A method for evaluating shear lag effect of a multi-span continuous rigid frame bridge in an initial damage state

Abstract

This invention discloses a method for assessing the shear lag effect under initial damage conditions in multi-span continuous rigid frame bridges, belonging to the field of structural mechanics analysis. The method includes: deploying displacement sensors to obtain measured deflection influence lines, calculating the difference between these lines and the theoretical deflection influence lines from the finite element method, identifying the damage location using the second derivative of the difference function, and quantifying the section stiffness degradation coefficient. Based on this, the stiffness degradation coefficient is introduced into the fourth-order parabolic shear lag warping displacement function and the energy variational functional to establish a shear lag control differential equation considering the influence of initial damage. A higher-order central difference method is used to construct a difference stiffness matrix and load array for different bridge span boundary conditions, and the analytical solution for bending normal stress is output after solving for the shear rotation displacement. This invention achieves real-time identification of initial damage and accurate stress assessment during construction, providing a basis for early warning of bridge construction safety.

Description

Technical Field

[0001] This invention belongs to the field of structural mechanics analysis, and in particular relates to a method for evaluating the shear lag effect under the initial damage state of a multi-span continuous rigid frame bridge. Background Technology

[0002] Multi-span continuous rigid frame bridges, due to their strong spanning capacity and high overall stiffness, have become the mainstream bridge type in mountainous and deep valley engineering projects, as well as in cross-river and cross-mountain projects. These bridges are generally constructed using the cantilever casting method with hanging baskets, resulting in a long construction period and complex stress on the main girder, which is a thin-walled spatial structure. During the cantilever casting stage, the concrete is in its early age and is susceptible to initial micro-cracks in the main girder section due to factors such as hydration heat temperature difference, early shrinkage and creep, and environmental humidity. Furthermore, frequent movement of the hanging baskets, asymmetrical loading of machinery, and local disturbances before closure can all impact the box girder, which has not yet reached its design strength, inducing local stiffness degradation. These initial damages during construction are often concealed; if they are not identified and quantified in a timely manner, they will accumulate as construction progresses, jeopardizing the bridge's alignment and service life.

[0003] Currently, existing construction monitoring methods primarily focus on macroscopic alignment measurements and surface discrete point monitoring, exhibiting a significant lag in initial damage identification. Furthermore, at the theoretical analysis level, existing technologies still rely on the plane section assumption or uncorrected shear lag models for stress prediction, failing to mechanistically couple the local stiffness degradation during construction with the spatial shear lag effect. The shear lag effect of thin-walled box girders is significantly influenced by the shear deformation of the flange plates, especially at the pier-girder fixed root section where stress concentration is prone to occur. When the main beam experiences stiffness degradation due to initial damage, the shear rotation angle distribution becomes distorted, resulting in a more severe actual shear lag effect than in the healthy state. Existing technologies lack a method to quantify initial damage in real time and simultaneously correct the shear lag analysis model, leading to a severe underestimation of the actual stress level in the damaged area and increasing the risk of cracking and collapse during construction. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a method for evaluating the shear lag effect under the initial damage state of a multi-span continuous rigid frame bridge, comprising: Based on the finite element reference model of the cantilever construction stage of a multi-span continuous rigid frame bridge, the theoretical deflection influence line equation of the main beam is obtained. Based on the vertical displacement data collected by the displacement sensors installed at the target monitoring point under the action of moving construction load, the equation of the influence line of the measured deflection is obtained by fitting. Based on the difference function between the theoretical deflection influence line equation and the measured deflection influence line equation, the initial damage location of the main beam is identified and the stiffness degradation coefficient of each nodal section is quantified. Based on the stiffness degradation coefficient, the shear lag control differential equation containing the stiffness degradation coefficient is discretized using the higher-order central difference method, and differential stiffness matrices are constructed for the boundary conditions of different bridge spans. The load array is constructed based on the initial bending moment of each node section and the stiffness degradation coefficient. Solve the system of linear algebraic equations based on the difference stiffness matrix and the load array to obtain the shear rotation displacement of each node; Based on the differential expression of the shear rotation displacement and the longitudinal normal stress superimposed with the prestressed axial force, the bending normal stress of the cross section is output.

[0005] Optionally, based on the finite element reference model of the cantilever construction stage of a multi-span continuous rigid frame bridge, the theoretical deflection influence line equation of the main beam is obtained, specifically including: A finite element reference model of the rod system for the segmented cantilever construction stage was established using structural analysis software, and the cross-sectional geometric property parameters at each discrete node were extracted. In the finite element reference model, the self-weight of the construction formwork is simulated as a moving load moving longitudinally along the main beam, and the theoretical deflection influence line equation of the target monitoring point is extracted.

[0006] Optionally, based on the vertical displacement data collected by displacement sensors installed at the target monitoring point under the action of moving construction load, the influence line equation of the measured deflection is fitted, specifically including: At the construction site, high-precision displacement sensors are installed at the target monitoring points, and the same moving construction load as in the finite element reference model is applied. The vertical displacement data of the measuring points are recorded in real time, and the influence line equation of the measured deflection under the initial damage state is obtained by fitting.

[0007] Optionally, based on the difference function between the theoretical deflection influence line equation and the measured deflection influence line equation, the initial damage location of the main beam is identified and the stiffness degradation coefficient of each nodal section is quantified, specifically including: Subtracting the measured deflection influence line equation from the theoretical deflection influence line equation yields the deflection influence line difference function; The second derivative of the difference function is taken, and the region where the second derivative is significantly non-zero is determined to have initial damage. The stiffness degradation coefficient of the node section is calculated based on the distance of the deflection displacement measuring point from the beam end, the calculated span of the bridge, and the aforementioned difference function.

[0008] Optionally, based on the stiffness degradation coefficient, the shear lag control differential equation containing the stiffness degradation coefficient is discretized using a higher-order central difference method, and differential stiffness matrices are constructed for the boundary conditions of different bridge spans, specifically including: The high-order central difference method is used to divide the grid nodes along the longitudinal direction of the main beam, and the difference expressions of the derivatives of the displacement function are established. For the simply supported-fixed boundary conditions of the side spans, a first differential stiffness matrix is ​​constructed, and for the double fixed boundary conditions of the middle span, a second differential stiffness matrix is ​​constructed. Calculate each element in the differential stiffness matrix based on the cross-sectional transverse width of each node, the material shear modulus, and the stiffness degradation coefficient.

[0009] Optionally, a higher-order central difference method is used to divide the grid nodes along the longitudinal direction of the main beam, and the difference expressions for the derivatives of the displacement function are established, specifically including: The calculation interval of the main beam is divided into multiple elements, with the element length as the step size; Based on the central difference scheme, difference expressions for the first, second, and third derivatives of the displacement function are established respectively, and the difference expressions are calculated based on the function values ​​of adjacent nodes.

[0010] Optionally, a load matrix is ​​constructed based on the initial bending moments of each node section and the stiffness degradation coefficient, specifically including: Extract the initial section bending moment of each node under the current construction condition from the finite element reference model; A forward difference scheme is used for the first node and a backward difference scheme is used for the last node. Combined with the stiffness degradation coefficient, a load matrix containing the load values ​​of each node is constructed.

[0011] Optionally, the shear rotation displacement of each node is obtained by solving a system of linear algebraic equations based on the difference stiffness matrix and the load array, specifically including: The relationship that the product of the difference stiffness matrix and the displacement matrix equals the load matrix is ​​transformed into a system of linear algebraic equations. The equations were solved using numerical calculation software to obtain the shear rotation displacement matrix of each node.

[0012] On the other hand, the present invention also provides an electronic device including a memory, a processor, and a computing program stored in the memory and executable on the processor, wherein the processor implements the method when executing the computing program.

[0013] On the other hand, the present invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the method.

[0014] Compared with the prior art, the present invention has the following advantages and technical effects: This invention utilizes the second derivative of the difference between the measured deflection influence line and the theoretical value to identify the initial damage location of the main girder during the cantilever construction stage of a multi-span continuous rigid frame bridge in real time, and accurately quantifies the section stiffness degradation coefficient, thus solving the problem of delayed damage identification in existing monitoring methods. Based on this, the stiffness degradation coefficient is introduced into the fourth-order parabolic shear lag warp displacement function and the energy variational functional to establish a shear lag control differential equation considering the influence of initial damage, overcoming the technical deficiency of existing theories in coupling stiffness degradation and stress redistribution. Combining the higher-order central difference method with differentiated stiffness matrices for different bridge span boundary conditions, this invention can efficiently solve for the shear rotation displacement under damaged conditions, outputting a high-precision analytical solution for bending normal stress, significantly improving the reliability of the assessment of the actual stress level in the damaged area, and providing a scientific basis for construction safety early warning and closure alignment control. Attached Figure Description

[0015] 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: Figure 1 This is a schematic diagram of the method flow according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the embedding of deflection displacement measuring points according to an embodiment of the present invention.

[0016] Figure 3 This is a schematic diagram of a bridge calculation model according to an embodiment of the present invention; Figure 4 This is a schematic diagram of a higher-order central difference method according to an embodiment of the present invention; Figure 5 This is a diagram showing the arrangement of the large cantilever in an embodiment of the present invention; Figure 6 This is a diagram showing the arrangement of the small cantilever in an embodiment of the present invention; Figure 7 This is a schematic diagram of a typical cross-section at the main pier support point according to an embodiment of the present invention; Figure 8 This is a schematic diagram of a typical cross-section at a straight segment according to an embodiment of the present invention; Figure 9 This is a finite difference partitioning diagram according to an embodiment of the present invention. Detailed Implementation

[0017] 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.

[0018] 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.

[0019] Example 1 like Figure 1 As shown in the figure, this embodiment provides a method for evaluating the shear lag effect of a multi-span continuous rigid frame bridge under initial damage conditions, including: Based on the finite element reference model of the cantilever construction stage of a multi-span continuous rigid frame bridge, the theoretical deflection influence line equation of the main beam is obtained. Based on the vertical displacement data collected by the displacement sensors installed at the target monitoring point under the action of moving construction load, the equation of the influence line of the measured deflection is obtained by fitting. Based on the difference function between the theoretical deflection influence line equation and the measured deflection influence line equation, the initial damage location of the main beam is identified and the stiffness degradation coefficient of each nodal section is quantified. Based on the stiffness degradation coefficient, the shear lag control differential equation containing the stiffness degradation coefficient is discretized using the higher-order central difference method, and differential stiffness matrices are constructed for the boundary conditions of different bridge spans. The load array is constructed based on the initial bending moment of each node section and the stiffness degradation coefficient. Solve the system of linear algebraic equations based on the difference stiffness matrix and the load array to obtain the shear rotation displacement of each node; Based on the differential expression of the shear rotation displacement and the longitudinal normal stress superimposed with the prestressed axial force, the bending normal stress of the cross section is output.

[0020] Specifically, it includes: Step 1: Establish a finite element baseline model and extract the theoretical deflection influence line; Using structural analysis software such as Midas / Civil, a finite element baseline model of the frame system for the segmented cantilever construction stage of this multi-span continuous rigid frame bridge was established. Discrete nodes were then extracted. The cross-sectional geometric parameters at the location include the section bending moment of inertia I, the sum of the bending moments of inertia of the top and bottom plates Is, the cross-sectional area Ai, the distance hi from the neutral axis to the centerline of the flange plate, and the half-width bi of the flange plate. Simultaneously, the self-weight P of the construction formwork is simulated as a moving load moving longitudinally along the main beam, and the theoretical deflection influence line equation of the target monitoring point is extracted. .

[0021] Step 2: Extraction of influence lines for quasi-static on-site tests and measured deflections; At the construction site, high-precision displacement sensors are installed at the target monitoring points, such as... Figure 2 As shown.

[0022] When the same moving construction load P is applied, the vertical displacement data of the measuring points are recorded in real time, and the influence line equation of the measured deflection under the initial damage state is obtained by fitting. Subtracting the measured value from the theoretical value yields the deflection influence line difference function. .

[0023] Step 3: Identify initial damage and quantify the stiffness degradation coefficient; Taking the second derivative of the difference function obtained in step 2, we get... If the second derivative of a certain interval is significantly non-zero, then the region is determined to have initial damage during the construction period.

[0024] based on Figure 3 The beam calculation model then shows the cross section of this node. Stiffness degradation coefficient : ; In the formula: is the stiffness degradation coefficient; E is the material elastic modulus; I is the bending moment of inertia of the section; as is the distance from the deflection displacement measuring point to the beam end; L is the calculated span of the bridge.

[0025] Step 4: Longitudinal differential separation of the main beam; Since the bottom slab of a continuous rigid frame bridge often exhibits a parabolic shape and is a variable cross-section beam, its parameters such as Is are functions of the longitudinal coordinate. To solve for parameters including the stiffness degradation coefficient... The complex variable-coefficient shear lag differential equation is solved using the higher-order central difference method.

[0026] The differential equation is discretized using the finite difference method, transforming it into a system of algebraic equations to obtain its approximate solution. In the specific solution process, the computation interval is first divided into difference intervals, and a central difference scheme is used for discretization derivation. The difference diagram is shown below. Figure 4 As shown.

[0027] element (can represent) , The difference expressions for the derivatives of each order of the (or other internal force functions) are as follows: First derivative: ; Second derivative: ; Third derivative: ; In the formula: Let be any difference point on the beam; The step size for dividing the grid.

[0028] Step 5: Construct the difference stiffness matrix considering stiffness degradation and boundary conditions. ; Multi-span continuous rigid frame bridges are statically indeterminate structures. Due to the different boundary conditions of the side spans and the middle span, they need to be calculated separately using the finite difference method. (1) Side span section; The boundary conditions at both ends of the side span are simply supported and fixed, respectively, and the section along the longitudinal direction of the main beam is divided into... Each unit is numbered, with the left hinged end numbered 0 and the right fixed end numbered 0. .

[0029] but The difference equation for time is: ; In the formula: It is a displacement function; The moving load is on the main beam.

[0030] but The difference equation at time is: ; when When the main beam section has a differential form, it can be represented by a differential stiffness matrix. It can be expressed in the form of: ; (2) The middle span section; The boundary conditions for the mid-span of a continuous rigid frame bridge are all fixed ends. Considering the boundary conditions on both sides, the stiffness matrix is... The expression is: ; The unknowns in the stiffness matrix are The formula for its calculation is: ; ; In the formula: The transverse width of the main beam section; This is the material's shear modulus.

[0031] Step 6: Construct a load array containing internal force and damage parameters. ; When a bridge is subjected to loads, its load matrix expression is as follows: ; Extract the initial section bending moment M of each node under the current construction condition. For intermediate nodes, use forward differencing; for end nodes, use backward differencing, combined with the stiffness degradation coefficient. Constructing a load array : when When using forward difference: ; when When using backward difference, we can obtain: ; The load array can be expressed as: ; Step 7: Solve for the longitudinal displacement matrix and output the bending normal stress; Transform the results obtained in steps 5 and 6 into a system of linear algebraic equations: ; In the formula: This is the difference stiffness matrix; It is a displacement matrix; For load array.

[0032] The equations were solved using numerical computation software such as MATLAB to obtain the shear rotation displacement matrix of each node. That is, to obtain the value of each node. .

[0033] The obtained displacement values Substituting the superimposed prestressed axial force The difference expression for longitudinal normal stress: ; Based on this expression, the output is the final bending normal stress of the cross section, taking into account prestress, local weakening of initial damage, and the spatial shear lag coupling effect of variable cross section. This will guide on-site prestressing tension control and structural safety assessment.

[0034] Example 2 This embodiment provides a method for evaluating the shear lag effect of a multi-span continuous rigid frame bridge under initial damage conditions, including: A prestressed concrete continuous rigid frame bridge with a span combination of 55+140+101m has a total span of 296m and a single-span clear width of 12.5m. It uses a variable cross-section prestressed concrete box girder as the superstructure, with double-limb thin-walled piers as the substructure. The entire bridge was constructed using a symmetrical cantilever casting process, with the girder structure divided into two T-shaped sections (No. 1 and No. 2) for simultaneous construction. The specific girder layout is as follows... Figure 5 and Figure 6 As shown.

[0035] The main girder has a single-box, single-cell, straight-web box girder cross-section. The girder height at the top support of pier #1 is 12m, and the height at the straight section of the side span is 3.5m, varying according to a 1.8-fold parabola. At pier #2, the girder height at the top support is 7m, and the height at the straight section of the side span is 3.5m, varying according to a 2-fold parabola. The box girder cross-section is a single-box, single-cell, straight-web box girder. The top slab width is 12.5m, the bottom width is 6.5m, and the cantilever length of the two side flanges is 3m.

[0036] Typical cross-sectional dimensions of the main pier support and straight section are as follows: Figure 7 , Figure 8 As shown.

[0037] Taking the small cantilever construction stage as an example, calculations were performed. Deflection displacement measuring points were arranged on the main beam, and deflection displacement data were extracted. To ensure convergence, the side span of the continuous rigid frame bridge was divided into 17 units along the longitudinal direction, each unit being 230cm in length. The fixed-end unit was numbered 0, and the unit numbers increased sequentially towards the simply supported end, with the simply supported end numbered 17. A total of 18 nodes were divided into the entire side span; for details, please refer to [link to relevant documentation]. Figure 9 A diagram showing the division of the data.

[0038] Using the section property analysis function of ANSYS finite element software, relevant parameters of each section from 0 to 17 were extracted and calculated. , The calculation results for the actual bridge are shown in the following tables, where Table 1 shows the cross-sectional calculation parameters and Table 2 shows the stiffness matrix. Calculation parameters, Table 3 shows the load array. calculate: Table 1

[0039] Table 2

[0040] The internal force values ​​of each section are extracted based on the Midas / Civil model, and the load array is calculated. Load array The values ​​are shown in Table 3.

[0041] Table 3

[0042] Table 4

[0043] It uses MATLAB software to solve the matrix simultaneously. The results. For safety reasons, this embodiment is intended for... Different difference processing methods were applied to solve the problem, and the most unfavorable value (i.e. the maximum value) in the calculation results was finally selected as the design basis.

[0044] extract Figure 9 The relevant calculation parameters of two typical sections, No. 1 and No. 11, were used to calculate the maximum shear lag coefficient of their top plate. The specific calculation results are shown in Table 4.

[0045] Table 4

[0046] The calculation error is around 6%, which is within the acceptable range for engineering projects and can provide a basis for bridge health assessment.

[0047] On the other hand, this embodiment also provides an electronic device, including a memory, a processor, and a computing program stored in the memory and executable on the processor, wherein the processor implements the method when executing the computing program.

[0048] On the other hand, this embodiment also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the method.

[0049] 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 evaluating the shear lag effect under initial damage conditions of a multi-span continuous rigid frame bridge, characterized in that, include: Based on the finite element reference model of the cantilever construction stage of a multi-span continuous rigid frame bridge, the theoretical deflection influence line equation of the main beam is obtained. Based on the vertical displacement data collected by the displacement sensors installed at the target monitoring point under the action of moving construction load, the equation of the influence line of the measured deflection is obtained by fitting. Based on the difference function between the theoretical deflection influence line equation and the measured deflection influence line equation, the initial damage location of the main beam is identified and the stiffness degradation coefficient of each nodal section is quantified. Based on the stiffness degradation coefficient, the shear lag control differential equation containing the stiffness degradation coefficient is discretized using the higher-order central difference method, and differential stiffness matrices are constructed for the boundary conditions of different bridge spans. The load array is constructed based on the initial bending moment of each node section and the stiffness degradation coefficient. Solve the system of linear algebraic equations based on the difference stiffness matrix and the load array to obtain the shear rotation displacement of each node; Based on the differential expression of the shear rotation displacement and the longitudinal normal stress superimposed with the prestressed axial force, the bending normal stress of the cross section is output.

2. The method according to claim 1, characterized in that, Based on the finite element reference model of the cantilever construction stage of a multi-span continuous rigid frame bridge, the theoretical deflection influence line equation of the main beam is obtained, specifically including: A finite element reference model of the rod system for the segmented cantilever construction stage was established using structural analysis software, and the cross-sectional geometric property parameters at each discrete node were extracted. In the finite element reference model, the self-weight of the construction formwork is simulated as a moving load moving longitudinally along the main beam, and the theoretical deflection influence line equation of the target monitoring point is extracted.

3. The method according to claim 1, characterized in that, Based on the vertical displacement data collected by displacement sensors installed at the target monitoring points under the action of moving construction loads, the influence line equation of the measured deflection is fitted, specifically including: At the construction site, high-precision displacement sensors are installed at the target monitoring points, and the same moving construction load as in the finite element reference model is applied. The vertical displacement data of the measuring points are recorded in real time, and the influence line equation of the measured deflection under the initial damage state is obtained by fitting.

4. The method according to claim 1, characterized in that, Based on the difference function between the theoretical deflection influence line equation and the measured deflection influence line equation, the initial damage location of the main beam is identified and the stiffness degradation coefficient of each nodal section is quantified, specifically including: Subtracting the measured deflection influence line equation from the theoretical deflection influence line equation yields the deflection influence line difference function; The second derivative of the difference function is taken, and the region where the second derivative is significantly non-zero is determined to have initial damage. The stiffness degradation coefficient of the node section is calculated based on the distance of the deflection displacement measuring point from the beam end, the calculated span of the bridge, and the aforementioned difference function.

5. The method according to claim 1, characterized in that, Based on the stiffness degradation coefficient, the shear lag control differential equation containing the stiffness degradation coefficient is discretized using the higher-order central difference method, and differential stiffness matrices are constructed for the boundary conditions of different bridge spans, specifically including: The high-order central difference method is used to divide the grid nodes along the longitudinal direction of the main beam, and the difference expressions of the derivatives of the displacement function are established. For the simply supported-fixed boundary conditions of the side spans, a first differential stiffness matrix is ​​constructed, and for the double fixed boundary conditions of the middle span, a second differential stiffness matrix is ​​constructed. Calculate each element in the differential stiffness matrix based on the cross-sectional transverse width of each node, the material shear modulus, and the stiffness degradation coefficient.

6. The method according to claim 5, characterized in that, A higher-order central difference method is used to divide the grid along the longitudinal direction of the main beam, and the difference expressions for the derivatives of the displacement function are established, specifically including: The calculation interval of the main beam is divided into multiple elements, with the element length as the step size; Based on the central difference scheme, difference expressions for the first, second, and third derivatives of the displacement function are established respectively, and the difference expressions are calculated based on the function values ​​of adjacent nodes.

7. The method according to claim 1, characterized in that, The load array is constructed based on the initial bending moments of each node section and the stiffness degradation coefficient, specifically including: Extract the initial section bending moment of each node under the current construction condition from the finite element reference model; A forward difference scheme is used for the first node and a backward difference scheme is used for the last node. Combined with the stiffness degradation coefficient, a load matrix containing the load values ​​of each node is constructed.

8. The method according to claim 1, characterized in that, Solving the system of linear algebraic equations based on the difference stiffness matrix and the load column matrix yields the shear rotation displacement of each node, specifically including: The relationship that the product of the difference stiffness matrix and the displacement matrix equals the load matrix is ​​transformed into a system of linear algebraic equations. The equations were solved using numerical calculation software to obtain the shear rotation displacement matrix of each node.

9. An electronic device comprising a memory, a processor, and a computing program stored in the memory and executable on the processor, characterized in that, When the processor executes the computing program, it implements the method of any one of claims 1-8.

10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the method of any one of claims 1-8.

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