Bridge swivel construction vulnerability evaluation method fusing spherical hinge probability performance

Abstract

The invention discloses a bridge swivel construction vulnerability evaluation method fusing spherical hinge probability performance, and relates to the technical field of bridge swivel construction. Selecting concrete strength as a random vector and obtaining an initial point set; subdividing and reforming the point set, and calculating an assignment probability; establishing a spherical hinge finite element model according to the adjusted point set and the concrete strength, and performing dynamic time-history analysis; the generalized probability density evolution equation is solved through a finite difference method, and the probability corresponding to the maximum stress of the surface of each spherical hinge is displayed through a stress probability density curve and a stress cumulative distribution function curve; defining a rotation system damage evaluation index; and calculating the probability corresponding to the damage evaluation index at each rotation speed. By introducing uncertain factors of concrete strength and combining the dynamic load effect of the spherical hinge influenced by the rotation speed in the rotation construction process, the damage probability of the spherical hinge during rotation can be quickly given, and the assessment precision and efficiency of the vulnerability of bridge rotation construction are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge rotation construction, and specifically, it is a vulnerability assessment method for bridge rotation construction that integrates the probabilistic performance of spherical hinges. Background Technique

[0002] As a new construction method, the bridge rotation construction technology has the advantages of short construction period, low cost, and small environmental impact, and has been widely used in bridge construction, especially in the construction of long-span bridges. In bridge rotation construction, the design and construction quality of the rotating system are crucial. Especially during the bridge rotation process, the performance of the spherical hinge of the rotating system directly affects the stability and safety of the construction. However, the current technological development of bridge rotation construction still faces a series of challenges, and the unpredictability of the concrete strength of the spherical hinge is one of them.

[0003] In bridge rotation construction, the concrete strength of the spherical hinge plays a crucial role in the stability of the rotation process. However, the actual strength of the spherical hinge concrete is not a fixed value but fluctuates within a certain range. Due to the influence of factors such as concrete mix ratio, pouring process, and curing conditions, there is a difference between the actual strength and the preset strength value in the specification. This strength fluctuation will affect the ability of the spherical hinge to bear the rotation load. Especially during the rotation process, the surface stress of the spherical hinge is not only affected by the self-weight of the bridge but also by the frictional surface stress caused by the rotation speed. In a complex construction environment, the problem of the uncertainty of the spherical hinge concrete strength is particularly prominent, and a more accurate assessment method needs to be designed for the influence of the rotation speed on the load.

[0004] Currently, there is no effective assessment method for the influence brought by the uncertainty of the spherical hinge concrete strength. Since the existing assessment methods often ignore the risks brought by the concrete strength fluctuation or rely too much on deterministic assumptions, it is difficult to accurately reflect the actual situation. In bridge rotation construction, due to the change of the rotation speed, it will cause the impact of dynamic load on the spherical hinge, resulting in the inability of the existing assessment methods to effectively consider this dynamic effect and difficult to provide an accurate vulnerability assessment. Therefore, there is an urgent need to design a spherical hinge vulnerability assessment method that can comprehensively consider the uncertainty of concrete strength and the influence of dynamic load of rotation speed to improve the safety and reliability of bridge rotation construction. Summary of the Invention

[0005] Aiming at the problem that the results obtained by the traditional method of performing deterministic analysis with a finite element to determine the concrete strength do not match the actual structural materials, the present invention provides a vulnerability assessment method for bridge rotation construction that integrates the probabilistic performance of spherical hinges. By introducing the uncertainty factor of concrete strength and combining the dynamic load effect of the spherical hinge affected by the rotation speed during the rotation construction process, it can quickly give the damage probability when the spherical hinge rotates, improving the assessment accuracy and efficiency of the vulnerability of bridge rotation construction.

[0006] To achieve the above object, the present invention adopts the following technical solution: A method for evaluating the vulnerability of bridge rotation construction by integrating the probabilistic performance of spherical hinges, comprising the following steps:

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

[0008] Select the concrete strength θ as the random vector, and use the Sobol sequence to select a uniform point set in the geometric space [0,1]: φ Basic =(x l ), l = 1, 2,..., n, where x l represents the coordinates of each point in the uniform point set, and n represents the number of divisions of the one-dimensional parameter space, n > 10;

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

[0010] Perform an inverse function transformation 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 the concrete strength θ is expressed as: φ n0 =(θ l ), l = 1, 2,..., n;

[0011] Step 2: Point set reorganization

[0012] Select the division boundary of the overall interval, and in the value range space Ω θ of the concrete strength θ, perform Voronoi set subdivision on the initial point set φ n0 to obtain n subdomains with θ l as the core Expressed as follows:

[0013]

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

[0015] Calculate the assigned probability P l of each subdivision unit, Adjust the initial point set φ n0 according to the following formula:

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

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

[0018] Obtain the adjusted point set φ n1 Denoted as: φ n1 =(θ′ l ), l = 1, 2,..., n, and perform Voronoi set subdivision on the adjusted point set φ n1 again to obtain n subdomains with θ′ l as the core And calculate the obtained probability P l ′,

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

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

[0021] Taking the rotational speed of the spherical hinge as the load level, denote the rotational speed as v i , i = 1, 2,..., I, where I represents the number of rotational speeds, perform I times of dynamic time history analysis on each spherical hinge finite element model, and record the maximum stress on the surface of the spherical hinge obtained from the i-th dynamic time history analysis

[0022] For the dynamic time history analysis results with v i as the rotational speed and θ′ l as the concrete strength, construct a virtual random process according to the extreme value, introduce a virtual time variable τ, and use a sine function as the virtual function, which is expressed as follows:

[0023]

[0024] In the formula, y represents the maximum stress on the surface of the spherical hinge of the virtual random process with θ′ l and τ as variables;

[0025] Step 4: Numerically solve the generalized probability density evolution equation using the finite difference method

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

[0027]

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

[0029] The (y~τ) plane is discretized and expressed as:

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

[0031] In the formula, y j = j·Δy, τ k = k·Δτ, Δy and Δτ are the spatial and temporal discretization step sizes respectively, and J and K are the corresponding discretization numbers;

[0032] The value of p l,i (y, θ′ l , τ) at the discretized grid point p l,i (y j , θ′ l , τ k ) is denoted as According to the Lax-Wendroff scheme, is discretized as:

[0033]

[0034] The boundary points are processed with periodic boundary conditions, that is:

[0035]

[0036] Among them:

[0037]

[0038] The initial value of:

[0039]

[0040] In the formula, δ(·) is the Dirac delta function;

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

[0042]

[0043] Then, the probability density function of the maximum stress on the spherical hinge surface considering the uncertainty of concrete strength under a certain rotational speed is expressed as follows:

[0044]

[0045] Furthermore, the stress probability density curve and the stress cumulative distribution function curve are plotted to visually display the probability corresponding to the maximum stress on each spherical hinge surface.

[0046] Step Five: Determination of the damage assessment index for the rotating system

[0047] Define \(y\in[0,y\) 0 ) as the undamaged state of the rotating system, \(y\in[y\) 0 ,y 1 ) as the slightly damaged state of the rotating system, \(y\in[y\) 1 ,y 2 ) as the moderately damaged state of the rotating system, \(y\in[y\) 2 ,y 3 ) as the severely damaged state of the rotating system, \(y\in[y\) 3 ,+\infty)\) as the completely damaged state of the rotating system, where: \(y\) 0 \( = 0.65\times y\) max ,y 1 \( = 0.75\times y\) max ,y 2 \( = 0.85\times y\) max ,y 3 \( = 0.95\times y\) max ,y max is the ultimate stress of the spherical hinge concrete obtained from the bridge design drawings.

[0048] Step Six: Calculate the vulnerability probability at each rotational speed

[0049] Based on the damage assessment index of the rotating system obtained in Step Five, calculate the probabilities of the rotating system in each damage state respectively: and are the probabilities corresponding to the undamaged state, slightly damaged state, moderately damaged state, severely damaged state, and completely damaged state of the rotating system in sequence.

[0050] Furthermore, in Step One, the distribution of the concrete strength \(\theta\) conforms to the lognormal distribution, expressed as:

[0051]

[0052]

[0053] In the formula, \(\varPhi\) represents the normal distribution function, \(\mu'\) 0 and \(\sigma'\) 0 are the mean and standard deviation of the concrete strength respectively.

[0054] Furthermore, in Step Three, the elastic modulus \(E\) and the shear modulus \(G\) are solved by the following formula:

[0055]

[0056] In the formula, ν represents the Poisson's ratio of concrete.

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

[0058] Compared with the prior art, the beneficial effects of the present invention are as follows: By establishing a finite element model for the uncertainty of spherical hinge concrete strength, considering different elastic moduli and shear moduli, conducting dynamic time history analysis of the contact stress of the spherical hinge at a specific rotational speed, recording the maximum stress on the surface of the spherical hinge, and using numerical calculation to solve the probability density evolution equation, the vulnerability assessment of the rotating system at each rotational speed is realized, and it has the following advantages:

[0059] 1. Considering the uncertainty of concrete strength: By introducing the uncertainty factor of concrete strength and combining the dynamic load effect of the spherical hinge affected by the rotational speed during the rotating construction process, the present invention establishes a more accurate method for assessing the vulnerability of the spherical hinge, breaking through the limitations of ignoring strength fluctuations and dynamic load effects in traditional assessment methods, being able to more realistically reflect the stress condition of the spherical hinge during construction, and providing a more reliable guarantee for bridge construction safety;

[0060] 2. Improving the assessment efficiency and being easy to operate: By constructing a damage state model of the spherical hinge based on uncertainty and dynamic load effect, the present invention realizes the rapid assessment of the vulnerability of the spherical hinge. Only a small number of definite concrete strength analyses are required to better simulate the stress-probability density curve, simplify the assessment process, greatly improve the assessment efficiency, and facilitate rapid decision-making by engineering and technical personnel at the construction site. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

[0063] Figure 3 is the stress cumulative distribution function curve diagram obtained in the embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0064] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a 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 those of ordinary skill in the art without making creative efforts shall fall within the protection scope of the present invention.

[0065] A vulnerability assessment method for bridge rotation construction that integrates the probabilistic performance of spherical hinges. Its process is combined with Figure 1 as shown, including the following steps:

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

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

[0068] φ Basic =(x l ), l = 1, 2,..., n

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

[0070] According to the "Code for Design of Concrete Structures" statistics, the distribution of the concrete strength θ conforms to the lognormal distribution, which is expressed as:

[0071]

[0072]

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

[0074] Perform an inverse function transformation 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 the concrete strength θ is expressed as: φ n0 =(θ l ), l = 1, 2,..., n;

[0075] Step 2: Point set reorganization

[0076] Select the division boundary of the overall interval. In the value range space Ω θ of the concrete strength θ, perform Voronoi set dissection on the initial point set φ n0 to obtain n sub-domains l with θ as the core, which is expressed as follows:

[0077]

[0078] In the formula, ‖·‖ 2Indicates the Euclidean distance.

[0079] Calculate the assignment probability P of each subdivision element l , According to the following formula, adjust the initial point set φ n0 as follows:

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

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

[0082] Obtain the adjusted point set φ n1 Denoted as: φ n1 =(θ′ l ), l = 1, 2,..., n, and perform Voronoi set subdivision on the adjusted point set φ n1 again to obtain n subdomains with θ′ l as the core and calculate the assignment probability P of each subdivision element l ′,

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

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

[0085]

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

[0087] Take the rotational speed of the spherical hinge as the load level, denote the rotational speed as v i , i = 1, 2,..., I, I represents the number of rotational speeds. Perform I times of dynamic time history analysis on each spherical hinge finite element model, that is, a total of n×I times of dynamic time history analysis. Record the maximum stress on the surface of the spherical hinge obtained from each dynamic time history analysis. Denote the maximum stress on the surface of the spherical hinge obtained from the i-th dynamic time history analysis as .

[0088] For the dynamic time history analysis results with v i as the rotational speed and θ′ l as the concrete strength, construct a virtual random process according to the extreme value, introduce the virtual time variable τ, and use the sine function as the virtual function, which is expressed as follows:

[0089]

[0090] In the formula, y represents the maximum stress on the spherical hinge surface of the virtual random process with θ′ l and τ as variables;

[0091] Step 4: Numerically solve the generalized probability density evolution equation by the finite difference method

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

[0093]

[0094] In the formula, p l,i (y, θ′ l , τ) represents the probability density function of the random variable y and the virtual time variable τ at a specific θ′ l ;

[0095] Discretize the (y~τ) plane, expressed as:

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

[0097] In the formula, y j = j·Δy, τ k = k·Δτ, Δy and Δτ are the spatial and temporal discretization steps respectively, and J and K are the corresponding discretization numbers.

[0098] For simplicity in subsequent representation, denote the value of p l,i (y, θ′ l , τ) at the discretized grid point p l,i (y j , θ′ l , τ k ) as

[0099] According to the Lax-Wendroff format, discretize as:

[0100]

[0101] To ensure the stability of the numerical solution, the boundary points are processed with periodic boundary conditions, that is:

[0102]

[0103] where:

[0104]

[0105] Initial value of:

[0106]

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

[0108]

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

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

[0111] Then the probability density function of the maximum stress on the spherical hinge surface considering the uncertainty of concrete strength at a determined rotational speed is expressed as follows:

[0112]

[0113] Furthermore, a stress probability density curve and a stress cumulative distribution function curve are plotted to visually display the probabilities corresponding to the maximum stress on each spherical hinge surface;

[0114] Step Five: Determination of the damage assessment index of the rotating system

[0115] Define y ∈ [0, y 0 ) as the non-damaged state of the rotating system, y ∈ [y 0 , y 1 ) as the slightly damaged state of the rotating system, y ∈ [y 1 , y 2 ) as the moderately damaged state of the rotating system, y ∈ [y 2 , y 3 ) as the severely damaged state of the rotating system, y ∈ [y 3 , +∞) as the completely damaged state of the rotating system, where:

[0116] y 0 = 0.65 × y max

[0117] y 1 = 0.75 × y max

[0118] y 2 = 0.85 × y max

[0119] y 3 = 0.95 × y max

[0120] Wherein, y max is the ultimate stress of the spherical hinge concrete obtained from the bridge design drawings;

[0121] Step Six: Calculate the fragility probability at each rotational speed

[0122] Based on the damage assessment index of the rotating system obtained in Step Five, calculate the probabilities of each damage state of the rotating system, which are expressed as follows:

[0123]

[0124] Wherein, and are the probabilities corresponding to the undamaged state, slightly damaged state, moderately damaged state, severely damaged state, and completely damaged state of the rotating system in sequence.

[0125] Example

[0126] This example is implemented for the special bridge of the Yinwu Connection Line. 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, use the Sobol sequence to select a uniform point set in the geometric space [0, 1], and take n = 63, then the obtained uniform point set φ Basic is shown in 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 C60 concrete strength are determined to be 64.9 MPa and 3.0 MPa respectively, then the lognormal distribution is expressed as:

[0131]

[0132] Perform the inverse function transformation on the uniform point set according to the probability distribution, and the obtained initial point set φ n0 is shown in Table 2:

[0133] Table 2 Initial point set φ n0

[0134]

[0135]

[0136] For the convenience of subsequent calculations, the initial point set φ n0 is sorted in ascending order, and the sorted initial point set φ n0 is shown in 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, the concrete strength θ, the boundary of the Voronoi set partition is the median between two points of the sorted initial point set φ n0 The 0.05% quantile and 99.34% quantile are used as the partition boundaries of the overall interval to obtain the partition boundary θ b is shown in Table 4:

[0140] Table 4 Partition 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 assigned probability of each partition unit:

[0145]

[0146] The assigned probability P of each partition unit is obtained l is shown in Table 5:

[0147] Table 5 Assigned probability P of each partition 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] Adjust the point set according to the following formula:

[0150]

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

[0152] Table 6 Probabilities 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 As shown in 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] Re - sub - divided assigned probability P l ′ As shown in Table 8:

[0158] Table 8 Re - sub - divided assigned probability 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. Establish 63 finite element models of spherical hinges according to the bridge design drawings, determine the concrete strength, elastic modulus and shear modulus of each model, conduct dynamic time - history analysis at a rotational speed of 0.5° / min, and record the maximum stress on the spherical hinge surface obtained from each dynamic time - history analysis as shown in Table 9:

[0161] Table 9 Maximum stress on the spherical hinge surface at a rotational 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 the virtual function as follows:

[0164]

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

[0166]

[0167] Where:

[0168]

[0169] The finally obtained stress probability density curve and stress cumulative distribution function curve are respectively combined with Figure 2 and Figure 3 as shown;

[0170] S5. For C60 concrete, take its maximum stress as 60 MPa, so the damage assessment index is defined as follows:

[0171] Non-damaged state 0MPa ≤ y < 39MPa Slightly damaged state 39MPa ≤ y < 45MPa Moderately damaged state 45MPa ≤ y < 51MPa Severely damaged state 51MPa ≤ y < 57MPa Completely damaged state y ≥ 57MPa

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

[0173]

[0174] The calculated results are: P 0 = 0.4532, P 1 = 0.3339, P 2 = 0.1974, P 3 = 0.0154, P 4 = 0.0001.

[0175] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and the present invention can be implemented in other forms of devices without departing from the spirit or basic characteristics of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-restrictive. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, all changes falling within the meaning and scope of the equivalent conditions of the claims are intended to be included in the present invention. Any reference signs in the claims should not be regarded as limiting the claimed claim.

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

Claims

1. A bridge rotation construction vulnerability assessment method integrating spherical joint probabilistic performance, characterized by: The following steps are involved: Step 1: Select a random vector and obtain an 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: φ 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 log-normal distribution, and the concrete strength distribution function F(θ) and the probability density function p(θ) of concrete strength distribution 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 value range space Ω of concrete strength θ θ In the example, for the initial point set φ n0 Perform Voronoi set subdivision and obtain l The core of n subdomains It is expressed as follows: In the formula, ‖·‖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 In the formula, I(·) is the indicative function, Get the adjusted point set φ n1 Expressed as: φ n1 =(θ′ l ), l=1,2,...,n, and for the adjusted point set φ n1 Perform Voronoi set subdivision again and obtain l The core of n subdomains And calculate the probability P of each subdivision unit l ′, Step 3: Establish finite element model and conduct 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 a virtual function, which is expressed as follows: In the formula, y represents the l The maximum stress on the spherical joint surface of the pseudo random process with τ as variables; 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: In the formula, 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: yy j ,τ=τ k ,j=0,1,2,...,J,k=0,1,2,...,K In the formula, y j =j·Δy,τ k = k·Δτ, Δy and Δτ are discrete steps in space and time, respectively, and J and K are corresponding discrete quantities; The 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: The 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 joint 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 Then the probability density function of the maximum stress on the spherical joint surface under the determined rotation speed considering the uncertainty of concrete strength is expressed as follows: Then, the stress probability density curve and stress cumulative distribution function curve are drawn to visualize the probability corresponding to the maximum stress on each spherical joint surface; Step 5: Determination of the damage assessment index of the rotating system 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: 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 probability of fragility at each rotation speed 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: 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. According to claim 1, a bridge rotation construction vulnerability assessment method integrating spherical joint probabilistic performance is characterized by: In the step 1, the distribution of concrete strength θ conforms to the log-normal distribution, which is 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 the probabilistic performance of spherical joints 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 the probabilistic performance of spherical joints according to claim 1 is characterized by: In step 4, in order to keep the numerical solution stable,

Images