A vulnerability assessment method for bridge rotation construction integrating probabilistic performance of spherical joints

Abstract

A bridge rotation construction vulnerability assessment method that integrates the probabilistic performance of spherical joints relates to the technical field of bridge rotation construction. Concrete strength is selected as a random vector and an initial point set is obtained; the point set is segmented and reorganized, and the assigned probability is calculated; a spherical joint finite element model is established according to the adjusted point set and concrete strength, and a dynamic time history analysis is performed; the generalized probability density evolution equation is numerically solved using the finite difference method, and the probability corresponding to the maximum stress on each spherical joint surface is displayed through a stress probability density curve and a stress cumulative distribution function curve; a rotation system damage assessment index is defined; and the probability corresponding to the damage assessment index at each rotation speed is calculated. By introducing the uncertainty factor of concrete strength and combining the dynamic load effect of the spherical joint affected by the rotation speed during the rotation construction process, the damage probability of the spherical joint during rotation can be quickly determined, thereby improving the accuracy and efficiency of the bridge rotation construction vulnerability assessment.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge rotation construction, and in particular to a bridge rotation construction vulnerability assessment method integrating the probabilistic performance of spherical joints. Background Art

[0002] As an emerging construction method, bridge rotation construction technology offers advantages such as short construction periods, low costs, and minimal environmental impact. It has been widely used in bridge construction, particularly for long-span bridges. The design and construction quality of the rotation system are crucial in bridge rotation construction. In particular, the performance of the spherical hinge during the rotation process directly impacts the stability and safety of the construction. However, the current development of bridge rotation construction technology still faces a number of challenges, one of which is the unpredictability of the strength of spherical hinge concrete.

[0003] During bridge rotation construction, the concrete strength of the spherical joint plays a vital role in the stability of the rotation process. However, the actual strength of the spherical joint concrete is not a fixed value, but fluctuates within a certain range. Due to factors such as the concrete mix ratio, pouring process, and curing conditions, there is a difference between the actual strength and the strength value preset in the specification. This strength fluctuation will affect the ability of the spherical joint to withstand rotational loads. In particular, during the rotation process, the surface stress of the spherical joint is affected not only by the deadweight of the bridge, but also by the friction surface stress caused by the rotation speed. In complex construction environments, the uncertainty of the spherical joint concrete strength is particularly prominent, and it is necessary to design a more accurate evaluation method based on the influence of rotation speed on load.

[0004] Currently, there is no effective assessment method for the impact of concrete strength uncertainty on spherical hinges. Existing assessment methods often overlook the risks associated with concrete strength fluctuations or rely too heavily on deterministic assumptions, failing to accurately reflect actual conditions. During bridge rotation construction, changes in rotational speed cause dynamic loads to impact the spherical hinge. Existing assessment methods are unable to effectively account for these dynamic effects, making it difficult to provide an accurate vulnerability assessment. Therefore, a spherical hinge vulnerability assessment method that comprehensively considers both concrete strength uncertainty and the effects of rotational speed dynamic loads is urgently needed to improve the safety and reliability of bridge rotation construction. Summary of the Invention

[0005] In order to address the problem that the traditional method of using finite element method to determine the concrete strength for deterministic analysis obtains results that are inconsistent with the actual structural materials, the present invention provides a bridge rotation construction vulnerability assessment method that integrates the probabilistic performance of spherical joints. By introducing the uncertainty factor of concrete strength and combining the dynamic load effect of the spherical joint affected by the rotation speed during the rotation construction process, the damage probability of the spherical joint during rotation can be quickly determined, thereby improving the accuracy and efficiency of the bridge rotation construction vulnerability assessment.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: a bridge rotation construction vulnerability assessment method integrating the probabilistic performance of spherical joints, comprising the following steps:

[0007] Step 1: Select a random vector and obtain the initial point set

[0008] The concrete strength θ is selected as a random vector, and a uniform point set is selected in the geometric space [0,1] using the Sobol sequence: Basic =(x l ), l=1,2,...,n,x l represents the coordinates of each point in the uniform point set, n represents the number of divisions of the single-dimensional parameter space, n>10;

[0009] The distribution of concrete strength θ conforms to the lognormal distribution, and the concrete strength distribution function F(θ) and the probability density function of concrete strength distribution p(θ) are obtained;

[0010] Make an inverse function change on the uniform point set according to the probability distribution, that is, take: θ l =F -1 (x l ), then the initial point set of concrete strength θ is expressed as: n0 =(θ l ), l=1,2,...,n;

[0011] Step 2: Point set reorganization

[0012] Select the boundary of the overall interval, in the range space Ω of concrete strength θ θ In the initial point set φ n0 Perform Voronoi set subdivision and obtain l n subdomains with the core It is expressed as follows:

[0013]

[0014] Where, ||·||2 represents the Euclid distance;

[0015] Calculate the probability P of each subdivision unit l , According to the following formula, the initial point set φ n0 Make adjustments:

[0016] Take l=2,...,n-1

[0017] Where I(·) is the indicator function,

[0018] Get the adjusted point set φ n1 Expressed as: φ n1 =(θ′ l ), l=1,2,...,n, and the adjusted point set φ n1 Perform Voronoi set subdivision again and get l n subdomains with the core And calculate the probability P of each subdivision unit l ′,

[0019] Step 3: Establish a finite element model and perform dynamic time history analysis

[0020] According to the bridge design drawings, according to the adjusted point set φ n1 According to different concrete strengths θ, n spherical joint finite element models are established, and each spherical joint finite element model considers different elastic modulus E and shear modulus G;

[0021] The rotation speed of the ball joint is used as the load level, and the rotation speed v is recorded as i , i=1,2,...,I, I represents the number of rotation speeds, perform I dynamic time history analysis on each spherical joint finite element model, and record the maximum stress on the spherical joint surface obtained from the i-th dynamic time history analysis

[0022] For v i is the rotation speed and θ′ l The dynamic time history analysis results of concrete strength are shown in the following table. A virtual random process is constructed based on the extreme value, a virtual time variable τ is introduced, and a sine function is used as the virtual function, which is expressed as follows:

[0023]

[0024] In the formula, y represents the l and τ are variables of the pseudo random process of the maximum stress on the spherical joint surface;

[0025] Step 4: Numerical solution of the generalized probability density evolution equation using the finite difference method

[0026] For a specific rotation speed v i , there are n probability density evolution equations, expressed as follows:

[0027]

[0028] Where p l,i (y,θ′ l ,τ) represents a specific θ′ l The probability density function of the random variable y and the virtual time variable τ;

[0029] Discretize the (y~τ) plane and express it as:

[0030] y=y j ,τ=τ k , j=0,1,2,...,J, k=0,1,2,...,K

[0031] Where y j =j·Δy,τ k = k·Δτ, Δy and Δτ are discrete steps in space and time, respectively, and J and K are the corresponding discrete quantities;

[0032] will p l,i (y,θ′ l ,τ) at the discretized grid point p l,i (y j ,θ′ l ,τ k ) is recorded as According to the Lax-Wendroff format, Discretized into:

[0033]

[0034] The boundary points are processed using periodic boundary conditions, namely:

[0035]

[0036] in:

[0037]

[0038] The initial value of :

[0039]

[0040] Where δ(·) is the Dirac delta function;

[0041] Calculate p l,i (y,θ′ l ,τ), the probability density function of the maximum stress on the spherical hinge surface under each specific concrete strength under the rotation speed is expressed as follows:

[0042]

[0043] The probability density function of the maximum stress on the spherical hinge surface under the determined rotation speed considering the uncertainty of concrete strength is expressed as follows:

[0044]

[0045] Then, a stress probability density curve and a stress cumulative distribution function curve are drawn to visually display the probability corresponding to the maximum stress on each spherical joint surface;

[0046] Step 5: Determine the damage assessment index of the rotating system

[0047] Define y∈[0,y0) as the state of no damage to the rotation system, y∈[y0,y1) as the state of slight damage to the rotation system, y∈[y1,y2) as the state of moderate damage to the rotation system, y∈[y2,y3) as the state of severe damage to the rotation system, and y∈[y3,+∞) as the state of complete damage to the rotation system, where: y0 = 0.65×y max , y1=0.75×y max , y2=0.85×y max , y3=0.95×y max ,y max is the ultimate stress of spherical hinge concrete obtained according to the bridge design drawings;

[0048] Step 6: Calculate the vulnerability probability at each rotation speed

[0049] Based on the damage assessment index of the rotating system obtained in step 5, the probability of each damage state of the rotating system is calculated: and They are the probabilities corresponding to the rotation system's no damage state, the rotation system's slightly damaged state, the rotation system's moderately damaged state, the rotation system's severely damaged state, and the rotation system's completely damaged state.

[0050] Furthermore, in step 1, the distribution of concrete strength θ conforms to the lognormal distribution, which can be expressed as:

[0051]

[0052]

[0053] Where Φ represents the normal distribution function, μ′0 and σ′0 are the mean and standard deviation of concrete strength, respectively.

[0054] Furthermore, in step 3, the elastic modulus E and the shear modulus G are solved by the following formula:

[0055]

[0056] Where ν represents the Poisson's ratio of concrete.

[0057] Furthermore, in step 4, in order to maintain the stability of the numerical solution,

[0058] Compared with the existing technology, the present invention has the following advantages: by establishing a finite element model for the uncertainty of spherical hinge concrete strength, considering different elastic moduli and shear moduli, conducting a dynamic time history analysis of the spherical hinge contact stress at specific rotational speeds, recording the maximum stress on the spherical hinge surface, and using numerical calculations to solve the probability density evolution equation, the present invention can evaluate the fragility of the rotating system at various rotational speeds. The present invention has the following advantages:

[0059] 1. Considering the uncertainty of concrete strength: This paper introduces the uncertainty factor of concrete strength and combines it with the dynamic load effect of the spherical hinge affected by the rotation speed during the rotation construction process to establish a more accurate spherical hinge vulnerability assessment method. This method breaks through the limitations of traditional assessment methods that ignore strength fluctuations and dynamic load effects. It can more realistically reflect the stress conditions of the spherical hinge during construction, providing more reliable protection for bridge construction safety.

[0060] 2. Improved evaluation efficiency and easy operation: The present invention realizes rapid evaluation of spherical joint vulnerability by constructing a spherical joint damage state model based on uncertainty and dynamic load effects. Only a small amount of determined concrete strength analysis is required to better simulate the stress-probability density curve, simplify the evaluation process, greatly improve the evaluation efficiency, and facilitate engineering and technical personnel to make quick decisions at the construction site. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 is a flow chart of the method of the present invention;

[0062] Figure 2 is a stress probability density curve obtained in the embodiment;

[0063] Figure 3 3 is a graph of the stress cumulative distribution function obtained in the embodiment. DETAILED DESCRIPTION

[0064] The technical solutions of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0065] A bridge rotation construction vulnerability assessment method integrating the probabilistic performance of spherical hinges. Figure 1 As shown, the following steps are included:

[0066] Step 1: Select a random vector and obtain the initial point set

[0067] The concrete strength θ is selected as a random vector, and a uniform point set is selected in the geometric space [0,1] using the Sobol sequence, which is expressed as follows:

[0068] φ Basic =(x l ), l=1,2,...,n

[0069] Where x l Represents the coordinates of each point in the uniform point set, n represents the number of divisions of the single-dimensional parameter space, and n>10.

[0070] According to the statistics of the Code for Design of Concrete Structures, the distribution of concrete strength θ conforms to the log-normal distribution, which can be expressed as:

[0071]

[0072]

[0073] Where F(θ) represents the concrete strength distribution function, Φ represents the normal distribution function, p(θ) represents the probability density function of concrete strength distribution, μ′0 and σ′0 are the mean and standard deviation of concrete strength, respectively.

[0074] Make an inverse function change on the uniform point set according to the probability distribution, that is, take: θ l =F -1 (x l ), then the initial point set of concrete strength θ is expressed as: n0 =(θ l ), l=1,2,...,n;

[0075] Step 2: Point set reorganization

[0076] Select the boundary of the overall interval, in the range space Ω of concrete strength θ θ In the initial point set φ n0 Perform Voronoi set subdivision and obtain l n subdomains with the core It is expressed as follows:

[0077]

[0078] Where ‖·‖2 represents the Euclid distance.

[0079] Calculate the probability P of each subdivision unit l , According to the following formula, the initial point set φ n0 Make adjustments:

[0080] Take l=2,...,n-1

[0081] Where I(·) is the indicator function,

[0082] Get the adjusted point set φ n1 Expressed as: φ n1 =(θ′ l ), l=1,2,...,n, and the adjusted point set φ n1 Perform Voronoi set subdivision again and get l n subdomains with the core And calculate the probability P of each subdivision unit l ′,

[0083] Step 3: Establish a finite element model and perform dynamic time history analysis

[0084] According to the bridge design drawings, according to the adjusted point set φ n1 According to different concrete strengths θ, n spherical joint finite element models are established. Each spherical joint finite element model considers different elastic modulus E and shear modulus G. The elastic modulus E and shear modulus G of each spherical joint finite element model are solved by the following formula:

[0085]

[0086] Where ν represents the Poisson's ratio of concrete.

[0087] The rotation speed of the ball joint is used as the load level, and the rotation speed v is recorded as i , i=1,2,...,I, I represents the number of rotation speeds, and the finite element model of each spherical joint is subjected to I dynamic time history analysis, that is, a total of n×I dynamic time history analysis is performed, and the maximum stress on the spherical joint surface obtained by each dynamic time history analysis is recorded. The maximum stress on the spherical joint surface obtained by the i-th dynamic time history analysis is expressed as express.

[0088] For v i is the rotation speed and θ′ l The dynamic time history analysis results of concrete strength are shown in the following table. A virtual random process is constructed based on the extreme value, a virtual time variable τ is introduced, and a sine function is used as the virtual function, which is expressed as follows:

[0089]

[0090] In the formula, y represents the l and τ are variables of the pseudo random process of the maximum stress on the spherical joint surface;

[0091] Step 4: Numerical solution of the generalized probability density evolution equation using the finite difference method

[0092] For a specific rotation speed v i , there are n probability density evolution equations, expressed as follows:

[0093]

[0094] Where p l,i (y,θ′ l ,τ) represents a specific θ′ l The probability density function of the random variable y and the dummy time variable τ.

[0095] Discretize the (y~τ) plane and express it as:

[0096] y=y j ,τ=τ k , j=0,1,2,...,J, k=0,1,2,...,K

[0097] Where y j =j·Δy,τ k =k·Δτ, Δy and Δτ are discrete steps in space and time respectively, and J and K are the corresponding discrete quantities respectively.

[0098] For the sake of simplicity, we will l,i (y,θ′ l ,τ) at the discretized grid point p l,i (y j ,θ′ l ,τ k ) is recorded as

[0099] According to the Lax-Wendroff format, Discretized into:

[0100]

[0101] In order to maintain the stability of the numerical solution, The boundary points are processed using periodic boundary conditions, namely:

[0102]

[0103] in:

[0104]

[0105] The initial value of :

[0106]

[0107] Where δ(·) is the Dirac delta function, which is expressed as follows:

[0108]

[0109] Calculate p l,i (y,θ′ l ,τ), the probability density function of the maximum stress on the spherical hinge surface under each specific concrete strength under the rotation speed is expressed as follows:

[0110] p l,i (y,θ′ l )=p l,i (y,θ′ l ,τ)| τ＝1

[0111] The probability density function of the maximum stress on the spherical hinge surface under the determined rotation speed considering the uncertainty of concrete strength is expressed as follows:

[0112]

[0113] Then, a stress probability density curve and a stress cumulative distribution function curve are drawn to visualize the probability corresponding to the maximum stress on each spherical joint surface;

[0114] Step 5: Determine the damage assessment index of the rotating system

[0115] Define y∈[0,y0) as the undamaged state of the rotation system, y∈[y0,y1) as the slightly damaged state of the rotation system, y∈[y1,y2) as the moderately damaged state of the rotation system, y∈[y2,y3) as the severely damaged state of the rotation system, and y∈[y3,+∞) as the completely damaged state of the rotation system, where:

[0116] y0=0.65×y max

[0117] y1=0.75×y max

[0118] y2=0.85×y max

[0119] y3=0.95×y max

[0120] Where y max is the ultimate stress of spherical hinge concrete obtained according to the bridge design drawings;

[0121] Step 6: Calculate the vulnerability probability at each rotation speed

[0122] The damage assessment index of the rotating system obtained in step 5 is used to calculate the probability of each damage state of the rotating system, which is expressed as follows:

[0123]

[0124] Where, and They are the probabilities corresponding to the rotation system's no damage state, the rotation system's slightly damaged state, the rotation system's moderately damaged state, the rotation system's severely damaged state, and the rotation system's completely damaged state.

[0125] Example

[0126] This embodiment is implemented for the Yinwu connecting line super-large bridge. The actual operation according to the method of the present invention is as follows:

[0127] S1. Select the concrete strength θ as a random vector, and use the Sobol sequence to select a uniform point set in the geometric space [0,1], taking n = 63, and the resulting uniform point set φ Basic See Table 1:

[0128] Table 1 Uniform point set φ Basic

[0129] 0.500000 0.750000 0.250000 0.375000 0.875000 0.625000 0.125000 0.187500 0.687500 0.937500 0.437500 0.312500 0.812500 0.562500 0.062500 0.093750 0.593750 0.843750 0.343750 0.468750 0.968750 0.718750 0.218750 0.156250 0.656250 0.906250 0.406250 0.281250 0.781250 0.531250 0.031250 0.046875 0.546875 0.796875 0.296875 0.421875 0.921875 0.671875 0.171875 0.234375 0.734375 0.984375 0.484375 0.359375 0.859375 0.609375 0.109375 0.078125 0.578125 0.828125 0.328125 0.453125 0.953125 0.703125 0.203125 0.140625 0.640625 0.890625 0.390625 0.265625 0.765625 0.515625 0.015625

[0130] According to the Code for Design of Concrete Structures, the mean and standard deviation of the strength of C60 concrete are 64.9 MPa and 3.0 MPa respectively. The lognormal distribution is expressed as:

[0131]

[0132] The initial point set φ is obtained by making an inverse function change on the uniform point set according to the probability distribution. n0 See Table 2:

[0133] Table 2 Initial point set φ n0

[0134]

[0135]

[0136] To facilitate subsequent calculations, the initial point set φ n0 Sort from small to large, the sorted initial point set φ n0 See Table 3:

[0137] Table 3 Sorted initial point set φ n0

[0138] 58.691 59.486 60.002 60.396 60.721 61.002 61.251 61.476 61.684 61.876 62.057 62.229 62.392 62.548 62.698 62.843 62.984 63.12 63.254 63.384 63.512 63.638 63.762 63.885 64.006 64.125 64.244 64.363 64.48 64.598 64.715 64.832 64.949 65.067 65.186 65.305 65.425 65.546 65.669 65.794 65.92 66.048 66.179 66.313 66.45 66.59 66.735 66.884 67.039 67.2 67.368 67.544 67.731 67.929 68.141 68.371 68.622 68.902 69.221 69.594 70.051 70.658 71.615

[0139] S2. Since there is only one-dimensional parameter concrete strength θ, the boundary of the Voronoi set is the sorted initial point set φ n0The median between the two points is used as the dividing boundary of the overall interval with the probability of 0.05% quantile and 99.34% quantile as the dividing boundary, and the dividing boundary θ is obtained. b See Table 4:

[0140] Table 4 Division boundary θ b

[0141] <![CDATA[θ b0 ]]> <![CDATA[θ b1 ]]> <![CDATA[θ b2 ]]> <![CDATA[θ b3 ]]> <![CDATA[θ b4 ]]> <![CDATA[θ b5 ]]> <![CDATA[θ b6 ]]> <![CDATA[θ b7 ]]> 58.452 59.089 59.744 60.199 60.559 60.862 61.127 61.364 <![CDATA[θ b8 ]]> <![CDATA[θ b9 ]]> <![CDATA[θ b10 ]]> <![CDATA[θ b11 ]]> <![CDATA[θ b12 ]]> <![CDATA[θ b13 ]]> <![CDATA[θ b14 ]]> <![CDATA[θ b15 ]]> 61.580 61.780 61.967 62.143 62.311 62.470 62.623 62.771 <![CDATA[θ b16 ]]> <![CDATA[θ b17 ]]> <![CDATA[θ b18 ]]> <![CDATA[θ b19 ]]> <![CDATA[θ b20 ]]> <![CDATA[θ b21 ]]> <![CDATA[θ b22 ]]> <![CDATA[θ b23 ]]> 62.914 63.052 63.187 63.319 63.448 63.575 63.700 63.824 <![CDATA[θ b24 ]]> <![CDATA[θ b25 ]]> <![CDATA[θ b26 ]]> <![CDATA[θ b27 ]]> <![CDATA[θ b28 ]]> <![CDATA[θ b29 ]]> <![CDATA[θ b30 ]]> <![CDATA[θ b31 ]]> 63.946 64.066 64.185 64.304 64.422 64.539 64.657 64.774 <![CDATA[θ b32 ]]> <![CDATA[θ b33 ]]> <![CDATA[θ b34 ]]> <![CDATA[θ b35 ]]> <![CDATA[θ b36 ]]> <![CDATA[θ b37 ]]> <![CDATA[θ b38 ]]> <![CDATA[θ b39 ]]> 64.891 65.008 65.127 65.246 65.365 65.486 65.608 65.732 <![CDATA[θ b40 ]]> <![CDATA[θ b41 ]]> <![CDATA[θ b42 ]]> <![CDATA[θ b43 ]]> <![CDATA[θ b44 ]]> <![CDATA[θ b45 ]]> <![CDATA[θ b46 ]]> <![CDATA[θ b47 ]]> 65.857 65.984 66.114 66.246 66.382 66.520 66.663 66.810 <![CDATA[θ b48 ]]> <![CDATA[θ b49 ]]> <![CDATA[θ b50 ]]> <![CDATA[θ b51 ]]> <![CDATA[θ b52 ]]> <![CDATA[θ b53 ]]> <![CDATA[θ b54 ]]> <![CDATA[θ b55 ]]> 66.962 67.120 67.284 67.456 67.638 67.830 68.035 68.256 <![CDATA[θ b56 ]]> <![CDATA[θ b57 ]]> <![CDATA[θ b58 ]]> <![CDATA[θ b59 ]]> <![CDATA[θ b60 ]]> <![CDATA[θ b61 ]]> <![CDATA[θ b62 ]]> <![CDATA[θ b63 ]]> 68.497 68.762 69.062 69.408 69.823 70.355 71.137 72.740

[0142] Then each subdomain interval:

[0143] Ω θl =(θ b(l-1) ,θ bl ), l=1,2,...,63

[0144] Calculate the probability of each subdivision unit:

[0145]

[0146] Get the probability P of each subdivision unit l See Table 5:

[0147] Table 5 The probability P of each subdivision unit l

[0148] 0.017330 0.016112 0.015822 0.015712 0.015689 0.015687 0.015646 0.015664 0.015647 0.015612 0.015675 0.015673 0.015629 0.015620 0.015620 0.015647 0.015605 0.015614 0.015627 0.015592 0.015635 0.015641 0.015673 0.015673 0.015576 0.015579 0.015686 0.015636 0.015626 0.015658 0.015599 0.015582 0.015609 0.015677 0.015654 0.015606 0.015597 0.015625 0.015687 0.015654 0.015589 0.015613 0.015656 0.015654 0.015607 0.015621 0.015630 0.015626 0.015647 0.015629 0.015607 0.015644 0.015662 0.015630 0.015649 0.015643 0.015643 0.015690 0.015709 0.015727 0.015822 0.016127 0.015920

[0149] Rescale the point set according to the following formula:

[0150]

[0151] The two boundary points are still the 0.05% quantile and the 99.34% quantile. The probabilities of each center point are shown in Table 6:

[0152] Table 6 Probability of each center point

[0153] 0.008471 0.024168 0.039555 0.054992 0.070601 0.086279 0.101699 0.11748 0.133 0.148314 0.164588 0.18024 0.195363 0.21087 0.22649 0.242529 0.257483 0.273245 0.2891 0.304044 0.320518 0.336282 0.352643 0.368316 0.381612 0.397265 0.415679 0.42999 0.445341 0.461911 0.47577 0.490833 0.507293 0.52518 0.540063 0.554013 0.569291 0.585938 0.60395 0.618333 0.631355 0.64794 0.66538 0.680949 0.694512 0.710756 0.726795 0.742235 0.75888 0.773636 0.788154 0.805666 0.822255 0.836205 0.852871 0.868187 0.88383 0.902175 0.918977 0.935757 0.957231 0.991811 0.995000

[0154] Adjusted point set φ n1 See Table 7:

[0155] Table 7 Adjusted point set φ n1

[0156] 58.060 59.180 59.780 60.217 60.572 60.873 61.132 61.370 61.585 61.780 61.974 62.150 62.312 62.470 62.623 62.774 62.911 63.051 63.188 63.314 63.450 63.578 63.709 63.832 63.936 64.057 64.197 64.306 64.422 64.546 64.650 64.763 64.887 65.021 65.134 65.240 65.357 65.486 65.626 65.740 65.845 65.980 66.124 66.256 66.374 66.518 66.664 66.809 66.971 67.121 67.273 67.466 67.660 67.832 68.051 68.268 68.510 68.826 69.158 69.549 70.192 72.436 73.025

[0157] The probability of obtaining the subdivision again is P l 'See Table 8:

[0158] Table 8 The probability of obtaining the subdivision again P l '

[0159] 0.002151 0.016473 0.015669 0.015667 0.015745 0.015598 0.015598 0.015704 0.015458 0.015767 0.015988 0.015440 0.015338 0.015561 0.015814 0.015510 0.015381 0.015812 0.015389 0.015670 0.016107 0.016085 0.016021 0.014501 0.014525 0.017010 0.016362 0.014874 0.015934 0.015179 0.014465 0.015795 0.017162 0.016371 0.014444 0.014619 0.016001 0.017316 0.016150 0.013749 0.014861 0.016979 0.016457 0.014594 0.014955 0.016128 0.015713 0.016044 0.015726 0.014650 0.016033 0.017080 0.015264 0.015352 0.015963 0.015508 0.017111 0.017575 0.016882 0.019633 0.033042 0.013158 0.000051

[0160] S3. Build 63 spherical joint finite element models based on the bridge design drawings. Determine the concrete strength, elastic modulus, and shear modulus of each model. Perform dynamic time history analysis at a rotation speed of 0.5° / min. Record the maximum surface stress of the spherical joint obtained from each dynamic time history analysis, as shown in Table 9:

[0161] Table 9 Maximum stress on the spherical joint surface at a rotation speed of 0.5° / min

[0162] 47.8651 47.8300 47.8889 47.7437 47.7419 47.4259 47.4139 47.0914 46.9829 46.3624 46.9081 46.6679 46.2674 46.2876 46.3758 46.2419 46.2364 45.7074 45.4051 45.5040 45.4354 45.2599 45.3519 44.9897 44.5675 44.7326 44.4348 44.9397 44.1831 43.7884 43.8366 43.7095 44.2543 43.8884 43.1483 43.4096 43.0939 43.5034 43.1711 43.3595 43.1280 42.7150 42.2196 42.8727 42.3337 42.6272 42.1612 41.8148 41.8921 42.2297 41.2335 41.1670 41.3570 41.0430 41.0007 40.7295 41.2834 40.8545 41.0101 40.3469 40.2877 40.4297 40.6117

[0163] Construct a virtual function as follows:

[0164]

[0165] S4. Take y∈[0,100], t∈[0,1], discretize y into 1000 parts, discretize t into 10000 parts, program the following formula, and perform program calculation.

[0166]

[0167] in:

[0168]

[0169] The final stress probability density curve and stress cumulative distribution function curve are combined Figure 2 and Figure 3 As shown;

[0170] The maximum stress of S5 and C60 concrete is 60 MPa, so the damage assessment index is defined as follows:

[0171] No damage status 0MPa≤y<39MPa Minor injury status 39MPa≤y<45MPa Moderate damage status 45MPa≤y<51MPa Severe injury status 51MPa≤y<57MPa Completely damaged state y≥57MPa

[0172] S6. Calculate the probability of each damage state of the rotating system:

[0173]

[0174] Calculation results show: P0=0.4532, P1=0.3339, P2=0.1974, P3=0.0154, P4=0.0001.

[0175] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other configurations without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims, not the foregoing description, and all variations coming within the meaning and range of equivalents of the claims are intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

[0176] In addition, it should be understood that although this specification is described in terms of implementation methods, not every implementation method contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment can also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A bridge rotation construction vulnerability assessment method integrating the probabilistic performance of spherical joints, characterized by: The following steps are involved: Step 1: Select a random vector and obtain the initial point set The concrete strength θ is selected as a random vector, and a uniform point set is selected in the geometric space [0,1] using the Sobol sequence: Basic =(x l ), l=1,2,...,n,x l represents the coordinates of each point in the uniform point set, n represents the number of divisions of the single-dimensional parameter space, n>10; The distribution of concrete strength θ conforms to the lognormal distribution, and the concrete strength distribution function F(θ) and the probability density function of concrete strength distribution p(θ) are obtained; Make an inverse function change on the uniform point set according to the probability distribution, that is, take: θ l =F -1 (x l ), then the initial point set of concrete strength θ is expressed as: n0 =(θ l ), l=1,2,...,n; Step 2: Point set reorganization Select the boundary of the overall interval, in the range space Ω of concrete strength θ θ In the initial point set φ n0 Perform Voronoi set subdivision and obtain l n subdomains with the core It is expressed as follows: Where, ‖·‖2 represents the Euclid distance; Calculate the probability P of each subdivision unit l , According to the following formula, the initial point set φ n0 Make adjustments: Take l=2,...,n-1 Where I(·) is the indicator function, Get the adjusted point set φ n1 Expressed as: φ n1 =(θ′ l ), l=1,2,...,n, and the adjusted point set φ n1 Perform Voronoi set subdivision again and obtain l n subdomains with the core And calculate the probability P of each subdivision unit l ′, Step 3: Establish a finite element model and perform dynamic time history analysis According to the bridge design drawings, according to the adjusted point set φ n1 According to different concrete strengths θ, n spherical joint finite element models are established, and each spherical joint finite element model considers different elastic modulus E and shear modulus G; The rotation speed of the ball joint is used as the load level, and the rotation speed v is recorded as i , i=1,2,...,I, I represents the number of rotation speeds, perform I dynamic time history analysis on each spherical joint finite element model, and record the maximum stress on the spherical joint surface obtained from the i-th dynamic time history analysis For v i is the rotation speed and θ′ l The dynamic time history analysis results of concrete strength are shown in the following table. A virtual random process is constructed based on the extreme value, a virtual time variable τ is introduced, and a sine function is used as the virtual function, which is expressed as follows: In the formula, y represents the l and τ are variables of the pseudo random process of the maximum stress on the spherical joint surface; Step 4: Numerical solution of the generalized probability density evolution equation using the finite difference method For a specific rotation speed v i , there are n probability density evolution equations, expressed as follows: Where p l,i (y,θ′ l ,τ) represents a specific θ′ l The probability density function of the random variable y and the virtual time variable τ; Discretize the (y~τ) plane and express it as: y=y j ,τ=τ k ,j=0,1,2,...,J,k=0,1,2,...,K Where y j =j·Δy,τ k = k·Δτ, Δy and Δτ are discrete steps in space and time, respectively, and J and K are the corresponding discrete quantities; will p l,i (y,θ′ l ,τ) at the discretized grid point p l,i (y j ,θ′ l ,τ k ) is recorded as According to the Lax-Wendroff format, Discretized into: The boundary points are processed using periodic boundary conditions, namely: in: Initial value of : Where δ(·) is the Dirac delta function; Calculate p l,i (y,θ′ l ,τ), the probability density function of the maximum stress on the spherical hinge surface under each specific concrete strength under the rotation speed is expressed as follows: p l,i (y,θ′ l )=p l,i (y,θ′ l ,t)| τ＝1 The probability density function of the maximum stress on the spherical hinge surface under the determined rotation speed considering the uncertainty of concrete strength is expressed as follows: Then, a stress probability density curve and a stress cumulative distribution function curve are drawn to visually display the probability corresponding to the maximum stress on each spherical joint surface; Step 5: Determine the damage assessment index of the rotating system Define y∈[0,y0) as the state of no damage to the rotation system, y∈[y0,y1) as the state of slight damage to the rotation system, y∈[y1,y2) as the state of moderate damage to the rotation system, y∈[y2,y3) as the state of severe damage to the rotation system, and y∈[y3,+∞) as the state of complete damage to the rotation system, where: y0 = 0.65×y max , y1=0.75×y max , y2=0.85×y max , y3=0.95×y max ,y max is the ultimate stress of spherical hinge concrete obtained according to the bridge design drawings; Step 6: Calculate the vulnerability probability at each rotation speed Based on the damage assessment index of the rotating system obtained in step 5, the probability of each damage state of the rotating system is calculated: P1 i 、 and They are the probabilities corresponding to the rotation system's no damage state, the rotation system's slightly damaged state, the rotation system's moderately damaged state, the rotation system's severely damaged state, and the rotation system's completely damaged state.

2. The bridge rotation construction vulnerability assessment method integrating spherical joint probabilistic performance according to claim 1 is characterized by: In step 1, the distribution of concrete strength θ conforms to the log-normal distribution, which can be expressed as: Where Φ represents the normal distribution function, μ′0 and σ′0 are the mean and standard deviation of concrete strength, respectively.

3. The bridge rotation construction vulnerability assessment method integrating spherical joint probabilistic performance according to claim 1 is characterized by: In step 3, the elastic modulus E and the shear modulus G are solved by the following formula: Where ν represents the Poisson's ratio of concrete.

4. The bridge rotation construction vulnerability assessment method integrating spherical joint probabilistic performance according to claim 1 is characterized by: In step 4, in order to maintain the stability of the numerical solution,

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