analysis of bridge piers.

[0004] If the fixed eccentricity is adopted, that is, the bending moment and axial force are assumed to be completely correlated for the reliability analysis of the bidirectional eccentric compression of the bridge pier, it is difficult to obtain accurate reliability indicators; in addition, when the bridge pier is subjected to bidirectional compression and bending, the coupling of the bending moments in the two directions reduces the bearing capacity of the bridge pier, and only considering the unidirectional eccentric compression will obtain unsafe calculation results. The bidirectional compression and bending bearing capacity of the bridge pier can be solved using the Ni Keqin formula or the load contour method. When using the Ni Keqin formula or the load contour method to establish the function function, it is difficult to solve the partial derivatives of the random variables due to the strong nonlinearity, and it is difficult to use the traditional first-order second-order moment method to solve the reliability index. Summary of the invention

[0005] The purpose of the present invention is to provide a pier compression-bending reliability analysis method considering bidirectional random eccentricity based on the above-mentioned deficiencies of the prior art. Firstly, the Ni Keqin formula is used as a functional function for solving the reliability index. Secondly, the material properties, geometric properties and external loads of the reinforced concrete pier are taken as random variables. The axial compressive bearing capacity and unidirectional eccentric bearing capacity required by the Ni Keqin formula are solved through the constitutive relationship between steel bars and concrete. Then, based on the physical meaning of the reliability index, the penalty function method is used to transform the reliability index solving problem into an unconstrained optimization problem. Finally, the simulated annealing genetic algorithm (SA-GA) is used to solve the reliability index.

[0006] The purpose of the present invention is achieved by the following technical solutions:

[0007] A pier bending reliability analysis method considering bidirectional random eccentricity, the specific steps are as follows:

[0008] Step 1: Solve the load effect of the bridge pier under the load.

[0009] Step 2: Determine the material properties, geometric properties, and external load distribution characteristics of reinforced concrete bridge piers.

[0010] Step 3: Calculate the axial compressive bearing capacity and uniaxial bending bearing capacity of the bridge pier.

[0011] Step 3.1: Using the constitutive relationship between steel and concrete, a set of discrete points on the unidirectional compression-bending failure curve is solved by changing the position of the neutral axis.

[0012] Step 3.2: Use cubic spline interpolation to construct a continuous uniaxial compression-bending curve passing through discrete points, and use the uniaxial compression-bending curve to solve the axial compressive bearing capacity and unidirectional compression-bending bearing capacity of the bridge pier.

[0013] Step 4: Establish a functional function through axial compressive bearing capacity, one-way bending bearing capacity and pier load effect.

[0014] Step 5: Establish a reliable index solution method based on SA-GA.

[0015] Step 5.1: Use the external penalty function method to transform the reliability index solution problem into an unconstrained optimization problem.

[0016]

[0017] Where β is the reliability index; r is the penalty factor, which is a real number greater than 0; N u0 is the axial compressive bearing capacity; N ux 、N uy The axial force N acts on the mutually perpendicular symmetry axes x and y and the corresponding eccentricity e is calculated x 、e y The design value of the unidirectional eccentric load-bearing capacity calculated later.

[0018] Step 5.2: Genetic Algorithm (GA) is used as the main process, and the simulated annealing algorithm (SA) is introduced to further adjust the optimization population and establish SA-GA to solve reliable indicators.

[0019] Furthermore, in step 1, the pier load effect can be solved by finite element modeling or other methods.

[0020] Furthermore, in step 3.1, the constitutive relationship between steel bars and concrete is as shown in equations (2) to (7),

[0021] For concrete:

[0022] When ε c When ≤ε0:

[0023]

[0024] When ε0<ε c ≤ε cu hour:

[0025] σ c =f c (3)

[0026]

[0027] ε0=0.002+0.5(f cu,k -50)×10 -5 (5)

[0028] ε cu =0.0033-(f cu,k -50)×10 -5 (6)

[0029] Where σ c The concrete compressive strain is ε c Concrete compressive stress at time f c is the design value of the concrete axial compressive strength; ε0 is the time when the concrete compressive stress reaches f c When the concrete compressive strain is ε0<0.002, ε0=0.002; ε cu is the ultimate compressive strain of concrete in the positive section. When it is in non-uniform compression and the calculated ε cu When >0.003, ε cu =0.003; when the axis is under pressure, ε cu =ε0;f cu,k is the standard value of the compressive strength of concrete cube; n is the coefficient. When the calculation coefficient n>2.0, n is taken as 2.0.

[0030] For rebar:

[0031]

[0032] Where E s is the elastic modulus of the steel bar, f y is the design value of steel bar tensile strength, ε s is the steel bar strain.

[0033] The one-way bending bearing capacity and axial compressive bearing capacity can be calculated by equations (8) to (9):

[0034] Axial force bearing capacity:

[0035]

[0036] Taking the moment at the section h / 2, the moment bearing capacity of the reinforced concrete section can be obtained:

[0037]

[0038] The first term on the right side of the equation is the concrete strain ε0<ε≤ε cu The first item is the axial and bending resistance caused by the concrete strain of 0<ε≤ε0; the second item is the axial and bending resistance caused by the concrete strain of 0<ε≤ε0; the third item is the axial and bending resistance caused by the reinforcement effect. ε0 =c-cε0 / ε cu is the x-axis coordinate corresponding to when the concrete strain reaches ε0, x εcu The concrete strain reaches ε cu The x-axis coordinate corresponding to the time, x εcu =0; ε(x) is the concrete strain at coordinate x, ε(x)=(cx)ε cu / c; σ s,k is the stress of the kth steel bar.

[0039] When c = 0, the reinforced concrete section is in an axial tension state, and the section bearing capacity is completely provided by the steel bars; when c increases until the stress at the lower edge of the concrete section reaches ε0, that is, c max =ε cu h0 / (ε cu -ε0), it can be considered that the cross section is under axial compression, and equations (8) and (9) become:

[0040]

[0041] Furthermore, in step 4, the functional function is as follows:

[0042]

[0043] Where R is the cross-sectional resistance of reinforced concrete piers, which is related to the material properties and geometric properties of reinforced concrete piers. The material properties include the compressive strength of concrete f c , steel bar tensile strength f y , elastic modulus of steel bar E s The geometric characteristics include the length h, width b, and cross-sectional area A of the kth steel bar of the reinforced concrete column. s,k , the position x of the kth reinforcement along the x-axis and y-axis s,k and s,k; F=(N,M x ,M y ) T , N is the axial force on the reinforced concrete column, M x 、M y are the bending moments in the x-axis and y-axis directions respectively.

[0044] The load effect of reinforced concrete piers can be solved by formula (13):

[0045]

[0046] In the formula, each component of the vector q represents an external load component. For the convenience of calculation, its standard value is normalized to 1; each component of the matrix k is the load effect generated when the corresponding load takes the standard value, e a,x 、e a,y The additional eccentricity in the eccentric direction x and y is the larger value of 20 mm and 1 / 30 of the cross-sectional dimension in the eccentric direction.

[0047] Furthermore, in step 5.2, crossover probability and mutation are two main parameters that affect GA performance. If the value is too large, the existing good model will be destroyed. If the value is too small, the speed of generating new individuals may be too slow. Therefore, according to the evolutionary algebra, and are adaptively adjusted. The formulas are shown in formulas (14) and (15).

[0048] P c =0.9-0.3k / K (14)

[0049] P m =0.2-0.19k / K(15)

[0050] Where P c , P m are the crossover probability and mutation probability respectively; k is the current number of iterations; K is the total number of evolutions. In SA-GA, SA is only nested in GA to improve the local optimization ability of GA, and the accuracy requirement is not high.

[0051] The advantages of the present invention are:

[0052] 1) The method of the present invention adopts the calculation method of the bidirectional compression-bending bearing capacity of reinforced concrete bridge piers based on Ni Keqin's formula as the functional function. The method takes into account the uncertainty of the axial force eccentricity, can accurately calculate the bidirectional compression-bending reliability index of the bridge pier, can check the design and guide the bridge construction, and has a wide range of applications;

[0053] 2) In the process of calculating the uniaxial bending capacity, the cubic spline interpolation method is used to visualize the relationship between the uniaxial bending moment capacity and the axial bearing capacity, which not only ensures the accuracy of the interpolation curve, but also facilitates the solution of the axial bearing capacity and bending moment capacity of the bridge pier during uniaxial bending;

[0054] 3) The optimization method is used to solve the reliability index, which avoids the problem of difficulty in convergence due to the complexity of the functional function when using the first-order second moment method to solve the reliability index. At the same time, it is faster than the Monte Carlo method. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 is a flow chart of the steps of the present invention;

[0056] Figure 2 This is a schematic diagram of the cross section of the bridge pier;

[0057] Figure 3 It is a flow chart of solving reliability index based on SA-GA in the present invention;

[0058] Figure 4 The failure points and failure surfaces of each working condition are displayed. DETAILED DESCRIPTION

[0059] The features of the present invention and other related features are further described in detail below through embodiments in conjunction with the accompanying drawings to facilitate understanding by those skilled in the art:

[0060] Example: This example adopts the pier bending reliability analysis method considering bidirectional random eccentricity to conduct reliability analysis on a reinforced concrete pier. The specific parameters of the reinforced concrete pier section are as follows: the pier length is 270cm, the pier width is 170cm, the concrete strength grade is C40, the longitudinal steel bars are HRB400 steel bars, the steel bar diameter is 28mm, and there are 104 steel bars in total. The pier section is as follows: Figure 2 shown.

[0061] The reliability index of the bridge pier under four load combinations is solved. Each working condition consists of three loads and the total load effect is the same. The load effects are shown in Table 1.

[0062] Table 1. Load effects of four working conditions

[0063]

[0064] Step 1: The load effects of bridge piers are shown in Table 1.

[0065] Step 2: The material, geometric properties and external load distribution characteristics of reinforced concrete piers are shown in Table 2.

[0066] Table 2. Material properties, geometric properties and load statistical properties of reinforced concrete columns

[0067]

[0068]

[0069] Step 3.1: Solve the discrete points on the uniaxial compression-bending failure curve through the constitutive relationship of reinforced concrete.

[0070] Step 3.2: Use cubic spline interpolation to construct a continuous uniaxial bending curve passing through discrete points, and use natural boundary conditions to form an interpolation curve.

[0071] Step 4: Create a function.

[0072] Step 5.1: Use the external penalty function method to transform the reliability index solution problem into an unconstrained optimization problem, where r is generally taken between 0.1 and 1.

[0073] Step 5.2: The established SA-GA-based reliable index solution method flow is as follows: Figure 3 As shown, the number of parents N p =8, SA initial temperature is 100℃.

[0074] The reliability index of the reinforced concrete column under four working conditions is solved by using the fixed eccentricity and random eccentricity reliability analysis methods based on Ni Keqin's formula. The results are shown in Table 3. As can be seen from Table 3, the reliability index without considering random eccentricity is higher than the reliability index considering random eccentricity. For a more intuitive observation, the failure points and failure surfaces of the four working conditions solved by the reliability analysis method considering bidirectional random eccentricity are plotted as follows: Figure 3 As shown. Figure 3 In the figure, the mean failure surface is the compression-bending failure surface drawn when the material properties and geometric properties take the mean value; the design value failure surface is the compression-bending failure surface drawn when the material properties and geometric properties take the design values; the failure surface is the compression-bending failure surface drawn when the material properties and geometric properties take the verification point values; the failure point is the load effect calculated after the external load component takes the verification point value and the corresponding calculated eccentricity is considered.

[0075] Table 3. Comparison of reliability index calculation results of two methods with and without considering random eccentricity

[0076]

[0077] Depend on Figure 4 It can be seen that the moment coupling under bidirectional compression and bending reduces the ultimate bearing capacity of the section, so it is unsafe to only consider unidirectional compression and bending when conducting reliability analysis. Although the initial eccentricity of each working condition is the same, the failure form of the column in each working condition is different due to the different load combinations of each working condition. Working conditions 1 and 3 are large eccentric compression failures, while working conditions 2 and 4 are small eccentric compression failures.

[0078] Although the above embodiments have described in detail the concepts and embodiments of the present invention with reference to the accompanying drawings, ordinary technicians in this field can recognize that various improvements and changes can still be made to the present invention without departing from the scope of the claims, so they are not described one by one here.

Claims

1. A pier bending reliability analysis method considering bidirectional random eccentricity, characterized by: The steps include: S1. Solve the load effect of bridge piers under load; S2. Determine the material properties, geometric properties, and external load distribution characteristics of the pier; S3, solving the axial compressive bearing capacity and uniaxial bending bearing capacity of the pier; S4, establishing a function function by combining the axial compressive bearing capacity, the one-way bending bearing capacity and the load effect of the pier; S5. Establish a reliability index solution method based on SA-GA, and use it to solve the reliability index.

2. The method for analyzing the pier bending reliability considering bidirectional random eccentricity according to claim 1 is characterized by: In S3, the bridge pier is a reinforced concrete pier. The constitutive relationship between steel bars and concrete is used to solve a set of discrete points on the unidirectional compression-bending failure curve by changing the position of the neutral axis. A continuous uniaxial compression-bending curve passing through the discrete points is constructed by cubic spline interpolation, and the uniaxial compression-bending curve is used to solve the axial compressive bearing capacity and the unidirectional compression-bending bearing capacity of the bridge pier.

3. The method for analyzing the pier bending reliability considering bidirectional random eccentricity according to claim 1 is characterized by: In S5, the external penalty function method is used to transform the reliability index solution problem into an unconstrained optimization problem; the genetic algorithm is used as the main process, the simulated annealing algorithm is introduced to further adjust the optimization population, and the SA-GA is established to solve the reliability index.

4. The method for analyzing the pier bending reliability considering bidirectional random eccentricity according to claim 4 is characterized by: The external penalty function is: In the formula, β is the reliability index; r is the penalty factor, which is a real number greater than 0; N u0 is the axial compressive bearing capacity; N ux 、N uy The axial force N acts on the mutually perpendicular symmetry axes x and y and the corresponding eccentricity e is calculated x 、e y The design value of the unidirectional eccentric load-bearing capacity calculated later.

5. The method for analyzing the pier bending reliability considering bidirectional random eccentricity according to claim 1 is characterized by: In S1, the pier load effect can be solved by finite element modeling.

6. The method for analyzing the pier bending reliability considering bidirectional random eccentricity according to claim 1 is characterized by: In S4, the functional function is: Where R is the cross-sectional resistance of reinforced concrete piers, which is related to the material properties and geometric properties of reinforced concrete piers. The material properties include the compressive strength of concrete f c , steel bar tensile strength f y , elastic modulus of steel bar E s The geometric characteristics include the length h, width b, and cross-sectional area A of the kth steel bar of the reinforced concrete column. s,k , the position x of the kth reinforcement along the x-axis and y-axis s,k and s,k ; F=(N,M x ,M y ) T , N is the axial force on the reinforced concrete column, M x 、M y are the bending moments in the x-axis and y-axis directions respectively.

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