CN117077272A - Axle coupling numerical value solution prediction method - Google Patents
Axle coupling numerical value solution prediction method Download PDFInfo
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Abstract
The application discloses a method for predicting an axle coupling numerical value, which belongs to the field of axle coupling analysis and comprises the following steps: constructing a bridge finite element model and a standard vehicle finite element model of each vehicle type; designing simulation calculation working conditions of a standard vehicle finite element model of each vehicle type by using a sampling method; calculating a corresponding axle coupling numerical solution; taking vehicle physical parameter information of a standard vehicle finite element model as input data, taking a corresponding axle coupling numerical solution as output data to form a plurality of training samples, classifying the training samples according to different vehicle weight ranges to form a plurality of sample sets, and respectively training out prediction models corresponding to different vehicle weight ranges; the dynamic weighing system measures the physical parameter information of the vehicles of the upper bridge vehicle and inputs the physical parameter information into a prediction model corresponding to the weight range of the vehicle, and the prediction model calculates and outputs the bridge structure response predicted value. The method can rapidly predict the response value of the bridge structure and can provide the response prediction of the bridge target position in a targeted manner.
Description
Technical Field
The application relates to the technical field of axle coupling analysis, in particular to an axle coupling numerical value de-prediction method.
Background
In the field of civil engineering, the structure numerical simulation can provide references in structural design, safety risk assessment and the like for engineers, and has very important engineering significance. For scientific theory research, the test component and loading equipment of civil engineering are larger, the cost of one test is higher, and the cost of theoretical research can be greatly reduced by numerical simulation.
In the monitoring of the daily operation of the bridge, the structural response of the bridge under the action of the vehicle load is very important for the bridge safety risk assessment. However, bridge structures are generally large in size, the number of sensors is limited, and all bridge structure sections or components cannot be monitored, so that auxiliary calculation, such as axle coupling analysis, is required by numerical simulation. The axle coupling analysis belongs to the structural dynamics category, and essentially solves a second-order motion differential equation set formed by a bridge structure and corresponding vehicle physical parameters, and usually adopts a numerical method of gradual integration to solve the dynamic response of the bridge structure.
Disclosure of Invention
The application aims to provide a bridge coupling numerical value solution prediction method, which can be used for rapidly predicting a bridge structure response value and can be used for pointedly providing a power response prediction of a bridge target position.
In order to achieve the above purpose, the application adopts the following technical scheme: an axle coupling numerical solution prediction method, comprising the steps of:
s1, constructing a bridge finite element model, and constructing a standard vehicle finite element model of a corresponding vehicle weight range according to different vehicle types;
s2, designing simulation calculation working conditions of the standard vehicle finite element model of each vehicle type by utilizing a sampling method, wherein variables of the simulation calculation working conditions comprise the vehicle type, the vehicle speed, the vehicle weight and the driving lane;
s3, combining the bridge finite element model and the standard vehicle finite element model, and calculating a corresponding axle coupling numerical solution, wherein the axle coupling numerical solution comprises various dynamic response data of a bridge target position, displacement data and internal force data of a bridge boundary condition position, and various dynamic response data of the bridge target position comprises displacement response, speed response and acceleration response of the bridge target position;
s4, taking vehicle physical parameter information of the standard vehicle finite element model under the simulation calculation working condition as input data, taking a corresponding axle coupling numerical value solution as output data, wherein the input data and the output data are matched one by one to form a plurality of training samples, classifying the training samples according to different vehicle weight ranges to form a plurality of sample sets, and training prediction models corresponding to different vehicle weight ranges based on the plurality of sample sets;
s5, measuring vehicle physical parameter information of the upper bridge vehicle by the dynamic weighing system, wherein the vehicle physical parameter information at least comprises a vehicle type and a vehicle weight, inputting the vehicle physical parameter information into the prediction model corresponding to the vehicle weight range, and calculating and outputting a bridge structure response predicted value by the prediction model corresponding to the vehicle weight range, wherein the bridge structure response predicted value comprises one or more of predicted displacement response of a bridge target position, predicted speed response of the bridge target position, predicted acceleration response of the bridge target position, predicted displacement data of a bridge boundary condition position and predicted internal force data of the bridge boundary condition position.
As a preferable mode, the specific steps of constructing the bridge finite element model in the step S1 are as follows: constructing a bridge finite element model according to the geometric dimension, material characteristics and support arrangement of a bridge structure in a construction drawing of the bridge; in the step S1, the specific steps of constructing the standard vehicle finite element model of the corresponding vehicle weight range according to different vehicle types are as follows: and respectively establishing standard vehicle finite element models corresponding to the two-axis vehicle, the three-axis vehicle, the four-axis vehicle, the five-axis vehicle and the six-axis vehicle according to the vehicle characteristics of different vehicle types.
As one preferable mode, the specific step of designing the simulation calculation condition of the standard vehicle finite element model of each vehicle model by using the sampling method in step S2 is as follows:
s21, determining corresponding vehicle speed and vehicle weight value ranges according to the standard vehicle finite element model of each vehicle type;
s22, designing the combined working conditions of different speeds, weights and driving lanes of the standard vehicle finite element models by using a sampling method.
Preferably, in step S3, a numerical solution for coupling the vehicle and the bridge is calculated by numerical integration.
As one preferable embodiment, the specific steps of using the vehicle physical parameter information as input data in step S4 are as follows: and (3) the physical parameter information of the vehicle is formed into an array, the influence line of the bridge under the action of each axle is calculated according to the vehicle type, and the axle weight of the vehicle is multiplied by the corresponding influence line to be used as input data of the prediction model.
Preferably, in step S4, the physical parameter information of the vehicle in the case of a vehicle weight of 10 tons or more is used as input data, and a sample set is divided for each 10 ton vehicle weight range.
As one preferable aspect, the step S5 specifically includes: and the dynamic weighing system measures the vehicle physical parameter information of the boarding vehicle, at least comprising the vehicle type, the vehicle weight and the axle weight distribution, uploads the vehicle physical parameter information to the terminal server, matrices the vehicle physical parameter information and calculates a bridge structure response predicted value by using the program of the prediction model.
Preferably, the prediction model is a BiLSTM model.
Preferably, the method for predicting the axle coupling value solution further comprises the steps of: and S6, defining a loss function comprising physical information constraint to train and optimize the prediction model.
As a preferred, the loss function is formulated as:
wherein the method comprises the steps of、/>、/>Respectively calculating acceleration response, speed response and displacement response of the bridge target position in the bridge structure response predicted values calculated and output by the prediction model in the step S5, +.>、/>、/>The acceleration response, the speed response and the displacement response of the bridge target position calculated in the step S3 are respectively calculated; d. f is displacement data and internal force data of bridge boundary condition positions in the bridge structure response predicted value calculated and output by the prediction model in the step S5, and n and m are the number of target positions and the number of boundary condition positions of the bridge respectively.
Compared with the prior art, the application has the beneficial effects that: (1) The prediction model can predict the numerical solution of the axle coupling differential equation, can directly calculate each dynamic response data of the bridge target position by using the measured vehicle load information, has high calculation speed, can meet the real-time and rapid calculation of the bridge structure response based on the digital twin technology, and is effectively applied to rapid evaluation of the bridge structure; (2) The prediction model provided by the application can predict the response of the target point position of the bridge structure according to the requirements, has more pertinence, and reasonably utilizes the calculation resources; (3) Furthermore, physical information constraint is added in the optimization loss function of the prediction model, so that the training and prediction of the prediction model are more accurate.
Drawings
FIG. 1 is a flow chart illustrating a method for predicting an axle coupling value according to an embodiment of the present application.
FIG. 2 is a schematic diagram of a bridge finite element model according to an embodiment of the present application.
FIG. 3 is a schematic diagram of a standard vehicle finite element model in accordance with one embodiment of the present application.
FIG. 4 is an example of a bridge displacement response value structure across an interrupted surface in an embodiment of the present application.
Fig. 5 is a schematic diagram of a displacement response of a bridge girder span in an embodiment of the present application, wherein a dotted line is a prediction result of a prediction model, and a solid line is a result obtained by solving an axle coupling motion differential equation.
Detailed Description
The present application will be further described with reference to the following specific embodiments, and it should be noted that, on the premise of no conflict, new embodiments may be formed by any combination of the embodiments or technical features described below.
In the description of the present application, it should be noted that, for the azimuth words such as terms "center", "lateral", "longitudinal", "length", "width", "thickness", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", "clockwise", "counterclockwise", etc., the azimuth and positional relationships are based on the azimuth or positional relationships shown in the drawings, it is merely for convenience of describing the present application and simplifying the description, and it is not to be construed as limiting the specific scope of protection of the present application that the device or element referred to must have a specific azimuth configuration and operation.
It should be noted that the terms "first," "second," and the like in the description and in the claims are used for distinguishing between similar objects and not necessarily for describing a particular sequential or chronological order.
The terms "comprises" and "comprising," along with any variations thereof, in the description and claims, are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those steps or elements but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.
In the mechanical analysis of bridge structures, a finite element model is generally built, and then a mechanical equation set is built for solving according to load, boundary conditions and analysis types. In axle coupling analysis, the external load change of a bridge structure is complex, and the external load is generally solved by adopting a numerical method of time domain gradual integration, namely a numerical integration method. In general, under the condition of considering structural nonlinearity, the numerical integration method for solving the second-order motion differential equation needs to continuously iterate and update the stiffness matrix to calculate the structural response of the bridge, and the calculation speed is low under a certain calculation resource condition and structural model scale until the iterative operation reaches the convergence of the result, so that the rapid calculation requirement of digital twin of the bridge cannot be met. For example, an overweight vehicle is identified 500m from an old bridge, and it is not possible to numerically analyze the old bridge in advance and evaluate the risk of the overweight vehicle passing the bridge. However, in the related art, although there is a method for predicting the response value of the bridge structure, the input data cannot be measured under the actual field condition, and the vehicle load under the actual condition cannot be represented, so as to form the mapping relationship from the load to the response of the bridge structure, so that the bridge response prediction method is limited to theory, and is difficult to be applied to the axle coupling analysis and the state evaluation of the actual bridge structure.
In order to solve the above problems, the present application provides a method for predicting an axle coupling value, as shown in fig. 1-5, comprising the steps of:
s1, constructing a bridge finite element model, and constructing a standard vehicle finite element model of a corresponding vehicle weight range according to different vehicle types;
s2, designing simulation calculation working conditions of a standard vehicle finite element model of each vehicle type by using a sampling method, wherein variables of the simulation calculation working conditions comprise the vehicle type, the vehicle speed, the vehicle weight and a driving lane, namely vehicle load parameter data;
s3, combining the bridge finite element model and the standard vehicle finite element model, performing axle coupling numerical calculation, and calculating a corresponding axle coupling numerical solution, wherein the axle coupling numerical solution is numerical simulation response data of the bridge and comprises various dynamic response data of a bridge target position, displacement data and internal force data of a bridge boundary condition position, and various dynamic response data of the bridge target position comprises displacement response, speed response and acceleration response of the bridge target position;
s4, taking vehicle physical parameter information of a standard vehicle finite element model under a simulation calculation working condition as input data, taking a corresponding axle coupling numerical value solution as output data, matching the input data with the output data one by one to form a plurality of training samples, classifying the training samples according to different vehicle weight ranges to form a plurality of sample sets, and training prediction models corresponding to different vehicle weight ranges based on the plurality of sample sets;
s5, measuring vehicle physical parameter information of the upper bridge vehicle by the dynamic weighing system, wherein the vehicle physical parameter information at least comprises a vehicle type and a vehicle weight, inputting the vehicle physical parameter information into the prediction model corresponding to the vehicle weight range, and calculating and outputting a bridge structure response predicted value by the prediction model corresponding to the vehicle weight range, wherein the bridge structure response predicted value comprises one or more of predicted displacement response of a bridge target position, predicted speed response of the bridge target position, predicted acceleration response of the bridge target position, predicted displacement data of a bridge boundary condition position and predicted internal force data of the bridge boundary condition position.
According to the method, a large number of simulated calculation conditions are designed in step S2 and step S3, corresponding axle coupling numerical solutions under different simulation calculation conditions are calculated through solving axle coupling motion differential equations, vehicle physical parameter information in a corresponding vehicle weight range and the calculated axle coupling numerical solutions are selected as sample sets for training of a prediction model according to a plurality of vehicle weight ranges divided in step S4, mapping relations between the vehicle physical parameter information and the axle coupling numerical solutions in each vehicle weight range are obtained through training, and finally in step S5, a dynamic weighing system measures vehicle physical parameter information of an upper bridge vehicle in an actual bridge structure and inputs the measured vehicle physical parameter information into the prediction model in the corresponding vehicle weight range, and the prediction model can rapidly calculate bridge structure response prediction values. The prediction model can learn different vehicle loads, namely the relation between different vehicle physical information parameters and bridge structure response, from the simulation structure response data in advance, and the model parameters learned in the prediction model can approximate the structural physical parameters of the bridge finite element model, so that the response of the bridge structure can be calculated rapidly in real time, and the method is applied to rapid evaluation of the bridge structure.
Meanwhile, the inventor finds that for simple bridge structures such as simply supported bridge, the bridge can be analyzed and evaluated by only the response of key parts, but in the prior art, the response of all nodes of the finite element model of the bridge is calculated according to the displacement boundary condition of the bridge by adopting a numerical integration method in the numerical simulation of the bridge, so that more calculation resources are wasted.
It should be pointed out that, the bridge coupling numerical value solution prediction method of the application is designed according to the actual traffic conditions of the bridge site, one of the purposes of rapidly predicting the bridge coupling numerical value prediction solution is to analyze the bridge in advance and rapidly and evaluate the risk that an overweight vehicle passes through the bridge, while under the actual traffic conditions of the bridge, the general passenger vehicles, passenger cars, vans and other vehicles have smaller vehicle weights, the influence on the bridge during passing is lower, the real need pay attention to the evaluation of vehicles with larger vehicle weights such as heavy trucks and overweight trucks, the situation that other vehicles around the bridge are generally kept a safe distance with the heavy/overweight vehicles during actual driving passing is considered, and the situation that a heavy/overweight vehicle and other vehicles run side by side is less occurs is meant that only one heavy/overweight vehicle passes through at a certain moment is generally considered, based on the bridge coupling numerical value prediction method of the application, the bridge coupling numerical value prediction method of the application only takes into consideration the situation that only one row of heavy/overtight vehicle is small, the bridge is required to be used as the prediction model of the bridge, the physical response of the bridge is required to be evaluated, and the physical response of the bridge is required to be calculated, and the predicted by the bridge coupling numerical value of the vehicle is required to be calculated, and the predicted by the physical state of the bridge is required to be calculated. Considering that the vehicles such as passenger cars, buses and vans have smaller vehicle weight and lower influence on bridges during bridge crossing, the application does not perform axle coupling numerical calculation and bridge structure response predicted value calculation on the vehicles.
In some embodiments, the specific steps of constructing the bridge finite element model in step S1 are: and constructing a bridge finite element model according to the geometric dimension, material characteristics and support arrangement of the bridge structure in the construction drawing of the bridge.
Fig. 2 illustrates a finite element model of a bridge.
In some embodiments, the specific steps of constructing the standard vehicle finite element model of the corresponding vehicle weight range according to different vehicle types in step S1 are as follows: and respectively establishing standard vehicle finite element models corresponding to the two-axis vehicle, the three-axis vehicle, the four-axis vehicle, the five-axis vehicle and the six-axis vehicle according to the vehicle characteristics of different vehicle types.
Fig. 3 illustrates a standard vehicle finite element model of a two-axle vehicle model.
In some embodiments, the specific steps for designing the simulation calculation conditions of the standard vehicle finite element model of each vehicle model by using the sampling method in step S2 are as follows:
s21, determining corresponding vehicle speed and vehicle weight value ranges according to standard vehicle finite element models of all vehicle types;
s22, designing the combined working conditions of different speeds, weights and driving lanes of each standard vehicle finite element model by using a sampling method.
In some embodiments, in step S3, a numerical solution for coupling the vehicle and the bridge is calculated by numerical integration.
In some embodiments, the specific steps of using the vehicle physical parameter information as the input data in step S4 are as follows: and matrixing the physical parameter information of the vehicle, calculating an influence line of the bridge under the action of each axle according to the vehicle type, and multiplying the axle weight of the vehicle by the corresponding influence line to serve as input data of the prediction model.
In some embodiments, taking the bridge mid-span position as the target position of the bridge, fig. 4 illustrates an example of a bridge mid-break surface displacement response numerical structure in one embodiment. The coordinate axes of the bridge are time, a cross-middle transverse point position and deflection respectively, wherein the deflection refers to linear displacement of the center of the cross section along the direction perpendicular to the axis when the bridge is bent and deformed, and the bridge belongs to displacement response of the bridge.
In some embodiments, in step S4, the vehicle physical parameter information in the case of a vehicle weight of 10 tons or more is used as input data, and a sample set is divided for every 10 tons of vehicle weight range. Specifically, the vehicle weight may be 10 tons or more, less than 20 tons as a vehicle weight range, 20 tons or more, less than 30 tons as a vehicle weight range, and so on. The application does not establish a corresponding prediction model and calculate the response predicted value of the bridge structure of lighter vehicles such as passenger cars, vans and the like passing through the bridge. More in line with the actual situation and saves the computing resources.
In some embodiments, step S5 specifically includes: the method comprises the steps of embedding a program and model parameters of a prediction model into a terminal server on a bridge site, measuring vehicle physical parameter information of an on-road vehicle by a dynamic weighing system, wherein the vehicle physical parameter information at least comprises vehicle type, vehicle weight and axle weight distribution, uploading the vehicle physical parameter information to the terminal server, matrixing the vehicle physical parameter information, and calculating a bridge structure response predicted value by using the program of the prediction model. In other words, the method adopts an edge calculation method, the response predicted value of the bridge structure can be directly and rapidly calculated by the terminal server on the bridge site, the axle coupling analysis is quicker, and the method combines the prediction model with a dynamic weighing system, so that the axle coupling numerical value solution prediction method is easy to apply to the existing bridge structure and can be applied to the ground.
In some embodiments, the predictive model is a BiLSTM model. The deep learning model supports parallel computing and GPU acceleration, and can be designed according to required structural response output, so that the computing efficiency is greatly improved.
In some embodiments, the axle coupling value solution prediction method further comprises the steps of: and S6, defining a loss function comprising physical information constraint to perform training optimization on the prediction model, so that the training optimization direction is more accurate, and the training data sample reduction is facilitated.
In some embodiments, the loss function is formulated as:
wherein the method comprises the steps of、/>、/>Respectively calculating acceleration response, speed response and displacement response of the bridge target position in the bridge structure response predicted values output by the prediction model in the step S5>、/>、/>Acceleration response, velocity response and displacement of the bridge target position calculated in step S3Responding; d. f is displacement data and internal force data of bridge boundary condition positions in bridge structure response predicted values calculated and output by the prediction model in the step S5, and n and m are the number of target positions of the bridge and the number of boundary condition positions respectively.
In order to verify the accuracy of the axle coupling numerical value solution prediction method, the method selects part of prediction results of the prediction model, and compares and verifies the bridge structure response prediction value with the axle coupling numerical value solution calculated by adopting a numerical integration method. Fig. 5 shows a comparison situation of displacement response values of a bridge girder midspan section under a specific simulation calculation working condition, wherein a dotted line is a bridge girder midspan deflection response result predicted by a prediction model, a solid line is a result obtained by solving an axle coupling motion differential equation by a numerical integration method, and the solid line is basically identical with the dotted line.
The foregoing has outlined the basic principles, features, and advantages of the present application. It will be understood by those skilled in the art that the present application is not limited to the embodiments described above, and that the above embodiments and descriptions are merely illustrative of the principles of the present application, and various changes and modifications may be made therein without departing from the spirit and scope of the application, which is defined by the appended claims. The scope of the application is defined by the appended claims and equivalents thereof.
Claims (10)
1. The axle coupling numerical value solution prediction method is characterized by comprising the following steps of:
s1, constructing a bridge finite element model, and constructing a standard vehicle finite element model of a corresponding vehicle weight range according to different vehicle types;
s2, designing simulation calculation working conditions of the standard vehicle finite element model of each vehicle type by utilizing a sampling method, wherein variables of the simulation calculation working conditions comprise the vehicle type, the vehicle speed, the vehicle weight and the driving lane;
s3, combining the bridge finite element model and the standard vehicle finite element model, and calculating a corresponding axle coupling numerical solution, wherein the axle coupling numerical solution comprises various dynamic response data of a bridge target position, displacement data and internal force data of a bridge boundary condition position, and various dynamic response data of the bridge target position comprises displacement response, speed response and acceleration response of the bridge target position;
s4, taking vehicle physical parameter information of the standard vehicle finite element model under the simulation calculation working condition as input data, taking a corresponding axle coupling numerical value solution as output data, wherein the input data and the output data are matched one by one to form a plurality of training samples, classifying the training samples according to different vehicle weight ranges to form a plurality of sample sets, and training prediction models corresponding to different vehicle weight ranges based on the plurality of sample sets;
s5, measuring vehicle physical parameter information of the upper bridge vehicle by the dynamic weighing system, wherein the vehicle physical parameter information at least comprises a vehicle type and a vehicle weight, inputting the vehicle physical parameter information into the prediction model corresponding to the vehicle weight range, and calculating and outputting a bridge structure response predicted value by the prediction model corresponding to the vehicle weight range, wherein the bridge structure response predicted value comprises one or more of predicted displacement response of a bridge target position, predicted speed response of the bridge target position, predicted acceleration response of the bridge target position, predicted displacement data of a bridge boundary condition position and predicted internal force data of the bridge boundary condition position.
2. The method for predicting the axle coupling value according to claim 1, wherein the specific steps of constructing the bridge finite element model in the step S1 are as follows: constructing a bridge finite element model according to the geometric dimension, material characteristics and support arrangement of a bridge structure in a construction drawing of the bridge; in the step S1, the specific steps of constructing the standard vehicle finite element model of the corresponding vehicle weight range according to different vehicle types are as follows: and respectively establishing standard vehicle finite element models corresponding to the two-axis vehicle, the three-axis vehicle, the four-axis vehicle, the five-axis vehicle and the six-axis vehicle according to the vehicle characteristics of different vehicle types.
3. The method for predicting the axle coupling numerical value according to claim 1, wherein the specific steps of designing the simulation calculation condition of the standard vehicle finite element model of each vehicle model by using the sampling method in the step S2 are as follows:
s21, determining corresponding vehicle speed and vehicle weight value ranges according to the standard vehicle finite element model of each vehicle type;
s22, designing the combined working conditions of different speeds, weights and driving lanes of the standard vehicle finite element models by using a sampling method.
4. The axle coupling value de-prediction method of claim 1, wherein: and in the step S3, calculating a vehicle-bridge coupling numerical solution by a numerical integration method.
5. The axle coupling value de-prediction method of claim 1, wherein: in step S4, the specific steps of using the vehicle physical parameter information of the standard vehicle finite element model under the simulation calculation working condition as input data are as follows: and matrixing the physical parameter information of the vehicle, calculating an influence line of the bridge under the action of each axle according to the vehicle type, and multiplying the axle weight of the vehicle by the corresponding influence line to serve as input data of the prediction model.
6. The axle coupling value de-prediction method of claim 1, wherein: in step S4, the physical parameter information of the vehicle is used as input data when the vehicle weight is greater than or equal to 10 tons, and a sample set is divided in a vehicle weight range of 10 tons.
7. The axle coupling value de-prediction method of claim 1, wherein: the step S5 specifically includes: and the dynamic weighing system measures the vehicle physical parameter information of the boarding vehicle, at least comprising the vehicle type, the vehicle weight and the axle weight distribution, uploads the vehicle physical parameter information to the terminal server, matrices the vehicle physical parameter information and calculates a bridge structure response predicted value by using the program of the prediction model.
8. The axle coupling value de-prediction method of claim 1, wherein: the prediction models are BiLSTM models.
9. The axle coupling value de-prediction method of any one of claims 1-8, further comprising the steps of: a penalty function including physical information constraints is defined to train the predictive model for optimization.
10. The axle coupling value de-prediction method of claim 9, wherein: the formula of the loss function is:
；
wherein the method comprises the steps of、/>、/>Respectively calculating acceleration response, speed response and displacement response of the bridge target position in the bridge structure response predicted values calculated and output by the prediction model in the step S5, +.>、/>、/>The acceleration response, the speed response and the displacement response of the bridge target position calculated in the step S3 are respectively calculated; d. f is the prediction model meter in step S5And calculating displacement data and internal force data of bridge boundary condition positions in the output bridge structure response predicted value, wherein n and m are the number of target positions of the bridge and the number of boundary condition positions respectively.
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