Friction pendulum support shock insulation continuous beam bridge substructure total probability reinforcement method under non-stationary earthquake effect

Abstract

The invention discloses a friction pendulum support shock insulation continuous beam bridge substructure total probability reinforcement method under the action of a non-stationary earthquake, which comprises the following steps: establishing a non-linear dynamic time-history analysis motion equation containing a friction pendulum support, converting the motion equation into a state equation, and solving an earthquake response coefficient matrix of a key part of a bridge substructure; based on the seismic response coefficient matrix, calculating a covariance matrix of the response quantity of the key part, and constructing a hyper-ellipsoid envelope surface of the bending moment-axial force response quantity; calculating a bending moment-axial force related envelope curve based on a reinforced concrete eccentric stress component theory; and performing iterative calculation by taking the assumed reinforcement ratio as an initial value, in each iteration, adjusting the bending moment-axial force related envelope curve, so that the bending moment-axial force related envelope curve is tangent to the hyper-ellipsoid envelope surface, and updating the reinforcement ratio according to a mathematical relationship at a tangent point until the reinforcement ratio converges, so as to obtain the reinforcement ratio of the hyper-ellipsoid. And taking the reinforcement ratio during convergence as the optimal reinforcement ratio of the key part. The method is high in calculation efficiency.

Description

Technical Field

[0001] This invention relates to the field of bridge seismic reinforcement design technology, and in particular to a method for full probability reinforcement of the substructure of a continuous beam bridge with friction pendulum bearings under non-stationary seismic loading. Background Technology

[0002] In high-intensity earthquake zones, seismic isolation bearings are required to ensure the safety of bridges under seismic loads. Friction pendulum bearings, as an effective seismic isolation device, have been rapidly promoted and widely used in modern bridge engineering. By extending the natural period of the structure and providing hysteretic energy dissipation, they can significantly reduce the seismic forces transmitted to the substructure of the bridge, thereby protecting key load-bearing components such as piers and foundations.

[0003] In bridge construction, the seismic reinforcement design of the substructure (such as piers) is a core aspect of ensuring structural safety. Currently, traditional reinforcement design methods based on codes are commonly used in engineering practice. These methods typically rely on deterministic analysis, which involves selecting several representative seismic ground motion records, performing nonlinear time history analysis, extracting the maximum values ​​of the structural response (such as bending moment and axial force), and then calculating reinforcement based on these values. However, this method has several significant drawbacks:

[0004] (1) Conservative and Independence Assumptions in Response Quantity Processing: Traditional methods for processing coupled response quantities such as bending moment and axial force typically employ a "rectangular envelope" approach, combining the maximum values ​​of bending moment and axial force separately. This method ignores the actual probabilistic correlation between bending moment and axial force. In actual seismic events, the maximum bending moment and maximum axial force rarely occur simultaneously. This conservative simplification leads to overly conservative reinforcement design, resulting in steel waste and increased engineering costs.

[0005] (2) Low computational efficiency: Although traditional probabilistic analysis methods based on Monte Carlo simulation can more accurately reflect the randomness of earthquake motion, they require tens of thousands of repeated numerical integrations of the structural motion equations (such as the Newmark-β method), which is extremely time-consuming and costly, making it difficult to apply routinely in actual engineering design cycles.

[0006] (3) Insufficient consideration of the characteristics of seismic isolation bridges: Although the application of friction pendulum bearings is becoming increasingly widespread, in-depth research on their unique bidirectional coupled nonlinear hysteresis characteristics (usually described by differential models such as Bouc-Wen) under non-stationary seismic excitation is still insufficient. Existing reinforcement design methods fail to fully utilize the stochastic response characteristics of seismic isolation systems under seismic loading, and lack a reinforcement design process that can accurately and efficiently coordinate structural safety and economy.

[0007] (4) Limitations of the full probability design method: In recent years, although some researchers have tried to apply the full probability theory to structural design to take into account the uncertainty of load and resistance, in the specific field of friction pendulum bearing seismic isolation bridges, the existing full probability method often has defects in constructing the response envelope surface.

[0008] Therefore, there is an urgent need in this field for a reinforcement design method that can simultaneously take into account computational efficiency, design accuracy, and economy. Summary of the Invention

[0009] For the reinforcement design of the substructure of multi-span continuous beam bridges with friction pendulum bearings under non-stationary seismic excitation, this invention proposes a full probability reinforcement method for the substructure of continuous beam bridges with friction pendulum bearings under non-stationary seismic loading. Compared with the reinforcement method in the code and the traditional full probability reinforcement method, it has higher computational efficiency and better accuracy.

[0010] To achieve the above objectives, this invention provides a method for the full probability reinforcement of the substructure of a continuous beam bridge with friction pendulum bearings under non-stationary seismic loading, comprising:

[0011] A nonlinear dynamic time history analysis equation of motion including the friction pendulum bearing was established. The Bouc-Wen model was used to simulate the bidirectional coupled hysteretic restoring force of the friction pendulum bearing. The equation of motion was transformed into a state equation, and the seismic response coefficient matrix of key parts of the bridge substructure was solved by the explicit time-domain method.

[0012] Based on the earthquake response coefficient matrix, the covariance matrix of the response of key parts is calculated, and the hyperellipsoidal envelope of the bending moment-axial force response is constructed.

[0013] Based on the cross-sectional dimensions, material strength, and assumed reinforcement ratio of the bridge substructure components, the moment-axial force related envelope curve is calculated using the theory of reinforced concrete eccentrically loaded members.

[0014] The reinforcement ratio is used as the initial value for iterative calculation. In each iteration, the moment-axial force related envelope curve is adjusted so that the moment-axial force related envelope curve is tangent to the hyperellipsoidal envelope surface. The reinforcement ratio is updated according to the mathematical relationship at the tangency point until the reinforcement ratio converges. The reinforcement ratio at the convergence point is taken as the optimal reinforcement ratio for the key parts.

[0015] Preferably, the restoring force of the seismic isolation bearing, based on the Bouc-Wen hysteresis model, is described as follows:

[0016] ;

[0017] In the formula, The coefficient of dynamic friction of the friction pendulum support; The vertical pressure on the support; Let be the radius of curvature of the sliding surface; , Hysteresis components of bidirectional coupling; , These are the restoring forces in the x and y directions, respectively. , These are the displacements in the x and y directions, respectively.

[0018] Preferably, the seismic response coefficient matrix of key components of the bridge substructure is solved using an explicit time-domain method, including:

[0019] Using the exponential matrix and discrete seismic excitation vector, the response coefficient matrix of the state vector is solved by recursive formula, and the coefficient row vectors corresponding to key parts are extracted.

[0020] Preferably, constructing the hyperellipsoidal envelope of the moment-axial force response includes:

[0021] Based on the seismic response coefficient matrix, the bending moment and axial force responses of key parts of the bridge substructure are extracted, the covariance matrix of the bending moment and axial force responses is calculated, and based on the covariance matrix and the stochastic characteristics of non-stationary seismic excitation, a hyperellipsoidal envelope of the bending moment-axial force response is constructed to characterize the joint probability distribution of bending moment and axial force in key parts.

[0022] Preferably, the hyperellipsoidal envelope of the moment-axial force response is:

[0023] ;

[0024] In the formula, Let be the covariance matrix of the structural response vector 𝑿(𝑡); The maximum peak value; Let X be the transpose of X. for The inverse matrix.

[0025] Preferably, the calculation of the moment-axial force related envelope curve includes:

[0026] When the relative height of the cross section is less than or equal to the height of the boundary relative compression zone, the member satisfies the characteristics of large eccentric compression. The calculation method is as follows:

[0027] ;

[0028] In the formula, The bending moment of the concrete member's cross section; The axial force on the normal section of the concrete member; To maximize the strength; This is the design value for the compressive strength of concrete; The width of the concrete section; The height of the concrete section; This is the design value of the compressive strength of the reinforcing steel in the compression zone; This represents the area of ​​the reinforcing steel bars in the compression zone. The height of the concrete section is calculated and expressed as follows: ; The thickness of the concrete protective layer in the compression zone;

[0029] When the relative height of the cross section is greater than the relative height of the limiting compression zone, the member satisfies the small eccentric compression characteristic, and the calculation method is as follows:

[0030] ;

[0031] In the formula, , All parameters are manually defined; This refers to the thickness of the concrete protective layer. This represents the cross-sectional area of ​​the reinforcing steel in the compression zone.

[0032] Preferably, when the cross-sectional shape of the concrete member is rectangular and the reinforcement is symmetrical, the relevant envelope curve of bending moment-axial force under small eccentric tension is:

[0033] ;

[0034] In the formula, This is the design value of the tensile strength of the reinforcing steel in the tension zone; This represents the cross-sectional area of ​​the reinforcing bars in the tension zone.

[0035] Preferably, updating the reinforcement ratio based on the mathematical relationship at the tangent point includes:

[0036] The slope of the moment-axial force correlation envelope curve and the slope of the hyperellipsoidal envelope surface with respect to axial force at the tangent point are equal, and the reinforcement ratio is updated based on the sensitivity of the reinforcement area at the corresponding tangent point to the moment.

[0037] Preferably, the convergence condition for the iterative calculation is that the difference in reinforcement ratio between two adjacent iterations is less than a preset allowable error.

[0038] Preferably, the method further includes a verification step, specifically: verifying the hyperellipsoidal envelope and the optimal reinforcement ratio through Monte Carlo simulation.

[0039] Compared with the prior art, the present invention has the following advantages and technical effects:

[0040] This invention provides a full-probability reinforcement method for the substructure of a friction pendulum bearing seismic-isolated continuous beam bridge under non-stationary seismic loading. Based on existing or artificially generated seismic motion data, it efficiently calculates the correlation between bending moment and axial force by the covariance between bending moment and axial force. This allows for more rational seismic reinforcement design for the substructure of friction pendulum bearing seismic-isolated bridges. Compared with standard methods, the time-domain explicit full-probability reinforcement method can significantly reduce the amount of steel reinforcement used and has higher computational efficiency, thereby saving on manufacturing costs and computational costs in engineering design. This invention helps to more accurately calculate the seismic reinforcement area of ​​the substructure of friction pendulum bearing continuous beam bridges, and has practical significance for reinforcement design in actual engineering projects. Attached Figure Description

[0041] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings:

[0042] Figure 1 This is a flowchart illustrating a method for full-probability reinforcement of the substructure of a continuous beam bridge with friction pendulum support under non-stationary seismic loading, according to an embodiment of the present invention.

[0043] Figure 2 This is a schematic diagram of the overall layout and key cross-sectional dimensions of the bridge structure according to an embodiment of the present invention;

[0044] Figure 3 This is a diagram showing the overall layout of the numerical model in an embodiment of the present invention.

[0045] Figure 4 The diagram shows the target response spectrum and seismic excitation acceleration curve in an embodiment of the present invention, wherein (a) is the target response spectrum curve and (b) is the non-stationary seismic excitation curve generated from the target response spectrum.

[0046] Figure 5 These are seismic response envelope curves under different methods according to embodiments of the present invention;

[0047] Figure 6 These are reinforcement diagrams for different methods in embodiments of the present invention;

[0048] Figure 7 This is a diagram showing the relationship between the moment-axial force envelope curves and elliptical envelopes for different reinforcement ratios in an embodiment of the present invention. Detailed Implementation

[0049] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0050] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0051] This embodiment proposes a method for the full probability reinforcement of the substructure of a continuous beam bridge with friction pendulum bearings under non-stationary seismic loading, such as... Figure 1 ,include:

[0052] A nonlinear dynamic time history analysis equation of motion including the friction pendulum bearing was established. The Bouc-Wen model was used to simulate the bidirectional coupled hysteretic restoring force of the friction pendulum bearing. The equation of motion was transformed into a state equation, and the seismic response coefficient matrix of key parts of the bridge substructure was solved by the explicit time-domain method.

[0053] Based on the earthquake response coefficient matrix, the covariance matrix of the response of key parts is calculated, and the hyperellipsoidal envelope of the bending moment-axial force response is constructed.

[0054] Based on the cross-sectional dimensions, material strength, and assumed reinforcement ratio of the bridge substructure components, the moment-axial force related envelope curve is calculated using the theory of reinforced concrete eccentrically loaded members.

[0055] The reinforcement ratio is used as the initial value for iterative calculation. In each iteration, the moment-axial force related envelope curve is adjusted so that the moment-axial force related envelope curve is tangent to the hyperellipsoidal envelope surface. The reinforcement ratio is updated according to the mathematical relationship at the tangency point until the reinforcement ratio converges. The reinforcement ratio at the convergence point is taken as the optimal reinforcement ratio for the key parts.

[0056] Furthermore, the equations of motion for the nonlinear dynamic time history analysis of seismically isolated bridges under horizontal seismic loading can be expressed as:

[0057] (1);

[0058] In the formula, for A column vector representing the absolute translational displacements of all free nodes in the structure; for 3D column vector, representing Forced ground displacement at each support node; for 3D column vector, representing Each support node is subjected to seismic forces consisting of three translational components of ground motion; This represents the matrix indicating the bidirectional horizontal restoring force position of the seismic isolation bearing. Non-zero positions in the matrix correspond to the degrees of freedom of the nodes connected to the seismic isolation bearing. for 3D column vector, representing The bidirectional horizontal restoring force column vector of each seismic isolation bearing This represents the relative displacement vector of the nodes connected to both ends of the seismic isolation bearing. The specific expression is related to the type of bearing selected for the seismic isolation bridge; These are the lumped mass matrix, damping matrix, and stiffness matrix, respectively; subscripts Indicates a free node; Indicates the support node; The quality of the free node; The mass of the supporting node; Damping between free nodes; Damping between the supporting node and the free node; Damping between the free node and the supporting node; Damping between support nodes; For the acceleration of the free node; The acceleration of the supporting node; The speed of the supporting node; The velocity of the free node; The stiffness between free nodes; The stiffness between the supporting node and the free node; The stiffness between the free node and the supporting node; This refers to the stiffness between the supporting nodes.

[0059] There are various models for the restoring force of seismic isolation bearings, including bilinear models, modified bilinear models, equivalent bilinear models, differential hysteretic models, and Bouc-Wen hysteretic models. Among these, the Bouc-Wen model is widely used because it can better simulate the hysteretic characteristics of seismic isolation bearings and is easy to solve together with the equations of motion. However, it should be noted that most existing seismic isolation bearing models assume that the vertical stiffness of the seismic isolation bearing is constant and do not consider the influence of vertical forces on the horizontal restoring force of the bearing. This embodiment also adopts this assumption when considering the mechanical model of the seismic isolation bearing.

[0060] The bidirectional nonlinear coupled horizontal restoring force model of the friction pendulum seismic isolation bearing can be expressed as:

[0061] (2);

[0062] In the formula, The coefficient of dynamic friction of the friction pendulum support; The vertical pressure on the support; Let be the radius of curvature of the sliding surface; , These are the restoring forces in the x and y directions, respectively. , These are the relative horizontal displacements of the top and bottom nodes connected to the seismic isolation bearing along the x and y directions, respectively. , The hysteresis components of the bidirectional coupling can be expressed as:

[0063] (3);

[0064] In the formula, This represents the elastic shear deformation of the friction pendulum support before sliding, and is typically taken as 0.13~0.5mm. , , The hysteresis curve shape of the friction pendulum support determines the hysteresis, which can usually be taken as... ; , The derivative of the hysteretic displacement component considering the bidirectional coupling effect of the support; , The relative horizontal velocity of the top and bottom nodes connected to the seismic isolation bearing along the x and y directions; , This represents the hysteresis characteristic component of the bidirectional coupling effect.

[0065] Define the state vector as The recursive formula for the state vector can be obtained as follows:

[0066] (4);

[0067] In the formula, It is an exponential matrix. The seismic excitation vector, ; ; for The seismic excitation vector at time t; for The seismic excitation vector at time t; Represented as:

[0068] (5);

[0069] In the formula, It is the identity matrix; It is an exponential matrix; It is the square of the inverse matrix of H; for ; For time step; .

[0070] Without loss of generality, based on the characteristics of seismic excitation, it is assumed that... and Based on equation (4), the explicit time-domain expression for the structure state vector can be obtained as follows:

[0071] (6);

[0072] In the formula, In response The coefficient matrix, where the coefficient vector The following closed-form formula can be used for calculation:

[0073] (7);

[0074] According to equation (7), only the coefficient vector of the first column can be obtained. Calculations and storage are required; all other coefficients can be obtained from... express.

[0075] It should be pointed out that the coefficient vector This represents the structure under a unit pulse excitation at time t1. The state vector at time t is the structural response state vector, so only one response time history analysis is needed to obtain the structural response state vector.

[0076] Generally, we only focus on the structural response of the joints, while the coefficient matrix solution is the response coefficient of all degrees of freedom of the structure. Therefore, after solving the coefficient matrix, we can extract the rows of the degrees of freedom corresponding to the key parts in the coefficient matrix for subsequent analysis, as shown in Equation (8):

[0077] (8);

[0078] In the formula, Represents the response state vector of key structural components. Represents the coefficient matrix The row vectors corresponding to the degrees of freedom of the key parts. This is the seismic excitation vector.

[0079] Furthermore, in practical engineering, structural members are generally rectangular in cross-section. To simplify construction and save steel, symmetrical reinforcement design is often used in the reinforcement design of the member's cross-section. According to reinforced concrete design theory, when the member's material, cross-sectional dimensions, and reinforcement ratio are determined, the axial force and bending moment that bring the eccentric member to its ultimate bearing capacity are not independent. The combination of bending moment and axial force that brings the member to its ultimate bearing capacity state is not unique. Therefore, when the material, cross-sectional dimensions, and reinforcement ratio of the member are determined, a series of combinations of axial force and bending moment that bring the member to its ultimate bearing capacity state can be obtained. These combinations are called the moment-axial force relationship of the eccentrically loaded member.

[0080] For a given reinforced concrete member, the failure mode varies depending on the magnitude and direction of the axial force it experiences. Furthermore, the fundamental formulas used to calculate the member's load-bearing capacity differ depending on the failure mode. Therefore, when calculating the moment-axial force relationship of an eccentrically loaded member, it is necessary to segment the member according to the magnitude of the axial force it experiences.

[0081] When a structural member is subjected to compression, according to the force equilibrium relationship, the concrete at the normal section of the member will experience compression. Depending on the relative height of the compression zone, there are two different types of stress in a compression member. When the relative height of the compression zone is less than or equal to the limit relative height of the compression zone... The component will satisfy the characteristics of large eccentric compression; when At that time, the component will satisfy the small eccentric compression characteristic; The height of the cross section relative to the compression zone. The height of the relative pressure zone is the boundary.

[0082] When the cross-section of a concrete member is rectangular and symmetrically reinforced, the relevant curve equation of bending moment-axial force under large eccentric compression can be expressed by equation (9):

[0083] (9);

[0084] In the formula, The bending moment of the concrete member's cross section; The axial force on the normal section of the concrete member; To maximize the strength; This is the design value for the compressive strength of concrete; The width of the concrete section; The height of the concrete section; This is the design value of the compressive strength of the reinforcing steel in the compression zone; This represents the area of ​​the reinforcing steel bars in the compression zone. The height of the concrete section is calculated and expressed as follows: ; The thickness of the concrete protective layer in the compression zone;

[0085] When a structural member fails under small eccentric compression, the concrete in the compression zone is crushed first, followed by the reinforcing steel in the compression zone. The stress on one side reaches the yield strength of the steel bar, while the stress on the other side reaches the yield strength of the steel bar. The concrete member may be subjected to compression or tension, but the reinforcement on that side does not reach the yield strength. When the cross-sectional shape of the concrete member is rectangular and the reinforcement is symmetrical, the relevant curve equation of bending moment-axial force under small eccentric compression can be expressed by equation (10).

[0086] (10);

[0087] In the formula, , All parameters are manually defined; This refers to the thickness of the concrete protective layer. This represents the cross-sectional area of ​​the reinforcing steel in the compression zone.

[0088] (11);

[0089] When a component is subjected to tensile force, the type of stress can be classified into large eccentric tension and small eccentric tension according to the location of the longitudinal tensile force along the cross-section. When the point of application of the axial tensile force is... The point of convergence and When the resultant force points are between the two points, the component will obey the force characteristics of small eccentric tension. If the point of application of the axial tensile force is not between the two points, the component will obey the force characteristics of small eccentric tension. The point of convergence and When the resultant force points are between the two points, the component will obey the force characteristics of large eccentric tension.

[0090] When the cross-sectional shape of a concrete member is rectangular and it is symmetrically reinforced, the relevant curve equation of bending moment-axial force under small eccentric tension can be expressed by equation (12):

[0091] (12);

[0092] In the formula, This is the design value of the tensile strength of the reinforcing steel in the tension zone; This represents the cross-sectional area of ​​the reinforcing bars in the tension zone.

[0093] When a member is subjected to a large eccentric tensile force, cracks appear in the cross-section, but the cracks do not penetrate the entire cross-section. At this time, there is still a compression zone in the normal cross-section of the member. For a rectangular cross-section with symmetrical reinforcement, regardless of whether the member is subjected to large or small eccentric tension, the bending moment and axial force have the same linear relationship, and the bending moment decreases as the axial force increases.

[0094] For any component, when the cross-sectional dimensions, material, steel reinforcement strength and cross-sectional reinforcement ratio of the component are given, the relevant curves of bending moment and axial force when the eccentric component is reinforced symmetrically can be determined by equations (9)-(12).

[0095] The envelopes of axial force and bending moment at the cross-section of all structural members during service can be obtained using the above method. Furthermore, when the cross-sectional dimensions, concrete strength grade, and reinforcement ratio of the member are known, the bending moment-axial force correlation curve of the member can be obtained through the bending moment-axial force correlation relationship.

[0096] For a building structure to be safe, reliable, and economical during use, all load-bearing components must meet their own load-bearing capacity requirements. Therefore, all combinations of axial force and bending moment in each component under load should lie inside the component's bending moment-axial force correlation curve; that is, the component's bending moment-axial force correlation curve should include the envelope curve of axial force and bending moment. Figure 7 It can be observed that when the envelope curve of axial force and bending moment of a member is determined, as the reinforcement ratio increases, countless bending moment-axial force correlation curves can be found to encompass it. For example, the three outer curves in Figure 7 can all encompass the envelope curve of axial force and bending moment. Therefore, these three curves can all satisfy the safety requirements of the member. However, in structural design, not only the safety of the structure should be considered, but also the economic efficiency of the project. Figure 7 This indicates that the outermost part of the moment-axial force correlation curve corresponds to a larger section reinforcement ratio, therefore, a higher reinforcement ratio should be selected. Figure 7 The reinforcement ratio corresponding to the second curve is taken as the design reinforcement ratio. The optimal reinforcement ratio should be such that the moment-axial force correlation curve corresponding to that reinforcement ratio is exactly tangent to the envelope curve of axial force and moment.

[0097] Furthermore, a hyperellipsoidal envelope of the moment-axial force response is constructed, including:

[0098] Based on the seismic response coefficient matrix, the bending moment and axial force responses of key parts of the bridge substructure are extracted, the covariance matrix of the bending moment and axial force responses is calculated, and based on the covariance matrix and the stochastic characteristics of non-stationary seismic excitation, a hyperellipsoidal envelope of the bending moment-axial force response is constructed to characterize the joint probability distribution of bending moment and axial force in key parts.

[0099] Specifically, for a multi-degree-of-freedom bridge system containing seismic isolation bearings, the number of degrees of freedom is... The system's dynamic response vector can be defined as follows: The number of single responses is It is the nodal displacement function, and the nodal displacement is calculated by equation (8), which is: Therefore, a single response quantity can be expressed as:

[0100] (13);

[0101] In the above formula, vector It is a function of structural stiffness and geometric properties, and a definition is given. matrix Then the system's response vector can be expressed as:

[0102] (14);

[0103] In the formula, Let Q be the transpose of Q.

[0104] To more clearly describe the structural response during earthquake loading, a Cartesian coordinate system with the same dimensions and number of response quantities can be chosen to represent the structural response space. The response vector... Direction vector in response space The projection on can be represented as:

[0105] (15);

[0106] In the formula, Let be the transpose of α.

[0107] Direction vector It is a unit vector, when the vector The When one element is 1 and all other elements are 0, For unit vectors In other words, It can be viewed as a response vector A linear combination of the elements. Substituting equation (10) into equation (11) yields:

[0108] (16);

[0109] In the above formula, For one A 3D vector is a vector representing the structural stiffness, geometry, and unit direction. The function.

[0110] for 3D random vector If all variables are independent and follow a standard normal distribution, then the sum of squares of the variables can be expressed as:

[0111] (17);

[0112] In the formula, Let be a random variable, and follow the rules of random variables. Chi-square distribution, denoted as Cumulative distribution function .

[0113] Determine an envelope in the standard normal space such that the random variable... The sample points should fall within this envelope as much as possible, and the sample distribution domain can be represented as... The probability that a sample point falls outside the domain is... Very small. (Based on the sample distribution domain) The defined envelope can be represented as Analyzing this equation reveals that the envelope surface... Change with change. This can be the square of the distance from the sample point to the origin of the standard normal space coordinate system. If the cumulative distribution function is directly used as the control function of the envelope surface, an elliptical envelope surface can be obtained in the two-dimensional standard normal space. Due to the polar axis symmetry phenomenon in the standard normal space, it can be guaranteed that the probability of each sample point appearing on the envelope surface is equal, which can be expressed as:

[0114] (18);

[0115] As can be seen from equation (14), changing the radius The size can be adjusted The probability that a random sample point falls within the envelope. For general... 3D random vector ,in It follows a multidimensional normal distribution, and its covariance matrix is ​​expressed as: The mean of each random variable is 0. Thus... 3D random vector The determined space can be transformed into the standard normal space using the following formula:

[0116] (19);

[0117] (20);

[0118] in, For the transformation vector, it simultaneously satisfies .

[0119] Substituting equation (16) into equation (14), we get:

[0120] (twenty one);

[0121] In the formula, The covariance matrix of the response. According to equation (17), by adjusting the radius... The magnitude of controls the probability that a sample point falls within the envelope. For a random normal variable with an expected value of 0... Its range of values ​​can be determined by the following expression:

[0122] (twenty two);

[0123] In the formula , For random variables The standard deviation.

[0124] In practical engineering, many physical phenomena are random processes, therefore they cannot be described using random variables or random vectors, but must be described using stochastic processes. For a structure under a non-stationary excitation with a mean of 0, the response vector of a certain degree of freedom... During a specific time period Within, the maximum peak value of the response vector can be expressed as ,in, It is a stochastic process The maximum peak coefficient within the time period It is the standard deviation of a stochastic process, and similarly, it is necessary to make the stochastic process... If the sample values ​​fall as close to the envelope as possible, then the equation of the hyperellipsoidal envelope is:

[0125] (twenty three);

[0126] In the formula, The covariance matrix of the structural response vector 𝑿(𝑡) can be expressed as:

[0127] (twenty four);

[0128] Further, updating the reinforcement ratio based on the mathematical relationship at the tangent point includes:

[0129] The slope of the moment-axial force correlation envelope curve and the slope of the hyperellipsoidal envelope surface with respect to axial force at the tangent point are equal, and the reinforcement ratio is updated based on the sensitivity of the reinforcement area at the corresponding tangent point to the moment.

[0130] Specifically, in a certain reinforcement area Below, the moment-axial force envelope curve intersects the elliptic curve at point... Tangent. At this point, the sensitivity of the envelope curve to the reinforced area can be written as:

[0131] (25);

[0132] In the formula, For symmetrical reinforcement area; For a given reinforcement area When, the bending moment component corresponding to the bending moment-axial force envelope curve calculated by equations (22) and (23) is obtained.

[0133] Because the elliptical envelope is tangent to the moment-axial force curve, the two curves are... Since the slopes of the forces N at each point are equal, then:

[0134] (26);

[0135] In the formula, The bending moment components of the elliptical envelope under the same axial force N;

[0136] No. In the next iteration, the axial force value that first intersects the elliptical envelope can be expressed as:

[0137] (27);

[0138] In the formula, Indicates the first The assumed reinforcement area in the next iteration;

[0139] At this point ,right By performing iterative calculations, we can obtain:

[0140] (28);

[0141] In the formula, For the first In the next iteration, the axial force value that first intersects the elliptical envelope; The reinforcement area is At that time, the bending moment component corresponding to the bending moment-axial force envelope curve; represents the bending moment component of the elliptical envelope under axial force N; For the first In the next iteration, the difference between the moment component corresponding to the moment-axial force envelope curve and the moment component of the elliptical envelope under the action of axial force N.

[0142] When the iteration satisfies When the reinforcement area converges to the optimal value, it can be considered that the reinforcement area has converged to the optimal value. At this point, the corresponding moment-axial force envelope is exactly tangent to the elliptical envelope:

[0143] (29);

[0144] In the formula, For the first The assumed reinforcement area in the next iteration; For the first The assumed reinforcement area in the next iteration; This is the allowable error.

[0145] To more clearly illustrate the technical solution of the present invention, specific embodiments are provided below for description:

[0146] This embodiment uses an urban viaduct in a high-intensity seismic zone as the engineering background. The site category is Class II, and the seismic intensity is VIII. The main girder structure is a 4×36m prestressed concrete continuous box girder system with uniform cross-section, employing a single-box, four-cell cross-section to meet the load-bearing requirements of a wide bridge deck. The girder height is 1.8m, the top slab width is 26m, and the bottom slab width is 16.8m. The web width at mid-span is 500mm, the top slab thickness is 240mm, and the bottom slab thickness is 220mm. At the support sections, the web width is 700mm, the top slab thickness is 240mm, and the bottom slab is thickened to 400mm to improve shear capacity. The substructure piers are double-column reinforced concrete rectangular piers with a cross-sectional dimension of 1.8×2m. There are no crossbeams or cap beams. The pier height is 9m, and the spacing between the double-column piers is 14m. The seismic isolation bearings are friction pendulum bearings with a design displacement capacity of [missing information]. mm. A schematic diagram of the overall layout and key section dimensions of the bridge structure is shown below. Figure 2 As shown.

[0147] The main beam concrete strength grade is C50, the pier concrete strength grade is C40, and the Poisson's ratio of the concrete materials is uniformly taken as... The self-weight of the beam is taken as 26 kN / m. 3 The self-weight of the bridge pier is taken as 25kN / m. 3 The bridge site is classified as a Class II site. The geological survey report reveals that the strata in the area are mainly composed of soft rock or hard soil composite strata, indicating favorable geological conditions.

[0148] A refined three-dimensional finite element model of a four-span friction pendulum supported seismic isolation continuous beam bridge was established based on the OpenSees platform. The overall layout diagram and detailed constitutive model schematics of each nonlinear component are shown below. Figure 3 As shown.

[0149] In the design of seismically isolated bridges, the main girder, as a capacity protection component, primarily bears vertical loads and transfers inertial forces to the seismic isolation bearings. Energy is concentrated and dissipated through these bearings, allowing the main girder to remain in an elastic state as a non-energy-dissipating component, preventing plastic deformation and ensuring the bridge's normal functionality. Based on historical earthquake damage records, damage to the bridge superstructure often stems from collisions between adjacent spans, abutment impacts, and girder collapses caused by excessive seismic displacement, rather than plastic failure of the main girder itself. Therefore, the elastic assumption of the main girder aligns with the actual seismic damage conditions of main girders in engineering projects. Furthermore, as a capacity protection component, the superstructure ensures that plastic deformation is concentrated in controllable isolation devices and piers. Therefore, considering computational accuracy and efficiency, this embodiment uses elastic beam-column elements in the OpenSees platform to simulate the main girder. A refined discretization strategy controls the modal frequency error to be less than 5%, and based on the principle of symmetry, the secondary dead load is applied as a uniformly distributed mass, considering its contribution to the main girder's mass. The total weight of the superstructure reaches 86786.6 kN. The structural node mass and gravity are applied to the nodes, with the node mass distributed as 50% of the sum of the masses of adjacent elements, and the structural load is evenly transferred to the top of the piers. The beam element section properties are calculated based on the main beam section dimensions and material mechanical properties.

[0150] In OpenSees, the mechanical behavior of friction pendulum bearings can be simulated using Single Friction Pendulum Bearing Element. Based on the characteristics of the PTFE-stainless steel sliding interface, the Coulomb friction model is used to simulate dynamic friction behavior. This embodiment assumes that the friction coefficient is independent of the sliding velocity, an assumption that has been experimentally verified to meet engineering accuracy requirements. During the modeling process, the actual bridge design parameters are used as a benchmark, and the friction coefficient of each pier's friction pendulum bearing is... The side piers (pier #1 and pier #5) have relatively low axial compression, so the design bearing capacity of the friction pendulum bearings is 7000kN, and the seismic isolation period is... The design bearing capacity of the central piers (piers #2, #3, and #4) is 17500 kN, and the seismic isolation period is... .

[0151] As the main lateral force resisting component of a bridge structure, the pier must withstand the coupled effects of bending moment, shear force, and axial force under seismic loading. Under strong earthquakes, its nonlinear behavior directly affects the overall seismic performance of the structure. Therefore, in nonlinear dynamic time-history analysis, to accurately simulate the nonlinear behavior of the pier and better capture its response under strong earthquakes, this embodiment constructs a plastic fiber section pier model based on the OpenSees platform. The flexibility method nonlinear beam-column element is used to simulate the bending-shear coupling effect of the pier column. The stress-strain relationship between confined and unconfined concrete is defined using the Mander concrete model. The Concrete02 material constitutive model is used to simulate the nonlinear stress-strain behavior of concrete under compression and tension; the Reinforcing Steel material constitutive model is used to simulate the stress-strain behavior of the longitudinal reinforcement of the pier. The values ​​of the Reinforcing Steel material constitutive parameters are based on the experimental study of the mechanical properties of HRB400 steel bars by Zhang Yaoting et al.; the stirrup spacing is 100mm. The pier fiber section is discretized into a 10cm×10cm grid, with element lengths of 1m along the pier height.

[0152] Earthquake ground motion is complex and random. To ensure the accuracy of seismic response results, the selection of earthquake ground motion should follow the three elements of earthquake ground motion when performing time history analysis on the structure: (1) earthquake ground motion amplitude, (2) spectral characteristics, and (3) effective duration, so as to ensure the rationality of the time history analysis results and make the analysis results more general. Since this embodiment optimizes the reinforcement ratio calculation method in the code, the earthquake target response spectrum specified in the code is used for comparative analysis. At the same time, considering the complexity of earthquake ground motion, this embodiment generates the target response spectrum (see Code for Seismic Design of Urban Bridges) based on seismic information such as the seismic intensity of the bridge site area, seismic grouping, site conditions, and epicentral distance, according to the Code for Seismic Design of Urban Bridges (CJJ 166-2011). Figure 4 As shown in (a), non-stationary seismic excitation is generated through the target response spectrum (see [reference]). Figure 4 (b)

[0153] The method presented in this embodiment utilizes random vibration theory and the explicit time-domain method to calculate the response of key components of a structure under seismic loading, thereby obtaining the covariance matrix of the response quantities of the key components. As derived above, given the covariance matrix of the response quantities of the key components, an elliptical envelope can be determined in the response space. In actual engineering reinforcement design, the optimal reinforcement ratio is obtained by iteratively solving the reinforcement area based on the specific values ​​of the response quantities of the key components.

[0154] The proposed time-domain explicit full-probability reinforcement method is used to... Figure 3The friction pendulum seismic isolation bearing continuous beam bridge shown is subjected to random vibration analysis under non-stationary seismic loading. The response elliptical envelope of the structure is obtained by equation (19). At the same time, the reference solution of the random vibration response is calculated by Monte Carlo simulation, and the motion equation is solved repeatedly by Newmark-β numerical integration. The number of earthquake samples in the Monte Carlo method is 1000, the time history analysis step size is 0.02s, and the total time history analysis length is 30s. The traditional Monte Carlo method requires calculation of the response of all degrees of freedom of the structure. Figure 5 The response quantities of different parts of the bridge were calculated by 1000 structural dynamic time history analyses. However, the explicit elliptical envelope method in the time domain only requires one response calculation for all degrees of freedom of the structure. Then, the coefficient matrix of the corresponding degrees of freedom of the characteristic parts can be extracted to obtain the response quantities of the key parts of the structure, which is very beneficial for structural response analysis and subsequent full probability reinforcement calculation analysis.

[0155] The bending moment and axial force at the top of the bridge pier are as follows Figure 5 As shown. From Figure 5 As can be seen, the time-domain explicit full probability reinforcement method agrees well with the traditional nonlinear time-history dynamic analysis method, indicating that the method in this embodiment has good accuracy.

[0156] Figure 6 These are the envelope curves of bending moment and axial force at the top section of pier #1 and pier #2 under non-stationary seismic loading. According to the code, the ultimate bending moment and ultimate axial force under seismic loading are obtained by averaging the values ​​after traditional nonlinear dynamic time history analysis, thus obtaining the rectangular response envelope. Since the code method does not consider the correlation between bending moment and axial force, the values ​​of ultimate bending moment and ultimate axial force are relatively conservative when designing reinforcement.

[0157] The method proposed by Menun and Der Kiureghian uses the maximum response matrix, not the covariance matrix, to determine the boundary of the elliptical envelope. Furthermore, the cross terms of the response matrix are given with reference to the diagonal elements, which may lead to inaccurate correlations between the obtained response quantities, resulting in incorrect magnitude and direction of the elliptical envelope. Figure 6 As can be seen, the elliptical envelope obtained by the literature method is tangent to both sides of the rectangular envelope determined by the standard method, and the elliptical envelope always falls inside the rectangular envelope. This not only reduces the cumulative probability of sample values, but also cannot guarantee that the probability of each point on the envelope is equal. Therefore, for the substructure of a bridge affected by multiple vibration modes, the cross-sectional reinforcement ratio obtained by the literature method may be too small, resulting in an unsafe situation.

[0158] The calculation results of the reinforcement ratio of the bridge substructure are shown in Table 1.

[0159] Table 1

[0160]

[0161] The data in Table 1 shows that the reinforcement ratio of the pier top section obtained by the method in this embodiment is always less than or equal to the result obtained by the standard method. Furthermore, the elliptical envelope of the literature method is determined by the maximum value of the response, resulting in a reinforcement ratio that is less than or equal to the result obtained by the method in this embodiment. Therefore, reinforcement according to the literature method can lead to insufficient reinforcement in some columns, increasing the risk of structural instability.

[0162] Regarding computational efficiency, Table 2 shows the time consumed by different methods. As can be seen from Table 2, the time-domain explicit full probability reinforcement method is significantly more efficient than the traditional full probability reinforcement method and the standard method based on traditional nonlinear dynamic time history analysis. The time-domain explicit full probability reinforcement method requires 267.58 seconds to calculate the reinforcement ratio of the bridge substructure. While the traditional full probability reinforcement method has a shorter calculation time for determining the reinforcement ratio, the total calculation time is 1255.83 seconds due to the complexity of the traditional nonlinear dynamic time history analysis.

[0163] Table 2

[0164]

[0165] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for full-probability reinforcement design of the substructure of a continuous beam bridge with friction pendulum bearings under non-stationary seismic loading, characterized in that, include: A nonlinear dynamic time history analysis equation of motion including the friction pendulum bearing was established. The Bouc-Wen model was used to simulate the bidirectional coupled hysteretic restoring force of the friction pendulum bearing. The equation of motion was transformed into a state equation, and the seismic response coefficient matrix of key parts of the bridge substructure was solved by the explicit time-domain method. Based on the earthquake response coefficient matrix, the covariance matrix of the response of key parts is calculated, and the hyperellipsoidal envelope of the bending moment-axial force response is constructed. Based on the cross-sectional dimensions, material strength, and assumed reinforcement ratio of the bridge substructure components, the moment-axial force related envelope curve is calculated using the theory of reinforced concrete eccentrically loaded members. The reinforcement ratio is used as the initial value for iterative calculation. In each iteration, the moment-axial force related envelope curve is adjusted so that the moment-axial force related envelope curve is tangent to the hyperellipsoidal envelope surface. The reinforcement ratio is updated according to the mathematical relationship at the tangency point until the reinforcement ratio converges. The reinforcement ratio at the convergence point is taken as the optimal reinforcement ratio for the key parts.

2. The method according to claim 1, characterized in that, The restoring force of the seismic isolation bearing, based on the Bouc-Wen hysteresis model, is as follows: ； In the formula, The coefficient of dynamic friction of the friction pendulum support; The vertical pressure on the support; Let be the radius of curvature of the sliding surface; , Hysteresis components of bidirectional coupling; , These are the restoring forces in the x and y directions, respectively. , These are the displacements in the x and y directions, respectively.

3. The method according to claim 1, characterized in that, The seismic response coefficient matrix of key components of the bridge substructure is solved using an explicit time-domain method, including: Using the exponential matrix and discrete seismic excitation vector, the response coefficient matrix of the state vector is solved by recursive formula, and the coefficient row vectors corresponding to key parts are extracted.

4. The method according to claim 1, characterized in that, Constructing the hyperellipsoidal envelope of the bending moment-axial force response includes: Based on the seismic response coefficient matrix, the bending moment and axial force responses of key parts of the bridge substructure are extracted, the covariance matrix of the bending moment and axial force responses is calculated, and based on the covariance matrix and the stochastic characteristics of non-stationary seismic excitation, a hyperellipsoidal envelope of the bending moment-axial force response is constructed to characterize the joint probability distribution of bending moment and axial force in key parts.

5. The method according to claim 4, characterized in that, The hyperellipsoidal envelope of the moment-axial force response is: ； In the formula, Let be the covariance matrix of the structural response vector 𝑿(𝑡); The maximum peak value; Let X be the transpose of X. for The inverse matrix.

6. The method according to claim 1, characterized in that, Calculate the moment-axial force related envelope curve, including: When the relative height of the cross section is less than or equal to the height of the boundary relative compression zone, the member satisfies the characteristics of large eccentric compression, and the calculation method is as follows: ； In the formula, The bending moment of the concrete member's cross section; The axial force on the normal section of the concrete member; To maximize the strength; This is the design value for the compressive strength of concrete; The width of the concrete section; The height of the concrete section; This is the design value of the compressive strength of the reinforcing steel in the compression zone; This represents the area of ​​the reinforcing steel bars in the compression zone. The height of the concrete section is calculated and expressed as follows: ; The thickness of the concrete protective layer in the compression zone; When the relative height of the cross section is greater than the relative height of the limiting compression zone, the member satisfies the small eccentric compression characteristic, and the calculation method is as follows: ； In the formula, , All parameters are manually defined; This refers to the thickness of the concrete protective layer. This represents the cross-sectional area of ​​the reinforcing steel in the compression zone.

7. The method according to claim 6, characterized in that, When a concrete member has a rectangular cross-section and is symmetrically reinforced, the relevant envelope curve of bending moment-axial force under small eccentric tension is: ； In the formula, This is the design value of the tensile strength of the reinforcing steel in the tension zone; This represents the cross-sectional area of ​​the reinforcing bars in the tension zone.

8. The method according to claim 1, characterized in that, Updating the reinforcement ratio based on the mathematical relationship at the tangent point includes: The slope of the moment-axial force correlation envelope curve and the slope of the hyperellipsoidal envelope surface with respect to axial force at the tangent point are equal, and the reinforcement ratio is updated based on the sensitivity of the reinforcement area at the corresponding tangent point to the moment.

9. The method according to claim 1, characterized in that, The convergence condition for the iterative calculation is that the difference in reinforcement ratio between two adjacent iterations is less than a preset allowable error.

10. The method according to claim 1, characterized in that, The method also includes a verification step, specifically: verifying the hyperellipsoidal envelope and the optimal reinforcement ratio through Monte Carlo simulation.

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