Method for updating section stiffness distribution of concrete box girder bridge

Abstract

The invention belongs to the technical field of structural safety detection, and discloses a concrete box girder bridge section stiffness distribution updating method which comprises the following steps: (1) acquiring strain data required for updating the section stiffness of a box girder; (2) establishing a stress-strain analysis theory; (3) discretizing the cross section of the box girder to establish a numerical model; and (4) updating the rigidity distribution of the cross section of the box girder. Based on the problems that the rigidity distribution of the cross section of the box girder bridge is uneven and the stress-strain distribution of the cross section is complex, a generalized girder theory and a model updating method are used for inverting the rigidity distribution condition in the cross section of the box girder, and the stress-strain in the cross section of the box girder when the rigidity is uneven is accurately calculated. According to the method, a rigidity updating strategy taking the inertia moment and the neutral axis as constraint conditions is applied, the section rigidity distribution can be well updated, and the rigidity updating calculation efficiency is improved. Meanwhile, the method considers the special deformation mode of the cross section of the box girder, obtains the proportion of various deformation modes in the total stress, and is more beneficial to the evaluation of the stress condition.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of structural safety detection, and relates to a concrete box girder bridge cross-section stiffness distribution updating method. BACKGROUND

[0002] Concrete box girder bridges are widely used in highway and railway bridges because the box section has high bending and torsional stiffness. Under the long-term coupling action of various loads, concrete box girder bridges may have phenomena such as local cracking, concrete spalling, and steel bar corrosion. Cracking, as one of the most typical diseases of concrete box girder bridges, has obvious regularity in the distribution on the box girder section, such as cracks in the midspan of the concrete box girder bridge usually concentrating on the web and the bottom plate. These phenomena cause uneven distribution of the internal stiffness of the box girder section, which has a certain influence on the accuracy of structural safety assessment. Stress analysis of the box girder bridge is an important link to ensure the safety of the bridge structure. The box girder bridge has torsion, distortion, lateral bending and other unique deformations due to the thin-walled closed section. Therefore, the stress state of the box girder section is complex, and in the analysis, it cannot be regarded as a traditional beam (Euler beam or Timoshenko beam). Meanwhile, the uneven distribution of the internal stiffness of the section further increases the difficulty of the analysis of the stress state of the section. Therefore, how to consider the complex stress state of the box girder bridge and the uneven degradation of the stiffness during operation in the stress analysis is a problem worthy of study.

[0003] The main girder cracking weakens the overall stiffness of the main girder cross section, and due to the unevenness of the cracking, the internal stiffness of the section is also unevenly distributed. In the midspan position, transverse cracks on the bottom plate and vertical cracks on the web usually appear; near the quarter span, oblique cracks on the web often appear. These cracks are structural cracks that have a serious impact on the overall performance of the structure and significantly affect its stiffness. Meanwhile, the reinforcement ratio inside the box girder section is not uniform, so the stiffness distribution of the box girder section is also affected. The box girder section belongs to a thin-walled closed section, and the deformation mode is quite different from the traditional beam theory. Due to the influence of torsion, distortion, lateral bending and other unique deformations, the stress state inside the section cannot satisfy the plane section assumption, and the deformation and stress state inside the section are relatively complex. With the uneven distribution of stiffness caused by cracking, the accurate analysis of the stress of the section is a relatively complex problem. For the problem of internal stress analysis of the box girder section, scholars have adopted many methods, such as the finite element method, the elastic theory method, etc. Among them, the generalized beam theory (GBT) is proposed as a method for analyzing thin-walled structures. This method is a semi-theoretical and semi-numerical method, which considers various unique deformation modes of the box girder section on the basis of the traditional beam theory, so it has great advantages in analyzing the complex stress state of the box girder section. For other hollow section single beams, this method is also applicable to the analysis of the internal stress and strain distribution.

[0004] There are two types of methods for the inversion of the bridge section stiffness, one is the model updating method based on optimization algorithm and finite element model, and the other is the theoretical method based on mechanical formula derivation. The model updating method is widely used in the field of bridge carrying capacity assessment, performance prediction, etc. due to its simplicity and association with the structure model. Ren et al. in 2011 (Response Surface–Based Finite-Element-Model Updating Using Structural Static Responses) updated the stiffness of each beam segment of a five-span continuous box girder bridge using the response surface method. Xie Weiping et al. in 2018 (Finite element model establishment, updating and analysis of wide steel box girder bridge based on modal test) completed the vibration characteristic identification and main girder stiffness updating of a wide steel box girder bridge based on vibration data. Lin et al. in 2009 (Dynamic finite element model updating of prestressed concrete continuous box-girder bridge) updated the model of a prestressed concrete continuous box girder bridge, and the main girder section stiffness was the main updating content. However, the current scholars' inversion of the stiffness of the box girder is mostly limited to the overall stiffness of the section, and the uneven distribution of the internal stiffness of the section is rarely involved. When the internal stiffness of the section is unevenly distributed, the stress distribution is naturally different from that of the uniform stiffness state. Therefore, the idea of model updating can be used to update the uneven stiffness of the internal section of the box girder, and the distribution of the section stiffness can be inverted. The updating idea is also applicable to other types of hollow section single beams.

[0005] In summary, when analyzing the stress of the concrete box girder bridge section and assessing its safety, the unevenness of the internal stiffness degradation of the box girder section and the complexity of the stress state of the box girder section should be considered. Therefore, it is of great significance to study a method for updating the distribution of the stiffness of the concrete box girder bridge section to evaluate the stress state and safety redundancy of the bridge section. SUMMARY

[0006] The purpose of the present application is to provide a method for updating the distribution of the stiffness of the concrete box girder bridge section.

[0007] The technical scheme of the present application is as follows:

[0008] A method for updating the distribution of the stiffness of the concrete box girder bridge section, the steps are as follows:

[0009] Step 1. Measure the updating data of the section;

[0010] The updating of the sectional stiffness distribution of concrete box girder bridges involves the inversion of local stiffness, and therefore requires local response quantities, i.e. strain data, which are obtained using static load tests or quasi-static load tests;

[0011] The strain data are collected using strain gauges with an accuracy of no less than 1 % and in accordance with the following installation principles: 1) the strain gauges are installed along the longitudinal direction of the bridge on the surface of the box girder; 2) at least one strain gauge is arranged at each joint of the box girder and at each end of each plate; 3) for plate members with a large length, at least one strain gauge is additionally arranged in the middle of the plate member, including the bottom plate and the top plate;

[0012] Step 2. Analysis of the stress and strain of the box girder section;

[0013] In the box girder section, a local coordinate system is established for any plate member , with the origin located at the end of the plate member, the x-axis parallel to the axial direction of the girder, the y-axis parallel to the tangential direction of the plate member and pointing in the counterclockwise direction, and the z-axis parallel to the normal direction of the plate member and pointing to the inside of the box girder; the axial displacement , the tangential displacement and the normal displacement of any point on the middle surface of the plate member along the three directions of the local coordinate system are defined, and the displacement vector of any point on the plate member is expressed in the following form:

[0014]

[0015] wherein d is the distance from the point to the middle surface of the plate member, , , are the first-order derivatives of , ; , , According to the generalized beam theory, the deformation modes are decomposed into a linear combination of deformation modes, which is expressed as:

[0016]

[0017] wherein n is the corresponding order of the deformation mode, , , , is the deformation mode displacement function with respect to y, is the amplitude function with respect to x; and the displacement-strain relationship of any point on the box girder section is expressed as:

[0018]

[0019] where, , , are the axial strain, transverse strain and shear strain, respectively, and the superscripts M and B denote the membrane strain component and the bending strain component, respectively; , denote the x and y direction curvatures at the point, is the torsion at the point, , denote the second derivatives of , , denote the third derivatives of , ,

[0020] According to the strain of equation (3) and the principle of minimum potential energy, the following control equation is derived:

[0021]

[0022]

[0023] where, is the cross-sectional stiffness matrix associated with the transverse extensional deformation, is the cross-sectional stiffness matrix associated with the longitudinal extensional deformation, is the cross-sectional stiffness matrix associated with the shear deformation, is the cross-sectional stiffness matrix associated with the longitudinal and transverse coupling deformation, is the load matrix, is the fourth derivative of x; the above matrices are obtained according to the following formula:

[0024]

[0025]

[0026]

[0027]

[0028]

[0029] where, , , , are the first, second, third and fourth derivatives of the corresponding matrix, respectively, Row Element of column, S is the cross-sectional area of the box girder, E is the elastic modulus, ν is the Poisson's ratio, G is the shear modulus, h is the wall thickness, , , respectively, the external force components in the x, y, z directions of the local coordinate system of the box girder plate; therefore, by obtaining the deformation mode of the box girder cross section, the structural analysis is completed, and the stress and strain distribution of the box girder cross section is obtained;

[0030] Step 3. Discretization and analysis of box girder cross section;

[0031] When using the GBT theory to analyze the stress of the box girder cross section, the box girder cross section needs to be discretized according to the characteristics of the box girder to obtain the deformation mode of the box girder cross section; the discretization of the box girder cross section needs to meet the requirements of stress fine analysis and the requirements of updating the stiffness distribution of the box girder cross section;

[0032] The following principles should be followed for cross section discretization: 1) each plate is divided into at least two units, including the top plate, the bottom plate, the web plate and the flange plate; 2) for plate with large length, increase the number of units, use 3-4 units; 3) the division length of each unit in the same plate is consistent; 4) the discretization length of all units in the entire box girder cross section is as close as possible;

[0033] Combined with step two, a complete numerical model of box girder bridge and cross section stress analysis method is formed, and the process is as follows:

[0034] (1) Discretize the box girder according to the discretization principle of the box girder cross section in step 3;

[0035] (2) separately apply unit displacement in x, y, z directions to each node on the box girder cross section, and the cross section shape is the initial shape function 、 、 ;

[0036] (3) Calculate the initial cross section matrix according to formula (6)-(9), and implement generalized orthogonalization to obtain the deformation mode with structural significance;

[0037] (4) Recalculate the cross section matrix according to the obtained deformation mode, and calculate the deformation state and stress and strain state of the box girder cross section according to formula (2)-(4);

[0038] Step 4. Updating the stiffness distribution of the box girder cross section;

[0039] The updating of the stiffness distribution of the box girder bridge cross section adopts the idea of model correction, and the objective function is established by the strain data obtained in step one, reflecting the difference between the actual stiffness change of the cross section and the theoretical state; the objective function of the cross section stiffness updating is as follows:

[0040]

[0041] In the formula, is the measured strain value, is the model calculated strain value; the model calculated strain value is the longitudinal bridge strain, including a thin film strain component and a bending strain component, and is obtained by the following formula:

[0042]

[0043] In addition, the existing conditions are used as additional constraints when updating the box girder section stiffness distribution, and the existing conditions include two aspects:

[0044] First, the overall stiffness of the box girder section is consistent with the prior inversion:

[0045]

[0046] In the formula, is the overall bending stiffness of the box girder section, which is obtained by using model correction or theoretical formula inversion according to the measured data; and are the inertia moment and area of the i-th plate element in the box girder section, respectively; is the distance from the centroid of the i-th plate element to the top surface of the box girder; is the stiffness variation coefficient of the i-th plate element, which is also a correction parameter; by updating so that the section stiffness distribution is consistent with the actual situation;

[0047] Second, the neutral axis of the box girder section is consistent with the measured strain calculation, that is:

[0048]

[0049] In the formula, and are the length and width of the i-th plate element in the box girder section, respectively, is the measured neutral axis of the box girder section, that is, the distance from the neutral axis to the top surface;

[0050] The stiffness variation coefficient is updated using an optimization algorithm under the condition of meeting the constraint condition , the objective function is minimized, that is, the process of updating the box girder bridge section stiffness distribution is completed, and a box girder bridge model with a stiffness distribution that matches the actual situation is obtained, which is used for accurate analysis and safety evaluation of the section stress.

[0051] The beneficial effects of the present application are:

[0052] 1. An updating method for the distribution of the sectional rigidity of a box girder bridge is proposed, which can fully consider the uneven changes in the sectional rigidity caused by the cracking and other deteriorations of the box girder section, obtain a more reasonable sectional rigidity distribution, and realize the accurate stress distribution acquisition and safety evaluation of the box girder section;

[0053] 2. A sectional discretization and stress-strain analysis method for a box girder bridge is proposed, which can consider the special deformation mode of the box girder section when the sectional rigidity distribution changes arbitrarily, obtain the accurate stress-strain distribution on the box girder section, and obtain the proportion of various deformation modes in the total stress;

[0054] 3. An updating strategy considering the inertia moment and neutral axis as constraint conditions is proposed for the updating of the sectional rigidity distribution of the box girder. The updating strategy can better update the sectional rigidity distribution and improve the calculation efficiency of the rigidity updating. BRIEF DESCRIPTION OF DRAWINGS

[0055] Figure 1 The figure shows the position of the measuring points of the method adopted by the present application;

[0056] Figure 2 The figure shows the reference diagram of the sectional discretization of the box girder of the method adopted by the present application;

[0057] Figure 3 The figure shows the flowchart of the implementation of the generalized beam theory of the method adopted by the present application;

[0058] Figure 4 The figure shows the flowchart of the implementation of the updating of the sectional rigidity distribution of the box girder of the method adopted by the present application;

[0059] Figure 5 The figure shows the box girder bridge simulated in the embodiment of the method of the present application: (a) front view; (b) A-A section;

[0060] Figure 6 The figure shows the deformation mode diagram of each order of the box girder section in the embodiment of the method of the present application; 1-13 are the corresponding order numbers;

[0061] Figure 7 The figure shows the strain comparison diagram of the box girder section in the embodiment of the method of the present application. DETAILED DESCRIPTION

[0062] The specific embodiments of the present application are further described below in combination with the drawings and technical solutions.

[0063] Implementation:

[0064] In the example, a 25m-span equal-section concrete box girder bridge is simulated, and the detailed information is shown in Table 1. Figure 5 The bridge model is established using the GBT theory, and the sectional discretization is performed according to the method of the present application. Figure 2As shown. The cross-sectional stiffness variations are set according to the plate members, with the top plate stiffness reduced by 10%, the web stiffness increased by 10%, and the bottom plate stiffness reduced by 30%. A 3t load is set at mid-span for simulation loading, and the results are extracted as follows. Figure 1 The longitudinal strain response at the indicated location.

[0065] Based on steps two and three, a numerical model of the box girder bridge based on GBT theory is established. First, the deformation modes of the cross-section can be obtained. The deformation of the box girder bridge cross-section is composed of these deformation modes in different proportions. The deformation modes involved in the deformation calculation are as follows: Figure 6 As shown, orders 1-4 represent overall deformation, order 5 represents distortion deformation, orders 6-9 represent local deformation, orders 10-11 represent overall shear deformation, and orders 12-13 represent local shear deformation. Following step four, the extracted longitudinal strain and the model's theoretical value are used to construct the objective function, which is then optimized using an optimization algorithm to update the cross-sectional stiffness distribution. During the update process, attention should be paid to the application of constraints. First, the overall cross-sectional stiffness is updated using conventional methods. The overall cross-sectional stiffness is inverted using a relatively mature model correction method, and the results show that the overall cross-sectional stiffness has decreased by 18.57%. Subsequently, based on the strain extraction results, the neutral axis is calculated to be located at 0.33m. After optimization iterations, the stiffness update results are shown in Table 1, where the stiffness coefficient = current stiffness / initial stiffness.

[0066] Table 1 Stiffness Update Results

[0067]

[0068] As shown in Table 1, after adopting the cross-sectional stiffness distribution update method proposed in this paper, the calculated stiffness of each plate of the box girder corresponds well with the set stiffness, laying the foundation for accurate cross-sectional stress-strain analysis. Figure 7 Table 2 shows the cross-sectional strain and extracted strain results before and after the stiffness update. Figure 7 It can be seen that the strain distribution differs significantly under different working conditions. Table 2 shows that after the stiffness reduction of the top and bottom plates, their strains increase by approximately 12.8% and 25.7%, respectively, with a clear correlation between the strain change and the stiffness adjustment. Compared to the overall section stiffness update method, the strain response under non-uniform stiffness distribution differs to some extent: the maximum strain difference in the top plate is approximately 9%, and in the bottom plate, approximately 1.9%. The specific degree of difference is related to the stiffness distribution pattern, reflecting the significant influence of stiffness non-uniformity on strain distribution. After updating using the proposed method, the section strain distribution agrees well with the measured values, indicating that considering section stiffness non-uniformity has significant advantages in stress-strain calculation and safety assessment of box girder bridges.

[0069] Table 2 Comparison of strain in box girders (unit: με)

[0070]

[0071] Note: Data in parentheses is the error from the set value.

Claims

1. A method for updating the distribution of the sectional rigidity of a concrete box girder bridge, characterized in that, The steps are as follows: Step 1. Measurement of concrete box girder bridge section updating data; The updating of the concrete box girder bridge section stiffness distribution involves the inversion of local stiffness, and therefore requires local response quantities, i.e. strain data, which are obtained using static load tests or quasi-static load tests; Step 2. Analysis of box girder section stress and strain; In the box girder section, a local coordinate system is established for any plate , of which the origin is at the end of the plate, the x-axis is parallel to the axial direction of the box girder, the y-axis is parallel to the tangential direction of the plate, pointing to the counterclockwise direction, and the z-axis is parallel to the normal direction of the plate, pointing to the inside of the box girder; the displacement of any point on the plate in the three directions of the local coordinate system is defined as: the axial displacement , the tangential displacement , and the normal displacement ; then the displacement vector of any point on the plate is expressed in the following form: ； wherein, is the distance of this point to the mid-plane of the plate, , is the first derivative of , , , is decomposed into a linear combination of deformation modes according to the generalized beam theory, expressed as:​ ； wherein, is the deformation mode corresponding order, , , is the deformation mode displacement function about y, is the amplitude function about x; the displacement-strain relationship of any point on the box girder section is expressed as: ； wherein , , are the axial strain, transverse strain and shear strain, respectively, and the superscripts M and B denote the membrane strain component and the bending strain component, respectively; , denote the x and y direction curvatures at the point, is the torsion at the point, , denote the second derivatives of the x and y direction curvatures, , denote the second derivatives of the x and y direction curvatures, denote the second derivatives of the x and y direction curvatures, , denote the second derivatives of the x and y direction curvatures, According to the strain of equation (3) and the principle of minimum potential energy, the following control equation is derived: ； ； wherein is the cross-sectional stiffness matrix related to transverse extensional deformation, is the cross-sectional stiffness matrix related to longitudinal extensional deformation, is the cross-sectional stiffness matrix related to shear deformation, is the cross-sectional stiffness matrix related to longitudinal and transverse coupling deformation, is the load matrix, is the fourth derivative of x with respect to x; the above matrix is obtained according to the following formula: ； ； ； ； ； wherein, , , , are elements in the matrix corresponding to the first row column, is the cross-sectional area of the box girder, E is the elastic modulus, is the Poisson's ratio, G is the shear modulus, h is the wall thickness, , , are the external force components of the plate in the local coordinate system x, y, z directions; therefore, by obtaining the deformation mode of the box girder section, the structural analysis is completed, and the stress and strain distribution of the box girder section is obtained; Step 3. Discretization and analysis of box girder section; When using the GBT theory to analyze the stress of the box girder section, it is necessary to discretize the box girder section according to its characteristics to obtain the deformation mode of the box girder section; the discretization of the box girder section must meet the requirements of both fine stress analysis and stiffness distribution updating of the box girder section; In combination with step two, a complete method for establishing a numerical model of a box girder bridge and analyzing the stress of the section is formed, and the process is as follows: (1) Discretize the box girder according to the discretization principle of the box girder section in step 3; (2) Apply unit displacement in x, y, z direction to each node of the box girder section respectively, and the section shape is the initial shape function , , ; (3) Calculate the initial section matrix according to equations (6)~(9) and implement generalized orthogonalization to obtain deformation modes with structural significance; (4) Recalculate the section matrix according to the obtained deformation modes, and calculate the deformation state and stress-strain state of the box girder section according to equations (2)~(4); Step 4. Updating of box girder section stiffness distribution; The updating of the box girder bridge section stiffness distribution uses the idea of model correction, and the objective function is established by the strain data obtained in step one, reflecting the difference between the actual stiffness change of the section and the theoretical state; the objective function for updating the section stiffness is as follows: ； wherein εmeas is the measured strain value, εmodel is the model calculated strain value; the model calculated strain value is the longitudinal bridge direction strain, including both the thin film strain component and the bending strain component, and is given by the equation: εmodel = εthin film + εbend ； In addition, existing conditions are used as additional constraints during the updating of the box girder section stiffness distribution, which includes two aspects: First, the overall stiffness of the box girder section is consistent with the previously inverted one: ； In the formula, is the overall bending stiffness of the box girder section, which is obtained by using model correction or theoretical formula inversion according to the measured data; and are the inertia moment and area of the i-th plate element in the box girder section, respectively; is the distance from the centroid of the i-th plate element to the top surface of the box girder; is the stiffness variation coefficient of the i-th plate element, which is also used as a correction parameter; by updating so that the cross-sectional stiffness distribution tends to be consistent with the actual situation; Second, the neutral axis of the box girder section is consistent with the one calculated according to the measured strain, i.e. ； In the formula, With L and W are the length and width of the i-th plate element in the box girder section, respectively, is the distance from the measured neutral axis of the box girder section to the top surface. Updating the stiffness variation coefficient using an optimization algorithm while satisfying the constraint conditions The process of minimizing the objective function, i.e. completing the updating of the stiffness distribution of the box girder bridge section, obtains a box girder bridge model with a stiffness distribution that matches the actual one, for accurate analysis of section stress and safety evaluation.

2. The method of claim 1, wherein, In step 1, the strain data collection uses a strain gauge with an accuracy of no less than 1 % and follows the following installation principles: 1) the strain gauge is installed on the surface of the box girder along the longitudinal direction of the bridge; 2) at least one strain gauge is arranged at each joint of the box girder and at each end of each plate; 3) for plate members with a relatively large length, at least one strain gauge needs to be additionally arranged at the middle part of the plate member, including the bottom plate and the top plate.

3. The method of claim 1, wherein, In step 3, the section discretization should follow the following principles: 1) Each plate piece is divided into at least two units, including the top plate, bottom plate, web plate, and flange plate; 2) For plate pieces with a longer length, increase the number of units to 3-4 units; 3) The division length of each unit in the same plate piece is consistent; 4) The discretization length of all units in the entire box girder section is as close as possible.

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