CN109282785B - Deformation monitoring method of elastic support plate based on strain monitoring - Google Patents
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Abstract
The invention discloses a deformation monitoring method of an elastic supporting plate based on strain monitoring, which expands the deformation monitoring theory of a beam to the deformation monitoring of a two-dimensional plate and deduces the monitoring theory of the bending deformation of the plate, and comprises the steps of utilizing a strain sensor to obtain the strain data of the plate, approximating the strain function of the plate under the condition of supporting the boundary of the elastic support at the opposite side, bringing the strain measurement results of different positions into the strain function to obtain the parameters of the strain function, obtaining a displacement function through the strain-displacement conversion relation, and obtaining the bending deformation value of the plate; the bending deformation of the plate at each moment is obtained by calculating the strain at each moment, so that the dynamic monitoring of the bending deformation of the plate can be realized. The invention can monitor the bending deformation of the plate with lower cost and simple equipment on the premise of ensuring the precision, and has important theoretical significance and engineering application value for structural health monitoring, structural safety assessment and the like.
Description
Technical Field
The invention relates to the technical field of civil engineering, in particular to a deformation monitoring method of an elastic supporting plate based on strain monitoring, wherein the monitoring result is used for evaluating the use state and the safety performance of the elastic supporting plate.
Background
In civil engineering, slab structures include floors, balconies, bridge decks, foundations and the like, and when external force is applied to the slabs, the slabs are deformed, and the deformation may affect the safety and durability of the slab structure, so that the slab deformation monitoring is necessary in some structures. Deformation monitoring is required to be carried out on the board in the construction and operation processes of some projects, and the basic theory of various boards is continuously developed and perfected. For small deflection sheets, the classical sheet theory based on Kirchhoff normal line assumption is the most common and effective. In the specific problem solving, a general analytic solution and a finite element solution are based on a basic theory and are solved in different angles.
The traditional measuring mode adopts a theodolite, a level and a dial indicator, but the methods are not suitable for manual measurement in many occasions, cannot carry out real-time monitoring and are inconvenient to use, time-consuming and labor-consuming. In recent years, some newer deflection measurement methods such as a GPS, a laser image, a communication tube method, a photoelectric imaging method, and the like are gradually emerging. However, these measurement methods require separate measurement equipment, increasing the amount of data and increasing the cost of monitoring. Especially for the structure under special environment, because of the complicated operating environment, it is often very difficult to directly install the displacement sensor to directly measure the deflection, and these methods can only measure the deformation of a certain or a certain limited point.
The method for solving the deformation problem of the elastic plate is various and can be roughly divided into three categories of analytical solutions, approximate solutions and numerical solutions. The general analytic solution of the elastic bending of the rectangular plate is complex, the existing methods have disadvantages in parameter determination and calculation speed, and the analytic method is difficult or even impossible to use in many cases. The reliability and convergence speed of the approximate solution are completely dependent on the approximate function assumed in advance for the basic variable, namely the quality of the test function. In a common numerical method, a finite element analysis method is effective for predicting the deformation of the plate, but the method is complex in calculation, large in calculation amount and long in time consumption, and a large calculation error can be generated because excessive units need to be divided to obtain a high-precision value. Most of the measuring methods for plate deformation in the civil engineering practical application are non-real-time monitoring, the monitoring means for the plate structure is deficient, and no deformation monitoring method is suitable for the elastic support plate in the civil engineering field at present.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, and provides the strain monitoring-based deformation monitoring method for the elastic support plate, which is simple, convenient and applicable and accurate in measurement, so as to realize dynamic and accurate identification and monitoring of the bending deformation of the plate.
The invention adopts the following technical scheme for solving the technical problems:
the invention relates to a deformation monitoring method of an elastic supporting plate based on strain monitoring, which is characterized by comprising the following steps of:
uniformly arranging N strain sensors on the elastic support plate as N strain measurement points, thereby obtaining strain data of the N strain measurement points, and recording the strain data as { epsilon ∈y1(x1,y1),εy2(x2,y2),…,εyi(xi,yi),…,εyN(xN,yN)}，εyi(xi,yi) Representing a position in the rectangular coordinate system O-XY at coordinate (x)i,yi) The strain value of the ith strain measuring point sensor in the Y-axis direction is 1,2, …, N is not less than 8;
εy(x,y)＝a1+a2x+a3y+a4xy+a5y2+a6xy2+a7y3+a8xy3(1)
in the formula (1), epsilony(X, Y) represents a strain value in a Y-axis direction at any point (X, Y) on the elastic support plate, X represents a value of any point in the rectangular coordinate system O-XY on an X-axis, and Y represents a value of any point in the rectangular coordinate system O-XY on a Y-axis; a is1,a2,a3,a4,a5,a6,a7,a8Are respectively 8 coefficients to be determined, and form a coefficient vector a ═ a1,a2,a3,a4,a5,a6,a7,a8]；
And 3, substituting the strain data of the N strain measuring points into the strain function to obtain an equation set consisting of N equations shown as the formula (2):
ε＝VaT(2)
in the formula (2), epsilon represents a strain vector, and epsilon ═ epsilony1(x1,y1),εy2(x2,y2),…,εyi(xi,yi),…,εyN(xN,yN)]T(ii) a V represents a strain gauge point coordinate matrix, an
aTRepresents a transpose of the coefficient vector a;
step 4, solving the equation set by using a least square method, and enabling an average error to be minimum, so as to obtain the coefficient vector a;
step 5, substituting the coefficient vector a into the strain function shown in the formula (1) to obtain the strain function corresponding to the elastic supporting plate;
and 6, obtaining a displacement function of the elastic supporting plate shown in the formula (4) by integration by using a strain-displacement relation of the elastic thin plate shown in the formula (3):
in the formula (3), z represents the distance from the elastic support plate surface strain sensor to half of the thickness of the elastic support plate; w (x, y) represents the linear displacement of any point (x, y) in the direction perpendicular to the O-XY plane;
in formula (4), f (x) represents a first-order polynomial, and f (x) bx + c; g (x) also represents a first order polynomial, and g (x) dx + e; b. c, d and e each represent undetermined coefficients determined by the monitored boundary conditions of the resilient support plate; the boundary conditions are boundary rotation angles and displacement conditions of the monitored elastic supporting plate;
and 5, substituting the coefficient vector a and the first-order polynomials f (x) and g (x) into the displacement function to obtain a deflection curve of the elastic supporting plate, so that the local transverse deformation of the elastic supporting plate is monitored.
Compared with the prior art, the invention has the beneficial effects that:
1. the invention aims at the defects of complex operation, large calculation amount, low calculation speed and the like in the existing plate deformation monitoring, and establishes a method for monitoring the local bending deformation of an elastic supporting plate based on strain monitoring.
2. According to the invention, the number and arrangement of strain measuring points are determined by researching the influence of the number and position of strain measurement on the monitoring of the transverse local bending deformation of the plate. The theory and the method for calculating the bending deformation and the local transverse bending deformation of the elastic supporting plate based on the strain are verified by utilizing numerical simulation and laboratory model test. By using the monitoring method, the bending deformation of the elastic supporting plate can be dynamically identified and monitored, and the bridge deck member is monitored at the key part of the bridge structure, so that the engineering application of monitoring the transverse local bending deformation of the bridge is realized, and the monitoring method has important significance for evaluating the operation state of the bridge.
3. The invention provides a simple and feasible monitoring method, which is reasonable in cost, high in precision and strong in applicability. In the bridge monitoring technology, the bridge strain test has low cost, is easy to operate and has high precision.
Drawings
FIG. 1 is a schematic diagram of a sensor arrangement of the present invention;
FIG. 2 is a schematic view of the present invention using ANSYS software to create deflection in a finite element model of a resilient support plate;
FIG. 3 is a schematic view of the deflection of an elastomeric support plate in a finite element model using ANSYS software;
FIG. 4 is a graphical representation of deflection calculations obtained by the present invention;
FIG. 5 is a graphical representation of deflection calculations obtained by the present invention;
reference numbers in the figures: 1 main beam/web plate for supporting elastic support plate; 2 strain sensors, a width of the panel monitored, b length of the panel monitored.
Detailed Description
In this embodiment, a method for monitoring bending deformation of a plate structure based on strain monitoring includes: the strain sensor is used for carrying out strain test on a plate structure to obtain strain data, a strain function is obtained according to the mutual function relation of the strain data, a displacement function is obtained according to the strain-displacement conversion relation, then a real-time flexible line of the plate structure is obtained according to the calculated displacement function, so that the monitoring of the bending deformation of the plate is realized, and the strain function is obtained according to the cross-correlation function of the strain data according to the following steps:
the elastic support plate is an opposite side elastic support plate, as shown in fig. 1, wherein 1 shows a main beam/web plate for supporting the elastic support plate; 2 denotes a strain sensor, a denotes the width of the monitored panel, b denotes the length of the monitored panel.
Uniformly arranging N strain sensors on the elastic support plate as N strain measurement points, thereby obtaining strain data of the N strain measurement points, and recording the strain data as { epsilon ∈y1(x1,y1),εy2(x2,y2),…,εyi(xi,yi),…,εyN(xN,yN)}，εyi(xi,yi) Expressed in coordinates (x) in a rectangular coordinate system O-XYi,yi) The strain value of the ith strain measuring point sensor in the Y-axis direction is 1,2, …, N is not less than 8;
εy(x,y)＝a1+a2x+a3y+a4xy+a5y2+a6xy2+a7y3+a8xy3(1)
in the formula (1), epsilony(X, Y) represents a strain value in a Y-axis direction at an arbitrary point (X, Y) on the elastic support plate, X represents a value in the X-axis direction of an arbitrary point in a rectangular coordinate system O-XY, and Y represents a value in the Y-axis direction of an arbitrary point in a rectangular coordinate system O-XY; a is1,a2,a3,a4,a5,a6,a7,a8Are respectively 8 coefficients to be determined, and form a coefficient vector a ═ a1,a2,a3,a4,a5,a6,a7,a8]；
And 3, substituting the strain data of the N strain measuring points into a strain function to obtain an equation set which is shown as the formula (2) and consists of N equations:
the system of equations can be expressed as equation (3):
ε＝VaT(3)
in the formula (3), epsilon represents a strain vector, and epsilon ═ epsilony1(x1,y1),εy2(x2,y2),…,εyi(xi,yi),…,εyN(xN,yN)]T(ii) a V represents a coordinate matrix of the strain measuring point and is obtained by the coordinates of the sensor and the formula (2),
aTrepresents a transpose of the coefficient vector a;
step 4, using least square method to make equation set epsilon be VaTSolving and minimizing the average error to obtain a coefficient vector a, wherein the average error is represented by an equation (4):
the approximate solution of the equation set can be obtained when the equation set of the formula (3) is not accurately solved by using the least square method, and the accuracy can meet the requirement.
Step 5, substituting the coefficient vector a into the strain function shown in the formula (1) to obtain the strain function corresponding to the elastic support plate;
and 6, calculating a displacement function of the elastic supporting plate shown in the formula (8) step by step through the integral of the formula (6) and the formula (7) by using the strain-displacement relation of the elastic thin plate shown in the formula (5):
in the formula (5), z represents the distance from the elastic support plate surface strain sensor to half of the thickness of the elastic support plate; w (x, y) represents the linear displacement of any point (x, y) in the direction perpendicular to the O-XY plane;
in formula (8), f (x) represents a first-order polynomial, and f (x) bx + c; g (x) also represents a first order polynomial, and g (x) dx + e; b. c, d and e each represent undetermined coefficients determined by the monitored boundary conditions of the resilient support plate; the boundary conditions are boundary rotation angle and displacement conditions of the monitored elastic supporting plate;
and 5, substituting the coefficient vector a and first-order polynomials f (x) and g (x) into a displacement function to obtain a deflection curve of the elastic supporting plate, thereby realizing the monitoring of the local transverse deformation of the elastic supporting plate.
In order to verify the effectiveness of the elastic support plate deformation monitoring method based on strain monitoring, a concrete slab with elastically supported sides is experimentally verified. The test strain sensor is arranged on the lower surface of the elastic supporting plate, a surface-mounted strain sensor is adopted, and a dial indicator is used for actually measuring displacement data. The upper surface of the test plate was subjected to concentrated force loading.
The length, width and height of the concrete panel are 800mm, 400mm and 20mm in sequence, C50 concrete is adopted, and the elastic modulus Ex of the concrete panel is 3.45 multiplied by 1010N/m2Poisson's ratio v is 0.2, density dens is 2700kg/m3. Elastically supporting the longitudinal opposite sides of the experimental plate and loading the plate. And simultaneously, establishing a finite element model of the elastic support plate by using ANSYS software, wherein the size, the elastic modulus, the Poisson ratio and the density of the concrete panel are the same as those of an experimental object. Elastic restraint is arranged on the longitudinal opposite sides, the elastic restraint directions are the z direction and the y direction, and the elastic modulus is 1.84 multiplied by 1012N/m2. The test loading was the same as the finite element simulation loading.
The strain sensor required by the test adopts a surface-mounted strain gauge, the sensor is arranged as shown in figure 1, the strain value of a measuring point and corresponding coordinates are substituted into a strain function, a least square method is adopted to minimize a target error to obtain a strain function coefficient, and the strain function coefficient is also a displacement function coefficient, so that a displacement function can be obtained. Substituting known boundary rotation angles into
The expression for f (x) can be found,
substituting known boundary shifts into
The expression g (x) can be obtained, and the deformation condition of the whole edition can be obtained through calculation.
The first test condition is as follows: the coordinate of the loading position on the coordinate system O-XY is (200, 400), the magnitude of the concentrated force is 200N, and the loading position is loaded on the upper surface of the experimental object. The finite element calculation results are shown in FIG. 2, and the patent method calculation results are shown in FIG. 3.
And (3) test working condition II: the coordinate of the loading position on the coordinate system O-XY is (300, 200), the magnitude of the concentrated force is 200N, and the loading position is loaded on the upper surface of the experimental object. The finite element calculation results are shown in FIG. 4, and the patent method calculation results are shown in FIG. 5.
TABLE 1
Table 1 is a comparative analysis of the experimentally measured data with the finite element calculations and the calculations of the present invention. It can be concluded that the error of the monitoring of the deformation of the resilient support plate based on strain monitoring is within 5%.
Claims (1)
1. A deformation monitoring method of an elastic supporting plate based on strain monitoring is characterized by comprising the following steps:
step 1, establishing a rectangular coordinate system O-XY of the elastic supporting plate by taking any vertex angle on the monitored elastic supporting plate as an origin O and two edges adjacent to the origin O as an X axis and a Y axis respectively;
uniformly arranging N strain sensors on the elastic support plate as N strain measurement points, thereby obtaining strain data of the N strain measurement points, and recording the strain data as { epsilon ∈y1(x1,y1),εy2(x2,y2),…,εyi(xi,yi),…,εyN(xN,yN)}，εyi(xi,yi) Representing a position in the rectangular coordinate system O-XY at coordinate (x)i,yi) The ith strain measuring point sensor is arranged on the Y axisThe strain value of the direction, i is 1,2, …, N is not less than 8;
step 2, assuming that the strain function of the elastic supporting plate under the condition of opposite-side elastic supporting is a cubic polynomial along the Y-axis direction and a cubic polynomial along the X-axis direction, so as to obtain the strain function shown in the formula (1):
εy(x,y)＝a1+a2x+a3y+a4xy+a5y2+a6xy2+a7y3+a8xy3(1)
in the formula (1), epsilony(X, Y) represents a strain value in a Y-axis direction at any point (X, Y) on the elastic support plate, X represents a value of any point in the rectangular coordinate system O-XY on an X-axis, and Y represents a value of any point in the rectangular coordinate system O-XY on a Y-axis; a is1,a2,a3,a4,a5,a6,a7,a8Are respectively 8 coefficients to be determined, and form a coefficient vector a ═ a1,a2,a3,a4,a5,a6,a7,a8]；
And 3, substituting the strain data of the N strain measuring points into the strain function to obtain an equation set consisting of N equations shown as the formula (2):
ε＝VaT(2)
in the formula (2), epsilon represents a strain vector, and epsilon ═ epsilony1(x1,y1),εy2(x2,y2),…,εyi(xi,yi),…,εyN(xN,yN)]T(ii) a V represents a strain gauge point coordinate matrix, an
aTRepresents a transpose of the coefficient vector a;
step 4, solving the equation set by using a least square method, and enabling an average error to be minimum, so as to obtain the coefficient vector a;
step 5, substituting the coefficient vector a into the strain function shown in the formula (1) to obtain the strain function corresponding to the elastic supporting plate;
and 6, obtaining a displacement function of the elastic supporting plate shown in the formula (4) by integration by using a strain-displacement relation of the elastic thin plate shown in the formula (3):
in the formula (3), z represents the distance from the elastic support plate surface strain sensor to half of the thickness of the elastic support plate; w (x, y) represents the linear displacement of any point (x, y) in the direction perpendicular to the O-XY plane;
in formula (4), f (x) represents a first-order polynomial, and f (x) bx + c; g (x) also represents a first order polynomial, and g (x) dx + e; b. c, d and e each represent undetermined coefficients determined by the monitored boundary conditions of the resilient support plate; the boundary conditions are boundary rotation angles and displacement conditions of the monitored elastic supporting plate;
and 5, substituting the coefficient vector a and the first-order polynomials f (x) and g (x) into the displacement function to obtain a deflection curve of the elastic supporting plate, so that the local transverse deformation of the elastic supporting plate is monitored.
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