A seismic reliability analysis method for double-groove aqueduct structures under strong earthquake action

Abstract

The present invention discloses a method for analyzing the seismic reliability of a double-groove aqueduct structure under strong earthquake action, including: S1, constructing a refined fiber beam element numerical model of the double-groove aqueduct structure; S2, based on the refined fiber beam element numerical model, conducting a nonlinear dynamic response analysis of the double-groove aqueduct structure; S3, based on the nonlinear dynamic response analysis process, simulating and analyzing the structural collapse of the double-groove aqueduct structure under earthquake action to determine the collapse judgment criterion for the overall instability of the double-groove aqueduct structure; S4, under random earthquake action, determining the overall seismic reliability of the double-groove aqueduct structure by using the probability density evolution method based on the collapse judgment criterion. The present invention establishes an overall collapse judgment criterion for the aqueduct structure under earthquake action from the perspective of energy absorption and dissipation, can accurately judge whether the aqueduct structure collapses, and can accurately give the critical collapse time, realizing the collapse analysis of reinforced concrete aqueduct structures under disastrous dynamic loads.

Description

Technical Field

[0001] The present invention belongs to the technical field of seismic resistance of aqueduct structures, and particularly relates to a method for analyzing the seismic reliability of double-groove aqueduct structures under strong earthquake actions. Background Technique

[0002] Large aqueduct structures will produce large deformations under strong earthquake actions, and some components will enter the plastic state. Their dynamic responses will be affected by material nonlinearity and geometric nonlinearity. The commonly used methods of linear elasticity, local component nonlinearity, and elastic-plastic separation in previous studies have certain limitations, and the results obtained often deviate greatly from the actual situation, resulting in a relatively conservative design of the seismic performance of aqueduct structures.

[0003] At present, some scholars have achieved fruitful research on the seismic reliability analysis of aqueduct structures. Wu Jianguo et al. proposed a sampling method for calculating the system reliability of aqueduct structures. According to the Metropolis criterion, a Markov chain was constructed to simulate samples, and by pre-sampling in the failure domain, the distribution information of the region with a large contribution to the failure probability was obtained, and then the failure probability of the structural system was calculated. An Xuwen and Zhu Tun suggested the inter-story displacement limit of the aqueduct truss based on the deformation failure criterion, and used the method combining Monte Carlo simulation and finite element analysis to study the reliability of the aqueduct truss. Ma and Chen

[44] established a reliability analysis model of the aqueduct system based on the main failure mode and the comprehensive correlation coefficient method, and discussed the system reliability of the multi-side walls of the pile beams of the Minghe Aqueduct. Zhang Duoxin et al. systematically summarized the research progress of aqueduct structure dynamics in the past decade, and pointed out that the dynamic reliability theory based on engineering structures has been studied in aqueduct engineering and shown promising prospects.

[0004] The above research provides important references for the seismic design of aqueduct structures. However, the existing literature studies have shown that the randomness of earthquake excitation will have a significant impact on the seismic response of engineering structures. In fact, due to the coupling of nonlinearity and earthquake randomness, the damage and failure of structures have great randomness, resulting in the position and failure mode of the structure's first failure being very different from the expected design. Therefore, it is more necessary to carry out seismic response and reliability analysis considering the randomness of earthquake excitation for the "top-heavy" aqueduct structures to more objectively reflect the response law and seismic reliability of aqueduct structures when encountering earthquake actions. Zhang Wei and Xu Jianguo started from the dynamic reliability theory and carried out nonlinear dynamic response analysis and reliability research on aqueduct structures under random earthquake ground motion excitation, and found that the randomness of earthquake excitation will have a significant impact on the nonlinear dynamic response law of aqueduct structures. Wang Zhou et al. established two generalized evolution spectrum models of fully non-stationary earthquake ground motion acceleration processes, and calculated the seismic reliability of aqueduct structures under random earthquake actions by combining the probability density evolution theory.

[0005] The above research results provide an important reference for the overall seismic reliability analysis of aqueduct structures under random earthquake actions. However, the reliability calculation mainly relies on the random simulation method, which has low calculation efficiency. Moreover, due to the lack of a unified structural failure criterion, empirical judgment is often relied on in deterministic analysis, and it is impossible to accurately and uniformly obtain the overall seismic reliability of aqueduct structures. Summary of the Invention

[0006] In view of the above deficiencies in the prior art, the seismic reliability analysis method for double-groove aqueduct structures under strong earthquake actions provided by the present invention solves the problem that the existing reliability calculation is based on the random simulation method and has low calculation efficiency.

[0007] In order to achieve the above invention object, the technical solution adopted by the present invention is as follows: A seismic reliability analysis method for double-groove aqueduct structures under strong earthquake actions, comprising the following steps:

[0008] S1. Construct a refined fiber beam element numerical model of the double-groove aqueduct structure;

[0009] S2. Based on the refined fiber beam element numerical model, perform a nonlinear dynamic response analysis on the double-groove aqueduct structure;

[0010] S3. Based on the nonlinear dynamic response analysis process, simulate and analyze the structural collapse of the double-groove aqueduct structure under earthquake actions, and determine the collapse criterion for the overall instability of the double-groove aqueduct structure;

[0011] S4. Under random earthquake actions, based on the collapse criterion, use the probability density evolution method to determine the overall seismic reliability of the double-groove aqueduct structure.

[0012] Further, the refined fiber beam element numerical model in step S1 satisfies the following conditions:

[0013] (1) Consider the influence of the coupled vibration of the lateral bending moment and warping deformation of the thin-walled aqueduct structure on the stiffness matrix of the thin-walled beam element;

[0014] (2) The deformation of the aqueduct interface satisfies the plane section assumption;

[0015] (3) The steel bars and concrete fibers are both in a uniaxial stress state, and are respectively represented by an elastoplastic tensile concrete constitutive model and a steel bar hardening constitutive model under cyclic loading.

[0016] Further, in the refined fiber beam element numerical model:

[0017] The displacement of any point within the model element is obtained by introducing a warping function, which is expressed as:

[0018]

[0019] where U(x, y, z) is the horizontal lateral displacement at any point, u(x) is the lateral horizontal displacement at the centroid of any point, θ z (x) is the rotation angle of the cross-section about the z-axis, ω(x) is the value of the cross-section warping function, θ y (x) is the rotation angle of the cross-section about the y-axis, θ x (x) is the rotation angle of the cross-section about the x-axis, v(x) is the longitudinal horizontal displacement at the centroid of any point, x is the coordinate value of the node on the x-axis, y is the coordinate value of the node on the y-axis, and z is the coordinate value of the node on the z-axis;

[0020] When the ferry structure enters the elastoplastic and large displacement deformation, the strain matrix considering the influence of nonlinearity and geometric nonlinearity on the stiffness matrix is:

[0021]

[0022] where [B0] is the linear small displacement strain matrix; [B L ({δ})] is the geometric nonlinear strain matrix, and δ is the nodal displacement array;

[0023] The stiffness matrix considering geometric nonlinear deformation is:

[0024]

[0025] where the subscript Ω is the entire integration region, [D] is the stiffness matrix, [k0] is the linear stiffness matrix, and [k L is the stiffness matrix of the nonlinear influence.

[0026] Furthermore, in the step S2, the nonlinear dynamic response analysis of the double flume structure includes the nonlinear response analysis of the displacement of the flume structure and the nonlinear response analysis of the internal forces at each cross-section of the flume structure;

[0027] Among them, the nonlinear response analysis of the displacement of the flume structure is the nonlinear response of the flume structure under different site types and different amplitude-modulated seismic waves; the nonlinear response value of the flume structure increases as the site condition weakens, and for different amplitude-modulated seismic waves, the displacement response of the flume structure continues to increase after entering the highly nonlinear state;

[0028] The nonlinear response analysis of the internal forces at each cross-section of the flume structure includes the calculation results of the nonlinear responses of the mid-span of the side span, the mid-span of the middle span, and the pier bottom cross-section of the flume structure.

[0029] Furthermore, in step S3, the incremental dynamic analysis method is adopted. By inputting the Northridge seismic wave with different amplitude values into the double-groove aqueduct structure, the structural collapse simulation is carried out based on the nonlinear dynamic response analysis process of the double-groove aqueduct structure, and the energy change of the double-groove aqueduct structure and the whole process of overall instability and collapse of the double-groove aqueduct structure are obtained during the simulation process.

[0030] Furthermore, the judgment criterion for the structural collapse in step S3 is as follows:

[0031] Under the action of seismic loads, when the effective characteristic energy of the whole double-groove aqueduct structure is less than the effective input energy, the structure maintains an overall stable state; conversely, if the effective characteristic energy of the structure exceeds the effective energy at a certain moment, the structure will show dynamic instability linearity from this moment and then cause collapse; it is expressed as:

[0032]

[0033] Among them, the effective characteristic energy E eff_intr (t) is the energy reflecting the structure's own properties generated during the self-vibration process of the structure caused by external load excitation, and its expression is:

[0034]

[0035] In the formula, Π(t) is the generalized complementary energy of the system at time t, f T (u,t) is the restoring force vector, u(t) is the nodal displacement vector, σ is the stress tensor, is the elastic strain rate tensor, and V is the integral region of the aqueduct structure;

[0036] The effective input energy E eff_inp (t) is the instantaneous energy input from the outside world into the structural system, and its expression is:

[0037]

[0038] In the formula, F T (t) is the external dynamic excitation vector, u(t) is the nodal displacement vector, is the nodal velocity vector, is the system damping force vector, E k (t) is the kinetic energy of the system at time t, E es (t) is the elastic strain energy of the system at time t, is the momentum of the system at time t;

[0039] The critical state judgment function for structural collapse is:

[0040] S c (t) = E eff_inp (t) - Eeff_intr (t)

[0041] Wherein, S c (t) is a critical state determination function. When S c (t) ≥ 0 holds for all t ≥ 0, the structure maintains dynamic stability; while when S c (t) < 0 appears for the first time, the structure will collapse due to instability.

[0042] Furthermore, the specific steps of step S4 are as follows:

[0043] S41. Construct a physical model of random earthquake motion;

[0044] S42. According to the physical model of random earthquake motion, randomly generate seismic excitations for the double-groove aqueduct structure;

[0045] S43. Based on the generated seismic excitations, use the probability density evolution method to solve the probability density evolution information of the double-groove aqueduct structure at different levels;

[0046] S44. Based on the probability density evolution information and combined with the dynamic reliability of the double-groove aqueduct structure based on the collapse judgment criterion, determine the overall reliability of the aqueduct structure at different levels, and then determine the overall seismic reliability of the double-groove aqueduct structure.

[0047] Furthermore, in step S44, the expression of the dynamic reliability R(t) of the double-groove aqueduct structure is:

[0048] R(t) = Pr{X(τ) ∈ Ω S , 0 ≤ τ ≤ t

[0049] Wherein, Pr{· is the symbol for calculating reliability, X is the physical quantity that causes the realization of the double-groove aqueduct structure, τ is time, and Ω S is the safety domain of the double-groove aqueduct structure.

[0050] Furthermore, in step S44, when determining the overall reliability of the aqueduct structure at different levels, the collapse judgment criterion of the aqueduct structure based on effective energy and pier top displacement angle is used as the physical mechanism for triggering probability dissipation.

[0051] The beneficial effects of the present invention are as follows:

[0052] (1) Based on the whole process and non-linear perspective, the present invention conducts research on the dynamic non-linear seismic response and overall seismic reliability of large-scale reinforced concrete double-groove aqueduct structures, which has important theoretical significance and engineering value. The present invention combines a refined structural element model and an efficient numerical algorithm, develops a fiber beam element subroutine suitable for simulating the non-linearity of reinforced concrete materials of large-scale double-groove aqueduct structures, and conducts non-linear dynamic time-history response analysis of double-groove aqueducts under different seismic excitations.

[0053] (2) From the perspective of energy absorption and dissipation, the present invention establishes an overall collapse judgment criterion for aqueduct structures under seismic action. This criterion can accurately determine whether the aqueduct structure collapses and can accurately give the critical collapse time, realizing the collapse analysis of reinforced concrete aqueduct structures under catastrophic dynamic loads.

[0054] (3) The present invention proposes two different levels of random seismic overall seismic reliability analysis methods for aqueduct structures. Under the design earthquake level of "no damage under minor earthquakes", the displacement angle limit at the top of the pier is used as the overall failure criterion of the structure; under the design earthquake level of "no collapse under major earthquakes", the overall failure criterion based on effective energy is used to qualitatively analyze the seismic reliability of the aqueduct structure. Description of the Drawings

[0055] Figure 1 It is a flowchart of the seismic reliability analysis method for the double-groove aqueduct structure under strong earthquake action in the embodiment of the present invention.

[0056] Figure 2 It is a fiber beam element model of the double-groove aqueduct in the embodiment of the present invention.

[0057] Figure 3 It is the time history curve of the displacement angle at the top of the pier in the embodiment of the present invention.

[0058] Figure 4 It is the stress-strain relationship curve of concrete elements at different positions in working condition 3 in the embodiment of the present invention.

[0059] Figure 5 It is the cross-section yield situation at 4 integration points of the bottom section of the pier in the embodiment of the present invention.

[0060] Figure 6 It is the time history curve of the displacement angle at the top of the pier under the amplitude modulation of 0.1g of 100 representative random ground motions in the embodiment of the present invention.

[0061] Figure 7 It is a comparison chart of the statistical mean and standard deviation of the displacement angle at the top of the pier in the embodiment of the present invention.

[0062] Figure 8 It is the probability density surface within a typical period in the embodiment of the present invention.

[0063] Figure 9 It is the probability density contour map in the embodiment of the present invention.

[0064] Figure 10 It is the probability density function at a typical moment in the embodiment of the present invention.

[0065] Figure 11 It is the overall seismic reliability curve of the aqueduct structure under different failure indices in the embodiment of the present invention.

[0066] Figure 12 The time history curves of the effective characteristic energy and the effective input energy of the aqueduct structure in the embodiments of the present invention.

[0067] Figure 13 The time history curve of the displacement angle at the top of the pier of the aqueduct structure in the embodiments of the present invention.

[0068] Figure 14 The reliability comparison of the aqueduct structure based on the energy collapse criterion and the limit value of the displacement angle at the top of the pier in the embodiments of the present invention. Specific implementation manners

[0069] The following describes the specific implementation manners of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation manners. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions and creations using the concept of the present invention are within the scope of protection.

[0070] Embodiment 1:

[0071] The embodiments of the present invention provide a method for analyzing the seismic reliability of a double-groove aqueduct structure under strong earthquake action, as Figure 1 shown, including the following steps:

[0072] S1. Construct a refined fiber beam element numerical model of the double-groove aqueduct structure;

[0073] S2. Based on the refined fiber beam element numerical model, perform a nonlinear dynamic response analysis on the double-groove aqueduct structure;

[0074] S3. Based on the nonlinear dynamic response analysis process, simulate and analyze the structural collapse of the double-groove aqueduct structure under earthquake action, and determine the collapse judgment criterion for the overall instability of the double-groove aqueduct structure;

[0075] S4. Under random earthquake action, based on the collapse judgment criterion, use the probability density evolution method to determine the overall seismic reliability of the double-groove aqueduct structure.

[0076] In step S1 of the embodiments of the present invention, the refined fiber beam element numerical model satisfies the following conditions:

[0077] (1) Consider the influence of the coupled vibration of the lateral bending moment and warping deformation of the thin-walled structure of the aqueduct on the stiffness matrix of the thin-walled beam element;

[0078] (2) The deformation of the aqueduct interface satisfies the plane section assumption;

[0079] (3) Both the steel bars and the concrete fibers are in a uniaxial stress state, and are respectively represented by an elastoplastic tensile concrete constitutive model and a steel bar hardening constitutive model under repeated loads.

[0080] In this embodiment, as Figure 2 shown in the refined fiber beam element numerical model, since the warping deformation has a great influence on the overall deformation of the aqueduct thin-walled structure; therefore, the displacement of any point within the model element is obtained by introducing a warping function, which is expressed as:

[0081]

[0082] In the formula, U(x, y, z) is the horizontal lateral displacement of any point, u(x) is the lateral horizontal displacement at the centroid of any point, θ z (x) is the rotation angle of the cross-section about the z-axis, ω(x) is the value of the cross-section warping function, θ y (x) is the rotation angle of the cross-section about the y-axis, θ x (x) is the rotation angle of the cross-section about the x-axis, v(x) is the longitudinal horizontal displacement at the centroid of any point, x is the node x-axis coordinate value, y is the node y-axis coordinate value, and z is the node z-axis coordinate value;

[0083] When the aqueduct structure enters the elastoplastic and large displacement deformation, considering the influence of nonlinearity and geometric nonlinearity on the stiffness matrix, the strain matrix is:

[0084]

[0085] In the formula, [B0] is the linear small displacement strain matrix; [B L ({δ})] is the geometric nonlinear strain matrix, and δ is the nodal displacement column matrix;

[0086] The stiffness matrix considering geometric nonlinear deformation is:

[0087]

[0088] In the formula, the subscript Ω is the entire integration region, [D] is the stiffness matrix, [k0] is the linear stiffness matrix, and [k L is the nonlinear influence stiffness matrix.

[0089] In step S2 of the embodiment of the present invention, the nonlinear dynamic response analysis of the double-channel aqueduct structure includes the nonlinear response analysis of the aqueduct structure displacement and the nonlinear response analysis of the internal forces at each cross-section of the aqueduct structure;

[0090] Among them, the nonlinear response analysis of the aqueduct structure displacement is the nonlinear response of the aqueduct structure under different site types and different amplitude-modulated seismic waves; the nonlinear response value of the aqueduct structure increases as the site conditions weaken, and for different amplitude-modulated seismic waves, the displacement response of the aqueduct structure continues to increase after entering a highly linear state; among them, different types of sites include hard sites, medium-hard sites, medium-soft sites, and soft sites;

[0091] The analysis of the nonlinear response of the internal forces at each cross-section of the aqueduct structure includes the calculation results of the nonlinear responses of the mid-span of the side span, the mid-span of the middle span, and the pier bottom cross-section of the aqueduct structure. The overall seismic performance of the aqueduct structure is reflected from two aspects: the macroscopic (displacement, internal force, etc.) and microscopic (stress time history curve, strain response, etc.) analysis results of the structure. The mid-span and pier bottom of the large double-channel aqueduct structure are its seismic weak links.

[0092] In step S3 of the embodiment of the present invention, based on the constructed refined fiber beam element numerical model of the aqueduct structure, the incremental dynamic analysis method is adopted. By inputting the Northridge seismic wave with different amplitude values into the double-channel aqueduct structure, the structure collapse simulation is carried out based on the nonlinear dynamic response analysis process of the double-channel aqueduct structure, and the energy change of the double-channel aqueduct structure and the whole process of the overall instability and collapse of the double-channel aqueduct structure are obtained.

[0093] In the process of the energy change of the double-channel aqueduct structure, through the method of energy monitoring, the elastic-plastic performance of the structure itself, the plastic cumulative effect of the structure under seismic loads, and the characteristics of seismic motion can be quickly and accurately understood. Moreover, information such as the total input energy of the structure under seismic action, the form of energy existence, and the distribution law of energy can also be mastered.

[0094] In step S3 of the embodiment of the present invention, the criterion for judging the structure collapse is:

[0095] Under the action of seismic loads, when the effective characteristic energy of the whole double-channel aqueduct structure is less than the effective input energy, the structure maintains an overall stable state; on the contrary, if the effective characteristic energy of the structure exceeds the effective energy at a certain moment, the structure will show dynamic instability linearity from this moment and then cause collapse; it is expressed as:

[0096]

[0097] Among them, the effective characteristic energy E eff_intr (t) is the energy reflecting the structure's own attributes generated during the vibration process of the structure caused by external load excitation, and its expression is:

[0098]

[0099] In the formula, Π(t) is the generalized complementary energy of the system at time t, f T(u, t) is the restoring force vector, u(t) is the nodal displacement vector, σ is the stress tensor, is the elastic strain rate tensor, V is the integral region of the aqueduct structure;

[0100] The effective input energy E eff_inp (t) is the instantaneous energy input from the outside world into the structural system, and its expression is:

[0101]

[0102] In the formula, F T (t) is the external dynamic excitation vector, u(t) is the nodal displacement vector, is the nodal velocity vector, is the system damping force vector, E k (t) is the kinetic energy of the system at time t, E es (t) is the elastic strain energy of the system at time t, is the momentum of the system at time t;

[0103] The critical state determination function for the structural collapse is:

[0104] S c (t) = E eff_inp (t) - E eff_intr (t)

[0105] In the formula, S c (t) is the critical state determination function. When S c (t) ≥ 0 holds for all t ≥ 0, the structure remains dynamically stable; while when S c (t) < 0 appears for the first time, the structure will undergo instability and collapse.

[0106] In addition, in the numerical simulation of structural collapse, the relevant energy indicators used in the above collapse criterion can be calculated from the structural internal forces, stresses, displacements and other indicators obtained through the nonlinear dynamic response analysis of the structure, so as to timely determine the stable state of the structural system. Except for the calculation work of the structural dynamic analysis itself, the above effective energy criterion does not increase the additional computational workload in the process of determining the structural instability and collapse, and has a high computational efficiency.

[0107] In the collapse determination criterion provided in the embodiments of the present invention, the effective characteristic energy of the aqueduct structure is the energy reflecting the structural own attributes generated during the self-vibration process of the structure caused by the external load excitation; the effective input energy is the instantaneous energy input from the outside world into the structural system. When the effective characteristic energy of the structure is always less than the effective input energy, the aqueduct structure remains in a dynamically stable state; while when at a certain moment, the effective characteristic energy of the aqueduct structure exceeds the effective input energy for the first time, the structure enters the dynamic instability state and further undergoes collapse damage.

[0108] In the process of the structural nonlinear development in step S4 of the embodiment of the present invention, the collapse failure mode will vary greatly due to the influence of earthquake randomness. Therefore, it is necessary to carry out the anti-collapse reliability analysis of the "top-heavy and bottom-light" aqueduct structure under random seismic action. Based on this, step S4 is specifically as follows:

[0109] S41. Construct a physical model of random earthquake motion;

[0110] S42. According to the physical model of random earthquake motion, randomly generate seismic excitations for the double-groove aqueduct structure;

[0111] S43. Based on the generated seismic excitations, use the probability density evolution method to solve the probability density evolution information of the double-groove aqueduct structure at different levels;

[0112] S44. Based on the probability density evolution information and combined with the dynamic reliability of the double-groove aqueduct structure based on the collapse judgment criterion, determine the overall reliability of the aqueduct structure at different levels, and then determine the overall seismic reliability of the double-groove aqueduct structure.

[0113] The basic physical quantities of the physical model of random earthquake motion in step S41 of this embodiment are observable and statistically measurable. For a specific site, the probability distribution of the equivalent site dominant circular frequency and damping ratio can be obtained through on-site tests, thereby greatly reducing the variation range of random earthquake motion. Based on this model, the anti-collapse reliability analysis of the aqueduct structure under random seismic excitation can be carried out.

[0114] In step S43 of this embodiment, the probability density evolution method provides a reference for the reliability solution of complex structures. In the probability density evolution theory, the motion equation of any multi-degree-of-freedom random dynamic system is expressed as:

[0115]

[0116] In the formula, Θ = (Θ1, Θ2,..., Θ n ) represents the random variables involved in the structural system, and n is the number of random variables; M represents the mass matrix of the system; C represents the damping matrix of the system; f int (·) represents the restoring force vector; f ext (·) represents the external dynamic load vector; u(t) respectively represent the acceleration, velocity and displacement vectors of the system.

[0117] Assume Z = (Z1, Z2,..., Z m ) T represents the physical response quantities of the system of interest, and m is the number of response quantities. According to the principle of probability conservation, the joint probability density function p of the augmented system (Z, Θ) zΘ(z, θ, t) satisfies the equation:

[0118]

[0119] Its initial conditions are:

[0120]

[0121] In the formula, δ(·) is the Dirac-δ function; z0 is the deterministic initial value.

[0122] By combining and solving the above equations, we can obtain:

[0123]

[0124] Among them, Ω θ is the probability space of random variables.

[0125] When only interested in a certain physical quantity, the above equation can be reduced to a one-dimensional partial differential equation:

[0126]

[0127] From the above evolution equation, it can be seen that the internal mechanism of probability density evolution is the evolution of the physical state of the system, and the above equation establishes the internal connection between the deterministic system and the random system.

[0128] When using the probability density evolution method for numerical solution in step S43 of this embodiment, for a complex random dynamic system, it is necessary to combine the physical equation with the probability density evolution equation, and obtain the probability density function of the response quantity of interest through numerical methods. To solve the generalized probability density evolution equation, it can be gradually realized through the following four steps:

[0129] (1) Probability space subdivision: In the probability space Ω θ of the basic random vector Θ, a series of discrete representative points θ q are selected by using number theory methods or an optimization point selection strategy based on GF-deviation, and at the same time, the assigned probability of each representative point is determined where V q represents the representative volume

[0130] (2) Solving the deterministic dynamic system. For the given Θ = θ q , solve the above motion equation to obtain the time derivative of the physical quantity of interest

[0131] (3) Solving the generalized probability density evolution equation. After selecting discrete representative points and determining the assigned probabilities in step (1), the joint probability density function p zΘ (z, θ, t) becomes:

[0132]

[0133] The corresponding initial condition formula becomes:

[0134]

[0135] That is, substituting the obtained in step (2) into the above formula), the joint probability density function P of (Z, Θ) can be solved zΘ (z, θ, t), q = 1, 2,..., n.

[0136] (4) Cumulative summation. Cumulating all the joint probability density functions obtained in step (3), the numerical solution of the joint probability density function P z (z, t) of Z(t) can be obtained:

[0137]

[0138] The above method combines the solution of a series of deterministic dynamic systems with the solution of the probability density evolution equation.

[0139] In step S44 of this embodiment, the expression of the dynamic reliability R(t) of the double-groove aqueduct structure is:

[0140] R(t) = Pr{X(τ) ∈ Ω S , 0 ≤ τ ≤ t

[0141] where Pr{· is the symbol for calculating reliability, X is the physical quantity that causes the realization of the double-groove aqueduct structure, τ is time, and Ω S is the safety domain of the double-groove aqueduct structure.

[0142] In step S44 of this embodiment, when determining the overall reliability of the aqueduct structure at different levels, the collapse judgment criterion of the aqueduct structure based on effective energy and the pier top displacement angle is used as the physical mechanism for triggering probability dissipation. For example, under the design seismic fortification level of "the structure remains undamaged under minor earthquakes", the overall reliability of the aqueduct structure under the goal of "the structure remains undamaged under minor earthquakes" can be obtained by setting the threshold of the maximum elastic pier top displacement angle and using whether the pier top displacement angle of the aqueduct structure exceeds the set value as the basis for collapse judgment; similarly, for the design seismic fortification level of "the structure does not collapse under major earthquakes", the overall collapse judgment criterion of the aqueduct structure based on effective energy is adopted to calculate the overall reliability of the aqueduct structure under the corresponding seismic fortification goal.

[0143] Example 2:

[0144] This embodiment of the present invention provides a verification example of the reliability of the collapse judgment criterion mentioned in the example:

[0145] Figure 3The time history curves of the displacement angles at the top of the aqueduct piers under four working conditions are given. It can be seen from Figure 3 that there is no drift phenomenon in the displacement at the top of the aqueduct structure under Working Conditions 1 and 2, indicating that the aqueduct structure does not experience instability and collapse under these two working conditions. However, it is found from the figure that there is a significant drift phenomenon in the time history of the displacement angles at the top of the aqueduct under Working Conditions 3 and 4, predicting that the aqueduct structure will experience instability and collapse due to the degradation of structural stiffness and the softening of materials. Referring to previous research experience, the start of the collapse of the aqueduct structure is roughly defined by the first exceeding 5% of the displacement angle at the top of the aqueduct. It can be seen from Figure 3 that the initial collapse time of the aqueduct structure under Working Condition 3 is approximately 5.8 s, and the initial collapse time of the aqueduct structure under Working Condition 4 is approximately 5.3 s. By comparison, it is found that both the judgment result based on the effective energy criterion and the collapse judgment criterion based on the displacement angle at the top of the pier can accurately determine whether the aqueduct structure collapses. When the limit value of the displacement angle at the top of the pier is defined as 5%, the predictions of the two judgment methods for the initial collapse time are also basically consistent, thus verifying the reliability of the collapse judgment method for the aqueduct structure based on the effective energy criterion. The above results show that whether it is the non-collapse working condition or the collapse working condition, the collapse judgment criterion based on the effective energy can accurately predict the dynamic stability of the aqueduct structure under earthquake action.

[0146] Taking Working Condition 3 as an example, the stress-strain relationship curves of the concrete elements at the bottom and top of the left pier and the middle pier are extracted respectively, as shown in Figure 4 . It can be seen that under Working Condition 3, a large number of concrete components in the structure experience plastic damage under earthquake action, and the increase in plastic energy dissipation is basically consistent with the change in plastic energy dissipation. By comparing the stress-strain relationship curves of the four parts, it can be seen that the middle pier of the aqueduct structure does not yield in compression at the initial stage of earthquake action, and the top of the pier is mainly damaged in compression. The bottom part of the pier is more likely to be damaged in tension at the initial stage of earthquake action.

[0147] It can be seen from Figure 4 that under the action of a disastrous earthquake, the concrete elements at the bottom of the left pier, the bottom of the middle pier, and the top of the middle pier are in tension and yield, while the compression does not reach the yield state; the concrete elements at the top of the left pier are in tension and compression and reach the yield state. There is a certain degree of yield failure in all four parts. The material stress-strain curve can further explain the collapse behavior of the aqueduct structure from the microscopic perspective of material yield.

[0148] During the numerical analysis process, when a certain material point in the element reaches the corresponding material yield criterion, this material point fails. When all the material points in the element yield and fail, this element also yields and fails. In order to reflect the nonlinear development of the structural element under earthquake action, the output of the control variable is programmed in Fortran language in the calculation program. When the cross-section of the element does not yield, the output result is 0, and when the cross-section of the element yields, the output result is 1.

[0149] Taking the bottom section of the pier of the aqueduct structure mentioned above as an example, Figure 5 It shows the yield conditions of the sections at four different integration points. It can be seen from the figure that the sections at the four integration points at the bottom of the pier of the aqueduct structure reach yield failure successively, and the time of their yield failure is slightly earlier than the time of the overall instability and collapse of the aqueduct structure. As mentioned above, the bottom part of the pier of the aqueduct structure is its seismic weak link, and it enters the nonlinear state earliest and has the strongest nonlinear degree under the action of earthquake. It can be seen that the damage and failure at the bottom of the pier are the direct reasons for the overall instability and collapse of the aqueduct structure.

[0150] Example 3:

[0151] This example provides a specific application example of the reliability analysis method in Example 1:

[0152] Adopt the number theory method to select 100 discrete representative points of random variables, generate 100 random ground motion samples with assigned probabilities, and input them into the numerical model of the aqueduct structure for random seismic excitation nonlinear response analysis to obtain the physical quantities of interest. In the following reliability analysis of the aqueduct structure, the aqueduct structure is divided into two working conditions for calculation, corresponding to failures at two levels respectively, and the corresponding failure criteria are selected for determination:

[0153] (1) Random seismic reliability analysis of the aqueduct structure under small earthquake action

[0154] During the analysis of the seismic reliability of the aqueduct structure under small earthquake action, 100 random ground motions with a peak acceleration of 0.1g are input into the aqueduct structure, and the dynamic nonlinear analysis method of the aqueduct structure established in Chapter 3 is adopted to obtain the physical response quantities of interest. Furthermore, based on the probability density evolution method, the time evolution of the probability density of the response quantity of the aqueduct structure is obtained.

[0155] Figure 6 The time history curve of the displacement angle at the top of the pier of the aqueduct structure under 0.1g random seismic excitation is given. It can be seen from the figure that even under the action of a small earthquake with a peak acceleration of 0.1g, the randomness of the seismic excitation also causes significant differences in the displacement response at the top of the pier of the aqueduct structure. The statistical mean and statistical standard deviation of the displacement angle response at the top of the pier of the aqueduct structure are as Figure 7 shown. Through comparative analysis, it is found that the variability of the displacement angle response at the top of the pier reaches more than 5 times. It can be seen that the influence caused by random seismic excitation cannot be ignored in the study of the overall seismic performance of the aqueduct structure.

[0156] The probability density evolution surface and the corresponding probability density contour map of the displacement angle response at the top of the pier of the aqueduct structure within a typical period are as Figure 8 and Figure 9As shown, the two extend like mountains and rivers respectively and fluctuate randomly. Obviously, based on the idea of probability density evolution analysis, the probability information of the structural response at any time can be directly obtained. These rich probability information can comprehensively describe the change of the overall dynamic response of the structure with time, and then be conveniently applied to the overall process reliability assessment of the structure. Figure 10 The probability density function curves of the displacement angle at the top of the pier of the aqueduct structure at three typical times are given. It can be seen that the probability density function curves are irregular in shape, and both the shape and the distribution width change with time due to the influence of the coupling of earthquake randomness and structural nonlinearity.

[0157] Based on the different-level reliability analysis methods of the aqueduct structure proposed in this paper, under the action of small earthquakes, taking whether the maximum displacement angle at the top of the pier of the aqueduct structure exceeds the threshold condition as the criterion for judging whether the overall structure fails, combined with the numerical solution of the probability density evolution equation, on the basis of obtaining the probability density information of the random dynamic response of the aqueduct structure, the time-varying curve of the seismic reliability of the aqueduct structure under different displacement angle threshold conditions at the top of the pier is shown in the figure. It can be found from the figure that the calculation result of the overall seismic reliability of the aqueduct structure is related to the selection of the displacement angle threshold at the top of the pier. The larger the selected displacement angle threshold at the top of the pier, the higher the overall seismic reliability of the aqueduct structure.

[0158] The limit value of the maximum displacement angle is widely used as an important control index in the seismic design of structures in the research on the seismic performance of various structures. Since the current seismic design code for hydraulic structures lacks relevant control indicators, referring to the seismic design code for buildings, under the action of 0.1g random earthquake, the displacement angle threshold at the top of the pier is taken as 0.125% in the linear elastic state and 1% in the nonlinear state. According to Figure 11 It can be known that the overall seismic reliability of this aqueduct structure calculated according to the displacement angle threshold at the top of the pier in the linear elastic state is 67.8%, while the overall seismic reliability of this aqueduct structure calculated according to the displacement angle threshold at the top of the pier in the nonlinear state is 100%.

[0159] (2) Random seismic reliability analysis of aqueduct structure under strong earthquakes

[0160] Based on the aforementioned different-level reliability analysis methods of the aqueduct structure, taking the energy in the aqueduct structure system as the physical quantity of interest, the seismic reliability of the aqueduct structure under the design earthquake intensity of "not collapsing under strong earthquakes" is obtained. In fact, when performing numerical calculations, the energy judgment criterion can be coupled with the seismic nonlinear response analysis of the structure, and the energy index in the structure collapse criterion can be calculated through Python scripts, and then the dynamic stability state of the structure system can be judged in real time. Limited by space, Figure 12The time - history curves of the effective characteristic energy and the effective input energy of the aqueduct structures for two samples are given. It can be seen that the effective characteristic energy in the aqueduct structure system of Sample 1 is always less than the effective input energy. According to the proposed collapse judgment criterion, the aqueduct structure in Sample 1 is overall stable. In Sample 2, the effective characteristic energy in the aqueduct structure system first exceeds the effective input energy around 15.4 s, and it can be inferred that the aqueduct structure in Sample 2 collapses around 15.4 s.

[0161] Similarly, in order to verify the discrimination results of the above - mentioned collapse judgment criterion for aqueduct structures based on effective energy, the time - history curves of the displacement angles at the top of the piers for Sample 1 and Sample 2 under corresponding working conditions are given, as Figure 13 shown. There is no obvious mutation in the displacement angle at the top of the aqueduct structure in Sample 1, indicating that the aqueduct structure is always in a dynamically stable state, which is consistent with the result based on the effective energy judgment criterion. In Sample 2, there is a large drift in the corresponding time - history curve of the displacement angle at the top of the aqueduct structure. When the maximum allowable displacement angle of the aqueduct structure is assumed to be 6%, and the time when the displacement angle first reaches 6% is around 15.4 s, it means that the aqueduct structure collapses from this moment. Obviously, whether the aqueduct structure in Sample 1 is dynamically stable or the aqueduct structure in Sample 2 collapses, the collapse judgment criterion for aqueduct structures based on the effective energy criterion can make an effective judgment.

[0162] There are no corresponding control parameters for aqueduct structures in the hydraulic seismic design code. Although the displacement angle at the top of the pier has been used as a control index to solve the reliability in reference to the seismic design code of bridge structures in previous studies, due to the lack of a unified reference standard, it is difficult to accurately obtain the dynamic disaster reliability of aqueduct structures under catastrophic seismic actions. In order to investigate the differences and connections between the two different methods, Figure 14 the seismic reliability curves of aqueduct structures obtained by two collapse judgment methods based on effective energy and the maximum displacement angle at the top of the pier are given. It should be noted that different failure thresholds are taken for the inter - story displacement angle for investigation. It can be seen that the dynamic reliability curve of the aqueduct structure based on the maximum displacement angle at the top of the pier changes significantly with the change of the failure threshold. Specifically, as the threshold of the displacement angle at the top of the pier decreases, the seismic reliability of the aqueduct structure decreases accordingly. From Figure 14 it can be known that it is difficult to accurately obtain the anti - collapse reliability of aqueduct structures by using the index of the maximum displacement angle at the top of the pier, while the collapse of the aqueduct structure can be accurately judged based on the effective energy index, so as to obtain the reliability curve of displacement, which lays a foundation for reasonably evaluating the anti - collapse reliability of aqueduct structures. Under the random seismic action with a peak acceleration of 1.5g, the non - collapse reliability of this aqueduct structure is 58.6%.

Claims

1. A seismic reliability analysis method for a double-groove aqueduct structure under strong earthquake action, characterized in that, It includes the following steps: S1. Construct a refined fiber beam element numerical model of a double - slot aqueduct structure; The refined fiber beam element numerical model satisfies the following conditions: (1) Consider the influence of the coupled vibration of the lateral bending moment and warping deformation of the thin - walled aqueduct structure on the stiffness matrix of the thin - walled beam element; (2) The deformations at the aqueduct interface all satisfy the plane - section assumption; (3) The steel bars and concrete fibers are both in a uniaxial stress state, and are respectively represented by an elastoplastic tensile concrete constitutive model and a steel bar hardening constitutive model under cyclic loading; In the refined fiber beam element numerical model: The displacement of any point within the model element is obtained by introducing a warping function, which is expressed as: In the formula, is the horizontal lateral displacement at any point, is the lateral horizontal displacement at the centroid of any point, is the rotation angle of the cross-section about the z-axis, is the value of the cross-section warping function, is the rotation angle of the cross-section about the y-axis, is the rotation angle of the cross-section about the x-axis, is the longitudinal horizontal displacement at the centroid of any point, is the node x axis coordinate value, is the node y axis coordinate value, is the node z axis coordinate value; When the ferry structure enters the elastoplastic and large displacement deformation, the strain matrix when considering the influence of nonlinearity and geometric nonlinearity on the stiffness matrix is as follows: In the formula, is the linear small displacement strain matrix; is the geometric nonlinear strain matrix, is the nodal displacement column array; Stiffness matrix considering geometric nonlinear deformation is as follows: In the formula, the subscript is the entire integration region, is the stiffness matrix, is the linear stiffness matrix, is the non - linear influence stiffness matrix; S2. Based on the refined fiber beam element numerical model, conduct a nonlinear dynamic response analysis of the double - slot aqueduct structure; S3. Based on the process of the nonlinear dynamic response analysis, simulate and analyze the structural collapse of the double - slot aqueduct structure under earthquake action, and determine the collapse judgment criterion for the overall instability of the double - slot aqueduct structure; S4. Under random earthquake action, based on the collapse judgment criterion, use the probability density evolution method to determine the overall seismic reliability of the double - slot aqueduct structure.

2. The seismic reliability analysis method for the double-groove aqueduct structure under strong earthquake action according to claim 1, characterized in that, In step S2, the nonlinear dynamic response analysis of the double - slot aqueduct structure includes the nonlinear response analysis of the displacement of the aqueduct structure and the nonlinear response analysis of the internal forces at each cross - section of the aqueduct structure; Among them, the nonlinear response analysis of the displacement of the aqueduct structure is the nonlinear response of the aqueduct structure under different site types and different amplitude - modulated seismic waves; the nonlinear response value of the aqueduct structure increases as the site conditions weaken, and for different amplitude - modulated seismic waves, the displacement response of the aqueduct structure continues to increase after entering a highly nonlinear state; The nonlinear response analysis of the internal forces at each cross - section of the aqueduct structure includes the calculation results of the nonlinear responses at the mid - span of the side - span, the mid - span of the middle - span, and the cross - section at the pier bottom of the aqueduct structure.

3. The seismic reliability analysis method for the double-groove aqueduct structure under strong earthquake action according to claim 1, characterized in that In the step S3, an incremental dynamic analysis method is adopted to input the Beiling seismic waves with different amplitude values to the double-groove aqueduct structure Northridge , and based on the nonlinear dynamic response analysis process of the double-groove aqueduct structure, structural collapse simulation is carried out to obtain the energy change of the double-groove aqueduct structure and the whole process of overall instability and collapse of the double-groove aqueduct structure during the simulation process.

4. The seismic reliability analysis method for the double-groove aqueduct structure under strong earthquake action according to claim 3, characterized in that, The collapse judgment criterion in step S3 is: Under earthquake load, when the effective characteristic energy of the overall double - slot aqueduct structure is less than the effective input energy, the structure maintains an overall stable state; conversely, if the effective characteristic energy of the structure exceeds the effective energy at a certain moment, the structure will experience dynamic instability linearity from this moment and then cause collapse; it is expressed as: Among them, the effective characteristic energy is the energy reflecting the structure's own properties generated during the self-vibration process of the structure caused by external load excitation, and its expression is: In the formula, is the generalized complementary energy of the system at time t, is the restoring force vector, is the nodal displacement vector, is the stress tensor, is the elastic strain rate tensor, is the integral region of the aqueduct structure; Effective input energy It is the instantaneous energy input from the outside to the structural system, and its expression is: wherein, is the external dynamic excitation vector, is the nodal displacement vector, is the nodal velocity vector, is the system damping force vector, is the system kinetic energy at time t, is the system elastic strain energy at time t, is the system momentum at time t; The critical state judgment function for structural collapse is: In the formula, is the critical state determination function. When holds for all , the structure remains dynamically stable; while when appears for the first time, the structure will undergo instability and collapse.

5. The seismic reliability analysis method for the double-groove aqueduct structure under strong earthquake action according to claim 3, characterized in that Step S4 is specifically as follows: S41. Construct a physical model of random seismic motion; S42. According to the physical model of random seismic motion, randomly generate seismic excitations for the double - slot aqueduct structure; S43. Based on the generated seismic excitations, use the probability density evolution method to solve the probability density evolution information of the double - slot aqueduct structure at different levels; S44. Based on the probability density evolution information and combined with the dynamic reliability of the double - slot aqueduct structure based on the collapse judgment criterion, determine the overall reliability of the aqueduct structure at different levels, and then determine the overall seismic reliability of the double - slot aqueduct structure.

6. The seismic reliability analysis method for the double-groove aqueduct structure under strong earthquake action according to claim 5, characterized in that In the step S44, the dynamic reliability of the double-groove aqueduct structure is expressed as: In the formula, is the symbol for calculating reliability, is the physical quantity that causes the realization of the double-groove aqueduct structure, is time, is the safety domain of the double-groove aqueduct structure.

7. The seismic reliability analysis method for the double-groove aqueduct structure under strong earthquake action according to claim 5, characterized in that In step S44, when determining the overall reliability of the aqueduct structure at different levels, the collapse judgment criterion of the aqueduct structure based on effective energy and pier - top displacement angle is used as the physical mechanism for triggering probability dissipation.

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